id	sid	tid	token	lemma	pos
ejde-673	1	1	electronic	electronic	ADJ
ejde-673	1	2	journal	journal	NOUN
ejde-673	1	3	of	of	ADP
ejde-673	1	4	differential	differential	ADJ
ejde-673	1	5	equations	equation	NOUN
ejde-673	1	6	,	,	PUNCT
ejde-673	1	7	vol	vol	NOUN
ejde-673	1	8	.	.	PUNCT
ejde-673	1	9	2025	2025	NUM
ejde-673	1	10	(	(	PUNCT
ejde-673	1	11	2025	2025	NUM
ejde-673	1	12	)	)	PUNCT
ejde-673	1	13	,	,	PUNCT
ejde-673	1	14	no	no	INTJ
ejde-673	1	15	.	.	NOUN
ejde-673	1	16	17	17	NUM
ejde-673	1	17	,	,	PUNCT
ejde-673	1	18	pp	pp	PROPN
ejde-673	1	19	.	.	PUNCT
ejde-673	2	1	1–30	1–30	PROPN
ejde-673	2	2	.	.	PUNCT
ejde-673	3	1	issn	issn	PROPN
ejde-673	3	2	:	:	PUNCT
ejde-673	3	3	1072	1072	NUM
ejde-673	3	4	-	-	SYM
ejde-673	3	5	6691	6691	NUM
ejde-673	3	6	.	.	PUNCT
ejde-673	4	1	url	url	PROPN
ejde-673	4	2	:	:	PUNCT
ejde-673	4	3	https://ejde.math.txstate.edu	https://ejde.math.txstate.edu	PROPN
ejde-673	4	4	,	,	PUNCT
ejde-673	4	5	https://ejde.math.unt.edu	https://ejde.math.unt.edu	PROPN
ejde-673	4	6	doi	doi	PROPN
ejde-673	4	7	:	:	PUNCT
ejde-673	4	8	10.58997	10.58997	NUM
ejde-673	4	9	/	/	SYM
ejde-673	4	10	ejde.2025.17	ejde.2025.17	PROPN
ejde-673	4	11	eigenvalue	eigenvalue	PROPN
ejde-673	4	12	problems	problem	NOUN
ejde-673	4	13	for	for	ADP
ejde-673	4	14	kirchhoff	kirchhoff	NOUN
ejde-673	4	15	-	-	PUNCT
ejde-673	4	16	type	type	NOUN
ejde-673	4	17	equations	equation	NOUN
ejde-673	4	18	in	in	ADP
ejde-673	4	19	variable	variable	ADJ
ejde-673	4	20	exponent	exponent	NOUN
ejde-673	4	21	sobolev	sobolev	NOUN
ejde-673	4	22	spaces	space	VERB
ejde-673	4	23	junichi	junichi	PROPN
ejde-673	4	24	aramaki	aramaki	PROPN
ejde-673	4	25	abstract	abstract	PROPN
ejde-673	4	26	.	.	PUNCT
ejde-673	5	1	in	in	ADP
ejde-673	5	2	this	this	DET
ejde-673	5	3	article	article	NOUN
ejde-673	5	4	,	,	PUNCT
ejde-673	5	5	we	we	PRON
ejde-673	5	6	consider	consider	VERB
ejde-673	5	7	an	an	DET
ejde-673	5	8	eigenvalue	eigenvalue	ADJ
ejde-673	5	9	problem	problem	NOUN
ejde-673	5	10	for	for	ADP
ejde-673	5	11	the	the	DET
ejde-673	5	12	kirchhofftype	kirchhofftype	NOUN
ejde-673	5	13	equation	equation	NOUN
ejde-673	5	14	containing	contain	VERB
ejde-673	5	15	p(·)-laplacian	p(·)-laplacian	PROPN
ejde-673	5	16	and	and	CCONJ
ejde-673	5	17	the	the	DET
ejde-673	5	18	mean	mean	ADJ
ejde-673	5	19	curvature	curvature	NOUN
ejde-673	5	20	operator	operator	NOUN
ejde-673	5	21	with	with	ADP
ejde-673	5	22	mixed	mixed	ADJ
ejde-673	5	23	boundary	boundary	ADJ
ejde-673	5	24	conditions	condition	NOUN
ejde-673	5	25	.	.	PUNCT
ejde-673	6	1	more	more	ADV
ejde-673	6	2	precisely	precisely	ADV
ejde-673	6	3	,	,	PUNCT
ejde-673	6	4	we	we	PRON
ejde-673	6	5	are	be	AUX
ejde-673	6	6	concerned	concerned	ADJ
ejde-673	6	7	with	with	ADP
ejde-673	6	8	the	the	DET
ejde-673	6	9	problem	problem	NOUN
ejde-673	6	10	with	with	ADP
ejde-673	6	11	the	the	DET
ejde-673	6	12	dirichlet	dirichlet	PROPN
ejde-673	6	13	condition	condition	NOUN
ejde-673	6	14	on	on	ADP
ejde-673	6	15	a	a	DET
ejde-673	6	16	part	part	NOUN
ejde-673	6	17	of	of	ADP
ejde-673	6	18	the	the	DET
ejde-673	6	19	boundary	boundary	NOUN
ejde-673	6	20	and	and	CCONJ
ejde-673	6	21	the	the	DET
ejde-673	6	22	steklov	steklov	ADJ
ejde-673	6	23	boundary	boundary	ADJ
ejde-673	6	24	condition	condition	NOUN
ejde-673	6	25	on	on	ADP
ejde-673	6	26	an	an	DET
ejde-673	6	27	another	another	DET
ejde-673	6	28	part	part	NOUN
ejde-673	6	29	of	of	ADP
ejde-673	6	30	the	the	DET
ejde-673	6	31	boundary	boundary	NOUN
ejde-673	6	32	.	.	PUNCT
ejde-673	7	1	we	we	PRON
ejde-673	7	2	show	show	VERB
ejde-673	7	3	that	that	SCONJ
ejde-673	7	4	the	the	DET
ejde-673	7	5	eigenvalue	eigenvalue	PROPN
ejde-673	7	6	problem	problem	NOUN
ejde-673	7	7	has	have	VERB
ejde-673	7	8	infinitely	infinitely	ADV
ejde-673	7	9	many	many	ADJ
ejde-673	7	10	eigenpairs	eigenpair	NOUN
ejde-673	7	11	by	by	ADP
ejde-673	7	12	using	use	VERB
ejde-673	7	13	the	the	DET
ejde-673	7	14	celebrated	celebrate	VERB
ejde-673	7	15	ljusternikschnirelmann	ljusternikschnirelmann	NOUN
ejde-673	7	16	principle	principle	NOUN
ejde-673	7	17	in	in	ADP
ejde-673	7	18	the	the	DET
ejde-673	7	19	calculus	calculus	NOUN
ejde-673	7	20	of	of	ADP
ejde-673	7	21	variation	variation	NOUN
ejde-673	7	22	.	.	PUNCT
ejde-673	8	1	moreover	moreover	ADV
ejde-673	8	2	,	,	PUNCT
ejde-673	8	3	we	we	PRON
ejde-673	8	4	derive	derive	VERB
ejde-673	8	5	that	that	SCONJ
ejde-673	8	6	in	in	ADP
ejde-673	8	7	a	a	DET
ejde-673	8	8	variable	variable	ADJ
ejde-673	8	9	exponent	exponent	NOUN
ejde-673	8	10	sobolev	sobolev	NOUN
ejde-673	8	11	space	space	NOUN
ejde-673	8	12	,	,	PUNCT
ejde-673	8	13	there	there	PRON
ejde-673	8	14	are	be	VERB
ejde-673	8	15	two	two	NUM
ejde-673	8	16	cases	case	NOUN
ejde-673	8	17	where	where	SCONJ
ejde-673	8	18	the	the	DET
ejde-673	8	19	infimum	infimum	NOUN
ejde-673	8	20	of	of	ADP
ejde-673	8	21	all	all	DET
ejde-673	8	22	eigenvalues	eigenvalue	NOUN
ejde-673	8	23	is	be	AUX
ejde-673	8	24	equal	equal	ADJ
ejde-673	8	25	to	to	ADP
ejde-673	8	26	zero	zero	NUM
ejde-673	8	27	and	and	CCONJ
ejde-673	8	28	is	be	AUX
ejde-673	8	29	positive	positive	ADJ
ejde-673	8	30	.	.	PUNCT
ejde-673	9	1	1	1	X
ejde-673	9	2	.	.	X
ejde-673	9	3	introduction	introduction	NOUN
ejde-673	9	4	in	in	ADP
ejde-673	9	5	this	this	DET
ejde-673	9	6	article	article	NOUN
ejde-673	9	7	,	,	PUNCT
ejde-673	9	8	we	we	PRON
ejde-673	9	9	consider	consider	VERB
ejde-673	9	10	the	the	DET
ejde-673	9	11	following	follow	VERB
ejde-673	9	12	eigenvalue	eigenvalue	PROPN
ejde-673	9	13	problem	problem	NOUN
ejde-673	9	14	with	with	ADP
ejde-673	9	15	mixed	mixed	ADJ
ejde-673	9	16	boundary	boundary	ADJ
ejde-673	9	17	conditions	condition	NOUN
ejde-673	9	18	−m	−m	NOUN
ejde-673	9	19	(	(	PUNCT
ejde-673	9	20	∫	∫	PROPN
ejde-673	9	21	ω	ω	NUM
ejde-673	9	22	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	9	23	)	)	PUNCT
ejde-673	9	24	)	)	PUNCT
ejde-673	9	25	dx	dx	PROPN
ejde-673	9	26	)	)	PUNCT
ejde-673	9	27	div[a(x,∇u(x	div[a(x,∇u(x	PROPN
ejde-673	9	28	)	)	PUNCT
ejde-673	9	29	)	)	PUNCT
ejde-673	9	30	]	]	PUNCT
ejde-673	10	1	=	=	PUNCT
ejde-673	10	2	0	0	NUM
ejde-673	10	3	in	in	ADP
ejde-673	10	4	ω	ω	PROPN
ejde-673	10	5	,	,	PUNCT
ejde-673	10	6	u(x	u(x	X
ejde-673	10	7	)	)	PUNCT
ejde-673	10	8	=	=	SYM
ejde-673	10	9	0	0	NUM
ejde-673	10	10	on	on	ADP
ejde-673	10	11	γ1	γ1	PROPN
ejde-673	10	12	,	,	PUNCT
ejde-673	10	13	m	m	PROPN
ejde-673	10	14	(	(	PUNCT
ejde-673	10	15	∫	∫	PROPN
ejde-673	10	16	ω	ω	NUM
ejde-673	10	17	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	10	18	)	)	PUNCT
ejde-673	10	19	)	)	PUNCT
ejde-673	10	20	dx	dx	PROPN
ejde-673	10	21	)	)	PUNCT
ejde-673	11	1	n(x	n(x	PROPN
ejde-673	11	2	)	)	PUNCT
ejde-673	11	3	·	·	PUNCT
ejde-673	12	1	a(x,∇u(x	a(x,∇u(x	NOUN
ejde-673	12	2	)	)	PUNCT
ejde-673	12	3	)	)	PUNCT
ejde-673	13	1	=	=	PUNCT
ejde-673	14	1	λg(x	λg(x	X
ejde-673	14	2	,	,	PUNCT
ejde-673	14	3	u(x	u(x	NOUN
ejde-673	14	4	)	)	PUNCT
ejde-673	14	5	)	)	PUNCT
ejde-673	14	6	on	on	ADP
ejde-673	14	7	γ2	γ2	PROPN
ejde-673	14	8	.	.	PUNCT
ejde-673	15	1	(	(	PUNCT
ejde-673	15	2	1.1	1.1	NUM
ejde-673	15	3	)	)	PUNCT
ejde-673	15	4	here	here	ADV
ejde-673	15	5	ω	ω	PROPN
ejde-673	15	6	is	be	AUX
ejde-673	15	7	a	a	DET
ejde-673	15	8	bounded	bounded	ADJ
ejde-673	15	9	domain	domain	NOUN
ejde-673	15	10	of	of	ADP
ejde-673	15	11	rn	rn	PROPN
ejde-673	15	12	(	(	PUNCT
ejde-673	15	13	n	n	CCONJ
ejde-673	15	14	≥	≥	NOUN
ejde-673	15	15	2	2	NUM
ejde-673	15	16	)	)	PUNCT
ejde-673	15	17	with	with	ADP
ejde-673	15	18	a	a	DET
ejde-673	15	19	lipschitz	lipschitz	NOUN
ejde-673	15	20	-	-	PUNCT
ejde-673	15	21	continuous	continuous	ADJ
ejde-673	15	22	(	(	PUNCT
ejde-673	15	23	c0,1	c0,1	NOUN
ejde-673	15	24	for	for	ADP
ejde-673	15	25	short	short	ADJ
ejde-673	15	26	)	)	PUNCT
ejde-673	15	27	boundary	boundary	ADJ
ejde-673	15	28	γ	γ	NOUN
ejde-673	15	29	satisfying	satisfy	VERB
ejde-673	15	30	that	that	SCONJ
ejde-673	15	31	γ1	γ1	NOUN
ejde-673	15	32	and	and	CCONJ
ejde-673	15	33	γ2	γ2	PROPN
ejde-673	15	34	are	be	AUX
ejde-673	15	35	disjoint	disjoint	ADJ
ejde-673	15	36	non	non	ADJ
ejde-673	15	37	-	-	ADJ
ejde-673	15	38	empty	empty	ADJ
ejde-673	15	39	open	open	ADJ
ejde-673	15	40	subsets	subset	NOUN
ejde-673	15	41	of	of	ADP
ejde-673	15	42	γ	γ	PROPN
ejde-673	15	43	such	such	ADJ
ejde-673	15	44	that	that	DET
ejde-673	15	45	γ1	γ1	PROPN
ejde-673	15	46	∪	∪	ADP
ejde-673	15	47	γ2	γ2	NOUN
ejde-673	15	48	=	=	SYM
ejde-673	15	49	γ	γ	X
ejde-673	15	50	,	,	PUNCT
ejde-673	15	51	(	(	PUNCT
ejde-673	15	52	1.2	1.2	NUM
ejde-673	15	53	)	)	PUNCT
ejde-673	15	54	and	and	CCONJ
ejde-673	15	55	the	the	DET
ejde-673	15	56	vector	vector	NOUN
ejde-673	15	57	field	field	NOUN
ejde-673	15	58	n	n	PRON
ejde-673	15	59	denotes	denote	VERB
ejde-673	15	60	the	the	DET
ejde-673	15	61	unit	unit	NOUN
ejde-673	15	62	,	,	PUNCT
ejde-673	15	63	outer	outer	ADJ
ejde-673	15	64	,	,	PUNCT
ejde-673	15	65	normal	normal	ADJ
ejde-673	15	66	vector	vector	NOUN
ejde-673	15	67	to	to	ADP
ejde-673	15	68	γ	γ	PROPN
ejde-673	15	69	.	.	PUNCT
ejde-673	16	1	furthermore	furthermore	ADV
ejde-673	16	2	,	,	PUNCT
ejde-673	16	3	a(x	a(x	PROPN
ejde-673	16	4	,	,	PUNCT
ejde-673	16	5	ξ	ξ	X
ejde-673	16	6	)	)	PUNCT
ejde-673	16	7	is	be	AUX
ejde-673	16	8	a	a	DET
ejde-673	16	9	carathéodory	carathéodory	NOUN
ejde-673	16	10	function	function	NOUN
ejde-673	16	11	on	on	ADP
ejde-673	16	12	ω	ω	NUM
ejde-673	16	13	×	×	NOUN
ejde-673	16	14	rn	rn	PROPN
ejde-673	16	15	satisfying	satisfy	VERB
ejde-673	16	16	some	some	DET
ejde-673	16	17	structure	structure	NOUN
ejde-673	16	18	conditions	condition	NOUN
ejde-673	16	19	associated	associate	VERB
ejde-673	16	20	with	with	ADP
ejde-673	16	21	an	an	DET
ejde-673	16	22	anisotropic	anisotropic	NOUN
ejde-673	16	23	exponent	exponent	NOUN
ejde-673	16	24	function	function	NOUN
ejde-673	16	25	p(x	p(x	PROPN
ejde-673	16	26	)	)	PUNCT
ejde-673	16	27	and	and	CCONJ
ejde-673	16	28	a(x	a(x	PROPN
ejde-673	16	29	,	,	PUNCT
ejde-673	16	30	ξ	ξ	X
ejde-673	16	31	)	)	PUNCT
ejde-673	16	32	is	be	AUX
ejde-673	16	33	a	a	DET
ejde-673	16	34	function	function	NOUN
ejde-673	16	35	satisfying	satisfy	VERB
ejde-673	16	36	∇ξa(x	∇ξa(x	NOUN
ejde-673	16	37	,	,	PUNCT
ejde-673	16	38	ξ	ξ	NOUN
ejde-673	16	39	)	)	PUNCT
ejde-673	16	40	=	=	SYM
ejde-673	16	41	a(x	a(x	NOUN
ejde-673	16	42	,	,	PUNCT
ejde-673	16	43	ξ	ξ	NOUN
ejde-673	16	44	)	)	PUNCT
ejde-673	16	45	.	.	PUNCT
ejde-673	17	1	here	here	ADV
ejde-673	17	2	we	we	PRON
ejde-673	17	3	say	say	VERB
ejde-673	17	4	that	that	SCONJ
ejde-673	17	5	a(x	a(x	NOUN
ejde-673	17	6	,	,	PUNCT
ejde-673	17	7	ξ	ξ	X
ejde-673	17	8	)	)	PUNCT
ejde-673	17	9	is	be	AUX
ejde-673	17	10	a	a	DET
ejde-673	17	11	carathéodory	carathéodory	NOUN
ejde-673	17	12	function	function	NOUN
ejde-673	17	13	on	on	ADP
ejde-673	17	14	ω	ω	NUM
ejde-673	17	15	×	×	PROPN
ejde-673	17	16	rn	rn	PROPN
ejde-673	17	17	,	,	PUNCT
ejde-673	17	18	if	if	SCONJ
ejde-673	17	19	for	for	ADP
ejde-673	17	20	a.e	a.e	PROPN
ejde-673	17	21	.	.	PUNCT
ejde-673	17	22	x	x	SYM
ejde-673	17	23	∈	∈	PROPN
ejde-673	17	24	ω	ω	PROPN
ejde-673	17	25	,	,	PUNCT
ejde-673	17	26	the	the	DET
ejde-673	17	27	map	map	NOUN
ejde-673	17	28	rn	rn	PROPN
ejde-673	17	29	∋	∋	NOUN
ejde-673	17	30	ξ	ξ	PROPN
ejde-673	17	31	7→	7→	PROPN
ejde-673	17	32	a(x	a(x	PROPN
ejde-673	17	33	,	,	PUNCT
ejde-673	17	34	ξ	ξ	X
ejde-673	17	35	)	)	PUNCT
ejde-673	17	36	is	be	AUX
ejde-673	17	37	continuous	continuous	ADJ
ejde-673	17	38	and	and	CCONJ
ejde-673	17	39	for	for	ADP
ejde-673	17	40	every	every	DET
ejde-673	17	41	ξ	ξ	PROPN
ejde-673	17	42	∈	∈	PROPN
ejde-673	17	43	rn	rn	PROPN
ejde-673	17	44	,	,	PUNCT
ejde-673	17	45	the	the	DET
ejde-673	17	46	map	map	NOUN
ejde-673	17	47	ω	ω	NUM
ejde-673	17	48	∋	∋	NOUN
ejde-673	17	49	x	x	X
ejde-673	17	50	7→	7→	NUM
ejde-673	17	51	a(x	a(x	PROPN
ejde-673	17	52	,	,	PUNCT
ejde-673	17	53	ξ	ξ	X
ejde-673	17	54	)	)	PUNCT
ejde-673	17	55	is	be	AUX
ejde-673	17	56	measurable	measurable	ADJ
ejde-673	17	57	on	on	ADP
ejde-673	17	58	ω	ω	PROPN
ejde-673	17	59	.	.	PUNCT
ejde-673	18	1	the	the	DET
ejde-673	18	2	operator	operator	NOUN
ejde-673	18	3	u	u	PROPN
ejde-673	18	4	7→	7→	NUM
ejde-673	18	5	div[a(x,∇u(x	div[a(x,∇u(x	NOUN
ejde-673	18	6	)	)	PUNCT
ejde-673	18	7	)	)	PUNCT
ejde-673	18	8	]	]	PUNCT
ejde-673	18	9	is	be	AUX
ejde-673	18	10	more	more	ADV
ejde-673	18	11	general	general	ADJ
ejde-673	18	12	than	than	ADP
ejde-673	18	13	the	the	DET
ejde-673	18	14	p(·)-laplacian	p(·)-laplacian	ADJ
ejde-673	18	15	∆p(x)u(x	∆p(x)u(x	PROPN
ejde-673	18	16	)	)	PUNCT
ejde-673	18	17	:	:	PUNCT
ejde-673	18	18	=	=	PUNCT
ejde-673	18	19	div[|∇u(x)|p(x)−2∇u(x	div[|∇u(x)|p(x)−2∇u(x	NOUN
ejde-673	18	20	)	)	PUNCT
ejde-673	18	21	]	]	PUNCT
ejde-673	18	22	and	and	CCONJ
ejde-673	18	23	the	the	DET
ejde-673	18	24	mean	mean	ADJ
ejde-673	18	25	curvature	curvature	NOUN
ejde-673	18	26	operator	operator	NOUN
ejde-673	18	27	div[(1	div[(1	PROPN
ejde-673	18	28	+	+	CCONJ
ejde-673	18	29	|∇u(x)|2)(p(x)−2)/2∇u(x	|∇u(x)|2)(p(x)−2)/2∇u(x	NOUN
ejde-673	18	30	)	)	PUNCT
ejde-673	18	31	]	]	PUNCT
ejde-673	18	32	.	.	PUNCT
ejde-673	19	1	this	this	DET
ejde-673	19	2	generality	generality	NOUN
ejde-673	19	3	brings	bring	VERB
ejde-673	19	4	about	about	ADP
ejde-673	19	5	difficulties	difficulty	NOUN
ejde-673	19	6	and	and	CCONJ
ejde-673	19	7	requires	require	VERB
ejde-673	19	8	2020	2020	NUM
ejde-673	19	9	mathematics	mathematic	NOUN
ejde-673	19	10	subject	subject	ADJ
ejde-673	19	11	classification	classification	NOUN
ejde-673	19	12	.	.	PUNCT
ejde-673	20	1	49r50	49r50	NUM
ejde-673	20	2	,	,	PUNCT
ejde-673	20	3	35a01	35a01	NUM
ejde-673	20	4	,	,	PUNCT
ejde-673	20	5	35j62	35j62	NUM
ejde-673	20	6	,	,	PUNCT
ejde-673	20	7	35j57	35j57	NUM
ejde-673	20	8	.	.	PUNCT
ejde-673	21	1	key	key	ADJ
ejde-673	21	2	words	word	NOUN
ejde-673	21	3	and	and	CCONJ
ejde-673	21	4	phrases	phrase	NOUN
ejde-673	21	5	.	.	PUNCT
ejde-673	22	1	eigenvalue	eigenvalue	PROPN
ejde-673	22	2	problem	problem	NOUN
ejde-673	22	3	;	;	PUNCT
ejde-673	22	4	kirchhoff	kirchhoff	NOUN
ejde-673	22	5	-	-	PUNCT
ejde-673	22	6	type	type	NOUN
ejde-673	22	7	operator	operator	NOUN
ejde-673	22	8	;	;	PUNCT
ejde-673	22	9	p(·)-laplacian	p(·)-laplacian	ADJ
ejde-673	22	10	;	;	PUNCT
ejde-673	22	11	mean	mean	ADJ
ejde-673	22	12	curvature	curvature	NOUN
ejde-673	22	13	operator	operator	NOUN
ejde-673	22	14	;	;	PUNCT
ejde-673	22	15	mixed	mixed	ADJ
ejde-673	22	16	boundary	boundary	ADJ
ejde-673	22	17	value	value	NOUN
ejde-673	22	18	problem	problem	NOUN
ejde-673	22	19	;	;	PUNCT
ejde-673	22	20	variable	variable	ADJ
ejde-673	22	21	exponent	exponent	NOUN
ejde-673	22	22	sobolev	sobolev	NOUN
ejde-673	22	23	space	space	NOUN
ejde-673	22	24	.	.	PUNCT
ejde-673	23	1	©	©	PROPN
ejde-673	23	2	2025	2025	NUM
ejde-673	23	3	.	.	PUNCT
ejde-673	24	1	this	this	DET
ejde-673	24	2	work	work	NOUN
ejde-673	24	3	is	be	AUX
ejde-673	24	4	licensed	license	VERB
ejde-673	24	5	under	under	ADP
ejde-673	24	6	a	a	DET
ejde-673	24	7	cc	cc	NOUN
ejde-673	24	8	by	by	ADP
ejde-673	24	9	4.0	4.0	NUM
ejde-673	24	10	license	license	NOUN
ejde-673	24	11	.	.	PUNCT
ejde-673	25	1	submitted	submit	VERB
ejde-673	25	2	april	april	PROPN
ejde-673	25	3	21	21	NUM
ejde-673	25	4	,	,	PUNCT
ejde-673	25	5	2024	2024	NUM
ejde-673	25	6	.	.	PUNCT
ejde-673	26	1	published	publish	VERB
ejde-673	26	2	february	february	PROPN
ejde-673	26	3	25	25	NUM
ejde-673	26	4	,	,	PUNCT
ejde-673	26	5	2025	2025	NUM
ejde-673	26	6	.	.	PUNCT
ejde-673	27	1	1	1	NUM
ejde-673	27	2	2	2	NUM
ejde-673	27	3	j.	j.	PROPN
ejde-673	27	4	aramaki	aramaki	PROPN
ejde-673	27	5	ejde-2025/17	ejde-2025/17	VERB
ejde-673	27	6	some	some	DET
ejde-673	27	7	conditions	condition	NOUN
ejde-673	27	8	.	.	PUNCT
ejde-673	28	1	the	the	DET
ejde-673	28	2	function	function	NOUN
ejde-673	28	3	m	m	PROPN
ejde-673	28	4	=	=	SYM
ejde-673	28	5	m(s	m(s	PROPN
ejde-673	28	6	)	)	PUNCT
ejde-673	28	7	defined	define	VERB
ejde-673	28	8	in	in	ADP
ejde-673	28	9	[	[	X
ejde-673	28	10	0,∞	0,∞	NOUN
ejde-673	28	11	)	)	PUNCT
ejde-673	28	12	satisfies	satisfy	VERB
ejde-673	28	13	the	the	DET
ejde-673	28	14	following	follow	VERB
ejde-673	28	15	condition	condition	NOUN
ejde-673	28	16	(	(	PUNCT
ejde-673	28	17	a1	a1	PROPN
ejde-673	28	18	)	)	PUNCT
ejde-673	28	19	m	m	VERB
ejde-673	28	20	:	:	PUNCT
ejde-673	29	1	[	[	X
ejde-673	29	2	0,∞	0,∞	NOUN
ejde-673	29	3	)	)	PUNCT
ejde-673	29	4	→	→	PUNCT
ejde-673	30	1	[	[	X
ejde-673	30	2	0,∞	0,∞	NUM
ejde-673	30	3	)	)	PUNCT
ejde-673	30	4	is	be	AUX
ejde-673	30	5	continuous	continuous	ADJ
ejde-673	30	6	and	and	CCONJ
ejde-673	30	7	monotone	monotone	ADJ
ejde-673	30	8	non	non	ADJ
ejde-673	30	9	-	-	ADJ
ejde-673	30	10	decreasing	decrease	VERB
ejde-673	30	11	,	,	PUNCT
ejde-673	30	12	and	and	CCONJ
ejde-673	30	13	there	there	PRON
ejde-673	30	14	exist	exist	VERB
ejde-673	30	15	0	0	NUM
ejde-673	30	16	<	<	X
ejde-673	30	17	m0	m0	PROPN
ejde-673	30	18	≤	≤	PROPN
ejde-673	30	19	m1	m1	PROPN
ejde-673	30	20	<	<	X
ejde-673	30	21	∞	∞	PROPN
ejde-673	30	22	and	and	CCONJ
ejde-673	30	23	k	k	PROPN
ejde-673	30	24	≥	≥	PROPN
ejde-673	30	25	l	l	NOUN
ejde-673	30	26	≥	≥	NUM
ejde-673	30	27	1	1	NUM
ejde-673	30	28	such	such	ADJ
ejde-673	30	29	that	that	PRON
ejde-673	30	30	m0s	m0s	PROPN
ejde-673	30	31	l−1	l−1	PROPN
ejde-673	30	32	≤	≤	NUM
ejde-673	30	33	m(s	m(s	PROPN
ejde-673	30	34	)	)	PUNCT
ejde-673	30	35	≤	≤	NOUN
ejde-673	30	36	m1(1	m1(1	ADJ
ejde-673	30	37	+	+	SYM
ejde-673	30	38	sk−1	sk−1	NOUN
ejde-673	30	39	)	)	PUNCT
ejde-673	30	40	for	for	ADP
ejde-673	30	41	s	s	PRON
ejde-673	30	42	≥	≥	NOUN
ejde-673	30	43	0	0	NUM
ejde-673	30	44	.	.	PUNCT
ejde-673	31	1	(	(	PUNCT
ejde-673	31	2	1.3	1.3	NUM
ejde-673	31	3	)	)	PUNCT
ejde-673	31	4	we	we	PRON
ejde-673	31	5	impose	impose	VERB
ejde-673	31	6	the	the	DET
ejde-673	31	7	mixed	mixed	ADJ
ejde-673	31	8	boundary	boundary	ADJ
ejde-673	31	9	conditions	condition	NOUN
ejde-673	31	10	,	,	PUNCT
ejde-673	31	11	that	that	ADV
ejde-673	31	12	is	is	ADV
ejde-673	31	13	,	,	PUNCT
ejde-673	31	14	the	the	DET
ejde-673	31	15	dirichlet	dirichlet	PROPN
ejde-673	31	16	condition	condition	NOUN
ejde-673	31	17	on	on	ADP
ejde-673	31	18	γ1	γ1	PROPN
ejde-673	31	19	and	and	CCONJ
ejde-673	31	20	the	the	DET
ejde-673	31	21	steklov	steklov	ADJ
ejde-673	31	22	condition	condition	NOUN
ejde-673	31	23	on	on	ADP
ejde-673	31	24	γ2	γ2	PROPN
ejde-673	31	25	.	.	PUNCT
ejde-673	32	1	the	the	DET
ejde-673	32	2	given	give	VERB
ejde-673	32	3	data	datum	NOUN
ejde-673	32	4	g	g	NOUN
ejde-673	32	5	:	:	PUNCT
ejde-673	32	6	γ2×r	γ2×r	PROPN
ejde-673	32	7	→	→	SYM
ejde-673	32	8	r	r	NOUN
ejde-673	32	9	is	be	AUX
ejde-673	32	10	a	a	DET
ejde-673	32	11	carathéodory	carathéodory	NOUN
ejde-673	32	12	function	function	NOUN
ejde-673	32	13	of	of	ADP
ejde-673	32	14	special	special	ADJ
ejde-673	32	15	type	type	NOUN
ejde-673	32	16	and	and	CCONJ
ejde-673	32	17	λ	λ	NOUN
ejde-673	32	18	is	be	AUX
ejde-673	32	19	a	a	DET
ejde-673	32	20	real	real	ADJ
ejde-673	32	21	number	number	NOUN
ejde-673	32	22	.	.	PUNCT
ejde-673	33	1	the	the	DET
ejde-673	33	2	study	study	NOUN
ejde-673	33	3	of	of	ADP
ejde-673	33	4	differential	differential	ADJ
ejde-673	33	5	equations	equation	NOUN
ejde-673	33	6	with	with	ADP
ejde-673	33	7	p(·)-growth	p(·)-growth	NOUN
ejde-673	33	8	conditions	condition	NOUN
ejde-673	33	9	is	be	AUX
ejde-673	33	10	a	a	DET
ejde-673	33	11	very	very	ADV
ejde-673	33	12	interesting	interesting	ADJ
ejde-673	33	13	topic	topic	NOUN
ejde-673	33	14	recently	recently	ADV
ejde-673	33	15	.	.	PUNCT
ejde-673	34	1	studying	study	VERB
ejde-673	34	2	such	such	ADJ
ejde-673	34	3	problem	problem	NOUN
ejde-673	34	4	stimulated	stimulate	VERB
ejde-673	34	5	its	its	PRON
ejde-673	34	6	application	application	NOUN
ejde-673	34	7	in	in	ADP
ejde-673	34	8	mathematical	mathematical	ADJ
ejde-673	34	9	physics	physics	NOUN
ejde-673	34	10	,	,	PUNCT
ejde-673	34	11	in	in	ADP
ejde-673	34	12	particular	particular	ADJ
ejde-673	34	13	,	,	PUNCT
ejde-673	34	14	in	in	ADP
ejde-673	34	15	elastic	elastic	ADJ
ejde-673	34	16	mechanics	mechanic	NOUN
ejde-673	34	17	(	(	PUNCT
ejde-673	34	18	zhikov	zhikov	PROPN
ejde-673	34	19	[	[	X
ejde-673	34	20	36	36	NUM
ejde-673	34	21	]	]	NUM
ejde-673	34	22	)	)	PUNCT
ejde-673	34	23	,	,	PUNCT
ejde-673	34	24	in	in	ADP
ejde-673	34	25	electrorheological	electrorheological	ADJ
ejde-673	34	26	fluids	fluid	NOUN
ejde-673	34	27	(	(	PUNCT
ejde-673	34	28	diening	diene	VERB
ejde-673	34	29	[	[	X
ejde-673	34	30	12	12	NUM
ejde-673	34	31	]	]	PUNCT
ejde-673	34	32	,	,	PUNCT
ejde-673	34	33	halsey	halsey	PROPN
ejde-673	35	1	[	[	X
ejde-673	35	2	21	21	NUM
ejde-673	35	3	]	]	PUNCT
ejde-673	35	4	,	,	PUNCT
ejde-673	35	5	mihăilescu	mihăilescu	PROPN
ejde-673	35	6	and	and	CCONJ
ejde-673	35	7	rădulescu	rădulescu	PROPN
ejde-673	36	1	[	[	X
ejde-673	36	2	28	28	NUM
ejde-673	36	3	]	]	PUNCT
ejde-673	36	4	,	,	PUNCT
ejde-673	36	5	růz̆ic̆ka	růz̆ic̆ka	PROPN
ejde-673	37	1	[	[	X
ejde-673	37	2	31	31	NUM
ejde-673	37	3	]	]	PUNCT
ejde-673	37	4	)	)	PUNCT
ejde-673	37	5	.	.	PUNCT
ejde-673	38	1	as	as	ADP
ejde-673	38	2	recent	recent	ADJ
ejde-673	38	3	works	work	NOUN
ejde-673	38	4	,	,	PUNCT
ejde-673	38	5	we	we	PRON
ejde-673	38	6	can	can	AUX
ejde-673	38	7	find	find	VERB
ejde-673	38	8	some	some	DET
ejde-673	38	9	interesting	interesting	ADJ
ejde-673	38	10	related	related	ADJ
ejde-673	38	11	articles	article	NOUN
ejde-673	38	12	.	.	PUNCT
ejde-673	39	1	see	see	VERB
ejde-673	39	2	alves	alves	PROPN
ejde-673	39	3	et	et	PROPN
ejde-673	39	4	al	al	PROPN
ejde-673	39	5	.	.	PUNCT
ejde-673	40	1	[	[	X
ejde-673	40	2	2	2	NUM
ejde-673	40	3	]	]	PUNCT
ejde-673	40	4	,	,	PUNCT
ejde-673	40	5	alves	alves	PROPN
ejde-673	40	6	and	and	CCONJ
ejde-673	40	7	tavares	tavare	NOUN
ejde-673	41	1	[	[	X
ejde-673	41	2	3	3	NUM
ejde-673	41	3	]	]	PUNCT
ejde-673	41	4	.	.	PUNCT
ejde-673	42	1	however	however	ADV
ejde-673	42	2	,	,	PUNCT
ejde-673	42	3	in	in	ADP
ejde-673	42	4	even	even	ADV
ejde-673	42	5	the	the	DET
ejde-673	42	6	case	case	NOUN
ejde-673	42	7	m	m	X
ejde-673	42	8	≡	≡	PROPN
ejde-673	42	9	1	1	NUM
ejde-673	42	10	,	,	PUNCT
ejde-673	42	11	as	as	SCONJ
ejde-673	42	12	we	we	PRON
ejde-673	42	13	only	only	ADV
ejde-673	42	14	find	find	VERB
ejde-673	42	15	a	a	DET
ejde-673	42	16	few	few	ADJ
ejde-673	42	17	papers	paper	NOUN
ejde-673	42	18	associate	associate	VERB
ejde-673	42	19	with	with	ADP
ejde-673	42	20	the	the	DET
ejde-673	42	21	problem	problem	NOUN
ejde-673	42	22	with	with	ADP
ejde-673	42	23	the	the	DET
ejde-673	42	24	mixed	mixed	ADJ
ejde-673	42	25	boundary	boundary	ADJ
ejde-673	42	26	condition	condition	NOUN
ejde-673	42	27	in	in	ADP
ejde-673	42	28	variable	variable	ADJ
ejde-673	42	29	exponent	exponent	NOUN
ejde-673	42	30	sobolev	sobolev	NOUN
ejde-673	42	31	space	space	NOUN
ejde-673	42	32	as	as	ADP
ejde-673	42	33	in	in	ADP
ejde-673	42	34	(	(	PUNCT
ejde-673	42	35	1.1	1.1	NUM
ejde-673	42	36	)	)	PUNCT
ejde-673	42	37	(	(	PUNCT
ejde-673	42	38	for	for	ADP
ejde-673	42	39	example	example	NOUN
ejde-673	42	40	,	,	PUNCT
ejde-673	42	41	aramaki	aramaki	NOUN
ejde-673	43	1	[	[	X
ejde-673	43	2	5	5	NUM
ejde-673	43	3	,	,	PUNCT
ejde-673	43	4	6	6	NUM
ejde-673	43	5	]	]	NUM
ejde-673	43	6	)	)	PUNCT
ejde-673	43	7	,	,	PUNCT
ejde-673	43	8	we	we	PRON
ejde-673	43	9	are	be	AUX
ejde-673	43	10	convinced	convince	VERB
ejde-673	43	11	of	of	ADP
ejde-673	43	12	the	the	DET
ejde-673	43	13	reason	reason	NOUN
ejde-673	43	14	for	for	ADP
ejde-673	43	15	existence	existence	NOUN
ejde-673	43	16	of	of	ADP
ejde-673	43	17	this	this	DET
ejde-673	43	18	article	article	NOUN
ejde-673	43	19	.	.	PUNCT
ejde-673	44	1	when	when	SCONJ
ejde-673	44	2	p(x	p(x	NOUN
ejde-673	44	3	)	)	PUNCT
ejde-673	44	4	≡	≡	PROPN
ejde-673	44	5	p	p	PROPN
ejde-673	44	6	(	(	PUNCT
ejde-673	44	7	a	a	DET
ejde-673	44	8	constant	constant	ADJ
ejde-673	44	9	)	)	PUNCT
ejde-673	44	10	,	,	PUNCT
ejde-673	44	11	there	there	PRON
ejde-673	44	12	are	be	VERB
ejde-673	44	13	many	many	ADJ
ejde-673	44	14	articles	article	NOUN
ejde-673	44	15	for	for	ADP
ejde-673	44	16	the	the	DET
ejde-673	44	17	p	p	NOUN
ejde-673	44	18	-	-	PUNCT
ejde-673	44	19	laplacian	laplacian	NOUN
ejde-673	44	20	.	.	PUNCT
ejde-673	45	1	for	for	ADP
ejde-673	45	2	example	example	NOUN
ejde-673	45	3	,	,	PUNCT
ejde-673	45	4	see	see	VERB
ejde-673	45	5	lê	lê	PROPN
ejde-673	46	1	[	[	X
ejde-673	46	2	24	24	NUM
ejde-673	46	3	]	]	PUNCT
ejde-673	46	4	,	,	PUNCT
ejde-673	46	5	anane	anane	PROPN
ejde-673	47	1	[	[	X
ejde-673	47	2	4	4	NUM
ejde-673	47	3	]	]	PUNCT
ejde-673	47	4	,	,	PUNCT
ejde-673	47	5	friedlander	friedlander	NOUN
ejde-673	47	6	[	[	X
ejde-673	47	7	20	20	NUM
ejde-673	47	8	]	]	PUNCT
ejde-673	47	9	.	.	PUNCT
ejde-673	48	1	for	for	ADP
ejde-673	48	2	the	the	DET
ejde-673	48	3	p	p	ADJ
ejde-673	48	4	-	-	PUNCT
ejde-673	48	5	laplacian	laplacian	ADJ
ejde-673	48	6	dirichlet	dirichlet	PROPN
ejde-673	48	7	eigenvalue	eigenvalue	PROPN
ejde-673	48	8	problem	problem	NOUN
ejde-673	48	9	:	:	PUNCT
ejde-673	48	10	−∆pu(x	−∆pu(x	PROPN
ejde-673	48	11	)	)	PUNCT
ejde-673	48	12	=	=	SYM
ejde-673	48	13	λ|u(x)|p−2u(x	λ|u(x)|p−2u(x	PROPN
ejde-673	48	14	)	)	PUNCT
ejde-673	48	15	in	in	ADP
ejde-673	48	16	ω	ω	PROPN
ejde-673	48	17	,	,	PUNCT
ejde-673	48	18	u(x	u(x	X
ejde-673	48	19	)	)	PUNCT
ejde-673	48	20	=	=	SYM
ejde-673	48	21	0	0	NUM
ejde-673	48	22	on	on	ADP
ejde-673	48	23	γ	γ	PROPN
ejde-673	48	24	,	,	PUNCT
ejde-673	48	25	we	we	PRON
ejde-673	48	26	can	can	AUX
ejde-673	48	27	see	see	VERB
ejde-673	48	28	that	that	SCONJ
ejde-673	48	29	the	the	DET
ejde-673	48	30	following	follow	VERB
ejde-673	48	31	properties	property	NOUN
ejde-673	48	32	hold	hold	VERB
ejde-673	48	33	.	.	PUNCT
ejde-673	49	1	(	(	PUNCT
ejde-673	49	2	1	1	X
ejde-673	49	3	)	)	PUNCT
ejde-673	49	4	there	there	PRON
ejde-673	49	5	exists	exist	VERB
ejde-673	49	6	a	a	DET
ejde-673	49	7	nondecreasing	nondecrease	VERB
ejde-673	49	8	sequence	sequence	NOUN
ejde-673	49	9	of	of	ADP
ejde-673	49	10	positive	positive	ADJ
ejde-673	49	11	eigenvalues	eigenvalue	NOUN
ejde-673	49	12	{	{	PUNCT
ejde-673	49	13	λn	λn	AUX
ejde-673	49	14	}	}	PUNCT
ejde-673	49	15	tending	tend	VERB
ejde-673	49	16	to	to	ADP
ejde-673	49	17	∞	∞	NUM
ejde-673	49	18	as	as	ADP
ejde-673	49	19	n	n	PROPN
ejde-673	49	20	→	→	SYM
ejde-673	49	21	∞.	∞.	PROPN
ejde-673	49	22	(	(	PUNCT
ejde-673	49	23	2	2	NUM
ejde-673	49	24	)	)	PUNCT
ejde-673	49	25	the	the	DET
ejde-673	49	26	first	first	PROPN
ejde-673	49	27	eigenvalue	eigenvalue	PROPN
ejde-673	49	28	λ1	λ1	PROPN
ejde-673	49	29	is	be	AUX
ejde-673	49	30	simple	simple	ADJ
ejde-673	49	31	and	and	CCONJ
ejde-673	49	32	only	only	ADV
ejde-673	49	33	eigenfunctions	eigenfunction	NOUN
ejde-673	49	34	associated	associate	VERB
ejde-673	49	35	with	with	ADP
ejde-673	49	36	λ1	λ1	PROPN
ejde-673	49	37	do	do	AUX
ejde-673	49	38	not	not	PART
ejde-673	49	39	change	change	VERB
ejde-673	49	40	sign	sign	NOUN
ejde-673	49	41	.	.	PUNCT
ejde-673	50	1	(	(	PUNCT
ejde-673	50	2	3	3	X
ejde-673	50	3	)	)	PUNCT
ejde-673	50	4	the	the	DET
ejde-673	50	5	set	set	NOUN
ejde-673	50	6	of	of	ADP
ejde-673	50	7	eigenvalues	eigenvalue	NOUN
ejde-673	50	8	is	be	AUX
ejde-673	50	9	closed	closed	ADJ
ejde-673	50	10	.	.	PUNCT
ejde-673	51	1	(	(	PUNCT
ejde-673	51	2	4	4	X
ejde-673	51	3	)	)	PUNCT
ejde-673	51	4	the	the	DET
ejde-673	51	5	first	first	PROPN
ejde-673	51	6	eigenvalue	eigenvalue	PROPN
ejde-673	51	7	λ1	λ1	PROPN
ejde-673	51	8	is	be	AUX
ejde-673	51	9	isolated	isolate	VERB
ejde-673	51	10	.	.	PUNCT
ejde-673	52	1	on	on	ADP
ejde-673	52	2	the	the	DET
ejde-673	52	3	contrary	contrary	NOUN
ejde-673	52	4	,	,	PUNCT
ejde-673	52	5	recently	recently	ADV
ejde-673	52	6	many	many	ADJ
ejde-673	52	7	authors	author	NOUN
ejde-673	52	8	study	study	VERB
ejde-673	52	9	the	the	DET
ejde-673	52	10	p(·)-laplacian	p(·)-laplacian	PROPN
ejde-673	52	11	.	.	PUNCT
ejde-673	53	1	in	in	ADP
ejde-673	53	2	particular	particular	ADJ
ejde-673	53	3	,	,	PUNCT
ejde-673	53	4	fan	fan	NOUN
ejde-673	54	1	[	[	X
ejde-673	54	2	15	15	NUM
ejde-673	54	3	]	]	PUNCT
ejde-673	54	4	has	have	AUX
ejde-673	54	5	studied	study	VERB
ejde-673	54	6	the	the	DET
ejde-673	54	7	eigenvalue	eigenvalue	PROPN
ejde-673	54	8	problem	problem	NOUN
ejde-673	54	9	for	for	ADP
ejde-673	54	10	the	the	DET
ejde-673	54	11	p(·)-laplacian	p(·)-laplacian	NOUN
ejde-673	54	12	with	with	ADP
ejde-673	54	13	zero	zero	NUM
ejde-673	54	14	neumann	neumann	PROPN
ejde-673	54	15	boundary	boundary	ADJ
ejde-673	54	16	condition	condition	NOUN
ejde-673	54	17	in	in	ADP
ejde-673	54	18	a	a	DET
ejde-673	54	19	bounded	bounded	ADJ
ejde-673	54	20	domain	domain	NOUN
ejde-673	54	21	,	,	PUNCT
ejde-673	54	22	and	and	CCONJ
ejde-673	54	23	fan	fan	NOUN
ejde-673	54	24	et	et	PROPN
ejde-673	54	25	al	al	PROPN
ejde-673	54	26	.	.	PUNCT
ejde-673	55	1	[	[	X
ejde-673	55	2	19	19	NUM
ejde-673	55	3	]	]	PUNCT
ejde-673	55	4	has	have	AUX
ejde-673	55	5	studied	study	VERB
ejde-673	55	6	the	the	DET
ejde-673	55	7	eigenvalue	eigenvalue	PROPN
ejde-673	55	8	problem	problem	NOUN
ejde-673	55	9	for	for	ADP
ejde-673	55	10	the	the	DET
ejde-673	55	11	p(·)-laplacian	p(·)-laplacian	ADJ
ejde-673	55	12	dirichlet	dirichlet	PROPN
ejde-673	55	13	problem	problem	NOUN
ejde-673	55	14	.	.	PUNCT
ejde-673	56	1	mihăilescu	mihăilescu	ADV
ejde-673	56	2	and	and	CCONJ
ejde-673	56	3	rădulescu	rădulescu	PROPN
ejde-673	56	4	[	[	X
ejde-673	56	5	29	29	NUM
ejde-673	56	6	]	]	PUNCT
ejde-673	56	7	have	have	AUX
ejde-673	56	8	studied	study	VERB
ejde-673	56	9	nonhomogeneous	nonhomogeneous	ADJ
ejde-673	56	10	quasilinear	quasilinear	NOUN
ejde-673	56	11	eigenvalue	eigenvalue	NOUN
ejde-673	56	12	problem	problem	NOUN
ejde-673	56	13	with	with	ADP
ejde-673	56	14	variable	variable	ADJ
ejde-673	56	15	exponent	exponent	NOUN
ejde-673	56	16	.	.	PUNCT
ejde-673	57	1	in	in	ADP
ejde-673	57	2	deng	deng	PROPN
ejde-673	57	3	[	[	X
ejde-673	57	4	11	11	NUM
ejde-673	57	5	]	]	PUNCT
ejde-673	57	6	,	,	PUNCT
ejde-673	57	7	the	the	DET
ejde-673	57	8	author	author	NOUN
ejde-673	57	9	treats	treat	VERB
ejde-673	57	10	only	only	ADV
ejde-673	57	11	the	the	DET
ejde-673	57	12	p(·)-laplacian	p(·)-laplacian	NOUN
ejde-673	57	13	in	in	ADP
ejde-673	57	14	the	the	DET
ejde-673	57	15	case	case	NOUN
ejde-673	57	16	γ1	γ1	NOUN
ejde-673	57	17	=	=	NOUN
ejde-673	57	18	∅	∅	NOUN
ejde-673	57	19	,	,	PUNCT
ejde-673	57	20	that	that	ADV
ejde-673	57	21	is	is	ADV
ejde-673	57	22	,	,	PUNCT
ejde-673	57	23	−∆p(x)u(x	−∆p(x)u(x	PROPN
ejde-673	57	24	)	)	PUNCT
ejde-673	57	25	+	+	NUM
ejde-673	57	26	|u(x)|p(x)−2u(x	|u(x)|p(x)−2u(x	NOUN
ejde-673	57	27	)	)	PUNCT
ejde-673	57	28	=	=	SYM
ejde-673	57	29	0	0	NUM
ejde-673	57	30	in	in	ADP
ejde-673	57	31	ω	ω	NUM
ejde-673	57	32	,	,	PUNCT
ejde-673	57	33	|∇u(x)|p(x)−2	|∇u(x)|p(x)−2	PROPN
ejde-673	57	34	∂u(x	∂u(x	PROPN
ejde-673	57	35	)	)	PUNCT
ejde-673	57	36	∂n	∂n	PROPN
ejde-673	57	37	=	=	SYM
ejde-673	57	38	λ|u(x)|p(x)−2u(x	λ|u(x)|p(x)−2u(x	PROPN
ejde-673	57	39	)	)	PUNCT
ejde-673	57	40	on	on	ADP
ejde-673	57	41	γ	γ	PROPN
ejde-673	57	42	.	.	PROPN
ejde-673	58	1	(	(	PUNCT
ejde-673	58	2	1.4	1.4	NUM
ejde-673	58	3	)	)	PUNCT
ejde-673	58	4	as	as	SCONJ
ejde-673	58	5	the	the	DET
ejde-673	58	6	author	author	NOUN
ejde-673	58	7	takes	take	VERB
ejde-673	58	8	the	the	DET
ejde-673	58	9	variable	variable	ADJ
ejde-673	58	10	exponent	exponent	NOUN
ejde-673	58	11	sobolev	sobolev	PROPN
ejde-673	58	12	space	space	PROPN
ejde-673	58	13	w	w	PROPN
ejde-673	58	14	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	58	15	)	)	PUNCT
ejde-673	58	16	as	as	ADP
ejde-673	58	17	the	the	DET
ejde-673	58	18	base	base	NOUN
ejde-673	58	19	space	space	NOUN
ejde-673	58	20	,	,	PUNCT
ejde-673	58	21	the	the	DET
ejde-673	58	22	second	second	ADJ
ejde-673	58	23	term	term	NOUN
ejde-673	58	24	in	in	ADP
ejde-673	58	25	the	the	DET
ejde-673	58	26	left	left	ADJ
ejde-673	58	27	-	-	PUNCT
ejde-673	58	28	hand	hand	NOUN
ejde-673	58	29	side	side	NOUN
ejde-673	58	30	of	of	ADP
ejde-673	58	31	the	the	DET
ejde-673	58	32	first	first	ADJ
ejde-673	58	33	equation	equation	NOUN
ejde-673	58	34	of	of	ADP
ejde-673	58	35	(	(	PUNCT
ejde-673	58	36	1.4	1.4	NUM
ejde-673	58	37	)	)	PUNCT
ejde-673	58	38	takes	take	VERB
ejde-673	58	39	the	the	DET
ejde-673	58	40	essential	essential	ADJ
ejde-673	58	41	role	role	NOUN
ejde-673	58	42	.	.	PUNCT
ejde-673	59	1	however	however	ADV
ejde-673	59	2	,	,	PUNCT
ejde-673	59	3	if	if	SCONJ
ejde-673	59	4	we	we	PRON
ejde-673	59	5	assume	assume	VERB
ejde-673	59	6	that	that	SCONJ
ejde-673	59	7	γ1	γ1	PROPN
ejde-673	59	8	̸=	̸=	PROPN
ejde-673	59	9	∅	∅	NOUN
ejde-673	59	10	,	,	PUNCT
ejde-673	59	11	we	we	PRON
ejde-673	59	12	can	can	AUX
ejde-673	59	13	delete	delete	VERB
ejde-673	59	14	such	such	DET
ejde-673	59	15	a	a	DET
ejde-673	59	16	term	term	NOUN
ejde-673	59	17	according	accord	VERB
ejde-673	59	18	to	to	ADP
ejde-673	59	19	the	the	DET
ejde-673	59	20	poincaré	poincaré	ADJ
ejde-673	59	21	type	type	NOUN
ejde-673	59	22	inequality	inequality	NOUN
ejde-673	59	23	due	due	ADP
ejde-673	59	24	to	to	ADP
ejde-673	59	25	ciarlet	ciarlet	VERB
ejde-673	59	26	and	and	CCONJ
ejde-673	59	27	dinca	dinca	VERB
ejde-673	60	1	[	[	X
ejde-673	60	2	10	10	NUM
ejde-673	60	3	]	]	PUNCT
ejde-673	60	4	.	.	PUNCT
ejde-673	61	1	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	61	2	eigenvalue	eigenvalue	VERB
ejde-673	61	3	problems	problem	NOUN
ejde-673	61	4	for	for	ADP
ejde-673	61	5	kirchhoff	kirchhoff	NOUN
ejde-673	61	6	-	-	PUNCT
ejde-673	61	7	type	type	NOUN
ejde-673	61	8	equations	equation	NOUN
ejde-673	61	9	3	3	NUM
ejde-673	61	10	for	for	ADP
ejde-673	61	11	physical	physical	ADJ
ejde-673	61	12	motivation	motivation	NOUN
ejde-673	61	13	to	to	ADP
ejde-673	61	14	the	the	DET
ejde-673	61	15	problem	problem	NOUN
ejde-673	61	16	(	(	PUNCT
ejde-673	61	17	1.1	1.1	NUM
ejde-673	61	18	)	)	PUNCT
ejde-673	61	19	,	,	PUNCT
ejde-673	61	20	we	we	PRON
ejde-673	61	21	consider	consider	VERB
ejde-673	61	22	the	the	DET
ejde-673	61	23	case	case	NOUN
ejde-673	61	24	where	where	SCONJ
ejde-673	61	25	γ	γ	PROPN
ejde-673	61	26	=	=	SYM
ejde-673	61	27	γ1	γ1	PROPN
ejde-673	61	28	and	and	CCONJ
ejde-673	61	29	p(x	p(x	PROPN
ejde-673	61	30	)	)	PUNCT
ejde-673	61	31	=	=	SYM
ejde-673	61	32	2	2	X
ejde-673	61	33	.	.	PUNCT
ejde-673	61	34	then	then	ADV
ejde-673	61	35	the	the	DET
ejde-673	61	36	equation	equation	NOUN
ejde-673	61	37	m(∥∇u∥2l2(ω))∆u(x	m(∥∇u∥2l2(ω))∆u(x	VERB
ejde-673	61	38	)	)	PUNCT
ejde-673	61	39	=	=	SYM
ejde-673	61	40	f(x	f(x	PROPN
ejde-673	61	41	,	,	PUNCT
ejde-673	61	42	u(x	u(x	NOUN
ejde-673	61	43	)	)	PUNCT
ejde-673	61	44	)	)	PUNCT
ejde-673	61	45	(	(	PUNCT
ejde-673	61	46	1.5	1.5	NUM
ejde-673	61	47	)	)	PUNCT
ejde-673	61	48	is	be	AUX
ejde-673	61	49	the	the	DET
ejde-673	61	50	kirchhoff	kirchhoff	NOUN
ejde-673	61	51	equation	equation	NOUN
ejde-673	61	52	which	which	PRON
ejde-673	61	53	arises	arise	VERB
ejde-673	61	54	in	in	ADP
ejde-673	61	55	nonlinear	nonlinear	ADJ
ejde-673	61	56	vibration	vibration	NOUN
ejde-673	61	57	,	,	PUNCT
ejde-673	61	58	namely	namely	ADV
ejde-673	61	59	utt	utt	ADJ
ejde-673	61	60	−m(∥∇u∥2l2(ω))∆u	−m(∥∇u∥2l2(ω))∆u	PUNCT
ejde-673	61	61	=	=	PUNCT
ejde-673	61	62	f(x	f(x	PROPN
ejde-673	61	63	,	,	PUNCT
ejde-673	61	64	u	u	NOUN
ejde-673	61	65	)	)	PUNCT
ejde-673	61	66	in	in	ADP
ejde-673	61	67	ω×	ω×	PROPN
ejde-673	61	68	(	(	PUNCT
ejde-673	61	69	0	0	NUM
ejde-673	61	70	,	,	PUNCT
ejde-673	61	71	t	t	NOUN
ejde-673	61	72	)	)	PUNCT
ejde-673	61	73	,	,	PUNCT
ejde-673	61	74	u	u	NOUN
ejde-673	61	75	=	=	NOUN
ejde-673	61	76	0	0	NUM
ejde-673	61	77	on	on	ADP
ejde-673	61	78	γ×	γ×	PROPN
ejde-673	61	79	(	(	PUNCT
ejde-673	61	80	0	0	NUM
ejde-673	61	81	,	,	PUNCT
ejde-673	61	82	t	t	NOUN
ejde-673	61	83	)	)	PUNCT
ejde-673	61	84	,	,	PUNCT
ejde-673	61	85	u(x	u(x	NOUN
ejde-673	61	86	,	,	PUNCT
ejde-673	61	87	0	0	NUM
ejde-673	61	88	)	)	PUNCT
ejde-673	61	89	=	=	SYM
ejde-673	61	90	u0(x	u0(x	NOUN
ejde-673	61	91	)	)	PUNCT
ejde-673	61	92	,	,	PUNCT
ejde-673	61	93	ut(x	ut(x	NOUN
ejde-673	61	94	,	,	PUNCT
ejde-673	61	95	0	0	NUM
ejde-673	61	96	)	)	PUNCT
ejde-673	61	97	=	=	SYM
ejde-673	62	1	u1(x	u1(x	NOUN
ejde-673	62	2	)	)	PUNCT
ejde-673	62	3	in	in	ADP
ejde-673	62	4	ω	ω	PROPN
ejde-673	62	5	.	.	PUNCT
ejde-673	63	1	(	(	PUNCT
ejde-673	63	2	1.6	1.6	NUM
ejde-673	63	3	)	)	PUNCT
ejde-673	63	4	equation	equation	NOUN
ejde-673	63	5	(	(	PUNCT
ejde-673	63	6	1.5	1.5	NUM
ejde-673	63	7	)	)	PUNCT
ejde-673	63	8	is	be	AUX
ejde-673	63	9	the	the	DET
ejde-673	63	10	stationary	stationary	ADJ
ejde-673	63	11	counterpart	counterpart	NOUN
ejde-673	63	12	of	of	ADP
ejde-673	63	13	(	(	PUNCT
ejde-673	63	14	1.6	1.6	NUM
ejde-673	63	15	)	)	PUNCT
ejde-673	63	16	.	.	PUNCT
ejde-673	64	1	such	such	DET
ejde-673	64	2	a	a	DET
ejde-673	64	3	hyperbolic	hyperbolic	ADJ
ejde-673	64	4	equation	equation	NOUN
ejde-673	64	5	is	be	AUX
ejde-673	64	6	a	a	DET
ejde-673	64	7	general	general	ADJ
ejde-673	64	8	version	version	NOUN
ejde-673	64	9	of	of	ADP
ejde-673	64	10	the	the	DET
ejde-673	64	11	kirchhoff	kirchhoff	NOUN
ejde-673	64	12	equation	equation	NOUN
ejde-673	64	13	ρutt	ρutt	PROPN
ejde-673	64	14	−	−	PROPN
ejde-673	65	1	(	(	PUNCT
ejde-673	65	2	ρ0	ρ0	PROPN
ejde-673	65	3	h	h	NOUN
ejde-673	65	4	+	+	CCONJ
ejde-673	65	5	e	e	X
ejde-673	65	6	2l	2l	NUM
ejde-673	65	7	∫	∫	PROPN
ejde-673	65	8	l	l	NOUN
ejde-673	65	9	0	0	NUM
ejde-673	65	10	∣∣∂u	∣∣∂u	PROPN
ejde-673	65	11	∂x	∂x	PROPN
ejde-673	65	12	∣∣2dx)∂2u	∣∣2dx)∂2u	ADJ
ejde-673	65	13	∂x2	∂x2	NOUN
ejde-673	65	14	=	=	SYM
ejde-673	65	15	0	0	NUM
ejde-673	65	16	presented	present	VERB
ejde-673	65	17	by	by	ADP
ejde-673	65	18	kirchhoff	kirchhoff	NOUN
ejde-673	66	1	[	[	X
ejde-673	66	2	22	22	NUM
ejde-673	66	3	]	]	PUNCT
ejde-673	66	4	.	.	PUNCT
ejde-673	67	1	this	this	DET
ejde-673	67	2	equation	equation	NOUN
ejde-673	67	3	extends	extend	VERB
ejde-673	67	4	the	the	DET
ejde-673	67	5	classical	classical	ADJ
ejde-673	67	6	d’alembert	d’alembert	NOUN
ejde-673	67	7	wave	wave	NOUN
ejde-673	67	8	equation	equation	NOUN
ejde-673	67	9	by	by	ADP
ejde-673	67	10	considering	consider	VERB
ejde-673	67	11	the	the	DET
ejde-673	67	12	effect	effect	NOUN
ejde-673	67	13	of	of	ADP
ejde-673	67	14	the	the	DET
ejde-673	67	15	changes	change	NOUN
ejde-673	67	16	in	in	ADP
ejde-673	67	17	the	the	DET
ejde-673	67	18	length	length	NOUN
ejde-673	67	19	of	of	ADP
ejde-673	67	20	the	the	DET
ejde-673	67	21	strings	string	NOUN
ejde-673	67	22	during	during	ADP
ejde-673	67	23	the	the	DET
ejde-673	67	24	vibrations	vibration	NOUN
ejde-673	67	25	,	,	PUNCT
ejde-673	67	26	where	where	SCONJ
ejde-673	67	27	l	l	NOUN
ejde-673	67	28	,	,	PUNCT
ejde-673	67	29	h	h	NOUN
ejde-673	67	30	,	,	PUNCT
ejde-673	67	31	e	e	PROPN
ejde-673	67	32	,	,	PUNCT
ejde-673	67	33	ρ	ρ	PROPN
ejde-673	67	34	and	and	CCONJ
ejde-673	67	35	ρ0	ρ0	PROPN
ejde-673	67	36	are	be	AUX
ejde-673	67	37	constants	constant	NOUN
ejde-673	67	38	.	.	PUNCT
ejde-673	68	1	in	in	ADP
ejde-673	68	2	afrouzu	afrouzu	ADJ
ejde-673	68	3	and	and	CCONJ
ejde-673	68	4	mirzapour	mirzapour	VERB
ejde-673	68	5	[	[	X
ejde-673	68	6	1	1	NUM
ejde-673	68	7	]	]	PUNCT
ejde-673	68	8	,	,	PUNCT
ejde-673	68	9	the	the	DET
ejde-673	68	10	authors	author	NOUN
ejde-673	68	11	studied	study	VERB
ejde-673	68	12	the	the	DET
ejde-673	68	13	p(·)-kirchhoff	p(·)-kirchhoff	NOUN
ejde-673	68	14	type	type	NOUN
ejde-673	68	15	eigenvalue	eigenvalue	PROPN
ejde-673	68	16	problem	problem	NOUN
ejde-673	68	17	−m	−m	NOUN
ejde-673	68	18	(	(	PUNCT
ejde-673	68	19	∫	∫	PROPN
ejde-673	68	20	ω	ω	PROPN
ejde-673	68	21	1	1	NUM
ejde-673	68	22	p(x	p(x	PROPN
ejde-673	68	23	)	)	PUNCT
ejde-673	68	24	|∇u(x)|p(x	|∇u(x)|p(x	PROPN
ejde-673	68	25	)	)	PUNCT
ejde-673	68	26	dx	dx	PROPN
ejde-673	68	27	)	)	PUNCT
ejde-673	69	1	∆p(x)u(x	∆p(x)u(x	PROPN
ejde-673	69	2	)	)	PUNCT
ejde-673	69	3	=	=	PUNCT
ejde-673	69	4	λ|u(x)|q(x)−2u(x	λ|u(x)|q(x)−2u(x	NOUN
ejde-673	69	5	)	)	PUNCT
ejde-673	69	6	in	in	ADP
ejde-673	69	7	ω	ω	PROPN
ejde-673	69	8	,	,	PUNCT
ejde-673	69	9	u(x	u(x	X
ejde-673	69	10	)	)	PUNCT
ejde-673	69	11	=	=	SYM
ejde-673	69	12	0	0	NUM
ejde-673	69	13	on	on	ADP
ejde-673	69	14	γ	γ	PROPN
ejde-673	69	15	.	.	PUNCT
ejde-673	69	16	(	(	PUNCT
ejde-673	69	17	1.7	1.7	NUM
ejde-673	69	18	)	)	PUNCT
ejde-673	69	19	they	they	PRON
ejde-673	69	20	derived	derive	VERB
ejde-673	69	21	the	the	DET
ejde-673	69	22	existence	existence	NOUN
ejde-673	69	23	of	of	ADP
ejde-673	69	24	a	a	DET
ejde-673	69	25	nontrivial	nontrivial	ADJ
ejde-673	69	26	weak	weak	ADJ
ejde-673	69	27	solution	solution	NOUN
ejde-673	69	28	under	under	ADP
ejde-673	69	29	some	some	DET
ejde-673	69	30	conditions	condition	NOUN
ejde-673	69	31	on	on	ADP
ejde-673	69	32	the	the	DET
ejde-673	69	33	functions	function	NOUN
ejde-673	69	34	m	m	PRON
ejde-673	69	35	,	,	PUNCT
ejde-673	69	36	p	p	X
ejde-673	69	37	(	(	PUNCT
ejde-673	69	38	·	·	PUNCT
ejde-673	69	39	)	)	PUNCT
ejde-673	69	40	,	,	PUNCT
ejde-673	69	41	q	q	X
ejde-673	69	42	(	(	PUNCT
ejde-673	69	43	·	·	PUNCT
ejde-673	69	44	)	)	PUNCT
ejde-673	69	45	and	and	CCONJ
ejde-673	69	46	a	a	DET
ejde-673	69	47	real	real	ADJ
ejde-673	69	48	number	number	NOUN
ejde-673	69	49	λ	λ	NOUN
ejde-673	69	50	.	.	PUNCT
ejde-673	70	1	mendéz	mendéz	PROPN
ejde-673	71	1	[	[	X
ejde-673	71	2	27	27	NUM
ejde-673	71	3	]	]	PUNCT
ejde-673	71	4	considered	consider	VERB
ejde-673	71	5	the	the	DET
ejde-673	71	6	problem	problem	NOUN
ejde-673	71	7	−m	−m	PROPN
ejde-673	71	8	(	(	PUNCT
ejde-673	71	9	∫	∫	PROPN
ejde-673	71	10	ω	ω	PROPN
ejde-673	71	11	|∇u(x)|p(x	|∇u(x)|p(x	PROPN
ejde-673	71	12	)	)	PUNCT
ejde-673	71	13	dx	dx	PROPN
ejde-673	71	14	)	)	PUNCT
ejde-673	71	15	div[p(x)|∇u(x)|p(x)−2∇u(x	div[p(x)|∇u(x)|p(x)−2∇u(x	PROPN
ejde-673	71	16	)	)	PUNCT
ejde-673	71	17	]	]	PUNCT
ejde-673	72	1	=	=	PUNCT
ejde-673	72	2	λp(x)|u(x)|p(x)−2u(x	λp(x)|u(x)|p(x)−2u(x	PROPN
ejde-673	72	3	)	)	PUNCT
ejde-673	72	4	in	in	ADP
ejde-673	72	5	ω	ω	PROPN
ejde-673	72	6	,	,	PUNCT
ejde-673	72	7	u(x	u(x	X
ejde-673	72	8	)	)	PUNCT
ejde-673	72	9	=	=	SYM
ejde-673	72	10	0	0	NUM
ejde-673	72	11	on	on	ADP
ejde-673	72	12	γ	γ	PROPN
ejde-673	72	13	.	.	PROPN
ejde-673	72	14	(	(	PUNCT
ejde-673	72	15	1.8	1.8	NUM
ejde-673	72	16	)	)	PUNCT
ejde-673	72	17	the	the	DET
ejde-673	72	18	author	author	NOUN
ejde-673	72	19	showed	show	VERB
ejde-673	72	20	that	that	SCONJ
ejde-673	72	21	for	for	ADP
ejde-673	72	22	any	any	DET
ejde-673	72	23	r	r	NOUN
ejde-673	72	24	>	>	X
ejde-673	72	25	0	0	NUM
ejde-673	72	26	,	,	PUNCT
ejde-673	72	27	there	there	PRON
ejde-673	72	28	exists	exist	VERB
ejde-673	72	29	a	a	DET
ejde-673	72	30	eigenpair	eigenpair	NOUN
ejde-673	72	31	(	(	PUNCT
ejde-673	72	32	u	u	NOUN
ejde-673	72	33	,	,	PUNCT
ejde-673	72	34	λ	λ	PROPN
ejde-673	72	35	)	)	PUNCT
ejde-673	72	36	∈	∈	PROPN
ejde-673	72	37	w	w	PROPN
ejde-673	72	38	1,p	1,p	PROPN
ejde-673	72	39	(	(	PUNCT
ejde-673	72	40	·	·	PUNCT
ejde-673	72	41	)	)	PUNCT
ejde-673	72	42	0	0	PUNCT
ejde-673	73	1	(	(	PUNCT
ejde-673	73	2	ω)×	ω)×	NOUN
ejde-673	73	3	r	r	NOUN
ejde-673	73	4	of	of	ADP
ejde-673	73	5	(	(	PUNCT
ejde-673	73	6	1.8	1.8	NUM
ejde-673	73	7	)	)	PUNCT
ejde-673	73	8	satisfying	satisfy	VERB
ejde-673	73	9	m̂	m̂	PROPN
ejde-673	73	10	(	(	PUNCT
ejde-673	73	11	∫	∫	PROPN
ejde-673	73	12	ω	ω	PROPN
ejde-673	73	13	|∇u(x)|p(x	|∇u(x)|p(x	PROPN
ejde-673	73	14	)	)	PUNCT
ejde-673	73	15	dx	dx	PROPN
ejde-673	73	16	)	)	PUNCT
ejde-673	74	1	=	=	PUNCT
ejde-673	74	2	r	r	NOUN
ejde-673	74	3	,	,	PUNCT
ejde-673	74	4	where	where	SCONJ
ejde-673	74	5	m̂(t	m̂(t	VERB
ejde-673	74	6	)	)	PUNCT
ejde-673	74	7	=	=	SYM
ejde-673	74	8	∫	∫	PROPN
ejde-673	74	9	t	t	PROPN
ejde-673	74	10	0	0	NUM
ejde-673	74	11	m(s	m(s	PROPN
ejde-673	74	12	)	)	PUNCT
ejde-673	74	13	ds	ds	NOUN
ejde-673	74	14	.	.	NOUN
ejde-673	75	1	in	in	ADP
ejde-673	75	2	this	this	DET
ejde-673	75	3	article	article	NOUN
ejde-673	75	4	,	,	PUNCT
ejde-673	75	5	we	we	PRON
ejde-673	75	6	extend	extend	VERB
ejde-673	75	7	these	these	DET
ejde-673	75	8	results	result	NOUN
ejde-673	75	9	to	to	ADP
ejde-673	75	10	a	a	DET
ejde-673	75	11	class	class	NOUN
ejde-673	75	12	of	of	ADP
ejde-673	75	13	operators	operator	NOUN
ejde-673	75	14	containing	contain	VERB
ejde-673	75	15	p(·)laplacian	p(·)laplacian	PROPN
ejde-673	75	16	and	and	CCONJ
ejde-673	75	17	the	the	DET
ejde-673	75	18	mean	mean	ADJ
ejde-673	75	19	curvature	curvature	NOUN
ejde-673	75	20	operator	operator	NOUN
ejde-673	75	21	.	.	PUNCT
ejde-673	76	1	the	the	DET
ejde-673	76	2	purpose	purpose	NOUN
ejde-673	76	3	of	of	ADP
ejde-673	76	4	this	this	DET
ejde-673	76	5	article	article	NOUN
ejde-673	76	6	is	be	AUX
ejde-673	76	7	to	to	PART
ejde-673	76	8	solve	solve	VERB
ejde-673	76	9	eigenvalue	eigenvalue	PROPN
ejde-673	76	10	problem	problem	NOUN
ejde-673	76	11	(	(	PUNCT
ejde-673	76	12	1.1	1.1	NUM
ejde-673	76	13	)	)	PUNCT
ejde-673	76	14	.	.	PUNCT
ejde-673	77	1	according	accord	VERB
ejde-673	77	2	to	to	ADP
ejde-673	77	3	some	some	DET
ejde-673	77	4	assumptions	assumption	NOUN
ejde-673	77	5	on	on	ADP
ejde-673	77	6	the	the	DET
ejde-673	77	7	given	give	VERB
ejde-673	77	8	function	function	NOUN
ejde-673	77	9	g	g	NOUN
ejde-673	77	10	,	,	PUNCT
ejde-673	77	11	we	we	PRON
ejde-673	77	12	use	use	VERB
ejde-673	77	13	the	the	DET
ejde-673	77	14	ljusternik	ljusternik	NOUN
ejde-673	77	15	-	-	PUNCT
ejde-673	77	16	schnirelmann	schnirelmann	ADJ
ejde-673	77	17	principle	principle	NOUN
ejde-673	77	18	in	in	ADP
ejde-673	77	19	the	the	DET
ejde-673	77	20	constrained	constrain	VERB
ejde-673	77	21	variational	variational	ADJ
ejde-673	77	22	method	method	NOUN
ejde-673	77	23	.	.	PUNCT
ejde-673	78	1	see	see	VERB
ejde-673	78	2	ljusternik	ljusternik	NOUN
ejde-673	78	3	and	and	CCONJ
ejde-673	78	4	schnirelmann	schnirelmann	X
ejde-673	79	1	[	[	X
ejde-673	79	2	25	25	NUM
ejde-673	79	3	]	]	PUNCT
ejde-673	79	4	and	and	CCONJ
ejde-673	79	5	szulkin	szulkin	ADJ
ejde-673	79	6	[	[	X
ejde-673	79	7	32	32	NUM
ejde-673	79	8	]	]	PUNCT
ejde-673	79	9	.	.	PUNCT
ejde-673	80	1	we	we	PRON
ejde-673	80	2	will	will	AUX
ejde-673	80	3	deal	deal	VERB
ejde-673	80	4	with	with	ADP
ejde-673	80	5	the	the	DET
ejde-673	80	6	mixed	mixed	ADJ
ejde-673	80	7	boundary	boundary	ADJ
ejde-673	80	8	value	value	NOUN
ejde-673	80	9	eigenvalue	eigenvalue	NOUN
ejde-673	80	10	problem	problem	NOUN
ejde-673	80	11	(	(	PUNCT
ejde-673	80	12	1.1	1.1	NUM
ejde-673	80	13	)	)	PUNCT
ejde-673	80	14	for	for	ADP
ejde-673	80	15	a	a	DET
ejde-673	80	16	class	class	NOUN
ejde-673	80	17	of	of	ADP
ejde-673	80	18	operators	operator	NOUN
ejde-673	80	19	involving	involve	VERB
ejde-673	80	20	the	the	DET
ejde-673	80	21	p(·)laplacian	p(·)laplacian	ADJ
ejde-673	80	22	and	and	CCONJ
ejde-673	80	23	the	the	DET
ejde-673	80	24	mean	mean	ADJ
ejde-673	80	25	curvature	curvature	NOUN
ejde-673	80	26	operator	operator	NOUN
ejde-673	80	27	which	which	PRON
ejde-673	80	28	seems	seem	VERB
ejde-673	80	29	to	to	PART
ejde-673	80	30	be	be	AUX
ejde-673	80	31	a	a	DET
ejde-673	80	32	new	new	ADJ
ejde-673	80	33	topic	topic	NOUN
ejde-673	80	34	.	.	PUNCT
ejde-673	81	1	we	we	PRON
ejde-673	81	2	will	will	AUX
ejde-673	81	3	show	show	VERB
ejde-673	81	4	that	that	SCONJ
ejde-673	81	5	there	there	PRON
ejde-673	81	6	exist	exist	VERB
ejde-673	81	7	infinitely	infinitely	ADV
ejde-673	81	8	many	many	ADJ
ejde-673	81	9	eigenvalues	eigenvalue	NOUN
ejde-673	81	10	{	{	PUNCT
ejde-673	81	11	λ(n	λ(n	PROPN
ejde-673	81	12	,	,	PUNCT
ejde-673	81	13	α	α	NOUN
ejde-673	81	14	)	)	PUNCT
ejde-673	81	15	}	}	PUNCT
ejde-673	81	16	tending	tend	VERB
ejde-673	81	17	to	to	ADP
ejde-673	81	18	∞	∞	NUM
ejde-673	81	19	as	as	ADP
ejde-673	81	20	n	n	PROPN
ejde-673	81	21	→	→	SYM
ejde-673	81	22	∞	∞	PROPN
ejde-673	81	23	for	for	ADP
ejde-673	81	24	any	any	DET
ejde-673	81	25	fixed	fixed	ADJ
ejde-673	81	26	α	α	PROPN
ejde-673	81	27	>	>	X
ejde-673	81	28	0	0	PROPN
ejde-673	81	29	.	.	PUNCT
ejde-673	82	1	moreover	moreover	ADV
ejde-673	82	2	,	,	PUNCT
ejde-673	82	3	we	we	PRON
ejde-673	82	4	will	will	AUX
ejde-673	82	5	derive	derive	VERB
ejde-673	82	6	that	that	SCONJ
ejde-673	82	7	under	under	ADP
ejde-673	82	8	some	some	DET
ejde-673	82	9	condition	condition	NOUN
ejde-673	82	10	,	,	PUNCT
ejde-673	82	11	the	the	DET
ejde-673	82	12	infimum	infimum	ADJ
ejde-673	82	13	λ∗	λ∗	NOUN
ejde-673	82	14	of	of	ADP
ejde-673	82	15	the	the	DET
ejde-673	82	16	set	set	NOUN
ejde-673	82	17	of	of	ADP
ejde-673	82	18	all	all	DET
ejde-673	82	19	eigenvalues	eigenvalue	NOUN
ejde-673	82	20	of	of	ADP
ejde-673	82	21	(	(	PUNCT
ejde-673	82	22	1.1	1.1	NUM
ejde-673	82	23	)	)	PUNCT
ejde-673	82	24	is	be	AUX
ejde-673	82	25	equal	equal	ADJ
ejde-673	82	26	to	to	ADP
ejde-673	82	27	zero	zero	NUM
ejde-673	82	28	,	,	PUNCT
ejde-673	82	29	so	so	SCONJ
ejde-673	82	30	there	there	PRON
ejde-673	82	31	does	do	AUX
ejde-673	82	32	not	not	PART
ejde-673	82	33	exist	exist	VERB
ejde-673	82	34	a	a	DET
ejde-673	82	35	principal	principal	ADJ
ejde-673	82	36	eigenvalue	eigenvalue	NOUN
ejde-673	82	37	and	and	CCONJ
ejde-673	82	38	the	the	DET
ejde-673	82	39	set	set	NOUN
ejde-673	82	40	of	of	ADP
ejde-673	82	41	eigenvalues	eigenvalue	NOUN
ejde-673	82	42	is	be	AUX
ejde-673	82	43	not	not	PART
ejde-673	82	44	closed	closed	ADJ
ejde-673	82	45	.	.	PUNCT
ejde-673	83	1	we	we	PRON
ejde-673	83	2	also	also	ADV
ejde-673	83	3	show	show	VERB
ejde-673	83	4	that	that	SCONJ
ejde-673	83	5	under	under	ADP
ejde-673	83	6	some	some	DET
ejde-673	83	7	condition	condition	NOUN
ejde-673	83	8	on	on	ADP
ejde-673	83	9	the	the	DET
ejde-673	83	10	function	function	NOUN
ejde-673	83	11	g	g	NOUN
ejde-673	83	12	and	and	CCONJ
ejde-673	83	13	variable	variable	ADJ
ejde-673	83	14	exponent	exponent	NOUN
ejde-673	83	15	function	function	NOUN
ejde-673	83	16	p	p	NOUN
ejde-673	83	17	in	in	ADP
ejde-673	83	18	(	(	PUNCT
ejde-673	83	19	1.1	1.1	NUM
ejde-673	83	20	)	)	PUNCT
ejde-673	83	21	,	,	PUNCT
ejde-673	83	22	there	there	PRON
ejde-673	83	23	is	be	VERB
ejde-673	83	24	a	a	DET
ejde-673	83	25	case	case	NOUN
ejde-673	83	26	where	where	SCONJ
ejde-673	83	27	λ∗	λ∗	NOUN
ejde-673	83	28	is	be	AUX
ejde-673	83	29	positive	positive	ADJ
ejde-673	83	30	.	.	PUNCT
ejde-673	84	1	this	this	DET
ejde-673	84	2	article	article	NOUN
ejde-673	84	3	is	be	AUX
ejde-673	84	4	organized	organize	VERB
ejde-673	84	5	as	as	SCONJ
ejde-673	84	6	follows	follow	VERB
ejde-673	84	7	.	.	PUNCT
ejde-673	85	1	in	in	ADP
ejde-673	85	2	section	section	NOUN
ejde-673	85	3	2	2	NUM
ejde-673	85	4	,	,	PUNCT
ejde-673	85	5	we	we	PRON
ejde-673	85	6	recall	recall	VERB
ejde-673	85	7	some	some	DET
ejde-673	85	8	results	result	NOUN
ejde-673	85	9	on	on	ADP
ejde-673	85	10	variable	variable	ADJ
ejde-673	85	11	exponent	exponent	NOUN
ejde-673	85	12	lebesgue	lebesgue	PROPN
ejde-673	85	13	-	-	PUNCT
ejde-673	85	14	sobolev	sobolev	NOUN
ejde-673	85	15	spaces	space	VERB
ejde-673	85	16	.	.	PUNCT
ejde-673	86	1	in	in	ADP
ejde-673	86	2	section	section	NOUN
ejde-673	86	3	3	3	NUM
ejde-673	86	4	,	,	PUNCT
ejde-673	86	5	we	we	PRON
ejde-673	86	6	give	give	VERB
ejde-673	86	7	the	the	DET
ejde-673	86	8	setting	setting	NOUN
ejde-673	86	9	of	of	ADP
ejde-673	86	10	problem	problem	NOUN
ejde-673	86	11	(	(	PUNCT
ejde-673	86	12	1.1	1.1	NUM
ejde-673	86	13	)	)	PUNCT
ejde-673	86	14	rigorously	rigorously	ADV
ejde-673	86	15	and	and	CCONJ
ejde-673	86	16	a	a	DET
ejde-673	86	17	main	main	ADJ
ejde-673	86	18	theorem	theorem	NOUN
ejde-673	86	19	(	(	PUNCT
ejde-673	86	20	theorems	theorem	NOUN
ejde-673	86	21	3.20	3.20	NUM
ejde-673	86	22	)	)	PUNCT
ejde-673	86	23	on	on	ADP
ejde-673	86	24	the	the	DET
ejde-673	86	25	eigenvalue	eigenvalue	PROPN
ejde-673	86	26	problem	problem	NOUN
ejde-673	86	27	(	(	PUNCT
ejde-673	86	28	1.1	1.1	NUM
ejde-673	86	29	)	)	PUNCT
ejde-673	86	30	in	in	ADP
ejde-673	86	31	which	which	PRON
ejde-673	86	32	we	we	PRON
ejde-673	86	33	show	show	VERB
ejde-673	86	34	the	the	DET
ejde-673	86	35	existence	existence	NOUN
ejde-673	86	36	of	of	ADP
ejde-673	86	37	infinitely	infinitely	ADV
ejde-673	86	38	many	many	ADJ
ejde-673	86	39	eigenpairs	eigenpair	NOUN
ejde-673	86	40	of	of	ADP
ejde-673	86	41	(	(	PUNCT
ejde-673	86	42	1.1	1.1	NUM
ejde-673	86	43	)	)	PUNCT
ejde-673	86	44	.	.	PUNCT
ejde-673	87	1	4	4	NUM
ejde-673	87	2	j.	j.	PROPN
ejde-673	87	3	aramaki	aramaki	PROPN
ejde-673	87	4	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	87	5	in	in	ADP
ejde-673	87	6	section	section	NOUN
ejde-673	87	7	4	4	NUM
ejde-673	87	8	,	,	PUNCT
ejde-673	87	9	we	we	PRON
ejde-673	87	10	present	present	VERB
ejde-673	87	11	some	some	DET
ejde-673	87	12	sufficient	sufficient	ADJ
ejde-673	87	13	conditions	condition	NOUN
ejde-673	87	14	for	for	ADP
ejde-673	87	15	the	the	DET
ejde-673	87	16	cases	case	NOUN
ejde-673	87	17	λ∗	λ∗	NOUN
ejde-673	87	18	=	=	SYM
ejde-673	87	19	0	0	PUNCT
ejde-673	87	20	and	and	CCONJ
ejde-673	87	21	λ∗	λ∗	PROPN
ejde-673	87	22	>	>	X
ejde-673	87	23	0	0	NUM
ejde-673	87	24	,	,	PUNCT
ejde-673	87	25	respectively	respectively	ADV
ejde-673	87	26	.	.	PUNCT
ejde-673	88	1	2	2	X
ejde-673	88	2	.	.	X
ejde-673	88	3	preliminaries	preliminary	NOUN
ejde-673	88	4	throughout	throughout	ADP
ejde-673	88	5	this	this	DET
ejde-673	88	6	article	article	NOUN
ejde-673	88	7	,	,	PUNCT
ejde-673	88	8	ω	ω	PROPN
ejde-673	88	9	is	be	AUX
ejde-673	88	10	a	a	DET
ejde-673	88	11	bounded	bounded	ADJ
ejde-673	88	12	domain	domain	NOUN
ejde-673	88	13	in	in	ADP
ejde-673	88	14	rn	rn	PROPN
ejde-673	88	15	(	(	PUNCT
ejde-673	88	16	n	n	CCONJ
ejde-673	88	17	≥	≥	NOUN
ejde-673	88	18	2	2	NUM
ejde-673	88	19	)	)	PUNCT
ejde-673	88	20	with	with	ADP
ejde-673	88	21	a	a	DET
ejde-673	88	22	c0,1boundary	c0,1boundary	ADJ
ejde-673	88	23	γ	γ	NOUN
ejde-673	88	24	and	and	CCONJ
ejde-673	88	25	ω	ω	PROPN
ejde-673	88	26	is	be	AUX
ejde-673	88	27	locally	locally	ADV
ejde-673	88	28	on	on	ADP
ejde-673	88	29	the	the	DET
ejde-673	88	30	same	same	ADJ
ejde-673	88	31	side	side	NOUN
ejde-673	88	32	of	of	ADP
ejde-673	88	33	γ	γ	PROPN
ejde-673	88	34	.	.	PUNCT
ejde-673	89	1	moreover	moreover	ADV
ejde-673	89	2	,	,	PUNCT
ejde-673	89	3	we	we	PRON
ejde-673	89	4	assume	assume	VERB
ejde-673	89	5	that	that	SCONJ
ejde-673	89	6	γ	γ	PROPN
ejde-673	89	7	satisfies	satisfie	NOUN
ejde-673	89	8	(	(	PUNCT
ejde-673	89	9	1.2	1.2	NUM
ejde-673	89	10	)	)	PUNCT
ejde-673	89	11	.	.	PUNCT
ejde-673	90	1	we	we	PRON
ejde-673	90	2	only	only	ADV
ejde-673	90	3	consider	consider	VERB
ejde-673	90	4	real	real	ADJ
ejde-673	90	5	vector	vector	NOUN
ejde-673	90	6	spaces	space	NOUN
ejde-673	90	7	of	of	ADP
ejde-673	90	8	real	real	ADJ
ejde-673	90	9	valued	value	VERB
ejde-673	90	10	functions	function	NOUN
ejde-673	90	11	over	over	ADP
ejde-673	90	12	r.	r.	PROPN
ejde-673	90	13	for	for	ADP
ejde-673	90	14	any	any	DET
ejde-673	90	15	space	space	NOUN
ejde-673	90	16	b	b	NOUN
ejde-673	90	17	,	,	PUNCT
ejde-673	90	18	we	we	PRON
ejde-673	90	19	denote	denote	VERB
ejde-673	90	20	bn	bn	ADP
ejde-673	90	21	by	by	ADP
ejde-673	90	22	the	the	DET
ejde-673	90	23	boldface	boldface	PROPN
ejde-673	90	24	character	character	PROPN
ejde-673	90	25	b.	b.	PROPN
ejde-673	90	26	hereafter	hereafter	PROPN
ejde-673	90	27	,	,	PUNCT
ejde-673	90	28	we	we	PRON
ejde-673	90	29	use	use	VERB
ejde-673	90	30	this	this	DET
ejde-673	90	31	character	character	NOUN
ejde-673	90	32	to	to	PART
ejde-673	90	33	denote	denote	VERB
ejde-673	90	34	vectors	vector	NOUN
ejde-673	90	35	and	and	CCONJ
ejde-673	90	36	vector	vector	NOUN
ejde-673	90	37	-	-	PUNCT
ejde-673	90	38	valued	value	VERB
ejde-673	90	39	functions	function	NOUN
ejde-673	90	40	,	,	PUNCT
ejde-673	90	41	and	and	CCONJ
ejde-673	90	42	we	we	PRON
ejde-673	90	43	denote	denote	VERB
ejde-673	90	44	the	the	DET
ejde-673	90	45	standard	standard	ADJ
ejde-673	90	46	inner	inner	ADJ
ejde-673	90	47	product	product	NOUN
ejde-673	90	48	of	of	ADP
ejde-673	90	49	vectors	vector	NOUN
ejde-673	90	50	a	a	DET
ejde-673	90	51	=	=	SYM
ejde-673	90	52	(	(	PUNCT
ejde-673	90	53	a1	a1	PROPN
ejde-673	90	54	,	,	PUNCT
ejde-673	90	55	.	.	PUNCT
ejde-673	90	56	.	.	PUNCT
ejde-673	90	57	.	.	PUNCT
ejde-673	91	1	,	,	PUNCT
ejde-673	91	2	an	an	PRON
ejde-673	91	3	)	)	PUNCT
ejde-673	91	4	and	and	CCONJ
ejde-673	91	5	b	b	X
ejde-673	91	6	=	=	SYM
ejde-673	91	7	(	(	PUNCT
ejde-673	91	8	b1	b1	PROPN
ejde-673	91	9	,	,	PUNCT
ejde-673	91	10	.	.	PUNCT
ejde-673	91	11	.	.	PUNCT
ejde-673	92	1	.	.	PUNCT
ejde-673	93	1	,	,	PUNCT
ejde-673	93	2	bn	bn	X
ejde-673	93	3	)	)	PUNCT
ejde-673	93	4	in	in	ADP
ejde-673	93	5	rn	rn	PROPN
ejde-673	93	6	by	by	ADP
ejde-673	93	7	a·b	a·b	NOUN
ejde-673	93	8	=	=	PUNCT
ejde-673	93	9	∑n	∑n	PROPN
ejde-673	93	10	i=1	i=1	PROPN
ejde-673	93	11	aibi	aibi	NOUN
ejde-673	93	12	and	and	CCONJ
ejde-673	93	13	|a|	|a|	PROPN
ejde-673	93	14	=	=	SYM
ejde-673	93	15	(	(	PUNCT
ejde-673	93	16	a	a	DET
ejde-673	93	17	·	·	PUNCT
ejde-673	93	18	a)1/2	a)1/2	PROPN
ejde-673	93	19	.	.	PUNCT
ejde-673	94	1	furthermore	furthermore	ADV
ejde-673	94	2	,	,	PUNCT
ejde-673	94	3	we	we	PRON
ejde-673	94	4	denote	denote	VERB
ejde-673	94	5	the	the	DET
ejde-673	94	6	dual	dual	ADJ
ejde-673	94	7	space	space	NOUN
ejde-673	94	8	of	of	ADP
ejde-673	94	9	b	b	NOUN
ejde-673	94	10	by	by	ADP
ejde-673	94	11	b∗	b∗	ADJ
ejde-673	94	12	and	and	CCONJ
ejde-673	94	13	the	the	DET
ejde-673	94	14	duality	duality	NOUN
ejde-673	94	15	bracket	bracket	NOUN
ejde-673	94	16	by	by	ADP
ejde-673	94	17	⟨	⟨	NOUN
ejde-673	94	18	·	·	NUM
ejde-673	94	19	,	,	PUNCT
ejde-673	94	20	·	·	PUNCT
ejde-673	94	21	⟩b∗,b	⟩b∗,b	INTJ
ejde-673	94	22	.	.	PUNCT
ejde-673	95	1	we	we	PRON
ejde-673	95	2	recall	recall	VERB
ejde-673	95	3	some	some	DET
ejde-673	95	4	well	well	ADV
ejde-673	95	5	-	-	PUNCT
ejde-673	95	6	known	know	VERB
ejde-673	95	7	results	result	NOUN
ejde-673	95	8	on	on	ADP
ejde-673	95	9	variable	variable	ADJ
ejde-673	95	10	exponent	exponent	NOUN
ejde-673	95	11	lebesgue	lebesgue	NOUN
ejde-673	95	12	and	and	CCONJ
ejde-673	95	13	sobolev	sobolev	NOUN
ejde-673	95	14	spaces	space	NOUN
ejde-673	95	15	.	.	PUNCT
ejde-673	96	1	see	see	VERB
ejde-673	96	2	fan	fan	NOUN
ejde-673	96	3	and	and	CCONJ
ejde-673	96	4	zhang	zhang	PROPN
ejde-673	97	1	[	[	X
ejde-673	97	2	17	17	NUM
ejde-673	97	3	]	]	PUNCT
ejde-673	97	4	,	,	PUNCT
ejde-673	97	5	kovác̆ik	kovác̆ik	PUNCT
ejde-673	97	6	and	and	CCONJ
ejde-673	97	7	rácosńık	rácosńık	PROPN
ejde-673	98	1	[	[	X
ejde-673	98	2	23	23	NUM
ejde-673	98	3	]	]	PUNCT
ejde-673	98	4	,	,	PUNCT
ejde-673	98	5	diening	diene	VERB
ejde-673	98	6	et	et	PROPN
ejde-673	98	7	al	al	PROPN
ejde-673	98	8	.	.	PUNCT
ejde-673	99	1	[	[	X
ejde-673	99	2	13	13	NUM
ejde-673	99	3	]	]	PUNCT
ejde-673	99	4	and	and	CCONJ
ejde-673	99	5	references	reference	NOUN
ejde-673	99	6	therein	therein	ADV
ejde-673	99	7	for	for	ADP
ejde-673	99	8	more	more	ADJ
ejde-673	99	9	details	detail	NOUN
ejde-673	99	10	.	.	PUNCT
ejde-673	100	1	we	we	PRON
ejde-673	100	2	consider	consider	VERB
ejde-673	100	3	some	some	DET
ejde-673	100	4	new	new	ADJ
ejde-673	100	5	properties	property	NOUN
ejde-673	100	6	on	on	ADP
ejde-673	100	7	variable	variable	ADJ
ejde-673	100	8	exponent	exponent	NOUN
ejde-673	100	9	lebesgue	lebesgue	NOUN
ejde-673	100	10	space	space	NOUN
ejde-673	100	11	.	.	PUNCT
ejde-673	101	1	we	we	PRON
ejde-673	101	2	define	define	VERB
ejde-673	101	3	c(ω	c(ω	NOUN
ejde-673	101	4	)	)	PUNCT
ejde-673	101	5	=	=	PRON
ejde-673	102	1	{	{	PUNCT
ejde-673	102	2	p	p	NOUN
ejde-673	102	3	is	be	AUX
ejde-673	102	4	a	a	DET
ejde-673	102	5	continuous	continuous	ADJ
ejde-673	102	6	function	function	NOUN
ejde-673	102	7	on	on	ADP
ejde-673	102	8	ω	ω	NOUN
ejde-673	102	9	}	}	PUNCT
ejde-673	102	10	,	,	PUNCT
ejde-673	102	11	and	and	CCONJ
ejde-673	102	12	for	for	ADP
ejde-673	102	13	any	any	DET
ejde-673	102	14	p	p	PROPN
ejde-673	102	15	∈	∈	PROPN
ejde-673	102	16	c(ω	c(ω	PROPN
ejde-673	102	17	)	)	PUNCT
ejde-673	102	18	,	,	PUNCT
ejde-673	102	19	put	put	VERB
ejde-673	102	20	p+	p+	NOUN
ejde-673	102	21	=	=	NOUN
ejde-673	102	22	p+(ω	p+(ω	PROPN
ejde-673	102	23	)	)	PUNCT
ejde-673	102	24	=	=	SYM
ejde-673	102	25	sup	sup	NOUN
ejde-673	102	26	x∈ω	x∈ω	NOUN
ejde-673	102	27	p(x	p(x	PROPN
ejde-673	102	28	)	)	PUNCT
ejde-673	102	29	=	=	SYM
ejde-673	102	30	max	max	PROPN
ejde-673	102	31	x∈ω	x∈ω	PROPN
ejde-673	102	32	p(x	p(x	PROPN
ejde-673	102	33	)	)	PUNCT
ejde-673	102	34	,	,	PUNCT
ejde-673	102	35	p−	p−	NOUN
ejde-673	102	36	=	=	SYM
ejde-673	102	37	p−(ω	p−(ω	NOUN
ejde-673	102	38	)	)	PUNCT
ejde-673	102	39	=	=	SYM
ejde-673	102	40	inf	inf	PROPN
ejde-673	102	41	x∈ω	x∈ω	NOUN
ejde-673	102	42	p(x	p(x	PROPN
ejde-673	102	43	)	)	PUNCT
ejde-673	102	44	=	=	SYM
ejde-673	102	45	min	min	NOUN
ejde-673	102	46	x∈ω	x∈ω	PROPN
ejde-673	102	47	p(x	p(x	PROPN
ejde-673	102	48	)	)	PUNCT
ejde-673	102	49	.	.	PUNCT
ejde-673	103	1	for	for	ADP
ejde-673	103	2	any	any	DET
ejde-673	103	3	p	p	PROPN
ejde-673	103	4	∈	∈	PROPN
ejde-673	103	5	c(ω	c(ω	PROPN
ejde-673	103	6	)	)	PUNCT
ejde-673	103	7	with	with	ADP
ejde-673	103	8	p−	p−	PRON
ejde-673	103	9	≥	≥	NOUN
ejde-673	103	10	1	1	NUM
ejde-673	103	11	and	and	CCONJ
ejde-673	103	12	for	for	ADP
ejde-673	103	13	any	any	DET
ejde-673	103	14	measurable	measurable	ADJ
ejde-673	103	15	function	function	NOUN
ejde-673	103	16	u	u	NOUN
ejde-673	103	17	on	on	ADP
ejde-673	103	18	ω	ω	PROPN
ejde-673	103	19	,	,	PUNCT
ejde-673	103	20	a	a	DET
ejde-673	103	21	modular	modular	NOUN
ejde-673	103	22	(	(	PUNCT
ejde-673	103	23	for	for	ADP
ejde-673	103	24	this	this	DET
ejde-673	103	25	notation	notation	NOUN
ejde-673	103	26	,	,	PUNCT
ejde-673	103	27	see	see	VERB
ejde-673	103	28	[	[	X
ejde-673	103	29	13	13	NUM
ejde-673	103	30	,	,	PUNCT
ejde-673	103	31	definition	definition	NOUN
ejde-673	103	32	2.1.1	2.1.1	NUM
ejde-673	103	33	]	]	NOUN
ejde-673	103	34	)	)	PUNCT
ejde-673	103	35	ρp	ρp	NOUN
ejde-673	103	36	(	(	PUNCT
ejde-673	103	37	·	·	PUNCT
ejde-673	103	38	)	)	PUNCT
ejde-673	104	1	=	=	PRON
ejde-673	104	2	ρp(·),ω	ρp(·),ω	PROPN
ejde-673	104	3	is	be	AUX
ejde-673	104	4	defined	define	VERB
ejde-673	104	5	by	by	ADP
ejde-673	104	6	ρp(·)(u	ρp(·)(u	NOUN
ejde-673	104	7	)	)	PUNCT
ejde-673	104	8	=	=	SYM
ejde-673	104	9	∫	∫	PROPN
ejde-673	104	10	ω	ω	PROPN
ejde-673	104	11	|u(x)|p(x	|u(x)|p(x	PROPN
ejde-673	104	12	)	)	PUNCT
ejde-673	104	13	dx	dx	PROPN
ejde-673	104	14	.	.	PUNCT
ejde-673	105	1	the	the	DET
ejde-673	105	2	variable	variable	ADJ
ejde-673	105	3	exponent	exponent	NOUN
ejde-673	105	4	lebesgue	lebesgue	NOUN
ejde-673	105	5	space	space	NOUN
ejde-673	105	6	is	be	AUX
ejde-673	105	7	defined	define	VERB
ejde-673	105	8	by	by	ADP
ejde-673	105	9	lp(·)(ω	lp(·)(ω	ADJ
ejde-673	105	10	)	)	PUNCT
ejde-673	105	11	=	=	PRON
ejde-673	106	1	{	{	PUNCT
ejde-673	106	2	u;u	u;u	X
ejde-673	106	3	:	:	PUNCT
ejde-673	106	4	ω	ω	NOUN
ejde-673	106	5	→	→	SYM
ejde-673	106	6	r	r	NOUN
ejde-673	106	7	is	be	AUX
ejde-673	106	8	a	a	DET
ejde-673	106	9	measurable	measurable	ADJ
ejde-673	106	10	function	function	NOUN
ejde-673	106	11	satisfying	satisfy	VERB
ejde-673	106	12	ρp(·)(u	ρp(·)(u	ADV
ejde-673	106	13	)	)	PUNCT
ejde-673	106	14	<	<	X
ejde-673	106	15	∞	∞	PROPN
ejde-673	106	16	}	}	PUNCT
ejde-673	106	17	equipped	equip	VERB
ejde-673	106	18	with	with	ADP
ejde-673	106	19	the	the	DET
ejde-673	106	20	(	(	PUNCT
ejde-673	106	21	luxemburg	luxemburg	PROPN
ejde-673	106	22	)	)	PUNCT
ejde-673	106	23	norm	norm	NOUN
ejde-673	106	24	∥u∥lp(·)(ω	∥u∥lp(·)(ω	NOUN
ejde-673	106	25	)	)	PUNCT
ejde-673	106	26	=	=	SYM
ejde-673	106	27	inf	inf	NOUN
ejde-673	106	28	{	{	PUNCT
ejde-673	106	29	τ	τ	PROPN
ejde-673	106	30	>	>	X
ejde-673	106	31	0	0	NUM
ejde-673	106	32	;	;	PUNCT
ejde-673	106	33	ρp	ρp	PRON
ejde-673	106	34	(	(	PUNCT
ejde-673	106	35	·	·	PUNCT
ejde-673	106	36	)	)	PUNCT
ejde-673	106	37	(	(	PUNCT
ejde-673	106	38	u	u	NOUN
ejde-673	106	39	τ	τ	PROPN
ejde-673	106	40	)	)	PUNCT
ejde-673	106	41	≤	≤	NUM
ejde-673	106	42	1	1	NUM
ejde-673	106	43	}	}	PUNCT
ejde-673	106	44	.	.	PUNCT
ejde-673	107	1	then	then	ADV
ejde-673	107	2	lp(·)(ω	lp(·)(ω	PROPN
ejde-673	107	3	)	)	PUNCT
ejde-673	107	4	is	be	AUX
ejde-673	107	5	a	a	DET
ejde-673	107	6	banach	banach	NOUN
ejde-673	107	7	space	space	NOUN
ejde-673	107	8	.	.	PUNCT
ejde-673	108	1	we	we	PRON
ejde-673	108	2	also	also	ADV
ejde-673	108	3	define	define	VERB
ejde-673	108	4	the	the	DET
ejde-673	108	5	sobolev	sobolev	NOUN
ejde-673	108	6	space	space	NOUN
ejde-673	108	7	w	w	PROPN
ejde-673	108	8	1,p(·)(ω	1,p(·)(ω	ADJ
ejde-673	108	9	)	)	PUNCT
ejde-673	108	10	=	=	PRON
ejde-673	108	11	{	{	PUNCT
ejde-673	108	12	u	u	NOUN
ejde-673	108	13	∈	∈	PROPN
ejde-673	108	14	lp(·)(ω	lp(·)(ω	PROPN
ejde-673	108	15	)	)	PUNCT
ejde-673	108	16	;	;	PUNCT
ejde-673	108	17	|∇u|	|∇u|	ADJ
ejde-673	108	18	∈	∈	PROPN
ejde-673	108	19	lp(·)(ω	lp(·)(ω	NOUN
ejde-673	108	20	)	)	PUNCT
ejde-673	108	21	}	}	PUNCT
ejde-673	108	22	,	,	PUNCT
ejde-673	108	23	where	where	SCONJ
ejde-673	108	24	∇	∇	PROPN
ejde-673	108	25	is	be	AUX
ejde-673	108	26	a	a	DET
ejde-673	108	27	gradient	gradient	NOUN
ejde-673	108	28	operator	operator	NOUN
ejde-673	108	29	,	,	PUNCT
ejde-673	108	30	that	that	ADV
ejde-673	108	31	is	is	ADV
ejde-673	108	32	,	,	PUNCT
ejde-673	108	33	∇u	∇u	PROPN
ejde-673	108	34	=	=	SYM
ejde-673	108	35	(	(	PUNCT
ejde-673	108	36	∂1u	∂1u	ADJ
ejde-673	108	37	,	,	PUNCT
ejde-673	108	38	.	.	PUNCT
ejde-673	108	39	.	.	PUNCT
ejde-673	108	40	.	.	PUNCT
ejde-673	109	1	,	,	PUNCT
ejde-673	109	2	∂nu	∂nu	PROPN
ejde-673	109	3	)	)	PUNCT
ejde-673	109	4	,	,	PUNCT
ejde-673	110	1	∂i	∂i	PROPN
ejde-673	110	2	=	=	SYM
ejde-673	110	3	∂/∂xi	∂/∂xi	PROPN
ejde-673	110	4	,	,	PUNCT
ejde-673	110	5	endowed	endow	VERB
ejde-673	110	6	with	with	ADP
ejde-673	110	7	the	the	DET
ejde-673	110	8	norm	norm	NOUN
ejde-673	110	9	∥u∥w	∥u∥w	VERB
ejde-673	110	10	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	110	11	)	)	PUNCT
ejde-673	110	12	=	=	SYM
ejde-673	110	13	∥u∥lp(·)(ω	∥u∥lp(·)(ω	ADJ
ejde-673	110	14	)	)	PUNCT
ejde-673	111	1	+	+	CCONJ
ejde-673	111	2	∥∇u∥lp(·)(ω	∥∇u∥lp(·)(ω	ADJ
ejde-673	111	3	)	)	PUNCT
ejde-673	111	4	,	,	PUNCT
ejde-673	111	5	and	and	CCONJ
ejde-673	111	6	∥∇u∥lp(·)(ω	∥∇u∥lp(·)(ω	PROPN
ejde-673	111	7	)	)	PUNCT
ejde-673	111	8	=	=	SYM
ejde-673	111	9	∥|∇u|∥lp(·)(ω	∥|∇u|∥lp(·)(ω	X
ejde-673	111	10	)	)	PUNCT
ejde-673	111	11	.	.	PUNCT
ejde-673	112	1	the	the	DET
ejde-673	112	2	following	follow	VERB
ejde-673	112	3	three	three	NUM
ejde-673	112	4	propositions	proposition	NOUN
ejde-673	112	5	are	be	AUX
ejde-673	112	6	well	well	ADV
ejde-673	112	7	known	know	VERB
ejde-673	112	8	(	(	PUNCT
ejde-673	112	9	see	see	VERB
ejde-673	112	10	[	[	X
ejde-673	112	11	19	19	NUM
ejde-673	112	12	]	]	X
ejde-673	112	13	,	,	PUNCT
ejde-673	112	14	fan	fan	NOUN
ejde-673	112	15	and	and	CCONJ
ejde-673	112	16	zhao	zhao	PROPN
ejde-673	113	1	[	[	X
ejde-673	113	2	18	18	NUM
ejde-673	113	3	]	]	PUNCT
ejde-673	113	4	,	,	PUNCT
ejde-673	113	5	zhao	zhao	PROPN
ejde-673	113	6	et	et	PROPN
ejde-673	113	7	al	al	PROPN
ejde-673	113	8	.	.	PUNCT
ejde-673	114	1	[	[	X
ejde-673	114	2	35	35	NUM
ejde-673	114	3	]	]	SYM
ejde-673	114	4	)	)	PUNCT
ejde-673	114	5	.	.	PUNCT
ejde-673	115	1	proposition	proposition	NOUN
ejde-673	115	2	2.1	2.1	NUM
ejde-673	115	3	.	.	PUNCT
ejde-673	116	1	let	let	VERB
ejde-673	116	2	p	p	PROPN
ejde-673	116	3	∈	∈	PROPN
ejde-673	116	4	c(ω	c(ω	PROPN
ejde-673	116	5	)	)	PUNCT
ejde-673	116	6	with	with	ADP
ejde-673	116	7	p−	p−	PRON
ejde-673	116	8	≥	≥	NOUN
ejde-673	116	9	1	1	NUM
ejde-673	116	10	,	,	PUNCT
ejde-673	116	11	and	and	CCONJ
ejde-673	116	12	let	let	VERB
ejde-673	116	13	u	u	NOUN
ejde-673	116	14	,	,	PUNCT
ejde-673	116	15	un	un	PROPN
ejde-673	116	16	∈	∈	PROPN
ejde-673	116	17	lp(·)(ω	lp(·)(ω	PROPN
ejde-673	116	18	)	)	PUNCT
ejde-673	116	19	(	(	PUNCT
ejde-673	116	20	n	n	NOUN
ejde-673	116	21	=	=	SYM
ejde-673	116	22	1	1	NUM
ejde-673	116	23	,	,	PUNCT
ejde-673	116	24	2	2	NUM
ejde-673	116	25	,	,	PUNCT
ejde-673	116	26	.	.	PUNCT
ejde-673	116	27	.	.	PUNCT
ejde-673	117	1	.	.	PUNCT
ejde-673	117	2	)	)	PUNCT
ejde-673	118	1	.	.	PUNCT
ejde-673	119	1	then	then	ADV
ejde-673	119	2	we	we	PRON
ejde-673	119	3	have	have	VERB
ejde-673	119	4	the	the	DET
ejde-673	119	5	following	follow	VERB
ejde-673	119	6	properties	property	NOUN
ejde-673	119	7	.	.	PUNCT
ejde-673	120	1	(	(	PUNCT
ejde-673	120	2	i	i	NOUN
ejde-673	120	3	)	)	PUNCT
ejde-673	120	4	∥u∥lp(·)(ω	∥u∥lp(·)(ω	PROPN
ejde-673	120	5	)	)	PUNCT
ejde-673	120	6	<	<	X
ejde-673	120	7	1(=	1(=	NUM
ejde-673	120	8	1	1	NUM
ejde-673	120	9	,	,	PUNCT
ejde-673	120	10	>	>	X
ejde-673	120	11	1	1	X
ejde-673	120	12	)	)	PUNCT
ejde-673	120	13	⇔	⇔	X
ejde-673	120	14	ρp(·)(u	ρp(·)(u	PROPN
ejde-673	120	15	)	)	PUNCT
ejde-673	120	16	<	<	X
ejde-673	120	17	1(=	1(=	NUM
ejde-673	120	18	1	1	NUM
ejde-673	120	19	,	,	PUNCT
ejde-673	120	20	>	>	X
ejde-673	120	21	1	1	NUM
ejde-673	120	22	)	)	PUNCT
ejde-673	120	23	.	.	PUNCT
ejde-673	121	1	(	(	PUNCT
ejde-673	121	2	ii	ii	NOUN
ejde-673	121	3	)	)	PUNCT
ejde-673	121	4	∥u∥lp(·)(ω	∥u∥lp(·)(ω	PROPN
ejde-673	121	5	)	)	PUNCT
ejde-673	121	6	>	>	SYM
ejde-673	121	7	1	1	NUM
ejde-673	121	8	⇒	⇒	NOUN
ejde-673	121	9	∥u∥p	∥u∥p	NOUN
ejde-673	122	1	−	−	PROPN
ejde-673	122	2	lp(·)(ω	lp(·)(ω	ADJ
ejde-673	122	3	)	)	PUNCT
ejde-673	122	4	≤	≤	NUM
ejde-673	122	5	ρp(·)(u	ρp(·)(u	ADJ
ejde-673	122	6	)	)	PUNCT
ejde-673	122	7	≤	≤	NUM
ejde-673	122	8	∥u∥p	∥u∥p	NOUN
ejde-673	122	9	+	+	CCONJ
ejde-673	122	10	lp(·)(ω	lp(·)(ω	ADJ
ejde-673	122	11	)	)	PUNCT
ejde-673	122	12	.	.	PUNCT
ejde-673	123	1	(	(	PUNCT
ejde-673	123	2	iii	iii	NOUN
ejde-673	123	3	)	)	PUNCT
ejde-673	123	4	∥u∥lp(·)(ω	∥u∥lp(·)(ω	NOUN
ejde-673	123	5	)	)	PUNCT
ejde-673	123	6	<	<	X
ejde-673	123	7	1	1	NUM
ejde-673	123	8	⇒	⇒	NOUN
ejde-673	123	9	∥u∥p	∥u∥p	NOUN
ejde-673	123	10	+	+	CCONJ
ejde-673	123	11	lp(·)(ω	lp(·)(ω	ADJ
ejde-673	123	12	)	)	PUNCT
ejde-673	123	13	≤	≤	NUM
ejde-673	123	14	ρp(·)(u	ρp(·)(u	ADJ
ejde-673	123	15	)	)	PUNCT
ejde-673	123	16	≤	≤	NUM
ejde-673	123	17	∥u∥p	∥u∥p	NOUN
ejde-673	123	18	−	−	PROPN
ejde-673	123	19	lp(·)(ω	lp(·)(ω	PROPN
ejde-673	123	20	)	)	PUNCT
ejde-673	123	21	.	.	PUNCT
ejde-673	124	1	(	(	PUNCT
ejde-673	124	2	iv	iv	X
ejde-673	124	3	)	)	PUNCT
ejde-673	124	4	limn→∞	limn→∞	PROPN
ejde-673	124	5	∥un	∥un	PROPN
ejde-673	124	6	−	−	PROPN
ejde-673	124	7	u∥lp(·)(ω	u∥lp(·)(ω	NOUN
ejde-673	124	8	)	)	PUNCT
ejde-673	124	9	=	=	SYM
ejde-673	124	10	0	0	NUM
ejde-673	124	11	⇔	⇔	X
ejde-673	124	12	limn→∞	limn→∞	PROPN
ejde-673	125	1	ρp(·)(un	ρp(·)(un	NOUN
ejde-673	125	2	−	−	PROPN
ejde-673	125	3	u	u	NOUN
ejde-673	125	4	)	)	PUNCT
ejde-673	125	5	=	=	SYM
ejde-673	125	6	0	0	X
ejde-673	125	7	.	.	NUM
ejde-673	125	8	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	125	9	eigenvalue	eigenvalue	VERB
ejde-673	125	10	problems	problem	NOUN
ejde-673	125	11	for	for	ADP
ejde-673	125	12	kirchhoff	kirchhoff	NOUN
ejde-673	125	13	-	-	PUNCT
ejde-673	125	14	type	type	NOUN
ejde-673	125	15	equations	equation	NOUN
ejde-673	125	16	5	5	NUM
ejde-673	125	17	(	(	PUNCT
ejde-673	125	18	v	v	NOUN
ejde-673	125	19	)	)	PUNCT
ejde-673	125	20	∥un∥lp(·)(ω	∥un∥lp(·)(ω	NOUN
ejde-673	125	21	)	)	PUNCT
ejde-673	125	22	→	→	SYM
ejde-673	125	23	∞	∞	PROPN
ejde-673	125	24	as	as	ADP
ejde-673	125	25	n	n	PROPN
ejde-673	125	26	→	→	SYM
ejde-673	125	27	∞	∞	PROPN
ejde-673	125	28	⇔	⇔	X
ejde-673	125	29	ρp(·)(un	ρp(·)(un	PROPN
ejde-673	125	30	)	)	PUNCT
ejde-673	125	31	→	→	SYM
ejde-673	125	32	∞	∞	PROPN
ejde-673	125	33	as	as	ADP
ejde-673	125	34	n	n	PROPN
ejde-673	125	35	→	→	SYM
ejde-673	125	36	∞.	∞.	PROPN
ejde-673	125	37	the	the	DET
ejde-673	125	38	following	follow	VERB
ejde-673	125	39	proposition	proposition	NOUN
ejde-673	125	40	is	be	AUX
ejde-673	125	41	a	a	DET
ejde-673	125	42	generalized	generalized	ADJ
ejde-673	125	43	hölder	hölder	NOUN
ejde-673	125	44	inequality	inequality	NOUN
ejde-673	125	45	.	.	PUNCT
ejde-673	126	1	proposition	proposition	NOUN
ejde-673	126	2	2.2	2.2	NUM
ejde-673	126	3	.	.	PUNCT
ejde-673	127	1	let	let	VERB
ejde-673	127	2	p	p	PROPN
ejde-673	127	3	∈	∈	PROPN
ejde-673	127	4	c+(ω	c+(ω	PROPN
ejde-673	127	5	)	)	PUNCT
ejde-673	127	6	,	,	PUNCT
ejde-673	127	7	where	where	SCONJ
ejde-673	127	8	c+(ω	c+(ω	X
ejde-673	127	9	)	)	PUNCT
ejde-673	127	10	:	:	PUNCT
ejde-673	128	1	=	=	PUNCT
ejde-673	128	2	{	{	PUNCT
ejde-673	128	3	p	p	X
ejde-673	128	4	∈	∈	PROPN
ejde-673	128	5	c(ω	c(ω	PROPN
ejde-673	128	6	)	)	PUNCT
ejde-673	128	7	;	;	PUNCT
ejde-673	128	8	p−	p−	X
ejde-673	128	9	>	>	X
ejde-673	128	10	1	1	NUM
ejde-673	128	11	}	}	PUNCT
ejde-673	128	12	.	.	PUNCT
ejde-673	129	1	for	for	ADP
ejde-673	129	2	any	any	DET
ejde-673	129	3	u	u	PROPN
ejde-673	129	4	∈	∈	PROPN
ejde-673	129	5	lp(·)(ω	lp(·)(ω	NOUN
ejde-673	129	6	)	)	PUNCT
ejde-673	129	7	and	and	CCONJ
ejde-673	129	8	v	v	ADP
ejde-673	129	9	∈	∈	PROPN
ejde-673	129	10	lp′(·)(ω	lp′(·)(ω	NOUN
ejde-673	129	11	)	)	PUNCT
ejde-673	129	12	,	,	PUNCT
ejde-673	129	13	we	we	PRON
ejde-673	129	14	have∫	have∫	VERB
ejde-673	129	15	ω	ω	NUM
ejde-673	129	16	|u(x)v(x)|dx	|u(x)v(x)|dx	X
ejde-673	129	17	≤	≤	NOUN
ejde-673	129	18	(	(	PUNCT
ejde-673	129	19	1	1	NUM
ejde-673	129	20	p−	p−	NOUN
ejde-673	129	21	+	+	NOUN
ejde-673	129	22	1	1	NUM
ejde-673	129	23	(	(	PUNCT
ejde-673	129	24	p′)−	p′)−	PROPN
ejde-673	129	25	)	)	PUNCT
ejde-673	129	26	∥u∥lp(·)(ω)∥v∥lp′(·)(ω	∥u∥lp(·)(ω)∥v∥lp′(·)(ω	PROPN
ejde-673	129	27	)	)	PUNCT
ejde-673	129	28	≤	≤	NOUN
ejde-673	129	29	2∥u∥lp(·)(ω)∥v∥lp′(·)(ω	2∥u∥lp(·)(ω)∥v∥lp′(·)(ω	NUM
ejde-673	129	30	)	)	PUNCT
ejde-673	129	31	.	.	PUNCT
ejde-673	130	1	here	here	ADV
ejde-673	130	2	and	and	CCONJ
ejde-673	130	3	from	from	ADP
ejde-673	130	4	now	now	ADV
ejde-673	130	5	on	on	ADV
ejde-673	130	6	,	,	PUNCT
ejde-673	130	7	for	for	ADP
ejde-673	130	8	any	any	DET
ejde-673	130	9	p	p	PROPN
ejde-673	130	10	∈	∈	PROPN
ejde-673	130	11	c+(ω	c+(ω	PROPN
ejde-673	130	12	)	)	PUNCT
ejde-673	130	13	,	,	PUNCT
ejde-673	130	14	p	p	NOUN
ejde-673	130	15	′	′	PROPN
ejde-673	130	16	(	(	PUNCT
ejde-673	130	17	·	·	PUNCT
ejde-673	130	18	)	)	PUNCT
ejde-673	130	19	denotes	denote	VERB
ejde-673	130	20	the	the	DET
ejde-673	130	21	conjugate	conjugate	ADJ
ejde-673	130	22	exponent	exponent	NOUN
ejde-673	130	23	of	of	ADP
ejde-673	130	24	p	p	X
ejde-673	130	25	(	(	PUNCT
ejde-673	130	26	·	·	PUNCT
ejde-673	130	27	)	)	PUNCT
ejde-673	130	28	,	,	PUNCT
ejde-673	130	29	that	that	ADV
ejde-673	130	30	is	is	ADV
ejde-673	130	31	,	,	PUNCT
ejde-673	130	32	p′(x	p′(x	NOUN
ejde-673	130	33	)	)	PUNCT
ejde-673	130	34	=	=	SYM
ejde-673	130	35	p(x)/(p(x)−	p(x)/(p(x)−	PROPN
ejde-673	130	36	1	1	NUM
ejde-673	130	37	)	)	PUNCT
ejde-673	130	38	for	for	ADP
ejde-673	130	39	x	x	PROPN
ejde-673	130	40	∈	∈	PROPN
ejde-673	130	41	ω	ω	PROPN
ejde-673	130	42	.	.	PUNCT
ejde-673	131	1	for	for	ADP
ejde-673	131	2	p	p	PROPN
ejde-673	131	3	∈	∈	PROPN
ejde-673	131	4	c+(ω	c+(ω	PROPN
ejde-673	131	5	)	)	PUNCT
ejde-673	131	6	,	,	PUNCT
ejde-673	131	7	we	we	PRON
ejde-673	131	8	define	define	VERB
ejde-673	131	9	,	,	PUNCT
ejde-673	131	10	for	for	ADP
ejde-673	131	11	x	x	PROPN
ejde-673	131	12	∈	∈	PROPN
ejde-673	131	13	ω	ω	PROPN
ejde-673	131	14	,	,	PUNCT
ejde-673	131	15	p∗(x	p∗(x	PROPN
ejde-673	131	16	)	)	PUNCT
ejde-673	131	17	=	=	NOUN
ejde-673	131	18	{	{	PUNCT
ejde-673	131	19	np(x	np(x	NOUN
ejde-673	131	20	)	)	PUNCT
ejde-673	131	21	n−p(x	n−p(x	NOUN
ejde-673	131	22	)	)	PUNCT
ejde-673	131	23	if	if	SCONJ
ejde-673	131	24	p(x	p(x	NOUN
ejde-673	131	25	)	)	PUNCT
ejde-673	131	26	<	<	X
ejde-673	131	27	n	n	X
ejde-673	131	28	,	,	PUNCT
ejde-673	131	29	∞	∞	PROPN
ejde-673	131	30	if	if	SCONJ
ejde-673	131	31	p(x	p(x	NOUN
ejde-673	131	32	)	)	PUNCT
ejde-673	131	33	≥	≥	NOUN
ejde-673	131	34	n.	n.	NOUN
ejde-673	131	35	proposition	proposition	NOUN
ejde-673	131	36	2.3	2.3	NUM
ejde-673	131	37	.	.	PUNCT
ejde-673	132	1	let	let	VERB
ejde-673	132	2	ω	ω	PRON
ejde-673	132	3	be	be	AUX
ejde-673	132	4	a	a	DET
ejde-673	132	5	bounded	bounded	ADJ
ejde-673	132	6	domain	domain	NOUN
ejde-673	132	7	of	of	ADP
ejde-673	132	8	rn	rn	PROPN
ejde-673	132	9	with	with	ADP
ejde-673	132	10	c0,1	c0,1	NOUN
ejde-673	132	11	-	-	PUNCT
ejde-673	132	12	boundary	boundary	NOUN
ejde-673	132	13	and	and	CCONJ
ejde-673	132	14	let	let	VERB
ejde-673	132	15	p	p	PROPN
ejde-673	132	16	∈	∈	PROPN
ejde-673	132	17	c+(ω	c+(ω	PROPN
ejde-673	132	18	)	)	PUNCT
ejde-673	132	19	.	.	PUNCT
ejde-673	133	1	then	then	ADV
ejde-673	133	2	we	we	PRON
ejde-673	133	3	have	have	VERB
ejde-673	133	4	the	the	DET
ejde-673	133	5	following	follow	VERB
ejde-673	133	6	properties	property	NOUN
ejde-673	133	7	.	.	PUNCT
ejde-673	134	1	(	(	PUNCT
ejde-673	134	2	i	i	NOUN
ejde-673	134	3	)	)	PUNCT
ejde-673	134	4	the	the	DET
ejde-673	134	5	spaces	space	NOUN
ejde-673	134	6	lp(·)(ω	lp(·)(ω	ADJ
ejde-673	134	7	)	)	PUNCT
ejde-673	134	8	and	and	CCONJ
ejde-673	134	9	w	w	PROPN
ejde-673	134	10	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	134	11	)	)	PUNCT
ejde-673	134	12	are	be	AUX
ejde-673	134	13	separable	separable	ADJ
ejde-673	134	14	,	,	PUNCT
ejde-673	134	15	reflexive	reflexive	ADJ
ejde-673	134	16	and	and	CCONJ
ejde-673	134	17	uniformly	uniformly	ADV
ejde-673	134	18	convex	convex	VERB
ejde-673	134	19	banach	banach	NOUN
ejde-673	134	20	spaces	space	VERB
ejde-673	134	21	.	.	PUNCT
ejde-673	135	1	(	(	PUNCT
ejde-673	135	2	ii	ii	NOUN
ejde-673	135	3	)	)	PUNCT
ejde-673	135	4	if	if	SCONJ
ejde-673	135	5	q	q	X
ejde-673	135	6	(	(	PUNCT
ejde-673	135	7	·	·	PUNCT
ejde-673	135	8	)	)	PUNCT
ejde-673	135	9	∈	∈	PROPN
ejde-673	135	10	c(ω	c(ω	PROPN
ejde-673	135	11	)	)	PUNCT
ejde-673	135	12	with	with	ADP
ejde-673	135	13	q−	q−	PROPN
ejde-673	135	14	≥	≥	NUM
ejde-673	135	15	1	1	NUM
ejde-673	135	16	satisfies	satisfie	NOUN
ejde-673	135	17	q(x	q(x	NOUN
ejde-673	135	18	)	)	PUNCT
ejde-673	135	19	≤	≤	NUM
ejde-673	135	20	p(x	p(x	PROPN
ejde-673	135	21	)	)	PUNCT
ejde-673	135	22	for	for	ADP
ejde-673	135	23	all	all	DET
ejde-673	135	24	x	x	SYM
ejde-673	135	25	∈	∈	PROPN
ejde-673	135	26	ω	ω	PROPN
ejde-673	135	27	,	,	PUNCT
ejde-673	135	28	then	then	ADV
ejde-673	135	29	w	w	PROPN
ejde-673	135	30	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	135	31	)	)	PUNCT
ejde-673	135	32	↪	↪	PROPN
ejde-673	135	33	→	→	SYM
ejde-673	135	34	w	w	NOUN
ejde-673	135	35	1,q(·)(ω	1,q(·)(ω	NUM
ejde-673	135	36	)	)	PUNCT
ejde-673	135	37	,	,	PUNCT
ejde-673	135	38	where	where	SCONJ
ejde-673	135	39	↪	↪	PROPN
ejde-673	135	40	→	→	SYM
ejde-673	135	41	means	mean	VERB
ejde-673	135	42	that	that	SCONJ
ejde-673	135	43	the	the	DET
ejde-673	135	44	embedding	embed	VERB
ejde-673	135	45	map	map	NOUN
ejde-673	135	46	is	be	AUX
ejde-673	135	47	continuous	continuous	ADJ
ejde-673	135	48	.	.	PUNCT
ejde-673	136	1	(	(	PUNCT
ejde-673	136	2	iii	iii	X
ejde-673	136	3	)	)	PUNCT
ejde-673	136	4	if	if	SCONJ
ejde-673	136	5	q(x	q(x	NOUN
ejde-673	136	6	)	)	PUNCT
ejde-673	136	7	∈	∈	PROPN
ejde-673	136	8	c(ω	c(ω	PROPN
ejde-673	136	9	)	)	PUNCT
ejde-673	136	10	with	with	ADP
ejde-673	136	11	q−	q−	PROPN
ejde-673	136	12	≥	≥	NUM
ejde-673	136	13	1	1	NUM
ejde-673	136	14	satisfies	satisfie	NOUN
ejde-673	136	15	that	that	SCONJ
ejde-673	136	16	q(x	q(x	NOUN
ejde-673	136	17	)	)	PUNCT
ejde-673	136	18	<	<	X
ejde-673	136	19	p∗(x	p∗(x	NOUN
ejde-673	136	20	)	)	PUNCT
ejde-673	136	21	for	for	ADP
ejde-673	136	22	all	all	DET
ejde-673	136	23	x	x	SYM
ejde-673	136	24	∈	∈	PROPN
ejde-673	136	25	ω	ω	PROPN
ejde-673	136	26	,	,	PUNCT
ejde-673	136	27	then	then	ADV
ejde-673	136	28	the	the	DET
ejde-673	136	29	embedding	embed	VERB
ejde-673	136	30	map	map	NOUN
ejde-673	136	31	w	w	ADP
ejde-673	136	32	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	136	33	)	)	PUNCT
ejde-673	136	34	↪	↪	PROPN
ejde-673	136	35	→	→	SYM
ejde-673	136	36	lq(·)(ω	lq(·)(ω	ADJ
ejde-673	136	37	)	)	PUNCT
ejde-673	136	38	is	be	AUX
ejde-673	136	39	compact	compact	ADJ
ejde-673	136	40	.	.	PUNCT
ejde-673	137	1	next	next	ADV
ejde-673	137	2	we	we	PRON
ejde-673	137	3	consider	consider	VERB
ejde-673	137	4	the	the	DET
ejde-673	137	5	trace	trace	NOUN
ejde-673	137	6	(	(	PUNCT
ejde-673	137	7	cf	cf	NOUN
ejde-673	137	8	.	.	PUNCT
ejde-673	138	1	fan	fan	NOUN
ejde-673	139	1	[	[	X
ejde-673	139	2	16	16	NUM
ejde-673	139	3	]	]	PUNCT
ejde-673	139	4	)	)	PUNCT
ejde-673	139	5	.	.	PUNCT
ejde-673	140	1	let	let	VERB
ejde-673	140	2	ω	ω	PRON
ejde-673	140	3	be	be	AUX
ejde-673	140	4	a	a	DET
ejde-673	140	5	bounded	bounded	ADJ
ejde-673	140	6	domain	domain	NOUN
ejde-673	140	7	of	of	ADP
ejde-673	140	8	rn	rn	PROPN
ejde-673	140	9	with	with	ADP
ejde-673	140	10	a	a	DET
ejde-673	140	11	c0,1	c0,1	NOUN
ejde-673	140	12	-	-	PUNCT
ejde-673	140	13	boundary	boundary	NOUN
ejde-673	140	14	γ	γ	NOUN
ejde-673	140	15	and	and	CCONJ
ejde-673	140	16	p	p	PROPN
ejde-673	140	17	∈	∈	PROPN
ejde-673	140	18	c(ω	c(ω	PROPN
ejde-673	140	19	)	)	PUNCT
ejde-673	140	20	with	with	ADP
ejde-673	140	21	p−	p−	PRON
ejde-673	140	22	≥	≥	NOUN
ejde-673	140	23	1	1	NUM
ejde-673	140	24	.	.	PUNCT
ejde-673	141	1	since	since	SCONJ
ejde-673	141	2	w	w	PROPN
ejde-673	141	3	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	141	4	)	)	PUNCT
ejde-673	141	5	⊂	⊂	PROPN
ejde-673	141	6	w	w	PROPN
ejde-673	141	7	1,1(ω	1,1(ω	PROPN
ejde-673	141	8	)	)	PUNCT
ejde-673	141	9	,	,	PUNCT
ejde-673	141	10	the	the	DET
ejde-673	141	11	trace	trace	NOUN
ejde-673	141	12	u	u	NOUN
ejde-673	141	13	∣∣	∣∣	NUM
ejde-673	141	14	γ	γ	X
ejde-673	141	15	to	to	ADP
ejde-673	141	16	γ	γ	NOUN
ejde-673	141	17	of	of	ADP
ejde-673	141	18	any	any	DET
ejde-673	141	19	function	function	NOUN
ejde-673	141	20	u	u	NOUN
ejde-673	141	21	in	in	ADP
ejde-673	141	22	w	w	PROPN
ejde-673	141	23	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	141	24	)	)	PUNCT
ejde-673	141	25	is	be	AUX
ejde-673	141	26	well	well	ADV
ejde-673	141	27	defined	define	VERB
ejde-673	141	28	as	as	ADP
ejde-673	141	29	a	a	DET
ejde-673	141	30	function	function	NOUN
ejde-673	141	31	in	in	ADP
ejde-673	141	32	l1(γ	l1(γ	NOUN
ejde-673	141	33	)	)	PUNCT
ejde-673	141	34	.	.	PUNCT
ejde-673	142	1	we	we	PRON
ejde-673	142	2	define	define	VERB
ejde-673	142	3	tr(w	tr(w	ADV
ejde-673	142	4	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	142	5	)	)	PUNCT
ejde-673	142	6	)	)	PUNCT
ejde-673	143	1	=	=	PRON
ejde-673	143	2	(	(	PUNCT
ejde-673	143	3	trw	trw	PROPN
ejde-673	143	4	1,p(·))(γ	1,p(·))(γ	NUM
ejde-673	143	5	)	)	PUNCT
ejde-673	143	6	=	=	SYM
ejde-673	143	7	{	{	PUNCT
ejde-673	143	8	f	f	PROPN
ejde-673	143	9	;	;	PUNCT
ejde-673	143	10	f	f	PROPN
ejde-673	143	11	is	be	AUX
ejde-673	143	12	the	the	DET
ejde-673	143	13	trace	trace	NOUN
ejde-673	143	14	to	to	ADP
ejde-673	143	15	γ	γ	NOUN
ejde-673	143	16	of	of	ADP
ejde-673	143	17	a	a	DET
ejde-673	143	18	function	function	NOUN
ejde-673	143	19	f	f	PROPN
ejde-673	143	20	∈	∈	PROPN
ejde-673	143	21	w	w	PROPN
ejde-673	143	22	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	143	23	)	)	PUNCT
ejde-673	143	24	}	}	PUNCT
ejde-673	143	25	equipped	equip	VERB
ejde-673	143	26	with	with	ADP
ejde-673	143	27	the	the	DET
ejde-673	143	28	norm	norm	NOUN
ejde-673	143	29	∥f∥(trw	∥f∥(trw	ADJ
ejde-673	143	30	1,p(·))(γ	1,p(·))(γ	NUM
ejde-673	143	31	)	)	PUNCT
ejde-673	143	32	=	=	NOUN
ejde-673	144	1	inf{∥f∥w	inf{∥f∥w	NOUN
ejde-673	144	2	1,p(·)(ω);f	1,p(·)(ω);f	NUM
ejde-673	144	3	∈	∈	PROPN
ejde-673	144	4	w	w	NOUN
ejde-673	144	5	1,p(·)(ω	1,p(·)(ω	ADJ
ejde-673	144	6	)	)	PUNCT
ejde-673	144	7	satisfying	satisfy	VERB
ejde-673	144	8	f	f	X
ejde-673	144	9	∣∣	∣∣	NUM
ejde-673	144	10	γ	γ	X
ejde-673	144	11	=	=	SYM
ejde-673	144	12	f	f	PROPN
ejde-673	144	13	}	}	PUNCT
ejde-673	144	14	for	for	ADP
ejde-673	144	15	f	f	PROPN
ejde-673	144	16	∈	∈	PROPN
ejde-673	144	17	(	(	PUNCT
ejde-673	144	18	trw	trw	PROPN
ejde-673	144	19	1,p(·))(γ	1,p(·))(γ	NUM
ejde-673	144	20	)	)	PUNCT
ejde-673	144	21	,	,	PUNCT
ejde-673	144	22	where	where	SCONJ
ejde-673	144	23	the	the	DET
ejde-673	144	24	infimum	infimum	NOUN
ejde-673	144	25	can	can	AUX
ejde-673	144	26	be	be	AUX
ejde-673	144	27	achieved	achieve	VERB
ejde-673	144	28	.	.	PUNCT
ejde-673	145	1	then	then	ADV
ejde-673	145	2	we	we	PRON
ejde-673	145	3	can	can	AUX
ejde-673	145	4	see	see	VERB
ejde-673	145	5	that	that	PRON
ejde-673	145	6	(	(	PUNCT
ejde-673	145	7	trw	trw	PROPN
ejde-673	145	8	1,p(·))(γ	1,p(·))(γ	NUM
ejde-673	145	9	)	)	PUNCT
ejde-673	145	10	is	be	AUX
ejde-673	145	11	a	a	DET
ejde-673	145	12	banach	banach	NOUN
ejde-673	145	13	space	space	NOUN
ejde-673	145	14	.	.	PUNCT
ejde-673	146	1	in	in	ADP
ejde-673	146	2	the	the	DET
ejde-673	146	3	later	later	NOUN
ejde-673	146	4	,	,	PUNCT
ejde-673	146	5	we	we	PRON
ejde-673	146	6	also	also	ADV
ejde-673	146	7	write	write	VERB
ejde-673	146	8	f	f	PROPN
ejde-673	146	9	∣∣	∣∣	X
ejde-673	146	10	γ	γ	X
ejde-673	146	11	=	=	SYM
ejde-673	146	12	g	g	PROPN
ejde-673	146	13	by	by	ADP
ejde-673	146	14	f	f	PROPN
ejde-673	146	15	=	=	SYM
ejde-673	146	16	g	g	PROPN
ejde-673	146	17	on	on	ADP
ejde-673	146	18	γ	γ	PROPN
ejde-673	146	19	.	.	PROPN
ejde-673	146	20	moreover	moreover	ADV
ejde-673	146	21	,	,	PUNCT
ejde-673	146	22	for	for	ADP
ejde-673	146	23	i	i	PROPN
ejde-673	146	24	=	=	SYM
ejde-673	146	25	1	1	NUM
ejde-673	146	26	,	,	PUNCT
ejde-673	146	27	2	2	NUM
ejde-673	146	28	,	,	PUNCT
ejde-673	146	29	we	we	PRON
ejde-673	146	30	denote	denote	VERB
ejde-673	146	31	(	(	PUNCT
ejde-673	146	32	trw	trw	PROPN
ejde-673	146	33	1,p(·))(γi	1,p(·))(γi	NUM
ejde-673	146	34	)	)	PUNCT
ejde-673	146	35	=	=	PRON
ejde-673	146	36	{	{	PUNCT
ejde-673	146	37	f	f	X
ejde-673	146	38	∣∣	∣∣	X
ejde-673	146	39	γi	γi	X
ejde-673	146	40	;	;	PUNCT
ejde-673	146	41	f	f	PROPN
ejde-673	146	42	∈	∈	PROPN
ejde-673	146	43	(	(	PUNCT
ejde-673	146	44	trw	trw	PROPN
ejde-673	146	45	1,p(·))(γ	1,p(·))(γ	NUM
ejde-673	146	46	)	)	PUNCT
ejde-673	146	47	}	}	PUNCT
ejde-673	146	48	equipped	equip	VERB
ejde-673	146	49	with	with	ADP
ejde-673	146	50	the	the	DET
ejde-673	146	51	norm	norm	NOUN
ejde-673	146	52	∥g∥(trw	∥g∥(trw	PROPN
ejde-673	146	53	1,p(·))(γi	1,p(·))(γi	PROPN
ejde-673	146	54	)	)	PUNCT
ejde-673	146	55	=	=	NOUN
ejde-673	146	56	inf{∥f∥(trw	inf{∥f∥(trw	NOUN
ejde-673	146	57	1,p(·))(γ	1,p(·))(γ	NUM
ejde-673	146	58	)	)	PUNCT
ejde-673	146	59	;	;	PUNCT
ejde-673	146	60	f	f	PROPN
ejde-673	146	61	∈	∈	PROPN
ejde-673	146	62	(	(	PUNCT
ejde-673	146	63	trw	trw	PROPN
ejde-673	146	64	1,p(·))(γ	1,p(·))(γ	NUM
ejde-673	146	65	)	)	PUNCT
ejde-673	146	66	satisfying	satisfy	VERB
ejde-673	146	67	f	f	X
ejde-673	146	68	∣∣	∣∣	X
ejde-673	146	69	γi	γi	X
ejde-673	146	70	=	=	SYM
ejde-673	146	71	g	g	NOUN
ejde-673	146	72	}	}	PUNCT
ejde-673	146	73	,	,	PUNCT
ejde-673	146	74	where	where	SCONJ
ejde-673	146	75	the	the	DET
ejde-673	146	76	infimum	infimum	NOUN
ejde-673	146	77	can	can	AUX
ejde-673	146	78	also	also	ADV
ejde-673	146	79	be	be	AUX
ejde-673	146	80	achieved	achieve	VERB
ejde-673	146	81	,	,	PUNCT
ejde-673	146	82	so	so	ADV
ejde-673	146	83	for	for	ADP
ejde-673	146	84	any	any	DET
ejde-673	146	85	g	g	PROPN
ejde-673	146	86	∈	∈	PROPN
ejde-673	146	87	(	(	PUNCT
ejde-673	146	88	trw	trw	PROPN
ejde-673	146	89	1,p(·))(γi	1,p(·))(γi	NUM
ejde-673	146	90	)	)	PUNCT
ejde-673	146	91	,	,	PUNCT
ejde-673	146	92	there	there	PRON
ejde-673	146	93	exists	exist	VERB
ejde-673	146	94	f	f	PROPN
ejde-673	146	95	∈	∈	PROPN
ejde-673	146	96	w	w	PROPN
ejde-673	146	97	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	146	98	)	)	PUNCT
ejde-673	146	99	such	such	ADJ
ejde-673	146	100	that	that	SCONJ
ejde-673	146	101	f	f	PROPN
ejde-673	146	102	∣∣	∣∣	X
ejde-673	146	103	γi	γi	X
ejde-673	146	104	=	=	SYM
ejde-673	146	105	g	g	PROPN
ejde-673	146	106	and	and	CCONJ
ejde-673	146	107	∥f∥w	∥f∥w	ADV
ejde-673	146	108	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	146	109	)	)	PUNCT
ejde-673	147	1	=	=	SYM
ejde-673	147	2	∥g∥(trw	∥g∥(trw	PROPN
ejde-673	147	3	1,p(·))(γi	1,p(·))(γi	NUM
ejde-673	147	4	)	)	PUNCT
ejde-673	147	5	.	.	PUNCT
ejde-673	148	1	let	let	VERB
ejde-673	148	2	q	q	PROPN
ejde-673	148	3	∈	∈	PROPN
ejde-673	148	4	c+(γ	c+(γ	PROPN
ejde-673	148	5	)	)	PUNCT
ejde-673	148	6	:	:	PUNCT
ejde-673	149	1	=	=	SYM
ejde-673	149	2	{	{	PUNCT
ejde-673	149	3	q	q	PROPN
ejde-673	149	4	∈	∈	PROPN
ejde-673	149	5	c(γ	c(γ	PROPN
ejde-673	149	6	)	)	PUNCT
ejde-673	149	7	;	;	PUNCT
ejde-673	149	8	q−	q−	PROPN
ejde-673	149	9	>	>	X
ejde-673	149	10	1	1	NUM
ejde-673	149	11	}	}	PUNCT
ejde-673	149	12	and	and	CCONJ
ejde-673	149	13	denote	denote	VERB
ejde-673	149	14	the	the	DET
ejde-673	149	15	surface	surface	NOUN
ejde-673	149	16	measure	measure	NOUN
ejde-673	149	17	on	on	ADP
ejde-673	149	18	γ	γ	NOUN
ejde-673	149	19	induced	induce	VERB
ejde-673	149	20	from	from	ADP
ejde-673	149	21	the	the	DET
ejde-673	149	22	lebesgue	lebesgue	NOUN
ejde-673	149	23	measure	measure	NOUN
ejde-673	149	24	dx	dx	PROPN
ejde-673	149	25	on	on	ADP
ejde-673	149	26	ω	ω	PROPN
ejde-673	149	27	by	by	ADP
ejde-673	149	28	dσx	dσx	NOUN
ejde-673	149	29	.	.	PUNCT
ejde-673	150	1	we	we	PRON
ejde-673	150	2	define	define	VERB
ejde-673	150	3	lq(·)(γ	lq(·)(γ	NOUN
ejde-673	150	4	)	)	PUNCT
ejde-673	151	1	=	=	PRON
ejde-673	151	2	{	{	PUNCT
ejde-673	151	3	u	u	NOUN
ejde-673	151	4	:	:	PUNCT
ejde-673	151	5	γ	γ	X
ejde-673	151	6	→	→	SYM
ejde-673	151	7	r	r	NOUN
ejde-673	151	8	is	be	AUX
ejde-673	151	9	a	a	DET
ejde-673	151	10	measurable	measurable	ADJ
ejde-673	151	11	function	function	NOUN
ejde-673	151	12	with	with	ADP
ejde-673	151	13	respect	respect	NOUN
ejde-673	151	14	to	to	ADP
ejde-673	151	15	dσx	dσx	PROPN
ejde-673	151	16	6	6	NUM
ejde-673	151	17	j.	j.	PROPN
ejde-673	151	18	aramaki	aramaki	PROPN
ejde-673	151	19	ejde-2025/17	ejde-2025/17	PROPN
ejde-673	151	20	satisfying	satisfy	VERB
ejde-673	151	21	∫	∫	PROPN
ejde-673	151	22	γ	γ	NOUN
ejde-673	151	23	|u(x)|q(x	|u(x)|q(x	ADV
ejde-673	151	24	)	)	PUNCT
ejde-673	151	25	dσx	dσx	NOUN
ejde-673	151	26	<	<	X
ejde-673	151	27	∞	∞	PUNCT
ejde-673	151	28	}	}	PUNCT
ejde-673	151	29	and	and	CCONJ
ejde-673	151	30	the	the	DET
ejde-673	151	31	norm	norm	NOUN
ejde-673	151	32	is	be	AUX
ejde-673	151	33	defined	define	VERB
ejde-673	151	34	by	by	ADP
ejde-673	151	35	∥u∥lq(·)(γ	∥u∥lq(·)(γ	PROPN
ejde-673	151	36	)	)	PUNCT
ejde-673	151	37	=	=	SYM
ejde-673	151	38	inf	inf	NOUN
ejde-673	151	39	{	{	PUNCT
ejde-673	151	40	τ	τ	PROPN
ejde-673	151	41	>	>	X
ejde-673	151	42	0	0	PROPN
ejde-673	151	43	;	;	PUNCT
ejde-673	151	44	∫	∫	PROPN
ejde-673	151	45	γ	γ	PROPN
ejde-673	151	46	∣∣u(x	∣∣u(x	PROPN
ejde-673	151	47	)	)	PUNCT
ejde-673	151	48	τ	τ	PROPN
ejde-673	151	49	∣∣q(x	∣∣q(x	PROPN
ejde-673	151	50	)	)	PUNCT
ejde-673	151	51	dσx	dσx	NOUN
ejde-673	151	52	≤	≤	NUM
ejde-673	151	53	1	1	NUM
ejde-673	151	54	}	}	PUNCT
ejde-673	151	55	,	,	PUNCT
ejde-673	151	56	and	and	CCONJ
ejde-673	151	57	we	we	PRON
ejde-673	151	58	also	also	ADV
ejde-673	151	59	define	define	VERB
ejde-673	151	60	a	a	DET
ejde-673	151	61	modular	modular	NOUN
ejde-673	151	62	on	on	ADP
ejde-673	151	63	lq(·)(γ	lq(·)(γ	NOUN
ejde-673	151	64	)	)	PUNCT
ejde-673	151	65	by	by	ADP
ejde-673	151	66	ρq(·),γ(u	ρq(·),γ(u	PROPN
ejde-673	151	67	)	)	PUNCT
ejde-673	152	1	=	=	SYM
ejde-673	152	2	∫	∫	PROPN
ejde-673	152	3	γ	γ	X
ejde-673	152	4	|u(x)|q(x	|u(x)|q(x	ADV
ejde-673	152	5	)	)	PUNCT
ejde-673	152	6	dσx	dσx	NOUN
ejde-673	152	7	.	.	PUNCT
ejde-673	153	1	similarly	similarly	ADV
ejde-673	153	2	as	as	ADP
ejde-673	153	3	proposition	proposition	NOUN
ejde-673	153	4	2.1	2.1	NUM
ejde-673	153	5	,	,	PUNCT
ejde-673	153	6	we	we	PRON
ejde-673	153	7	have	have	VERB
ejde-673	153	8	the	the	DET
ejde-673	153	9	following	follow	VERB
ejde-673	153	10	proposition	proposition	NOUN
ejde-673	153	11	.	.	PUNCT
ejde-673	154	1	proposition	proposition	NOUN
ejde-673	154	2	2.4	2.4	NUM
ejde-673	154	3	.	.	PUNCT
ejde-673	155	1	let	let	VERB
ejde-673	155	2	q	q	PROPN
ejde-673	155	3	∈	∈	PROPN
ejde-673	155	4	c(γ	c(γ	PROPN
ejde-673	155	5	)	)	PUNCT
ejde-673	155	6	with	with	ADP
ejde-673	155	7	q−	q−	PROPN
ejde-673	155	8	≥	≥	NUM
ejde-673	155	9	1	1	NUM
ejde-673	155	10	,	,	PUNCT
ejde-673	155	11	and	and	CCONJ
ejde-673	155	12	let	let	VERB
ejde-673	155	13	u	u	NOUN
ejde-673	155	14	,	,	PUNCT
ejde-673	155	15	un	un	PROPN
ejde-673	155	16	∈	∈	PROPN
ejde-673	155	17	lq(·)(γ	lq(·)(γ	NOUN
ejde-673	155	18	)	)	PUNCT
ejde-673	155	19	.	.	PUNCT
ejde-673	156	1	then	then	ADV
ejde-673	156	2	we	we	PRON
ejde-673	156	3	have	have	VERB
ejde-673	156	4	the	the	DET
ejde-673	156	5	following	follow	VERB
ejde-673	156	6	properties	property	NOUN
ejde-673	156	7	.	.	PUNCT
ejde-673	157	1	(	(	PUNCT
ejde-673	157	2	i	i	NOUN
ejde-673	157	3	)	)	PUNCT
ejde-673	157	4	∥u∥lq(·)(γ	∥u∥lq(·)(γ	PROPN
ejde-673	157	5	)	)	PUNCT
ejde-673	157	6	<	<	X
ejde-673	158	1	1(=	1(=	NUM
ejde-673	158	2	1	1	NUM
ejde-673	158	3	,	,	PUNCT
ejde-673	158	4	>	>	X
ejde-673	158	5	1	1	X
ejde-673	158	6	)	)	PUNCT
ejde-673	158	7	⇔	⇔	X
ejde-673	158	8	ρq(·),γ(u	ρq(·),γ(u	PROPN
ejde-673	158	9	)	)	PUNCT
ejde-673	158	10	<	<	X
ejde-673	159	1	1(=	1(=	NUM
ejde-673	159	2	1	1	NUM
ejde-673	159	3	,	,	PUNCT
ejde-673	159	4	>	>	X
ejde-673	159	5	1	1	NUM
ejde-673	159	6	)	)	PUNCT
ejde-673	159	7	.	.	PUNCT
ejde-673	160	1	(	(	PUNCT
ejde-673	160	2	ii	ii	NOUN
ejde-673	160	3	)	)	PUNCT
ejde-673	160	4	∥u∥lq(·)(γ	∥u∥lq(·)(γ	PROPN
ejde-673	160	5	)	)	PUNCT
ejde-673	160	6	>	>	SYM
ejde-673	160	7	1	1	NUM
ejde-673	160	8	⇒	⇒	NOUN
ejde-673	160	9	∥u∥q	∥u∥q	ADJ
ejde-673	160	10	−	−	ADP
ejde-673	160	11	lq(·)(γ	lq(·)(γ	NOUN
ejde-673	160	12	)	)	PUNCT
ejde-673	160	13	≤	≤	NUM
ejde-673	160	14	ρq(·),γ(u	ρq(·),γ(u	NOUN
ejde-673	160	15	)	)	PUNCT
ejde-673	160	16	≤	≤	PUNCT
ejde-673	161	1	∥u∥q	∥u∥q	ADJ
ejde-673	161	2	+	+	NUM
ejde-673	161	3	lq(·)(γ	lq(·)(γ	NOUN
ejde-673	161	4	)	)	PUNCT
ejde-673	161	5	.	.	PUNCT
ejde-673	162	1	(	(	PUNCT
ejde-673	162	2	iii	iii	X
ejde-673	162	3	)	)	PUNCT
ejde-673	162	4	∥u∥lq(·)(γ	∥u∥lq(·)(γ	NOUN
ejde-673	162	5	)	)	PUNCT
ejde-673	162	6	<	<	X
ejde-673	162	7	1	1	NUM
ejde-673	162	8	⇒	⇒	NOUN
ejde-673	162	9	∥u∥q	∥u∥q	VERB
ejde-673	162	10	+	+	CCONJ
ejde-673	162	11	lq(·)(γ	lq(·)(γ	NOUN
ejde-673	162	12	)	)	PUNCT
ejde-673	162	13	≤	≤	NOUN
ejde-673	162	14	ρq(·),γ(u	ρq(·),γ(u	NOUN
ejde-673	162	15	)	)	PUNCT
ejde-673	162	16	≤	≤	PUNCT
ejde-673	163	1	∥u∥q	∥u∥q	ADV
ejde-673	163	2	−	−	NOUN
ejde-673	163	3	lq(·)(γ	lq(·)(γ	NOUN
ejde-673	163	4	)	)	PUNCT
ejde-673	163	5	.	.	PUNCT
ejde-673	164	1	(	(	PUNCT
ejde-673	164	2	iv	iv	X
ejde-673	164	3	)	)	PUNCT
ejde-673	164	4	∥un∥lq(·)(γ	∥un∥lq(·)(γ	NOUN
ejde-673	164	5	)	)	PUNCT
ejde-673	164	6	→	→	SYM
ejde-673	164	7	0	0	NUM
ejde-673	164	8	⇔	⇔	NUM
ejde-673	164	9	ρq(·),γ(un	ρq(·),γ(un	PROPN
ejde-673	164	10	)	)	PUNCT
ejde-673	164	11	→	→	SYM
ejde-673	164	12	0	0	NUM
ejde-673	164	13	.	.	PUNCT
ejde-673	165	1	(	(	PUNCT
ejde-673	165	2	v	v	NOUN
ejde-673	165	3	)	)	PUNCT
ejde-673	165	4	∥un∥lq(·)(γ	∥un∥lq(·)(γ	NOUN
ejde-673	165	5	)	)	PUNCT
ejde-673	165	6	→	→	SYM
ejde-673	165	7	∞	∞	PROPN
ejde-673	165	8	⇔	⇔	X
ejde-673	165	9	ρq(·),γ(un	ρq(·),γ(un	PROPN
ejde-673	165	10	)	)	PUNCT
ejde-673	165	11	→	→	PUNCT
ejde-673	165	12	∞.	∞.	PROPN
ejde-673	165	13	the	the	DET
ejde-673	165	14	hölder	hölder	NOUN
ejde-673	165	15	inequality	inequality	NOUN
ejde-673	165	16	also	also	ADV
ejde-673	165	17	holds	hold	VERB
ejde-673	165	18	for	for	ADP
ejde-673	165	19	functions	function	NOUN
ejde-673	165	20	on	on	ADP
ejde-673	165	21	γ	γ	PROPN
ejde-673	165	22	.	.	PROPN
ejde-673	165	23	proposition	proposition	NOUN
ejde-673	165	24	2.5	2.5	NUM
ejde-673	165	25	.	.	PUNCT
ejde-673	166	1	let	let	VERB
ejde-673	166	2	q	q	PROPN
ejde-673	166	3	∈	∈	PROPN
ejde-673	166	4	c(γ	c(γ	PROPN
ejde-673	166	5	)	)	PUNCT
ejde-673	166	6	with	with	ADP
ejde-673	166	7	q−	q−	PROPN
ejde-673	166	8	>	>	X
ejde-673	167	1	1	1	NUM
ejde-673	167	2	.	.	PUNCT
ejde-673	168	1	then	then	ADV
ejde-673	168	2	the	the	DET
ejde-673	168	3	following	follow	VERB
ejde-673	168	4	inequality	inequality	NOUN
ejde-673	168	5	holds.∫	holds.∫	NUM
ejde-673	168	6	γ	γ	PROPN
ejde-673	168	7	|f(x)g(x)|dσx	|f(x)g(x)|dσx	NOUN
ejde-673	168	8	≤	≤	NOUN
ejde-673	168	9	2∥f∥lq(·)(γ)∥g∥lq′(·)(γ	2∥f∥lq(·)(γ)∥g∥lq′(·)(γ	NUM
ejde-673	168	10	)	)	PUNCT
ejde-673	168	11	for	for	ADP
ejde-673	168	12	all	all	DET
ejde-673	168	13	f	f	PROPN
ejde-673	168	14	∈	∈	PROPN
ejde-673	168	15	lq(·)(γ	lq(·)(γ	NOUN
ejde-673	168	16	)	)	PUNCT
ejde-673	168	17	,	,	PUNCT
ejde-673	168	18	g	g	PROPN
ejde-673	168	19	∈	∈	PROPN
ejde-673	168	20	lq′(·)(γ	lq′(·)(γ	NOUN
ejde-673	168	21	)	)	PUNCT
ejde-673	168	22	.	.	PUNCT
ejde-673	169	1	proposition	proposition	NOUN
ejde-673	169	2	2.6	2.6	NUM
ejde-673	169	3	.	.	PUNCT
ejde-673	170	1	let	let	VERB
ejde-673	170	2	ω	ω	PRON
ejde-673	170	3	be	be	AUX
ejde-673	170	4	a	a	DET
ejde-673	170	5	bounded	bounded	ADJ
ejde-673	170	6	domain	domain	NOUN
ejde-673	170	7	of	of	ADP
ejde-673	170	8	rn	rn	PROPN
ejde-673	170	9	with	with	ADP
ejde-673	170	10	a	a	DET
ejde-673	170	11	c0,1	c0,1	NOUN
ejde-673	170	12	-	-	PUNCT
ejde-673	170	13	boundary	boundary	NOUN
ejde-673	170	14	γ	γ	NOUN
ejde-673	170	15	and	and	CCONJ
ejde-673	170	16	let	let	VERB
ejde-673	170	17	p	p	PROPN
ejde-673	170	18	∈	∈	PROPN
ejde-673	170	19	c+(ω	c+(ω	PROPN
ejde-673	170	20	)	)	PUNCT
ejde-673	170	21	.	.	PUNCT
ejde-673	171	1	if	if	SCONJ
ejde-673	171	2	f	f	PROPN
ejde-673	171	3	∈	∈	PROPN
ejde-673	171	4	(	(	PUNCT
ejde-673	171	5	trw	trw	PROPN
ejde-673	171	6	1,p(·))(γ	1,p(·))(γ	NUM
ejde-673	171	7	)	)	PUNCT
ejde-673	171	8	,	,	PUNCT
ejde-673	171	9	then	then	ADV
ejde-673	171	10	f	f	PROPN
ejde-673	171	11	∈	∈	PROPN
ejde-673	171	12	lp(·)(γ	lp(·)(γ	PROPN
ejde-673	171	13	)	)	PUNCT
ejde-673	171	14	and	and	CCONJ
ejde-673	171	15	there	there	PRON
ejde-673	171	16	exists	exist	VERB
ejde-673	171	17	a	a	DET
ejde-673	171	18	constant	constant	ADJ
ejde-673	171	19	c	c	NOUN
ejde-673	171	20	>	>	X
ejde-673	171	21	0	0	NUM
ejde-673	171	22	such	such	ADJ
ejde-673	171	23	that	that	DET
ejde-673	171	24	∥f∥lp(·)(γ	∥f∥lp(·)(γ	NOUN
ejde-673	171	25	)	)	PUNCT
ejde-673	171	26	≤	≤	NUM
ejde-673	171	27	c∥f∥(trw	c∥f∥(trw	NOUN
ejde-673	171	28	1,p(·))(γ	1,p(·))(γ	NUM
ejde-673	171	29	)	)	PUNCT
ejde-673	171	30	.	.	PUNCT
ejde-673	172	1	in	in	ADP
ejde-673	172	2	particular	particular	ADJ
ejde-673	172	3	,	,	PUNCT
ejde-673	172	4	if	if	SCONJ
ejde-673	172	5	f	f	PROPN
ejde-673	172	6	∈	∈	PROPN
ejde-673	172	7	(	(	PUNCT
ejde-673	172	8	trw	trw	PROPN
ejde-673	172	9	1,p(·))(γ	1,p(·))(γ	NUM
ejde-673	172	10	)	)	PUNCT
ejde-673	172	11	,	,	PUNCT
ejde-673	172	12	then	then	ADV
ejde-673	172	13	f	f	PROPN
ejde-673	172	14	∈	∈	PROPN
ejde-673	172	15	lp(·)(γi	lp(·)(γi	PROPN
ejde-673	172	16	)	)	PUNCT
ejde-673	172	17	and	and	CCONJ
ejde-673	172	18	∥f∥lp(·)(γi	∥f∥lp(·)(γi	NUM
ejde-673	172	19	)	)	PUNCT
ejde-673	172	20	≤	≤	NUM
ejde-673	172	21	c∥f∥(trw	c∥f∥(trw	NOUN
ejde-673	172	22	1,p(·))(γ	1,p(·))(γ	NOUN
ejde-673	172	23	)	)	PUNCT
ejde-673	172	24	for	for	ADP
ejde-673	172	25	i	i	PROPN
ejde-673	172	26	=	=	SYM
ejde-673	172	27	1	1	NUM
ejde-673	172	28	,	,	PUNCT
ejde-673	172	29	2	2	NUM
ejde-673	172	30	.	.	X
ejde-673	172	31	for	for	ADP
ejde-673	172	32	p	p	PROPN
ejde-673	172	33	∈	∈	PROPN
ejde-673	172	34	c+(ω	c+(ω	PROPN
ejde-673	172	35	)	)	PUNCT
ejde-673	172	36	,	,	PUNCT
ejde-673	172	37	we	we	PRON
ejde-673	172	38	define	define	VERB
ejde-673	172	39	,	,	PUNCT
ejde-673	172	40	for	for	ADP
ejde-673	172	41	x	x	PROPN
ejde-673	172	42	∈	∈	PROPN
ejde-673	172	43	ω	ω	PROPN
ejde-673	172	44	,	,	PUNCT
ejde-673	172	45	p∂(x	p∂(x	NOUN
ejde-673	172	46	)	)	PUNCT
ejde-673	172	47	=	=	PRON
ejde-673	172	48	{	{	PUNCT
ejde-673	172	49	(	(	PUNCT
ejde-673	172	50	n−1)p(x	n−1)p(x	NUM
ejde-673	172	51	)	)	PUNCT
ejde-673	172	52	n−p(x	n−p(x	PROPN
ejde-673	172	53	)	)	PUNCT
ejde-673	172	54	if	if	SCONJ
ejde-673	172	55	p(x	p(x	NOUN
ejde-673	172	56	)	)	PUNCT
ejde-673	172	57	<	<	X
ejde-673	172	58	n	n	X
ejde-673	172	59	,	,	PUNCT
ejde-673	172	60	∞	∞	PROPN
ejde-673	172	61	if	if	SCONJ
ejde-673	172	62	p(x	p(x	NOUN
ejde-673	172	63	)	)	PUNCT
ejde-673	172	64	≥	≥	NOUN
ejde-673	172	65	n.	n.	NOUN
ejde-673	172	66	the	the	DET
ejde-673	172	67	next	next	ADJ
ejde-673	172	68	proposition	proposition	NOUN
ejde-673	172	69	follows	follow	VERB
ejde-673	172	70	from	from	ADP
ejde-673	172	71	yao	yao	PROPN
ejde-673	173	1	[	[	X
ejde-673	173	2	33	33	NUM
ejde-673	173	3	,	,	PUNCT
ejde-673	173	4	proposition	proposition	NOUN
ejde-673	173	5	2.6	2.6	NUM
ejde-673	173	6	]	]	PUNCT
ejde-673	173	7	.	.	PUNCT
ejde-673	174	1	proposition	proposition	NOUN
ejde-673	174	2	2.7	2.7	NUM
ejde-673	174	3	.	.	PUNCT
ejde-673	175	1	let	let	VERB
ejde-673	175	2	p	p	PROPN
ejde-673	175	3	∈	∈	PROPN
ejde-673	175	4	c+(ω	c+(ω	PROPN
ejde-673	175	5	)	)	PUNCT
ejde-673	175	6	.	.	PUNCT
ejde-673	176	1	then	then	ADV
ejde-673	176	2	if	if	SCONJ
ejde-673	176	3	q	q	PROPN
ejde-673	176	4	∈	∈	PROPN
ejde-673	176	5	c+(γ	c+(γ	PROPN
ejde-673	176	6	)	)	PUNCT
ejde-673	176	7	satisfies	satisfie	NOUN
ejde-673	176	8	q(x	q(x	NOUN
ejde-673	176	9	)	)	PUNCT
ejde-673	176	10	<	<	X
ejde-673	176	11	p∂(x	p∂(x	PROPN
ejde-673	176	12	)	)	PUNCT
ejde-673	176	13	for	for	ADP
ejde-673	176	14	all	all	DET
ejde-673	176	15	x	x	SYM
ejde-673	176	16	∈	∈	PROPN
ejde-673	176	17	γ	γ	X
ejde-673	176	18	,	,	PUNCT
ejde-673	176	19	then	then	ADV
ejde-673	176	20	the	the	DET
ejde-673	176	21	trace	trace	NOUN
ejde-673	176	22	mapping	mapping	NOUN
ejde-673	176	23	w	w	NOUN
ejde-673	176	24	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	176	25	)	)	PUNCT
ejde-673	176	26	→	→	SYM
ejde-673	176	27	lq(·)(γ	lq(·)(γ	NOUN
ejde-673	176	28	)	)	PUNCT
ejde-673	176	29	is	be	AUX
ejde-673	176	30	well	well	ADV
ejde-673	176	31	-	-	PUNCT
ejde-673	176	32	defined	define	VERB
ejde-673	176	33	and	and	CCONJ
ejde-673	176	34	compact	compact	ADJ
ejde-673	176	35	.	.	PUNCT
ejde-673	177	1	in	in	ADP
ejde-673	177	2	particular	particular	ADJ
ejde-673	177	3	,	,	PUNCT
ejde-673	177	4	the	the	DET
ejde-673	177	5	trace	trace	NOUN
ejde-673	177	6	mapping	mapping	NOUN
ejde-673	177	7	w	w	NOUN
ejde-673	177	8	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	177	9	)	)	PUNCT
ejde-673	177	10	→	→	SYM
ejde-673	177	11	lp(·)(γ	lp(·)(γ	NOUN
ejde-673	177	12	)	)	PUNCT
ejde-673	177	13	is	be	AUX
ejde-673	177	14	compact	compact	ADJ
ejde-673	177	15	and	and	CCONJ
ejde-673	177	16	there	there	PRON
ejde-673	177	17	exists	exist	VERB
ejde-673	177	18	a	a	DET
ejde-673	177	19	constant	constant	ADJ
ejde-673	177	20	c	c	NOUN
ejde-673	177	21	>	>	X
ejde-673	177	22	0	0	NUM
ejde-673	177	23	such	such	ADJ
ejde-673	177	24	that	that	SCONJ
ejde-673	177	25	∥u∥lp(·)(γ	∥u∥lp(·)(γ	NOUN
ejde-673	177	26	)	)	PUNCT
ejde-673	177	27	≤	≤	NOUN
ejde-673	177	28	c∥u∥w	c∥u∥w	VERB
ejde-673	177	29	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	177	30	)	)	PUNCT
ejde-673	177	31	for	for	ADP
ejde-673	177	32	u	u	PROPN
ejde-673	177	33	∈	∈	PROPN
ejde-673	177	34	w	w	NOUN
ejde-673	177	35	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	177	36	)	)	PUNCT
ejde-673	177	37	.	.	PUNCT
ejde-673	178	1	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	178	2	eigenvalue	eigenvalue	VERB
ejde-673	178	3	problems	problem	NOUN
ejde-673	178	4	for	for	ADP
ejde-673	178	5	kirchhoff	kirchhoff	NOUN
ejde-673	178	6	-	-	PUNCT
ejde-673	178	7	type	type	NOUN
ejde-673	178	8	equations	equation	NOUN
ejde-673	178	9	7	7	NUM
ejde-673	178	10	now	now	ADV
ejde-673	178	11	we	we	PRON
ejde-673	178	12	consider	consider	VERB
ejde-673	178	13	the	the	DET
ejde-673	178	14	weighted	weight	VERB
ejde-673	178	15	variable	variable	ADJ
ejde-673	178	16	exponent	exponent	NOUN
ejde-673	178	17	lebesgue	lebesgue	NOUN
ejde-673	178	18	space	space	NOUN
ejde-673	178	19	.	.	PUNCT
ejde-673	179	1	let	let	VERB
ejde-673	179	2	p	p	PROPN
ejde-673	179	3	∈	∈	PROPN
ejde-673	179	4	c(ω	c(ω	PROPN
ejde-673	179	5	)	)	PUNCT
ejde-673	179	6	with	with	ADP
ejde-673	179	7	p−	p−	PRON
ejde-673	179	8	≥	≥	NOUN
ejde-673	179	9	1	1	NUM
ejde-673	179	10	and	and	CCONJ
ejde-673	179	11	let	let	VERB
ejde-673	179	12	a(x	a(x	NOUN
ejde-673	179	13	)	)	PUNCT
ejde-673	179	14	be	be	AUX
ejde-673	179	15	a	a	DET
ejde-673	179	16	measurable	measurable	ADJ
ejde-673	179	17	function	function	NOUN
ejde-673	179	18	on	on	ADP
ejde-673	179	19	ω	ω	PROPN
ejde-673	179	20	with	with	ADP
ejde-673	179	21	a(x	a(x	NOUN
ejde-673	179	22	)	)	PUNCT
ejde-673	179	23	>	>	X
ejde-673	179	24	0	0	NUM
ejde-673	180	1	a.e	a.e	PROPN
ejde-673	180	2	.	.	PUNCT
ejde-673	180	3	x	x	SYM
ejde-673	180	4	∈	∈	PROPN
ejde-673	180	5	ω	ω	X
ejde-673	180	6	.	.	PUNCT
ejde-673	181	1	we	we	PRON
ejde-673	181	2	define	define	VERB
ejde-673	181	3	a	a	DET
ejde-673	181	4	modular	modular	NOUN
ejde-673	181	5	ρ(p(·),a(·))(u	ρ(p(·),a(·))(u	NOUN
ejde-673	181	6	)	)	PUNCT
ejde-673	181	7	=	=	SYM
ejde-673	182	1	∫	∫	PROPN
ejde-673	182	2	ω	ω	PROPN
ejde-673	182	3	a(x)|u(x)|p(x	a(x)|u(x)|p(x	PROPN
ejde-673	182	4	)	)	PUNCT
ejde-673	182	5	dx	dx	PROPN
ejde-673	182	6	for	for	ADP
ejde-673	182	7	any	any	DET
ejde-673	182	8	measurable	measurable	ADJ
ejde-673	182	9	function	function	NOUN
ejde-673	182	10	u	u	NOUN
ejde-673	182	11	in	in	ADP
ejde-673	182	12	ω	ω	PROPN
ejde-673	182	13	.	.	PUNCT
ejde-673	183	1	then	then	ADV
ejde-673	183	2	the	the	DET
ejde-673	183	3	weighted	weight	VERB
ejde-673	183	4	lebesgue	lebesgue	NOUN
ejde-673	183	5	space	space	NOUN
ejde-673	183	6	is	be	AUX
ejde-673	183	7	defined	define	VERB
ejde-673	183	8	by	by	ADP
ejde-673	183	9	l	l	PROPN
ejde-673	183	10	p	p	X
ejde-673	183	11	(	(	PUNCT
ejde-673	183	12	·	·	PUNCT
ejde-673	183	13	)	)	PUNCT
ejde-673	183	14	a(·)(ω	a(·)(ω	NOUN
ejde-673	183	15	)	)	PUNCT
ejde-673	184	1	=	=	PRON
ejde-673	184	2	{	{	PUNCT
ejde-673	184	3	u	u	NOUN
ejde-673	184	4	is	be	AUX
ejde-673	184	5	a	a	DET
ejde-673	184	6	measurable	measurable	ADJ
ejde-673	184	7	function	function	NOUN
ejde-673	184	8	on	on	ADP
ejde-673	184	9	ω	ω	NUM
ejde-673	184	10	satisfying	satisfy	VERB
ejde-673	184	11	ρ(p(·),a(·))(u	ρ(p(·),a(·))(u	ADP
ejde-673	184	12	)	)	PUNCT
ejde-673	184	13	<	<	X
ejde-673	184	14	∞	∞	PROPN
ejde-673	184	15	}	}	PUNCT
ejde-673	184	16	equipped	equip	VERB
ejde-673	184	17	with	with	ADP
ejde-673	184	18	the	the	DET
ejde-673	184	19	norm	norm	NOUN
ejde-673	184	20	∥u∥	∥u∥	NOUN
ejde-673	184	21	l	l	NOUN
ejde-673	184	22	p	p	X
ejde-673	184	23	(	(	PUNCT
ejde-673	184	24	·	·	PUNCT
ejde-673	184	25	)	)	PUNCT
ejde-673	184	26	a(·)(ω	a(·)(ω	NOUN
ejde-673	184	27	)	)	PUNCT
ejde-673	184	28	=	=	SYM
ejde-673	184	29	inf	inf	NOUN
ejde-673	184	30	{	{	PUNCT
ejde-673	184	31	τ	τ	PROPN
ejde-673	184	32	>	>	X
ejde-673	184	33	0	0	PROPN
ejde-673	184	34	;	;	PUNCT
ejde-673	184	35	∫	∫	PROPN
ejde-673	184	36	ω	ω	PROPN
ejde-673	184	37	a(x	a(x	PROPN
ejde-673	184	38	)	)	PUNCT
ejde-673	184	39	∣∣u(x	∣∣u(x	PROPN
ejde-673	184	40	)	)	PUNCT
ejde-673	184	41	τ	τ	PROPN
ejde-673	184	42	∣∣p(x	∣∣p(x	PROPN
ejde-673	184	43	)	)	PUNCT
ejde-673	184	44	dx	dx	PROPN
ejde-673	184	45	≤	≤	NUM
ejde-673	184	46	1	1	NUM
ejde-673	184	47	}	}	PUNCT
ejde-673	184	48	.	.	PUNCT
ejde-673	185	1	then	then	ADV
ejde-673	185	2	l	l	PROPN
ejde-673	185	3	p	p	X
ejde-673	185	4	(	(	PUNCT
ejde-673	185	5	·	·	PUNCT
ejde-673	185	6	)	)	PUNCT
ejde-673	185	7	a(·)(ω	a(·)(ω	NOUN
ejde-673	185	8	)	)	PUNCT
ejde-673	185	9	is	be	AUX
ejde-673	185	10	a	a	DET
ejde-673	185	11	banach	banach	NOUN
ejde-673	185	12	space	space	NOUN
ejde-673	185	13	.	.	PUNCT
ejde-673	186	1	we	we	PRON
ejde-673	186	2	have	have	VERB
ejde-673	186	3	the	the	DET
ejde-673	186	4	following	follow	VERB
ejde-673	186	5	proposition	proposition	NOUN
ejde-673	186	6	(	(	PUNCT
ejde-673	186	7	cf	cf	NOUN
ejde-673	186	8	.	.	PUNCT
ejde-673	187	1	fan	fan	PROPN
ejde-673	188	1	[	[	X
ejde-673	188	2	14	14	NUM
ejde-673	188	3	,	,	PUNCT
ejde-673	188	4	proposition	proposition	NOUN
ejde-673	188	5	2.5	2.5	NUM
ejde-673	188	6	]	]	PUNCT
ejde-673	188	7	)	)	PUNCT
ejde-673	188	8	.	.	PUNCT
ejde-673	189	1	proposition	proposition	NOUN
ejde-673	189	2	2.8	2.8	NUM
ejde-673	189	3	.	.	PUNCT
ejde-673	190	1	let	let	VERB
ejde-673	190	2	p	p	PROPN
ejde-673	190	3	∈	∈	PROPN
ejde-673	190	4	c(ω	c(ω	PROPN
ejde-673	190	5	)	)	PUNCT
ejde-673	190	6	with	with	ADP
ejde-673	190	7	p−	p−	PRON
ejde-673	190	8	≥	≥	NOUN
ejde-673	190	9	1	1	NUM
ejde-673	190	10	.	.	PUNCT
ejde-673	191	1	for	for	ADP
ejde-673	191	2	u	u	PROPN
ejde-673	191	3	,	,	PUNCT
ejde-673	191	4	un	un	PROPN
ejde-673	191	5	∈	∈	PROPN
ejde-673	191	6	l	l	NOUN
ejde-673	191	7	p	p	X
ejde-673	191	8	(	(	PUNCT
ejde-673	191	9	·	·	PUNCT
ejde-673	191	10	)	)	PUNCT
ejde-673	191	11	a(·)(ω	a(·)(ω	NOUN
ejde-673	191	12	)	)	PUNCT
ejde-673	191	13	,	,	PUNCT
ejde-673	191	14	we	we	PRON
ejde-673	191	15	have	have	VERB
ejde-673	191	16	the	the	DET
ejde-673	191	17	following	following	NOUN
ejde-673	191	18	.	.	PUNCT
ejde-673	192	1	(	(	PUNCT
ejde-673	192	2	i	i	NOUN
ejde-673	192	3	)	)	PUNCT
ejde-673	192	4	for	for	ADP
ejde-673	192	5	u	u	NOUN
ejde-673	192	6	̸=	̸=	PROPN
ejde-673	192	7	0	0	NUM
ejde-673	192	8	,	,	PUNCT
ejde-673	192	9	∥u∥	∥u∥	X
ejde-673	192	10	l	l	NOUN
ejde-673	192	11	p	p	X
ejde-673	192	12	(	(	PUNCT
ejde-673	192	13	·	·	PUNCT
ejde-673	192	14	)	)	PUNCT
ejde-673	192	15	a(·)(ω	a(·)(ω	NOUN
ejde-673	192	16	)	)	PUNCT
ejde-673	192	17	=	=	SYM
ejde-673	192	18	τ	τ	PROPN
ejde-673	192	19	⇔	⇔	PROPN
ejde-673	192	20	ρ(p(·),a	ρ(p(·),a	PROPN
ejde-673	192	21	(	(	PUNCT
ejde-673	192	22	·	·	PUNCT
ejde-673	192	23	)	)	PUNCT
ejde-673	192	24	)	)	PUNCT
ejde-673	193	1	(	(	PUNCT
ejde-673	193	2	u	u	NOUN
ejde-673	193	3	τ	τ	PROPN
ejde-673	193	4	)	)	PUNCT
ejde-673	193	5	=	=	SYM
ejde-673	194	1	1	1	X
ejde-673	194	2	.	.	PUNCT
ejde-673	194	3	(	(	PUNCT
ejde-673	194	4	ii	ii	NOUN
ejde-673	194	5	)	)	PUNCT
ejde-673	194	6	∥u∥	∥u∥	NOUN
ejde-673	195	1	l	l	X
ejde-673	195	2	p	p	X
ejde-673	195	3	(	(	PUNCT
ejde-673	195	4	·	·	PUNCT
ejde-673	195	5	)	)	PUNCT
ejde-673	195	6	a(·)(ω	a(·)(ω	NOUN
ejde-673	195	7	)	)	PUNCT
ejde-673	196	1	<	<	X
ejde-673	196	2	1(=	1(=	NUM
ejde-673	196	3	1	1	NUM
ejde-673	196	4	,	,	PUNCT
ejde-673	196	5	>	>	X
ejde-673	196	6	1	1	X
ejde-673	196	7	)	)	PUNCT
ejde-673	196	8	⇔	⇔	X
ejde-673	196	9	ρ(p(·),a(·))(u	ρ(p(·),a(·))(u	NOUN
ejde-673	196	10	)	)	PUNCT
ejde-673	196	11	<	<	X
ejde-673	197	1	1(=	1(=	NUM
ejde-673	197	2	1	1	NUM
ejde-673	197	3	,	,	PUNCT
ejde-673	197	4	>	>	X
ejde-673	197	5	1	1	NUM
ejde-673	197	6	)	)	PUNCT
ejde-673	197	7	.	.	PUNCT
ejde-673	198	1	(	(	PUNCT
ejde-673	198	2	iii	iii	X
ejde-673	198	3	)	)	PUNCT
ejde-673	198	4	∥u∥	∥u∥	NOUN
ejde-673	199	1	l	l	X
ejde-673	199	2	p	p	X
ejde-673	199	3	(	(	PUNCT
ejde-673	199	4	·	·	PUNCT
ejde-673	199	5	)	)	PUNCT
ejde-673	199	6	a(·)(ω	a(·)(ω	NOUN
ejde-673	199	7	)	)	PUNCT
ejde-673	199	8	>	>	SYM
ejde-673	199	9	1	1	NUM
ejde-673	199	10	⇒	⇒	NOUN
ejde-673	199	11	∥u∥p	∥u∥p	NOUN
ejde-673	199	12	−	−	PROPN
ejde-673	199	13	l	l	NOUN
ejde-673	199	14	p	p	X
ejde-673	199	15	(	(	PUNCT
ejde-673	199	16	·	·	PUNCT
ejde-673	199	17	)	)	PUNCT
ejde-673	199	18	a(·)(ω	a(·)(ω	NOUN
ejde-673	199	19	)	)	PUNCT
ejde-673	199	20	≤	≤	NUM
ejde-673	199	21	ρ(p(·),a(·))(u	ρ(p(·),a(·))(u	X
ejde-673	199	22	)	)	PUNCT
ejde-673	199	23	≤	≤	NOUN
ejde-673	199	24	∥u∥p	∥u∥p	NOUN
ejde-673	199	25	+	+	X
ejde-673	199	26	l	l	NOUN
ejde-673	199	27	p	p	X
ejde-673	199	28	(	(	PUNCT
ejde-673	199	29	·	·	PUNCT
ejde-673	199	30	)	)	PUNCT
ejde-673	199	31	a(·)(ω	a(·)(ω	NOUN
ejde-673	199	32	)	)	PUNCT
ejde-673	199	33	.	.	PUNCT
ejde-673	200	1	(	(	PUNCT
ejde-673	200	2	iv	iv	X
ejde-673	200	3	)	)	PUNCT
ejde-673	200	4	∥u∥	∥u∥	NOUN
ejde-673	201	1	l	l	X
ejde-673	201	2	p	p	X
ejde-673	201	3	(	(	PUNCT
ejde-673	201	4	·	·	PUNCT
ejde-673	201	5	)	)	PUNCT
ejde-673	201	6	a(·)(ω	a(·)(ω	NOUN
ejde-673	201	7	)	)	PUNCT
ejde-673	201	8	<	<	X
ejde-673	201	9	1	1	NUM
ejde-673	201	10	⇒	⇒	NOUN
ejde-673	201	11	∥u∥p	∥u∥p	NOUN
ejde-673	201	12	+	+	X
ejde-673	201	13	l	l	NOUN
ejde-673	201	14	p	p	X
ejde-673	201	15	(	(	PUNCT
ejde-673	201	16	·	·	PUNCT
ejde-673	201	17	)	)	PUNCT
ejde-673	201	18	a(·)(ω	a(·)(ω	NOUN
ejde-673	201	19	)	)	PUNCT
ejde-673	201	20	≤	≤	NUM
ejde-673	201	21	ρ(p(·),a(·))(u	ρ(p(·),a(·))(u	X
ejde-673	201	22	)	)	PUNCT
ejde-673	201	23	≤	≤	NUM
ejde-673	201	24	∥u∥p	∥u∥p	NOUN
ejde-673	201	25	−	−	PROPN
ejde-673	201	26	l	l	NOUN
ejde-673	201	27	p	p	X
ejde-673	201	28	(	(	PUNCT
ejde-673	201	29	·	·	PUNCT
ejde-673	201	30	)	)	PUNCT
ejde-673	201	31	a(·)(ω	a(·)(ω	NOUN
ejde-673	201	32	)	)	PUNCT
ejde-673	201	33	.	.	PUNCT
ejde-673	202	1	(	(	PUNCT
ejde-673	202	2	v	v	NOUN
ejde-673	202	3	)	)	PUNCT
ejde-673	202	4	limn→∞	limn→∞	PROPN
ejde-673	202	5	∥un	∥un	PROPN
ejde-673	202	6	−	−	PROPN
ejde-673	202	7	u∥	u∥	SYM
ejde-673	203	1	l	l	NOUN
ejde-673	203	2	p	p	X
ejde-673	203	3	(	(	PUNCT
ejde-673	203	4	·	·	PUNCT
ejde-673	203	5	)	)	PUNCT
ejde-673	203	6	a(·)(ω	a(·)(ω	NOUN
ejde-673	203	7	)	)	PUNCT
ejde-673	204	1	=	=	SYM
ejde-673	204	2	0	0	NUM
ejde-673	204	3	⇔	⇔	X
ejde-673	204	4	limn→∞	limn→∞	PROPN
ejde-673	204	5	ρ(p(·),a(·))(un	ρ(p(·),a(·))(un	PROPN
ejde-673	204	6	−	−	PROPN
ejde-673	204	7	u	u	NOUN
ejde-673	204	8	)	)	PUNCT
ejde-673	204	9	=	=	SYM
ejde-673	204	10	0	0	X
ejde-673	204	11	.	.	PUNCT
ejde-673	204	12	(	(	PUNCT
ejde-673	204	13	vi	vi	NOUN
ejde-673	204	14	)	)	PUNCT
ejde-673	204	15	∥un∥lp	∥un∥lp	NOUN
ejde-673	204	16	(	(	PUNCT
ejde-673	204	17	·	·	PUNCT
ejde-673	204	18	)	)	PUNCT
ejde-673	204	19	a(·)(ω	a(·)(ω	NOUN
ejde-673	204	20	)	)	PUNCT
ejde-673	204	21	→	→	SYM
ejde-673	204	22	∞	∞	PROPN
ejde-673	204	23	as	as	ADP
ejde-673	204	24	n	n	PROPN
ejde-673	204	25	→	→	SYM
ejde-673	204	26	∞	∞	PROPN
ejde-673	204	27	⇔	⇔	PROPN
ejde-673	204	28	ρ(p(·),a(·))(un	ρ(p(·),a(·))(un	PROPN
ejde-673	204	29	)	)	PUNCT
ejde-673	204	30	→	→	SYM
ejde-673	204	31	∞	∞	PROPN
ejde-673	204	32	as	as	ADP
ejde-673	204	33	n	n	PROPN
ejde-673	204	34	→	→	SYM
ejde-673	204	35	∞.	∞.	PROPN
ejde-673	204	36	the	the	DET
ejde-673	204	37	author	author	NOUN
ejde-673	204	38	of	of	ADP
ejde-673	204	39	[	[	X
ejde-673	204	40	14	14	NUM
ejde-673	204	41	]	]	PUNCT
ejde-673	204	42	also	also	ADV
ejde-673	204	43	derived	derive	VERB
ejde-673	204	44	the	the	DET
ejde-673	204	45	following	follow	VERB
ejde-673	204	46	proposition	proposition	NOUN
ejde-673	204	47	(	(	PUNCT
ejde-673	204	48	cf	cf	NOUN
ejde-673	204	49	.	.	PUNCT
ejde-673	205	1	[	[	X
ejde-673	205	2	14	14	NUM
ejde-673	205	3	,	,	PUNCT
ejde-673	205	4	theorem	theorem	VERB
ejde-673	205	5	2.1	2.1	NUM
ejde-673	205	6	]	]	PUNCT
ejde-673	205	7	)	)	PUNCT
ejde-673	205	8	.	.	PUNCT
ejde-673	206	1	proposition	proposition	NOUN
ejde-673	206	2	2.9	2.9	NUM
ejde-673	206	3	.	.	PUNCT
ejde-673	207	1	let	let	VERB
ejde-673	207	2	ω	ω	NUM
ejde-673	207	3	be	be	AUX
ejde-673	207	4	a	a	DET
ejde-673	207	5	bounded	bounded	ADJ
ejde-673	207	6	domain	domain	NOUN
ejde-673	207	7	of	of	ADP
ejde-673	207	8	rn	rn	PROPN
ejde-673	207	9	with	with	ADP
ejde-673	207	10	a	a	DET
ejde-673	207	11	c0,1	c0,1	NOUN
ejde-673	207	12	-	-	PUNCT
ejde-673	207	13	boundary	boundary	NOUN
ejde-673	207	14	and	and	CCONJ
ejde-673	207	15	p	p	NOUN
ejde-673	207	16	∈	∈	PROPN
ejde-673	207	17	c+(ω	c+(ω	PROPN
ejde-673	207	18	)	)	PUNCT
ejde-673	207	19	.	.	PUNCT
ejde-673	208	1	moreover	moreover	ADV
ejde-673	208	2	,	,	PUNCT
ejde-673	208	3	let	let	VERB
ejde-673	208	4	a	a	DET
ejde-673	208	5	∈	∈	NOUN
ejde-673	208	6	lα(·)(ω	lα(·)(ω	NOUN
ejde-673	208	7	)	)	PUNCT
ejde-673	208	8	satisfy	satisfy	VERB
ejde-673	208	9	a(x	a(x	NOUN
ejde-673	208	10	)	)	PUNCT
ejde-673	208	11	>	>	X
ejde-673	208	12	0	0	NUM
ejde-673	209	1	a.e	a.e	PROPN
ejde-673	209	2	.	.	PUNCT
ejde-673	209	3	x	x	PUNCT
ejde-673	209	4	∈	∈	PROPN
ejde-673	209	5	ω	ω	PROPN
ejde-673	209	6	and	and	CCONJ
ejde-673	209	7	α	α	PRON
ejde-673	209	8	∈	∈	PROPN
ejde-673	209	9	c+(ω	c+(ω	PROPN
ejde-673	209	10	)	)	PUNCT
ejde-673	209	11	.	.	PUNCT
ejde-673	210	1	if	if	SCONJ
ejde-673	210	2	q	q	PROPN
ejde-673	210	3	∈	∈	PROPN
ejde-673	210	4	c(ω	c(ω	PROPN
ejde-673	210	5	)	)	PUNCT
ejde-673	210	6	satisfies	satisfy	VERB
ejde-673	210	7	1	1	NUM
ejde-673	210	8	≤	≤	NUM
ejde-673	210	9	q(x	q(x	NOUN
ejde-673	210	10	)	)	PUNCT
ejde-673	210	11	<	<	X
ejde-673	210	12	α(x)−	α(x)−	PROPN
ejde-673	210	13	1	1	NUM
ejde-673	210	14	α(x	α(x	NOUN
ejde-673	210	15	)	)	PUNCT
ejde-673	210	16	p∗(x	p∗(x	PROPN
ejde-673	210	17	)	)	PUNCT
ejde-673	210	18	for	for	ADP
ejde-673	210	19	all	all	DET
ejde-673	210	20	x	x	SYM
ejde-673	210	21	∈	∈	PROPN
ejde-673	210	22	ω	ω	PROPN
ejde-673	210	23	,	,	PUNCT
ejde-673	210	24	then	then	ADV
ejde-673	210	25	the	the	DET
ejde-673	210	26	embedding	embed	VERB
ejde-673	210	27	map	map	NOUN
ejde-673	210	28	w	w	ADP
ejde-673	210	29	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	210	30	)	)	PUNCT
ejde-673	210	31	↪	↪	PROPN
ejde-673	210	32	→	→	SYM
ejde-673	210	33	l	l	NOUN
ejde-673	210	34	q	q	X
ejde-673	210	35	(	(	PUNCT
ejde-673	210	36	·	·	PUNCT
ejde-673	210	37	)	)	PUNCT
ejde-673	210	38	a(·)(ω	a(·)(ω	NOUN
ejde-673	210	39	)	)	PUNCT
ejde-673	210	40	is	be	AUX
ejde-673	210	41	compact	compact	ADJ
ejde-673	210	42	.	.	PUNCT
ejde-673	211	1	similarly	similarly	ADV
ejde-673	211	2	,	,	PUNCT
ejde-673	211	3	let	let	VERB
ejde-673	211	4	q	q	X
ejde-673	211	5	∈	∈	PROPN
ejde-673	211	6	c(γ	c(γ	PROPN
ejde-673	211	7	)	)	PUNCT
ejde-673	211	8	with	with	ADP
ejde-673	211	9	q−	q−	PROPN
ejde-673	211	10	≥	≥	NUM
ejde-673	211	11	1	1	NUM
ejde-673	211	12	and	and	CCONJ
ejde-673	211	13	let	let	VERB
ejde-673	211	14	b(x	b(x	NOUN
ejde-673	211	15	)	)	PUNCT
ejde-673	211	16	be	be	AUX
ejde-673	211	17	a	a	DET
ejde-673	211	18	measurable	measurable	ADJ
ejde-673	211	19	function	function	NOUN
ejde-673	211	20	with	with	ADP
ejde-673	211	21	respect	respect	NOUN
ejde-673	211	22	to	to	ADP
ejde-673	211	23	dσx	dσx	NOUN
ejde-673	211	24	on	on	ADP
ejde-673	211	25	γ	γ	NOUN
ejde-673	211	26	with	with	ADP
ejde-673	211	27	b(x	b(x	NOUN
ejde-673	211	28	)	)	PUNCT
ejde-673	211	29	>	>	X
ejde-673	211	30	0	0	NUM
ejde-673	212	1	σ	σ	PROPN
ejde-673	212	2	-	-	PUNCT
ejde-673	212	3	a.e	a.e	PROPN
ejde-673	212	4	.	.	PUNCT
ejde-673	212	5	x	x	SYM
ejde-673	212	6	∈	∈	PROPN
ejde-673	212	7	γ	γ	X
ejde-673	212	8	.	.	PUNCT
ejde-673	213	1	we	we	PRON
ejde-673	213	2	define	define	VERB
ejde-673	213	3	a	a	DET
ejde-673	213	4	modular	modular	ADJ
ejde-673	213	5	ρ(q(·),b(·)),γ(u	ρ(q(·),b(·)),γ(u	NOUN
ejde-673	213	6	)	)	PUNCT
ejde-673	214	1	=	=	SYM
ejde-673	214	2	∫	∫	PROPN
ejde-673	214	3	γ	γ	X
ejde-673	214	4	b(x)|u(x)|q(x)dσx	b(x)|u(x)|q(x)dσx	PROPN
ejde-673	214	5	.	.	PUNCT
ejde-673	215	1	then	then	ADV
ejde-673	215	2	the	the	DET
ejde-673	215	3	weighted	weight	VERB
ejde-673	215	4	lebesgue	lebesgue	NOUN
ejde-673	215	5	space	space	NOUN
ejde-673	215	6	on	on	ADP
ejde-673	215	7	γ	γ	X
ejde-673	215	8	is	be	AUX
ejde-673	215	9	defined	define	VERB
ejde-673	215	10	by	by	ADP
ejde-673	215	11	l	l	PROPN
ejde-673	215	12	q	q	PROPN
ejde-673	215	13	(	(	PUNCT
ejde-673	215	14	·	·	PUNCT
ejde-673	215	15	)	)	PUNCT
ejde-673	215	16	b(·)(γ	b(·)(γ	NOUN
ejde-673	215	17	)	)	PUNCT
ejde-673	216	1	=	=	PRON
ejde-673	216	2	{	{	PUNCT
ejde-673	216	3	u	u	NOUN
ejde-673	216	4	is	be	AUX
ejde-673	216	5	a	a	DET
ejde-673	216	6	σ	σ	NOUN
ejde-673	216	7	-	-	PUNCT
ejde-673	216	8	measurable	measurable	ADJ
ejde-673	216	9	function	function	NOUN
ejde-673	216	10	on	on	ADP
ejde-673	216	11	γ	γ	X
ejde-673	216	12	satisfying	satisfy	VERB
ejde-673	216	13	ρ(q(·),b(·)),γ(u	ρ(q(·),b(·)),γ(u	NOUN
ejde-673	216	14	)	)	PUNCT
ejde-673	216	15	<	<	X
ejde-673	216	16	∞	∞	PROPN
ejde-673	216	17	}	}	PUNCT
ejde-673	216	18	equipped	equip	VERB
ejde-673	216	19	with	with	ADP
ejde-673	216	20	the	the	DET
ejde-673	216	21	norm	norm	NOUN
ejde-673	216	22	∥u∥	∥u∥	NOUN
ejde-673	216	23	l	l	NOUN
ejde-673	216	24	q	q	X
ejde-673	216	25	(	(	PUNCT
ejde-673	216	26	·	·	PUNCT
ejde-673	216	27	)	)	PUNCT
ejde-673	216	28	b(·)(γ	b(·)(γ	NOUN
ejde-673	216	29	)	)	PUNCT
ejde-673	216	30	=	=	SYM
ejde-673	216	31	inf	inf	NOUN
ejde-673	216	32	{	{	PUNCT
ejde-673	216	33	τ	τ	PROPN
ejde-673	216	34	>	>	X
ejde-673	216	35	0	0	PROPN
ejde-673	216	36	;	;	PUNCT
ejde-673	216	37	∫	∫	PROPN
ejde-673	216	38	γ	γ	X
ejde-673	216	39	b(x	b(x	X
ejde-673	216	40	)	)	PUNCT
ejde-673	216	41	∣∣u(x	∣∣u(x	PROPN
ejde-673	216	42	)	)	PUNCT
ejde-673	216	43	τ	τ	PROPN
ejde-673	216	44	∣∣q(x	∣∣q(x	PROPN
ejde-673	216	45	)	)	PUNCT
ejde-673	216	46	dσx	dσx	NOUN
ejde-673	216	47	≤	≤	NUM
ejde-673	216	48	1	1	NUM
ejde-673	216	49	}	}	PUNCT
ejde-673	216	50	.	.	PUNCT
ejde-673	217	1	then	then	ADV
ejde-673	217	2	l	l	NOUN
ejde-673	217	3	q	q	ADJ
ejde-673	217	4	(	(	PUNCT
ejde-673	217	5	·	·	PUNCT
ejde-673	217	6	)	)	PUNCT
ejde-673	217	7	b(·)(γ	b(·)(γ	NOUN
ejde-673	217	8	)	)	PUNCT
ejde-673	217	9	is	be	AUX
ejde-673	217	10	a	a	DET
ejde-673	217	11	banach	banach	NOUN
ejde-673	217	12	space	space	NOUN
ejde-673	217	13	.	.	PUNCT
ejde-673	218	1	8	8	NUM
ejde-673	218	2	j.	j.	PROPN
ejde-673	218	3	aramaki	aramaki	PROPN
ejde-673	218	4	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	218	5	proposition	proposition	NOUN
ejde-673	218	6	2.10	2.10	NUM
ejde-673	218	7	.	.	PUNCT
ejde-673	219	1	let	let	VERB
ejde-673	219	2	q	q	PROPN
ejde-673	219	3	∈	∈	PROPN
ejde-673	219	4	c(γ	c(γ	PROPN
ejde-673	219	5	)	)	PUNCT
ejde-673	219	6	with	with	ADP
ejde-673	219	7	q−	q−	PROPN
ejde-673	219	8	≥	≥	NUM
ejde-673	219	9	1	1	NUM
ejde-673	219	10	.	.	PUNCT
ejde-673	220	1	for	for	ADP
ejde-673	220	2	u	u	PROPN
ejde-673	220	3	,	,	PUNCT
ejde-673	220	4	un	un	PROPN
ejde-673	220	5	∈	∈	PROPN
ejde-673	220	6	l	l	NOUN
ejde-673	220	7	q	q	X
ejde-673	220	8	(	(	PUNCT
ejde-673	220	9	·	·	PUNCT
ejde-673	220	10	)	)	PUNCT
ejde-673	220	11	b(·)(γ	b(·)(γ	NOUN
ejde-673	220	12	)	)	PUNCT
ejde-673	220	13	,	,	PUNCT
ejde-673	220	14	we	we	PRON
ejde-673	220	15	have	have	VERB
ejde-673	220	16	the	the	DET
ejde-673	220	17	following	following	NOUN
ejde-673	220	18	.	.	PUNCT
ejde-673	221	1	(	(	PUNCT
ejde-673	221	2	i	i	NOUN
ejde-673	221	3	)	)	PUNCT
ejde-673	221	4	∥u∥	∥u∥	NOUN
ejde-673	221	5	l	l	X
ejde-673	221	6	q	q	X
ejde-673	221	7	(	(	PUNCT
ejde-673	221	8	·	·	PUNCT
ejde-673	221	9	)	)	PUNCT
ejde-673	221	10	b(·)(γ	b(·)(γ	NOUN
ejde-673	221	11	)	)	PUNCT
ejde-673	221	12	<	<	X
ejde-673	222	1	1(=	1(=	NUM
ejde-673	222	2	1	1	NUM
ejde-673	222	3	,	,	PUNCT
ejde-673	222	4	>	>	X
ejde-673	222	5	1	1	X
ejde-673	222	6	)	)	PUNCT
ejde-673	222	7	⇔	⇔	X
ejde-673	222	8	ρ(q(·),b(·)),γ(u	ρ(q(·),b(·)),γ(u	NOUN
ejde-673	222	9	)	)	PUNCT
ejde-673	222	10	<	<	X
ejde-673	222	11	1(=	1(=	NUM
ejde-673	222	12	1	1	NUM
ejde-673	222	13	,	,	PUNCT
ejde-673	222	14	>	>	X
ejde-673	222	15	1	1	NUM
ejde-673	222	16	)	)	PUNCT
ejde-673	222	17	.	.	PUNCT
ejde-673	223	1	(	(	PUNCT
ejde-673	223	2	ii	ii	NOUN
ejde-673	223	3	)	)	PUNCT
ejde-673	223	4	∥u∥	∥u∥	NOUN
ejde-673	223	5	l	l	X
ejde-673	223	6	q	q	X
ejde-673	223	7	(	(	PUNCT
ejde-673	223	8	·	·	PUNCT
ejde-673	223	9	)	)	PUNCT
ejde-673	223	10	b(·)(γ	b(·)(γ	NOUN
ejde-673	223	11	)	)	PUNCT
ejde-673	223	12	>	>	SYM
ejde-673	223	13	1	1	NUM
ejde-673	223	14	⇒	⇒	NOUN
ejde-673	223	15	∥u∥q	∥u∥q	NUM
ejde-673	223	16	−	−	VERB
ejde-673	223	17	l	l	NOUN
ejde-673	223	18	q	q	X
ejde-673	223	19	(	(	PUNCT
ejde-673	223	20	·	·	PUNCT
ejde-673	223	21	)	)	PUNCT
ejde-673	223	22	b(·)(γ	b(·)(γ	NOUN
ejde-673	223	23	)	)	PUNCT
ejde-673	223	24	≤	≤	NOUN
ejde-673	223	25	ρ(q(·),b(·)),γ(u	ρ(q(·),b(·)),γ(u	NOUN
ejde-673	223	26	)	)	PUNCT
ejde-673	223	27	≤	≤	PUNCT
ejde-673	224	1	∥u∥q	∥u∥q	ADJ
ejde-673	225	1	+	+	NUM
ejde-673	225	2	l	l	NOUN
ejde-673	225	3	q	q	X
ejde-673	225	4	(	(	PUNCT
ejde-673	225	5	·	·	PUNCT
ejde-673	225	6	)	)	PUNCT
ejde-673	225	7	b(·)(γ	b(·)(γ	NOUN
ejde-673	225	8	)	)	PUNCT
ejde-673	225	9	.	.	PUNCT
ejde-673	226	1	(	(	PUNCT
ejde-673	226	2	iii	iii	X
ejde-673	226	3	)	)	PUNCT
ejde-673	226	4	∥u∥	∥u∥	NOUN
ejde-673	226	5	l	l	X
ejde-673	226	6	q	q	X
ejde-673	226	7	(	(	PUNCT
ejde-673	226	8	·	·	PUNCT
ejde-673	226	9	)	)	PUNCT
ejde-673	226	10	b(·)(γ	b(·)(γ	NOUN
ejde-673	226	11	)	)	PUNCT
ejde-673	227	1	<	<	X
ejde-673	227	2	1	1	NUM
ejde-673	227	3	⇒	⇒	NOUN
ejde-673	227	4	∥u∥q	∥u∥q	VERB
ejde-673	228	1	+	+	NUM
ejde-673	228	2	l	l	NOUN
ejde-673	228	3	q	q	X
ejde-673	228	4	(	(	PUNCT
ejde-673	228	5	·	·	PUNCT
ejde-673	228	6	)	)	PUNCT
ejde-673	228	7	b(·)(γ	b(·)(γ	NOUN
ejde-673	228	8	)	)	PUNCT
ejde-673	228	9	≤	≤	NOUN
ejde-673	228	10	ρ(q(·),b(·)),γ(u	ρ(q(·),b(·)),γ(u	NOUN
ejde-673	228	11	)	)	PUNCT
ejde-673	229	1	≤	≤	PUNCT
ejde-673	230	1	∥u∥q	∥u∥q	NUM
ejde-673	230	2	−	−	NOUN
ejde-673	230	3	l	l	NOUN
ejde-673	230	4	q	q	X
ejde-673	230	5	(	(	PUNCT
ejde-673	230	6	·	·	PUNCT
ejde-673	230	7	)	)	PUNCT
ejde-673	230	8	b(·)(γ	b(·)(γ	NOUN
ejde-673	230	9	)	)	PUNCT
ejde-673	230	10	.	.	PUNCT
ejde-673	231	1	(	(	PUNCT
ejde-673	231	2	iv	iv	X
ejde-673	231	3	)	)	PUNCT
ejde-673	231	4	limn→∞	limn→∞	PROPN
ejde-673	231	5	∥un	∥un	PROPN
ejde-673	231	6	−	−	PROPN
ejde-673	231	7	u∥	u∥	PROPN
ejde-673	231	8	l	l	NOUN
ejde-673	231	9	q	q	X
ejde-673	231	10	(	(	PUNCT
ejde-673	231	11	·	·	PUNCT
ejde-673	231	12	)	)	PUNCT
ejde-673	231	13	b(·)(γ	b(·)(γ	NOUN
ejde-673	231	14	)	)	PUNCT
ejde-673	232	1	=	=	SYM
ejde-673	232	2	0	0	NUM
ejde-673	233	1	⇔	⇔	X
ejde-673	233	2	limn→∞	limn→∞	PROPN
ejde-673	233	3	ρ(q(·),b(·)),γ(un	ρ(q(·),b(·)),γ(un	X
ejde-673	233	4	−	−	NUM
ejde-673	233	5	u	u	NOUN
ejde-673	233	6	)	)	PUNCT
ejde-673	233	7	=	=	SYM
ejde-673	233	8	0	0	X
ejde-673	233	9	.	.	PUNCT
ejde-673	234	1	(	(	PUNCT
ejde-673	234	2	v	v	NOUN
ejde-673	234	3	)	)	PUNCT
ejde-673	234	4	∥un∥lq	∥un∥lq	PROPN
ejde-673	234	5	(	(	PUNCT
ejde-673	234	6	·	·	PUNCT
ejde-673	234	7	)	)	PUNCT
ejde-673	234	8	b(·)(γ	b(·)(γ	NOUN
ejde-673	234	9	)	)	PUNCT
ejde-673	234	10	→	→	SYM
ejde-673	234	11	∞	∞	PROPN
ejde-673	234	12	as	as	ADP
ejde-673	234	13	n	n	PROPN
ejde-673	234	14	→	→	SYM
ejde-673	234	15	∞	∞	PROPN
ejde-673	234	16	⇔	⇔	X
ejde-673	234	17	ρ(q(·),b(·)),γ(un	ρ(q(·),b(·)),γ(un	PROPN
ejde-673	234	18	)	)	PUNCT
ejde-673	234	19	→	→	SYM
ejde-673	234	20	∞	∞	PROPN
ejde-673	234	21	as	as	ADP
ejde-673	234	22	n	n	PROPN
ejde-673	234	23	→	→	SYM
ejde-673	234	24	∞.	∞.	PROPN
ejde-673	234	25	the	the	DET
ejde-673	234	26	following	follow	VERB
ejde-673	234	27	proposition	proposition	NOUN
ejde-673	234	28	plays	play	VERB
ejde-673	234	29	an	an	DET
ejde-673	234	30	important	important	ADJ
ejde-673	234	31	role	role	NOUN
ejde-673	234	32	in	in	ADP
ejde-673	234	33	the	the	DET
ejde-673	234	34	present	present	ADJ
ejde-673	234	35	paper	paper	NOUN
ejde-673	234	36	.	.	PUNCT
ejde-673	235	1	proposition	proposition	NOUN
ejde-673	235	2	2.11	2.11	NUM
ejde-673	235	3	.	.	PUNCT
ejde-673	236	1	let	let	VERB
ejde-673	236	2	ω	ω	PRON
ejde-673	236	3	be	be	AUX
ejde-673	236	4	a	a	DET
ejde-673	236	5	bounded	bounded	ADJ
ejde-673	236	6	domain	domain	NOUN
ejde-673	236	7	of	of	ADP
ejde-673	236	8	rn	rn	PROPN
ejde-673	236	9	with	with	ADP
ejde-673	236	10	a	a	DET
ejde-673	236	11	c0,1	c0,1	NOUN
ejde-673	236	12	-	-	PUNCT
ejde-673	236	13	boundary	boundary	NOUN
ejde-673	236	14	γ	γ	NOUN
ejde-673	236	15	and	and	CCONJ
ejde-673	236	16	let	let	VERB
ejde-673	236	17	p	p	PROPN
ejde-673	236	18	∈	∈	PROPN
ejde-673	236	19	c+(ω	c+(ω	PROPN
ejde-673	236	20	)	)	PUNCT
ejde-673	236	21	.	.	PUNCT
ejde-673	237	1	assume	assume	VERB
ejde-673	237	2	that	that	SCONJ
ejde-673	237	3	0	0	NUM
ejde-673	237	4	<	<	X
ejde-673	237	5	b	b	X
ejde-673	237	6	∈	∈	PROPN
ejde-673	237	7	lβ(·)(γ	lβ(·)(γ	NOUN
ejde-673	237	8	)	)	PUNCT
ejde-673	237	9	,	,	PUNCT
ejde-673	237	10	β	β	PROPN
ejde-673	237	11	∈	∈	PROPN
ejde-673	237	12	c+(γ	c+(γ	PROPN
ejde-673	237	13	)	)	PUNCT
ejde-673	237	14	.	.	PUNCT
ejde-673	238	1	if	if	SCONJ
ejde-673	238	2	r	r	NOUN
ejde-673	238	3	∈	∈	PROPN
ejde-673	238	4	c(γ	c(γ	PROPN
ejde-673	238	5	)	)	PUNCT
ejde-673	238	6	satisfies	satisfy	VERB
ejde-673	238	7	1	1	NUM
ejde-673	238	8	≤	≤	NUM
ejde-673	238	9	r(x	r(x	PROPN
ejde-673	238	10	)	)	PUNCT
ejde-673	238	11	<	<	X
ejde-673	238	12	β(x)−	β(x)−	PROPN
ejde-673	238	13	1	1	NUM
ejde-673	238	14	β(x	β(x	NOUN
ejde-673	238	15	)	)	PUNCT
ejde-673	238	16	p∂(x	p∂(x	NOUN
ejde-673	238	17	)	)	PUNCT
ejde-673	238	18	for	for	ADP
ejde-673	238	19	all	all	DET
ejde-673	238	20	x	x	SYM
ejde-673	238	21	∈	∈	PROPN
ejde-673	238	22	γ	γ	X
ejde-673	238	23	,	,	PUNCT
ejde-673	238	24	then	then	ADV
ejde-673	238	25	the	the	DET
ejde-673	238	26	embedding	embed	VERB
ejde-673	238	27	map	map	NOUN
ejde-673	238	28	w	w	ADP
ejde-673	238	29	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	238	30	)	)	PUNCT
ejde-673	238	31	↪	↪	PROPN
ejde-673	238	32	→	→	SYM
ejde-673	238	33	l	l	NOUN
ejde-673	238	34	r	r	NOUN
ejde-673	238	35	(	(	PUNCT
ejde-673	238	36	·	·	PUNCT
ejde-673	238	37	)	)	PUNCT
ejde-673	238	38	b(·)(γ	b(·)(γ	NOUN
ejde-673	238	39	)	)	PUNCT
ejde-673	238	40	is	be	AUX
ejde-673	238	41	compact	compact	ADJ
ejde-673	238	42	.	.	PUNCT
ejde-673	239	1	proof	proof	NOUN
ejde-673	239	2	.	.	PUNCT
ejde-673	240	1	let	let	VERB
ejde-673	240	2	u	u	PRON
ejde-673	240	3	∈	∈	PROPN
ejde-673	240	4	w	w	NOUN
ejde-673	240	5	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	240	6	)	)	PUNCT
ejde-673	240	7	.	.	PUNCT
ejde-673	241	1	set	set	VERB
ejde-673	241	2	h(x	h(x	PROPN
ejde-673	241	3	)	)	PUNCT
ejde-673	242	1	=	=	PUNCT
ejde-673	242	2	β′(x)r(x	β′(x)r(x	NOUN
ejde-673	242	3	)	)	PUNCT
ejde-673	242	4	.	.	PUNCT
ejde-673	243	1	from	from	ADP
ejde-673	243	2	the	the	DET
ejde-673	243	3	hypothesis	hypothesis	NOUN
ejde-673	243	4	,	,	PUNCT
ejde-673	243	5	we	we	PRON
ejde-673	243	6	have	have	VERB
ejde-673	243	7	h(x	h(x	PROPN
ejde-673	243	8	)	)	PUNCT
ejde-673	243	9	<	<	X
ejde-673	244	1	p∂(x	p∂(x	PROPN
ejde-673	244	2	)	)	PUNCT
ejde-673	244	3	for	for	ADP
ejde-673	244	4	all	all	DET
ejde-673	244	5	x	x	SYM
ejde-673	244	6	∈	∈	PROPN
ejde-673	244	7	γ	γ	X
ejde-673	244	8	.	.	PUNCT
ejde-673	244	9	by	by	ADP
ejde-673	244	10	proposition	proposition	NOUN
ejde-673	244	11	2.7	2.7	NUM
ejde-673	244	12	,	,	PUNCT
ejde-673	244	13	the	the	DET
ejde-673	244	14	embedding	embed	VERB
ejde-673	244	15	map	map	NOUN
ejde-673	244	16	w	w	ADP
ejde-673	244	17	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	244	18	)	)	PUNCT
ejde-673	244	19	↪	↪	PROPN
ejde-673	244	20	→	→	SYM
ejde-673	244	21	lh(·)(γ	lh(·)(γ	NOUN
ejde-673	244	22	)	)	PUNCT
ejde-673	244	23	is	be	AUX
ejde-673	244	24	compact	compact	ADJ
ejde-673	244	25	.	.	PUNCT
ejde-673	245	1	since	since	SCONJ
ejde-673	245	2	|u(x)|r(x	|u(x)|r(x	X
ejde-673	245	3	)	)	PUNCT
ejde-673	245	4	∈	∈	PROPN
ejde-673	245	5	lβ′(·)(γ	lβ′(·)(γ	NOUN
ejde-673	245	6	)	)	PUNCT
ejde-673	245	7	,	,	PUNCT
ejde-673	245	8	it	it	PRON
ejde-673	245	9	follows	follow	VERB
ejde-673	245	10	from	from	ADP
ejde-673	245	11	the	the	DET
ejde-673	245	12	hölder	hölder	NOUN
ejde-673	245	13	inequality	inequality	NOUN
ejde-673	245	14	(	(	PUNCT
ejde-673	245	15	proposition	proposition	NOUN
ejde-673	245	16	2.5	2.5	NUM
ejde-673	245	17	)	)	PUNCT
ejde-673	245	18	that∫	that∫	NOUN
ejde-673	245	19	γ	γ	PROPN
ejde-673	245	20	b(x)|u(x)|r(x	b(x)|u(x)|r(x	PROPN
ejde-673	245	21	)	)	PUNCT
ejde-673	245	22	dσx	dσx	NOUN
ejde-673	245	23	≤	≤	ADJ
ejde-673	245	24	2∥b∥lβ(·)(γ)∥|u|r(·)∥lβ′(·)(γ	2∥b∥lβ(·)(γ)∥|u|r(·)∥lβ′(·)(γ	NUM
ejde-673	245	25	)	)	PUNCT
ejde-673	246	1	<	<	X
ejde-673	246	2	∞.	∞.	PROPN
ejde-673	246	3	hence	hence	ADV
ejde-673	246	4	w	w	PROPN
ejde-673	246	5	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	246	6	)	)	PUNCT
ejde-673	247	1	⊂	⊂	X
ejde-673	247	2	l	l	NOUN
ejde-673	247	3	r	r	X
ejde-673	247	4	(	(	PUNCT
ejde-673	247	5	·	·	PUNCT
ejde-673	247	6	)	)	PUNCT
ejde-673	247	7	b(·)(γ	b(·)(γ	NOUN
ejde-673	247	8	)	)	PUNCT
ejde-673	247	9	.	.	PUNCT
ejde-673	248	1	we	we	PRON
ejde-673	248	2	show	show	VERB
ejde-673	248	3	that	that	SCONJ
ejde-673	248	4	the	the	DET
ejde-673	248	5	embedding	embed	VERB
ejde-673	248	6	w	w	NOUN
ejde-673	248	7	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	248	8	)	)	PUNCT
ejde-673	248	9	↪	↪	PROPN
ejde-673	248	10	→	→	SYM
ejde-673	248	11	l	l	NOUN
ejde-673	248	12	r	r	NOUN
ejde-673	248	13	(	(	PUNCT
ejde-673	248	14	·	·	PUNCT
ejde-673	248	15	)	)	PUNCT
ejde-673	248	16	b(·)(γ	b(·)(γ	NOUN
ejde-673	248	17	)	)	PUNCT
ejde-673	248	18	is	be	AUX
ejde-673	248	19	compact	compact	ADJ
ejde-673	248	20	.	.	PUNCT
ejde-673	249	1	let	let	VERB
ejde-673	249	2	un	un	PROPN
ejde-673	249	3	→	→	SYM
ejde-673	249	4	0	0	NUM
ejde-673	249	5	weakly	weakly	ADV
ejde-673	249	6	in	in	ADP
ejde-673	249	7	w	w	NOUN
ejde-673	249	8	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	249	9	)	)	PUNCT
ejde-673	249	10	.	.	PUNCT
ejde-673	250	1	then	then	ADV
ejde-673	250	2	un	un	PROPN
ejde-673	250	3	→	→	SYM
ejde-673	250	4	0	0	NUM
ejde-673	250	5	strongly	strongly	ADV
ejde-673	250	6	in	in	ADP
ejde-673	250	7	lh(·)(γ	lh(·)(γ	PROPN
ejde-673	250	8	)	)	PUNCT
ejde-673	250	9	.	.	PUNCT
ejde-673	251	1	since	since	SCONJ
ejde-673	251	2	ρβ′(·),γ(|un|r	ρβ′(·),γ(|un|r	NOUN
ejde-673	251	3	(	(	PUNCT
ejde-673	251	4	·	·	PUNCT
ejde-673	251	5	)	)	PUNCT
ejde-673	251	6	)	)	PUNCT
ejde-673	252	1	=	=	SYM
ejde-673	252	2	∫	∫	PROPN
ejde-673	252	3	γ	γ	PROPN
ejde-673	252	4	|un(x)|r(x)β	|un(x)|r(x)β	PROPN
ejde-673	252	5	′(x)dσx	′(x)dσx	NOUN
ejde-673	252	6	=	=	SYM
ejde-673	252	7	∫	∫	PROPN
ejde-673	252	8	γ	γ	X
ejde-673	252	9	|un(x)|h(x	|un(x)|h(x	PROPN
ejde-673	252	10	)	)	PUNCT
ejde-673	252	11	dσx	dσx	NOUN
ejde-673	252	12	→	→	SYM
ejde-673	252	13	0	0	NUM
ejde-673	252	14	,	,	PUNCT
ejde-673	252	15	we	we	PRON
ejde-673	252	16	have	have	VERB
ejde-673	252	17	∥|un|r(·)∥lβ′(·)(γ	∥|un|r(·)∥lβ′(·)(γ	ADJ
ejde-673	252	18	)	)	PUNCT
ejde-673	252	19	→	→	SYM
ejde-673	252	20	0	0	NUM
ejde-673	252	21	from	from	ADP
ejde-673	252	22	proposition	proposition	NOUN
ejde-673	252	23	2.10	2.10	NUM
ejde-673	252	24	(	(	PUNCT
ejde-673	252	25	iv	iv	NUM
ejde-673	252	26	)	)	PUNCT
ejde-673	252	27	.	.	PUNCT
ejde-673	253	1	therefore,∫	therefore,∫	PUNCT
ejde-673	253	2	γ	γ	PROPN
ejde-673	253	3	b(x)|un(x)|r(x	b(x)|un(x)|r(x	PROPN
ejde-673	253	4	)	)	PUNCT
ejde-673	253	5	dσx	dσx	NOUN
ejde-673	253	6	≤	≤	NUM
ejde-673	253	7	2∥b∥lr(·)(γ)∥|un|r(·)∥lβ′(·)(γ	2∥b∥lr(·)(γ)∥|un|r(·)∥lβ′(·)(γ	NUM
ejde-673	253	8	)	)	PUNCT
ejde-673	253	9	→	→	SYM
ejde-673	253	10	0	0	X
ejde-673	253	11	.	.	PUNCT
ejde-673	254	1	thus	thus	ADV
ejde-673	254	2	it	it	PRON
ejde-673	254	3	also	also	ADV
ejde-673	254	4	follows	follow	VERB
ejde-673	254	5	from	from	ADP
ejde-673	254	6	proposition	proposition	NOUN
ejde-673	254	7	2.10	2.10	NUM
ejde-673	254	8	(	(	PUNCT
ejde-673	254	9	iv	iv	X
ejde-673	254	10	)	)	PUNCT
ejde-673	254	11	that	that	SCONJ
ejde-673	254	12	∥un∥lr	∥un∥lr	NOUN
ejde-673	254	13	(	(	PUNCT
ejde-673	254	14	·	·	PUNCT
ejde-673	254	15	)	)	PUNCT
ejde-673	254	16	b(·)(γ	b(·)(γ	NOUN
ejde-673	254	17	)	)	PUNCT
ejde-673	254	18	→	→	SYM
ejde-673	254	19	0	0	NUM
ejde-673	254	20	,	,	PUNCT
ejde-673	254	21	sow	sow	VERB
ejde-673	254	22	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	254	23	)	)	PUNCT
ejde-673	254	24	↪	↪	PROPN
ejde-673	254	25	→	→	SYM
ejde-673	254	26	l	l	NOUN
ejde-673	254	27	r	r	NOUN
ejde-673	254	28	(	(	PUNCT
ejde-673	254	29	·	·	PUNCT
ejde-673	254	30	)	)	PUNCT
ejde-673	254	31	b(·)(γ	b(·)(γ	NOUN
ejde-673	254	32	)	)	PUNCT
ejde-673	254	33	is	be	AUX
ejde-673	254	34	compact	compact	ADJ
ejde-673	254	35	.	.	PUNCT
ejde-673	255	1	□	□	PUNCT
ejde-673	255	2	now	now	ADV
ejde-673	255	3	we	we	PRON
ejde-673	255	4	consider	consider	VERB
ejde-673	255	5	the	the	DET
ejde-673	255	6	nemytskii	nemytskii	ADJ
ejde-673	255	7	operator	operator	NOUN
ejde-673	255	8	.	.	PUNCT
ejde-673	256	1	proposition	proposition	NOUN
ejde-673	256	2	2.12	2.12	NUM
ejde-673	256	3	.	.	PUNCT
ejde-673	257	1	let	let	VERB
ejde-673	257	2	q	q	PROPN
ejde-673	257	3	∈	∈	PROPN
ejde-673	257	4	c(ω	c(ω	PROPN
ejde-673	257	5	)	)	PUNCT
ejde-673	257	6	with	with	ADP
ejde-673	257	7	q−	q−	PROPN
ejde-673	257	8	≥	≥	NUM
ejde-673	257	9	1	1	NUM
ejde-673	257	10	and	and	CCONJ
ejde-673	257	11	a	a	DET
ejde-673	257	12	be	be	AUX
ejde-673	257	13	a	a	DET
ejde-673	257	14	measurable	measurable	ADJ
ejde-673	257	15	function	function	NOUN
ejde-673	257	16	with	with	ADP
ejde-673	257	17	a(x	a(x	NOUN
ejde-673	257	18	)	)	PUNCT
ejde-673	257	19	>	>	X
ejde-673	257	20	0	0	PUNCT
ejde-673	258	1	for	for	ADP
ejde-673	258	2	a.e	a.e	PROPN
ejde-673	258	3	.	.	PUNCT
ejde-673	258	4	x	x	SYM
ejde-673	258	5	∈	∈	PROPN
ejde-673	258	6	ω	ω	X
ejde-673	258	7	.	.	PUNCT
ejde-673	258	8	assume	assume	VERB
ejde-673	258	9	that	that	SCONJ
ejde-673	258	10	(	(	PUNCT
ejde-673	258	11	1	1	X
ejde-673	258	12	)	)	PUNCT
ejde-673	258	13	a	a	DET
ejde-673	258	14	function	function	NOUN
ejde-673	259	1	f	f	X
ejde-673	259	2	(	(	PUNCT
ejde-673	259	3	x	x	PROPN
ejde-673	259	4	,	,	PUNCT
ejde-673	259	5	t	t	PROPN
ejde-673	259	6	)	)	PUNCT
ejde-673	259	7	is	be	AUX
ejde-673	259	8	a	a	DET
ejde-673	259	9	carathéodory	carathéodory	NOUN
ejde-673	259	10	function	function	NOUN
ejde-673	259	11	on	on	ADP
ejde-673	259	12	ω×	ω×	PROPN
ejde-673	259	13	r.	r.	X
ejde-673	259	14	(	(	PUNCT
ejde-673	259	15	2	2	NUM
ejde-673	259	16	)	)	PUNCT
ejde-673	259	17	the	the	DET
ejde-673	259	18	growth	growth	NOUN
ejde-673	259	19	condition	condition	NOUN
ejde-673	259	20	holds	hold	VERB
ejde-673	259	21	:	:	PUNCT
ejde-673	259	22	there	there	PRON
ejde-673	259	23	exist	exist	VERB
ejde-673	259	24	c	c	PROPN
ejde-673	259	25	∈	∈	PROPN
ejde-673	259	26	lq1(·)(ω	lq1(·)(ω	PROPN
ejde-673	259	27	)	)	PUNCT
ejde-673	259	28	with	with	ADP
ejde-673	259	29	c(x	c(x	NOUN
ejde-673	259	30	)	)	PUNCT
ejde-673	259	31	≥	≥	NOUN
ejde-673	259	32	0	0	NUM
ejde-673	260	1	a.e	a.e	PROPN
ejde-673	260	2	.	.	PUNCT
ejde-673	260	3	x	x	SYM
ejde-673	260	4	∈	∈	PROPN
ejde-673	260	5	ω	ω	PROPN
ejde-673	260	6	,	,	PUNCT
ejde-673	260	7	q1	q1	PROPN
ejde-673	260	8	∈	∈	PROPN
ejde-673	260	9	c(ω	c(ω	PROPN
ejde-673	260	10	)	)	PUNCT
ejde-673	260	11	with	with	ADP
ejde-673	260	12	q−1	q−1	PROPN
ejde-673	260	13	≥	≥	NUM
ejde-673	260	14	1	1	NUM
ejde-673	260	15	and	and	CCONJ
ejde-673	260	16	a	a	DET
ejde-673	260	17	constant	constant	ADJ
ejde-673	260	18	c1	c1	NOUN
ejde-673	260	19	>	>	X
ejde-673	260	20	0	0	NUM
ejde-673	261	1	such	such	ADJ
ejde-673	261	2	that	that	SCONJ
ejde-673	261	3	|f	|f	PROPN
ejde-673	261	4	(	(	PUNCT
ejde-673	261	5	x	x	X
ejde-673	261	6	,	,	PUNCT
ejde-673	261	7	t)|	t)|	ADJ
ejde-673	261	8	≤	≤	ADJ
ejde-673	261	9	c(x	c(x	NOUN
ejde-673	261	10	)	)	PUNCT
ejde-673	261	11	+	+	CCONJ
ejde-673	261	12	c1a(x	c1a(x	NOUN
ejde-673	261	13	)	)	PUNCT
ejde-673	261	14	1	1	NUM
ejde-673	261	15	/	/	SYM
ejde-673	261	16	q1(x)|t|q(x)/q1(x	q1(x)|t|q(x)/q1(x	NOUN
ejde-673	261	17	)	)	PUNCT
ejde-673	261	18	for	for	ADP
ejde-673	261	19	a.e	a.e	PROPN
ejde-673	261	20	.	.	PUNCT
ejde-673	261	21	x	x	PUNCT
ejde-673	261	22	∈	∈	PROPN
ejde-673	261	23	ω	ω	NOUN
ejde-673	261	24	and	and	CCONJ
ejde-673	261	25	all	all	DET
ejde-673	261	26	t	t	PROPN
ejde-673	261	27	∈	∈	PROPN
ejde-673	261	28	r.	r.	PROPN
ejde-673	261	29	ejde-2025/17	ejde-2025/17	X
ejde-673	261	30	eigenvalue	eigenvalue	NOUN
ejde-673	261	31	problems	problem	NOUN
ejde-673	261	32	for	for	ADP
ejde-673	261	33	kirchhoff	kirchhoff	NOUN
ejde-673	261	34	-	-	PUNCT
ejde-673	261	35	type	type	NOUN
ejde-673	261	36	equations	equation	NOUN
ejde-673	261	37	9	9	NUM
ejde-673	261	38	then	then	ADV
ejde-673	261	39	the	the	DET
ejde-673	261	40	nemytskii	nemytskii	ADJ
ejde-673	261	41	operator	operator	NOUN
ejde-673	261	42	nf	nf	NOUN
ejde-673	261	43	:	:	PUNCT
ejde-673	261	44	l	l	NOUN
ejde-673	261	45	q	q	X
ejde-673	261	46	(	(	PUNCT
ejde-673	261	47	·	·	PUNCT
ejde-673	261	48	)	)	PUNCT
ejde-673	261	49	a(·)(ω	a(·)(ω	NOUN
ejde-673	261	50	)	)	PUNCT
ejde-673	262	1	∋	∋	NOUN
ejde-673	262	2	u	u	NOUN
ejde-673	262	3	7→	7→	PROPN
ejde-673	262	4	f	f	X
ejde-673	262	5	(	(	PUNCT
ejde-673	262	6	x	x	NOUN
ejde-673	262	7	,	,	PUNCT
ejde-673	262	8	u(x	u(x	NOUN
ejde-673	262	9	)	)	PUNCT
ejde-673	262	10	)	)	PUNCT
ejde-673	262	11	∈	∈	PROPN
ejde-673	262	12	lq1(·)(ω	lq1(·)(ω	NOUN
ejde-673	262	13	)	)	PUNCT
ejde-673	262	14	is	be	AUX
ejde-673	262	15	continuous	continuous	ADJ
ejde-673	262	16	and	and	CCONJ
ejde-673	262	17	there	there	PRON
ejde-673	262	18	exists	exist	VERB
ejde-673	262	19	a	a	DET
ejde-673	262	20	constant	constant	ADJ
ejde-673	262	21	c	c	NOUN
ejde-673	262	22	>	>	X
ejde-673	262	23	0	0	NUM
ejde-673	262	24	such	such	ADJ
ejde-673	262	25	that	that	SCONJ
ejde-673	262	26	ρq1(·)(nf	ρq1(·)(nf	PROPN
ejde-673	262	27	(	(	PUNCT
ejde-673	262	28	u	u	NOUN
ejde-673	262	29	)	)	PUNCT
ejde-673	262	30	)	)	PUNCT
ejde-673	262	31	≤	≤	NOUN
ejde-673	263	1	c(ρq1(·)(c	c(ρq1(·)(c	ADJ
ejde-673	263	2	)	)	PUNCT
ejde-673	263	3	+	+	NUM
ejde-673	264	1	ρ(q(·),a(·))(u	ρ(q(·),a(·))(u	NUM
ejde-673	264	2	)	)	PUNCT
ejde-673	264	3	)	)	PUNCT
ejde-673	264	4	for	for	ADP
ejde-673	264	5	all	all	DET
ejde-673	264	6	u	u	NOUN
ejde-673	264	7	∈	∈	PROPN
ejde-673	264	8	l	l	NOUN
ejde-673	264	9	q	q	X
ejde-673	264	10	(	(	PUNCT
ejde-673	264	11	·	·	PUNCT
ejde-673	264	12	)	)	PUNCT
ejde-673	264	13	a(·)(ω	a(·)(ω	NOUN
ejde-673	264	14	)	)	PUNCT
ejde-673	264	15	.	.	PUNCT
ejde-673	265	1	in	in	ADP
ejde-673	265	2	particular	particular	ADJ
ejde-673	265	3	,	,	PUNCT
ejde-673	265	4	if	if	SCONJ
ejde-673	265	5	q1(x	q1(x	NOUN
ejde-673	265	6	)	)	PUNCT
ejde-673	265	7	≡	≡	PROPN
ejde-673	265	8	1	1	NUM
ejde-673	265	9	,	,	PUNCT
ejde-673	265	10	then	then	ADV
ejde-673	265	11	nf	nf	VERB
ejde-673	265	12	:	:	PUNCT
ejde-673	265	13	l	l	NOUN
ejde-673	265	14	q	q	X
ejde-673	265	15	(	(	PUNCT
ejde-673	265	16	·	·	PUNCT
ejde-673	265	17	)	)	PUNCT
ejde-673	265	18	a(·)(ω	a(·)(ω	NOUN
ejde-673	265	19	)	)	PUNCT
ejde-673	265	20	→	→	SYM
ejde-673	265	21	l1(ω	l1(ω	X
ejde-673	265	22	)	)	PUNCT
ejde-673	265	23	is	be	AUX
ejde-673	265	24	continuous	continuous	ADJ
ejde-673	265	25	.	.	PUNCT
ejde-673	266	1	for	for	ADP
ejde-673	266	2	a	a	DET
ejde-673	266	3	proof	proof	NOUN
ejde-673	266	4	of	of	ADP
ejde-673	266	5	the	the	DET
ejde-673	266	6	above	above	ADJ
ejde-673	266	7	proposition	proposition	NOUN
ejde-673	266	8	,	,	PUNCT
ejde-673	266	9	see	see	VERB
ejde-673	266	10	aramaki	aramaki	NOUN
ejde-673	267	1	[	[	X
ejde-673	267	2	9	9	NUM
ejde-673	267	3	,	,	PUNCT
ejde-673	267	4	proposition	proposition	NOUN
ejde-673	267	5	7	7	NUM
ejde-673	267	6	]	]	PUNCT
ejde-673	267	7	.	.	PUNCT
ejde-673	268	1	the	the	DET
ejde-673	268	2	proposition	proposition	NOUN
ejde-673	268	3	is	be	AUX
ejde-673	268	4	an	an	DET
ejde-673	268	5	extension	extension	NOUN
ejde-673	268	6	of	of	ADP
ejde-673	268	7	[	[	X
ejde-673	268	8	6	6	NUM
ejde-673	268	9	,	,	PUNCT
ejde-673	268	10	proposition	proposition	NOUN
ejde-673	268	11	2.12	2.12	NUM
ejde-673	268	12	]	]	PUNCT
ejde-673	268	13	.	.	PUNCT
ejde-673	269	1	similarly	similarly	ADV
ejde-673	269	2	we	we	PRON
ejde-673	269	3	have	have	VERB
ejde-673	269	4	the	the	DET
ejde-673	269	5	following	follow	VERB
ejde-673	269	6	proposition	proposition	NOUN
ejde-673	269	7	.	.	PUNCT
ejde-673	270	1	proposition	proposition	NOUN
ejde-673	270	2	2.13	2.13	NUM
ejde-673	270	3	.	.	PUNCT
ejde-673	271	1	let	let	VERB
ejde-673	271	2	r	r	NOUN
ejde-673	271	3	∈	∈	PROPN
ejde-673	271	4	c(γ2	c(γ2	NOUN
ejde-673	271	5	)	)	PUNCT
ejde-673	271	6	with	with	ADP
ejde-673	271	7	r−	r−	PROPN
ejde-673	271	8	≥	≥	NUM
ejde-673	271	9	1	1	NUM
ejde-673	271	10	and	and	CCONJ
ejde-673	271	11	b	b	NOUN
ejde-673	271	12	be	be	AUX
ejde-673	271	13	a	a	DET
ejde-673	271	14	σ	σ	NOUN
ejde-673	271	15	-	-	PUNCT
ejde-673	271	16	measurable	measurable	ADJ
ejde-673	271	17	function	function	NOUN
ejde-673	271	18	with	with	ADP
ejde-673	271	19	b(x	b(x	NOUN
ejde-673	271	20	)	)	PUNCT
ejde-673	271	21	>	>	X
ejde-673	271	22	0	0	NUM
ejde-673	272	1	σ	σ	PROPN
ejde-673	272	2	-	-	PUNCT
ejde-673	272	3	a.e	a.e	PROPN
ejde-673	272	4	.	.	PUNCT
ejde-673	272	5	x	x	SYM
ejde-673	272	6	∈	∈	PROPN
ejde-673	272	7	γ2	γ2	PROPN
ejde-673	272	8	.	.	PUNCT
ejde-673	273	1	assume	assume	VERB
ejde-673	273	2	that	that	SCONJ
ejde-673	273	3	(	(	PUNCT
ejde-673	273	4	1	1	X
ejde-673	273	5	)	)	PUNCT
ejde-673	273	6	the	the	DET
ejde-673	273	7	function	function	NOUN
ejde-673	273	8	h(x	h(x	PROPN
ejde-673	273	9	,	,	PUNCT
ejde-673	273	10	t	t	PROPN
ejde-673	273	11	)	)	PUNCT
ejde-673	273	12	is	be	AUX
ejde-673	273	13	a	a	DET
ejde-673	273	14	carathéodory	carathéodory	NOUN
ejde-673	273	15	function	function	NOUN
ejde-673	273	16	on	on	ADP
ejde-673	273	17	γ2	γ2	PROPN
ejde-673	273	18	×	×	PROPN
ejde-673	273	19	r.	r.	PROPN
ejde-673	273	20	(	(	PUNCT
ejde-673	273	21	2	2	X
ejde-673	273	22	)	)	PUNCT
ejde-673	273	23	the	the	DET
ejde-673	273	24	growth	growth	NOUN
ejde-673	273	25	condition	condition	NOUN
ejde-673	273	26	holds	hold	VERB
ejde-673	273	27	:	:	PUNCT
ejde-673	273	28	there	there	PRON
ejde-673	273	29	exist	exist	VERB
ejde-673	273	30	d	d	PROPN
ejde-673	273	31	∈	∈	PROPN
ejde-673	273	32	lr1(·)(γ2	lr1(·)(γ2	PROPN
ejde-673	273	33	)	)	PUNCT
ejde-673	273	34	with	with	ADP
ejde-673	273	35	d(x	d(x	PROPN
ejde-673	273	36	)	)	PUNCT
ejde-673	273	37	≥	≥	NOUN
ejde-673	273	38	0	0	NUM
ejde-673	274	1	σ	σ	PROPN
ejde-673	274	2	-	-	PUNCT
ejde-673	274	3	a.e	a.e	PROPN
ejde-673	274	4	.	.	PUNCT
ejde-673	274	5	x	x	SYM
ejde-673	274	6	∈	∈	PROPN
ejde-673	274	7	γ2	γ2	NOUN
ejde-673	274	8	,	,	PUNCT
ejde-673	274	9	r1	r1	PROPN
ejde-673	274	10	∈	∈	PROPN
ejde-673	274	11	c(γ2	c(γ2	NOUN
ejde-673	274	12	)	)	PUNCT
ejde-673	274	13	with	with	ADP
ejde-673	274	14	r1	r1	PROPN
ejde-673	274	15	≥	≥	NUM
ejde-673	274	16	1	1	NUM
ejde-673	274	17	,	,	PUNCT
ejde-673	274	18	and	and	CCONJ
ejde-673	274	19	a	a	DET
ejde-673	274	20	constant	constant	ADJ
ejde-673	274	21	d1	d1	NOUN
ejde-673	274	22	>	>	X
ejde-673	274	23	0	0	NUM
ejde-673	275	1	such	such	ADJ
ejde-673	275	2	that	that	SCONJ
ejde-673	275	3	|h(x	|h(x	PROPN
ejde-673	275	4	,	,	PUNCT
ejde-673	275	5	t)|	t)|	ADJ
ejde-673	275	6	≤	≤	ADJ
ejde-673	275	7	d(x	d(x	NOUN
ejde-673	275	8	)	)	PUNCT
ejde-673	275	9	+	+	X
ejde-673	275	10	d1b(x	d1b(x	ADJ
ejde-673	275	11	)	)	PUNCT
ejde-673	275	12	1	1	NUM
ejde-673	275	13	/	/	SYM
ejde-673	275	14	r1(x)|t|r(x)/r1(x	r1(x)|t|r(x)/r1(x	NOUN
ejde-673	275	15	)	)	PUNCT
ejde-673	275	16	for	for	ADP
ejde-673	275	17	σ	σ	PROPN
ejde-673	275	18	-	-	PROPN
ejde-673	275	19	a.e	a.e	PROPN
ejde-673	275	20	.	.	PUNCT
ejde-673	275	21	x	x	SYM
ejde-673	275	22	∈	∈	PROPN
ejde-673	275	23	γ2	γ2	NOUN
ejde-673	275	24	and	and	CCONJ
ejde-673	275	25	all	all	DET
ejde-673	275	26	t	t	PROPN
ejde-673	275	27	∈	∈	PROPN
ejde-673	275	28	r.	r.	PROPN
ejde-673	275	29	then	then	ADV
ejde-673	275	30	the	the	DET
ejde-673	275	31	nemytskii	nemytskii	ADJ
ejde-673	275	32	operator	operator	NOUN
ejde-673	275	33	nh	nh	PROPN
ejde-673	275	34	:	:	PUNCT
ejde-673	275	35	l	l	NOUN
ejde-673	275	36	r	r	X
ejde-673	275	37	(	(	PUNCT
ejde-673	275	38	·	·	PUNCT
ejde-673	275	39	)	)	PUNCT
ejde-673	275	40	b(·)(γ2	b(·)(γ2	NOUN
ejde-673	275	41	)	)	PUNCT
ejde-673	275	42	∋	∋	NOUN
ejde-673	275	43	v	v	ADP
ejde-673	275	44	7→	7→	NUM
ejde-673	275	45	h(x	h(x	PROPN
ejde-673	275	46	,	,	PUNCT
ejde-673	275	47	v(x	v(x	PROPN
ejde-673	275	48	)	)	PUNCT
ejde-673	275	49	)	)	PUNCT
ejde-673	276	1	∈	∈	PROPN
ejde-673	276	2	lr1(·)(γ2	lr1(·)(γ2	PROPN
ejde-673	276	3	)	)	PUNCT
ejde-673	276	4	is	be	AUX
ejde-673	276	5	continuous	continuous	ADJ
ejde-673	276	6	and	and	CCONJ
ejde-673	276	7	there	there	PRON
ejde-673	276	8	exists	exist	VERB
ejde-673	276	9	a	a	DET
ejde-673	276	10	constant	constant	ADJ
ejde-673	276	11	c	c	NOUN
ejde-673	276	12	>	>	X
ejde-673	276	13	0	0	NUM
ejde-673	276	14	such	such	ADJ
ejde-673	276	15	that	that	SCONJ
ejde-673	276	16	ρr1(·),γ2	ρr1(·),γ2	PROPN
ejde-673	276	17	(	(	PUNCT
ejde-673	276	18	nh(v	nh(v	NOUN
ejde-673	276	19	)	)	PUNCT
ejde-673	276	20	)	)	PUNCT
ejde-673	277	1	≤	≤	NUM
ejde-673	277	2	c(ρr1(·),γ2	c(ρr1(·),γ2	NOUN
ejde-673	277	3	(	(	PUNCT
ejde-673	277	4	d	d	NOUN
ejde-673	277	5	)	)	PUNCT
ejde-673	277	6	+	+	NUM
ejde-673	277	7	ρ(r(·),b(·)),γ2	ρ(r(·),b(·)),γ2	NOUN
ejde-673	277	8	(	(	PUNCT
ejde-673	277	9	v	v	NOUN
ejde-673	277	10	)	)	PUNCT
ejde-673	277	11	)	)	PUNCT
ejde-673	277	12	for	for	ADP
ejde-673	277	13	all	all	DET
ejde-673	277	14	v	v	ADP
ejde-673	277	15	∈	∈	NOUN
ejde-673	277	16	l	l	NOUN
ejde-673	277	17	r	r	X
ejde-673	277	18	(	(	PUNCT
ejde-673	277	19	·	·	PUNCT
ejde-673	277	20	)	)	PUNCT
ejde-673	277	21	b(·)(γ2	b(·)(γ2	NOUN
ejde-673	277	22	)	)	PUNCT
ejde-673	277	23	.	.	PUNCT
ejde-673	278	1	in	in	ADP
ejde-673	278	2	particular	particular	ADJ
ejde-673	278	3	,	,	PUNCT
ejde-673	278	4	if	if	SCONJ
ejde-673	278	5	r1(x	r1(x	NOUN
ejde-673	278	6	)	)	PUNCT
ejde-673	278	7	≡	≡	PROPN
ejde-673	278	8	1	1	NUM
ejde-673	278	9	,	,	PUNCT
ejde-673	278	10	then	then	ADV
ejde-673	278	11	nh	nh	PROPN
ejde-673	278	12	:	:	PUNCT
ejde-673	278	13	l	l	NOUN
ejde-673	278	14	r	r	X
ejde-673	278	15	(	(	PUNCT
ejde-673	278	16	·	·	PUNCT
ejde-673	278	17	)	)	PUNCT
ejde-673	278	18	b(·)(γ2	b(·)(γ2	NOUN
ejde-673	278	19	)	)	PUNCT
ejde-673	278	20	→	→	SYM
ejde-673	278	21	l1(γ2	l1(γ2	NOUN
ejde-673	278	22	)	)	PUNCT
ejde-673	278	23	is	be	AUX
ejde-673	278	24	continuous	continuous	ADJ
ejde-673	278	25	.	.	PUNCT
ejde-673	279	1	now	now	ADV
ejde-673	279	2	we	we	PRON
ejde-673	279	3	define	define	VERB
ejde-673	279	4	the	the	DET
ejde-673	279	5	space	space	NOUN
ejde-673	279	6	x	x	PUNCT
ejde-673	280	1	=	=	PUNCT
ejde-673	280	2	{	{	PUNCT
ejde-673	280	3	v	v	NUM
ejde-673	280	4	∈	∈	NOUN
ejde-673	280	5	w	w	NOUN
ejde-673	280	6	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	280	7	)	)	PUNCT
ejde-673	280	8	;	;	PUNCT
ejde-673	280	9	v	v	X
ejde-673	280	10	=	=	SYM
ejde-673	280	11	0	0	NUM
ejde-673	280	12	on	on	ADP
ejde-673	280	13	γ1	γ1	PROPN
ejde-673	280	14	}	}	PUNCT
ejde-673	280	15	.	.	PUNCT
ejde-673	281	1	(	(	PUNCT
ejde-673	281	2	2.1	2.1	NUM
ejde-673	281	3	)	)	PUNCT
ejde-673	281	4	then	then	ADV
ejde-673	281	5	it	it	PRON
ejde-673	281	6	is	be	AUX
ejde-673	281	7	clear	clear	ADJ
ejde-673	281	8	that	that	SCONJ
ejde-673	281	9	x	x	PRON
ejde-673	281	10	is	be	AUX
ejde-673	281	11	a	a	DET
ejde-673	281	12	closed	closed	ADJ
ejde-673	281	13	subspace	subspace	NOUN
ejde-673	281	14	of	of	ADP
ejde-673	281	15	w	w	PROPN
ejde-673	281	16	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	281	17	)	)	PUNCT
ejde-673	281	18	,	,	PUNCT
ejde-673	281	19	so	so	CCONJ
ejde-673	281	20	x	x	PUNCT
ejde-673	281	21	is	be	AUX
ejde-673	281	22	a	a	DET
ejde-673	281	23	reflexive	reflexive	ADJ
ejde-673	281	24	and	and	CCONJ
ejde-673	281	25	separable	separable	ADJ
ejde-673	281	26	banach	banach	NOUN
ejde-673	281	27	space	space	NOUN
ejde-673	281	28	.	.	PUNCT
ejde-673	282	1	we	we	PRON
ejde-673	282	2	can	can	AUX
ejde-673	282	3	see	see	VERB
ejde-673	282	4	the	the	DET
ejde-673	282	5	following	follow	VERB
ejde-673	282	6	poincaré-type	poincaré-type	NOUN
ejde-673	282	7	inequality	inequality	NOUN
ejde-673	282	8	(	(	PUNCT
ejde-673	282	9	cf	cf	NOUN
ejde-673	282	10	.	.	PUNCT
ejde-673	283	1	[	[	X
ejde-673	283	2	10	10	NUM
ejde-673	283	3	]	]	NUM
ejde-673	283	4	)	)	PUNCT
ejde-673	283	5	.	.	PUNCT
ejde-673	284	1	proposition	proposition	NOUN
ejde-673	284	2	2.14	2.14	NUM
ejde-673	284	3	.	.	PUNCT
ejde-673	285	1	let	let	VERB
ejde-673	285	2	ω	ω	PRON
ejde-673	285	3	be	be	AUX
ejde-673	285	4	a	a	DET
ejde-673	285	5	bounded	bounded	ADJ
ejde-673	285	6	domain	domain	NOUN
ejde-673	285	7	of	of	ADP
ejde-673	285	8	rn	rn	PROPN
ejde-673	285	9	with	with	ADP
ejde-673	285	10	a	a	DET
ejde-673	285	11	c0,1	c0,1	NOUN
ejde-673	285	12	-	-	PUNCT
ejde-673	285	13	boundary	boundary	NOUN
ejde-673	285	14	and	and	CCONJ
ejde-673	285	15	let	let	VERB
ejde-673	285	16	p	p	PROPN
ejde-673	285	17	∈	∈	PROPN
ejde-673	285	18	c+(ω	c+(ω	PROPN
ejde-673	285	19	)	)	PUNCT
ejde-673	285	20	.	.	PUNCT
ejde-673	286	1	then	then	ADV
ejde-673	286	2	there	there	PRON
ejde-673	286	3	exists	exist	VERB
ejde-673	286	4	a	a	DET
ejde-673	286	5	constant	constant	ADJ
ejde-673	286	6	c	c	NOUN
ejde-673	286	7	=	=	SYM
ejde-673	286	8	c(ω	c(ω	PROPN
ejde-673	286	9	,	,	PUNCT
ejde-673	286	10	n	n	CCONJ
ejde-673	286	11	,	,	PUNCT
ejde-673	286	12	p	p	NOUN
ejde-673	286	13	)	)	PUNCT
ejde-673	286	14	>	>	X
ejde-673	286	15	0	0	NUM
ejde-673	286	16	such	such	ADJ
ejde-673	286	17	that	that	PRON
ejde-673	286	18	∥u∥lp(·)(ω	∥u∥lp(·)(ω	NOUN
ejde-673	286	19	)	)	PUNCT
ejde-673	286	20	≤	≤	NOUN
ejde-673	286	21	c∥∇u∥lp(·)(ω	c∥∇u∥lp(·)(ω	PROPN
ejde-673	286	22	)	)	PUNCT
ejde-673	286	23	for	for	ADP
ejde-673	286	24	all	all	DET
ejde-673	286	25	u	u	PROPN
ejde-673	286	26	∈	∈	NOUN
ejde-673	286	27	x.	x.	NOUN
ejde-673	286	28	in	in	ADP
ejde-673	286	29	particular	particular	ADJ
ejde-673	286	30	,	,	PUNCT
ejde-673	286	31	∥∇u∥lp(·)(ω	∥∇u∥lp(·)(ω	PROPN
ejde-673	286	32	)	)	PUNCT
ejde-673	286	33	is	be	AUX
ejde-673	286	34	equivalent	equivalent	ADJ
ejde-673	286	35	to	to	PART
ejde-673	286	36	∥u∥w	∥u∥w	VERB
ejde-673	286	37	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	286	38	)	)	PUNCT
ejde-673	286	39	for	for	ADP
ejde-673	286	40	u	u	PROPN
ejde-673	286	41	∈	∈	PROPN
ejde-673	286	42	x.	x.	NOUN
ejde-673	286	43	for	for	ADP
ejde-673	286	44	a	a	DET
ejde-673	286	45	proof	proof	NOUN
ejde-673	286	46	of	of	ADP
ejde-673	286	47	the	the	DET
ejde-673	286	48	above	above	ADJ
ejde-673	286	49	proposition	proposition	NOUN
ejde-673	286	50	see	see	VERB
ejde-673	286	51	[	[	X
ejde-673	286	52	5	5	NUM
ejde-673	286	53	,	,	PUNCT
ejde-673	286	54	lemma	lemma	PROPN
ejde-673	286	55	2.5	2.5	NUM
ejde-673	286	56	]	]	PUNCT
ejde-673	286	57	.	.	PUNCT
ejde-673	287	1	thus	thus	ADV
ejde-673	287	2	we	we	PRON
ejde-673	287	3	can	can	AUX
ejde-673	287	4	define	define	VERB
ejde-673	287	5	the	the	DET
ejde-673	287	6	norm	norm	NOUN
ejde-673	287	7	on	on	ADP
ejde-673	287	8	x	x	PUNCT
ejde-673	287	9	so	so	SCONJ
ejde-673	287	10	that	that	SCONJ
ejde-673	287	11	∥v∥x	∥v∥x	X
ejde-673	287	12	=	=	SYM
ejde-673	287	13	∥∇v∥lp(·)(ω	∥∇v∥lp(·)(ω	NOUN
ejde-673	287	14	)	)	PUNCT
ejde-673	287	15	for	for	ADP
ejde-673	287	16	v	v	NUM
ejde-673	287	17	∈	∈	PROPN
ejde-673	287	18	x	x	X
ejde-673	287	19	,	,	PUNCT
ejde-673	287	20	(	(	PUNCT
ejde-673	287	21	2.2	2.2	NUM
ejde-673	287	22	)	)	PUNCT
ejde-673	287	23	which	which	PRON
ejde-673	287	24	is	be	AUX
ejde-673	287	25	equivalent	equivalent	ADJ
ejde-673	287	26	to	to	ADP
ejde-673	287	27	∥v∥w	∥v∥w	PROPN
ejde-673	287	28	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	287	29	)	)	PUNCT
ejde-673	287	30	from	from	ADP
ejde-673	287	31	proposition	proposition	NOUN
ejde-673	287	32	2.14	2.14	NUM
ejde-673	287	33	.	.	PUNCT
ejde-673	288	1	3	3	X
ejde-673	288	2	.	.	NOUN
ejde-673	288	3	assumptions	assumption	NOUN
ejde-673	288	4	and	and	CCONJ
ejde-673	288	5	main	main	ADJ
ejde-673	288	6	theorem	theorem	NOUN
ejde-673	288	7	let	let	VERB
ejde-673	288	8	p	p	PROPN
ejde-673	288	9	∈	∈	PROPN
ejde-673	288	10	c+(ω	c+(ω	PROPN
ejde-673	288	11	)	)	PUNCT
ejde-673	288	12	be	be	AUX
ejde-673	288	13	fixed	fix	VERB
ejde-673	288	14	.	.	PUNCT
ejde-673	289	1	assume	assume	VERB
ejde-673	289	2	that	that	SCONJ
ejde-673	289	3	the	the	DET
ejde-673	289	4	following	following	NOUN
ejde-673	289	5	:	:	PUNCT
ejde-673	289	6	(	(	PUNCT
ejde-673	289	7	a2	a2	PROPN
ejde-673	289	8	)	)	PUNCT
ejde-673	289	9	a	a	DET
ejde-673	289	10	:	:	PUNCT
ejde-673	289	11	ω	ω	NUM
ejde-673	289	12	×	×	PROPN
ejde-673	289	13	rn	rn	PROPN
ejde-673	289	14	→	→	PROPN
ejde-673	289	15	r	r	NOUN
ejde-673	289	16	is	be	AUX
ejde-673	289	17	a	a	DET
ejde-673	289	18	function	function	NOUN
ejde-673	289	19	satisfying	satisfy	VERB
ejde-673	289	20	that	that	SCONJ
ejde-673	289	21	for	for	ADP
ejde-673	289	22	a.e	a.e	PROPN
ejde-673	289	23	.	.	PUNCT
ejde-673	289	24	x	x	SYM
ejde-673	289	25	∈	∈	PROPN
ejde-673	289	26	ω	ω	PROPN
ejde-673	289	27	,	,	PUNCT
ejde-673	289	28	the	the	DET
ejde-673	289	29	function	function	NOUN
ejde-673	289	30	a(x	a(x	NOUN
ejde-673	289	31	,	,	PUNCT
ejde-673	289	32	·	·	PUNCT
ejde-673	289	33	)	)	PUNCT
ejde-673	289	34	:	:	PUNCT
ejde-673	289	35	rn	rn	PROPN
ejde-673	289	36	∋	∋	NOUN
ejde-673	289	37	ξ	ξ	PROPN
ejde-673	289	38	7→	7→	PUNCT
ejde-673	289	39	a(x	a(x	PROPN
ejde-673	289	40	,	,	PUNCT
ejde-673	289	41	ξ	ξ	X
ejde-673	289	42	)	)	PUNCT
ejde-673	289	43	is	be	AUX
ejde-673	289	44	of	of	ADP
ejde-673	289	45	c1	c1	NOUN
ejde-673	289	46	-	-	PUNCT
ejde-673	289	47	class	class	NOUN
ejde-673	289	48	,	,	PUNCT
ejde-673	289	49	and	and	CCONJ
ejde-673	289	50	for	for	ADP
ejde-673	289	51	all	all	DET
ejde-673	289	52	ξ	ξ	PROPN
ejde-673	289	53	∈	∈	PROPN
ejde-673	289	54	rn	rn	PROPN
ejde-673	289	55	,	,	PUNCT
ejde-673	289	56	the	the	DET
ejde-673	289	57	function	function	NOUN
ejde-673	289	58	a	a	PRON
ejde-673	289	59	(	(	PUNCT
ejde-673	289	60	·	·	PUNCT
ejde-673	289	61	,	,	PUNCT
ejde-673	289	62	ξ	ξ	X
ejde-673	289	63	)	)	PUNCT
ejde-673	289	64	:	:	PUNCT
ejde-673	290	1	ω	ω	NUM
ejde-673	290	2	∋	∋	NOUN
ejde-673	290	3	x	x	X
ejde-673	290	4	7→	7→	NUM
ejde-673	290	5	a(x	a(x	PROPN
ejde-673	290	6	,	,	PUNCT
ejde-673	290	7	ξ	ξ	X
ejde-673	290	8	)	)	PUNCT
ejde-673	290	9	is	be	AUX
ejde-673	290	10	measurable	measurable	ADJ
ejde-673	290	11	.	.	PUNCT
ejde-673	291	1	moreover	moreover	ADV
ejde-673	291	2	,	,	PUNCT
ejde-673	291	3	suppose	suppose	VERB
ejde-673	291	4	that	that	SCONJ
ejde-673	291	5	a(x,0	a(x,0	PROPN
ejde-673	291	6	)	)	PUNCT
ejde-673	292	1	=	=	SYM
ejde-673	292	2	0	0	PUNCT
ejde-673	293	1	and	and	CCONJ
ejde-673	293	2	put	put	VERB
ejde-673	293	3	a(x	a(x	NOUN
ejde-673	293	4	,	,	PUNCT
ejde-673	293	5	ξ	ξ	NOUN
ejde-673	293	6	)	)	PUNCT
ejde-673	293	7	=	=	SYM
ejde-673	293	8	∇ξa(x	∇ξa(x	PROPN
ejde-673	293	9	,	,	PUNCT
ejde-673	293	10	ξ	ξ	NOUN
ejde-673	293	11	)	)	PUNCT
ejde-673	293	12	.	.	PUNCT
ejde-673	294	1	then	then	ADV
ejde-673	294	2	a(x	a(x	PROPN
ejde-673	294	3	,	,	PUNCT
ejde-673	294	4	ξ	ξ	X
ejde-673	294	5	)	)	PUNCT
ejde-673	294	6	is	be	AUX
ejde-673	294	7	a	a	DET
ejde-673	294	8	carathéodory	carathéodory	NOUN
ejde-673	294	9	function	function	NOUN
ejde-673	294	10	.	.	PUNCT
ejde-673	295	1	for	for	ADP
ejde-673	295	2	items	item	NOUN
ejde-673	295	3	(	(	PUNCT
ejde-673	295	4	a3)–(a5	a3)–(a5	ADJ
ejde-673	295	5	)	)	PUNCT
ejde-673	295	6	,	,	PUNCT
ejde-673	295	7	c	c	X
ejde-673	295	8	,	,	PUNCT
ejde-673	295	9	k0	k0	PROPN
ejde-673	295	10	,	,	PUNCT
ejde-673	295	11	k1	k1	X
ejde-673	295	12	>	>	SYM
ejde-673	295	13	0	0	NUM
ejde-673	295	14	denote	denote	NOUN
ejde-673	295	15	constants	constant	NOUN
ejde-673	295	16	,	,	PUNCT
ejde-673	295	17	h0	h0	NOUN
ejde-673	295	18	∈	∈	PROPN
ejde-673	295	19	lp′(·)(ω	lp′(·)(ω	NOUN
ejde-673	295	20	)	)	PUNCT
ejde-673	295	21	is	be	AUX
ejde-673	295	22	a	a	DET
ejde-673	295	23	non	non	ADJ
ejde-673	295	24	-	-	ADJ
ejde-673	295	25	negative	negative	ADJ
ejde-673	295	26	function	function	NOUN
ejde-673	295	27	,	,	PUNCT
ejde-673	295	28	and	and	CCONJ
ejde-673	295	29	h1	h1	PROPN
ejde-673	295	30	∈	∈	PROPN
ejde-673	295	31	l1	l1	PROPN
ejde-673	295	32	loc(ω	loc(ω	PROPN
ejde-673	295	33	)	)	PUNCT
ejde-673	295	34	with	with	ADP
ejde-673	295	35	h1(x	h1(x	NOUN
ejde-673	295	36	)	)	PUNCT
ejde-673	295	37	≥	≥	NOUN
ejde-673	295	38	1	1	NUM
ejde-673	295	39	for	for	ADP
ejde-673	295	40	a.e	a.e	PROPN
ejde-673	295	41	.	.	PUNCT
ejde-673	295	42	x	x	SYM
ejde-673	295	43	∈	∈	PROPN
ejde-673	295	44	ω	ω	PROPN
ejde-673	295	45	.	.	PROPN
ejde-673	295	46	10	10	NUM
ejde-673	295	47	j.	j.	PROPN
ejde-673	295	48	aramaki	aramaki	PROPN
ejde-673	295	49	ejde-2025/17	ejde-2025/17	X
ejde-673	295	50	(	(	PUNCT
ejde-673	295	51	a3	a3	NOUN
ejde-673	295	52	)	)	PUNCT
ejde-673	295	53	|a(x	|a(x	PROPN
ejde-673	295	54	,	,	PUNCT
ejde-673	295	55	ξ)|	ξ)|	ADJ
ejde-673	295	56	≤	≤	PROPN
ejde-673	295	57	c(h0(x	c(h0(x	PROPN
ejde-673	295	58	)	)	PUNCT
ejde-673	296	1	+	+	SYM
ejde-673	296	2	h1(x)|ξ|p(x)−1	h1(x)|ξ|p(x)−1	NOUN
ejde-673	296	3	)	)	PUNCT
ejde-673	296	4	for	for	ADP
ejde-673	296	5	all	all	DET
ejde-673	296	6	ξ	ξ	PROPN
ejde-673	296	7	∈	∈	PROPN
ejde-673	296	8	rn	rn	PROPN
ejde-673	296	9	and	and	CCONJ
ejde-673	296	10	a.e	a.e	PROPN
ejde-673	296	11	.	.	PROPN
ejde-673	296	12	x	x	SYM
ejde-673	296	13	∈	∈	PROPN
ejde-673	296	14	ω	ω	PROPN
ejde-673	296	15	.	.	PUNCT
ejde-673	296	16	(	(	PUNCT
ejde-673	296	17	a4	a4	INTJ
ejde-673	296	18	)	)	PUNCT
ejde-673	296	19	a	a	PRON
ejde-673	296	20	is	be	AUX
ejde-673	296	21	p(·)-uniformly	p(·)-uniformly	ADV
ejde-673	296	22	convex	convex	ADJ
ejde-673	296	23	,	,	PUNCT
ejde-673	296	24	that	that	ADV
ejde-673	296	25	is	is	ADV
ejde-673	296	26	,	,	PUNCT
ejde-673	296	27	a	a	DET
ejde-673	296	28	(	(	PUNCT
ejde-673	296	29	x	x	NOUN
ejde-673	296	30	,	,	PUNCT
ejde-673	296	31	ξ	ξ	PROPN
ejde-673	296	32	+	+	SYM
ejde-673	296	33	η	η	PROPN
ejde-673	296	34	2	2	NUM
ejde-673	296	35	)	)	PUNCT
ejde-673	296	36	+	+	CCONJ
ejde-673	296	37	k1h1(x)|ξ	k1h1(x)|ξ	PROPN
ejde-673	296	38	−	−	PROPN
ejde-673	296	39	η|p(x	η|p(x	PROPN
ejde-673	296	40	)	)	PUNCT
ejde-673	296	41	≤	≤	NUM
ejde-673	296	42	1	1	NUM
ejde-673	296	43	2	2	NUM
ejde-673	296	44	a(x	a(x	NOUN
ejde-673	296	45	,	,	PUNCT
ejde-673	296	46	ξ	ξ	NOUN
ejde-673	296	47	)	)	PUNCT
ejde-673	296	48	+	+	CCONJ
ejde-673	296	49	1	1	NUM
ejde-673	296	50	2	2	NUM
ejde-673	296	51	a(x	a(x	PROPN
ejde-673	296	52	,	,	PUNCT
ejde-673	296	53	η	η	NOUN
ejde-673	296	54	)	)	PUNCT
ejde-673	296	55	for	for	ADP
ejde-673	296	56	all	all	DET
ejde-673	296	57	ξ	ξ	PROPN
ejde-673	296	58	,	,	PUNCT
ejde-673	296	59	η	η	PROPN
ejde-673	296	60	∈	∈	PROPN
ejde-673	296	61	rn	rn	PROPN
ejde-673	296	62	and	and	CCONJ
ejde-673	296	63	a.e	a.e	PROPN
ejde-673	296	64	.	.	PROPN
ejde-673	296	65	x	x	SYM
ejde-673	296	66	∈	∈	PROPN
ejde-673	296	67	ω	ω	PROPN
ejde-673	296	68	.	.	PUNCT
ejde-673	297	1	(	(	PUNCT
ejde-673	297	2	a5	a5	PROPN
ejde-673	297	3	)	)	PUNCT
ejde-673	297	4	k0h1(x)|ξ|p(x	k0h1(x)|ξ|p(x	PROPN
ejde-673	297	5	)	)	PUNCT
ejde-673	297	6	≤	≤	PUNCT
ejde-673	297	7	a(x	a(x	PROPN
ejde-673	297	8	,	,	PUNCT
ejde-673	297	9	ξ	ξ	NOUN
ejde-673	297	10	)	)	PUNCT
ejde-673	297	11	·	·	PUNCT
ejde-673	298	1	ξ	ξ	X
ejde-673	298	2	≤	≤	NUM
ejde-673	298	3	p(x)a(x	p(x)a(x	NOUN
ejde-673	298	4	,	,	PUNCT
ejde-673	298	5	ξ	ξ	NOUN
ejde-673	298	6	)	)	PUNCT
ejde-673	298	7	for	for	ADP
ejde-673	298	8	all	all	DET
ejde-673	298	9	ξ	ξ	PROPN
ejde-673	298	10	∈	∈	PROPN
ejde-673	298	11	rn	rn	PROPN
ejde-673	298	12	and	and	CCONJ
ejde-673	298	13	a.e	a.e	PROPN
ejde-673	298	14	.	.	PROPN
ejde-673	298	15	x	x	SYM
ejde-673	298	16	∈	∈	PROPN
ejde-673	298	17	ω	ω	PROPN
ejde-673	298	18	.	.	PUNCT
ejde-673	299	1	(	(	PUNCT
ejde-673	299	2	a6	a6	NOUN
ejde-673	299	3	)	)	PUNCT
ejde-673	299	4	(	(	PUNCT
ejde-673	299	5	a(x	a(x	PROPN
ejde-673	299	6	,	,	PUNCT
ejde-673	299	7	ξ)−	ξ)−	PROPN
ejde-673	299	8	a(x	a(x	PROPN
ejde-673	299	9	,	,	PUNCT
ejde-673	299	10	η	η	NOUN
ejde-673	299	11	)	)	PUNCT
ejde-673	299	12	)	)	PUNCT
ejde-673	299	13	·	·	PUNCT
ejde-673	300	1	(	(	PUNCT
ejde-673	300	2	ξ−	ξ−	PROPN
ejde-673	300	3	η	η	PROPN
ejde-673	300	4	)	)	PUNCT
ejde-673	300	5	>	>	X
ejde-673	300	6	0	0	PUNCT
ejde-673	300	7	for	for	ADP
ejde-673	300	8	all	all	DET
ejde-673	300	9	ξ	ξ	PROPN
ejde-673	300	10	,	,	PUNCT
ejde-673	300	11	η	η	PROPN
ejde-673	300	12	∈	∈	PROPN
ejde-673	300	13	rn	rn	PROPN
ejde-673	300	14	with	with	ADP
ejde-673	300	15	ξ	ξ	PROPN
ejde-673	300	16	̸=	̸=	PROPN
ejde-673	300	17	η	η	PROPN
ejde-673	300	18	and	and	CCONJ
ejde-673	300	19	a.e	a.e	PROPN
ejde-673	300	20	.	.	PROPN
ejde-673	300	21	x	x	SYM
ejde-673	300	22	∈	∈	PROPN
ejde-673	300	23	ω	ω	PROPN
ejde-673	300	24	.	.	PUNCT
ejde-673	300	25	(	(	PUNCT
ejde-673	300	26	a7	a7	PROPN
ejde-673	300	27	)	)	PUNCT
ejde-673	300	28	a(x,−ξ	a(x,−ξ	PROPN
ejde-673	300	29	)	)	PUNCT
ejde-673	300	30	=	=	PUNCT
ejde-673	300	31	a(x	a(x	NOUN
ejde-673	300	32	,	,	PUNCT
ejde-673	300	33	ξ	ξ	NOUN
ejde-673	300	34	)	)	PUNCT
ejde-673	300	35	for	for	ADP
ejde-673	300	36	all	all	DET
ejde-673	300	37	ξ	ξ	PROPN
ejde-673	300	38	∈	∈	PROPN
ejde-673	300	39	rn	rn	PROPN
ejde-673	300	40	and	and	CCONJ
ejde-673	300	41	a.e	a.e	PROPN
ejde-673	300	42	.	.	PROPN
ejde-673	300	43	x	x	SYM
ejde-673	300	44	∈	∈	PROPN
ejde-673	300	45	ω	ω	PROPN
ejde-673	300	46	.	.	PUNCT
ejde-673	300	47	remark	remark	PROPN
ejde-673	300	48	3.1	3.1	NUM
ejde-673	300	49	.	.	PUNCT
ejde-673	301	1	(	(	PUNCT
ejde-673	301	2	i	i	NOUN
ejde-673	301	3	)	)	PUNCT
ejde-673	301	4	the	the	DET
ejde-673	301	5	condition	condition	NOUN
ejde-673	301	6	(	(	PUNCT
ejde-673	301	7	a3	a3	NOUN
ejde-673	301	8	)	)	PUNCT
ejde-673	301	9	is	be	AUX
ejde-673	301	10	more	more	ADV
ejde-673	301	11	general	general	ADJ
ejde-673	301	12	than	than	ADP
ejde-673	301	13	that	that	PRON
ejde-673	301	14	of	of	ADP
ejde-673	301	15	mashiyev	mashiyev	NOUN
ejde-673	301	16	et	et	PROPN
ejde-673	301	17	al	al	PROPN
ejde-673	301	18	.	.	PUNCT
ejde-673	302	1	[	[	X
ejde-673	302	2	26	26	NUM
ejde-673	302	3	]	]	PUNCT
ejde-673	302	4	who	who	PRON
ejde-673	302	5	considered	consider	VERB
ejde-673	302	6	the	the	DET
ejde-673	302	7	case	case	NOUN
ejde-673	302	8	h1(x	h1(x	NOUN
ejde-673	302	9	)	)	PUNCT
ejde-673	302	10	≡	≡	PROPN
ejde-673	302	11	1	1	X
ejde-673	302	12	.	.	PUNCT
ejde-673	303	1	in	in	ADP
ejde-673	303	2	our	our	PRON
ejde-673	303	3	case	case	NOUN
ejde-673	303	4	,	,	PUNCT
ejde-673	303	5	to	to	PART
ejde-673	303	6	overcome	overcome	VERB
ejde-673	303	7	this	this	PRON
ejde-673	303	8	we	we	PRON
ejde-673	303	9	have	have	VERB
ejde-673	303	10	to	to	PART
ejde-673	303	11	consider	consider	VERB
ejde-673	303	12	the	the	DET
ejde-673	303	13	space	space	NOUN
ejde-673	303	14	y	y	NOUN
ejde-673	303	15	defined	define	VERB
ejde-673	303	16	by	by	ADP
ejde-673	303	17	(	(	PUNCT
ejde-673	303	18	3.2	3.2	NUM
ejde-673	303	19	)	)	PUNCT
ejde-673	303	20	later	later	ADV
ejde-673	303	21	as	as	ADP
ejde-673	303	22	a	a	DET
ejde-673	303	23	basic	basic	ADJ
ejde-673	303	24	space	space	NOUN
ejde-673	303	25	rather	rather	ADV
ejde-673	303	26	than	than	ADP
ejde-673	303	27	the	the	DET
ejde-673	303	28	space	space	NOUN
ejde-673	303	29	x	x	PUNCT
ejde-673	303	30	defined	define	VERB
ejde-673	303	31	by	by	ADP
ejde-673	303	32	(	(	PUNCT
ejde-673	303	33	2.1	2.1	NUM
ejde-673	303	34	)	)	PUNCT
ejde-673	303	35	.	.	PUNCT
ejde-673	304	1	(	(	PUNCT
ejde-673	304	2	ii	ii	NOUN
ejde-673	304	3	)	)	PUNCT
ejde-673	304	4	(	(	PUNCT
ejde-673	304	5	a5	a5	PROPN
ejde-673	304	6	)	)	PUNCT
ejde-673	304	7	implies	imply	VERB
ejde-673	304	8	that	that	SCONJ
ejde-673	304	9	a	a	PRON
ejde-673	304	10	is	be	AUX
ejde-673	304	11	p(·)-sub	p(·)-sub	ADJ
ejde-673	304	12	-	-	ADJ
ejde-673	304	13	homogeneous	homogeneous	ADJ
ejde-673	304	14	,	,	PUNCT
ejde-673	304	15	that	that	ADV
ejde-673	304	16	is	is	ADV
ejde-673	304	17	,	,	PUNCT
ejde-673	304	18	a(x	a(x	PROPN
ejde-673	304	19	,	,	PUNCT
ejde-673	304	20	sξ	sξ	ADJ
ejde-673	304	21	)	)	PUNCT
ejde-673	304	22	≤	≤	PROPN
ejde-673	304	23	a(x	a(x	NOUN
ejde-673	304	24	,	,	PUNCT
ejde-673	304	25	ξ)sp(x	ξ)sp(x	NOUN
ejde-673	304	26	)	)	PUNCT
ejde-673	304	27	for	for	ADP
ejde-673	304	28	each	each	DET
ejde-673	304	29	ξ	ξ	PROPN
ejde-673	304	30	∈	∈	PROPN
ejde-673	304	31	rn	rn	PROPN
ejde-673	304	32	,	,	PUNCT
ejde-673	304	33	a.e	a.e	PROPN
ejde-673	304	34	.	.	PROPN
ejde-673	304	35	x	x	SYM
ejde-673	304	36	∈	∈	PROPN
ejde-673	304	37	ω	ω	PROPN
ejde-673	304	38	and	and	CCONJ
ejde-673	304	39	s	s	X
ejde-673	304	40	≥	≥	NOUN
ejde-673	304	41	1	1	NUM
ejde-673	304	42	.	.	PUNCT
ejde-673	305	1	(	(	PUNCT
ejde-673	305	2	3.1	3.1	NUM
ejde-673	305	3	)	)	PUNCT
ejde-673	305	4	for	for	ADP
ejde-673	305	5	a	a	DET
ejde-673	305	6	proof	proof	NOUN
ejde-673	305	7	,	,	PUNCT
ejde-673	305	8	see	see	VERB
ejde-673	305	9	aramaki	aramaki	NOUN
ejde-673	306	1	[	[	X
ejde-673	306	2	7	7	NUM
ejde-673	306	3	,	,	PUNCT
ejde-673	306	4	(	(	PUNCT
ejde-673	306	5	4.14	4.14	NUM
ejde-673	306	6	)	)	PUNCT
ejde-673	306	7	]	]	PUNCT
ejde-673	306	8	.	.	PUNCT
ejde-673	306	9	example	example	NOUN
ejde-673	307	1	3.2	3.2	NUM
ejde-673	307	2	.	.	PUNCT
ejde-673	308	1	let	let	AUX
ejde-673	308	2	(	(	PUNCT
ejde-673	308	3	i	i	NOUN
ejde-673	308	4	)	)	PUNCT
ejde-673	308	5	a(x	a(x	PROPN
ejde-673	308	6	,	,	PUNCT
ejde-673	308	7	ξ	ξ	NOUN
ejde-673	308	8	)	)	PUNCT
ejde-673	308	9	=	=	SYM
ejde-673	308	10	h(x	h(x	PROPN
ejde-673	308	11	)	)	PUNCT
ejde-673	308	12	p(x	p(x	PROPN
ejde-673	308	13	)	)	PUNCT
ejde-673	308	14	|ξ|	|ξ|	PROPN
ejde-673	308	15	p(x	p(x	PROPN
ejde-673	308	16	)	)	PUNCT
ejde-673	308	17	with	with	ADP
ejde-673	308	18	p−	p−	PRON
ejde-673	308	19	≥	≥	NOUN
ejde-673	308	20	2	2	NUM
ejde-673	308	21	,	,	PUNCT
ejde-673	308	22	h	h	NOUN
ejde-673	308	23	∈	∈	PROPN
ejde-673	308	24	l1	l1	PROPN
ejde-673	308	25	loc(ω	loc(ω	PROPN
ejde-673	308	26	)	)	PUNCT
ejde-673	308	27	satisfying	satisfy	VERB
ejde-673	308	28	h(x	h(x	PROPN
ejde-673	308	29	)	)	PUNCT
ejde-673	308	30	≥	≥	NOUN
ejde-673	309	1	1	1	NUM
ejde-673	309	2	a.e	a.e	PROPN
ejde-673	309	3	.	.	PUNCT
ejde-673	309	4	x	x	SYM
ejde-673	309	5	∈	∈	PROPN
ejde-673	309	6	ω	ω	PROPN
ejde-673	309	7	.	.	PUNCT
ejde-673	309	8	(	(	PUNCT
ejde-673	309	9	ii	ii	NOUN
ejde-673	309	10	)	)	PUNCT
ejde-673	309	11	a(x	a(x	PROPN
ejde-673	309	12	,	,	PUNCT
ejde-673	309	13	ξ	ξ	NOUN
ejde-673	309	14	)	)	PUNCT
ejde-673	309	15	=	=	SYM
ejde-673	309	16	h(x	h(x	PROPN
ejde-673	309	17	)	)	PUNCT
ejde-673	309	18	p(x	p(x	PROPN
ejde-673	309	19	)	)	PUNCT
ejde-673	309	20	(	(	PUNCT
ejde-673	309	21	(	(	PUNCT
ejde-673	309	22	1	1	NUM
ejde-673	309	23	+	+	CCONJ
ejde-673	309	24	|ξ|2)p(x)/2	|ξ|2)p(x)/2	PROPN
ejde-673	309	25	−	−	NOUN
ejde-673	309	26	1	1	NUM
ejde-673	309	27	)	)	PUNCT
ejde-673	309	28	with	with	ADP
ejde-673	309	29	p−	p−	PRON
ejde-673	309	30	≥	≥	NOUN
ejde-673	309	31	2	2	NUM
ejde-673	309	32	,	,	PUNCT
ejde-673	309	33	h	h	NOUN
ejde-673	309	34	∈	∈	PROPN
ejde-673	309	35	lp′(·)(ω	lp′(·)(ω	VERB
ejde-673	309	36	)	)	PUNCT
ejde-673	309	37	satisfying	satisfy	VERB
ejde-673	309	38	h(x	h(x	PROPN
ejde-673	309	39	)	)	PUNCT
ejde-673	309	40	≥	≥	NOUN
ejde-673	309	41	1	1	NUM
ejde-673	309	42	a.e	a.e	PROPN
ejde-673	309	43	.	.	PUNCT
ejde-673	309	44	x	x	SYM
ejde-673	310	1	∈	∈	PROPN
ejde-673	310	2	ω	ω	PROPN
ejde-673	310	3	.	.	PUNCT
ejde-673	311	1	then	then	ADV
ejde-673	311	2	a(x	a(x	PROPN
ejde-673	311	3	,	,	PUNCT
ejde-673	311	4	ξ	ξ	NOUN
ejde-673	311	5	)	)	PUNCT
ejde-673	311	6	and	and	CCONJ
ejde-673	311	7	a(x	a(x	PROPN
ejde-673	311	8	,	,	PUNCT
ejde-673	311	9	ξ	ξ	NOUN
ejde-673	311	10	)	)	PUNCT
ejde-673	311	11	=	=	SYM
ejde-673	311	12	∇ξa(x	∇ξa(x	PROPN
ejde-673	311	13	,	,	PUNCT
ejde-673	311	14	ξ	ξ	NOUN
ejde-673	311	15	)	)	PUNCT
ejde-673	311	16	of	of	ADP
ejde-673	311	17	(	(	PUNCT
ejde-673	311	18	i	i	NOUN
ejde-673	311	19	)	)	PUNCT
ejde-673	311	20	and	and	CCONJ
ejde-673	311	21	(	(	PUNCT
ejde-673	311	22	ii	ii	NOUN
ejde-673	311	23	)	)	PUNCT
ejde-673	311	24	satisfy	satisfy	NOUN
ejde-673	311	25	(	(	PUNCT
ejde-673	311	26	a2)–(a7	a2)–(a7	NOUN
ejde-673	311	27	)	)	PUNCT
ejde-673	311	28	.	.	PUNCT
ejde-673	312	1	remark	remark	VERB
ejde-673	312	2	3.3	3.3	NUM
ejde-673	312	3	.	.	PUNCT
ejde-673	313	1	in	in	ADP
ejde-673	313	2	example	example	NOUN
ejde-673	313	3	3.2	3.2	NUM
ejde-673	313	4	,	,	PUNCT
ejde-673	313	5	when	when	SCONJ
ejde-673	313	6	h(x	h(x	PROPN
ejde-673	313	7	)	)	PUNCT
ejde-673	313	8	≡	≡	PROPN
ejde-673	313	9	1	1	NUM
ejde-673	313	10	,	,	PUNCT
ejde-673	313	11	(	(	PUNCT
ejde-673	313	12	i	i	NOUN
ejde-673	313	13	)	)	PUNCT
ejde-673	313	14	corresponds	correspond	VERB
ejde-673	313	15	to	to	ADP
ejde-673	313	16	the	the	DET
ejde-673	313	17	p(·)-laplacian	p(·)-laplacian	ADJ
ejde-673	313	18	and	and	CCONJ
ejde-673	313	19	(	(	PUNCT
ejde-673	313	20	ii	ii	NOUN
ejde-673	313	21	)	)	PUNCT
ejde-673	313	22	corresponds	correspond	VERB
ejde-673	313	23	to	to	ADP
ejde-673	313	24	the	the	DET
ejde-673	313	25	prescribed	prescribe	VERB
ejde-673	313	26	mean	mean	NOUN
ejde-673	313	27	curvature	curvature	NOUN
ejde-673	313	28	operator	operator	NOUN
ejde-673	313	29	for	for	ADP
ejde-673	313	30	nonparametric	nonparametric	NOUN
ejde-673	313	31	surface	surface	NOUN
ejde-673	313	32	.	.	PUNCT
ejde-673	314	1	for	for	ADP
ejde-673	314	2	the	the	DET
ejde-673	314	3	function	function	NOUN
ejde-673	314	4	h1	h1	PROPN
ejde-673	314	5	∈	∈	PROPN
ejde-673	314	6	l1	l1	PROPN
ejde-673	314	7	loc(ω	loc(ω	PROPN
ejde-673	314	8	)	)	PUNCT
ejde-673	314	9	with	with	ADP
ejde-673	314	10	h1(x	h1(x	NOUN
ejde-673	314	11	)	)	PUNCT
ejde-673	314	12	≥	≥	NOUN
ejde-673	314	13	1	1	NUM
ejde-673	314	14	for	for	ADP
ejde-673	314	15	a.e	a.e	PROPN
ejde-673	314	16	.	.	PUNCT
ejde-673	314	17	x	x	SYM
ejde-673	314	18	∈	∈	PROPN
ejde-673	314	19	ω	ω	NOUN
ejde-673	314	20	,	,	PUNCT
ejde-673	314	21	we	we	PRON
ejde-673	314	22	define	define	VERB
ejde-673	314	23	a	a	DET
ejde-673	314	24	modular	modular	NOUN
ejde-673	314	25	on	on	ADP
ejde-673	314	26	x	x	PUNCT
ejde-673	314	27	by	by	ADP
ejde-673	314	28	ρ̃(p(·),h1(·))(v	ρ̃(p(·),h1(·))(v	X
ejde-673	314	29	)	)	PUNCT
ejde-673	315	1	=	=	SYM
ejde-673	315	2	∫	∫	PROPN
ejde-673	315	3	ω	ω	NUM
ejde-673	315	4	h1(x)|∇v(x)|p(x	h1(x)|∇v(x)|p(x	PROPN
ejde-673	315	5	)	)	PUNCT
ejde-673	315	6	dx	dx	PROPN
ejde-673	315	7	for	for	ADP
ejde-673	315	8	v	v	NUM
ejde-673	315	9	∈	∈	PROPN
ejde-673	315	10	x	x	NOUN
ejde-673	315	11	,	,	PUNCT
ejde-673	315	12	where	where	SCONJ
ejde-673	315	13	the	the	DET
ejde-673	315	14	space	space	NOUN
ejde-673	315	15	x	x	PUNCT
ejde-673	315	16	is	be	AUX
ejde-673	315	17	defined	define	VERB
ejde-673	315	18	by	by	ADP
ejde-673	315	19	(	(	PUNCT
ejde-673	315	20	2.1	2.1	NUM
ejde-673	315	21	)	)	PUNCT
ejde-673	315	22	.	.	PUNCT
ejde-673	316	1	we	we	PRON
ejde-673	316	2	define	define	VERB
ejde-673	316	3	our	our	PRON
ejde-673	316	4	basic	basic	ADJ
ejde-673	316	5	space	space	NOUN
ejde-673	316	6	y	y	PROPN
ejde-673	316	7	=	=	SYM
ejde-673	316	8	y	y	PROPN
ejde-673	316	9	(	(	PUNCT
ejde-673	316	10	ω	ω	NOUN
ejde-673	316	11	)	)	PUNCT
ejde-673	316	12	=	=	PRON
ejde-673	316	13	{	{	PUNCT
ejde-673	316	14	v	v	NUM
ejde-673	316	15	∈	∈	PROPN
ejde-673	316	16	x	x	X
ejde-673	316	17	;	;	PUNCT
ejde-673	316	18	ρ̃(p(·),h1(·))(v	ρ̃(p(·),h1(·))(v	X
ejde-673	316	19	)	)	PUNCT
ejde-673	316	20	<	<	X
ejde-673	316	21	∞	∞	PROPN
ejde-673	316	22	}	}	PUNCT
ejde-673	316	23	(	(	PUNCT
ejde-673	316	24	3.2	3.2	NUM
ejde-673	316	25	)	)	PUNCT
ejde-673	316	26	equipped	equip	VERB
ejde-673	316	27	with	with	ADP
ejde-673	316	28	the	the	DET
ejde-673	316	29	norm	norm	NOUN
ejde-673	316	30	∥v∥y	∥v∥y	ADV
ejde-673	316	31	=	=	SYM
ejde-673	316	32	inf	inf	NOUN
ejde-673	316	33	{	{	PUNCT
ejde-673	316	34	τ	τ	PROPN
ejde-673	316	35	>	>	X
ejde-673	316	36	0	0	NUM
ejde-673	316	37	;	;	PUNCT
ejde-673	316	38	ρ̃(p(·),h1	ρ̃(p(·),h1	NUM
ejde-673	316	39	(	(	PUNCT
ejde-673	316	40	·	·	PUNCT
ejde-673	316	41	)	)	PUNCT
ejde-673	316	42	)	)	PUNCT
ejde-673	317	1	(	(	PUNCT
ejde-673	317	2	v	v	X
ejde-673	317	3	τ	τ	X
ejde-673	317	4	)	)	PUNCT
ejde-673	317	5	≤	≤	NUM
ejde-673	317	6	1	1	NUM
ejde-673	317	7	}	}	PUNCT
ejde-673	317	8	.	.	PUNCT
ejde-673	318	1	proposition	proposition	NOUN
ejde-673	318	2	3.4	3.4	NUM
ejde-673	318	3	.	.	PUNCT
ejde-673	319	1	the	the	DET
ejde-673	319	2	space	space	NOUN
ejde-673	319	3	(	(	PUNCT
ejde-673	319	4	y	y	NOUN
ejde-673	319	5	,	,	PUNCT
ejde-673	319	6	∥	∥	X
ejde-673	319	7	·	·	PUNCT
ejde-673	319	8	∥y	∥y	NOUN
ejde-673	319	9	)	)	PUNCT
ejde-673	319	10	is	be	AUX
ejde-673	319	11	a	a	DET
ejde-673	319	12	separable	separable	ADJ
ejde-673	319	13	and	and	CCONJ
ejde-673	319	14	reflexive	reflexive	ADJ
ejde-673	319	15	banach	banach	NOUN
ejde-673	319	16	space	space	NOUN
ejde-673	319	17	.	.	PUNCT
ejde-673	320	1	for	for	ADP
ejde-673	320	2	a	a	DET
ejde-673	320	3	proof	proof	NOUN
ejde-673	320	4	of	of	ADP
ejde-673	320	5	the	the	DET
ejde-673	320	6	above	above	ADJ
ejde-673	320	7	propositon	propositon	NOUN
ejde-673	320	8	see	see	VERB
ejde-673	320	9	aramaki	aramaki	NOUN
ejde-673	320	10	[	[	X
ejde-673	320	11	8	8	NUM
ejde-673	320	12	,	,	PUNCT
ejde-673	320	13	proposition	proposition	NOUN
ejde-673	320	14	3.4	3.4	NUM
ejde-673	320	15	]	]	PUNCT
ejde-673	320	16	.	.	PUNCT
ejde-673	321	1	we	we	PRON
ejde-673	321	2	note	note	VERB
ejde-673	321	3	that	that	SCONJ
ejde-673	321	4	c∞	c∞	PROPN
ejde-673	321	5	0	0	NUM
ejde-673	321	6	(	(	PUNCT
ejde-673	321	7	ω	ω	NOUN
ejde-673	321	8	)	)	PUNCT
ejde-673	321	9	⊂	⊂	PROPN
ejde-673	322	1	y	y	PROPN
ejde-673	322	2	.	.	PUNCT
ejde-673	323	1	since	since	SCONJ
ejde-673	323	2	h1(x	h1(x	NOUN
ejde-673	323	3	)	)	PUNCT
ejde-673	323	4	≥	≥	NOUN
ejde-673	323	5	1	1	NUM
ejde-673	323	6	a.e	a.e	PROPN
ejde-673	323	7	.	.	PUNCT
ejde-673	323	8	x	x	SYM
ejde-673	323	9	∈	∈	PROPN
ejde-673	323	10	ω	ω	NUM
ejde-673	323	11	,	,	PUNCT
ejde-673	323	12	it	it	PRON
ejde-673	323	13	follows	follow	VERB
ejde-673	323	14	that	that	SCONJ
ejde-673	323	15	ρ̃(p(·),h1(·))(v	ρ̃(p(·),h1(·))(v	X
ejde-673	323	16	)	)	PUNCT
ejde-673	323	17	=	=	PUNCT
ejde-673	323	18	ρp(·)(h	ρp(·)(h	PROPN
ejde-673	323	19	1	1	NUM
ejde-673	323	20	/	/	SYM
ejde-673	323	21	p	p	X
ejde-673	323	22	(	(	PUNCT
ejde-673	323	23	·	·	PUNCT
ejde-673	323	24	)	)	PUNCT
ejde-673	323	25	1	1	NUM
ejde-673	323	26	|∇v|	|∇v|	NOUN
ejde-673	323	27	)	)	PUNCT
ejde-673	323	28	≥	≥	NOUN
ejde-673	323	29	ρp(·)(|∇v|	ρp(·)(|∇v|	NOUN
ejde-673	323	30	)	)	PUNCT
ejde-673	323	31	for	for	ADP
ejde-673	323	32	v	v	ADP
ejde-673	323	33	∈	∈	PROPN
ejde-673	323	34	y	y	PROPN
ejde-673	323	35	and	and	CCONJ
ejde-673	323	36	∥v∥y	∥v∥y	ADV
ejde-673	323	37	=	=	PUNCT
ejde-673	323	38	∥h1	∥h1	PROPN
ejde-673	323	39	/	/	SYM
ejde-673	323	40	p	p	X
ejde-673	323	41	(	(	PUNCT
ejde-673	323	42	·	·	PUNCT
ejde-673	323	43	)	)	PUNCT
ejde-673	323	44	1	1	NUM
ejde-673	323	45	∇v∥lp(·)(ω	∇v∥lp(·)(ω	PROPN
ejde-673	323	46	)	)	PUNCT
ejde-673	323	47	≥	≥	NOUN
ejde-673	323	48	∥∇v∥lp(·)(ω	∥∇v∥lp(·)(ω	NOUN
ejde-673	323	49	)	)	PUNCT
ejde-673	323	50	=	=	SYM
ejde-673	323	51	∥v∥x	∥v∥x	PROPN
ejde-673	323	52	for	for	ADP
ejde-673	323	53	v	v	NOUN
ejde-673	323	54	∈	∈	PROPN
ejde-673	323	55	y.	y.	NOUN
ejde-673	323	56	(	(	PUNCT
ejde-673	323	57	3.3	3.3	NUM
ejde-673	323	58	)	)	PUNCT
ejde-673	323	59	from	from	ADP
ejde-673	323	60	(	(	PUNCT
ejde-673	323	61	3.3	3.3	NUM
ejde-673	323	62	)	)	PUNCT
ejde-673	323	63	and	and	CCONJ
ejde-673	323	64	proposition	proposition	NOUN
ejde-673	323	65	2.1	2.1	NUM
ejde-673	323	66	,	,	PUNCT
ejde-673	323	67	we	we	PRON
ejde-673	323	68	have	have	VERB
ejde-673	323	69	the	the	DET
ejde-673	323	70	following	follow	VERB
ejde-673	323	71	proposition	proposition	NOUN
ejde-673	323	72	.	.	PUNCT
ejde-673	324	1	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	324	2	eigenvalue	eigenvalue	VERB
ejde-673	324	3	problems	problem	NOUN
ejde-673	324	4	for	for	ADP
ejde-673	324	5	kirchhoff	kirchhoff	NOUN
ejde-673	324	6	-	-	PUNCT
ejde-673	324	7	type	type	NOUN
ejde-673	324	8	equations	equation	NOUN
ejde-673	324	9	11	11	NUM
ejde-673	324	10	proposition	proposition	NOUN
ejde-673	324	11	3.5	3.5	NUM
ejde-673	324	12	.	.	PUNCT
ejde-673	325	1	let	let	VERB
ejde-673	325	2	p	p	PROPN
ejde-673	325	3	∈	∈	PROPN
ejde-673	325	4	c+(ω	c+(ω	PROPN
ejde-673	325	5	)	)	PUNCT
ejde-673	325	6	and	and	CCONJ
ejde-673	325	7	let	let	VERB
ejde-673	325	8	u	u	NOUN
ejde-673	325	9	,	,	PUNCT
ejde-673	325	10	un	un	PROPN
ejde-673	325	11	∈	∈	PROPN
ejde-673	325	12	y	y	PROPN
ejde-673	325	13	(	(	PUNCT
ejde-673	325	14	n	n	NOUN
ejde-673	325	15	=	=	SYM
ejde-673	325	16	1	1	NUM
ejde-673	325	17	,	,	PUNCT
ejde-673	325	18	2	2	NUM
ejde-673	325	19	,	,	PUNCT
ejde-673	325	20	.	.	PUNCT
ejde-673	325	21	.	.	PUNCT
ejde-673	326	1	.	.	PUNCT
ejde-673	326	2	)	)	PUNCT
ejde-673	327	1	.	.	PUNCT
ejde-673	328	1	then	then	ADV
ejde-673	328	2	the	the	DET
ejde-673	328	3	following	follow	VERB
ejde-673	328	4	properties	property	NOUN
ejde-673	328	5	hold	hold	VERB
ejde-673	328	6	:	:	PUNCT
ejde-673	328	7	(	(	PUNCT
ejde-673	328	8	i	i	NOUN
ejde-673	328	9	)	)	PUNCT
ejde-673	328	10	y	y	PROPN
ejde-673	328	11	↪	↪	PROPN
ejde-673	328	12	→	→	SYM
ejde-673	328	13	x	x	X
ejde-673	328	14	and	and	CCONJ
ejde-673	328	15	∥u∥x	∥u∥x	ADJ
ejde-673	328	16	≤	≤	NUM
ejde-673	328	17	∥u∥y	∥u∥y	NOUN
ejde-673	328	18	.	.	PUNCT
ejde-673	329	1	(	(	PUNCT
ejde-673	329	2	ii	ii	NOUN
ejde-673	329	3	)	)	PUNCT
ejde-673	329	4	∥u∥y	∥u∥y	NOUN
ejde-673	329	5	>	>	X
ejde-673	329	6	1(=	1(=	NUM
ejde-673	329	7	1	1	NUM
ejde-673	329	8	,	,	PUNCT
ejde-673	329	9	<	<	X
ejde-673	329	10	1	1	X
ejde-673	329	11	)	)	PUNCT
ejde-673	329	12	⇔	⇔	X
ejde-673	329	13	ρ̃(p(·),h1(·))(u	ρ̃(p(·),h1(·))(u	PROPN
ejde-673	329	14	)	)	PUNCT
ejde-673	329	15	>	>	X
ejde-673	330	1	1(=	1(=	NUM
ejde-673	330	2	1	1	NUM
ejde-673	330	3	,	,	PUNCT
ejde-673	330	4	<	<	X
ejde-673	330	5	1	1	NUM
ejde-673	330	6	)	)	PUNCT
ejde-673	330	7	.	.	PUNCT
ejde-673	331	1	(	(	PUNCT
ejde-673	331	2	iii	iii	NOUN
ejde-673	331	3	)	)	PUNCT
ejde-673	331	4	∥u∥y	∥u∥y	NOUN
ejde-673	331	5	>	>	X
ejde-673	331	6	1	1	NUM
ejde-673	331	7	⇒	⇒	NOUN
ejde-673	331	8	∥u∥p	∥u∥p	NOUN
ejde-673	331	9	−	−	PROPN
ejde-673	331	10	y	y	PROPN
ejde-673	331	11	≤	≤	NUM
ejde-673	331	12	ρ̃(p(·),h1(·))(u	ρ̃(p(·),h1(·))(u	NOUN
ejde-673	331	13	)	)	PUNCT
ejde-673	331	14	≤	≤	NUM
ejde-673	331	15	∥u∥p	∥u∥p	NOUN
ejde-673	331	16	+	+	X
ejde-673	331	17	y	y	PROPN
ejde-673	331	18	.	.	PUNCT
ejde-673	332	1	(	(	PUNCT
ejde-673	332	2	iv	iv	X
ejde-673	332	3	)	)	PUNCT
ejde-673	332	4	∥u∥y	∥u∥y	PUNCT
ejde-673	332	5	<	<	X
ejde-673	332	6	1	1	NUM
ejde-673	332	7	⇒	⇒	NOUN
ejde-673	332	8	∥u∥p	∥u∥p	NOUN
ejde-673	332	9	+	+	CCONJ
ejde-673	332	10	y	y	PROPN
ejde-673	332	11	≤	≤	NUM
ejde-673	332	12	ρ̃(p(·),h1(·))(u	ρ̃(p(·),h1(·))(u	NOUN
ejde-673	332	13	)	)	PUNCT
ejde-673	332	14	≤	≤	NUM
ejde-673	332	15	∥u∥p	∥u∥p	NOUN
ejde-673	333	1	−	−	PROPN
ejde-673	333	2	y	y	PROPN
ejde-673	333	3	.	.	PUNCT
ejde-673	334	1	(	(	PUNCT
ejde-673	334	2	v	v	NOUN
ejde-673	334	3	)	)	PUNCT
ejde-673	334	4	limn→∞	limn→∞	PROPN
ejde-673	334	5	∥un	∥un	PROPN
ejde-673	334	6	−	−	PROPN
ejde-673	334	7	u∥y	u∥y	ADV
ejde-673	334	8	=	=	SYM
ejde-673	334	9	0	0	NUM
ejde-673	334	10	⇔	⇔	X
ejde-673	334	11	limn→∞	limn→∞	PROPN
ejde-673	335	1	ρ̃(p(·),h1(·))(un	ρ̃(p(·),h1(·))(un	PROPN
ejde-673	335	2	−	−	PROPN
ejde-673	335	3	u	u	NOUN
ejde-673	335	4	)	)	PUNCT
ejde-673	335	5	=	=	SYM
ejde-673	335	6	0	0	X
ejde-673	335	7	.	.	PUNCT
ejde-673	335	8	(	(	PUNCT
ejde-673	335	9	vi	vi	NOUN
ejde-673	335	10	)	)	PUNCT
ejde-673	335	11	∥un∥y	∥un∥y	X
ejde-673	335	12	→	→	SYM
ejde-673	335	13	∞	∞	PROPN
ejde-673	335	14	as	as	ADP
ejde-673	335	15	n	n	PROPN
ejde-673	335	16	→	→	SYM
ejde-673	335	17	∞	∞	PROPN
ejde-673	335	18	⇔	⇔	X
ejde-673	335	19	ρ̃(p(·),h1(·))(un	ρ̃(p(·),h1(·))(un	PROPN
ejde-673	335	20	)	)	PUNCT
ejde-673	335	21	→	→	SYM
ejde-673	335	22	∞	∞	PROPN
ejde-673	335	23	as	as	ADP
ejde-673	335	24	n	n	PROPN
ejde-673	335	25	→	→	SYM
ejde-673	335	26	∞.	∞.	PROPN
ejde-673	335	27	we	we	PRON
ejde-673	335	28	assume	assume	VERB
ejde-673	335	29	that	that	SCONJ
ejde-673	335	30	the	the	DET
ejde-673	335	31	function	function	NOUN
ejde-673	335	32	g	g	NOUN
ejde-673	335	33	in	in	ADP
ejde-673	335	34	(	(	PUNCT
ejde-673	335	35	1.1	1.1	NUM
ejde-673	335	36	)	)	PUNCT
ejde-673	335	37	satisfies	satisfie	NOUN
ejde-673	335	38	(	(	PUNCT
ejde-673	335	39	a8	a8	PROPN
ejde-673	335	40	)	)	PUNCT
ejde-673	335	41	the	the	DET
ejde-673	335	42	function	function	NOUN
ejde-673	335	43	g(x	g(x	PROPN
ejde-673	335	44	,	,	PUNCT
ejde-673	335	45	t	t	PROPN
ejde-673	335	46	)	)	PUNCT
ejde-673	335	47	is	be	AUX
ejde-673	335	48	of	of	ADP
ejde-673	335	49	the	the	DET
ejde-673	335	50	form	form	NOUN
ejde-673	335	51	g(x	g(x	NOUN
ejde-673	335	52	,	,	PUNCT
ejde-673	335	53	t	t	PROPN
ejde-673	335	54	)	)	PUNCT
ejde-673	335	55	=	=	PUNCT
ejde-673	336	1	b(x)|t|r(x)−2	b(x)|t|r(x)−2	PROPN
ejde-673	336	2	t	t	PROPN
ejde-673	336	3	,	,	PUNCT
ejde-673	336	4	where	where	SCONJ
ejde-673	336	5	b	b	NOUN
ejde-673	336	6	satisfies	satisfy	VERB
ejde-673	336	7	0	0	PUNCT
ejde-673	336	8	<	<	X
ejde-673	336	9	b	b	X
ejde-673	336	10	∈	∈	PROPN
ejde-673	336	11	lβ(·)(γ2	lβ(·)(γ2	NOUN
ejde-673	336	12	)	)	PUNCT
ejde-673	336	13	with	with	ADP
ejde-673	336	14	β	β	PROPN
ejde-673	336	15	∈	∈	PROPN
ejde-673	336	16	c+(γ2	c+(γ2	NOUN
ejde-673	336	17	)	)	PUNCT
ejde-673	336	18	,	,	PUNCT
ejde-673	336	19	and	and	CCONJ
ejde-673	336	20	r	r	NOUN
ejde-673	336	21	∈	∈	PROPN
ejde-673	336	22	c+(γ2	c+(γ2	NOUN
ejde-673	336	23	)	)	PUNCT
ejde-673	336	24	satisfies	satisfy	VERB
ejde-673	336	25	r(x	r(x	PROPN
ejde-673	336	26	)	)	PUNCT
ejde-673	336	27	<	<	X
ejde-673	336	28	β(x)−	β(x)−	PROPN
ejde-673	336	29	1	1	NUM
ejde-673	336	30	β(x	β(x	NOUN
ejde-673	336	31	)	)	PUNCT
ejde-673	336	32	p∂(x	p∂(x	NOUN
ejde-673	336	33	)	)	PUNCT
ejde-673	336	34	for	for	ADP
ejde-673	336	35	all	all	DET
ejde-673	336	36	x	x	PROPN
ejde-673	336	37	∈	∈	PROPN
ejde-673	336	38	γ2	γ2	NOUN
ejde-673	336	39	.	.	PUNCT
ejde-673	337	1	if	if	SCONJ
ejde-673	337	2	we	we	PRON
ejde-673	337	3	define	define	VERB
ejde-673	337	4	g(x	g(x	PROPN
ejde-673	337	5	,	,	PUNCT
ejde-673	337	6	t	t	PROPN
ejde-673	337	7	)	)	PUNCT
ejde-673	337	8	=	=	SYM
ejde-673	338	1	∫	∫	PROPN
ejde-673	338	2	t	t	NOUN
ejde-673	338	3	0	0	NUM
ejde-673	339	1	g(x	g(x	PROPN
ejde-673	339	2	,	,	PUNCT
ejde-673	339	3	s	s	X
ejde-673	339	4	)	)	PUNCT
ejde-673	339	5	ds	ds	ADJ
ejde-673	339	6	,	,	PUNCT
ejde-673	339	7	then	then	ADV
ejde-673	339	8	g(x	g(x	PROPN
ejde-673	339	9	,	,	PUNCT
ejde-673	339	10	t	t	PROPN
ejde-673	339	11	)	)	PUNCT
ejde-673	339	12	=	=	SYM
ejde-673	339	13	b(x	b(x	NOUN
ejde-673	339	14	)	)	PUNCT
ejde-673	339	15	r(x	r(x	PROPN
ejde-673	339	16	)	)	PUNCT
ejde-673	339	17	|t|	|t|	PROPN
ejde-673	339	18	r(x	r(x	PROPN
ejde-673	339	19	)	)	PUNCT
ejde-673	340	1	,	,	PUNCT
ejde-673	340	2	so	so	SCONJ
ejde-673	340	3	we	we	PRON
ejde-673	340	4	have	have	VERB
ejde-673	340	5	r(x)g(x	r(x)g(x	ADJ
ejde-673	340	6	,	,	PUNCT
ejde-673	340	7	t	t	NOUN
ejde-673	340	8	)	)	PUNCT
ejde-673	340	9	=	=	SYM
ejde-673	340	10	b(x)|t|r(x	b(x)|t|r(x	PROPN
ejde-673	340	11	)	)	PUNCT
ejde-673	341	1	=	=	SYM
ejde-673	341	2	g(x	g(x	NOUN
ejde-673	341	3	,	,	PUNCT
ejde-673	341	4	t)t	t)t	X
ejde-673	341	5	>	>	X
ejde-673	341	6	0	0	PUNCT
ejde-673	341	7	(	(	PUNCT
ejde-673	341	8	3.4	3.4	NUM
ejde-673	341	9	)	)	PUNCT
ejde-673	341	10	for	for	ADP
ejde-673	341	11	σ	σ	PROPN
ejde-673	341	12	-	-	PROPN
ejde-673	341	13	a.e	a.e	PROPN
ejde-673	341	14	.	.	PUNCT
ejde-673	341	15	x	x	SYM
ejde-673	341	16	∈	∈	PROPN
ejde-673	341	17	γ2	γ2	NOUN
ejde-673	341	18	and	and	CCONJ
ejde-673	341	19	all	all	DET
ejde-673	341	20	0	0	NUM
ejde-673	341	21	̸=	̸=	PROPN
ejde-673	341	22	t	t	PROPN
ejde-673	341	23	∈	∈	PROPN
ejde-673	341	24	r.	r.	PROPN
ejde-673	341	25	now	now	ADV
ejde-673	341	26	we	we	PRON
ejde-673	341	27	introduce	introduce	VERB
ejde-673	341	28	the	the	DET
ejde-673	341	29	notion	notion	NOUN
ejde-673	341	30	of	of	ADP
ejde-673	341	31	a	a	DET
ejde-673	341	32	weak	weak	ADJ
ejde-673	341	33	solution	solution	NOUN
ejde-673	341	34	and	and	CCONJ
ejde-673	341	35	an	an	DET
ejde-673	341	36	eigenfunction	eigenfunction	NOUN
ejde-673	341	37	for	for	ADP
ejde-673	341	38	the	the	DET
ejde-673	341	39	problem	problem	NOUN
ejde-673	341	40	(	(	PUNCT
ejde-673	341	41	1.1	1.1	NUM
ejde-673	341	42	)	)	PUNCT
ejde-673	341	43	.	.	PUNCT
ejde-673	342	1	definition	definition	NOUN
ejde-673	342	2	3.6	3.6	NUM
ejde-673	342	3	.	.	PUNCT
ejde-673	343	1	(	(	PUNCT
ejde-673	343	2	i	i	NOUN
ejde-673	343	3	)	)	PUNCT
ejde-673	343	4	we	we	PRON
ejde-673	343	5	say	say	VERB
ejde-673	343	6	that	that	SCONJ
ejde-673	343	7	a	a	DET
ejde-673	343	8	pair	pair	NOUN
ejde-673	343	9	(	(	PUNCT
ejde-673	343	10	u	u	NOUN
ejde-673	343	11	,	,	PUNCT
ejde-673	343	12	λ	λ	PROPN
ejde-673	343	13	)	)	PUNCT
ejde-673	343	14	∈	∈	PROPN
ejde-673	343	15	y	y	NOUN
ejde-673	343	16	×	×	NOUN
ejde-673	343	17	r	r	NOUN
ejde-673	343	18	is	be	AUX
ejde-673	343	19	a	a	DET
ejde-673	343	20	weak	weak	ADJ
ejde-673	343	21	solution	solution	NOUN
ejde-673	343	22	of	of	ADP
ejde-673	343	23	(	(	PUNCT
ejde-673	343	24	1.1	1.1	NUM
ejde-673	343	25	)	)	PUNCT
ejde-673	343	26	,	,	PUNCT
ejde-673	343	27	if	if	SCONJ
ejde-673	343	28	m	m	PROPN
ejde-673	343	29	(	(	PUNCT
ejde-673	343	30	∫	∫	PROPN
ejde-673	343	31	ω	ω	NUM
ejde-673	343	32	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	343	33	)	)	PUNCT
ejde-673	343	34	)	)	PUNCT
ejde-673	343	35	dx	dx	PROPN
ejde-673	343	36	)	)	PUNCT
ejde-673	343	37	∫	∫	PROPN
ejde-673	344	1	ω	ω	NUM
ejde-673	344	2	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	344	3	)	)	PUNCT
ejde-673	344	4	)	)	PUNCT
ejde-673	344	5	·	·	SYM
ejde-673	344	6	∇v(x	∇v(x	NUM
ejde-673	344	7	)	)	PUNCT
ejde-673	344	8	dx	dx	PROPN
ejde-673	345	1	=	=	SYM
ejde-673	345	2	λ	λ	PROPN
ejde-673	345	3	∫	∫	PROPN
ejde-673	345	4	γ2	γ2	PROPN
ejde-673	345	5	g(x	g(x	PROPN
ejde-673	345	6	,	,	PUNCT
ejde-673	345	7	u(x))v(x)dσx	u(x))v(x)dσx	NOUN
ejde-673	345	8	(	(	PUNCT
ejde-673	345	9	3.5	3.5	NUM
ejde-673	345	10	)	)	PUNCT
ejde-673	345	11	for	for	ADP
ejde-673	345	12	all	all	PRON
ejde-673	345	13	v	v	ADP
ejde-673	345	14	∈	∈	PROPN
ejde-673	345	15	y	y	NOUN
ejde-673	345	16	.	.	PUNCT
ejde-673	346	1	(	(	PUNCT
ejde-673	346	2	ii	ii	NOUN
ejde-673	346	3	)	)	PUNCT
ejde-673	346	4	such	such	DET
ejde-673	346	5	a	a	DET
ejde-673	346	6	pair	pair	NOUN
ejde-673	346	7	(	(	PUNCT
ejde-673	346	8	u	u	NOUN
ejde-673	346	9	,	,	PUNCT
ejde-673	346	10	λ	λ	PROPN
ejde-673	346	11	)	)	PUNCT
ejde-673	346	12	∈	∈	PROPN
ejde-673	346	13	y	y	PROPN
ejde-673	346	14	×r	×r	VERB
ejde-673	346	15	with	with	ADP
ejde-673	346	16	u	u	NOUN
ejde-673	346	17	̸=	̸=	PROPN
ejde-673	346	18	0	0	NUM
ejde-673	346	19	is	be	AUX
ejde-673	346	20	called	call	VERB
ejde-673	346	21	an	an	DET
ejde-673	346	22	eigenpair	eigenpair	NOUN
ejde-673	346	23	,	,	PUNCT
ejde-673	346	24	λ	λ	PROPN
ejde-673	346	25	is	be	AUX
ejde-673	346	26	called	call	VERB
ejde-673	346	27	an	an	DET
ejde-673	346	28	eigenvalue	eigenvalue	NOUN
ejde-673	346	29	and	and	CCONJ
ejde-673	346	30	u	u	NOUN
ejde-673	346	31	is	be	AUX
ejde-673	346	32	called	call	VERB
ejde-673	346	33	an	an	DET
ejde-673	346	34	associated	associated	ADJ
ejde-673	346	35	eigenfunction	eigenfunction	NOUN
ejde-673	346	36	.	.	PUNCT
ejde-673	347	1	if	if	SCONJ
ejde-673	347	2	we	we	PRON
ejde-673	347	3	define	define	VERB
ejde-673	347	4	a	a	DET
ejde-673	347	5	function	function	NOUN
ejde-673	347	6	associated	associate	VERB
ejde-673	347	7	with	with	ADP
ejde-673	347	8	the	the	DET
ejde-673	347	9	function	function	NOUN
ejde-673	347	10	m	m	VERB
ejde-673	347	11	by	by	ADP
ejde-673	347	12	m̂(t	m̂(t	ADJ
ejde-673	347	13	)	)	PUNCT
ejde-673	347	14	=	=	SYM
ejde-673	347	15	∫	∫	PROPN
ejde-673	347	16	t	t	PROPN
ejde-673	347	17	0	0	NUM
ejde-673	347	18	m(s	m(s	PROPN
ejde-673	347	19	)	)	PUNCT
ejde-673	347	20	ds	ds	NOUN
ejde-673	347	21	for	for	ADP
ejde-673	347	22	t	t	PROPN
ejde-673	347	23	≥	≥	NUM
ejde-673	347	24	0	0	NUM
ejde-673	347	25	,	,	PUNCT
ejde-673	347	26	then	then	ADV
ejde-673	347	27	we	we	PRON
ejde-673	347	28	see	see	VERB
ejde-673	347	29	that	that	DET
ejde-673	347	30	m̂	m̂	PROPN
ejde-673	347	31	∈	∈	PROPN
ejde-673	347	32	c1([0,∞	c1([0,∞	NOUN
ejde-673	347	33	)	)	PUNCT
ejde-673	347	34	)	)	PUNCT
ejde-673	347	35	and	and	CCONJ
ejde-673	347	36	satisfies	satisfy	VERB
ejde-673	347	37	m0	m0	PROPN
ejde-673	347	38	l	l	PROPN
ejde-673	347	39	tl	tl	PROPN
ejde-673	347	40	≤	≤	NOUN
ejde-673	347	41	m̂(t	m̂(t	NOUN
ejde-673	347	42	)	)	PUNCT
ejde-673	347	43	≤	≤	NOUN
ejde-673	347	44	m1	m1	NOUN
ejde-673	347	45	(	(	PUNCT
ejde-673	347	46	t+	t+	ADP
ejde-673	347	47	1	1	NUM
ejde-673	347	48	k	k	PROPN
ejde-673	347	49	tk	tk	PROPN
ejde-673	347	50	)	)	PUNCT
ejde-673	347	51	for	for	ADP
ejde-673	347	52	t	t	PROPN
ejde-673	347	53	≥	≥	NOUN
ejde-673	347	54	0	0	NUM
ejde-673	347	55	.	.	PUNCT
ejde-673	348	1	(	(	PUNCT
ejde-673	348	2	3.6	3.6	NUM
ejde-673	348	3	)	)	PUNCT
ejde-673	348	4	moreover	moreover	ADV
ejde-673	348	5	,	,	PUNCT
ejde-673	348	6	since	since	SCONJ
ejde-673	348	7	m̂	m̂	PROPN
ejde-673	348	8	′(t	′(t	PART
ejde-673	348	9	)	)	PUNCT
ejde-673	348	10	=	=	SYM
ejde-673	348	11	m(t	m(t	NOUN
ejde-673	348	12	)	)	PUNCT
ejde-673	348	13	is	be	AUX
ejde-673	348	14	monotone	monotone	ADJ
ejde-673	348	15	non	non	ADJ
ejde-673	348	16	-	-	ADJ
ejde-673	348	17	decreasing	decrease	VERB
ejde-673	348	18	and	and	CCONJ
ejde-673	348	19	satisfies	satisfie	NOUN
ejde-673	348	20	(	(	PUNCT
ejde-673	348	21	1.3	1.3	NUM
ejde-673	348	22	)	)	PUNCT
ejde-673	348	23	,	,	PUNCT
ejde-673	348	24	m̂(t	m̂(t	NOUN
ejde-673	348	25	)	)	PUNCT
ejde-673	348	26	is	be	AUX
ejde-673	348	27	convex	convex	ADJ
ejde-673	348	28	and	and	CCONJ
ejde-673	348	29	strictly	strictly	ADV
ejde-673	348	30	monotone	monotone	ADJ
ejde-673	348	31	increasing	increase	VERB
ejde-673	348	32	on	on	ADP
ejde-673	348	33	[	[	X
ejde-673	348	34	0,∞	0,∞	NOUN
ejde-673	348	35	)	)	PUNCT
ejde-673	348	36	.	.	PUNCT
ejde-673	349	1	we	we	PRON
ejde-673	349	2	define	define	VERB
ejde-673	349	3	functionals	functional	NOUN
ejde-673	349	4	on	on	ADP
ejde-673	349	5	y	y	NOUN
ejde-673	349	6	by	by	ADP
ejde-673	349	7	φ(u	φ(u	NOUN
ejde-673	349	8	)	)	PUNCT
ejde-673	350	1	=	=	SYM
ejde-673	350	2	∫	∫	PROPN
ejde-673	350	3	ω	ω	NUM
ejde-673	350	4	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	350	5	)	)	PUNCT
ejde-673	350	6	)	)	PUNCT
ejde-673	350	7	dx	dx	PROPN
ejde-673	350	8	,	,	PUNCT
ejde-673	350	9	ψ(u	ψ(u	PROPN
ejde-673	350	10	)	)	PUNCT
ejde-673	350	11	=	=	SYM
ejde-673	350	12	m̂(φ(u	m̂(φ(u	NOUN
ejde-673	350	13	)	)	PUNCT
ejde-673	350	14	)	)	PUNCT
ejde-673	350	15	,	,	PUNCT
ejde-673	350	16	k(u	k(u	X
ejde-673	350	17	)	)	PUNCT
ejde-673	351	1	=	=	SYM
ejde-673	351	2	∫	∫	PROPN
ejde-673	351	3	γ2	γ2	PROPN
ejde-673	351	4	g(x	g(x	PROPN
ejde-673	351	5	,	,	PUNCT
ejde-673	351	6	u(x))dσx	u(x))dσx	NUM
ejde-673	351	7	(	(	PUNCT
ejde-673	351	8	3.7	3.7	NUM
ejde-673	351	9	)	)	PUNCT
ejde-673	351	10	for	for	ADP
ejde-673	351	11	u	u	PROPN
ejde-673	351	12	∈	∈	PROPN
ejde-673	351	13	y	y	PROPN
ejde-673	351	14	.	.	PUNCT
ejde-673	352	1	it	it	PRON
ejde-673	352	2	follows	follow	VERB
ejde-673	352	3	from	from	ADP
ejde-673	352	4	(	(	PUNCT
ejde-673	352	5	a7	a7	PROPN
ejde-673	352	6	)	)	PUNCT
ejde-673	352	7	and	and	CCONJ
ejde-673	352	8	(	(	PUNCT
ejde-673	352	9	a8	a8	PROPN
ejde-673	352	10	)	)	PUNCT
ejde-673	352	11	that	that	SCONJ
ejde-673	352	12	φ	φ	PROPN
ejde-673	352	13	,	,	PUNCT
ejde-673	352	14	ψ	ψ	PROPN
ejde-673	352	15	and	and	CCONJ
ejde-673	352	16	k	k	PROPN
ejde-673	352	17	are	be	AUX
ejde-673	352	18	even	even	ADV
ejde-673	352	19	functionals	functional	NOUN
ejde-673	352	20	,	,	PUNCT
ejde-673	352	21	that	that	ADV
ejde-673	352	22	is	is	ADV
ejde-673	352	23	,	,	PUNCT
ejde-673	352	24	φ(−u	φ(−u	ADV
ejde-673	352	25	)	)	PUNCT
ejde-673	352	26	=	=	SYM
ejde-673	352	27	φ(u	φ(u	NOUN
ejde-673	352	28	)	)	PUNCT
ejde-673	352	29	,	,	PUNCT
ejde-673	352	30	ψ(−u	ψ(−u	NOUN
ejde-673	352	31	)	)	PUNCT
ejde-673	352	32	=	=	SYM
ejde-673	352	33	ψ(u	ψ(u	PROPN
ejde-673	352	34	)	)	PUNCT
ejde-673	352	35	and	and	CCONJ
ejde-673	352	36	k(−u	k(−u	X
ejde-673	352	37	)	)	PUNCT
ejde-673	352	38	=	=	SYM
ejde-673	352	39	k(u	k(u	X
ejde-673	352	40	)	)	PUNCT
ejde-673	352	41	for	for	ADP
ejde-673	352	42	all	all	PRON
ejde-673	352	43	u	u	PROPN
ejde-673	352	44	∈	∈	PROPN
ejde-673	352	45	y	y	PROPN
ejde-673	352	46	.	.	PUNCT
ejde-673	353	1	12	12	NUM
ejde-673	353	2	j.	j.	PROPN
ejde-673	353	3	aramaki	aramaki	PROPN
ejde-673	353	4	ejde-2025/17	ejde-2025/17	PROPN
ejde-673	353	5	lemma	lemma	PROPN
ejde-673	353	6	3.7	3.7	NUM
ejde-673	353	7	.	.	PUNCT
ejde-673	354	1	(	(	PUNCT
ejde-673	354	2	i	i	NOUN
ejde-673	354	3	)	)	PUNCT
ejde-673	354	4	we	we	PRON
ejde-673	354	5	have	have	VERB
ejde-673	354	6	k0	k0	PROPN
ejde-673	354	7	p+	p+	PART
ejde-673	354	8	ρ̃(p(·),h1(·))(u	ρ̃(p(·),h1(·))(u	PROPN
ejde-673	354	9	)	)	PUNCT
ejde-673	354	10	≤	≤	NOUN
ejde-673	354	11	φ(u	φ(u	NOUN
ejde-673	354	12	)	)	PUNCT
ejde-673	354	13	≤	≤	NUM
ejde-673	354	14	c(2∥h0∥lp′(·)(ω)∥∇u∥lp(·)(ω	c(2∥h0∥lp′(·)(ω)∥∇u∥lp(·)(ω	NOUN
ejde-673	354	15	)	)	PUNCT
ejde-673	355	1	+	+	CCONJ
ejde-673	356	1	ρ̃(p(·),h1(·))(u	ρ̃(p(·),h1(·))(u	NOUN
ejde-673	356	2	)	)	PUNCT
ejde-673	356	3	)	)	PUNCT
ejde-673	356	4	for	for	ADP
ejde-673	356	5	u	u	PROPN
ejde-673	356	6	∈	∈	PROPN
ejde-673	356	7	y	y	PROPN
ejde-673	356	8	,	,	PUNCT
ejde-673	356	9	where	where	SCONJ
ejde-673	356	10	c	c	PROPN
ejde-673	356	11	and	and	CCONJ
ejde-673	356	12	k0	k0	PROPN
ejde-673	356	13	are	be	AUX
ejde-673	356	14	the	the	DET
ejde-673	356	15	constants	constant	NOUN
ejde-673	356	16	in	in	ADP
ejde-673	356	17	(	(	PUNCT
ejde-673	356	18	a.1	a.1	NOUN
ejde-673	356	19	)	)	PUNCT
ejde-673	356	20	and	and	CCONJ
ejde-673	356	21	(	(	PUNCT
ejde-673	356	22	a5	a5	PROPN
ejde-673	356	23	)	)	PUNCT
ejde-673	356	24	.	.	PUNCT
ejde-673	357	1	(	(	PUNCT
ejde-673	357	2	ii	ii	X
ejde-673	357	3	)	)	PUNCT
ejde-673	357	4	we	we	PRON
ejde-673	357	5	have	have	VERB
ejde-673	357	6	φ	φ	PROPN
ejde-673	357	7	(	(	PUNCT
ejde-673	357	8	u+	u+	NOUN
ejde-673	357	9	v	v	ADP
ejde-673	357	10	2	2	NUM
ejde-673	357	11	)	)	PUNCT
ejde-673	357	12	+	+	CCONJ
ejde-673	358	1	k1ρ̃(p(·),h1(·))(u−	k1ρ̃(p(·),h1(·))(u−	PRON
ejde-673	358	2	v	v	NOUN
ejde-673	358	3	)	)	PUNCT
ejde-673	358	4	≤	≤	NUM
ejde-673	358	5	1	1	NUM
ejde-673	358	6	2	2	NUM
ejde-673	358	7	φ(u	φ(u	NOUN
ejde-673	358	8	)	)	PUNCT
ejde-673	358	9	+	+	CCONJ
ejde-673	358	10	1	1	NUM
ejde-673	358	11	2	2	NUM
ejde-673	358	12	φ(v	φ(v	NOUN
ejde-673	358	13	)	)	PUNCT
ejde-673	358	14	for	for	ADP
ejde-673	358	15	all	all	DET
ejde-673	358	16	u	u	NOUN
ejde-673	358	17	,	,	PUNCT
ejde-673	358	18	v	v	NOUN
ejde-673	358	19	∈	∈	PROPN
ejde-673	358	20	y	y	NOUN
ejde-673	358	21	,	,	PUNCT
ejde-673	358	22	where	where	SCONJ
ejde-673	358	23	k1	k1	PROPN
ejde-673	358	24	is	be	AUX
ejde-673	358	25	the	the	DET
ejde-673	358	26	constant	constant	ADJ
ejde-673	358	27	in	in	ADP
ejde-673	358	28	(	(	PUNCT
ejde-673	358	29	a4	a4	NOUN
ejde-673	358	30	)	)	PUNCT
ejde-673	358	31	.	.	PUNCT
ejde-673	359	1	in	in	ADP
ejde-673	359	2	particular	particular	ADJ
ejde-673	359	3	,	,	PUNCT
ejde-673	359	4	φ	φ	PROPN
ejde-673	359	5	is	be	AUX
ejde-673	359	6	convex	convex	PROPN
ejde-673	359	7	,	,	PUNCT
ejde-673	359	8	that	that	ADV
ejde-673	359	9	is	is	ADV
ejde-673	359	10	,	,	PUNCT
ejde-673	359	11	φ((1−	φ((1−	NUM
ejde-673	359	12	τ)u+	τ)u+	X
ejde-673	359	13	τv	τv	NOUN
ejde-673	359	14	)	)	PUNCT
ejde-673	359	15	≤	≤	NOUN
ejde-673	359	16	(	(	PUNCT
ejde-673	359	17	1−	1−	NUM
ejde-673	359	18	τ)φ(u	τ)φ(u	NOUN
ejde-673	359	19	)	)	PUNCT
ejde-673	359	20	+	+	NUM
ejde-673	359	21	τφ(v	τφ(v	NOUN
ejde-673	359	22	)	)	PUNCT
ejde-673	359	23	for	for	ADP
ejde-673	359	24	all	all	DET
ejde-673	359	25	u	u	NOUN
ejde-673	359	26	,	,	PUNCT
ejde-673	359	27	v	v	NOUN
ejde-673	359	28	∈	∈	PROPN
ejde-673	359	29	y	y	PROPN
ejde-673	359	30	and	and	CCONJ
ejde-673	359	31	τ	τ	PROPN
ejde-673	359	32	∈	∈	PROPN
ejde-673	360	1	[	[	X
ejde-673	360	2	0	0	NUM
ejde-673	360	3	,	,	PUNCT
ejde-673	360	4	1	1	NUM
ejde-673	360	5	]	]	PUNCT
ejde-673	360	6	.	.	PUNCT
ejde-673	361	1	proof	proof	NOUN
ejde-673	361	2	.	.	PUNCT
ejde-673	362	1	(	(	PUNCT
ejde-673	362	2	i	i	NOUN
ejde-673	362	3	)	)	PUNCT
ejde-673	362	4	easily	easily	ADV
ejde-673	362	5	follows	follow	VERB
ejde-673	362	6	from	from	ADP
ejde-673	362	7	(	(	PUNCT
ejde-673	362	8	a5	a5	PROPN
ejde-673	362	9	)	)	PUNCT
ejde-673	362	10	and	and	CCONJ
ejde-673	362	11	the	the	DET
ejde-673	362	12	hölder	hölder	NOUN
ejde-673	362	13	inequality	inequality	NOUN
ejde-673	362	14	(	(	PUNCT
ejde-673	362	15	proposition	proposition	NOUN
ejde-673	362	16	2.2	2.2	NUM
ejde-673	362	17	)	)	PUNCT
ejde-673	362	18	.	.	PUNCT
ejde-673	363	1	(	(	PUNCT
ejde-673	363	2	ii	ii	NOUN
ejde-673	363	3	)	)	PUNCT
ejde-673	363	4	easily	easily	ADV
ejde-673	363	5	follows	follow	VERB
ejde-673	363	6	from	from	ADP
ejde-673	363	7	(	(	PUNCT
ejde-673	363	8	a4	a4	NOUN
ejde-673	363	9	)	)	PUNCT
ejde-673	363	10	and	and	CCONJ
ejde-673	363	11	the	the	DET
ejde-673	363	12	continuity	continuity	NOUN
ejde-673	363	13	of	of	ADP
ejde-673	363	14	a(x	a(x	PROPN
ejde-673	363	15	,	,	PUNCT
ejde-673	363	16	ξ	ξ	NOUN
ejde-673	363	17	)	)	PUNCT
ejde-673	363	18	with	with	ADP
ejde-673	363	19	respect	respect	NOUN
ejde-673	363	20	to	to	ADP
ejde-673	363	21	ξ	ξ	PROPN
ejde-673	363	22	.	.	PUNCT
ejde-673	364	1	□	□	PUNCT
ejde-673	364	2	the	the	DET
ejde-673	364	3	functional	functional	ADJ
ejde-673	364	4	ψ	ψ	X
ejde-673	364	5	defined	define	VERB
ejde-673	364	6	by	by	ADP
ejde-673	364	7	(	(	PUNCT
ejde-673	364	8	3.7	3.7	NUM
ejde-673	364	9	)	)	PUNCT
ejde-673	364	10	is	be	AUX
ejde-673	364	11	a	a	DET
ejde-673	364	12	continuous	continuous	ADJ
ejde-673	364	13	modular	modular	NOUN
ejde-673	364	14	on	on	ADP
ejde-673	364	15	a	a	DET
ejde-673	364	16	real	real	ADJ
ejde-673	364	17	banach	banach	NOUN
ejde-673	364	18	space	space	NOUN
ejde-673	364	19	y	y	NOUN
ejde-673	364	20	in	in	ADP
ejde-673	364	21	the	the	DET
ejde-673	364	22	sense	sense	NOUN
ejde-673	364	23	of	of	ADP
ejde-673	364	24	[	[	X
ejde-673	364	25	13	13	NUM
ejde-673	364	26	,	,	PUNCT
ejde-673	364	27	definition	definition	NOUN
ejde-673	364	28	2.1.11	2.1.11	NUM
ejde-673	364	29	]	]	PUNCT
ejde-673	364	30	,	,	PUNCT
ejde-673	364	31	that	that	ADV
ejde-673	364	32	is	is	ADV
ejde-673	364	33	,	,	PUNCT
ejde-673	364	34	ψ	ψ	X
ejde-673	364	35	has	have	VERB
ejde-673	364	36	the	the	DET
ejde-673	364	37	following	follow	VERB
ejde-673	364	38	properties	property	NOUN
ejde-673	364	39	:	:	PUNCT
ejde-673	364	40	(	(	PUNCT
ejde-673	364	41	a	a	X
ejde-673	364	42	)	)	PUNCT
ejde-673	364	43	ψ(0	ψ(0	NOUN
ejde-673	364	44	)	)	PUNCT
ejde-673	364	45	=	=	SYM
ejde-673	365	1	0	0	X
ejde-673	365	2	.	.	PUNCT
ejde-673	366	1	this	this	PRON
ejde-673	366	2	easily	easily	ADV
ejde-673	366	3	follows	follow	VERB
ejde-673	366	4	from	from	ADP
ejde-673	366	5	a(x,0	a(x,0	PROPN
ejde-673	366	6	)	)	PUNCT
ejde-673	367	1	=	=	SYM
ejde-673	367	2	0	0	NUM
ejde-673	368	1	and	and	CCONJ
ejde-673	368	2	the	the	DET
ejde-673	368	3	definition	definition	NOUN
ejde-673	368	4	of	of	ADP
ejde-673	368	5	m̂	m̂	PROPN
ejde-673	368	6	.	.	PUNCT
ejde-673	369	1	(	(	PUNCT
ejde-673	369	2	b	b	NOUN
ejde-673	369	3	)	)	PUNCT
ejde-673	369	4	ψ(−u	ψ(−u	NOUN
ejde-673	369	5	)	)	PUNCT
ejde-673	370	1	=	=	SYM
ejde-673	370	2	ψ(u	ψ(u	PROPN
ejde-673	370	3	)	)	PUNCT
ejde-673	370	4	for	for	ADP
ejde-673	370	5	every	every	DET
ejde-673	370	6	u	u	PROPN
ejde-673	370	7	∈	∈	PROPN
ejde-673	370	8	y	y	PROPN
ejde-673	370	9	.	.	PUNCT
ejde-673	371	1	this	this	PRON
ejde-673	371	2	follows	follow	VERB
ejde-673	371	3	from	from	ADP
ejde-673	371	4	(	(	PUNCT
ejde-673	371	5	a7	a7	PROPN
ejde-673	371	6	)	)	PUNCT
ejde-673	371	7	.	.	PUNCT
ejde-673	372	1	(	(	PUNCT
ejde-673	372	2	c	c	X
ejde-673	372	3	)	)	PUNCT
ejde-673	372	4	ψ	ψ	NOUN
ejde-673	372	5	is	be	AUX
ejde-673	372	6	convex	convex	ADJ
ejde-673	372	7	.	.	PUNCT
ejde-673	373	1	indeed	indeed	ADV
ejde-673	373	2	,	,	PUNCT
ejde-673	373	3	since	since	SCONJ
ejde-673	373	4	m̂	m̂	PROPN
ejde-673	373	5	is	be	AUX
ejde-673	373	6	convex	convex	ADJ
ejde-673	373	7	and	and	CCONJ
ejde-673	373	8	strictly	strictly	ADV
ejde-673	373	9	monotone	monotone	ADJ
ejde-673	373	10	increasing	increasing	NOUN
ejde-673	373	11	and	and	CCONJ
ejde-673	373	12	φ	φ	PROPN
ejde-673	373	13	is	be	AUX
ejde-673	373	14	convex	convex	PROPN
ejde-673	373	15	,	,	PUNCT
ejde-673	373	16	for	for	ADP
ejde-673	373	17	any	any	DET
ejde-673	373	18	u	u	NOUN
ejde-673	373	19	,	,	PUNCT
ejde-673	373	20	v	v	PROPN
ejde-673	373	21	∈	∈	PROPN
ejde-673	373	22	y	y	PROPN
ejde-673	373	23	and	and	CCONJ
ejde-673	373	24	τ	τ	PROPN
ejde-673	373	25	∈	∈	PROPN
ejde-673	374	1	[	[	X
ejde-673	374	2	0	0	NUM
ejde-673	374	3	,	,	PUNCT
ejde-673	374	4	1	1	NUM
ejde-673	374	5	]	]	PUNCT
ejde-673	374	6	we	we	PRON
ejde-673	374	7	have	have	VERB
ejde-673	374	8	ψ((1−	ψ((1−	X
ejde-673	374	9	τ)u+	τ)u+	X
ejde-673	374	10	τv	τv	NUM
ejde-673	374	11	)	)	PUNCT
ejde-673	374	12	=	=	SYM
ejde-673	374	13	m̂(φ((1−	m̂(φ((1−	X
ejde-673	374	14	τ)u+	τ)u+	X
ejde-673	374	15	τv	τv	NOUN
ejde-673	374	16	)	)	PUNCT
ejde-673	374	17	)	)	PUNCT
ejde-673	374	18	≤	≤	NOUN
ejde-673	375	1	m̂((1−	m̂((1−	PROPN
ejde-673	375	2	τ)φ(u	τ)φ(u	NOUN
ejde-673	375	3	)	)	PUNCT
ejde-673	375	4	+	+	NUM
ejde-673	375	5	τφ(v	τφ(v	NOUN
ejde-673	375	6	)	)	PUNCT
ejde-673	375	7	)	)	PUNCT
ejde-673	376	1	≤	≤	NOUN
ejde-673	376	2	(	(	PUNCT
ejde-673	376	3	1−	1−	NUM
ejde-673	376	4	τ)ψ(u	τ)ψ(u	NOUN
ejde-673	376	5	)	)	PUNCT
ejde-673	376	6	+	+	NUM
ejde-673	376	7	τψ(v	τψ(v	NUM
ejde-673	376	8	)	)	PUNCT
ejde-673	376	9	.	.	PUNCT
ejde-673	377	1	(	(	PUNCT
ejde-673	377	2	d	d	X
ejde-673	377	3	)	)	PUNCT
ejde-673	377	4	the	the	DET
ejde-673	377	5	function	function	NOUN
ejde-673	377	6	[	[	X
ejde-673	377	7	0,∞	0,∞	NOUN
ejde-673	377	8	)	)	PUNCT
ejde-673	377	9	∋	∋	NOUN
ejde-673	377	10	λ	λ	PROPN
ejde-673	377	11	7→	7→	PROPN
ejde-673	377	12	ψ(λu	ψ(λu	PROPN
ejde-673	377	13	)	)	PUNCT
ejde-673	377	14	is	be	AUX
ejde-673	377	15	continuous	continuous	ADJ
ejde-673	377	16	for	for	ADP
ejde-673	377	17	every	every	DET
ejde-673	377	18	u	u	PROPN
ejde-673	377	19	∈	∈	PROPN
ejde-673	377	20	y	y	PROPN
ejde-673	377	21	.	.	PUNCT
ejde-673	378	1	indeed	indeed	ADV
ejde-673	378	2	,	,	PUNCT
ejde-673	378	3	let	let	VERB
ejde-673	378	4	[	[	PRON
ejde-673	378	5	0,∞	0,∞	NUM
ejde-673	378	6	)	)	PUNCT
ejde-673	378	7	∋	∋	NOUN
ejde-673	378	8	λn	λn	NOUN
ejde-673	378	9	→	→	NOUN
ejde-673	378	10	λ0	λ0	NOUN
ejde-673	378	11	as	as	ADP
ejde-673	378	12	n	n	NOUN
ejde-673	378	13	→	→	SYM
ejde-673	378	14	∞.	∞.	PROPN
ejde-673	378	15	here	here	ADV
ejde-673	378	16	we	we	PRON
ejde-673	378	17	can	can	AUX
ejde-673	378	18	assume	assume	VERB
ejde-673	378	19	that	that	SCONJ
ejde-673	378	20	0	0	NUM
ejde-673	378	21	≤	≤	NUM
ejde-673	378	22	λn	λn	NOUN
ejde-673	378	23	≤	≤	NOUN
ejde-673	378	24	λ0	λ0	NOUN
ejde-673	378	25	+1	+1	NOUN
ejde-673	378	26	for	for	ADP
ejde-673	378	27	large	large	ADJ
ejde-673	378	28	n	n	CCONJ
ejde-673	378	29	∈	∈	PROPN
ejde-673	378	30	n.	n.	NOUN
ejde-673	378	31	from	from	ADP
ejde-673	378	32	(	(	PUNCT
ejde-673	378	33	a.0	a.0	ADJ
ejde-673	378	34	)	)	PUNCT
ejde-673	378	35	and	and	CCONJ
ejde-673	378	36	(	(	PUNCT
ejde-673	378	37	a5	a5	PROPN
ejde-673	378	38	)	)	PUNCT
ejde-673	378	39	,	,	PUNCT
ejde-673	378	40	we	we	PRON
ejde-673	378	41	have	have	VERB
ejde-673	378	42	|a(x	|a(x	NOUN
ejde-673	378	43	,	,	PUNCT
ejde-673	378	44	λn∇u(x))|	λn∇u(x))|	X
ejde-673	378	45	≤	≤	X
ejde-673	378	46	c(λ0	c(λ0	NOUN
ejde-673	378	47	+	+	CCONJ
ejde-673	378	48	1)h0(x)|∇u(x)|+	1)h0(x)|∇u(x)|+	NUM
ejde-673	378	49	c(λ0	c(λ0	NOUN
ejde-673	379	1	+	+	CCONJ
ejde-673	379	2	1)p	1)p	NUM
ejde-673	379	3	+	+	NUM
ejde-673	379	4	h1(x)|∇u(x)|p(x	h1(x)|∇u(x)|p(x	PROPN
ejde-673	379	5	)	)	PUNCT
ejde-673	379	6	.	.	PUNCT
ejde-673	380	1	since	since	SCONJ
ejde-673	380	2	h0	h0	PROPN
ejde-673	380	3	∈	∈	PROPN
ejde-673	380	4	lp′(·)(ω	lp′(·)(ω	NOUN
ejde-673	380	5	)	)	PUNCT
ejde-673	380	6	and	and	CCONJ
ejde-673	380	7	|∇u(·)|	|∇u(·)|	NUM
ejde-673	380	8	∈	∈	PROPN
ejde-673	380	9	lp(·)(ω	lp(·)(ω	NOUN
ejde-673	380	10	)	)	PUNCT
ejde-673	380	11	and	and	CCONJ
ejde-673	380	12	u	u	PROPN
ejde-673	380	13	∈	∈	PROPN
ejde-673	380	14	y	y	PROPN
ejde-673	380	15	,	,	PUNCT
ejde-673	380	16	the	the	DET
ejde-673	380	17	right	right	ADJ
ejde-673	380	18	-	-	PUNCT
ejde-673	380	19	hand	hand	NOUN
ejde-673	380	20	side	side	NOUN
ejde-673	380	21	in	in	ADP
ejde-673	380	22	the	the	DET
ejde-673	380	23	above	above	ADJ
ejde-673	380	24	inequality	inequality	NOUN
ejde-673	380	25	is	be	AUX
ejde-673	380	26	an	an	DET
ejde-673	380	27	integrable	integrable	ADJ
ejde-673	380	28	function	function	NOUN
ejde-673	380	29	independent	independent	ADJ
ejde-673	380	30	of	of	ADP
ejde-673	380	31	n.	n.	NOUN
ejde-673	380	32	clearly	clearly	ADV
ejde-673	380	33	,	,	PUNCT
ejde-673	380	34	we	we	PRON
ejde-673	380	35	see	see	VERB
ejde-673	380	36	that	that	SCONJ
ejde-673	380	37	a(x	a(x	NOUN
ejde-673	380	38	,	,	PUNCT
ejde-673	380	39	λn∇u(x	λn∇u(x	NOUN
ejde-673	380	40	)	)	PUNCT
ejde-673	380	41	)	)	PUNCT
ejde-673	380	42	→	→	SYM
ejde-673	380	43	a(x	a(x	NOUN
ejde-673	380	44	,	,	PUNCT
ejde-673	380	45	λ0∇u(x	λ0∇u(x	NOUN
ejde-673	380	46	)	)	PUNCT
ejde-673	380	47	)	)	PUNCT
ejde-673	380	48	as	as	ADP
ejde-673	380	49	n	n	PROPN
ejde-673	380	50	→	→	SYM
ejde-673	380	51	∞	∞	PROPN
ejde-673	380	52	for	for	ADP
ejde-673	380	53	a.e	a.e	PROPN
ejde-673	380	54	.	.	PUNCT
ejde-673	380	55	x	x	SYM
ejde-673	380	56	∈	∈	PROPN
ejde-673	380	57	ω	ω	X
ejde-673	380	58	.	.	PUNCT
ejde-673	381	1	by	by	ADP
ejde-673	381	2	the	the	DET
ejde-673	381	3	lebesgue	lebesgue	NOUN
ejde-673	381	4	dominated	dominate	VERB
ejde-673	381	5	convergent	convergent	NOUN
ejde-673	381	6	theorem	theorem	VERB
ejde-673	381	7	,	,	PUNCT
ejde-673	381	8	we	we	PRON
ejde-673	381	9	see	see	VERB
ejde-673	381	10	that	that	PRON
ejde-673	381	11	φ(λnu	φ(λnu	VERB
ejde-673	381	12	)	)	PUNCT
ejde-673	381	13	→	→	SYM
ejde-673	381	14	φ(λ0u	φ(λ0u	X
ejde-673	381	15	)	)	PUNCT
ejde-673	381	16	as	as	ADP
ejde-673	381	17	n	n	PROPN
ejde-673	381	18	→	→	SYM
ejde-673	381	19	∞	∞	PROPN
ejde-673	381	20	,	,	PUNCT
ejde-673	381	21	so	so	SCONJ
ejde-673	381	22	ψ(λnu	ψ(λnu	PROPN
ejde-673	381	23	)	)	PUNCT
ejde-673	381	24	→	→	PUNCT
ejde-673	381	25	ψ(λ0u	ψ(λ0u	ADV
ejde-673	381	26	)	)	PUNCT
ejde-673	381	27	.	.	PUNCT
ejde-673	382	1	(	(	PUNCT
ejde-673	382	2	e	e	X
ejde-673	382	3	)	)	PUNCT
ejde-673	382	4	ψ(u	ψ(u	PROPN
ejde-673	382	5	)	)	PUNCT
ejde-673	383	1	=	=	SYM
ejde-673	383	2	0	0	NUM
ejde-673	383	3	implies	imply	VERB
ejde-673	383	4	u	u	NOUN
ejde-673	383	5	=	=	PROPN
ejde-673	383	6	0	0	NUM
ejde-673	383	7	.	.	PUNCT
ejde-673	384	1	indeed	indeed	ADV
ejde-673	384	2	,	,	PUNCT
ejde-673	384	3	if	if	SCONJ
ejde-673	384	4	ψ(u	ψ(u	PRON
ejde-673	384	5	)	)	PUNCT
ejde-673	384	6	=	=	PUNCT
ejde-673	384	7	0	0	NUM
ejde-673	384	8	,	,	PUNCT
ejde-673	384	9	then	then	ADV
ejde-673	384	10	φ(u	φ(u	NOUN
ejde-673	384	11	)	)	PUNCT
ejde-673	384	12	=	=	SYM
ejde-673	384	13	0	0	X
ejde-673	384	14	.	.	PUNCT
ejde-673	385	1	hence	hence	ADV
ejde-673	385	2	it	it	PRON
ejde-673	385	3	follows	follow	VERB
ejde-673	385	4	from	from	ADP
ejde-673	385	5	(	(	PUNCT
ejde-673	385	6	a5	a5	PROPN
ejde-673	385	7	)	)	PUNCT
ejde-673	385	8	and	and	CCONJ
ejde-673	385	9	the	the	DET
ejde-673	385	10	poincaré-type	poincaré-type	NOUN
ejde-673	385	11	inequality	inequality	NOUN
ejde-673	385	12	(	(	PUNCT
ejde-673	385	13	proposition	proposition	NOUN
ejde-673	385	14	2.14	2.14	NUM
ejde-673	385	15	)	)	PUNCT
ejde-673	385	16	that	that	SCONJ
ejde-673	385	17	u	u	NOUN
ejde-673	385	18	=	=	NOUN
ejde-673	385	19	0	0	PROPN
ejde-673	385	20	.	.	PUNCT
ejde-673	385	21	.	.	PUNCT
ejde-673	386	1	thus	thus	ADV
ejde-673	386	2	we	we	PRON
ejde-673	386	3	can	can	AUX
ejde-673	386	4	define	define	VERB
ejde-673	386	5	a	a	DET
ejde-673	386	6	modular	modular	ADJ
ejde-673	386	7	space	space	NOUN
ejde-673	386	8	yψ	yψ	ADV
ejde-673	386	9	=	=	PUNCT
ejde-673	386	10	{	{	PUNCT
ejde-673	386	11	u	u	NOUN
ejde-673	386	12	∈	∈	PROPN
ejde-673	386	13	y	y	PROPN
ejde-673	386	14	;	;	PUNCT
ejde-673	386	15	lim	lim	PROPN
ejde-673	386	16	τ→0	τ→0	PUNCT
ejde-673	386	17	ψ(τu	ψ(τu	PROPN
ejde-673	386	18	)	)	PUNCT
ejde-673	386	19	=	=	SYM
ejde-673	386	20	0	0	X
ejde-673	386	21	}	}	PUNCT
ejde-673	386	22	=	=	SYM
ejde-673	386	23	{	{	PUNCT
ejde-673	386	24	u	u	NOUN
ejde-673	386	25	∈	∈	PROPN
ejde-673	386	26	y	y	PROPN
ejde-673	386	27	;	;	PUNCT
ejde-673	386	28	ψ(τu	ψ(τu	NOUN
ejde-673	386	29	)	)	PUNCT
ejde-673	386	30	<	<	X
ejde-673	387	1	∞	∞	PROPN
ejde-673	388	1	for	for	ADP
ejde-673	388	2	some	some	DET
ejde-673	388	3	τ	τ	PROPN
ejde-673	388	4	>	>	X
ejde-673	388	5	0	0	NUM
ejde-673	388	6	}	}	PUNCT
ejde-673	388	7	with	with	ADP
ejde-673	388	8	the	the	DET
ejde-673	388	9	luxemburg	luxemburg	PROPN
ejde-673	388	10	norm	norm	NOUN
ejde-673	388	11	∥u∥ψ	∥u∥ψ	PROPN
ejde-673	388	12	=	=	SYM
ejde-673	388	13	inf	inf	PROPN
ejde-673	388	14	{	{	PUNCT
ejde-673	388	15	τ	τ	PROPN
ejde-673	388	16	>	>	X
ejde-673	388	17	0;ψ	0;ψ	NUM
ejde-673	388	18	(	(	PUNCT
ejde-673	388	19	u	u	NOUN
ejde-673	388	20	τ	τ	PROPN
ejde-673	388	21	)	)	PUNCT
ejde-673	388	22	≤	≤	NUM
ejde-673	388	23	1	1	NUM
ejde-673	388	24	}	}	PUNCT
ejde-673	388	25	for	for	ADP
ejde-673	388	26	u	u	PROPN
ejde-673	388	27	∈	∈	PROPN
ejde-673	388	28	yφ	yφ	PROPN
ejde-673	388	29	.	.	PUNCT
ejde-673	389	1	then	then	ADV
ejde-673	389	2	(	(	PUNCT
ejde-673	389	3	yψ	yψ	NOUN
ejde-673	389	4	,	,	PUNCT
ejde-673	389	5	∥	∥	PROPN
ejde-673	389	6	·	·	PUNCT
ejde-673	389	7	∥ψ	∥ψ	NOUN
ejde-673	389	8	)	)	PUNCT
ejde-673	389	9	is	be	AUX
ejde-673	389	10	a	a	DET
ejde-673	389	11	normed	normed	ADJ
ejde-673	389	12	linear	linear	ADJ
ejde-673	389	13	space	space	NOUN
ejde-673	389	14	over	over	ADP
ejde-673	389	15	r	r	NOUN
ejde-673	389	16	from	from	ADP
ejde-673	389	17	[	[	X
ejde-673	389	18	13	13	NUM
ejde-673	389	19	,	,	PUNCT
ejde-673	389	20	theorem	theorem	VERB
ejde-673	389	21	2.1.7	2.1.7	NUM
ejde-673	389	22	]	]	PUNCT
ejde-673	389	23	.	.	PUNCT
ejde-673	390	1	clearly	clearly	ADV
ejde-673	390	2	we	we	PRON
ejde-673	390	3	see	see	VERB
ejde-673	390	4	that	that	PRON
ejde-673	390	5	yψ	yψ	ADP
ejde-673	390	6	=	=	SYM
ejde-673	390	7	y	y	PROPN
ejde-673	390	8	,	,	PUNCT
ejde-673	390	9	and	and	CCONJ
ejde-673	390	10	the	the	DET
ejde-673	390	11	norms	norm	NOUN
ejde-673	390	12	∥	∥	X
ejde-673	390	13	·	·	PUNCT
ejde-673	390	14	∥ψ	∥ψ	NOUN
ejde-673	390	15	and	and	CCONJ
ejde-673	390	16	∥	∥	NOUN
ejde-673	390	17	·	·	PUNCT
ejde-673	391	1	∥y	∥y	NOUN
ejde-673	391	2	are	be	AUX
ejde-673	391	3	equivalent	equivalent	ADJ
ejde-673	391	4	(	(	PUNCT
ejde-673	391	5	cf	cf	NOUN
ejde-673	391	6	.	.	PUNCT
ejde-673	392	1	[	[	X
ejde-673	392	2	8	8	NUM
ejde-673	392	3	,	,	PUNCT
ejde-673	392	4	lemma	lemma	PROPN
ejde-673	392	5	4.3	4.3	NUM
ejde-673	392	6	]	]	NOUN
ejde-673	392	7	)	)	PUNCT
ejde-673	392	8	.	.	PUNCT
ejde-673	393	1	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	393	2	eigenvalue	eigenvalue	VERB
ejde-673	393	3	problems	problem	NOUN
ejde-673	393	4	for	for	ADP
ejde-673	393	5	kirchhoff	kirchhoff	NOUN
ejde-673	393	6	-	-	PUNCT
ejde-673	393	7	type	type	NOUN
ejde-673	393	8	equations	equation	NOUN
ejde-673	393	9	13	13	NUM
ejde-673	393	10	from	from	ADP
ejde-673	393	11	now	now	ADV
ejde-673	393	12	on	on	ADV
ejde-673	393	13	,	,	PUNCT
ejde-673	393	14	we	we	PRON
ejde-673	393	15	denote	denote	VERB
ejde-673	393	16	a	a	DET
ejde-673	393	17	∨	∨	NOUN
ejde-673	393	18	b	b	NOUN
ejde-673	393	19	=	=	SYM
ejde-673	393	20	max{a	max{a	PROPN
ejde-673	393	21	,	,	PUNCT
ejde-673	393	22	b	b	NOUN
ejde-673	393	23	}	}	PUNCT
ejde-673	393	24	and	and	CCONJ
ejde-673	393	25	a	a	DET
ejde-673	393	26	∧	∧	PROPN
ejde-673	393	27	b	b	NOUN
ejde-673	393	28	=	=	SYM
ejde-673	393	29	min{a	min{a	PROPN
ejde-673	393	30	,	,	PUNCT
ejde-673	393	31	b	b	NOUN
ejde-673	393	32	}	}	PUNCT
ejde-673	393	33	for	for	ADP
ejde-673	393	34	any	any	DET
ejde-673	393	35	real	real	ADJ
ejde-673	393	36	numbers	number	NOUN
ejde-673	393	37	a	a	PRON
ejde-673	393	38	and	and	CCONJ
ejde-673	393	39	b.	b.	PROPN
ejde-673	393	40	since	since	SCONJ
ejde-673	393	41	φ(u	φ(u	NOUN
ejde-673	393	42	)	)	PUNCT
ejde-673	393	43	≥	≥	PROPN
ejde-673	393	44	k0	k0	PROPN
ejde-673	393	45	p+	p+	PROPN
ejde-673	393	46	∫	∫	PROPN
ejde-673	393	47	ω	ω	PROPN
ejde-673	393	48	h1(x)|∇u(x)|p(x	h1(x)|∇u(x)|p(x	PROPN
ejde-673	393	49	)	)	PUNCT
ejde-673	393	50	dx	dx	PROPN
ejde-673	393	51	≥	≥	PROPN
ejde-673	393	52	k0	k0	PROPN
ejde-673	393	53	p+	p+	PROPN
ejde-673	393	54	(	(	PUNCT
ejde-673	393	55	∥u∥p	∥u∥p	NOUN
ejde-673	393	56	+	+	CCONJ
ejde-673	393	57	y	y	PROPN
ejde-673	393	58	∧	∧	PROPN
ejde-673	393	59	∥u∥p	∥u∥p	NOUN
ejde-673	393	60	−	−	PROPN
ejde-673	393	61	y	y	PROPN
ejde-673	393	62	)	)	PUNCT
ejde-673	393	63	,	,	PUNCT
ejde-673	393	64	it	it	PRON
ejde-673	393	65	follows	follow	VERB
ejde-673	393	66	from	from	ADP
ejde-673	393	67	(	(	PUNCT
ejde-673	393	68	3.6	3.6	NUM
ejde-673	393	69	)	)	PUNCT
ejde-673	393	70	that	that	PRON
ejde-673	393	71	ψ(u	ψ(u	PRON
ejde-673	393	72	)	)	PUNCT
ejde-673	393	73	=	=	SYM
ejde-673	393	74	m̂(φ(u	m̂(φ(u	NOUN
ejde-673	393	75	)	)	PUNCT
ejde-673	393	76	)	)	PUNCT
ejde-673	393	77	≥	≥	PROPN
ejde-673	394	1	m0	m0	PROPN
ejde-673	394	2	l	l	PROPN
ejde-673	395	1	(	(	PUNCT
ejde-673	395	2	k0	k0	PROPN
ejde-673	395	3	p+	p+	PROPN
ejde-673	395	4	(	(	PUNCT
ejde-673	395	5	∥u∥p	∥u∥p	NOUN
ejde-673	395	6	+	+	CCONJ
ejde-673	395	7	y	y	PROPN
ejde-673	395	8	∧	∧	PROPN
ejde-673	395	9	∥u∥p	∥u∥p	NOUN
ejde-673	395	10	−	−	PROPN
ejde-673	395	11	y	y	PROPN
ejde-673	395	12	)	)	PUNCT
ejde-673	395	13	)	)	PUNCT
ejde-673	395	14	l	l	NOUN
ejde-673	395	15	,	,	PUNCT
ejde-673	395	16	(	(	PUNCT
ejde-673	395	17	3.8	3.8	NUM
ejde-673	395	18	)	)	PUNCT
ejde-673	395	19	lemma	lemma	PROPN
ejde-673	395	20	3.8	3.8	NUM
ejde-673	395	21	.	.	PUNCT
ejde-673	396	1	if	if	SCONJ
ejde-673	396	2	un	un	PROPN
ejde-673	396	3	→	→	SYM
ejde-673	396	4	u	u	PROPN
ejde-673	396	5	weakly	weakly	ADV
ejde-673	396	6	in	in	ADP
ejde-673	396	7	y	y	PROPN
ejde-673	396	8	and	and	CCONJ
ejde-673	396	9	ψ(un	ψ(un	PROPN
ejde-673	396	10	)	)	PUNCT
ejde-673	396	11	→	→	SYM
ejde-673	396	12	ψ(u	ψ(u	PROPN
ejde-673	396	13	)	)	PUNCT
ejde-673	396	14	as	as	ADP
ejde-673	396	15	n	n	PROPN
ejde-673	396	16	→	→	SYM
ejde-673	396	17	∞	∞	PROPN
ejde-673	396	18	,	,	PUNCT
ejde-673	396	19	then	then	ADV
ejde-673	396	20	we	we	PRON
ejde-673	396	21	have	have	VERB
ejde-673	396	22	ψ	ψ	X
ejde-673	396	23	(	(	PUNCT
ejde-673	396	24	un−u	un−u	NOUN
ejde-673	396	25	2	2	NUM
ejde-673	396	26	)	)	PUNCT
ejde-673	396	27	→	→	SYM
ejde-673	396	28	0	0	NUM
ejde-673	396	29	as	as	ADP
ejde-673	396	30	n	n	NOUN
ejde-673	396	31	→	→	SYM
ejde-673	396	32	∞.	∞.	PROPN
ejde-673	396	33	in	in	ADP
ejde-673	396	34	particular	particular	ADJ
ejde-673	396	35	,	,	PUNCT
ejde-673	396	36	un	un	PROPN
ejde-673	396	37	→	→	SYM
ejde-673	396	38	u	u	PROPN
ejde-673	396	39	strongly	strongly	ADV
ejde-673	396	40	in	in	ADP
ejde-673	396	41	y	y	PROPN
ejde-673	396	42	as	as	ADP
ejde-673	396	43	n	n	PROPN
ejde-673	396	44	→	→	SYM
ejde-673	396	45	∞.	∞.	PROPN
ejde-673	396	46	proof	proof	NOUN
ejde-673	396	47	.	.	PUNCT
ejde-673	397	1	let	let	VERB
ejde-673	397	2	un	un	PROPN
ejde-673	397	3	→	→	SYM
ejde-673	397	4	u	u	NOUN
ejde-673	397	5	weakly	weakly	ADV
ejde-673	397	6	in	in	ADP
ejde-673	397	7	y	y	PROPN
ejde-673	397	8	and	and	CCONJ
ejde-673	397	9	ψ(un	ψ(un	PROPN
ejde-673	397	10	)	)	PUNCT
ejde-673	397	11	→	→	SYM
ejde-673	397	12	ψ(u	ψ(u	PROPN
ejde-673	397	13	)	)	PUNCT
ejde-673	397	14	as	as	ADP
ejde-673	397	15	n	n	PROPN
ejde-673	397	16	→	→	SYM
ejde-673	397	17	∞.	∞.	PROPN
ejde-673	397	18	then	then	ADV
ejde-673	397	19	,	,	PUNCT
ejde-673	397	20	if	if	SCONJ
ejde-673	397	21	we	we	PRON
ejde-673	397	22	use	use	VERB
ejde-673	397	23	[	[	X
ejde-673	397	24	13	13	NUM
ejde-673	397	25	,	,	PUNCT
ejde-673	397	26	lemma	lemma	PROPN
ejde-673	397	27	2.4.17	2.4.17	NUM
ejde-673	397	28	]	]	X
ejde-673	397	29	(	(	PUNCT
ejde-673	397	30	cf	cf	NOUN
ejde-673	397	31	.	.	PUNCT
ejde-673	397	32	aramaki	aramaki	PROPN
ejde-673	398	1	[	[	X
ejde-673	398	2	9	9	NUM
ejde-673	398	3	,	,	PUNCT
ejde-673	398	4	lemma	lemma	PROPN
ejde-673	398	5	20	20	NUM
ejde-673	398	6	]	]	PUNCT
ejde-673	398	7	)	)	PUNCT
ejde-673	398	8	,	,	PUNCT
ejde-673	398	9	then	then	ADV
ejde-673	398	10	we	we	PRON
ejde-673	398	11	can	can	AUX
ejde-673	398	12	show	show	VERB
ejde-673	398	13	that	that	SCONJ
ejde-673	398	14	ψ	ψ	X
ejde-673	398	15	(	(	PUNCT
ejde-673	398	16	un−u	un−u	NOUN
ejde-673	398	17	2	2	NUM
ejde-673	398	18	)	)	PUNCT
ejde-673	398	19	→	→	SYM
ejde-673	398	20	0	0	NUM
ejde-673	398	21	as	as	ADP
ejde-673	398	22	n	n	NUM
ejde-673	398	23	→	→	SYM
ejde-673	398	24	∞	∞	PROPN
ejde-673	398	25	,	,	PUNCT
ejde-673	398	26	so	so	SCONJ
ejde-673	398	27	un	un	PROPN
ejde-673	398	28	→	→	SYM
ejde-673	398	29	u	u	PROPN
ejde-673	398	30	strongly	strongly	ADV
ejde-673	398	31	in	in	ADP
ejde-673	398	32	y	y	NOUN
ejde-673	398	33	using	use	VERB
ejde-673	398	34	(	(	PUNCT
ejde-673	398	35	3.8	3.8	NUM
ejde-673	398	36	)	)	PUNCT
ejde-673	398	37	.	.	PUNCT
ejde-673	399	1	□	□	PUNCT
ejde-673	399	2	first	first	ADV
ejde-673	399	3	we	we	PRON
ejde-673	399	4	list	list	VERB
ejde-673	399	5	the	the	DET
ejde-673	399	6	properties	property	NOUN
ejde-673	399	7	of	of	ADP
ejde-673	399	8	ψ	ψ	PROPN
ejde-673	399	9	.	.	PUNCT
ejde-673	399	10	proposition	proposition	NOUN
ejde-673	399	11	3.9	3.9	NUM
ejde-673	399	12	.	.	PUNCT
ejde-673	400	1	(	(	PUNCT
ejde-673	400	2	i	i	NOUN
ejde-673	400	3	)	)	PUNCT
ejde-673	400	4	ψ	ψ	X
ejde-673	400	5	is	be	AUX
ejde-673	400	6	coercive	coercive	ADJ
ejde-673	400	7	,	,	PUNCT
ejde-673	400	8	that	that	ADV
ejde-673	400	9	is	is	ADV
ejde-673	400	10	,	,	PUNCT
ejde-673	400	11	ψ(u	ψ(u	PROPN
ejde-673	400	12	)	)	PUNCT
ejde-673	400	13	→	→	SYM
ejde-673	400	14	∞	∞	NUM
ejde-673	400	15	as	as	ADP
ejde-673	400	16	∥u∥y	∥u∥y	NOUN
ejde-673	400	17	→	→	SYM
ejde-673	400	18	∞.	∞.	PROPN
ejde-673	400	19	(	(	PUNCT
ejde-673	400	20	ii	ii	NOUN
ejde-673	400	21	)	)	PUNCT
ejde-673	400	22	ψ	ψ	NOUN
ejde-673	400	23	is	be	AUX
ejde-673	400	24	sequentially	sequentially	ADV
ejde-673	400	25	weakly	weakly	ADV
ejde-673	400	26	lower	lower	ADV
ejde-673	400	27	-	-	PUNCT
ejde-673	400	28	semicontinuous	semicontinuous	ADJ
ejde-673	400	29	on	on	ADP
ejde-673	400	30	y	y	PROPN
ejde-673	400	31	.	.	PUNCT
ejde-673	401	1	(	(	PUNCT
ejde-673	401	2	iii	iii	X
ejde-673	401	3	)	)	PUNCT
ejde-673	401	4	ψ	ψ	X
ejde-673	401	5	∈	∈	PROPN
ejde-673	401	6	c1(y	c1(y	PROPN
ejde-673	401	7	,	,	PUNCT
ejde-673	401	8	r	r	NOUN
ejde-673	401	9	)	)	PUNCT
ejde-673	401	10	and	and	CCONJ
ejde-673	401	11	the	the	DET
ejde-673	401	12	fréchet	fréchet	NOUN
ejde-673	401	13	derivative	derivative	ADJ
ejde-673	401	14	ψ′	ψ′	NUM
ejde-673	401	15	of	of	ADP
ejde-673	401	16	ψ	ψ	X
ejde-673	401	17	satisfies	satisfie	NOUN
ejde-673	401	18	⟨ψ′(u	⟨ψ′(u	PROPN
ejde-673	401	19	)	)	PUNCT
ejde-673	401	20	,	,	PUNCT
ejde-673	401	21	v⟩y	v⟩y	PROPN
ejde-673	401	22	∗,y	∗,y	PROPN
ejde-673	401	23	=	=	SYM
ejde-673	401	24	m(φ(u	m(φ(u	NOUN
ejde-673	401	25	)	)	PUNCT
ejde-673	401	26	)	)	PUNCT
ejde-673	402	1	∫	∫	PROPN
ejde-673	402	2	ω	ω	NUM
ejde-673	402	3	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	402	4	)	)	PUNCT
ejde-673	402	5	)	)	PUNCT
ejde-673	402	6	·	·	SYM
ejde-673	402	7	∇v(x	∇v(x	NUM
ejde-673	402	8	)	)	PUNCT
ejde-673	402	9	dx	dx	PROPN
ejde-673	402	10	for	for	ADP
ejde-673	402	11	u	u	PROPN
ejde-673	402	12	,	,	PUNCT
ejde-673	402	13	v	v	ADP
ejde-673	402	14	∈	∈	PROPN
ejde-673	402	15	y.	y.	NOUN
ejde-673	402	16	(	(	PUNCT
ejde-673	402	17	3.9	3.9	NUM
ejde-673	402	18	)	)	PUNCT
ejde-673	402	19	(	(	PUNCT
ejde-673	402	20	iv	iv	X
ejde-673	402	21	)	)	PUNCT
ejde-673	402	22	ψ	ψ	NOUN
ejde-673	402	23	∈	∈	PROPN
ejde-673	402	24	wy	wy	PROPN
ejde-673	402	25	,	,	PUNCT
ejde-673	402	26	that	that	ADV
ejde-673	402	27	is	is	ADV
ejde-673	402	28	,	,	PUNCT
ejde-673	402	29	if	if	SCONJ
ejde-673	402	30	un	un	PROPN
ejde-673	402	31	→	→	SYM
ejde-673	402	32	u	u	NOUN
ejde-673	402	33	weakly	weakly	ADV
ejde-673	402	34	in	in	ADP
ejde-673	402	35	y	y	PROPN
ejde-673	402	36	and	and	CCONJ
ejde-673	402	37	lim	lim	PROPN
ejde-673	402	38	infn→∞	infn→∞	PROPN
ejde-673	402	39	ψ(un	ψ(un	PROPN
ejde-673	402	40	)	)	PUNCT
ejde-673	402	41	≤	≤	NOUN
ejde-673	402	42	ψ(u	ψ(u	PROPN
ejde-673	402	43	)	)	PUNCT
ejde-673	402	44	,	,	PUNCT
ejde-673	402	45	then	then	ADV
ejde-673	402	46	the	the	DET
ejde-673	402	47	sequence	sequence	NOUN
ejde-673	402	48	{	{	PUNCT
ejde-673	402	49	un	un	PROPN
ejde-673	402	50	}	}	PUNCT
ejde-673	402	51	has	have	VERB
ejde-673	402	52	a	a	DET
ejde-673	402	53	strongly	strongly	ADV
ejde-673	402	54	convergent	convergent	ADJ
ejde-673	402	55	subsequence	subsequence	NOUN
ejde-673	402	56	.	.	PUNCT
ejde-673	403	1	(	(	PUNCT
ejde-673	403	2	v	v	NOUN
ejde-673	403	3	)	)	PUNCT
ejde-673	403	4	ψ	ψ	NOUN
ejde-673	403	5	is	be	AUX
ejde-673	403	6	bounded	bound	VERB
ejde-673	403	7	on	on	ADP
ejde-673	403	8	every	every	DET
ejde-673	403	9	bounded	bound	VERB
ejde-673	403	10	subset	subset	NOUN
ejde-673	403	11	of	of	ADP
ejde-673	403	12	y	y	PROPN
ejde-673	403	13	.	.	PUNCT
ejde-673	404	1	proof	proof	NOUN
ejde-673	404	2	.	.	PUNCT
ejde-673	405	1	(	(	PUNCT
ejde-673	405	2	i	i	NOUN
ejde-673	405	3	)	)	PUNCT
ejde-673	405	4	follows	follow	VERB
ejde-673	405	5	from	from	ADP
ejde-673	405	6	(	(	PUNCT
ejde-673	405	7	3.8	3.8	NUM
ejde-673	405	8	)	)	PUNCT
ejde-673	405	9	.	.	PUNCT
ejde-673	406	1	(	(	PUNCT
ejde-673	406	2	ii	ii	NOUN
ejde-673	406	3	)	)	PUNCT
ejde-673	406	4	follows	follow	VERB
ejde-673	406	5	from	from	ADP
ejde-673	406	6	aramaki	aramaki	NOUN
ejde-673	407	1	[	[	X
ejde-673	407	2	7	7	NUM
ejde-673	407	3	,	,	PUNCT
ejde-673	407	4	proposition	proposition	NOUN
ejde-673	407	5	4.4	4.4	NUM
ejde-673	407	6	]	]	PUNCT
ejde-673	407	7	and	and	CCONJ
ejde-673	407	8	the	the	DET
ejde-673	407	9	fact	fact	NOUN
ejde-673	407	10	that	that	SCONJ
ejde-673	407	11	m̂	m̂	NOUN
ejde-673	407	12	is	be	AUX
ejde-673	407	13	monotone	monotone	ADJ
ejde-673	407	14	increasing	increase	VERB
ejde-673	407	15	and	and	CCONJ
ejde-673	407	16	continuous	continuous	ADJ
ejde-673	407	17	.	.	PUNCT
ejde-673	408	1	(	(	PUNCT
ejde-673	408	2	iii	iii	NOUN
ejde-673	408	3	)	)	PUNCT
ejde-673	408	4	follows	follow	VERB
ejde-673	408	5	from	from	ADP
ejde-673	408	6	[	[	X
ejde-673	408	7	7	7	NUM
ejde-673	408	8	,	,	PUNCT
ejde-673	408	9	proposition	proposition	NOUN
ejde-673	408	10	4.1	4.1	NUM
ejde-673	408	11	]	]	PUNCT
ejde-673	408	12	and	and	CCONJ
ejde-673	408	13	m̂	m̂	PROPN
ejde-673	408	14	∈	∈	PROPN
ejde-673	408	15	c1([0,∞	c1([0,∞	NOUN
ejde-673	408	16	)	)	PUNCT
ejde-673	408	17	)	)	PUNCT
ejde-673	408	18	.	.	PUNCT
ejde-673	409	1	(	(	PUNCT
ejde-673	409	2	iv	iv	X
ejde-673	409	3	)	)	PUNCT
ejde-673	409	4	let	let	VERB
ejde-673	409	5	un	un	PROPN
ejde-673	409	6	→	→	SYM
ejde-673	409	7	u	u	NOUN
ejde-673	409	8	weakly	weakly	ADV
ejde-673	409	9	in	in	ADP
ejde-673	409	10	y	y	PROPN
ejde-673	409	11	and	and	CCONJ
ejde-673	409	12	lim	lim	PROPN
ejde-673	409	13	infn→∞	infn→∞	PROPN
ejde-673	409	14	ψ(un	ψ(un	PROPN
ejde-673	409	15	)	)	PUNCT
ejde-673	409	16	≤	≤	NOUN
ejde-673	409	17	ψ(u	ψ(u	PROPN
ejde-673	409	18	)	)	PUNCT
ejde-673	409	19	.	.	PUNCT
ejde-673	410	1	since	since	SCONJ
ejde-673	410	2	ψ	ψ	NOUN
ejde-673	410	3	is	be	AUX
ejde-673	410	4	sequentially	sequentially	ADV
ejde-673	410	5	weakly	weakly	ADV
ejde-673	410	6	lower	low	ADJ
ejde-673	410	7	semi	semi	ADJ
ejde-673	410	8	-	-	ADJ
ejde-673	410	9	continuous	continuous	ADJ
ejde-673	410	10	,	,	PUNCT
ejde-673	410	11	ψ(u	ψ(u	PROPN
ejde-673	410	12	)	)	PUNCT
ejde-673	410	13	≤	≤	NOUN
ejde-673	410	14	lim	lim	PROPN
ejde-673	410	15	infn→∞	infn→∞	PROPN
ejde-673	410	16	ψ(un	ψ(un	PROPN
ejde-673	410	17	)	)	PUNCT
ejde-673	410	18	,	,	PUNCT
ejde-673	410	19	so	so	SCONJ
ejde-673	410	20	that	that	SCONJ
ejde-673	410	21	lim	lim	PROPN
ejde-673	410	22	infn→∞	infn→∞	PROPN
ejde-673	410	23	ψ(un	ψ(un	PROPN
ejde-673	410	24	)	)	PUNCT
ejde-673	410	25	=	=	SYM
ejde-673	410	26	ψ(u	ψ(u	PROPN
ejde-673	410	27	)	)	PUNCT
ejde-673	410	28	.	.	PUNCT
ejde-673	411	1	hence	hence	ADV
ejde-673	411	2	there	there	PRON
ejde-673	411	3	exists	exist	VERB
ejde-673	411	4	a	a	DET
ejde-673	411	5	subsequence	subsequence	NOUN
ejde-673	411	6	{	{	PUNCT
ejde-673	411	7	un′	un′	NOUN
ejde-673	411	8	}	}	PUNCT
ejde-673	411	9	of	of	ADP
ejde-673	411	10	{	{	PUNCT
ejde-673	411	11	un	un	PROPN
ejde-673	411	12	}	}	PUNCT
ejde-673	411	13	such	such	ADJ
ejde-673	411	14	that	that	SCONJ
ejde-673	411	15	limn′→∞	limn′→∞	NOUN
ejde-673	411	16	ψ(un′	ψ(un′	NOUN
ejde-673	411	17	)	)	PUNCT
ejde-673	411	18	=	=	SYM
ejde-673	411	19	ψ(u	ψ(u	PROPN
ejde-673	411	20	)	)	PUNCT
ejde-673	411	21	.	.	PUNCT
ejde-673	412	1	by	by	ADP
ejde-673	412	2	lemma	lemma	PROPN
ejde-673	412	3	3.8	3.8	NUM
ejde-673	412	4	,	,	PUNCT
ejde-673	412	5	we	we	PRON
ejde-673	412	6	see	see	VERB
ejde-673	412	7	that	that	SCONJ
ejde-673	412	8	un′	un′	PROPN
ejde-673	412	9	→	→	SYM
ejde-673	412	10	u	u	NOUN
ejde-673	412	11	strongly	strongly	ADV
ejde-673	412	12	in	in	ADP
ejde-673	412	13	y	y	PROPN
ejde-673	412	14	.	.	PUNCT
ejde-673	413	1	(	(	PUNCT
ejde-673	413	2	v	v	NOUN
ejde-673	413	3	)	)	PUNCT
ejde-673	413	4	follows	follow	VERB
ejde-673	413	5	from	from	ADP
ejde-673	413	6	lemma	lemma	PROPN
ejde-673	413	7	3.7	3.7	NUM
ejde-673	413	8	(	(	PUNCT
ejde-673	413	9	i	i	NOUN
ejde-673	413	10	)	)	PUNCT
ejde-673	413	11	and	and	CCONJ
ejde-673	413	12	(	(	PUNCT
ejde-673	413	13	3.6	3.6	NUM
ejde-673	413	14	)	)	PUNCT
ejde-673	413	15	.	.	PUNCT
ejde-673	414	1	□	□	PUNCT
ejde-673	414	2	next	next	ADV
ejde-673	414	3	we	we	PRON
ejde-673	414	4	derive	derive	VERB
ejde-673	414	5	the	the	DET
ejde-673	414	6	properties	property	NOUN
ejde-673	414	7	of	of	ADP
ejde-673	414	8	ψ′.	ψ′.	NOUN
ejde-673	414	9	proposition	proposition	NOUN
ejde-673	414	10	3.10	3.10	NUM
ejde-673	414	11	.	.	PUNCT
ejde-673	415	1	(	(	PUNCT
ejde-673	415	2	i	i	NOUN
ejde-673	415	3	)	)	PUNCT
ejde-673	415	4	ψ′	ψ′	VERB
ejde-673	415	5	is	be	AUX
ejde-673	415	6	strictly	strictly	ADV
ejde-673	415	7	monotone	monotone	ADJ
ejde-673	415	8	in	in	ADP
ejde-673	415	9	y	y	PROPN
ejde-673	415	10	,	,	PUNCT
ejde-673	415	11	that	that	ADV
ejde-673	415	12	is	is	ADV
ejde-673	415	13	,	,	PUNCT
ejde-673	415	14	⟨ψ′(u)−ψ′(v	⟨ψ′(u)−ψ′(v	NOUN
ejde-673	415	15	)	)	PUNCT
ejde-673	415	16	,	,	PUNCT
ejde-673	415	17	u−	u−	PROPN
ejde-673	415	18	v⟩y	v⟩y	PROPN
ejde-673	415	19	∗,y	∗,y	PROPN
ejde-673	415	20	>	>	X
ejde-673	415	21	0	0	PUNCT
ejde-673	415	22	for	for	ADP
ejde-673	415	23	all	all	DET
ejde-673	415	24	u	u	NOUN
ejde-673	415	25	,	,	PUNCT
ejde-673	415	26	v	v	PROPN
ejde-673	415	27	∈	∈	PROPN
ejde-673	415	28	y	y	NOUN
ejde-673	415	29	with	with	ADP
ejde-673	415	30	u	u	NOUN
ejde-673	415	31	̸=	̸=	PROPN
ejde-673	415	32	v.	v.	CCONJ
ejde-673	415	33	moreover	moreover	ADV
ejde-673	415	34	,	,	PUNCT
ejde-673	415	35	ψ′	ψ′	PUNCT
ejde-673	415	36	is	be	AUX
ejde-673	415	37	bounded	bound	VERB
ejde-673	415	38	on	on	ADP
ejde-673	415	39	every	every	DET
ejde-673	415	40	bounded	bound	VERB
ejde-673	415	41	subset	subset	NOUN
ejde-673	415	42	of	of	ADP
ejde-673	415	43	y	y	PROPN
ejde-673	415	44	and	and	CCONJ
ejde-673	415	45	coercive	coercive	ADJ
ejde-673	415	46	in	in	ADP
ejde-673	415	47	the	the	DET
ejde-673	415	48	sense	sense	NOUN
ejde-673	415	49	that	that	SCONJ
ejde-673	415	50	lim	lim	PROPN
ejde-673	415	51	∥u∥y	∥u∥y	PRON
ejde-673	415	52	→∞	→∞	PROPN
ejde-673	415	53	⟨ψ′(u	⟨ψ′(u	PROPN
ejde-673	415	54	)	)	PUNCT
ejde-673	415	55	,	,	PUNCT
ejde-673	415	56	u⟩y	u⟩y	NUM
ejde-673	415	57	∗,y	∗,y	NOUN
ejde-673	415	58	∥u∥y	∥u∥y	NOUN
ejde-673	415	59	=	=	PUNCT
ejde-673	415	60	∞.	∞.	PROPN
ejde-673	415	61	(	(	PUNCT
ejde-673	415	62	ii	ii	NOUN
ejde-673	415	63	)	)	PUNCT
ejde-673	415	64	ψ′	ψ′	PUNCT
ejde-673	415	65	is	be	AUX
ejde-673	415	66	of	of	ADP
ejde-673	415	67	(	(	PUNCT
ejde-673	415	68	s+)-type	s+)-type	PROPN
ejde-673	415	69	,	,	PUNCT
ejde-673	415	70	that	that	ADV
ejde-673	415	71	is	is	ADV
ejde-673	415	72	,	,	PUNCT
ejde-673	415	73	if	if	SCONJ
ejde-673	415	74	un	un	PROPN
ejde-673	415	75	→	→	SYM
ejde-673	415	76	u	u	NOUN
ejde-673	415	77	weakly	weakly	ADV
ejde-673	415	78	in	in	ADP
ejde-673	415	79	y	y	PROPN
ejde-673	415	80	and	and	CCONJ
ejde-673	415	81	lim	lim	PROPN
ejde-673	415	82	sup	sup	PROPN
ejde-673	415	83	n→∞	n→∞	NUM
ejde-673	415	84	⟨ψ′(un	⟨ψ′(un	PROPN
ejde-673	415	85	)	)	PUNCT
ejde-673	415	86	,	,	PUNCT
ejde-673	415	87	un	un	PROPN
ejde-673	415	88	−	−	PROPN
ejde-673	415	89	u⟩y	u⟩y	NUM
ejde-673	415	90	∗.y	∗.y	PROPN
ejde-673	415	91	≤	≤	NUM
ejde-673	415	92	0	0	NUM
ejde-673	415	93	,	,	PUNCT
ejde-673	415	94	then	then	ADV
ejde-673	415	95	un	un	PROPN
ejde-673	415	96	→	→	SYM
ejde-673	415	97	u	u	PROPN
ejde-673	415	98	strongly	strongly	ADV
ejde-673	415	99	in	in	ADP
ejde-673	415	100	y	y	PROPN
ejde-673	415	101	.	.	PUNCT
ejde-673	416	1	(	(	PUNCT
ejde-673	416	2	iii	iii	X
ejde-673	416	3	)	)	PUNCT
ejde-673	416	4	the	the	DET
ejde-673	416	5	mapping	mapping	NOUN
ejde-673	416	6	ψ′	ψ′	PUNCT
ejde-673	416	7	:	:	PUNCT
ejde-673	416	8	y	y	PROPN
ejde-673	416	9	→	→	SYM
ejde-673	416	10	y	y	PROPN
ejde-673	416	11	∗	∗	NOUN
ejde-673	416	12	is	be	AUX
ejde-673	416	13	a	a	DET
ejde-673	416	14	homeomorphism	homeomorphism	NOUN
ejde-673	416	15	.	.	PUNCT
ejde-673	417	1	14	14	NUM
ejde-673	417	2	j.	j.	PROPN
ejde-673	417	3	aramaki	aramaki	PROPN
ejde-673	417	4	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	417	5	proof	proof	NOUN
ejde-673	417	6	.	.	PUNCT
ejde-673	418	1	(	(	PUNCT
ejde-673	418	2	i	i	NOUN
ejde-673	418	3	)	)	PUNCT
ejde-673	418	4	in	in	ADP
ejde-673	418	5	general	general	ADJ
ejde-673	418	6	,	,	PUNCT
ejde-673	418	7	when	when	SCONJ
ejde-673	418	8	a	a	DET
ejde-673	418	9	functional	functional	ADJ
ejde-673	418	10	f	f	NOUN
ejde-673	418	11	:	:	PUNCT
ejde-673	418	12	y	y	PROPN
ejde-673	418	13	→	→	PUNCT
ejde-673	418	14	r	r	NOUN
ejde-673	418	15	is	be	AUX
ejde-673	418	16	of	of	ADP
ejde-673	418	17	c1	c1	NOUN
ejde-673	418	18	-	-	PUNCT
ejde-673	418	19	class	class	NOUN
ejde-673	418	20	,	,	PUNCT
ejde-673	418	21	f	f	PROPN
ejde-673	418	22	is	be	AUX
ejde-673	418	23	strictly	strictly	ADV
ejde-673	418	24	convex	convex	ADJ
ejde-673	418	25	if	if	SCONJ
ejde-673	418	26	and	and	CCONJ
ejde-673	418	27	only	only	ADV
ejde-673	418	28	if	if	SCONJ
ejde-673	418	29	f	f	PROPN
ejde-673	419	1	′	′	NUM
ejde-673	419	2	:	:	PUNCT
ejde-673	419	3	y	y	PROPN
ejde-673	419	4	→	→	SYM
ejde-673	419	5	y	y	PROPN
ejde-673	419	6	∗	∗	NOUN
ejde-673	419	7	is	be	AUX
ejde-673	419	8	strictly	strictly	ADV
ejde-673	419	9	monotone	monotone	ADJ
ejde-673	419	10	(	(	PUNCT
ejde-673	419	11	cf	cf	NOUN
ejde-673	419	12	.	.	PUNCT
ejde-673	420	1	zeidler	zeidler	PROPN
ejde-673	421	1	[	[	X
ejde-673	421	2	34	34	NUM
ejde-673	421	3	,	,	PUNCT
ejde-673	421	4	proposition	proposition	NOUN
ejde-673	421	5	25.10	25.10	NUM
ejde-673	421	6	]	]	PUNCT
ejde-673	421	7	)	)	PUNCT
ejde-673	421	8	,	,	PUNCT
ejde-673	421	9	that	that	ADV
ejde-673	421	10	is	is	ADV
ejde-673	421	11	,	,	PUNCT
ejde-673	421	12	⟨f	⟨f	X
ejde-673	422	1	′(u)−	′(u)−	PROPN
ejde-673	422	2	f	f	PROPN
ejde-673	422	3	′(v	′(v	PROPN
ejde-673	422	4	)	)	PUNCT
ejde-673	422	5	,	,	PUNCT
ejde-673	422	6	u−	u−	PROPN
ejde-673	422	7	v⟩y	v⟩y	PROPN
ejde-673	422	8	∗,y	∗,y	PROPN
ejde-673	422	9	>	>	X
ejde-673	422	10	0	0	PUNCT
ejde-673	422	11	for	for	ADP
ejde-673	422	12	all	all	DET
ejde-673	422	13	u	u	NOUN
ejde-673	422	14	,	,	PUNCT
ejde-673	422	15	v	v	PROPN
ejde-673	422	16	∈	∈	PROPN
ejde-673	422	17	y	y	NOUN
ejde-673	422	18	with	with	ADP
ejde-673	422	19	u	u	NOUN
ejde-673	422	20	̸=	̸=	PROPN
ejde-673	422	21	v.	v.	ADP
ejde-673	422	22	from	from	ADP
ejde-673	422	23	(	(	PUNCT
ejde-673	422	24	a6	a6	NOUN
ejde-673	422	25	)	)	PUNCT
ejde-673	422	26	,	,	PUNCT
ejde-673	422	27	⟨φ′(u)−φ′(v	⟨φ′(u)−φ′(v	NOUN
ejde-673	422	28	)	)	PUNCT
ejde-673	422	29	,	,	PUNCT
ejde-673	422	30	u−v⟩y	u−v⟩y	PROPN
ejde-673	422	31	∗,y	∗,y	PROPN
ejde-673	422	32	=	=	SYM
ejde-673	422	33	∫	∫	PROPN
ejde-673	422	34	ω	ω	PROPN
ejde-673	422	35	(	(	PUNCT
ejde-673	422	36	a(x,∇u(x)−a(x,∇v(x)))·(∇u(x)−∇v(x	a(x,∇u(x)−a(x,∇v(x)))·(∇u(x)−∇v(x	PROPN
ejde-673	422	37	)	)	PUNCT
ejde-673	422	38	)	)	PUNCT
ejde-673	422	39	dx	dx	PROPN
ejde-673	422	40	>	>	X
ejde-673	422	41	0	0	PUNCT
ejde-673	423	1	for	for	ADP
ejde-673	423	2	all	all	DET
ejde-673	423	3	u	u	NOUN
ejde-673	423	4	,	,	PUNCT
ejde-673	423	5	v	v	PROPN
ejde-673	423	6	∈	∈	PROPN
ejde-673	423	7	y	y	NOUN
ejde-673	423	8	with	with	ADP
ejde-673	423	9	u	u	NOUN
ejde-673	423	10	̸=	̸=	PROPN
ejde-673	423	11	v	v	NOUN
ejde-673	423	12	,	,	PUNCT
ejde-673	423	13	so	so	SCONJ
ejde-673	423	14	φ′	φ′	NUM
ejde-673	423	15	is	be	AUX
ejde-673	423	16	strictly	strictly	ADV
ejde-673	423	17	monotone	monotone	ADJ
ejde-673	423	18	in	in	ADP
ejde-673	423	19	y	y	PROPN
ejde-673	423	20	,	,	PUNCT
ejde-673	423	21	so	so	ADV
ejde-673	423	22	φ	φ	PROPN
ejde-673	423	23	is	be	AUX
ejde-673	423	24	strictly	strictly	ADV
ejde-673	423	25	convex	convex	ADJ
ejde-673	423	26	.	.	PUNCT
ejde-673	424	1	the	the	DET
ejde-673	424	2	function	function	NOUN
ejde-673	424	3	m̂	m̂	PROPN
ejde-673	424	4	is	be	AUX
ejde-673	424	5	strictly	strictly	ADV
ejde-673	424	6	monotone	monotone	ADJ
ejde-673	424	7	increasing	increase	VERB
ejde-673	424	8	and	and	CCONJ
ejde-673	424	9	convex	convex	NOUN
ejde-673	424	10	.	.	PUNCT
ejde-673	425	1	hence	hence	ADV
ejde-673	425	2	for	for	ADP
ejde-673	425	3	u	u	NOUN
ejde-673	425	4	,	,	PUNCT
ejde-673	425	5	v	v	AUX
ejde-673	425	6	∈	∈	PROPN
ejde-673	425	7	y	y	NOUN
ejde-673	425	8	with	with	ADP
ejde-673	425	9	u	u	NOUN
ejde-673	425	10	̸=	̸=	PROPN
ejde-673	425	11	v	v	NOUN
ejde-673	425	12	and	and	CCONJ
ejde-673	425	13	τ	τ	PROPN
ejde-673	425	14	∈	∈	PROPN
ejde-673	425	15	(	(	PUNCT
ejde-673	425	16	0	0	NUM
ejde-673	425	17	,	,	PUNCT
ejde-673	425	18	1	1	NUM
ejde-673	425	19	)	)	PUNCT
ejde-673	425	20	,	,	PUNCT
ejde-673	425	21	since	since	SCONJ
ejde-673	425	22	φ((1−	φ((1−	NUM
ejde-673	425	23	τ)u+	τ)u+	X
ejde-673	425	24	τv	τv	NUM
ejde-673	425	25	)	)	PUNCT
ejde-673	425	26	<	<	X
ejde-673	425	27	(	(	PUNCT
ejde-673	425	28	1−	1−	NUM
ejde-673	425	29	τ)φ(u	τ)φ(u	NOUN
ejde-673	425	30	)	)	PUNCT
ejde-673	425	31	+	+	NUM
ejde-673	425	32	τφ(v	τφ(v	NOUN
ejde-673	425	33	)	)	PUNCT
ejde-673	425	34	,	,	PUNCT
ejde-673	425	35	we	we	PRON
ejde-673	425	36	have	have	VERB
ejde-673	425	37	m̂(φ(1−	m̂(φ(1−	PROPN
ejde-673	425	38	τ)u+	τ)u+	PUNCT
ejde-673	425	39	τv	τv	NOUN
ejde-673	425	40	)	)	PUNCT
ejde-673	425	41	)	)	PUNCT
ejde-673	426	1	<	<	X
ejde-673	426	2	m̂((1−	m̂((1−	PROPN
ejde-673	426	3	τ)φ(u	τ)φ(u	NOUN
ejde-673	426	4	)	)	PUNCT
ejde-673	426	5	+	+	NUM
ejde-673	426	6	τφ(v	τφ(v	NOUN
ejde-673	426	7	)	)	PUNCT
ejde-673	426	8	)	)	PUNCT
ejde-673	426	9	≤	≤	NOUN
ejde-673	426	10	(	(	PUNCT
ejde-673	426	11	1−	1−	NUM
ejde-673	426	12	τ)m̂(φ(u	τ)m̂(φ(u	NUM
ejde-673	426	13	)	)	PUNCT
ejde-673	426	14	)	)	PUNCT
ejde-673	427	1	+	+	CCONJ
ejde-673	427	2	τm̂(φ(v	τm̂(φ(v	NOUN
ejde-673	427	3	)	)	PUNCT
ejde-673	427	4	)	)	PUNCT
ejde-673	427	5	,	,	PUNCT
ejde-673	427	6	so	so	ADV
ejde-673	427	7	ψ((1−	ψ((1−	PRON
ejde-673	427	8	τ)u+	τ)u+	X
ejde-673	427	9	τv	τv	NUM
ejde-673	427	10	)	)	PUNCT
ejde-673	427	11	<	<	X
ejde-673	427	12	(	(	PUNCT
ejde-673	427	13	1−	1−	NUM
ejde-673	427	14	τ)ψ(u	τ)ψ(u	NOUN
ejde-673	427	15	)	)	PUNCT
ejde-673	427	16	+	+	NUM
ejde-673	427	17	τψ(v	τψ(v	NUM
ejde-673	427	18	)	)	PUNCT
ejde-673	427	19	.	.	PUNCT
ejde-673	428	1	thus	thus	ADV
ejde-673	428	2	ψ	ψ	X
ejde-673	428	3	is	be	AUX
ejde-673	428	4	strictly	strictly	ADV
ejde-673	428	5	convex	convex	ADJ
ejde-673	428	6	,	,	PUNCT
ejde-673	428	7	so	so	ADV
ejde-673	428	8	ψ′	ψ′	PROPN
ejde-673	428	9	(	(	PUNCT
ejde-673	428	10	·	·	PUNCT
ejde-673	428	11	)	)	PUNCT
ejde-673	429	1	=	=	SYM
ejde-673	429	2	m(φ(·))φ′	m(φ(·))φ′	NOUN
ejde-673	429	3	(	(	PUNCT
ejde-673	429	4	·	·	PUNCT
ejde-673	429	5	)	)	PUNCT
ejde-673	429	6	is	be	AUX
ejde-673	429	7	strictly	strictly	ADV
ejde-673	429	8	monotone	monotone	ADJ
ejde-673	429	9	in	in	ADP
ejde-673	429	10	y	y	PROPN
ejde-673	429	11	.	.	PUNCT
ejde-673	430	1	it	it	PRON
ejde-673	430	2	follows	follow	VERB
ejde-673	430	3	from	from	ADP
ejde-673	430	4	the	the	DET
ejde-673	430	5	hölder	hölder	NOUN
ejde-673	430	6	inequality	inequality	NOUN
ejde-673	430	7	(	(	PUNCT
ejde-673	430	8	proposition	proposition	NOUN
ejde-673	430	9	2.2	2.2	NUM
ejde-673	430	10	)	)	PUNCT
ejde-673	430	11	and	and	CCONJ
ejde-673	430	12	proposition	proposition	NOUN
ejde-673	430	13	3.5	3.5	NUM
ejde-673	430	14	(	(	PUNCT
ejde-673	430	15	i	i	NOUN
ejde-673	430	16	)	)	PUNCT
ejde-673	430	17	that	that	PRON
ejde-673	430	18	|⟨ψ′(u	|⟨ψ′(u	VERB
ejde-673	430	19	)	)	PUNCT
ejde-673	430	20	,	,	PUNCT
ejde-673	430	21	v⟩y	v⟩y	PROPN
ejde-673	430	22	∗,y	∗,y	NOUN
ejde-673	430	23	|	|	NOUN
ejde-673	430	24	=	=	SYM
ejde-673	430	25	m(φ(u	m(φ(u	NOUN
ejde-673	430	26	)	)	PUNCT
ejde-673	430	27	)	)	PUNCT
ejde-673	430	28	∣∣	∣∣	NUM
ejde-673	431	1	∫	∫	PROPN
ejde-673	431	2	ω	ω	NUM
ejde-673	431	3	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	431	4	)	)	PUNCT
ejde-673	431	5	)	)	PUNCT
ejde-673	431	6	·	·	SYM
ejde-673	431	7	∇v(x	∇v(x	NUM
ejde-673	431	8	)	)	PUNCT
ejde-673	431	9	dx	dx	PROPN
ejde-673	432	1	∣∣	∣∣	NUM
ejde-673	432	2	≤	≤	X
ejde-673	432	3	cm(φ(u	cm(φ(u	NOUN
ejde-673	432	4	)	)	PUNCT
ejde-673	432	5	)	)	PUNCT
ejde-673	433	1	∫	∫	PROPN
ejde-673	433	2	ω	ω	INTJ
ejde-673	433	3	(	(	PUNCT
ejde-673	433	4	h0(x)|∇v(x)|+	h0(x)|∇v(x)|+	NOUN
ejde-673	433	5	h1(x)|∇u(x)|p(x)−1|∇v(x)|	h1(x)|∇u(x)|p(x)−1|∇v(x)|	NOUN
ejde-673	433	6	)	)	PUNCT
ejde-673	433	7	dx	dx	PROPN
ejde-673	433	8	=	=	PUNCT
ejde-673	433	9	cm(φ(u	cm(φ(u	NOUN
ejde-673	433	10	)	)	PUNCT
ejde-673	433	11	)	)	PUNCT
ejde-673	434	1	∫	∫	PROPN
ejde-673	434	2	ω	ω	INTJ
ejde-673	434	3	(	(	PUNCT
ejde-673	434	4	h0(x)|∇v(x)|+	h0(x)|∇v(x)|+	NOUN
ejde-673	434	5	h1(x	h1(x	NOUN
ejde-673	434	6	)	)	PUNCT
ejde-673	434	7	1	1	NUM
ejde-673	434	8	/	/	SYM
ejde-673	434	9	p′(x)|∇u(x)|p(x)−1h1(x	p′(x)|∇u(x)|p(x)−1h1(x	NOUN
ejde-673	434	10	)	)	PUNCT
ejde-673	434	11	1	1	NUM
ejde-673	434	12	/	/	SYM
ejde-673	434	13	p(x)|∇v(x)|	p(x)|∇v(x)|	NOUN
ejde-673	434	14	)	)	PUNCT
ejde-673	434	15	dx	dx	PROPN
ejde-673	434	16	≤	≤	NUM
ejde-673	434	17	2cm1(1	2cm1(1	NUM
ejde-673	434	18	+	+	NUM
ejde-673	434	19	φ(u)k−1)(∥h0∥lp′(·)(ω)∥v∥y	φ(u)k−1)(∥h0∥lp′(·)(ω)∥v∥y	NOUN
ejde-673	434	20	+	+	CCONJ
ejde-673	434	21	∥h1	∥h1	PROPN
ejde-673	434	22	/	/	SYM
ejde-673	434	23	p′	p′	NOUN
ejde-673	434	24	(	(	PUNCT
ejde-673	434	25	·	·	PUNCT
ejde-673	434	26	)	)	PUNCT
ejde-673	434	27	1	1	NUM
ejde-673	434	28	|∇u|p(·)−1∥lp′(·)(ω)∥h	|∇u|p(·)−1∥lp′(·)(ω)∥h	NOUN
ejde-673	434	29	1	1	NUM
ejde-673	434	30	/	/	SYM
ejde-673	434	31	p	p	X
ejde-673	434	32	(	(	PUNCT
ejde-673	434	33	·	·	PUNCT
ejde-673	434	34	)	)	PUNCT
ejde-673	434	35	1	1	NUM
ejde-673	434	36	|∇v|∥lp(·)(ω	|∇v|∥lp(·)(ω	NOUN
ejde-673	434	37	)	)	PUNCT
ejde-673	434	38	=	=	PUNCT
ejde-673	435	1	2cm1(1	2cm1(1	NUM
ejde-673	435	2	+	+	SYM
ejde-673	435	3	φ(u)k−1)(∥h0∥lp′(·)(ω	φ(u)k−1)(∥h0∥lp′(·)(ω	ADJ
ejde-673	435	4	)	)	PUNCT
ejde-673	436	1	+	+	CCONJ
ejde-673	436	2	∥h1	∥h1	PROPN
ejde-673	436	3	/	/	SYM
ejde-673	436	4	p′	p′	NOUN
ejde-673	436	5	(	(	PUNCT
ejde-673	436	6	·	·	PUNCT
ejde-673	436	7	)	)	PUNCT
ejde-673	436	8	1	1	NUM
ejde-673	436	9	|∇u|p(·)−1∥lp′(·)(ω))∥v∥y	|∇u|p(·)−1∥lp′(·)(ω))∥v∥y	NOUN
ejde-673	436	10	for	for	ADP
ejde-673	436	11	all	all	PRON
ejde-673	436	12	v	v	ADP
ejde-673	436	13	∈	∈	PROPN
ejde-673	436	14	y	y	NOUN
ejde-673	436	15	.	.	PUNCT
ejde-673	437	1	hence	hence	ADV
ejde-673	437	2	we	we	PRON
ejde-673	437	3	have	have	VERB
ejde-673	437	4	∥ψ′(u)∥y	∥ψ′(u)∥y	PROPN
ejde-673	437	5	∗	∗	NOUN
ejde-673	437	6	≤	≤	NOUN
ejde-673	437	7	2cm1(1	2cm1(1	NUM
ejde-673	437	8	+	+	SYM
ejde-673	437	9	φ(u)k−1)(∥h0∥lp′(·)(ω	φ(u)k−1)(∥h0∥lp′(·)(ω	ADJ
ejde-673	437	10	)	)	PUNCT
ejde-673	437	11	+	+	CCONJ
ejde-673	437	12	∥h1	∥h1	PROPN
ejde-673	437	13	/	/	SYM
ejde-673	437	14	p′	p′	NOUN
ejde-673	437	15	(	(	PUNCT
ejde-673	437	16	·	·	PUNCT
ejde-673	437	17	)	)	PUNCT
ejde-673	437	18	1	1	NUM
ejde-673	437	19	|∇u|p(·)−1∥lp′(·)(ω	|∇u|p(·)−1∥lp′(·)(ω	ADJ
ejde-673	437	20	)	)	PUNCT
ejde-673	437	21	)	)	PUNCT
ejde-673	437	22	.	.	PUNCT
ejde-673	438	1	here	here	ADV
ejde-673	438	2	we	we	PRON
ejde-673	438	3	note	note	VERB
ejde-673	438	4	that	that	SCONJ
ejde-673	438	5	φ(u)k−1	φ(u)k−1	X
ejde-673	438	6	≤	≤	ADJ
ejde-673	438	7	ck−1(2∥h0∥lp′(·)(ω)∥u∥y	ck−1(2∥h0∥lp′(·)(ω)∥u∥y	ADJ
ejde-673	438	8	+	+	CCONJ
ejde-673	438	9	∥u∥p	∥u∥p	NOUN
ejde-673	438	10	+	+	CCONJ
ejde-673	438	11	y	y	PROPN
ejde-673	438	12	∨	∨	NUM
ejde-673	438	13	∥u∥p	∥u∥p	NOUN
ejde-673	438	14	−	−	PROPN
ejde-673	438	15	y	y	PROPN
ejde-673	438	16	)	)	PUNCT
ejde-673	438	17	k−1	k−1	PROPN
ejde-673	438	18	,	,	PUNCT
ejde-673	438	19	ρp′(·)(h	ρp′(·)(h	NUM
ejde-673	438	20	1	1	NUM
ejde-673	438	21	/	/	SYM
ejde-673	438	22	p′	p′	PROPN
ejde-673	438	23	(	(	PUNCT
ejde-673	438	24	·	·	PUNCT
ejde-673	438	25	)	)	PUNCT
ejde-673	438	26	1	1	NUM
ejde-673	438	27	|∇u|p(·)−1	|∇u|p(·)−1	NOUN
ejde-673	438	28	)	)	PUNCT
ejde-673	438	29	=	=	SYM
ejde-673	438	30	∫	∫	PROPN
ejde-673	438	31	ω	ω	NUM
ejde-673	438	32	h1(x)|∇u(x)|p(x	h1(x)|∇u(x)|p(x	PROPN
ejde-673	438	33	)	)	PUNCT
ejde-673	438	34	dx	dx	PROPN
ejde-673	438	35	≤	≤	PROPN
ejde-673	438	36	∥u∥p	∥u∥p	NOUN
ejde-673	438	37	+	+	CCONJ
ejde-673	438	38	y	y	PROPN
ejde-673	438	39	∨	∨	NUM
ejde-673	438	40	∥u∥p−y	∥u∥p−y	PRON
ejde-673	438	41	.	.	PUNCT
ejde-673	439	1	if	if	SCONJ
ejde-673	439	2	∥u∥	∥u∥	NOUN
ejde-673	439	3	≤	≤	X
ejde-673	439	4	m	m	VERB
ejde-673	439	5	,	,	PUNCT
ejde-673	439	6	then	then	ADV
ejde-673	439	7	it	it	PRON
ejde-673	439	8	is	be	AUX
ejde-673	439	9	clear	clear	ADJ
ejde-673	439	10	that	that	SCONJ
ejde-673	439	11	there	there	PRON
ejde-673	439	12	exists	exist	VERB
ejde-673	439	13	a	a	DET
ejde-673	439	14	constant	constant	ADJ
ejde-673	439	15	c(a1	c(a1	NOUN
ejde-673	439	16	)	)	PUNCT
ejde-673	439	17	>	>	X
ejde-673	439	18	0	0	NUM
ejde-673	439	19	such	such	ADJ
ejde-673	439	20	that	that	SCONJ
ejde-673	439	21	∥ψ′(u)∥y	∥ψ′(u)∥y	PRON
ejde-673	439	22	∗	∗	NOUN
ejde-673	439	23	≤	≤	NUM
ejde-673	439	24	c(a1	c(a1	NOUN
ejde-673	439	25	)	)	PUNCT
ejde-673	439	26	,	,	PUNCT
ejde-673	439	27	so	so	ADV
ejde-673	439	28	ψ′	ψ′	PROPN
ejde-673	439	29	is	be	AUX
ejde-673	439	30	bounded	bound	VERB
ejde-673	439	31	on	on	ADP
ejde-673	439	32	every	every	DET
ejde-673	439	33	bounded	bound	VERB
ejde-673	439	34	subset	subset	NOUN
ejde-673	439	35	of	of	ADP
ejde-673	439	36	y	y	PROPN
ejde-673	439	37	.	.	PUNCT
ejde-673	440	1	let	let	VERB
ejde-673	440	2	∥u∥y	∥u∥y	PRON
ejde-673	440	3	>	>	X
ejde-673	440	4	1	1	X
ejde-673	440	5	.	.	PUNCT
ejde-673	441	1	then	then	ADV
ejde-673	441	2	from	from	ADP
ejde-673	441	3	(	(	PUNCT
ejde-673	441	4	a1	a1	NOUN
ejde-673	441	5	)	)	PUNCT
ejde-673	441	6	and	and	CCONJ
ejde-673	441	7	(	(	PUNCT
ejde-673	441	8	a5	a5	PROPN
ejde-673	441	9	)	)	PUNCT
ejde-673	441	10	,	,	PUNCT
ejde-673	441	11	⟨ψ′(u	⟨ψ′(u	PROPN
ejde-673	441	12	)	)	PUNCT
ejde-673	441	13	,	,	PUNCT
ejde-673	441	14	u⟩y	u⟩y	NUM
ejde-673	441	15	∗,y	∗,y	NOUN
ejde-673	441	16	=	=	SYM
ejde-673	441	17	m(φ(u	m(φ(u	NOUN
ejde-673	441	18	)	)	PUNCT
ejde-673	441	19	)	)	PUNCT
ejde-673	442	1	∫	∫	PROPN
ejde-673	442	2	ω	ω	NUM
ejde-673	442	3	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	442	4	)	)	PUNCT
ejde-673	442	5	·	·	PUNCT
ejde-673	442	6	∇u(x	∇u(x	NOUN
ejde-673	442	7	)	)	PUNCT
ejde-673	442	8	dx	dx	PROPN
ejde-673	442	9	≥	≥	PROPN
ejde-673	442	10	k0m(φ(u	k0m(φ(u	NOUN
ejde-673	442	11	)	)	PUNCT
ejde-673	442	12	)	)	PUNCT
ejde-673	443	1	∫	∫	PROPN
ejde-673	443	2	ω	ω	NUM
ejde-673	443	3	h1(x)|∇u(x)|p(x	h1(x)|∇u(x)|p(x	PROPN
ejde-673	443	4	)	)	PUNCT
ejde-673	443	5	dx	dx	PROPN
ejde-673	443	6	≥	≥	PROPN
ejde-673	443	7	kl0	kl0	PROPN
ejde-673	443	8	(	(	PUNCT
ejde-673	443	9	p+)l−1	p+)l−1	NUM
ejde-673	443	10	m0∥u∥(l−1)p−	m0∥u∥(l−1)p−	NUM
ejde-673	443	11	y	y	PROPN
ejde-673	443	12	∥u∥p	∥u∥p	NOUN
ejde-673	443	13	−	−	PROPN
ejde-673	443	14	y	y	PROPN
ejde-673	443	15	ejde-2025/17	ejde-2025/17	X
ejde-673	443	16	eigenvalue	eigenvalue	VERB
ejde-673	443	17	problems	problem	NOUN
ejde-673	443	18	for	for	ADP
ejde-673	443	19	kirchhoff	kirchhoff	NOUN
ejde-673	443	20	-	-	PUNCT
ejde-673	443	21	type	type	NOUN
ejde-673	443	22	equations	equation	NOUN
ejde-673	443	23	15	15	NUM
ejde-673	443	24	=	=	SYM
ejde-673	443	25	m0k	m0k	NOUN
ejde-673	443	26	l	l	NOUN
ejde-673	443	27	0	0	NUM
ejde-673	443	28	(	(	PUNCT
ejde-673	443	29	p+)l−1	p+)l−1	NUM
ejde-673	443	30	∥u∥lp	∥u∥lp	NUM
ejde-673	443	31	−	−	PROPN
ejde-673	443	32	y	y	PROPN
ejde-673	443	33	.	.	PUNCT
ejde-673	444	1	since	since	SCONJ
ejde-673	444	2	lp−	lp−	PROPN
ejde-673	444	3	>	>	X
ejde-673	444	4	1	1	NUM
ejde-673	444	5	,	,	PUNCT
ejde-673	444	6	this	this	PRON
ejde-673	444	7	implies	imply	VERB
ejde-673	444	8	the	the	DET
ejde-673	444	9	coerciveness	coerciveness	NOUN
ejde-673	444	10	of	of	ADP
ejde-673	444	11	ψ′.	ψ′.	PROPN
ejde-673	444	12	(	(	PUNCT
ejde-673	444	13	ii	ii	NOUN
ejde-673	444	14	)	)	PUNCT
ejde-673	444	15	let	let	VERB
ejde-673	444	16	un	un	PROPN
ejde-673	444	17	→	→	SYM
ejde-673	444	18	u	u	NOUN
ejde-673	444	19	weakly	weakly	ADV
ejde-673	444	20	in	in	ADP
ejde-673	444	21	y	y	PROPN
ejde-673	444	22	and	and	CCONJ
ejde-673	444	23	lim	lim	PROPN
ejde-673	444	24	supn→∞⟨ψ′(un	supn→∞⟨ψ′(un	PROPN
ejde-673	444	25	)	)	PUNCT
ejde-673	444	26	,	,	PUNCT
ejde-673	444	27	un	un	PROPN
ejde-673	444	28	−	−	PROPN
ejde-673	444	29	u⟩y	u⟩y	NUM
ejde-673	444	30	∗,y	∗,y	PROPN
ejde-673	444	31	≤	≤	NUM
ejde-673	444	32	0	0	NUM
ejde-673	444	33	.	.	PUNCT
ejde-673	445	1	since	since	SCONJ
ejde-673	445	2	ψ′	ψ′	PROPN
ejde-673	445	3	is	be	AUX
ejde-673	445	4	monotone	monotone	ADJ
ejde-673	445	5	from	from	ADP
ejde-673	445	6	(	(	PUNCT
ejde-673	445	7	i	i	NOUN
ejde-673	445	8	)	)	PUNCT
ejde-673	445	9	,	,	PUNCT
ejde-673	445	10	⟨ψ′(un)−ψ′(u	⟨ψ′(un)−ψ′(u	PROPN
ejde-673	445	11	)	)	PUNCT
ejde-673	445	12	,	,	PUNCT
ejde-673	445	13	un	un	PROPN
ejde-673	445	14	−	−	PROPN
ejde-673	445	15	u⟩y	u⟩y	NUM
ejde-673	445	16	∗,y	∗,y	PROPN
ejde-673	445	17	≥	≥	NOUN
ejde-673	445	18	0	0	NUM
ejde-673	445	19	.	.	PUNCT
ejde-673	446	1	hence	hence	ADV
ejde-673	446	2	0	0	NUM
ejde-673	446	3	≤	≤	PROPN
ejde-673	446	4	lim	lim	PROPN
ejde-673	446	5	inf	inf	PROPN
ejde-673	446	6	n→∞	n→∞	X
ejde-673	446	7	⟨ψ′(un)−ψ′(u	⟨ψ′(un)−ψ′(u	NUM
ejde-673	446	8	)	)	PUNCT
ejde-673	446	9	,	,	PUNCT
ejde-673	446	10	un	un	PROPN
ejde-673	446	11	−	−	PROPN
ejde-673	446	12	u⟩y	u⟩y	NUM
ejde-673	446	13	∗,y	∗,y	PROPN
ejde-673	446	14	=	=	SYM
ejde-673	446	15	lim	lim	PROPN
ejde-673	446	16	inf	inf	PROPN
ejde-673	446	17	n→∞	n→∞	NUM
ejde-673	446	18	⟨ψ′(un	⟨ψ′(un	PROPN
ejde-673	446	19	)	)	PUNCT
ejde-673	446	20	,	,	PUNCT
ejde-673	446	21	un	un	PROPN
ejde-673	446	22	−	−	PROPN
ejde-673	446	23	u⟩y	u⟩y	NUM
ejde-673	446	24	∗,y	∗,y	PROPN
ejde-673	446	25	≤	≤	NUM
ejde-673	446	26	lim	lim	PROPN
ejde-673	446	27	sup	sup	PROPN
ejde-673	446	28	n→∞	n→∞	NUM
ejde-673	446	29	⟨ψ′(un	⟨ψ′(un	PROPN
ejde-673	446	30	)	)	PUNCT
ejde-673	446	31	,	,	PUNCT
ejde-673	446	32	un	un	PROPN
ejde-673	446	33	−	−	PROPN
ejde-673	446	34	u⟩y	u⟩y	NUM
ejde-673	446	35	∗,y	∗,y	PROPN
ejde-673	446	36	≤	≤	NUM
ejde-673	446	37	0	0	NUM
ejde-673	446	38	.	.	PUNCT
ejde-673	447	1	therefore	therefore	ADV
ejde-673	447	2	,	,	PUNCT
ejde-673	447	3	limn→∞	limn→∞	PROPN
ejde-673	447	4	m(φ(un))⟨φ′(un	m(φ(un))⟨φ′(un	PROPN
ejde-673	447	5	)	)	PUNCT
ejde-673	447	6	,	,	PUNCT
ejde-673	448	1	un−u⟩y	un−u⟩y	NOUN
ejde-673	448	2	∗,y	∗,y	PROPN
ejde-673	448	3	=	=	SYM
ejde-673	448	4	0	0	NUM
ejde-673	448	5	.	.	PUNCT
ejde-673	449	1	since	since	SCONJ
ejde-673	449	2	un	un	PROPN
ejde-673	449	3	→	→	SYM
ejde-673	449	4	u	u	PROPN
ejde-673	449	5	weakly	weakly	ADV
ejde-673	449	6	in	in	ADP
ejde-673	449	7	y	y	PROPN
ejde-673	449	8	,	,	PUNCT
ejde-673	449	9	the	the	DET
ejde-673	449	10	sequence	sequence	NOUN
ejde-673	449	11	{	{	PUNCT
ejde-673	449	12	∥un∥y	∥un∥y	X
ejde-673	449	13	}	}	PUNCT
ejde-673	449	14	is	be	AUX
ejde-673	449	15	bounded	bound	VERB
ejde-673	449	16	.	.	PUNCT
ejde-673	450	1	hence	hence	ADV
ejde-673	450	2	since	since	SCONJ
ejde-673	450	3	m(φ(un	m(φ(un	NOUN
ejde-673	450	4	)	)	PUNCT
ejde-673	450	5	)	)	PUNCT
ejde-673	450	6	is	be	AUX
ejde-673	450	7	bounded	bound	VERB
ejde-673	450	8	from	from	ADP
ejde-673	450	9	lemma	lemma	PROPN
ejde-673	450	10	3.7	3.7	NUM
ejde-673	450	11	(	(	PUNCT
ejde-673	450	12	i	i	NOUN
ejde-673	450	13	)	)	PUNCT
ejde-673	450	14	,	,	PUNCT
ejde-673	450	15	we	we	PRON
ejde-673	450	16	have	have	VERB
ejde-673	450	17	limn→∞	limn→∞	PROPN
ejde-673	450	18	m(φ(un))⟨φ′(u	m(φ(un))⟨φ′(u	NOUN
ejde-673	450	19	)	)	PUNCT
ejde-673	450	20	,	,	PUNCT
ejde-673	450	21	un	un	PROPN
ejde-673	451	1	−	−	PROPN
ejde-673	451	2	u⟩y	u⟩y	NUM
ejde-673	451	3	∗,y	∗,y	PROPN
ejde-673	451	4	=	=	SYM
ejde-673	451	5	0	0	NUM
ejde-673	451	6	.	.	PUNCT
ejde-673	452	1	therefore	therefore	ADV
ejde-673	452	2	,	,	PUNCT
ejde-673	452	3	lim	lim	PROPN
ejde-673	452	4	n→∞	n→∞	NUM
ejde-673	452	5	m(φ(un))⟨φ′(un)−	m(φ(un))⟨φ′(un)−	PROPN
ejde-673	452	6	φ′(u	φ′(u	PROPN
ejde-673	452	7	)	)	PUNCT
ejde-673	452	8	,	,	PUNCT
ejde-673	452	9	un	un	PROPN
ejde-673	452	10	−	−	PROPN
ejde-673	452	11	u⟩y	u⟩y	NUM
ejde-673	452	12	∗,y	∗,y	PROPN
ejde-673	452	13	=	=	SYM
ejde-673	452	14	0	0	X
ejde-673	452	15	.	.	PUNCT
ejde-673	453	1	thereby	thereby	ADV
ejde-673	453	2	,	,	PUNCT
ejde-673	453	3	since	since	SCONJ
ejde-673	453	4	m(φ(un	m(φ(un	NOUN
ejde-673	453	5	)	)	PUNCT
ejde-673	453	6	)	)	PUNCT
ejde-673	453	7	≥	≥	NOUN
ejde-673	453	8	0	0	NUM
ejde-673	453	9	and	and	CCONJ
ejde-673	453	10	⟨φ′(un	⟨φ′(un	PROPN
ejde-673	453	11	)	)	PUNCT
ejde-673	453	12	−	−	PROPN
ejde-673	453	13	φ′(u	φ′(u	NOUN
ejde-673	453	14	)	)	PUNCT
ejde-673	453	15	,	,	PUNCT
ejde-673	453	16	un	un	PROPN
ejde-673	453	17	−	−	PROPN
ejde-673	453	18	u⟩y	u⟩y	NUM
ejde-673	453	19	∗,y	∗,y	PROPN
ejde-673	453	20	≥	≥	NUM
ejde-673	453	21	0	0	NUM
ejde-673	453	22	,	,	PUNCT
ejde-673	453	23	we	we	PRON
ejde-673	453	24	obtain	obtain	VERB
ejde-673	453	25	that	that	SCONJ
ejde-673	453	26	limn→∞	limn→∞	PROPN
ejde-673	453	27	m(φ(un	m(φ(un	X
ejde-673	453	28	)	)	PUNCT
ejde-673	453	29	)	)	PUNCT
ejde-673	454	1	=	=	SYM
ejde-673	454	2	0	0	NUM
ejde-673	454	3	or	or	CCONJ
ejde-673	454	4	limn→∞⟨φ′(un	limn→∞⟨φ′(un	PROPN
ejde-673	454	5	)	)	PUNCT
ejde-673	454	6	−	−	PROPN
ejde-673	454	7	φ′(u	φ′(u	NOUN
ejde-673	454	8	)	)	PUNCT
ejde-673	454	9	,	,	PUNCT
ejde-673	454	10	un	un	PROPN
ejde-673	455	1	−	−	PROPN
ejde-673	455	2	u⟩y	u⟩y	NUM
ejde-673	455	3	∗,y	∗,y	PROPN
ejde-673	455	4	=	=	SYM
ejde-673	455	5	0	0	X
ejde-673	455	6	.	.	PUNCT
ejde-673	456	1	indeed	indeed	ADV
ejde-673	456	2	,	,	PUNCT
ejde-673	456	3	if	if	SCONJ
ejde-673	456	4	we	we	PRON
ejde-673	456	5	put	put	VERB
ejde-673	456	6	an	an	DET
ejde-673	456	7	=	=	ADJ
ejde-673	456	8	m(φ(un	m(φ(un	NOUN
ejde-673	456	9	)	)	PUNCT
ejde-673	456	10	)	)	PUNCT
ejde-673	456	11	and	and	CCONJ
ejde-673	456	12	bn	bn	X
ejde-673	456	13	=	=	SYM
ejde-673	456	14	⟨φ′(un	⟨φ′(un	PROPN
ejde-673	456	15	)	)	PUNCT
ejde-673	456	16	−	−	PROPN
ejde-673	456	17	φ′(u	φ′(u	NOUN
ejde-673	456	18	)	)	PUNCT
ejde-673	456	19	,	,	PUNCT
ejde-673	456	20	un	un	PROPN
ejde-673	456	21	−	−	PROPN
ejde-673	456	22	u⟩y	u⟩y	NUM
ejde-673	456	23	∗,y	∗,y	PROPN
ejde-673	456	24	,	,	PUNCT
ejde-673	456	25	then	then	ADV
ejde-673	456	26	it	it	PRON
ejde-673	456	27	suffices	suffice	VERB
ejde-673	456	28	to	to	PART
ejde-673	456	29	derive	derive	VERB
ejde-673	456	30	that	that	SCONJ
ejde-673	456	31	an	an	DET
ejde-673	456	32	≥	≥	NOUN
ejde-673	456	33	0	0	NUM
ejde-673	456	34	,	,	PUNCT
ejde-673	456	35	bn	bn	X
ejde-673	456	36	≥	≥	NOUN
ejde-673	456	37	0	0	NUM
ejde-673	456	38	and	and	CCONJ
ejde-673	456	39	limn→∞	limn→∞	PROPN
ejde-673	456	40	anbn	anbn	PROPN
ejde-673	456	41	=	=	SYM
ejde-673	456	42	0	0	NUM
ejde-673	456	43	implies	imply	VERB
ejde-673	456	44	that	that	SCONJ
ejde-673	456	45	limn→∞	limn→∞	PROPN
ejde-673	456	46	an	an	DET
ejde-673	456	47	=	=	SYM
ejde-673	456	48	0	0	NUM
ejde-673	456	49	or	or	CCONJ
ejde-673	456	50	limn→∞	limn→∞	PRON
ejde-673	456	51	bn	bn	X
ejde-673	456	52	=	=	SYM
ejde-673	456	53	0	0	PROPN
ejde-673	456	54	.	.	PUNCT
ejde-673	457	1	for	for	ADP
ejde-673	457	2	any	any	DET
ejde-673	457	3	subsequence	subsequence	NOUN
ejde-673	457	4	{	{	PUNCT
ejde-673	457	5	n′	n′	NOUN
ejde-673	457	6	}	}	PUNCT
ejde-673	457	7	of	of	ADP
ejde-673	457	8	n	n	CCONJ
ejde-673	457	9	,	,	PUNCT
ejde-673	457	10	we	we	PRON
ejde-673	457	11	have	have	VERB
ejde-673	457	12	limn′→∞	limn′→∞	NOUN
ejde-673	457	13	an′bn′	an′bn′	NOUN
ejde-673	457	14	=	=	X
ejde-673	458	1	0	0	X
ejde-673	458	2	.	.	PUNCT
ejde-673	459	1	if	if	SCONJ
ejde-673	459	2	limn′→∞	limn′→∞	NOUN
ejde-673	459	3	an′	an′	NOUN
ejde-673	459	4	does	do	AUX
ejde-673	459	5	not	not	PART
ejde-673	459	6	exist	exist	VERB
ejde-673	459	7	or	or	CCONJ
ejde-673	459	8	exists	exist	VERB
ejde-673	459	9	and	and	CCONJ
ejde-673	459	10	is	be	AUX
ejde-673	459	11	equal	equal	ADJ
ejde-673	459	12	to	to	ADP
ejde-673	459	13	a	a	DET
ejde-673	459	14	positive	positive	ADJ
ejde-673	459	15	number	number	NOUN
ejde-673	459	16	,	,	PUNCT
ejde-673	459	17	then	then	ADV
ejde-673	459	18	there	there	PRON
ejde-673	459	19	exist	exist	VERB
ejde-673	459	20	ε0	ε0	PROPN
ejde-673	459	21	>	>	X
ejde-673	459	22	0	0	PUNCT
ejde-673	460	1	and	and	CCONJ
ejde-673	460	2	a	a	DET
ejde-673	460	3	subsequence	subsequence	NOUN
ejde-673	460	4	{	{	PUNCT
ejde-673	460	5	an′′	an′′	NOUN
ejde-673	460	6	}	}	PUNCT
ejde-673	460	7	of	of	ADP
ejde-673	460	8	{	{	PUNCT
ejde-673	460	9	an′	an′	PROPN
ejde-673	460	10	}	}	PUNCT
ejde-673	460	11	such	such	ADJ
ejde-673	460	12	that	that	SCONJ
ejde-673	460	13	an′′	an′′	PROPN
ejde-673	460	14	≥	≥	PRON
ejde-673	460	15	ε0	ε0	PROPN
ejde-673	460	16	for	for	ADP
ejde-673	460	17	any	any	DET
ejde-673	460	18	an′′	an′′	PROPN
ejde-673	460	19	.	.	PUNCT
ejde-673	461	1	hence	hence	ADV
ejde-673	461	2	we	we	PRON
ejde-673	461	3	have	have	VERB
ejde-673	461	4	an′′bn′′	an′′bn′′	PROPN
ejde-673	461	5	≥	≥	NUM
ejde-673	461	6	ε0bn′′	ε0bn′′	X
ejde-673	461	7	≥	≥	NOUN
ejde-673	461	8	0	0	NUM
ejde-673	461	9	.	.	PUNCT
ejde-673	462	1	since	since	SCONJ
ejde-673	462	2	limn′′→∞	limn′′→∞	NOUN
ejde-673	462	3	an′′bn′′	an′′bn′′	PROPN
ejde-673	462	4	=	=	SYM
ejde-673	462	5	0	0	NUM
ejde-673	462	6	,	,	PUNCT
ejde-673	462	7	we	we	PRON
ejde-673	462	8	see	see	VERB
ejde-673	462	9	that	that	SCONJ
ejde-673	462	10	limn′′→∞	limn′′→∞	NOUN
ejde-673	462	11	bn′′	bn′′	NOUN
ejde-673	462	12	=	=	SYM
ejde-673	462	13	0	0	NUM
ejde-673	462	14	,	,	PUNCT
ejde-673	462	15	so	so	SCONJ
ejde-673	462	16	according	accord	VERB
ejde-673	462	17	to	to	ADP
ejde-673	462	18	the	the	DET
ejde-673	462	19	convergent	convergent	NOUN
ejde-673	462	20	principal	principal	NOUN
ejde-673	462	21	we	we	PRON
ejde-673	462	22	have	have	VERB
ejde-673	462	23	limn→∞	limn→∞	PROPN
ejde-673	462	24	bn	bn	X
ejde-673	462	25	=	=	SYM
ejde-673	462	26	0	0	PROPN
ejde-673	462	27	.	.	PUNCT
ejde-673	463	1	if	if	SCONJ
ejde-673	463	2	limn′→∞	limn′→∞	NOUN
ejde-673	463	3	an′	an′	NOUN
ejde-673	464	1	=	=	X
ejde-673	464	2	0	0	NUM
ejde-673	464	3	for	for	ADP
ejde-673	464	4	any	any	DET
ejde-673	464	5	subsequence	subsequence	NOUN
ejde-673	464	6	{	{	PUNCT
ejde-673	464	7	an′	an′	PROPN
ejde-673	464	8	}	}	PUNCT
ejde-673	464	9	,	,	PUNCT
ejde-673	464	10	then	then	ADV
ejde-673	464	11	we	we	PRON
ejde-673	464	12	clearly	clearly	ADV
ejde-673	464	13	have	have	VERB
ejde-673	464	14	limn→∞	limn→∞	PROPN
ejde-673	464	15	an	an	DET
ejde-673	464	16	=	=	NOUN
ejde-673	464	17	0	0	NUM
ejde-673	464	18	.	.	PUNCT
ejde-673	465	1	when	when	SCONJ
ejde-673	465	2	m(φ(un	m(φ(un	NOUN
ejde-673	465	3	)	)	PUNCT
ejde-673	465	4	)	)	PUNCT
ejde-673	465	5	→	→	SYM
ejde-673	465	6	0	0	NUM
ejde-673	465	7	as	as	ADP
ejde-673	465	8	n	n	NUM
ejde-673	465	9	→	→	SYM
ejde-673	465	10	∞	∞	PROPN
ejde-673	465	11	,	,	PUNCT
ejde-673	465	12	we	we	PRON
ejde-673	465	13	have	have	VERB
ejde-673	465	14	φ(un	φ(un	NUM
ejde-673	465	15	)	)	PUNCT
ejde-673	465	16	→	→	SYM
ejde-673	465	17	0	0	X
ejde-673	465	18	=	=	SYM
ejde-673	465	19	φ(0	φ(0	ADJ
ejde-673	465	20	)	)	PUNCT
ejde-673	465	21	.	.	PUNCT
ejde-673	466	1	by	by	ADP
ejde-673	466	2	lemma	lemma	PROPN
ejde-673	466	3	3.8	3.8	NUM
ejde-673	466	4	with	with	ADP
ejde-673	466	5	m	m	PROPN
ejde-673	466	6	≡	≡	PROPN
ejde-673	466	7	1	1	NUM
ejde-673	466	8	,	,	PUNCT
ejde-673	466	9	un	un	PROPN
ejde-673	466	10	→	→	SYM
ejde-673	466	11	0	0	NUM
ejde-673	466	12	strongly	strongly	ADV
ejde-673	466	13	in	in	ADP
ejde-673	466	14	y	y	PROPN
ejde-673	466	15	(	(	PUNCT
ejde-673	466	16	in	in	ADP
ejde-673	466	17	this	this	DET
ejde-673	466	18	case	case	NOUN
ejde-673	466	19	we	we	PRON
ejde-673	466	20	necessarily	necessarily	ADV
ejde-673	466	21	have	have	VERB
ejde-673	466	22	u	u	NOUN
ejde-673	466	23	=	=	NOUN
ejde-673	466	24	0	0	NUM
ejde-673	466	25	)	)	PUNCT
ejde-673	466	26	.	.	PUNCT
ejde-673	467	1	when	when	SCONJ
ejde-673	467	2	lim	lim	PROPN
ejde-673	467	3	n→∞	n→∞	X
ejde-673	467	4	⟨φ′(un)−	⟨φ′(un)−	PUNCT
ejde-673	467	5	φ′(u	φ′(u	PROPN
ejde-673	467	6	)	)	PUNCT
ejde-673	467	7	,	,	PUNCT
ejde-673	467	8	un	un	PROPN
ejde-673	467	9	−	−	PROPN
ejde-673	467	10	u⟩y	u⟩y	NUM
ejde-673	467	11	∗,y	∗,y	PROPN
ejde-673	467	12	=	=	SYM
ejde-673	467	13	lim	lim	PROPN
ejde-673	467	14	n→∞	n→∞	NUM
ejde-673	468	1	⟨φ′(un	⟨φ′(un	PROPN
ejde-673	468	2	)	)	PUNCT
ejde-673	468	3	,	,	PUNCT
ejde-673	468	4	un	un	PROPN
ejde-673	469	1	−	−	PROPN
ejde-673	469	2	u⟩y	u⟩y	NUM
ejde-673	469	3	∗,y	∗,y	PROPN
ejde-673	469	4	=	=	SYM
ejde-673	469	5	0	0	NUM
ejde-673	469	6	,	,	PUNCT
ejde-673	469	7	since	since	SCONJ
ejde-673	469	8	φ′	φ′	NUM
ejde-673	469	9	is	be	AUX
ejde-673	469	10	of	of	ADP
ejde-673	469	11	(	(	PUNCT
ejde-673	469	12	s+)-type	s+)-type	PROPN
ejde-673	469	13	(	(	PUNCT
ejde-673	469	14	cf	cf	NOUN
ejde-673	469	15	.	.	PUNCT
ejde-673	470	1	[	[	X
ejde-673	470	2	9	9	NUM
ejde-673	470	3	,	,	PUNCT
ejde-673	470	4	proposition	proposition	NOUN
ejde-673	470	5	21	21	NUM
ejde-673	470	6	(	(	PUNCT
ejde-673	470	7	ii	ii	NOUN
ejde-673	470	8	)	)	PUNCT
ejde-673	470	9	]	]	PUNCT
ejde-673	470	10	)	)	PUNCT
ejde-673	470	11	,	,	PUNCT
ejde-673	470	12	we	we	PRON
ejde-673	470	13	have	have	VERB
ejde-673	470	14	un	un	PROPN
ejde-673	470	15	→	→	SYM
ejde-673	470	16	u	u	PRON
ejde-673	470	17	strongly	strongly	ADV
ejde-673	470	18	in	in	ADP
ejde-673	470	19	y	y	PROPN
ejde-673	470	20	.	.	PUNCT
ejde-673	471	1	(	(	PUNCT
ejde-673	471	2	iii	iii	NOUN
ejde-673	471	3	)	)	PUNCT
ejde-673	471	4	since	since	SCONJ
ejde-673	471	5	ψ′	ψ′	PROPN
ejde-673	471	6	is	be	AUX
ejde-673	471	7	strictly	strictly	ADV
ejde-673	471	8	monotone	monotone	ADJ
ejde-673	471	9	from	from	ADP
ejde-673	471	10	(	(	PUNCT
ejde-673	471	11	i	i	NOUN
ejde-673	471	12	)	)	PUNCT
ejde-673	471	13	,	,	PUNCT
ejde-673	471	14	ψ′	ψ′	PUNCT
ejde-673	471	15	is	be	AUX
ejde-673	471	16	injective	injective	ADJ
ejde-673	471	17	.	.	PUNCT
ejde-673	472	1	we	we	PRON
ejde-673	472	2	show	show	VERB
ejde-673	472	3	that	that	PRON
ejde-673	472	4	ψ′	ψ′	PUNCT
ejde-673	472	5	:	:	PUNCT
ejde-673	472	6	y	y	PROPN
ejde-673	472	7	→	→	SYM
ejde-673	472	8	y	y	PROPN
ejde-673	472	9	∗	∗	NOUN
ejde-673	472	10	is	be	AUX
ejde-673	472	11	surjective	surjective	ADJ
ejde-673	472	12	.	.	PUNCT
ejde-673	473	1	let	let	VERB
ejde-673	473	2	w	w	PROPN
ejde-673	473	3	∈	∈	PROPN
ejde-673	473	4	y	y	PROPN
ejde-673	473	5	∗.	∗.	AUX
ejde-673	473	6	define	define	VERB
ejde-673	473	7	a	a	DET
ejde-673	473	8	functional	functional	ADJ
ejde-673	473	9	on	on	ADP
ejde-673	473	10	y	y	PROPN
ejde-673	473	11	by	by	ADP
ejde-673	473	12	φ(u	φ(u	NOUN
ejde-673	473	13	)	)	PUNCT
ejde-673	473	14	:	:	PUNCT
ejde-673	473	15	=	=	PUNCT
ejde-673	473	16	ψ(u)−	ψ(u)−	PROPN
ejde-673	473	17	⟨w	⟨w	ADV
ejde-673	473	18	,	,	PUNCT
ejde-673	473	19	u⟩y	u⟩y	NUM
ejde-673	473	20	∗,y	∗,y	PROPN
ejde-673	473	21	for	for	ADP
ejde-673	473	22	u	u	PROPN
ejde-673	473	23	∈	∈	PROPN
ejde-673	473	24	y.	y.	NOUN
ejde-673	473	25	from	from	ADP
ejde-673	473	26	(	(	PUNCT
ejde-673	473	27	a1	a1	PROPN
ejde-673	473	28	)	)	PUNCT
ejde-673	473	29	and	and	CCONJ
ejde-673	473	30	lemma	lemma	PROPN
ejde-673	473	31	3.7	3.7	NUM
ejde-673	473	32	(	(	PUNCT
ejde-673	473	33	i	i	NOUN
ejde-673	473	34	)	)	PUNCT
ejde-673	473	35	,	,	PUNCT
ejde-673	473	36	for	for	ADP
ejde-673	473	37	∥u∥y	∥u∥y	X
ejde-673	473	38	>	>	X
ejde-673	473	39	1	1	NUM
ejde-673	473	40	,	,	PUNCT
ejde-673	473	41	we	we	PRON
ejde-673	473	42	see	see	VERB
ejde-673	473	43	that	that	SCONJ
ejde-673	473	44	φ(u	φ(u	NOUN
ejde-673	473	45	)	)	PUNCT
ejde-673	473	46	≥	≥	NOUN
ejde-673	473	47	m̂(φ(u))−	m̂(φ(u))−	NUM
ejde-673	473	48	⟨w	⟨w	NOUN
ejde-673	473	49	,	,	PUNCT
ejde-673	473	50	u⟩y	u⟩y	PROPN
ejde-673	473	51	∗,y	∗,y	PROPN
ejde-673	473	52	≥	≥	NUM
ejde-673	473	53	(	(	PUNCT
ejde-673	473	54	k0	k0	PROPN
ejde-673	473	55	p+	p+	PROPN
ejde-673	473	56	)	)	PUNCT
ejde-673	474	1	l	l	NOUN
ejde-673	474	2	∥u∥lp	∥u∥lp	NUM
ejde-673	474	3	−	−	PROPN
ejde-673	474	4	y	y	PROPN
ejde-673	474	5	−	−	PROPN
ejde-673	474	6	∥w∥y	∥w∥y	VERB
ejde-673	474	7	∗∥u∥y	∗∥u∥y	PRON
ejde-673	474	8	.	.	PUNCT
ejde-673	475	1	since	since	SCONJ
ejde-673	475	2	lp−	lp−	PROPN
ejde-673	475	3	>	>	X
ejde-673	475	4	1	1	NUM
ejde-673	475	5	,	,	PUNCT
ejde-673	475	6	φ	φ	PROPN
ejde-673	475	7	is	be	AUX
ejde-673	475	8	coercive	coercive	ADJ
ejde-673	475	9	.	.	PUNCT
ejde-673	476	1	since	since	SCONJ
ejde-673	476	2	ψ	ψ	NOUN
ejde-673	476	3	is	be	AUX
ejde-673	476	4	sequentially	sequentially	ADV
ejde-673	476	5	weakly	weakly	ADV
ejde-673	476	6	lower	low	ADJ
ejde-673	476	7	semi	semi	ADJ
ejde-673	476	8	-	-	ADJ
ejde-673	476	9	continuous	continuous	ADJ
ejde-673	476	10	,	,	PUNCT
ejde-673	476	11	φ	φ	PROPN
ejde-673	476	12	is	be	AUX
ejde-673	476	13	so	so	ADV
ejde-673	476	14	.	.	PUNCT
ejde-673	477	1	if	if	SCONJ
ejde-673	477	2	we	we	PRON
ejde-673	477	3	put	put	VERB
ejde-673	477	4	γ	γ	NOUN
ejde-673	477	5	=	=	PUNCT
ejde-673	477	6	infu∈y	infu∈y	ADJ
ejde-673	477	7	φ(u	φ(u	NOUN
ejde-673	477	8	)	)	PUNCT
ejde-673	477	9	(	(	PUNCT
ejde-673	477	10	<	<	X
ejde-673	477	11	∞	∞	NUM
ejde-673	477	12	)	)	PUNCT
ejde-673	477	13	,	,	PUNCT
ejde-673	477	14	then	then	ADV
ejde-673	477	15	there	there	PRON
ejde-673	477	16	exists	exist	VERB
ejde-673	477	17	a	a	DET
ejde-673	477	18	sequence	sequence	NOUN
ejde-673	477	19	{	{	PUNCT
ejde-673	477	20	un	un	PROPN
ejde-673	477	21	}	}	PUNCT
ejde-673	477	22	⊂	⊂	PROPN
ejde-673	477	23	y	y	PROPN
ejde-673	477	24	such	such	ADJ
ejde-673	477	25	that	that	SCONJ
ejde-673	477	26	γ	γ	PROPN
ejde-673	477	27	=	=	SYM
ejde-673	477	28	limn→∞	limn→∞	PRON
ejde-673	477	29	φ(un	φ(un	NUM
ejde-673	477	30	)	)	PUNCT
ejde-673	477	31	.	.	PUNCT
ejde-673	478	1	since	since	SCONJ
ejde-673	478	2	φ	φ	PROPN
ejde-673	478	3	is	be	AUX
ejde-673	478	4	coercive	coercive	ADJ
ejde-673	478	5	,	,	PUNCT
ejde-673	478	6	the	the	DET
ejde-673	478	7	sequence	sequence	NOUN
ejde-673	478	8	{	{	PUNCT
ejde-673	478	9	un	un	PROPN
ejde-673	478	10	}	}	PUNCT
ejde-673	478	11	is	be	AUX
ejde-673	478	12	bounded	bound	VERB
ejde-673	478	13	.	.	PUNCT
ejde-673	479	1	since	since	SCONJ
ejde-673	479	2	y	y	PROPN
ejde-673	479	3	is	be	AUX
ejde-673	479	4	a	a	DET
ejde-673	479	5	reflexive	reflexive	ADJ
ejde-673	479	6	banach	banach	NOUN
ejde-673	479	7	space	space	NOUN
ejde-673	479	8	,	,	PUNCT
ejde-673	479	9	there	there	PRON
ejde-673	479	10	exist	exist	VERB
ejde-673	479	11	a	a	DET
ejde-673	479	12	subsequence	subsequence	NOUN
ejde-673	479	13	{	{	PUNCT
ejde-673	479	14	un′	un′	NOUN
ejde-673	479	15	}	}	PUNCT
ejde-673	479	16	of	of	ADP
ejde-673	479	17	{	{	PUNCT
ejde-673	479	18	un	un	PROPN
ejde-673	479	19	}	}	PUNCT
ejde-673	479	20	and	and	CCONJ
ejde-673	479	21	u0	u0	PROPN
ejde-673	479	22	∈	∈	PROPN
ejde-673	479	23	y	y	PROPN
ejde-673	479	24	such	such	ADJ
ejde-673	479	25	that	that	SCONJ
ejde-673	479	26	un′	un′	PROPN
ejde-673	479	27	→	→	SYM
ejde-673	479	28	u0	u0	ADJ
ejde-673	479	29	weakly	weakly	ADJ
ejde-673	479	30	in	in	ADP
ejde-673	479	31	y	y	PROPN
ejde-673	479	32	,	,	PUNCT
ejde-673	479	33	so	so	ADV
ejde-673	479	34	φ(u0	φ(u0	NOUN
ejde-673	479	35	)	)	PUNCT
ejde-673	479	36	≤	≤	PROPN
ejde-673	479	37	lim	lim	PROPN
ejde-673	479	38	infn′→∞	infn′→∞	PROPN
ejde-673	479	39	φ(un′	φ(un′	NOUN
ejde-673	479	40	)	)	PUNCT
ejde-673	480	1	=	=	SYM
ejde-673	480	2	γ	γ	X
ejde-673	480	3	.	.	PUNCT
ejde-673	481	1	this	this	PRON
ejde-673	481	2	implies	imply	VERB
ejde-673	481	3	that	that	SCONJ
ejde-673	481	4	γ	γ	PROPN
ejde-673	481	5	>	>	X
ejde-673	481	6	−∞	−∞	PROPN
ejde-673	481	7	and	and	CCONJ
ejde-673	481	8	u0	u0	PROPN
ejde-673	481	9	is	be	AUX
ejde-673	481	10	a	a	DET
ejde-673	481	11	minimizer	minimizer	NOUN
ejde-673	481	12	of	of	ADP
ejde-673	481	13	φ	φ	PROPN
ejde-673	481	14	,	,	PUNCT
ejde-673	481	15	so	so	ADV
ejde-673	481	16	φ′(u0	φ′(u0	ADJ
ejde-673	481	17	)	)	PUNCT
ejde-673	482	1	=	=	SYM
ejde-673	482	2	0	0	NUM
ejde-673	482	3	,	,	PUNCT
ejde-673	482	4	i.e.	i.e.	X
ejde-673	482	5	,	,	PUNCT
ejde-673	482	6	ψ′(u0	ψ′(u0	NOUN
ejde-673	482	7	)	)	PUNCT
ejde-673	483	1	=	=	SYM
ejde-673	483	2	w.	w.	PROPN
ejde-673	483	3	therefore	therefore	ADV
ejde-673	483	4	,	,	PUNCT
ejde-673	483	5	ψ′	ψ′	PROPN
ejde-673	483	6	has	have	VERB
ejde-673	483	7	an	an	DET
ejde-673	483	8	inverse	inverse	NOUN
ejde-673	483	9	operator	operator	NOUN
ejde-673	483	10	(	(	PUNCT
ejde-673	483	11	ψ′)−1	ψ′)−1	X
ejde-673	483	12	:	:	PUNCT
ejde-673	483	13	y	y	NOUN
ejde-673	483	14	∗	∗	NOUN
ejde-673	483	15	→	→	SYM
ejde-673	483	16	y	y	PROPN
ejde-673	483	17	.	.	PUNCT
ejde-673	484	1	we	we	PRON
ejde-673	484	2	show	show	VERB
ejde-673	484	3	that	that	SCONJ
ejde-673	484	4	(	(	PUNCT
ejde-673	484	5	ψ′)−1	ψ′)−1	NOUN
ejde-673	484	6	is	be	AUX
ejde-673	484	7	continuous	continuous	ADJ
ejde-673	484	8	.	.	PUNCT
ejde-673	485	1	let	let	VERB
ejde-673	485	2	fn	fn	VERB
ejde-673	485	3	→	→	SYM
ejde-673	485	4	f	f	PROPN
ejde-673	485	5	in	in	ADP
ejde-673	485	6	y	y	PROPN
ejde-673	485	7	∗	∗	NOUN
ejde-673	485	8	as	as	ADP
ejde-673	485	9	n	n	PROPN
ejde-673	485	10	→	→	SYM
ejde-673	485	11	∞.	∞.	PROPN
ejde-673	485	12	then	then	ADV
ejde-673	485	13	there	there	PRON
ejde-673	485	14	exist	exist	VERB
ejde-673	485	15	un	un	PROPN
ejde-673	485	16	,	,	PUNCT
ejde-673	485	17	u	u	PROPN
ejde-673	485	18	∈	∈	PROPN
ejde-673	485	19	y	y	PROPN
ejde-673	485	20	such	such	ADJ
ejde-673	485	21	that	that	PRON
ejde-673	485	22	ψ′(un	ψ′(un	PROPN
ejde-673	485	23	)	)	PUNCT
ejde-673	485	24	=	=	SYM
ejde-673	485	25	fn	fn	NOUN
ejde-673	485	26	and	and	CCONJ
ejde-673	485	27	ψ′(u	ψ′(u	PRON
ejde-673	485	28	)	)	PUNCT
ejde-673	486	1	=	=	SYM
ejde-673	486	2	f	f	PROPN
ejde-673	486	3	.	.	PUNCT
ejde-673	487	1	then	then	ADV
ejde-673	487	2	{	{	PUNCT
ejde-673	487	3	un	un	PROPN
ejde-673	487	4	}	}	PUNCT
ejde-673	487	5	is	be	AUX
ejde-673	487	6	bounded	bound	VERB
ejde-673	487	7	in	in	ADP
ejde-673	487	8	y	y	PROPN
ejde-673	487	9	.	.	PUNCT
ejde-673	488	1	indeed	indeed	ADV
ejde-673	488	2	,	,	PUNCT
ejde-673	488	3	if	if	SCONJ
ejde-673	488	4	{	{	PUNCT
ejde-673	488	5	un	un	ADJ
ejde-673	488	6	}	}	PUNCT
ejde-673	488	7	is	be	AUX
ejde-673	488	8	16	16	NUM
ejde-673	488	9	j.	j.	PROPN
ejde-673	488	10	aramaki	aramaki	PROPN
ejde-673	488	11	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	488	12	unbounded	unbounded	ADJ
ejde-673	488	13	,	,	PUNCT
ejde-673	488	14	then	then	ADV
ejde-673	488	15	there	there	PRON
ejde-673	488	16	exists	exist	VERB
ejde-673	488	17	a	a	DET
ejde-673	488	18	subsequence	subsequence	NOUN
ejde-673	488	19	{	{	PUNCT
ejde-673	488	20	un′	un′	NOUN
ejde-673	488	21	}	}	PUNCT
ejde-673	488	22	of	of	ADP
ejde-673	488	23	{	{	PUNCT
ejde-673	488	24	un	un	PROPN
ejde-673	488	25	}	}	PUNCT
ejde-673	488	26	such	such	ADJ
ejde-673	488	27	that	that	SCONJ
ejde-673	488	28	∥un′∥y	∥un′∥y	PROPN
ejde-673	488	29	→	→	SYM
ejde-673	488	30	∞	∞	PROPN
ejde-673	488	31	as	as	ADP
ejde-673	488	32	n′	n′	PROPN
ejde-673	488	33	→	→	SYM
ejde-673	488	34	∞.	∞.	PROPN
ejde-673	488	35	hence	hence	ADV
ejde-673	488	36	⟨ψ′(un′	⟨ψ′(un′	PROPN
ejde-673	488	37	)	)	PUNCT
ejde-673	488	38	,	,	PUNCT
ejde-673	488	39	un′⟩y	un′⟩y	NOUN
ejde-673	488	40	∗,y	∗,y	NUM
ejde-673	488	41	=	=	SYM
ejde-673	488	42	⟨fn′	⟨fn′	NOUN
ejde-673	488	43	,	,	PUNCT
ejde-673	488	44	un′⟩y	un′⟩y	PROPN
ejde-673	488	45	∗,y	∗,y	PROPN
ejde-673	488	46	≤	≤	NUM
ejde-673	488	47	∥fn′∥y	∥fn′∥y	PROPN
ejde-673	489	1	∗∥un′∥y	∗∥un′∥y	NOUN
ejde-673	489	2	≤	≤	ADJ
ejde-673	489	3	c∥un′∥y	c∥un′∥y	NOUN
ejde-673	489	4	for	for	ADP
ejde-673	489	5	some	some	DET
ejde-673	489	6	constant	constant	ADJ
ejde-673	489	7	c	c	NOUN
ejde-673	489	8	>	>	X
ejde-673	489	9	0	0	X
ejde-673	489	10	.	.	PUNCT
ejde-673	490	1	this	this	PRON
ejde-673	490	2	contradict	contradict	VERB
ejde-673	490	3	the	the	DET
ejde-673	490	4	coerciveness	coerciveness	NOUN
ejde-673	490	5	of	of	ADP
ejde-673	490	6	ψ′.	ψ′.	PROPN
ejde-673	490	7	since	since	SCONJ
ejde-673	490	8	y	y	PROPN
ejde-673	490	9	is	be	AUX
ejde-673	490	10	a	a	DET
ejde-673	490	11	reflexive	reflexive	ADJ
ejde-673	490	12	banach	banach	NOUN
ejde-673	490	13	space	space	NOUN
ejde-673	490	14	,	,	PUNCT
ejde-673	490	15	there	there	PRON
ejde-673	490	16	exist	exist	VERB
ejde-673	490	17	a	a	DET
ejde-673	490	18	subsequence	subsequence	NOUN
ejde-673	490	19	(	(	PUNCT
ejde-673	490	20	still	still	ADV
ejde-673	490	21	denoted	denote	VERB
ejde-673	490	22	by	by	ADP
ejde-673	490	23	{	{	PUNCT
ejde-673	490	24	un′	un′	NOUN
ejde-673	490	25	}	}	PUNCT
ejde-673	490	26	)	)	PUNCT
ejde-673	490	27	and	and	CCONJ
ejde-673	490	28	u0	u0	PROPN
ejde-673	490	29	∈	∈	PROPN
ejde-673	490	30	y	y	PROPN
ejde-673	490	31	such	such	ADJ
ejde-673	490	32	that	that	SCONJ
ejde-673	490	33	un′	un′	PROPN
ejde-673	490	34	→	→	SYM
ejde-673	490	35	u0	u0	ADJ
ejde-673	490	36	weakly	weakly	ADJ
ejde-673	490	37	in	in	ADP
ejde-673	490	38	y	y	PROPN
ejde-673	490	39	.	.	PUNCT
ejde-673	491	1	hence	hence	ADV
ejde-673	491	2	lim	lim	PROPN
ejde-673	491	3	n′→∞	n′→∞	PROPN
ejde-673	491	4	⟨ψ′(un′	⟨ψ′(un′	PROPN
ejde-673	491	5	)	)	PUNCT
ejde-673	491	6	,	,	PUNCT
ejde-673	491	7	un′	un′	PROPN
ejde-673	491	8	−	−	PROPN
ejde-673	491	9	u0⟩y	u0⟩y	NOUN
ejde-673	492	1	∗,y	∗,y	PROPN
ejde-673	492	2	=	=	SYM
ejde-673	492	3	lim	lim	PROPN
ejde-673	492	4	n′→∞	n′→∞	PROPN
ejde-673	492	5	⟨ψ′(un′)−ψ′(u	⟨ψ′(un′)−ψ′(u	PROPN
ejde-673	492	6	)	)	PUNCT
ejde-673	493	1	,	,	PUNCT
ejde-673	493	2	un′	un′	PROPN
ejde-673	493	3	−	−	PROPN
ejde-673	493	4	u0⟩y	u0⟩y	NOUN
ejde-673	493	5	∗,y	∗,y	PROPN
ejde-673	493	6	=	=	SYM
ejde-673	493	7	lim	lim	PROPN
ejde-673	493	8	n′→∞	n′→∞	PROPN
ejde-673	493	9	⟨fn′	⟨fn′	VERB
ejde-673	493	10	−	−	PROPN
ejde-673	493	11	f	f	PROPN
ejde-673	493	12	,	,	PUNCT
ejde-673	493	13	un′	un′	PROPN
ejde-673	493	14	−	−	PROPN
ejde-673	493	15	u0⟩y	u0⟩y	NOUN
ejde-673	493	16	∗,y	∗,y	NOUN
ejde-673	493	17	=	=	SYM
ejde-673	493	18	0	0	X
ejde-673	493	19	.	.	PUNCT
ejde-673	494	1	since	since	SCONJ
ejde-673	494	2	ψ′	ψ′	PROPN
ejde-673	494	3	is	be	AUX
ejde-673	494	4	of	of	ADP
ejde-673	494	5	(	(	PUNCT
ejde-673	494	6	s+)-type	s+)-type	PROPN
ejde-673	494	7	,	,	PUNCT
ejde-673	494	8	we	we	PRON
ejde-673	494	9	see	see	VERB
ejde-673	494	10	that	that	SCONJ
ejde-673	494	11	un′	un′	PROPN
ejde-673	494	12	→	→	SYM
ejde-673	494	13	u0	u0	ADJ
ejde-673	494	14	strongly	strongly	ADV
ejde-673	494	15	in	in	ADP
ejde-673	494	16	y	y	PROPN
ejde-673	494	17	.	.	PUNCT
ejde-673	495	1	according	accord	VERB
ejde-673	495	2	to	to	ADP
ejde-673	495	3	the	the	DET
ejde-673	495	4	continuity	continuity	NOUN
ejde-673	495	5	of	of	ADP
ejde-673	495	6	ψ′	ψ′	PROPN
ejde-673	495	7	,	,	PUNCT
ejde-673	495	8	ψ′(un′	ψ′(un′	PROPN
ejde-673	495	9	)	)	PUNCT
ejde-673	495	10	=	=	SYM
ejde-673	495	11	fn′	fn′	PROPN
ejde-673	495	12	→	→	SYM
ejde-673	495	13	f	f	NOUN
ejde-673	495	14	=	=	SYM
ejde-673	495	15	ψ′(u0	ψ′(u0	NOUN
ejde-673	495	16	)	)	PUNCT
ejde-673	495	17	=	=	PUNCT
ejde-673	496	1	ψ′(u	ψ′(u	NOUN
ejde-673	496	2	)	)	PUNCT
ejde-673	496	3	,	,	PUNCT
ejde-673	496	4	so	so	SCONJ
ejde-673	496	5	we	we	PRON
ejde-673	496	6	have	have	VERB
ejde-673	496	7	u0	u0	ADJ
ejde-673	496	8	=	=	NOUN
ejde-673	496	9	u	u	NOUN
ejde-673	496	10	from	from	ADP
ejde-673	496	11	the	the	DET
ejde-673	496	12	injectiveness	injectiveness	NOUN
ejde-673	496	13	of	of	ADP
ejde-673	496	14	ψ′.	ψ′.	NOUN
ejde-673	496	15	by	by	ADP
ejde-673	496	16	the	the	DET
ejde-673	496	17	convergent	convergent	NOUN
ejde-673	496	18	principle	principle	NOUN
ejde-673	496	19	(	(	PUNCT
ejde-673	496	20	cf	cf	NOUN
ejde-673	496	21	.	.	PUNCT
ejde-673	497	1	[	[	X
ejde-673	497	2	34	34	NUM
ejde-673	497	3	,	,	PUNCT
ejde-673	497	4	theorem	theorem	VERB
ejde-673	497	5	10.13	10.13	NUM
ejde-673	497	6	(	(	PUNCT
ejde-673	497	7	i	i	NOUN
ejde-673	497	8	)	)	PUNCT
ejde-673	497	9	]	]	PUNCT
ejde-673	497	10	)	)	PUNCT
ejde-673	497	11	,	,	PUNCT
ejde-673	497	12	for	for	ADP
ejde-673	497	13	full	full	ADJ
ejde-673	497	14	sequence	sequence	NOUN
ejde-673	497	15	{	{	PUNCT
ejde-673	497	16	un	un	PROPN
ejde-673	497	17	}	}	PUNCT
ejde-673	497	18	,	,	PUNCT
ejde-673	497	19	un	un	PROPN
ejde-673	497	20	→	→	SYM
ejde-673	497	21	u	u	PROPN
ejde-673	497	22	strongly	strongly	ADV
ejde-673	497	23	in	in	ADP
ejde-673	497	24	y	y	PROPN
ejde-673	497	25	,	,	PUNCT
ejde-673	497	26	that	that	ADV
ejde-673	497	27	is	is	ADV
ejde-673	497	28	,	,	PUNCT
ejde-673	497	29	(	(	PUNCT
ejde-673	497	30	ψ′)−1(fn	ψ′)−1(fn	NOUN
ejde-673	497	31	)	)	PUNCT
ejde-673	497	32	→	→	SYM
ejde-673	497	33	(	(	PUNCT
ejde-673	497	34	ψ′)−1(f	ψ′)−1(f	PROPN
ejde-673	497	35	)	)	PUNCT
ejde-673	497	36	as	as	ADP
ejde-673	497	37	n	n	PROPN
ejde-673	497	38	→	→	SYM
ejde-673	497	39	∞.	∞.	PROPN
ejde-673	497	40	□	□	PUNCT
ejde-673	497	41	for	for	ADP
ejde-673	497	42	the	the	DET
ejde-673	497	43	functional	functional	ADJ
ejde-673	497	44	k	k	NOUN
ejde-673	497	45	defined	define	VERB
ejde-673	497	46	by	by	ADP
ejde-673	497	47	(	(	PUNCT
ejde-673	497	48	3.7	3.7	NUM
ejde-673	497	49	)	)	PUNCT
ejde-673	497	50	,	,	PUNCT
ejde-673	497	51	we	we	PRON
ejde-673	497	52	have	have	VERB
ejde-673	497	53	the	the	DET
ejde-673	497	54	following	follow	VERB
ejde-673	497	55	proposition	proposition	NOUN
ejde-673	497	56	.	.	PUNCT
ejde-673	498	1	proposition	proposition	NOUN
ejde-673	498	2	3.11	3.11	NUM
ejde-673	498	3	.	.	PUNCT
ejde-673	499	1	under	under	ADP
ejde-673	499	2	hypotheses	hypothesis	NOUN
ejde-673	499	3	(	(	PUNCT
ejde-673	499	4	a8	a8	PROPN
ejde-673	499	5	)	)	PUNCT
ejde-673	499	6	,	,	PUNCT
ejde-673	499	7	we	we	PRON
ejde-673	499	8	have	have	VERB
ejde-673	499	9	the	the	DET
ejde-673	499	10	following	following	NOUN
ejde-673	499	11	.	.	PUNCT
ejde-673	500	1	(	(	PUNCT
ejde-673	500	2	i	i	NOUN
ejde-673	500	3	)	)	PUNCT
ejde-673	500	4	k	k	PROPN
ejde-673	500	5	∈	∈	PROPN
ejde-673	500	6	c1(y	c1(y	PROPN
ejde-673	500	7	,	,	PUNCT
ejde-673	500	8	r	r	NOUN
ejde-673	500	9	)	)	PUNCT
ejde-673	500	10	and	and	CCONJ
ejde-673	500	11	⟨k	⟨k	PRON
ejde-673	500	12	′(u	′(u	NOUN
ejde-673	500	13	)	)	PUNCT
ejde-673	500	14	,	,	PUNCT
ejde-673	500	15	v⟩y	v⟩y	PROPN
ejde-673	500	16	∗,y	∗,y	PROPN
ejde-673	500	17	=	=	SYM
ejde-673	500	18	∫	∫	PROPN
ejde-673	500	19	γ2	γ2	PROPN
ejde-673	500	20	g(x	g(x	PROPN
ejde-673	500	21	,	,	PUNCT
ejde-673	500	22	u(x))v(x	u(x))v(x	NOUN
ejde-673	500	23	)	)	PUNCT
ejde-673	500	24	dσx	dσx	NOUN
ejde-673	500	25	for	for	ADP
ejde-673	500	26	u	u	NOUN
ejde-673	500	27	,	,	PUNCT
ejde-673	500	28	v	v	ADP
ejde-673	500	29	∈	∈	PROPN
ejde-673	500	30	y.	y.	NOUN
ejde-673	500	31	(	(	PUNCT
ejde-673	500	32	3.10	3.10	NUM
ejde-673	500	33	)	)	PUNCT
ejde-673	500	34	(	(	PUNCT
ejde-673	500	35	ii	ii	NOUN
ejde-673	500	36	)	)	PUNCT
ejde-673	501	1	k	k	X
ejde-673	501	2	is	be	AUX
ejde-673	501	3	sequentially	sequentially	ADV
ejde-673	501	4	weakly	weakly	ADV
ejde-673	501	5	continuous	continuous	ADJ
ejde-673	501	6	in	in	ADP
ejde-673	501	7	y	y	PROPN
ejde-673	501	8	.	.	PUNCT
ejde-673	502	1	(	(	PUNCT
ejde-673	502	2	iii	iii	X
ejde-673	502	3	)	)	PUNCT
ejde-673	503	1	k	k	NOUN
ejde-673	503	2	′	′	NUM
ejde-673	503	3	:	:	PUNCT
ejde-673	504	1	y	y	PROPN
ejde-673	504	2	→	→	SYM
ejde-673	504	3	y	y	PROPN
ejde-673	504	4	∗	∗	NOUN
ejde-673	504	5	is	be	AUX
ejde-673	504	6	weakly	weakly	ADV
ejde-673	504	7	-	-	PUNCT
ejde-673	504	8	strongly	strongly	ADV
ejde-673	504	9	continuous	continuous	ADJ
ejde-673	504	10	,	,	PUNCT
ejde-673	504	11	that	that	ADV
ejde-673	504	12	is	is	ADV
ejde-673	504	13	,	,	PUNCT
ejde-673	504	14	if	if	SCONJ
ejde-673	504	15	un	un	PROPN
ejde-673	504	16	→	→	SYM
ejde-673	504	17	u	u	PROPN
ejde-673	504	18	weakly	weakly	ADV
ejde-673	504	19	in	in	ADP
ejde-673	504	20	y	y	PROPN
ejde-673	504	21	as	as	ADP
ejde-673	504	22	n	n	PROPN
ejde-673	504	23	→	→	SYM
ejde-673	504	24	∞	∞	PROPN
ejde-673	504	25	,	,	PUNCT
ejde-673	504	26	then	then	ADV
ejde-673	504	27	k	k	PROPN
ejde-673	504	28	′(un	′(un	PROPN
ejde-673	504	29	)	)	PUNCT
ejde-673	504	30	→	→	SYM
ejde-673	504	31	k	k	X
ejde-673	504	32	′(u	′(u	NOUN
ejde-673	504	33	)	)	PUNCT
ejde-673	504	34	strongly	strongly	ADV
ejde-673	504	35	in	in	ADP
ejde-673	504	36	y	y	PROPN
ejde-673	504	37	∗	∗	NOUN
ejde-673	504	38	as	as	ADP
ejde-673	504	39	n	n	PROPN
ejde-673	504	40	→	→	SYM
ejde-673	504	41	∞.	∞.	PROPN
ejde-673	504	42	proof	proof	NOUN
ejde-673	504	43	.	.	PUNCT
ejde-673	505	1	(	(	PUNCT
ejde-673	505	2	i	i	NOUN
ejde-673	505	3	)	)	PUNCT
ejde-673	505	4	and	and	CCONJ
ejde-673	505	5	(	(	PUNCT
ejde-673	505	6	ii	ii	NOUN
ejde-673	505	7	)	)	PUNCT
ejde-673	505	8	follows	follow	VERB
ejde-673	505	9	from	from	ADP
ejde-673	505	10	aramaki	aramaki	NOUN
ejde-673	506	1	[	[	X
ejde-673	506	2	7	7	NUM
ejde-673	506	3	,	,	PUNCT
ejde-673	506	4	proposition	proposition	NOUN
ejde-673	506	5	4.2	4.2	NUM
ejde-673	506	6	,	,	PUNCT
ejde-673	506	7	proposition	proposition	NOUN
ejde-673	506	8	4.4	4.4	NUM
ejde-673	506	9	]	]	PUNCT
ejde-673	506	10	.	.	PUNCT
ejde-673	507	1	so	so	ADV
ejde-673	507	2	we	we	PRON
ejde-673	507	3	only	only	ADV
ejde-673	507	4	verify	verify	VERB
ejde-673	507	5	(	(	PUNCT
ejde-673	507	6	iii	iii	NOUN
ejde-673	507	7	)	)	PUNCT
ejde-673	507	8	.	.	PUNCT
ejde-673	508	1	let	let	VERB
ejde-673	508	2	un	un	PROPN
ejde-673	508	3	→	→	SYM
ejde-673	508	4	u	u	NOUN
ejde-673	508	5	weakly	weakly	ADV
ejde-673	508	6	in	in	ADP
ejde-673	508	7	y	y	PROPN
ejde-673	508	8	.	.	PUNCT
ejde-673	509	1	then	then	ADV
ejde-673	509	2	⟨k	⟨k	PROPN
ejde-673	509	3	′(un)−k	′(un)−k	NOUN
ejde-673	509	4	′(u	′(u	NOUN
ejde-673	509	5	)	)	PUNCT
ejde-673	509	6	,	,	PUNCT
ejde-673	509	7	v⟩y	v⟩y	PROPN
ejde-673	509	8	∗,y	∗,y	PROPN
ejde-673	509	9	=	=	SYM
ejde-673	509	10	∫	∫	PROPN
ejde-673	509	11	γ2	γ2	PROPN
ejde-673	509	12	(	(	PUNCT
ejde-673	509	13	g(x	g(x	PROPN
ejde-673	509	14	,	,	PUNCT
ejde-673	509	15	un(x))−	un(x))−	ADJ
ejde-673	509	16	g(x	g(x	NOUN
ejde-673	509	17	,	,	PUNCT
ejde-673	509	18	u(x)))v(x)dσx	u(x)))v(x)dσx	NOUN
ejde-673	509	19	for	for	ADP
ejde-673	509	20	v	v	NOUN
ejde-673	509	21	∈	∈	PROPN
ejde-673	509	22	y.	y.	NOUN
ejde-673	509	23	from	from	ADP
ejde-673	509	24	proposition	proposition	NOUN
ejde-673	509	25	2.11	2.11	NUM
ejde-673	509	26	and	and	CCONJ
ejde-673	509	27	(	(	PUNCT
ejde-673	509	28	a8	a8	PROPN
ejde-673	509	29	)	)	PUNCT
ejde-673	509	30	,	,	PUNCT
ejde-673	509	31	the	the	DET
ejde-673	509	32	embeddingw	embeddingw	NOUN
ejde-673	509	33	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	509	34	)	)	PUNCT
ejde-673	509	35	↪	↪	PROPN
ejde-673	509	36	→	→	SYM
ejde-673	509	37	l	l	NOUN
ejde-673	509	38	r	r	NOUN
ejde-673	509	39	(	(	PUNCT
ejde-673	509	40	·	·	PUNCT
ejde-673	509	41	)	)	PUNCT
ejde-673	509	42	b(·)(γ2	b(·)(γ2	NOUN
ejde-673	509	43	)	)	PUNCT
ejde-673	509	44	is	be	AUX
ejde-673	509	45	compact	compact	ADJ
ejde-673	509	46	.	.	PUNCT
ejde-673	510	1	since	since	SCONJ
ejde-673	510	2	y	y	PROPN
ejde-673	510	3	↪	↪	PROPN
ejde-673	510	4	→	→	SYM
ejde-673	510	5	x	x	SYM
ejde-673	510	6	↪	↪	PROPN
ejde-673	510	7	→	→	SYM
ejde-673	510	8	w	w	NOUN
ejde-673	510	9	1,p(·)(ω	1,p(·)(ω	NUM
ejde-673	510	10	)	)	PUNCT
ejde-673	510	11	,	,	PUNCT
ejde-673	510	12	there	there	PRON
ejde-673	510	13	exists	exist	VERB
ejde-673	510	14	a	a	DET
ejde-673	510	15	constant	constant	ADJ
ejde-673	510	16	c	c	NOUN
ejde-673	510	17	>	>	X
ejde-673	510	18	0	0	NUM
ejde-673	510	19	such	such	ADJ
ejde-673	510	20	that	that	SCONJ
ejde-673	510	21	∥v∥	∥v∥	ADJ
ejde-673	510	22	l	l	NOUN
ejde-673	510	23	r	r	NOUN
ejde-673	510	24	(	(	PUNCT
ejde-673	510	25	·	·	PUNCT
ejde-673	510	26	)	)	PUNCT
ejde-673	510	27	b(·)(γ2	b(·)(γ2	NOUN
ejde-673	510	28	)	)	PUNCT
ejde-673	510	29	≤	≤	PUNCT
ejde-673	510	30	c∥v∥y	c∥v∥y	VERB
ejde-673	510	31	for	for	ADP
ejde-673	510	32	all	all	DET
ejde-673	510	33	v	v	NOUN
ejde-673	510	34	∈	∈	NOUN
ejde-673	510	35	y.	y.	NOUN
ejde-673	510	36	by	by	ADP
ejde-673	510	37	the	the	DET
ejde-673	510	38	hölder	hölder	NOUN
ejde-673	510	39	inequality	inequality	NOUN
ejde-673	510	40	(	(	PUNCT
ejde-673	510	41	proposition	proposition	NOUN
ejde-673	510	42	2.5	2.5	NUM
ejde-673	510	43	)	)	PUNCT
ejde-673	510	44	,	,	PUNCT
ejde-673	510	45	for	for	ADP
ejde-673	510	46	any	any	DET
ejde-673	510	47	v	v	NOUN
ejde-673	510	48	∈	∈	PROPN
ejde-673	510	49	y	y	NOUN
ejde-673	510	50	,	,	PUNCT
ejde-673	510	51	we	we	PRON
ejde-673	510	52	have	have	VERB
ejde-673	510	53	|⟨k	|⟨k	NUM
ejde-673	510	54	′(un)−k	′(un)−k	VERB
ejde-673	510	55	′(u	′(u	NOUN
ejde-673	510	56	)	)	PUNCT
ejde-673	510	57	,	,	PUNCT
ejde-673	510	58	v⟩y	v⟩y	PROPN
ejde-673	510	59	∗,y	∗,y	PROPN
ejde-673	510	60	|	|	ADV
ejde-673	510	61	≤	≤	NUM
ejde-673	510	62	∫	∫	PROPN
ejde-673	510	63	γ2	γ2	PROPN
ejde-673	510	64	b(x)−1	b(x)−1	PROPN
ejde-673	510	65	/	/	SYM
ejde-673	510	66	r(x)|g(x	r(x)|g(x	NOUN
ejde-673	510	67	,	,	PUNCT
ejde-673	510	68	un(x))−	un(x))−	ADJ
ejde-673	510	69	g(x	g(x	NOUN
ejde-673	510	70	,	,	PUNCT
ejde-673	510	71	u(x))|b(x)1	u(x))|b(x)1	ADJ
ejde-673	510	72	/	/	SYM
ejde-673	510	73	r(x)|v(x)|dσx	r(x)|v(x)|dσx	NOUN
ejde-673	510	74	≤	≤	NUM
ejde-673	510	75	2∥b(·)−1	2∥b(·)−1	NUM
ejde-673	510	76	/	/	SYM
ejde-673	510	77	r(·)|g	r(·)|g	PROPN
ejde-673	510	78	(	(	PUNCT
ejde-673	510	79	·	·	PUNCT
ejde-673	510	80	,	,	PUNCT
ejde-673	510	81	un(·))−	un(·))−	PROPN
ejde-673	510	82	g	g	PROPN
ejde-673	510	83	(	(	PUNCT
ejde-673	510	84	·	·	PUNCT
ejde-673	510	85	,	,	PUNCT
ejde-673	510	86	u(·))|∥lr′(·)(γ2	u(·))|∥lr′(·)(γ2	ADJ
ejde-673	510	87	)	)	PUNCT
ejde-673	510	88	∥b(·)1	∥b(·)1	NOUN
ejde-673	510	89	/	/	SYM
ejde-673	510	90	r(·)|v(·)|∥lr(·)(γ2	r(·)|v(·)|∥lr(·)(γ2	NOUN
ejde-673	510	91	)	)	PUNCT
ejde-673	510	92	.	.	PUNCT
ejde-673	511	1	since	since	SCONJ
ejde-673	511	2	∥b(·)1	∥b(·)1	NOUN
ejde-673	511	3	/	/	SYM
ejde-673	511	4	r(·)v(·)∥lr(·)(γ2	r(·)v(·)∥lr(·)(γ2	NOUN
ejde-673	511	5	)	)	PUNCT
ejde-673	511	6	=	=	PUNCT
ejde-673	511	7	∥v∥	∥v∥	NUM
ejde-673	511	8	l	l	NOUN
ejde-673	511	9	r	r	NOUN
ejde-673	511	10	(	(	PUNCT
ejde-673	511	11	·	·	PUNCT
ejde-673	511	12	)	)	PUNCT
ejde-673	511	13	b(·)(γ2	b(·)(γ2	NOUN
ejde-673	511	14	)	)	PUNCT
ejde-673	511	15	≤	≤	PUNCT
ejde-673	511	16	c∥v∥y	c∥v∥y	VERB
ejde-673	511	17	,	,	PUNCT
ejde-673	511	18	we	we	PRON
ejde-673	511	19	have	have	VERB
ejde-673	511	20	∥k	∥k	PROPN
ejde-673	511	21	′(un)−k	′(un)−k	NOUN
ejde-673	511	22	′(u)∥y	′(u)∥y	NOUN
ejde-673	511	23	∗	∗	NOUN
ejde-673	511	24	≤	≤	NUM
ejde-673	511	25	2c∥b(·)−1	2c∥b(·)−1	NUM
ejde-673	511	26	/	/	SYM
ejde-673	511	27	r(·)|g	r(·)|g	PROPN
ejde-673	511	28	(	(	PUNCT
ejde-673	511	29	·	·	PUNCT
ejde-673	511	30	,	,	PUNCT
ejde-673	511	31	un(·))−	un(·))−	PROPN
ejde-673	511	32	g	g	PROPN
ejde-673	511	33	(	(	PUNCT
ejde-673	511	34	·	·	PUNCT
ejde-673	511	35	,	,	PUNCT
ejde-673	511	36	u(·))|∥lr′(·)(γ2	u(·))|∥lr′(·)(γ2	NOUN
ejde-673	511	37	)	)	PUNCT
ejde-673	511	38	.	.	PUNCT
ejde-673	512	1	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	512	2	eigenvalue	eigenvalue	VERB
ejde-673	512	3	problems	problem	NOUN
ejde-673	512	4	for	for	ADP
ejde-673	512	5	kirchhoff	kirchhoff	NOUN
ejde-673	512	6	-	-	PUNCT
ejde-673	512	7	type	type	NOUN
ejde-673	512	8	equations	equation	NOUN
ejde-673	512	9	17	17	NUM
ejde-673	512	10	we	we	PRON
ejde-673	512	11	want	want	VERB
ejde-673	512	12	to	to	PART
ejde-673	512	13	show	show	VERB
ejde-673	512	14	that	that	SCONJ
ejde-673	512	15	∥k	∥k	PROPN
ejde-673	512	16	′(un	′(un	PROPN
ejde-673	512	17	)	)	PUNCT
ejde-673	512	18	−	−	PROPN
ejde-673	513	1	k	k	PROPN
ejde-673	513	2	′(u)∥y	′(u)∥y	PROPN
ejde-673	513	3	∗	∗	NOUN
ejde-673	513	4	→	→	SYM
ejde-673	513	5	0	0	NUM
ejde-673	513	6	as	as	ADP
ejde-673	513	7	n	n	PROPN
ejde-673	513	8	→	→	PUNCT
ejde-673	513	9	∞.	∞.	PROPN
ejde-673	513	10	by	by	ADP
ejde-673	513	11	proposition	proposition	NOUN
ejde-673	513	12	2.4	2.4	NUM
ejde-673	513	13	(	(	PUNCT
ejde-673	513	14	iv	iv	NUM
ejde-673	513	15	)	)	PUNCT
ejde-673	513	16	,	,	PUNCT
ejde-673	513	17	it	it	PRON
ejde-673	513	18	suffices	suffice	VERB
ejde-673	513	19	to	to	PART
ejde-673	513	20	show	show	VERB
ejde-673	513	21	that	that	SCONJ
ejde-673	514	1	ρr′(·),γ2	ρr′(·),γ2	PROPN
ejde-673	514	2	(	(	PUNCT
ejde-673	514	3	b(·)−1	b(·)−1	NOUN
ejde-673	514	4	/	/	SYM
ejde-673	514	5	r(·)g	r(·)g	NOUN
ejde-673	514	6	(	(	PUNCT
ejde-673	514	7	·	·	PUNCT
ejde-673	514	8	,	,	PUNCT
ejde-673	514	9	un(·))−	un(·))−	PROPN
ejde-673	514	10	b(·)−1	b(·)−1	NOUN
ejde-673	514	11	/	/	SYM
ejde-673	514	12	r(·)g	r(·)g	NOUN
ejde-673	514	13	(	(	PUNCT
ejde-673	514	14	·	·	PUNCT
ejde-673	514	15	,	,	PUNCT
ejde-673	514	16	u	u	NOUN
ejde-673	514	17	(	(	PUNCT
ejde-673	514	18	·	·	PUNCT
ejde-673	514	19	)	)	PUNCT
ejde-673	514	20	)	)	PUNCT
ejde-673	514	21	)	)	PUNCT
ejde-673	514	22	→	→	SYM
ejde-673	514	23	0	0	NUM
ejde-673	514	24	as	as	ADP
ejde-673	514	25	n	n	PROPN
ejde-673	514	26	→	→	SYM
ejde-673	514	27	∞.	∞.	PROPN
ejde-673	514	28	(	(	PUNCT
ejde-673	514	29	3.11	3.11	NUM
ejde-673	514	30	)	)	PUNCT
ejde-673	514	31	we	we	PRON
ejde-673	514	32	can	can	AUX
ejde-673	514	33	see	see	VERB
ejde-673	514	34	that	that	SCONJ
ejde-673	514	35	ρr′(·),γ2	ρr′(·),γ2	PROPN
ejde-673	514	36	(	(	PUNCT
ejde-673	514	37	b(·)−1	b(·)−1	NOUN
ejde-673	514	38	/	/	SYM
ejde-673	514	39	r(·)g	r(·)g	NOUN
ejde-673	514	40	(	(	PUNCT
ejde-673	514	41	·	·	PUNCT
ejde-673	514	42	,	,	PUNCT
ejde-673	514	43	un(·))−	un(·))−	PROPN
ejde-673	514	44	b(·)−1	b(·)−1	NOUN
ejde-673	514	45	/	/	SYM
ejde-673	514	46	r(·)g	r(·)g	NOUN
ejde-673	514	47	(	(	PUNCT
ejde-673	514	48	·	·	PUNCT
ejde-673	514	49	,	,	PUNCT
ejde-673	514	50	u	u	NOUN
ejde-673	514	51	(	(	PUNCT
ejde-673	514	52	·	·	PUNCT
ejde-673	514	53	)	)	PUNCT
ejde-673	514	54	)	)	PUNCT
ejde-673	514	55	)	)	PUNCT
ejde-673	515	1	=	=	SYM
ejde-673	515	2	∫	∫	PROPN
ejde-673	515	3	γ2	γ2	PROPN
ejde-673	515	4	b(x)−r′(x)/r(x)|g(x	b(x)−r′(x)/r(x)|g(x	PROPN
ejde-673	515	5	,	,	PUNCT
ejde-673	515	6	un(x))−	un(x))−	ADJ
ejde-673	515	7	g(x	g(x	NOUN
ejde-673	515	8	,	,	PUNCT
ejde-673	515	9	u(x))|r	u(x))|r	PROPN
ejde-673	515	10	′(x)dσx	′(x)dσx	NOUN
ejde-673	515	11	.	.	PUNCT
ejde-673	516	1	since	since	SCONJ
ejde-673	516	2	un	un	PROPN
ejde-673	516	3	→	→	SYM
ejde-673	516	4	u	u	PROPN
ejde-673	516	5	weakly	weakly	ADV
ejde-673	516	6	in	in	ADP
ejde-673	516	7	y	y	PROPN
ejde-673	516	8	and	and	CCONJ
ejde-673	516	9	the	the	DET
ejde-673	516	10	embedding	embed	VERB
ejde-673	516	11	map	map	NOUN
ejde-673	516	12	y	y	PROPN
ejde-673	516	13	↪	↪	PROPN
ejde-673	516	14	→	→	SYM
ejde-673	516	15	l	l	NOUN
ejde-673	516	16	r	r	NOUN
ejde-673	516	17	(	(	PUNCT
ejde-673	516	18	·	·	PUNCT
ejde-673	516	19	)	)	PUNCT
ejde-673	516	20	b(·)(γ2	b(·)(γ2	NOUN
ejde-673	516	21	)	)	PUNCT
ejde-673	516	22	is	be	AUX
ejde-673	516	23	compact	compact	ADJ
ejde-673	516	24	,	,	PUNCT
ejde-673	516	25	we	we	PRON
ejde-673	516	26	can	can	AUX
ejde-673	516	27	see	see	VERB
ejde-673	516	28	that	that	DET
ejde-673	516	29	un	un	PROPN
ejde-673	516	30	→	→	SYM
ejde-673	516	31	u	u	PRON
ejde-673	516	32	strongly	strongly	ADV
ejde-673	516	33	in	in	ADP
ejde-673	516	34	l	l	NOUN
ejde-673	516	35	r	r	X
ejde-673	516	36	(	(	PUNCT
ejde-673	516	37	·	·	PUNCT
ejde-673	516	38	)	)	PUNCT
ejde-673	516	39	b(·)(γ2	b(·)(γ2	NOUN
ejde-673	516	40	)	)	PUNCT
ejde-673	516	41	.	.	PUNCT
ejde-673	517	1	from	from	ADP
ejde-673	517	2	[	[	X
ejde-673	517	3	6	6	NUM
ejde-673	517	4	,	,	PUNCT
ejde-673	517	5	theorem	theorem	ADJ
ejde-673	517	6	a.1	a.1	NOUN
ejde-673	517	7	]	]	PUNCT
ejde-673	517	8	,	,	PUNCT
ejde-673	517	9	there	there	PRON
ejde-673	517	10	exist	exist	VERB
ejde-673	517	11	a	a	DET
ejde-673	517	12	subsequence	subsequence	NOUN
ejde-673	517	13	{	{	PUNCT
ejde-673	517	14	un′	un′	NOUN
ejde-673	517	15	}	}	PUNCT
ejde-673	517	16	of	of	ADP
ejde-673	517	17	{	{	PUNCT
ejde-673	517	18	un	un	PROPN
ejde-673	517	19	}	}	PUNCT
ejde-673	517	20	and	and	CCONJ
ejde-673	517	21	f	f	PROPN
ejde-673	517	22	∈	∈	PROPN
ejde-673	517	23	lr(·)(γ2	lr(·)(γ2	NOUN
ejde-673	517	24	)	)	PUNCT
ejde-673	517	25	such	such	ADJ
ejde-673	517	26	that	that	SCONJ
ejde-673	517	27	b(x)1	b(x)1	NOUN
ejde-673	517	28	/	/	SYM
ejde-673	517	29	r(x)un′(x	r(x)un′(x	NOUN
ejde-673	517	30	)	)	PUNCT
ejde-673	517	31	→	→	PUNCT
ejde-673	517	32	b(x)1	b(x)1	NOUN
ejde-673	517	33	/	/	SYM
ejde-673	517	34	r(x)u(x	r(x)u(x	NOUN
ejde-673	517	35	)	)	PUNCT
ejde-673	517	36	σ	σ	PROPN
ejde-673	517	37	-	-	PUNCT
ejde-673	517	38	a.e	a.e	PROPN
ejde-673	517	39	.	.	PUNCT
ejde-673	517	40	x	x	SYM
ejde-673	517	41	∈	∈	PROPN
ejde-673	517	42	γ2	γ2	NOUN
ejde-673	517	43	and	and	CCONJ
ejde-673	517	44	|b(x)1	|b(x)1	ADJ
ejde-673	517	45	/	/	SYM
ejde-673	517	46	r(x)un′(x)|	r(x)un′(x)|	NOUN
ejde-673	517	47	≤	≤	NUM
ejde-673	517	48	f(x	f(x	PROPN
ejde-673	517	49	)	)	PUNCT
ejde-673	517	50	for	for	ADP
ejde-673	517	51	σ	σ	PROPN
ejde-673	517	52	-	-	PROPN
ejde-673	517	53	a.e	a.e	PROPN
ejde-673	517	54	.	.	PUNCT
ejde-673	517	55	x	x	SYM
ejde-673	517	56	∈	∈	PROPN
ejde-673	517	57	γ2	γ2	NOUN
ejde-673	517	58	.	.	PUNCT
ejde-673	518	1	since	since	SCONJ
ejde-673	518	2	b(x	b(x	VERB
ejde-673	518	3	)	)	PUNCT
ejde-673	518	4	>	>	X
ejde-673	518	5	0	0	NUM
ejde-673	518	6	σ	σ	PROPN
ejde-673	518	7	-	-	PUNCT
ejde-673	518	8	a.e	a.e	PROPN
ejde-673	518	9	.	.	PUNCT
ejde-673	518	10	x	x	SYM
ejde-673	518	11	∈	∈	PROPN
ejde-673	518	12	γ2	γ2	NOUN
ejde-673	518	13	,	,	PUNCT
ejde-673	518	14	un′(x	un′(x	NOUN
ejde-673	518	15	)	)	PUNCT
ejde-673	518	16	→	→	SYM
ejde-673	518	17	u(x	u(x	PROPN
ejde-673	518	18	)	)	PUNCT
ejde-673	518	19	σ	σ	PROPN
ejde-673	518	20	-	-	PUNCT
ejde-673	518	21	a.e	a.e	PROPN
ejde-673	518	22	.	.	PUNCT
ejde-673	518	23	x	x	SYM
ejde-673	518	24	∈	∈	PROPN
ejde-673	518	25	γ2	γ2	NOUN
ejde-673	518	26	,	,	PUNCT
ejde-673	518	27	so	so	SCONJ
ejde-673	518	28	we	we	PRON
ejde-673	518	29	see	see	VERB
ejde-673	518	30	that	that	SCONJ
ejde-673	518	31	g(x	g(x	NOUN
ejde-673	518	32	,	,	PUNCT
ejde-673	518	33	un′(x	un′(x	NOUN
ejde-673	518	34	)	)	PUNCT
ejde-673	518	35	)	)	PUNCT
ejde-673	519	1	→	→	PUNCT
ejde-673	519	2	g(x	g(x	X
ejde-673	519	3	,	,	PUNCT
ejde-673	519	4	u(x	u(x	NOUN
ejde-673	519	5	)	)	PUNCT
ejde-673	519	6	)	)	PUNCT
ejde-673	520	1	σ	σ	PROPN
ejde-673	520	2	-	-	PUNCT
ejde-673	520	3	a.e	a.e	PROPN
ejde-673	520	4	.	.	PUNCT
ejde-673	520	5	x	x	SYM
ejde-673	520	6	∈	∈	PROPN
ejde-673	520	7	γ2	γ2	NOUN
ejde-673	520	8	.	.	PUNCT
ejde-673	521	1	from	from	ADP
ejde-673	521	2	(	(	PUNCT
ejde-673	521	3	a8	a8	PROPN
ejde-673	521	4	)	)	PUNCT
ejde-673	521	5	,	,	PUNCT
ejde-673	521	6	we	we	PRON
ejde-673	521	7	have	have	VERB
ejde-673	521	8	b(x)−r′(x)/r(x)|g(x	b(x)−r′(x)/r(x)|g(x	VERB
ejde-673	521	9	,	,	PUNCT
ejde-673	521	10	un′(x)−	un′(x)−	PROPN
ejde-673	521	11	g(x	g(x	PROPN
ejde-673	521	12	,	,	PUNCT
ejde-673	521	13	u(x))|r	u(x))|r	PROPN
ejde-673	521	14	′(x	′(x	NOUN
ejde-673	521	15	)	)	PUNCT
ejde-673	521	16	≤	≤	NUM
ejde-673	521	17	b(x)−r′(x)/r(x)(b(x)|un′(x)|r(x)−1	b(x)−r′(x)/r(x)(b(x)|un′(x)|r(x)−1	NOUN
ejde-673	521	18	+	+	CCONJ
ejde-673	521	19	b(x)|u(x)|r(x)−1)r	b(x)|u(x)|r(x)−1)r	NOUN
ejde-673	521	20	′(x	′(x	NOUN
ejde-673	521	21	)	)	PUNCT
ejde-673	521	22	≤	≤	NUM
ejde-673	521	23	b(x)r	b(x)r	NOUN
ejde-673	521	24	′(x)−r′(x)/r(x)(|un′(x)|r(x	′(x)−r′(x)/r(x)(|un′(x)|r(x	NOUN
ejde-673	521	25	)	)	PUNCT
ejde-673	521	26	+	+	NUM
ejde-673	521	27	|u(x)|r(x	|u(x)|r(x	X
ejde-673	521	28	)	)	PUNCT
ejde-673	521	29	)	)	PUNCT
ejde-673	522	1	≤	≤	NUM
ejde-673	522	2	b(x)(|un′(x)|r(x	b(x)(|un′(x)|r(x	NOUN
ejde-673	522	3	)	)	PUNCT
ejde-673	522	4	+	+	NUM
ejde-673	522	5	|u(x)|r(x	|u(x)|r(x	X
ejde-673	522	6	)	)	PUNCT
ejde-673	522	7	)	)	PUNCT
ejde-673	523	1	≤	≤	NOUN
ejde-673	524	1	2f(x)r(x	2f(x)r(x	NUM
ejde-673	524	2	)	)	PUNCT
ejde-673	524	3	.	.	PUNCT
ejde-673	525	1	the	the	DET
ejde-673	525	2	last	last	ADJ
ejde-673	525	3	term	term	NOUN
ejde-673	525	4	is	be	AUX
ejde-673	525	5	an	an	DET
ejde-673	525	6	integrable	integrable	ADJ
ejde-673	525	7	function	function	NOUN
ejde-673	525	8	in	in	ADP
ejde-673	525	9	ω	ω	PROPN
ejde-673	525	10	independent	independent	ADJ
ejde-673	525	11	of	of	ADP
ejde-673	525	12	n′.	n′.	PRON
ejde-673	525	13	thus	thus	ADV
ejde-673	525	14	by	by	ADP
ejde-673	525	15	the	the	DET
ejde-673	525	16	lebesgue	lebesgue	NOUN
ejde-673	525	17	dominated	dominate	VERB
ejde-673	525	18	convergence	convergence	NOUN
ejde-673	525	19	theorem	theorem	VERB
ejde-673	525	20	,	,	PUNCT
ejde-673	525	21	we	we	PRON
ejde-673	525	22	have	have	VERB
ejde-673	525	23	ρr′(·),γ2	ρr′(·),γ2	PROPN
ejde-673	525	24	(	(	PUNCT
ejde-673	525	25	b(·)−1	b(·)−1	VERB
ejde-673	525	26	/	/	SYM
ejde-673	525	27	r(·)g	r(·)g	NOUN
ejde-673	525	28	(	(	PUNCT
ejde-673	525	29	·	·	PUNCT
ejde-673	525	30	,	,	PUNCT
ejde-673	525	31	un′(·))−	un′(·))−	ADJ
ejde-673	525	32	b(·)−1	b(·)−1	NOUN
ejde-673	525	33	/	/	SYM
ejde-673	525	34	r(·)g	r(·)g	NOUN
ejde-673	525	35	(	(	PUNCT
ejde-673	525	36	·	·	PUNCT
ejde-673	525	37	,	,	PUNCT
ejde-673	525	38	u	u	NOUN
ejde-673	525	39	(	(	PUNCT
ejde-673	525	40	·	·	PUNCT
ejde-673	525	41	)	)	PUNCT
ejde-673	525	42	)	)	PUNCT
ejde-673	525	43	)	)	PUNCT
ejde-673	526	1	→	→	SYM
ejde-673	526	2	0	0	NUM
ejde-673	526	3	as	as	ADP
ejde-673	526	4	n′	n′	PROPN
ejde-673	526	5	→	→	PUNCT
ejde-673	526	6	∞.	∞.	PROPN
ejde-673	526	7	from	from	ADP
ejde-673	526	8	the	the	DET
ejde-673	526	9	convergent	convergent	NOUN
ejde-673	526	10	principle	principle	NOUN
ejde-673	527	1	[	[	X
ejde-673	527	2	34	34	NUM
ejde-673	527	3	,	,	PUNCT
ejde-673	527	4	proposition	proposition	NOUN
ejde-673	527	5	10.13	10.13	NUM
ejde-673	527	6	]	]	PUNCT
ejde-673	527	7	,	,	PUNCT
ejde-673	527	8	we	we	PRON
ejde-673	527	9	see	see	VERB
ejde-673	527	10	that	that	SCONJ
ejde-673	527	11	(	(	PUNCT
ejde-673	527	12	3.11	3.11	NUM
ejde-673	527	13	)	)	PUNCT
ejde-673	527	14	holds	hold	VERB
ejde-673	527	15	,	,	PUNCT
ejde-673	527	16	so	so	ADV
ejde-673	527	17	∥k	∥k	PROPN
ejde-673	527	18	′(un)−k	′(un)−k	VERB
ejde-673	527	19	′(u)∥y	′(u)∥y	NOUN
ejde-673	527	20	∗	∗	NOUN
ejde-673	527	21	→	→	SYM
ejde-673	527	22	0	0	NUM
ejde-673	527	23	as	as	ADP
ejde-673	527	24	n	n	NOUN
ejde-673	527	25	→	→	SYM
ejde-673	527	26	∞.	∞.	PROPN
ejde-673	527	27	□	□	PUNCT
ejde-673	527	28	remark	remark	NOUN
ejde-673	527	29	3.12	3.12	NUM
ejde-673	527	30	.	.	PUNCT
ejde-673	528	1	from	from	ADP
ejde-673	528	2	(	(	PUNCT
ejde-673	528	3	3.9	3.9	NUM
ejde-673	528	4	)	)	PUNCT
ejde-673	528	5	,	,	PUNCT
ejde-673	528	6	(	(	PUNCT
ejde-673	528	7	3.10	3.10	NUM
ejde-673	528	8	)	)	PUNCT
ejde-673	528	9	and	and	CCONJ
ejde-673	528	10	definition	definition	NOUN
ejde-673	528	11	3.6	3.6	NUM
ejde-673	528	12	,	,	PUNCT
ejde-673	528	13	we	we	PRON
ejde-673	528	14	can	can	AUX
ejde-673	528	15	see	see	VERB
ejde-673	528	16	that	that	PRON
ejde-673	528	17	(	(	PUNCT
ejde-673	528	18	u	u	NOUN
ejde-673	528	19	,	,	PUNCT
ejde-673	528	20	λ	λ	PROPN
ejde-673	528	21	)	)	PUNCT
ejde-673	528	22	∈	∈	PROPN
ejde-673	528	23	y	y	PROPN
ejde-673	528	24	×r	×r	X
ejde-673	528	25	is	be	AUX
ejde-673	528	26	a	a	DET
ejde-673	528	27	weak	weak	ADJ
ejde-673	528	28	solution	solution	NOUN
ejde-673	528	29	of	of	ADP
ejde-673	528	30	(	(	PUNCT
ejde-673	528	31	1.1	1.1	NUM
ejde-673	528	32	)	)	PUNCT
ejde-673	529	1	if	if	SCONJ
ejde-673	529	2	and	and	CCONJ
ejde-673	529	3	only	only	ADV
ejde-673	529	4	if	if	SCONJ
ejde-673	529	5	ψ′(u	ψ′(u	PRON
ejde-673	529	6	)	)	PUNCT
ejde-673	529	7	=	=	SYM
ejde-673	529	8	λk	λk	PRON
ejde-673	529	9	′(u	′(u	NOUN
ejde-673	529	10	)	)	PUNCT
ejde-673	529	11	.	.	PUNCT
ejde-673	530	1	(	(	PUNCT
ejde-673	530	2	3.12	3.12	NUM
ejde-673	530	3	)	)	PUNCT
ejde-673	530	4	in	in	ADP
ejde-673	530	5	particular	particular	ADJ
ejde-673	530	6	,	,	PUNCT
ejde-673	530	7	we	we	PRON
ejde-673	530	8	have	have	VERB
ejde-673	530	9	⟨ψ′(u	⟨ψ′(u	PROPN
ejde-673	530	10	)	)	PUNCT
ejde-673	530	11	,	,	PUNCT
ejde-673	530	12	u⟩y	u⟩y	NUM
ejde-673	530	13	∗,y	∗,y	PROPN
ejde-673	530	14	=	=	SYM
ejde-673	530	15	λ⟨k	λ⟨k	NUM
ejde-673	530	16	′(u	′(u	NOUN
ejde-673	530	17	)	)	PUNCT
ejde-673	530	18	,	,	PUNCT
ejde-673	530	19	u⟩y	u⟩y	NUM
ejde-673	530	20	∗,y	∗,y	NOUN
ejde-673	530	21	.	.	PUNCT
ejde-673	531	1	if	if	SCONJ
ejde-673	531	2	(	(	PUNCT
ejde-673	531	3	u	u	NOUN
ejde-673	531	4	,	,	PUNCT
ejde-673	531	5	λ	λ	X
ejde-673	531	6	)	)	PUNCT
ejde-673	531	7	is	be	AUX
ejde-673	531	8	an	an	DET
ejde-673	531	9	eigenpair	eigenpair	NOUN
ejde-673	531	10	of	of	ADP
ejde-673	531	11	(	(	PUNCT
ejde-673	531	12	1.1	1.1	NUM
ejde-673	531	13	)	)	PUNCT
ejde-673	531	14	,	,	PUNCT
ejde-673	531	15	then	then	ADV
ejde-673	531	16	from	from	ADP
ejde-673	531	17	(	(	PUNCT
ejde-673	531	18	a5	a5	PROPN
ejde-673	531	19	)	)	PUNCT
ejde-673	531	20	,	,	PUNCT
ejde-673	531	21	(	(	PUNCT
ejde-673	531	22	a1	a1	NOUN
ejde-673	531	23	)	)	PUNCT
ejde-673	531	24	and	and	CCONJ
ejde-673	531	25	(	(	PUNCT
ejde-673	531	26	a8)it	a8)it	PROPN
ejde-673	531	27	follows	follow	VERB
ejde-673	531	28	that	that	SCONJ
ejde-673	531	29	⟨ψ′(u	⟨ψ′(u	PROPN
ejde-673	531	30	)	)	PUNCT
ejde-673	531	31	,	,	PUNCT
ejde-673	531	32	u⟩y	u⟩y	NUM
ejde-673	531	33	∗,y	∗,y	NOUN
ejde-673	531	34	=	=	SYM
ejde-673	531	35	m(φ(u	m(φ(u	NOUN
ejde-673	531	36	)	)	PUNCT
ejde-673	531	37	)	)	PUNCT
ejde-673	532	1	∫	∫	PROPN
ejde-673	532	2	ω	ω	NUM
ejde-673	532	3	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	532	4	)	)	PUNCT
ejde-673	532	5	)	)	PUNCT
ejde-673	532	6	·	·	PUNCT
ejde-673	532	7	∇u(x	∇u(x	NOUN
ejde-673	532	8	)	)	PUNCT
ejde-673	532	9	dx	dx	PROPN
ejde-673	532	10	≥	≥	PROPN
ejde-673	532	11	m0	m0	PROPN
ejde-673	532	12	(	(	PUNCT
ejde-673	532	13	∫	∫	PROPN
ejde-673	532	14	ω	ω	NUM
ejde-673	532	15	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	532	16	)	)	PUNCT
ejde-673	532	17	)	)	PUNCT
ejde-673	532	18	dx	dx	PROPN
ejde-673	532	19	)	)	PUNCT
ejde-673	533	1	l−1	l−1	PROPN
ejde-673	533	2	∫	∫	PROPN
ejde-673	533	3	ω	ω	NUM
ejde-673	533	4	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	533	5	)	)	PUNCT
ejde-673	533	6	)	)	PUNCT
ejde-673	533	7	·	·	PUNCT
ejde-673	533	8	∇u(x	∇u(x	NOUN
ejde-673	533	9	)	)	PUNCT
ejde-673	533	10	dx	dx	PROPN
ejde-673	533	11	≥	≥	PROPN
ejde-673	533	12	m0	m0	PROPN
ejde-673	533	13	(	(	PUNCT
ejde-673	533	14	∫	∫	PROPN
ejde-673	533	15	ω	ω	PROPN
ejde-673	533	16	1	1	NUM
ejde-673	533	17	p(x	p(x	PROPN
ejde-673	533	18	)	)	PUNCT
ejde-673	533	19	a(x,∇u(x	a(x,∇u(x	NOUN
ejde-673	533	20	)	)	PUNCT
ejde-673	533	21	)	)	PUNCT
ejde-673	533	22	·	·	PUNCT
ejde-673	533	23	∇u(x	∇u(x	NOUN
ejde-673	533	24	)	)	PUNCT
ejde-673	533	25	dx	dx	PROPN
ejde-673	533	26	)	)	PUNCT
ejde-673	534	1	l−1	l−1	PROPN
ejde-673	534	2	∫	∫	PROPN
ejde-673	534	3	ω	ω	NUM
ejde-673	534	4	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	534	5	)	)	PUNCT
ejde-673	534	6	)	)	PUNCT
ejde-673	534	7	·	·	PUNCT
ejde-673	534	8	∇u(x	∇u(x	NOUN
ejde-673	534	9	)	)	PUNCT
ejde-673	534	10	dx	dx	PROPN
ejde-673	534	11	≥	≥	PROPN
ejde-673	534	12	m0	m0	PROPN
ejde-673	534	13	(	(	PUNCT
ejde-673	534	14	p+)l−1	p+)l−1	NUM
ejde-673	534	15	(	(	PUNCT
ejde-673	534	16	∫	∫	PROPN
ejde-673	534	17	ω	ω	NUM
ejde-673	534	18	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	534	19	)	)	PUNCT
ejde-673	534	20	)	)	PUNCT
ejde-673	534	21	·	·	PUNCT
ejde-673	534	22	∇u(x	∇u(x	NOUN
ejde-673	534	23	)	)	PUNCT
ejde-673	534	24	dx	dx	PROPN
ejde-673	534	25	)	)	PUNCT
ejde-673	535	1	l	l	PROPN
ejde-673	535	2	18	18	NUM
ejde-673	535	3	j.	j.	PROPN
ejde-673	535	4	aramaki	aramaki	PROPN
ejde-673	535	5	ejde-2025/17	ejde-2025/17	X
ejde-673	535	6	≥	≥	NUM
ejde-673	535	7	m0k	m0k	PROPN
ejde-673	535	8	l	l	NOUN
ejde-673	535	9	0	0	NUM
ejde-673	535	10	(	(	PUNCT
ejde-673	535	11	p+)l−1	p+)l−1	NUM
ejde-673	535	12	(	(	PUNCT
ejde-673	535	13	∫	∫	PROPN
ejde-673	535	14	ω	ω	PROPN
ejde-673	535	15	h1(x)|∇u(x)|p(x	h1(x)|∇u(x)|p(x	PROPN
ejde-673	535	16	)	)	PUNCT
ejde-673	535	17	dx	dx	PROPN
ejde-673	535	18	)	)	PUNCT
ejde-673	535	19	l	l	NOUN
ejde-673	535	20	≥	≥	NOUN
ejde-673	535	21	m0k	m0k	PROPN
ejde-673	535	22	l	l	X
ejde-673	535	23	0	0	NUM
ejde-673	535	24	(	(	PUNCT
ejde-673	535	25	p+)l−1	p+)l−1	NUM
ejde-673	535	26	(	(	PUNCT
ejde-673	535	27	∥u∥p	∥u∥p	NOUN
ejde-673	535	28	+	+	CCONJ
ejde-673	535	29	y	y	PROPN
ejde-673	535	30	∧	∧	PROPN
ejde-673	535	31	∥u∥p	∥u∥p	NOUN
ejde-673	535	32	−	−	PROPN
ejde-673	535	33	y	y	PROPN
ejde-673	535	34	)	)	PUNCT
ejde-673	535	35	l	l	NOUN
ejde-673	535	36	>	>	PUNCT
ejde-673	535	37	0	0	PUNCT
ejde-673	535	38	and	and	CCONJ
ejde-673	535	39	from	from	ADP
ejde-673	535	40	(	(	PUNCT
ejde-673	535	41	3.12	3.12	NUM
ejde-673	535	42	)	)	PUNCT
ejde-673	535	43	and	and	CCONJ
ejde-673	535	44	(	(	PUNCT
ejde-673	535	45	3.4	3.4	NUM
ejde-673	535	46	)	)	PUNCT
ejde-673	535	47	,	,	PUNCT
ejde-673	535	48	⟨k	⟨k	PRON
ejde-673	535	49	′(u	′(u	NOUN
ejde-673	535	50	)	)	PUNCT
ejde-673	535	51	,	,	PUNCT
ejde-673	535	52	u⟩y	u⟩y	NUM
ejde-673	535	53	∗,y	∗,y	PROPN
ejde-673	535	54	=	=	SYM
ejde-673	535	55	∫	∫	PROPN
ejde-673	535	56	γ2	γ2	PROPN
ejde-673	535	57	g(x	g(x	PROPN
ejde-673	535	58	,	,	PUNCT
ejde-673	535	59	u(x))u(x)dσx	u(x))u(x)dσx	VERB
ejde-673	535	60	>	>	X
ejde-673	535	61	0	0	NUM
ejde-673	535	62	,	,	PUNCT
ejde-673	535	63	so	so	SCONJ
ejde-673	535	64	we	we	PRON
ejde-673	535	65	have	have	VERB
ejde-673	535	66	λ	λ	NOUN
ejde-673	535	67	=	=	SYM
ejde-673	535	68	⟨ψ′(u	⟨ψ′(u	PROPN
ejde-673	535	69	)	)	PUNCT
ejde-673	535	70	,	,	PUNCT
ejde-673	535	71	u⟩y	u⟩y	NUM
ejde-673	535	72	∗,y	∗,y	PROPN
ejde-673	535	73	⟨k	⟨k	NOUN
ejde-673	535	74	′(u	′(u	NOUN
ejde-673	535	75	)	)	PUNCT
ejde-673	535	76	,	,	PUNCT
ejde-673	535	77	u⟩y	u⟩y	NUM
ejde-673	535	78	∗,y	∗,y	PROPN
ejde-673	535	79	>	>	X
ejde-673	535	80	0	0	NUM
ejde-673	535	81	.	.	PUNCT
ejde-673	536	1	(	(	PUNCT
ejde-673	536	2	3.13	3.13	NUM
ejde-673	536	3	)	)	PUNCT
ejde-673	536	4	this	this	PRON
ejde-673	536	5	means	mean	VERB
ejde-673	536	6	that	that	SCONJ
ejde-673	536	7	any	any	DET
ejde-673	536	8	eigenvalue	eigenvalue	NOUN
ejde-673	536	9	of	of	ADP
ejde-673	536	10	problem	problem	NOUN
ejde-673	536	11	(	(	PUNCT
ejde-673	536	12	1.1	1.1	NUM
ejde-673	536	13	)	)	PUNCT
ejde-673	536	14	is	be	AUX
ejde-673	536	15	positive	positive	ADJ
ejde-673	536	16	.	.	PUNCT
ejde-673	537	1	to	to	PART
ejde-673	537	2	solve	solve	VERB
ejde-673	537	3	the	the	DET
ejde-673	537	4	eigenvalue	eigenvalue	PROPN
ejde-673	537	5	problem	problem	NOUN
ejde-673	537	6	(	(	PUNCT
ejde-673	537	7	3.12	3.12	NUM
ejde-673	537	8	)	)	PUNCT
ejde-673	537	9	,	,	PUNCT
ejde-673	537	10	we	we	PRON
ejde-673	537	11	apply	apply	VERB
ejde-673	537	12	the	the	DET
ejde-673	537	13	constrained	constrain	VERB
ejde-673	537	14	variational	variational	ADJ
ejde-673	537	15	method	method	NOUN
ejde-673	537	16	.	.	PUNCT
ejde-673	538	1	we	we	PRON
ejde-673	538	2	take	take	VERB
ejde-673	538	3	ψ	ψ	PRON
ejde-673	538	4	as	as	ADP
ejde-673	538	5	an	an	DET
ejde-673	538	6	objective	objective	ADJ
ejde-673	538	7	functional	functional	NOUN
ejde-673	538	8	and	and	CCONJ
ejde-673	538	9	k	k	PROPN
ejde-673	538	10	as	as	ADP
ejde-673	538	11	a	a	DET
ejde-673	538	12	constraint	constraint	NOUN
ejde-673	538	13	functional	functional	ADJ
ejde-673	538	14	.	.	PUNCT
ejde-673	539	1	for	for	ADP
ejde-673	539	2	any	any	DET
ejde-673	539	3	fixed	fixed	ADJ
ejde-673	539	4	α	α	PROPN
ejde-673	539	5	>	>	X
ejde-673	539	6	0	0	NUM
ejde-673	539	7	,	,	PUNCT
ejde-673	539	8	put	put	VERB
ejde-673	539	9	mα	mα	NOUN
ejde-673	539	10	=	=	PUNCT
ejde-673	539	11	{	{	PUNCT
ejde-673	539	12	u	u	NOUN
ejde-673	539	13	∈	∈	PROPN
ejde-673	539	14	y	y	PROPN
ejde-673	539	15	;	;	PUNCT
ejde-673	539	16	k(u	k(u	X
ejde-673	539	17	)	)	PUNCT
ejde-673	539	18	=	=	SYM
ejde-673	539	19	α	α	X
ejde-673	539	20	}	}	PUNCT
ejde-673	539	21	.	.	PUNCT
ejde-673	540	1	(	(	PUNCT
ejde-673	540	2	3.14	3.14	NUM
ejde-673	540	3	)	)	PUNCT
ejde-673	540	4	if	if	SCONJ
ejde-673	540	5	u	u	PROPN
ejde-673	540	6	∈	∈	PROPN
ejde-673	540	7	mα	mα	PROPN
ejde-673	540	8	,	,	PUNCT
ejde-673	540	9	then	then	ADV
ejde-673	540	10	from	from	ADP
ejde-673	540	11	(	(	PUNCT
ejde-673	540	12	a8	a8	PROPN
ejde-673	540	13	)	)	PUNCT
ejde-673	540	14	,	,	PUNCT
ejde-673	540	15	⟨k	⟨k	PRON
ejde-673	540	16	′(u	′(u	NOUN
ejde-673	540	17	)	)	PUNCT
ejde-673	540	18	,	,	PUNCT
ejde-673	540	19	u⟩y	u⟩y	NUM
ejde-673	540	20	∗,y	∗,y	PROPN
ejde-673	540	21	=	=	SYM
ejde-673	540	22	∫	∫	PROPN
ejde-673	540	23	γ2	γ2	PROPN
ejde-673	540	24	g(x	g(x	PROPN
ejde-673	540	25	,	,	PUNCT
ejde-673	540	26	u(x))u(x)dσx	u(x))u(x)dσx	VERB
ejde-673	540	27	≥	≥	NUM
ejde-673	540	28	r−	r−	PROPN
ejde-673	540	29	∫	∫	PROPN
ejde-673	540	30	γ2	γ2	PROPN
ejde-673	540	31	g(x	g(x	PROPN
ejde-673	540	32	,	,	PUNCT
ejde-673	540	33	u(x))dσx	u(x))dσx	X
ejde-673	540	34	=	=	PUNCT
ejde-673	540	35	r−k(u	r−k(u	X
ejde-673	540	36	)	)	PUNCT
ejde-673	540	37	=	=	SYM
ejde-673	540	38	r−	r−	PROPN
ejde-673	540	39	α	α	PROPN
ejde-673	540	40	>	>	X
ejde-673	540	41	0	0	NUM
ejde-673	540	42	,	,	PUNCT
ejde-673	540	43	(	(	PUNCT
ejde-673	540	44	3.15	3.15	NUM
ejde-673	540	45	)	)	PUNCT
ejde-673	540	46	sok	sok	NOUN
ejde-673	540	47	′(u	′(u	NOUN
ejde-673	540	48	)	)	PUNCT
ejde-673	540	49	̸=	̸=	PROPN
ejde-673	540	50	0	0	NUM
ejde-673	540	51	.	.	PUNCT
ejde-673	541	1	hencemα	hencemα	PROPN
ejde-673	541	2	is	be	AUX
ejde-673	541	3	a	a	DET
ejde-673	541	4	c1	c1	NOUN
ejde-673	541	5	-	-	PUNCT
ejde-673	541	6	submanifold	submanifold	NOUN
ejde-673	541	7	of	of	ADP
ejde-673	541	8	y	y	PROPN
ejde-673	541	9	with	with	ADP
ejde-673	541	10	codimension	codimension	NOUN
ejde-673	541	11	one	one	NUM
ejde-673	541	12	.	.	PUNCT
ejde-673	542	1	moreover	moreover	ADV
ejde-673	542	2	,	,	PUNCT
ejde-673	542	3	mα	mα	PROPN
ejde-673	542	4	is	be	AUX
ejde-673	542	5	weakly	weakly	ADV
ejde-673	542	6	closed	closed	ADJ
ejde-673	542	7	subset	subset	NOUN
ejde-673	542	8	of	of	ADP
ejde-673	542	9	y	y	PROPN
ejde-673	542	10	.	.	PUNCT
ejde-673	543	1	indeed	indeed	ADV
ejde-673	543	2	,	,	PUNCT
ejde-673	543	3	let	let	VERB
ejde-673	543	4	uj	uj	PROPN
ejde-673	543	5	∈	∈	PROPN
ejde-673	543	6	mα	mα	PROPN
ejde-673	543	7	and	and	CCONJ
ejde-673	543	8	uj	uj	PROPN
ejde-673	543	9	→	→	SYM
ejde-673	543	10	u	u	NOUN
ejde-673	543	11	weakly	weakly	ADV
ejde-673	543	12	in	in	ADP
ejde-673	543	13	y	y	PROPN
ejde-673	543	14	as	as	ADP
ejde-673	543	15	j	j	PROPN
ejde-673	543	16	→	→	SYM
ejde-673	543	17	∞.	∞.	PROPN
ejde-673	543	18	since	since	SCONJ
ejde-673	543	19	k	k	PROPN
ejde-673	543	20	is	be	AUX
ejde-673	543	21	sequentially	sequentially	ADV
ejde-673	543	22	weakly	weakly	ADV
ejde-673	543	23	continuous	continuous	ADJ
ejde-673	543	24	from	from	ADP
ejde-673	543	25	proposition	proposition	NOUN
ejde-673	543	26	3.11	3.11	NUM
ejde-673	543	27	(	(	PUNCT
ejde-673	543	28	ii	ii	NOUN
ejde-673	543	29	)	)	PUNCT
ejde-673	543	30	,	,	PUNCT
ejde-673	543	31	α	α	X
ejde-673	543	32	=	=	PUNCT
ejde-673	543	33	k(uj	k(uj	NOUN
ejde-673	543	34	)	)	PUNCT
ejde-673	543	35	→	→	SYM
ejde-673	543	36	k(u	k(u	NOUN
ejde-673	543	37	)	)	PUNCT
ejde-673	543	38	,	,	PUNCT
ejde-673	543	39	so	so	CCONJ
ejde-673	543	40	u	u	PROPN
ejde-673	543	41	∈	∈	PROPN
ejde-673	543	42	mα	mα	PROPN
ejde-673	543	43	.	.	PUNCT
ejde-673	544	1	it	it	PRON
ejde-673	544	2	is	be	AUX
ejde-673	544	3	well	well	ADV
ejde-673	544	4	known	know	VERB
ejde-673	544	5	that	that	SCONJ
ejde-673	544	6	when	when	SCONJ
ejde-673	544	7	u	u	PROPN
ejde-673	544	8	∈	∈	PROPN
ejde-673	544	9	mα	mα	PROPN
ejde-673	544	10	,	,	PUNCT
ejde-673	544	11	a	a	DET
ejde-673	544	12	pair	pair	NOUN
ejde-673	544	13	(	(	PUNCT
ejde-673	544	14	u	u	NOUN
ejde-673	544	15	,	,	PUNCT
ejde-673	544	16	λ	λ	PROPN
ejde-673	544	17	)	)	PUNCT
ejde-673	544	18	∈	∈	PROPN
ejde-673	544	19	y	y	PROPN
ejde-673	544	20	×	×	NOUN
ejde-673	544	21	r	r	NOUN
ejde-673	544	22	solves	solve	NOUN
ejde-673	544	23	(	(	PUNCT
ejde-673	544	24	3.12	3.12	NUM
ejde-673	544	25	)	)	PUNCT
ejde-673	544	26	if	if	SCONJ
ejde-673	545	1	and	and	CCONJ
ejde-673	545	2	only	only	ADV
ejde-673	545	3	if	if	SCONJ
ejde-673	545	4	u	u	NOUN
ejde-673	545	5	is	be	AUX
ejde-673	545	6	a	a	DET
ejde-673	545	7	critical	critical	ADJ
ejde-673	545	8	point	point	NOUN
ejde-673	545	9	of	of	ADP
ejde-673	545	10	ψ	ψ	NOUN
ejde-673	545	11	with	with	ADP
ejde-673	545	12	respect	respect	NOUN
ejde-673	545	13	to	to	ADP
ejde-673	545	14	mα	mα	PROPN
ejde-673	545	15	,	,	PUNCT
ejde-673	545	16	that	that	ADV
ejde-673	545	17	is	is	ADV
ejde-673	545	18	,	,	PUNCT
ejde-673	545	19	⟨ψ′(u	⟨ψ′(u	PROPN
ejde-673	545	20	)	)	PUNCT
ejde-673	545	21	,	,	PUNCT
ejde-673	545	22	h⟩y	h⟩y	X
ejde-673	545	23	∗,y	∗,y	NOUN
ejde-673	545	24	=	=	SYM
ejde-673	545	25	0	0	NUM
ejde-673	545	26	for	for	ADP
ejde-673	545	27	all	all	DET
ejde-673	545	28	h	h	NOUN
ejde-673	545	29	∈	∈	PROPN
ejde-673	545	30	tumα	tumα	NOUN
ejde-673	545	31	,	,	PUNCT
ejde-673	545	32	(	(	PUNCT
ejde-673	545	33	see	see	VERB
ejde-673	545	34	for	for	ADP
ejde-673	545	35	example	example	NOUN
ejde-673	545	36	[	[	X
ejde-673	545	37	34	34	NUM
ejde-673	545	38	,	,	PUNCT
ejde-673	545	39	proposition	proposition	NOUN
ejde-673	545	40	43.21	43.21	NUM
ejde-673	545	41	]	]	PUNCT
ejde-673	545	42	)	)	PUNCT
ejde-673	545	43	.	.	PUNCT
ejde-673	546	1	here	here	ADV
ejde-673	546	2	tumα	tumα	PROPN
ejde-673	546	3	is	be	AUX
ejde-673	546	4	the	the	DET
ejde-673	546	5	tangent	tangent	ADJ
ejde-673	546	6	space	space	NOUN
ejde-673	546	7	of	of	ADP
ejde-673	546	8	mα	mα	PROPN
ejde-673	546	9	at	at	ADP
ejde-673	546	10	u	u	PROPN
ejde-673	546	11	∈	∈	PROPN
ejde-673	546	12	mα	mα	NOUN
ejde-673	547	1	and	and	CCONJ
ejde-673	547	2	we	we	PRON
ejde-673	547	3	can	can	AUX
ejde-673	547	4	see	see	VERB
ejde-673	547	5	that	that	DET
ejde-673	547	6	tumα	tumα	NOUN
ejde-673	547	7	=	=	SYM
ejde-673	547	8	ker(k	ker(k	PROPN
ejde-673	547	9	′(u	′(u	NOUN
ejde-673	547	10	)	)	PUNCT
ejde-673	547	11	)	)	PUNCT
ejde-673	548	1	=	=	PRON
ejde-673	548	2	{	{	PUNCT
ejde-673	548	3	v	v	NUM
ejde-673	548	4	∈	∈	X
ejde-673	548	5	y	y	PROPN
ejde-673	548	6	;	;	PUNCT
ejde-673	548	7	⟨k	⟨k	X
ejde-673	548	8	′(u	′(u	NOUN
ejde-673	548	9	)	)	PUNCT
ejde-673	548	10	,	,	PUNCT
ejde-673	548	11	v⟩y	v⟩y	PROPN
ejde-673	548	12	∗,y	∗,y	PROPN
ejde-673	548	13	=	=	SYM
ejde-673	548	14	0	0	NUM
ejde-673	548	15	}	}	PUNCT
ejde-673	548	16	.	.	PUNCT
ejde-673	549	1	let	let	VERB
ejde-673	549	2	p	p	NOUN
ejde-673	549	3	:	:	PUNCT
ejde-673	549	4	y	y	PROPN
ejde-673	549	5	→	→	PUNCT
ejde-673	549	6	tumα	tumα	NOUN
ejde-673	549	7	be	be	AUX
ejde-673	549	8	the	the	DET
ejde-673	549	9	natural	natural	ADJ
ejde-673	549	10	projection	projection	NOUN
ejde-673	549	11	.	.	PUNCT
ejde-673	550	1	note	note	VERB
ejde-673	550	2	that	that	SCONJ
ejde-673	550	3	the	the	DET
ejde-673	550	4	bounded	bounded	ADJ
ejde-673	550	5	linear	linear	PROPN
ejde-673	550	6	map	map	NOUN
ejde-673	550	7	k	k	PROPN
ejde-673	550	8	′(u	′(u	NOUN
ejde-673	550	9	)	)	PUNCT
ejde-673	550	10	:	:	PUNCT
ejde-673	551	1	y	y	X
ejde-673	551	2	→	→	PUNCT
ejde-673	551	3	r	r	NOUN
ejde-673	551	4	is	be	AUX
ejde-673	551	5	surjective	surjective	ADJ
ejde-673	551	6	.	.	PUNCT
ejde-673	552	1	we	we	PRON
ejde-673	552	2	denote	denote	VERB
ejde-673	552	3	the	the	DET
ejde-673	552	4	restriction	restriction	NOUN
ejde-673	552	5	of	of	ADP
ejde-673	552	6	ψ	ψ	X
ejde-673	552	7	to	to	PART
ejde-673	552	8	mα	mα	VERB
ejde-673	552	9	by	by	ADP
ejde-673	552	10	ψ̃	ψ̃	PROPN
ejde-673	552	11	=	=	SYM
ejde-673	552	12	ψ	ψ	X
ejde-673	552	13	∣∣	∣∣	X
ejde-673	552	14	mα	mα	PROPN
ejde-673	552	15	and	and	CCONJ
ejde-673	552	16	the	the	DET
ejde-673	552	17	derivative	derivative	ADJ
ejde-673	552	18	dψ̃(u	dψ̃(u	NOUN
ejde-673	552	19	)	)	PUNCT
ejde-673	552	20	∈	∈	PROPN
ejde-673	552	21	y	y	PROPN
ejde-673	552	22	∗	∗	NOUN
ejde-673	552	23	of	of	ADP
ejde-673	552	24	ψ̃	ψ̃	PROPN
ejde-673	552	25	at	at	ADP
ejde-673	552	26	u	u	PROPN
ejde-673	552	27	∈	∈	PROPN
ejde-673	552	28	mα	mα	NOUN
ejde-673	552	29	can	can	AUX
ejde-673	552	30	be	be	AUX
ejde-673	552	31	defined	define	VERB
ejde-673	552	32	by	by	ADP
ejde-673	552	33	⟨dψ̃(u	⟨dψ̃(u	PROPN
ejde-673	552	34	)	)	PUNCT
ejde-673	552	35	,	,	PUNCT
ejde-673	552	36	v⟩y	v⟩y	PROPN
ejde-673	552	37	∗,y	∗,y	PROPN
ejde-673	552	38	=	=	SYM
ejde-673	552	39	⟨ψ′(u	⟨ψ′(u	PROPN
ejde-673	552	40	)	)	PUNCT
ejde-673	552	41	,	,	PUNCT
ejde-673	552	42	pv⟩y	pv⟩y	PROPN
ejde-673	552	43	∗,y	∗,y	PROPN
ejde-673	552	44	for	for	ADP
ejde-673	552	45	v	v	NOUN
ejde-673	552	46	∈	∈	PROPN
ejde-673	552	47	y	y	PROPN
ejde-673	552	48	.	.	PUNCT
ejde-673	553	1	for	for	ADP
ejde-673	553	2	u	u	PROPN
ejde-673	553	3	∈	∈	PROPN
ejde-673	553	4	mα	mα	PROPN
ejde-673	553	5	,	,	PUNCT
ejde-673	553	6	put	put	VERB
ejde-673	553	7	w	w	NOUN
ejde-673	553	8	=	=	PUNCT
ejde-673	553	9	(	(	PUNCT
ejde-673	553	10	ψ′)−1(k	ψ′)−1(k	NOUN
ejde-673	553	11	′(u	′(u	NOUN
ejde-673	553	12	)	)	PUNCT
ejde-673	553	13	)	)	PUNCT
ejde-673	553	14	.	.	PUNCT
ejde-673	554	1	then	then	ADV
ejde-673	554	2	since	since	SCONJ
ejde-673	554	3	we	we	PRON
ejde-673	554	4	have	have	VERB
ejde-673	554	5	(	(	PUNCT
ejde-673	554	6	3.15	3.15	NUM
ejde-673	554	7	)	)	PUNCT
ejde-673	554	8	,	,	PUNCT
ejde-673	554	9	we	we	PRON
ejde-673	554	10	see	see	VERB
ejde-673	554	11	that	that	SCONJ
ejde-673	554	12	k	k	PROPN
ejde-673	554	13	′(u	′(u	NOUN
ejde-673	554	14	)	)	PUNCT
ejde-673	554	15	̸=	̸=	PROPN
ejde-673	554	16	0	0	NUM
ejde-673	554	17	.	.	PUNCT
ejde-673	555	1	from	from	ADP
ejde-673	555	2	(	(	PUNCT
ejde-673	555	3	a7	a7	PROPN
ejde-673	555	4	)	)	PUNCT
ejde-673	555	5	,	,	PUNCT
ejde-673	555	6	the	the	DET
ejde-673	555	7	functional	functional	ADJ
ejde-673	555	8	ψ	ψ	NOUN
ejde-673	555	9	is	be	AUX
ejde-673	555	10	even	even	ADV
ejde-673	555	11	,	,	PUNCT
ejde-673	555	12	so	so	ADV
ejde-673	555	13	ψ′	ψ′	PROPN
ejde-673	555	14	is	be	AUX
ejde-673	555	15	odd	odd	ADJ
ejde-673	555	16	and	and	CCONJ
ejde-673	555	17	so	so	ADV
ejde-673	555	18	ψ′(0	ψ′(0	PROPN
ejde-673	555	19	)	)	PUNCT
ejde-673	555	20	=	=	SYM
ejde-673	556	1	0	0	X
ejde-673	556	2	.	.	PUNCT
ejde-673	557	1	since	since	SCONJ
ejde-673	557	2	(	(	PUNCT
ejde-673	557	3	ψ′)−1	ψ′)−1	NOUN
ejde-673	557	4	is	be	AUX
ejde-673	557	5	injective	injective	ADJ
ejde-673	557	6	,	,	PUNCT
ejde-673	557	7	we	we	PRON
ejde-673	557	8	have	have	VERB
ejde-673	557	9	w	w	ADP
ejde-673	557	10	̸=	̸=	PROPN
ejde-673	557	11	0	0	NUM
ejde-673	557	12	.	.	PUNCT
ejde-673	558	1	from	from	ADP
ejde-673	558	2	strict	strict	ADJ
ejde-673	558	3	monotonicity	monotonicity	NOUN
ejde-673	558	4	of	of	ADP
ejde-673	558	5	ψ′	ψ′	PROPN
ejde-673	558	6	(	(	PUNCT
ejde-673	558	7	proposition	proposition	NOUN
ejde-673	558	8	3.11	3.11	NUM
ejde-673	558	9	(	(	PUNCT
ejde-673	558	10	i	i	NOUN
ejde-673	558	11	)	)	PUNCT
ejde-673	558	12	)	)	PUNCT
ejde-673	558	13	,	,	PUNCT
ejde-673	558	14	⟨k	⟨k	PRON
ejde-673	558	15	′(u	′(u	NOUN
ejde-673	558	16	)	)	PUNCT
ejde-673	558	17	,	,	PUNCT
ejde-673	558	18	w⟩y	w⟩y	X
ejde-673	558	19	∗,y	∗,y	PROPN
ejde-673	558	20	=	=	SYM
ejde-673	558	21	⟨k	⟨k	PRON
ejde-673	558	22	′(u	′(u	NOUN
ejde-673	558	23	)	)	PUNCT
ejde-673	558	24	,	,	PUNCT
ejde-673	558	25	(	(	PUNCT
ejde-673	558	26	ψ′)−1(k	ψ′)−1(k	ADP
ejde-673	558	27	′(u))⟩y	′(u))⟩y	PROPN
ejde-673	558	28	∗,y	∗,y	NOUN
ejde-673	558	29	=	=	SYM
ejde-673	558	30	⟨ψ′(w	⟨ψ′(w	PROPN
ejde-673	558	31	)	)	PUNCT
ejde-673	558	32	,	,	PUNCT
ejde-673	559	1	w⟩y	w⟩y	X
ejde-673	559	2	∗,y	∗,y	PROPN
ejde-673	559	3	>	>	X
ejde-673	559	4	0	0	NUM
ejde-673	559	5	.	.	PUNCT
ejde-673	560	1	(	(	PUNCT
ejde-673	560	2	3.16	3.16	NUM
ejde-673	560	3	)	)	PUNCT
ejde-673	560	4	hence	hence	ADV
ejde-673	560	5	since	since	SCONJ
ejde-673	560	6	w	w	NOUN
ejde-673	560	7	=	=	PUNCT
ejde-673	560	8	(	(	PUNCT
ejde-673	560	9	ψ′)−1(k	ψ′)−1(k	NOUN
ejde-673	560	10	′(u	′(u	NOUN
ejde-673	560	11	)	)	PUNCT
ejde-673	560	12	)	)	PUNCT
ejde-673	561	1	̸∈	̸∈	PROPN
ejde-673	561	2	tumα	tumα	PROPN
ejde-673	561	3	,	,	PUNCT
ejde-673	561	4	we	we	PRON
ejde-673	561	5	can	can	AUX
ejde-673	561	6	see	see	VERB
ejde-673	561	7	that	that	SCONJ
ejde-673	561	8	y	y	PROPN
ejde-673	561	9	=	=	PUNCT
ejde-673	561	10	tumα	tumα	PROPN
ejde-673	561	11	⊕	⊕	PROPN
ejde-673	561	12	{	{	PUNCT
ejde-673	561	13	β(ψ′)−1(k	β(ψ′)−1(k	ADP
ejde-673	561	14	′(u));β	′(u));β	VERB
ejde-673	561	15	∈	∈	NOUN
ejde-673	561	16	r	r	NOUN
ejde-673	561	17	}	}	PUNCT
ejde-673	561	18	.	.	PUNCT
ejde-673	562	1	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	562	2	eigenvalue	eigenvalue	VERB
ejde-673	562	3	problems	problem	NOUN
ejde-673	562	4	for	for	ADP
ejde-673	562	5	kirchhoff	kirchhoff	NOUN
ejde-673	562	6	-	-	PUNCT
ejde-673	562	7	type	type	NOUN
ejde-673	562	8	equations	equation	NOUN
ejde-673	562	9	19	19	NUM
ejde-673	562	10	for	for	ADP
ejde-673	562	11	every	every	DET
ejde-673	562	12	v	v	NOUN
ejde-673	562	13	∈	∈	PROPN
ejde-673	562	14	y	y	NOUN
ejde-673	562	15	,	,	PUNCT
ejde-673	562	16	there	there	PRON
ejde-673	562	17	exists	exist	VERB
ejde-673	562	18	a	a	DET
ejde-673	562	19	unique	unique	ADJ
ejde-673	562	20	β	β	X
ejde-673	562	21	∈	∈	NOUN
ejde-673	562	22	r	r	NOUN
ejde-673	562	23	such	such	DET
ejde-673	562	24	that	that	DET
ejde-673	562	25	v	v	NOUN
ejde-673	562	26	=	=	SYM
ejde-673	563	1	pv	pv	NOUN
ejde-673	564	1	+	+	CCONJ
ejde-673	564	2	β(ψ′)−1(k	β(ψ′)−1(k	ADP
ejde-673	564	3	′(u	′(u	NOUN
ejde-673	564	4	)	)	PUNCT
ejde-673	564	5	)	)	PUNCT
ejde-673	564	6	.	.	PUNCT
ejde-673	565	1	since	since	SCONJ
ejde-673	565	2	pv	pv	PROPN
ejde-673	565	3	∈	∈	PROPN
ejde-673	565	4	tumα	tumα	NOUN
ejde-673	565	5	=	=	SYM
ejde-673	565	6	ker(k	ker(k	PROPN
ejde-673	565	7	′(u	′(u	NOUN
ejde-673	565	8	)	)	PUNCT
ejde-673	565	9	)	)	PUNCT
ejde-673	565	10	,	,	PUNCT
ejde-673	565	11	we	we	PRON
ejde-673	565	12	have	have	VERB
ejde-673	565	13	⟨k	⟨k	NOUN
ejde-673	565	14	′(u	′(u	NOUN
ejde-673	565	15	)	)	PUNCT
ejde-673	565	16	,	,	PUNCT
ejde-673	565	17	v⟩y	v⟩y	PROPN
ejde-673	565	18	∗,y	∗,y	PROPN
ejde-673	565	19	=	=	SYM
ejde-673	565	20	β⟨k	β⟨k	NUM
ejde-673	565	21	′(u	′(u	NOUN
ejde-673	565	22	)	)	PUNCT
ejde-673	565	23	,	,	PUNCT
ejde-673	565	24	(	(	PUNCT
ejde-673	565	25	ψ′)−1(k	ψ′)−1(k	PROPN
ejde-673	565	26	′(u))⟩y	′(u))⟩y	PROPN
ejde-673	565	27	∗,y	∗,y	PROPN
ejde-673	565	28	.	.	PUNCT
ejde-673	566	1	thus	thus	ADV
ejde-673	566	2	from	from	ADP
ejde-673	566	3	(	(	PUNCT
ejde-673	566	4	3.14	3.14	NUM
ejde-673	566	5	)	)	PUNCT
ejde-673	566	6	,	,	PUNCT
ejde-673	566	7	we	we	PRON
ejde-673	566	8	can	can	AUX
ejde-673	566	9	write	write	VERB
ejde-673	566	10	β	β	X
ejde-673	566	11	=	=	SYM
ejde-673	566	12	⟨k	⟨k	PRON
ejde-673	566	13	′(u	′(u	NOUN
ejde-673	566	14	)	)	PUNCT
ejde-673	566	15	,	,	PUNCT
ejde-673	566	16	v⟩y	v⟩y	PROPN
ejde-673	566	17	∗,y	∗,y	PROPN
ejde-673	566	18	⟨k	⟨k	NOUN
ejde-673	566	19	′(u	′(u	NOUN
ejde-673	566	20	)	)	PUNCT
ejde-673	566	21	,	,	PUNCT
ejde-673	566	22	(	(	PUNCT
ejde-673	566	23	ψ′)−1(k	ψ′)−1(k	PROPN
ejde-673	566	24	′(u))⟩y	′(u))⟩y	PROPN
ejde-673	566	25	∗,y	∗,y	PROPN
ejde-673	566	26	.	.	PUNCT
ejde-673	567	1	now	now	ADV
ejde-673	567	2	we	we	PRON
ejde-673	567	3	have	have	VERB
ejde-673	567	4	⟨dψ̃(u	⟨dψ̃(u	PROPN
ejde-673	567	5	)	)	PUNCT
ejde-673	567	6	,	,	PUNCT
ejde-673	567	7	v⟩y	v⟩y	PROPN
ejde-673	567	8	∗,y	∗,y	PROPN
ejde-673	567	9	=	=	SYM
ejde-673	567	10	⟨ψ′(u	⟨ψ′(u	PROPN
ejde-673	567	11	)	)	PUNCT
ejde-673	567	12	,	,	PUNCT
ejde-673	567	13	pv⟩y	pv⟩y	PROPN
ejde-673	567	14	∗,y	∗,y	PROPN
ejde-673	567	15	=	=	SYM
ejde-673	567	16	⟨ψ′(u	⟨ψ′(u	PROPN
ejde-673	567	17	)	)	PUNCT
ejde-673	567	18	,	,	PUNCT
ejde-673	567	19	v⟩y	v⟩y	PROPN
ejde-673	567	20	∗,y	∗,y	PROPN
ejde-673	567	21	−	−	PROPN
ejde-673	567	22	〈	〈	PROPN
ejde-673	567	23	ψ′(u	ψ′(u	NOUN
ejde-673	567	24	)	)	PUNCT
ejde-673	567	25	,	,	PUNCT
ejde-673	567	26	⟨k	⟨k	PRON
ejde-673	567	27	′(u	′(u	NOUN
ejde-673	567	28	)	)	PUNCT
ejde-673	567	29	,	,	PUNCT
ejde-673	567	30	v⟩y	v⟩y	PROPN
ejde-673	567	31	∗,y	∗,y	PROPN
ejde-673	567	32	⟨k	⟨k	NOUN
ejde-673	567	33	′(u	′(u	NOUN
ejde-673	567	34	)	)	PUNCT
ejde-673	567	35	,	,	PUNCT
ejde-673	567	36	(	(	PUNCT
ejde-673	567	37	ψ′)−1(k	ψ′)−1(k	ADJ
ejde-673	567	38	′(u))⟩y	′(u))⟩y	PROPN
ejde-673	567	39	∗,y	∗,y	PROPN
ejde-673	567	40	(	(	PUNCT
ejde-673	567	41	ψ′)−1(k	ψ′)−1(k	NOUN
ejde-673	567	42	′(u	′(u	NOUN
ejde-673	567	43	)	)	PUNCT
ejde-673	567	44	)	)	PUNCT
ejde-673	567	45	〉	〉	NOUN
ejde-673	568	1	y	y	PROPN
ejde-673	568	2	∗,y	∗,y	PROPN
ejde-673	568	3	=	=	SYM
ejde-673	568	4	〈	〈	PROPN
ejde-673	568	5	ψ′(u)−	ψ′(u)−	PROPN
ejde-673	568	6	⟨ψ′(u	⟨ψ′(u	PROPN
ejde-673	568	7	)	)	PUNCT
ejde-673	568	8	,	,	PUNCT
ejde-673	568	9	(	(	PUNCT
ejde-673	568	10	ψ′)−1(k	ψ′)−1(k	ADJ
ejde-673	568	11	′(u))⟩y	′(u))⟩y	PROPN
ejde-673	568	12	∗,y	∗,y	PROPN
ejde-673	568	13	⟨k	⟨k	PROPN
ejde-673	568	14	′(u	′(u	NOUN
ejde-673	568	15	)	)	PUNCT
ejde-673	568	16	,	,	PUNCT
ejde-673	568	17	(	(	PUNCT
ejde-673	568	18	ψ′)−1(k	ψ′)−1(k	PROPN
ejde-673	568	19	′(u))⟩y	′(u))⟩y	PROPN
ejde-673	568	20	∗,y	∗,y	PROPN
ejde-673	568	21	k	k	PROPN
ejde-673	568	22	′(u	′(u	NOUN
ejde-673	568	23	)	)	PUNCT
ejde-673	568	24	,	,	PUNCT
ejde-673	568	25	v	v	X
ejde-673	568	26	〉	〉	NOUN
ejde-673	568	27	y	y	PROPN
ejde-673	568	28	∗,y	∗,y	PROPN
ejde-673	568	29	for	for	ADP
ejde-673	568	30	all	all	DET
ejde-673	568	31	v	v	NOUN
ejde-673	568	32	∈	∈	NOUN
ejde-673	568	33	y.	y.	NOUN
ejde-673	569	1	thus	thus	ADV
ejde-673	569	2	we	we	PRON
ejde-673	569	3	have	have	VERB
ejde-673	569	4	dψ̃(u	dψ̃(u	ADJ
ejde-673	569	5	)	)	PUNCT
ejde-673	569	6	=	=	SYM
ejde-673	569	7	ψ′(u)−	ψ′(u)−	DET
ejde-673	569	8	λ(u)k	λ(u)k	PROPN
ejde-673	569	9	′(u	′(u	NOUN
ejde-673	569	10	)	)	PUNCT
ejde-673	569	11	,	,	PUNCT
ejde-673	570	1	where	where	SCONJ
ejde-673	570	2	λ(u	λ(u	NUM
ejde-673	570	3	)	)	PUNCT
ejde-673	570	4	=	=	SYM
ejde-673	570	5	⟨ψ′(u	⟨ψ′(u	PROPN
ejde-673	570	6	)	)	PUNCT
ejde-673	570	7	,	,	PUNCT
ejde-673	570	8	(	(	PUNCT
ejde-673	570	9	ψ′)−1(k	ψ′)−1(k	ADJ
ejde-673	570	10	′(u))⟩y	′(u))⟩y	PROPN
ejde-673	570	11	∗,y	∗,y	PROPN
ejde-673	570	12	⟨k	⟨k	PROPN
ejde-673	570	13	′(u	′(u	NOUN
ejde-673	570	14	)	)	PUNCT
ejde-673	570	15	,	,	PUNCT
ejde-673	570	16	(	(	PUNCT
ejde-673	570	17	ψ′)−1(k	ψ′)−1(k	PROPN
ejde-673	570	18	′(u))⟩y	′(u))⟩y	PROPN
ejde-673	570	19	∗,y	∗,y	PROPN
ejde-673	570	20	.	.	PUNCT
ejde-673	571	1	proposition	proposition	NOUN
ejde-673	571	2	3.13	3.13	NUM
ejde-673	571	3	.	.	PUNCT
ejde-673	572	1	for	for	ADP
ejde-673	572	2	each	each	DET
ejde-673	572	3	α	α	PROPN
ejde-673	572	4	>	>	X
ejde-673	572	5	0	0	PROPN
ejde-673	572	6	,	,	PUNCT
ejde-673	572	7	the	the	DET
ejde-673	572	8	functional	functional	ADJ
ejde-673	572	9	ψ̃	ψ̃	PROPN
ejde-673	572	10	:	:	PUNCT
ejde-673	572	11	mα	mα	PROPN
ejde-673	572	12	→	→	SYM
ejde-673	572	13	r	r	NOUN
ejde-673	572	14	satisfies	satisfie	NOUN
ejde-673	572	15	(	(	PUNCT
ejde-673	572	16	ps)ccondition	ps)ccondition	NOUN
ejde-673	572	17	for	for	ADP
ejde-673	572	18	any	any	DET
ejde-673	572	19	c	c	PROPN
ejde-673	572	20	∈	∈	PROPN
ejde-673	572	21	r	r	NOUN
ejde-673	572	22	,	,	PUNCT
ejde-673	572	23	that	that	ADV
ejde-673	572	24	is	is	ADV
ejde-673	572	25	,	,	PUNCT
ejde-673	572	26	if	if	SCONJ
ejde-673	572	27	any	any	DET
ejde-673	572	28	sequence	sequence	NOUN
ejde-673	572	29	{	{	PUNCT
ejde-673	572	30	un	un	PROPN
ejde-673	572	31	}	}	PUNCT
ejde-673	572	32	⊂	⊂	NOUN
ejde-673	572	33	mα	mα	ADP
ejde-673	572	34	such	such	ADJ
ejde-673	572	35	that	that	DET
ejde-673	572	36	ψ̃(un	ψ̃(un	NOUN
ejde-673	572	37	)	)	PUNCT
ejde-673	572	38	→	→	SYM
ejde-673	572	39	c	c	NOUN
ejde-673	572	40	and	and	CCONJ
ejde-673	572	41	∥dψ̃(un)∥y	∥dψ̃(un)∥y	PROPN
ejde-673	572	42	∗	∗	NOUN
ejde-673	572	43	→	→	SYM
ejde-673	572	44	0	0	NUM
ejde-673	572	45	as	as	ADP
ejde-673	572	46	n	n	NUM
ejde-673	572	47	→	→	SYM
ejde-673	572	48	∞	∞	PROPN
ejde-673	572	49	,	,	PUNCT
ejde-673	572	50	then	then	ADV
ejde-673	572	51	{	{	PUNCT
ejde-673	572	52	un	un	PROPN
ejde-673	572	53	}	}	PUNCT
ejde-673	572	54	contains	contain	VERB
ejde-673	572	55	a	a	DET
ejde-673	572	56	convergent	convergent	NOUN
ejde-673	572	57	subsequence	subsequence	NOUN
ejde-673	572	58	.	.	PUNCT
ejde-673	573	1	proof	proof	NOUN
ejde-673	573	2	.	.	PUNCT
ejde-673	574	1	let	let	AUX
ejde-673	574	2	{	{	PUNCT
ejde-673	574	3	un	un	ADJ
ejde-673	574	4	}	}	PUNCT
ejde-673	574	5	⊂	⊂	NOUN
ejde-673	574	6	mα	mα	PROPN
ejde-673	574	7	satisfy	satisfy	VERB
ejde-673	574	8	that	that	DET
ejde-673	574	9	ψ̃(un	ψ̃(un	NOUN
ejde-673	574	10	)	)	PUNCT
ejde-673	574	11	→	→	SYM
ejde-673	574	12	c	c	NOUN
ejde-673	574	13	and	and	CCONJ
ejde-673	574	14	dψ̃(un	dψ̃(un	PROPN
ejde-673	574	15	)	)	PUNCT
ejde-673	574	16	→	→	SYM
ejde-673	574	17	0	0	NUM
ejde-673	574	18	in	in	ADP
ejde-673	574	19	y	y	PROPN
ejde-673	574	20	∗	∗	NOUN
ejde-673	574	21	as	as	ADP
ejde-673	574	22	n	n	PROPN
ejde-673	574	23	→	→	SYM
ejde-673	574	24	∞.	∞.	PROPN
ejde-673	574	25	then	then	ADV
ejde-673	574	26	since	since	SCONJ
ejde-673	574	27	from	from	ADP
ejde-673	574	28	(	(	PUNCT
ejde-673	574	29	3.6	3.6	NUM
ejde-673	574	30	)	)	PUNCT
ejde-673	574	31	and	and	CCONJ
ejde-673	574	32	(	(	PUNCT
ejde-673	574	33	a5	a5	PROPN
ejde-673	574	34	)	)	PUNCT
ejde-673	574	35	,	,	PUNCT
ejde-673	574	36	ψ̃(un	ψ̃(un	NOUN
ejde-673	574	37	)	)	PUNCT
ejde-673	574	38	=	=	SYM
ejde-673	574	39	m̂(φ(un	m̂(φ(un	NOUN
ejde-673	574	40	)	)	PUNCT
ejde-673	574	41	)	)	PUNCT
ejde-673	574	42	≥	≥	PROPN
ejde-673	574	43	m0	m0	PROPN
ejde-673	574	44	l	l	PROPN
ejde-673	575	1	(	(	PUNCT
ejde-673	575	2	k0	k0	PROPN
ejde-673	575	3	p+	p+	PROPN
ejde-673	575	4	∫	∫	PROPN
ejde-673	575	5	ω	ω	PROPN
ejde-673	575	6	h1(x)|∇un(x)|p(x	h1(x)|∇un(x)|p(x	PROPN
ejde-673	575	7	)	)	PUNCT
ejde-673	575	8	dx	dx	PROPN
ejde-673	575	9	)	)	PUNCT
ejde-673	576	1	l	l	NOUN
ejde-673	576	2	≥	≥	PROPN
ejde-673	576	3	m0	m0	PROPN
ejde-673	576	4	l	l	PROPN
ejde-673	576	5	(	(	PUNCT
ejde-673	576	6	k0	k0	PROPN
ejde-673	576	7	p+	p+	PROPN
ejde-673	576	8	∥un∥p	∥un∥p	PROPN
ejde-673	577	1	+	+	CCONJ
ejde-673	577	2	y	y	PROPN
ejde-673	577	3	∧	∧	PROPN
ejde-673	577	4	∥un∥p	∥un∥p	NUM
ejde-673	578	1	−	−	PROPN
ejde-673	578	2	y	y	PROPN
ejde-673	578	3	)	)	PUNCT
ejde-673	578	4	l	l	NOUN
ejde-673	578	5	,	,	PUNCT
ejde-673	578	6	{	{	PUNCT
ejde-673	578	7	un	un	PROPN
ejde-673	578	8	}	}	PUNCT
ejde-673	578	9	is	be	AUX
ejde-673	578	10	bounded	bound	VERB
ejde-673	578	11	in	in	ADP
ejde-673	578	12	y	y	PROPN
ejde-673	578	13	.	.	PUNCT
ejde-673	579	1	since	since	SCONJ
ejde-673	579	2	y	y	PROPN
ejde-673	579	3	is	be	AUX
ejde-673	579	4	a	a	DET
ejde-673	579	5	reflexive	reflexive	ADJ
ejde-673	579	6	banach	banach	NOUN
ejde-673	579	7	space	space	NOUN
ejde-673	579	8	from	from	ADP
ejde-673	579	9	proposition	proposition	NOUN
ejde-673	579	10	3.4	3.4	NUM
ejde-673	579	11	,	,	PUNCT
ejde-673	579	12	there	there	PRON
ejde-673	579	13	exist	exist	VERB
ejde-673	579	14	a	a	DET
ejde-673	579	15	subsequence	subsequence	NOUN
ejde-673	579	16	{	{	PUNCT
ejde-673	579	17	un′	un′	NOUN
ejde-673	579	18	}	}	PUNCT
ejde-673	579	19	of	of	ADP
ejde-673	579	20	{	{	PUNCT
ejde-673	579	21	un	un	PROPN
ejde-673	579	22	}	}	PUNCT
ejde-673	579	23	and	and	CCONJ
ejde-673	579	24	u0	u0	PROPN
ejde-673	579	25	∈	∈	PROPN
ejde-673	579	26	y	y	PROPN
ejde-673	579	27	such	such	ADJ
ejde-673	579	28	that	that	SCONJ
ejde-673	579	29	un′	un′	PROPN
ejde-673	579	30	→	→	SYM
ejde-673	579	31	u0	u0	ADJ
ejde-673	579	32	weakly	weakly	ADJ
ejde-673	579	33	in	in	ADP
ejde-673	579	34	y	y	PROPN
ejde-673	579	35	.	.	PUNCT
ejde-673	580	1	by	by	ADP
ejde-673	580	2	proposition	proposition	NOUN
ejde-673	580	3	3.11	3.11	NUM
ejde-673	580	4	(	(	PUNCT
ejde-673	580	5	ii	ii	NOUN
ejde-673	580	6	)	)	PUNCT
ejde-673	580	7	and	and	CCONJ
ejde-673	580	8	(	(	PUNCT
ejde-673	580	9	iii	iii	NOUN
ejde-673	580	10	)	)	PUNCT
ejde-673	580	11	,	,	PUNCT
ejde-673	580	12	k	k	PROPN
ejde-673	580	13	′(un′	′(un′	PROPN
ejde-673	580	14	)	)	PUNCT
ejde-673	580	15	→	→	SYM
ejde-673	580	16	k	k	PROPN
ejde-673	580	17	′(u0	′(u0	NOUN
ejde-673	580	18	)	)	PUNCT
ejde-673	580	19	in	in	ADP
ejde-673	580	20	y	y	PROPN
ejde-673	580	21	∗	∗	NOUN
ejde-673	580	22	and	and	CCONJ
ejde-673	580	23	k(un′	k(un′	NOUN
ejde-673	580	24	)	)	PUNCT
ejde-673	580	25	→	→	SYM
ejde-673	580	26	k(u0	k(u0	PROPN
ejde-673	580	27	)	)	PUNCT
ejde-673	580	28	as	as	ADP
ejde-673	580	29	n	n	PROPN
ejde-673	580	30	→	→	SYM
ejde-673	580	31	∞.	∞.	PROPN
ejde-673	580	32	thereby	thereby	ADV
ejde-673	580	33	,	,	PUNCT
ejde-673	580	34	u0	u0	PROPN
ejde-673	580	35	∈	∈	PROPN
ejde-673	580	36	mα	mα	PROPN
ejde-673	580	37	.	.	PUNCT
ejde-673	580	38	put	put	VERB
ejde-673	580	39	wn′	wn′	NOUN
ejde-673	580	40	=	=	PUNCT
ejde-673	580	41	(	(	PUNCT
ejde-673	580	42	ψ′)−1(k	ψ′)−1(k	NOUN
ejde-673	580	43	′(un′	′(un′	NOUN
ejde-673	580	44	)	)	PUNCT
ejde-673	580	45	)	)	PUNCT
ejde-673	580	46	.	.	PUNCT
ejde-673	581	1	since	since	SCONJ
ejde-673	581	2	k	k	PROPN
ejde-673	581	3	′(un′	′(un′	PROPN
ejde-673	581	4	)	)	PUNCT
ejde-673	581	5	→	→	SYM
ejde-673	581	6	k	k	PROPN
ejde-673	581	7	′(u0	′(u0	NOUN
ejde-673	581	8	)	)	PUNCT
ejde-673	581	9	̸=	̸=	NOUN
ejde-673	581	10	0	0	NUM
ejde-673	581	11	in	in	ADP
ejde-673	581	12	y	y	PROPN
ejde-673	581	13	∗	∗	NOUN
ejde-673	581	14	from	from	ADP
ejde-673	581	15	(	(	PUNCT
ejde-673	581	16	3.15	3.15	NUM
ejde-673	581	17	)	)	PUNCT
ejde-673	581	18	,	,	PUNCT
ejde-673	581	19	we	we	PRON
ejde-673	581	20	see	see	VERB
ejde-673	581	21	that	that	SCONJ
ejde-673	581	22	wn′	wn′	PROPN
ejde-673	581	23	→	→	SYM
ejde-673	581	24	w0	w0	PROPN
ejde-673	581	25	̸=	̸=	PROPN
ejde-673	581	26	0	0	NUM
ejde-673	581	27	in	in	ADP
ejde-673	581	28	y	y	PROPN
ejde-673	581	29	,	,	PUNCT
ejde-673	581	30	where	where	SCONJ
ejde-673	581	31	w0	w0	PROPN
ejde-673	581	32	=	=	SYM
ejde-673	581	33	(	(	PUNCT
ejde-673	581	34	ψ′)−1(k	ψ′)−1(k	NOUN
ejde-673	581	35	′(u0	′(u0	NOUN
ejde-673	581	36	)	)	PUNCT
ejde-673	581	37	)	)	PUNCT
ejde-673	581	38	.	.	PUNCT
ejde-673	582	1	thus	thus	ADV
ejde-673	582	2	⟨k	⟨k	DET
ejde-673	582	3	′(un′	′(un′	NOUN
ejde-673	582	4	)	)	PUNCT
ejde-673	582	5	,	,	PUNCT
ejde-673	582	6	(	(	PUNCT
ejde-673	582	7	ψ′)−1(k	ψ′)−1(k	ADP
ejde-673	582	8	′(un′))⟩y	′(un′))⟩y	ADJ
ejde-673	582	9	∗,y	∗,y	NOUN
ejde-673	582	10	=	=	SYM
ejde-673	582	11	⟨ψ′(wn′	⟨ψ′(wn′	PROPN
ejde-673	582	12	)	)	PUNCT
ejde-673	582	13	,	,	PUNCT
ejde-673	582	14	wn′⟩y	wn′⟩y	PROPN
ejde-673	582	15	∗,y	∗,y	PROPN
ejde-673	582	16	→	→	SYM
ejde-673	582	17	⟨ψ′(w0	⟨ψ′(w0	PROPN
ejde-673	582	18	)	)	PUNCT
ejde-673	582	19	,	,	PUNCT
ejde-673	582	20	w0⟩y	w0⟩y	NOUN
ejde-673	582	21	∗,y	∗,y	PROPN
ejde-673	582	22	>	>	X
ejde-673	582	23	0	0	NUM
ejde-673	582	24	.	.	PUNCT
ejde-673	583	1	(	(	PUNCT
ejde-673	583	2	3.17	3.17	NUM
ejde-673	583	3	)	)	PUNCT
ejde-673	583	4	on	on	ADP
ejde-673	583	5	the	the	DET
ejde-673	583	6	other	other	ADJ
ejde-673	583	7	hand	hand	NOUN
ejde-673	583	8	,	,	PUNCT
ejde-673	583	9	|⟨ψ′(un′	|⟨ψ′(un′	PROPN
ejde-673	583	10	)	)	PUNCT
ejde-673	583	11	,	,	PUNCT
ejde-673	583	12	(	(	PUNCT
ejde-673	583	13	ψ′)−1(k	ψ′)−1(k	PUNCT
ejde-673	583	14	′(un′))⟩y	′(un′))⟩y	ADJ
ejde-673	583	15	∗,y	∗,y	NOUN
ejde-673	583	16	|	|	NOUN
ejde-673	583	17	=	=	SYM
ejde-673	583	18	|⟨ψ′(un′	|⟨ψ′(un′	PROPN
ejde-673	583	19	)	)	PUNCT
ejde-673	583	20	,	,	PUNCT
ejde-673	583	21	wn′⟩y	wn′⟩y	PROPN
ejde-673	583	22	∗,y	∗,y	PROPN
ejde-673	583	23	|	|	ADV
ejde-673	583	24	≤	≤	NUM
ejde-673	583	25	∥ψ′(un′)∥y	∥ψ′(un′)∥y	NOUN
ejde-673	583	26	∗∥wn′∥y	∗∥wn′∥y	NUM
ejde-673	583	27	.	.	PUNCT
ejde-673	584	1	20	20	NUM
ejde-673	584	2	j.	j.	PROPN
ejde-673	584	3	aramaki	aramaki	PROPN
ejde-673	584	4	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	584	5	since	since	SCONJ
ejde-673	584	6	un′	un′	PROPN
ejde-673	584	7	→	→	SYM
ejde-673	584	8	u0	u0	ADJ
ejde-673	584	9	weakly	weakly	ADV
ejde-673	584	10	in	in	ADP
ejde-673	584	11	y	y	PROPN
ejde-673	584	12	,	,	PUNCT
ejde-673	584	13	we	we	PRON
ejde-673	584	14	see	see	VERB
ejde-673	584	15	that	that	SCONJ
ejde-673	584	16	{	{	PUNCT
ejde-673	584	17	un′	un′	NOUN
ejde-673	584	18	}	}	PUNCT
ejde-673	584	19	is	be	AUX
ejde-673	584	20	bounded	bound	VERB
ejde-673	584	21	in	in	ADP
ejde-673	584	22	y	y	PROPN
ejde-673	584	23	,	,	PUNCT
ejde-673	584	24	so	so	ADV
ejde-673	584	25	by	by	ADP
ejde-673	584	26	proposition	proposition	NOUN
ejde-673	584	27	3.10	3.10	NUM
ejde-673	584	28	(	(	PUNCT
ejde-673	584	29	i	i	NOUN
ejde-673	584	30	)	)	PUNCT
ejde-673	584	31	,	,	PUNCT
ejde-673	584	32	∥ψ′(un′)∥y	∥ψ′(un′)∥y	NOUN
ejde-673	584	33	∗	∗	NOUN
ejde-673	584	34	is	be	AUX
ejde-673	584	35	bounded	bound	VERB
ejde-673	584	36	.	.	PUNCT
ejde-673	585	1	hence	hence	ADV
ejde-673	585	2	,	,	PUNCT
ejde-673	585	3	there	there	PRON
ejde-673	585	4	exists	exist	VERB
ejde-673	585	5	a	a	DET
ejde-673	585	6	constant	constant	ADJ
ejde-673	585	7	c2	c2	PROPN
ejde-673	585	8	>	>	X
ejde-673	585	9	0	0	NUM
ejde-673	586	1	such	such	ADJ
ejde-673	586	2	that	that	DET
ejde-673	586	3	|⟨ψ′(un′	|⟨ψ′(un′	NOUN
ejde-673	586	4	)	)	PUNCT
ejde-673	586	5	,	,	PUNCT
ejde-673	586	6	(	(	PUNCT
ejde-673	586	7	ψ′)−1(k	ψ′)−1(k	PUNCT
ejde-673	586	8	′(un′))⟩y	′(un′))⟩y	ADJ
ejde-673	586	9	∗,y	∗,y	PROPN
ejde-673	586	10	|	|	ADV
ejde-673	586	11	≤	≤	PROPN
ejde-673	586	12	c2	c2	PROPN
ejde-673	586	13	.	.	PUNCT
ejde-673	587	1	(	(	PUNCT
ejde-673	587	2	3.18	3.18	NUM
ejde-673	587	3	)	)	PUNCT
ejde-673	587	4	from	from	ADP
ejde-673	587	5	(	(	PUNCT
ejde-673	587	6	3.17	3.17	NUM
ejde-673	587	7	)	)	PUNCT
ejde-673	587	8	and	and	CCONJ
ejde-673	587	9	(	(	PUNCT
ejde-673	587	10	3.18	3.18	NUM
ejde-673	587	11	)	)	PUNCT
ejde-673	587	12	,	,	PUNCT
ejde-673	587	13	{	{	PUNCT
ejde-673	587	14	λ(un′	λ(un′	NOUN
ejde-673	587	15	)	)	PUNCT
ejde-673	587	16	}	}	PUNCT
ejde-673	587	17	is	be	AUX
ejde-673	587	18	bounded	bound	VERB
ejde-673	587	19	in	in	ADP
ejde-673	587	20	r.	r.	PROPN
ejde-673	587	21	passing	pass	VERB
ejde-673	587	22	to	to	ADP
ejde-673	587	23	a	a	DET
ejde-673	587	24	subsequence	subsequence	NOUN
ejde-673	587	25	,	,	PUNCT
ejde-673	587	26	we	we	PRON
ejde-673	587	27	may	may	AUX
ejde-673	587	28	assume	assume	VERB
ejde-673	587	29	that	that	SCONJ
ejde-673	587	30	λ(un′	λ(un′	NOUN
ejde-673	587	31	)	)	PUNCT
ejde-673	587	32	→	→	SYM
ejde-673	587	33	λ0	λ0	NOUN
ejde-673	587	34	for	for	ADP
ejde-673	587	35	some	some	DET
ejde-673	587	36	λ0	λ0	NOUN
ejde-673	587	37	∈	∈	PROPN
ejde-673	587	38	r.	r.	NOUN
ejde-673	587	39	since	since	SCONJ
ejde-673	587	40	dψ̃(un′	dψ̃(un′	PROPN
ejde-673	587	41	)	)	PUNCT
ejde-673	587	42	→	→	SYM
ejde-673	587	43	0	0	NUM
ejde-673	587	44	in	in	ADP
ejde-673	587	45	y	y	PROPN
ejde-673	587	46	∗	∗	NOUN
ejde-673	587	47	,	,	PUNCT
ejde-673	587	48	we	we	PRON
ejde-673	587	49	see	see	VERB
ejde-673	587	50	that	that	DET
ejde-673	587	51	φ′(un′	φ′(un′	NOUN
ejde-673	587	52	)	)	PUNCT
ejde-673	588	1	−	−	PROPN
ejde-673	588	2	λ(un′)k	λ(un′)k	PROPN
ejde-673	589	1	′(un′	′(un′	PROPN
ejde-673	589	2	)	)	PUNCT
ejde-673	589	3	→	→	SYM
ejde-673	589	4	0	0	NUM
ejde-673	589	5	as	as	ADP
ejde-673	589	6	n′	n′	PROPN
ejde-673	589	7	→	→	SYM
ejde-673	589	8	∞.	∞.	PROPN
ejde-673	589	9	hence	hence	ADV
ejde-673	589	10	,	,	PUNCT
ejde-673	589	11	since	since	SCONJ
ejde-673	589	12	k	k	PROPN
ejde-673	589	13	′(un′	′(un′	PROPN
ejde-673	589	14	)	)	PUNCT
ejde-673	589	15	→	→	SYM
ejde-673	589	16	k	k	PROPN
ejde-673	589	17	′(u0	′(u0	NOUN
ejde-673	589	18	)	)	PUNCT
ejde-673	589	19	in	in	ADP
ejde-673	589	20	y	y	PROPN
ejde-673	589	21	∗	∗	NOUN
ejde-673	589	22	,	,	PUNCT
ejde-673	589	23	ψ′(un′	ψ′(un′	NOUN
ejde-673	589	24	)	)	PUNCT
ejde-673	589	25	=	=	SYM
ejde-673	589	26	(	(	PUNCT
ejde-673	589	27	ψ′(un′)−	ψ′(un′)−	PROPN
ejde-673	589	28	λ(un′)k	λ(un′)k	PROPN
ejde-673	589	29	′(un′	′(un′	PROPN
ejde-673	589	30	)	)	PUNCT
ejde-673	589	31	)	)	PUNCT
ejde-673	590	1	+	+	CCONJ
ejde-673	590	2	λ(un′)k	λ(un′)k	PROPN
ejde-673	590	3	′(un′	′(un′	NOUN
ejde-673	590	4	)	)	PUNCT
ejde-673	590	5	→	→	PUNCT
ejde-673	590	6	λ0k	λ0k	NOUN
ejde-673	590	7	′(u0	′(u0	NOUN
ejde-673	590	8	)	)	PUNCT
ejde-673	590	9	in	in	ADP
ejde-673	590	10	y	y	PROPN
ejde-673	590	11	∗	∗	NOUN
ejde-673	590	12	as	as	ADP
ejde-673	590	13	n′	n′	PROPN
ejde-673	590	14	→	→	SYM
ejde-673	590	15	∞.	∞.	PROPN
ejde-673	590	16	therefore	therefore	ADV
ejde-673	590	17	,	,	PUNCT
ejde-673	590	18	we	we	PRON
ejde-673	590	19	see	see	VERB
ejde-673	590	20	that	that	SCONJ
ejde-673	590	21	un′	un′	PROPN
ejde-673	590	22	→	→	SYM
ejde-673	590	23	(	(	PUNCT
ejde-673	590	24	ψ′)−1(λ0k	ψ′)−1(λ0k	NOUN
ejde-673	590	25	′(u0	′(u0	NOUN
ejde-673	590	26	)	)	PUNCT
ejde-673	590	27	)	)	PUNCT
ejde-673	590	28	strongly	strongly	ADV
ejde-673	590	29	in	in	ADP
ejde-673	590	30	y	y	PROPN
ejde-673	590	31	as	as	ADP
ejde-673	590	32	n′	n′	PROPN
ejde-673	590	33	→	→	SYM
ejde-673	590	34	∞.	∞.	PROPN
ejde-673	590	35	□	□	PUNCT
ejde-673	590	36	here	here	ADV
ejde-673	590	37	we	we	PRON
ejde-673	590	38	recall	recall	VERB
ejde-673	590	39	the	the	DET
ejde-673	590	40	notion	notion	NOUN
ejde-673	590	41	of	of	ADP
ejde-673	590	42	“	"	PUNCT
ejde-673	590	43	genus	genus	PROPN
ejde-673	590	44	”	"	PUNCT
ejde-673	590	45	which	which	PRON
ejde-673	590	46	wass	wass	VERB
ejde-673	590	47	introduced	introduce	VERB
ejde-673	590	48	in	in	ADP
ejde-673	590	49	rabinowitz	rabinowitz	NOUN
ejde-673	591	1	[	[	X
ejde-673	591	2	30	30	NUM
ejde-673	591	3	,	,	PUNCT
ejde-673	591	4	chapter	chapter	NOUN
ejde-673	591	5	7	7	NUM
ejde-673	591	6	]	]	PUNCT
ejde-673	591	7	or	or	CCONJ
ejde-673	591	8	[	[	X
ejde-673	591	9	34	34	NUM
ejde-673	591	10	,	,	PUNCT
ejde-673	591	11	section	section	NOUN
ejde-673	591	12	44.3	44.3	NUM
ejde-673	591	13	]	]	PUNCT
ejde-673	591	14	.	.	PUNCT
ejde-673	592	1	let	let	VERB
ejde-673	592	2	e	e	PRON
ejde-673	592	3	be	be	AUX
ejde-673	592	4	a	a	DET
ejde-673	592	5	real	real	ADJ
ejde-673	592	6	banach	banach	NOUN
ejde-673	592	7	space	space	NOUN
ejde-673	592	8	and	and	CCONJ
ejde-673	592	9	let	let	VERB
ejde-673	592	10	e	e	PRON
ejde-673	592	11	denote	denote	VERB
ejde-673	592	12	the	the	DET
ejde-673	592	13	family	family	NOUN
ejde-673	592	14	of	of	ADP
ejde-673	592	15	subsets	subset	NOUN
ejde-673	592	16	a	a	DET
ejde-673	592	17	⊂	⊂	PROPN
ejde-673	592	18	e	e	X
ejde-673	592	19	\	\	X
ejde-673	592	20	{	{	PUNCT
ejde-673	592	21	0	0	NUM
ejde-673	592	22	}	}	PUNCT
ejde-673	592	23	such	such	ADJ
ejde-673	592	24	that	that	SCONJ
ejde-673	592	25	a	a	PRON
ejde-673	592	26	is	be	AUX
ejde-673	592	27	closed	close	VERB
ejde-673	592	28	in	in	ADP
ejde-673	592	29	e	e	NOUN
ejde-673	592	30	and	and	CCONJ
ejde-673	592	31	symmetric	symmetric	ADJ
ejde-673	592	32	with	with	ADP
ejde-673	592	33	respect	respect	NOUN
ejde-673	592	34	to	to	ADP
ejde-673	592	35	0	0	NUM
ejde-673	592	36	,	,	PUNCT
ejde-673	592	37	that	that	ADV
ejde-673	592	38	is	is	ADV
ejde-673	592	39	,	,	PUNCT
ejde-673	592	40	x	x	SYM
ejde-673	592	41	∈	∈	PROPN
ejde-673	592	42	a	a	PRON
ejde-673	592	43	implies	imply	VERB
ejde-673	592	44	−x	−x	PROPN
ejde-673	592	45	∈	∈	PROPN
ejde-673	592	46	a.	a.	NOUN
ejde-673	592	47	for	for	ADP
ejde-673	592	48	∅	∅	NOUN
ejde-673	592	49	̸=	̸=	PROPN
ejde-673	592	50	a	a	DET
ejde-673	592	51	∈	∈	PROPN
ejde-673	592	52	e	e	NOUN
ejde-673	592	53	,	,	PUNCT
ejde-673	592	54	define	define	VERB
ejde-673	592	55	the	the	DET
ejde-673	592	56	genus	genus	NOUN
ejde-673	592	57	of	of	ADP
ejde-673	592	58	a	a	PRON
ejde-673	592	59	to	to	PART
ejde-673	592	60	be	be	AUX
ejde-673	592	61	n	n	PRON
ejde-673	592	62	≥	≥	NUM
ejde-673	592	63	1	1	NUM
ejde-673	592	64	(	(	PUNCT
ejde-673	592	65	denoted	denote	VERB
ejde-673	592	66	by	by	ADP
ejde-673	592	67	γ(a	γ(a	NOUN
ejde-673	592	68	)	)	PUNCT
ejde-673	592	69	=	=	SYM
ejde-673	592	70	n	n	CCONJ
ejde-673	592	71	)	)	PUNCT
ejde-673	592	72	if	if	SCONJ
ejde-673	592	73	there	there	PRON
ejde-673	592	74	is	be	VERB
ejde-673	592	75	a	a	DET
ejde-673	592	76	map	map	NOUN
ejde-673	592	77	φ	φ	PROPN
ejde-673	592	78	∈	∈	PROPN
ejde-673	592	79	c(a	c(a	PROPN
ejde-673	592	80	,	,	PUNCT
ejde-673	592	81	rn	rn	PROPN
ejde-673	592	82	\	\	PROPN
ejde-673	592	83	{	{	PUNCT
ejde-673	592	84	0	0	NUM
ejde-673	592	85	}	}	PUNCT
ejde-673	592	86	)	)	PUNCT
ejde-673	592	87	with	with	ADP
ejde-673	592	88	φ	φ	PROPN
ejde-673	592	89	odd	odd	ADJ
ejde-673	592	90	and	and	CCONJ
ejde-673	592	91	n	n	PRON
ejde-673	592	92	is	be	AUX
ejde-673	592	93	the	the	DET
ejde-673	592	94	smallest	small	ADJ
ejde-673	592	95	integer	integer	NOUN
ejde-673	592	96	with	with	ADP
ejde-673	592	97	this	this	DET
ejde-673	592	98	property	property	NOUN
ejde-673	592	99	.	.	PUNCT
ejde-673	593	1	when	when	SCONJ
ejde-673	593	2	there	there	PRON
ejde-673	593	3	does	do	AUX
ejde-673	593	4	not	not	PART
ejde-673	593	5	exist	exist	VERB
ejde-673	593	6	a	a	DET
ejde-673	593	7	finite	finite	NOUN
ejde-673	593	8	such	such	ADJ
ejde-673	593	9	n	n	NUM
ejde-673	593	10	,	,	PUNCT
ejde-673	593	11	set	set	VERB
ejde-673	593	12	γ(a	γ(a	NOUN
ejde-673	593	13	)	)	PUNCT
ejde-673	594	1	=	=	SYM
ejde-673	594	2	∞.	∞.	PROPN
ejde-673	594	3	finally	finally	ADV
ejde-673	594	4	set	set	VERB
ejde-673	594	5	γ(∅	γ(∅	NOUN
ejde-673	594	6	)	)	PUNCT
ejde-673	594	7	=	=	PUNCT
ejde-673	595	1	0	0	X
ejde-673	595	2	.	.	PUNCT
ejde-673	596	1	the	the	DET
ejde-673	596	2	main	main	ADJ
ejde-673	596	3	properties	property	NOUN
ejde-673	596	4	of	of	ADP
ejde-673	596	5	genus	genus	NOUN
ejde-673	596	6	will	will	AUX
ejde-673	596	7	be	be	AUX
ejde-673	596	8	listed	list	VERB
ejde-673	596	9	in	in	ADP
ejde-673	596	10	the	the	DET
ejde-673	596	11	next	next	ADJ
ejde-673	596	12	proposition	proposition	NOUN
ejde-673	596	13	.	.	PUNCT
ejde-673	597	1	proposition	proposition	NOUN
ejde-673	597	2	3.14	3.14	NUM
ejde-673	597	3	.	.	PUNCT
ejde-673	598	1	let	let	VERB
ejde-673	598	2	a	a	DET
ejde-673	598	3	,	,	PUNCT
ejde-673	598	4	b	b	PROPN
ejde-673	598	5	∈	∈	PROPN
ejde-673	598	6	e.	e.	PROPN
ejde-673	598	7	then	then	ADV
ejde-673	598	8	the	the	DET
ejde-673	598	9	following	follow	VERB
ejde-673	598	10	properties	property	NOUN
ejde-673	598	11	hold	hold	VERB
ejde-673	598	12	.	.	PUNCT
ejde-673	599	1	(	(	PUNCT
ejde-673	599	2	i	i	NOUN
ejde-673	599	3	)	)	PUNCT
ejde-673	599	4	if	if	SCONJ
ejde-673	599	5	there	there	PRON
ejde-673	599	6	exits	exit	VERB
ejde-673	599	7	an	an	DET
ejde-673	599	8	odd	odd	ADJ
ejde-673	599	9	map	map	NOUN
ejde-673	599	10	f	f	PROPN
ejde-673	599	11	∈	∈	PROPN
ejde-673	599	12	c(a	c(a	PROPN
ejde-673	599	13	,	,	PUNCT
ejde-673	599	14	b	b	NOUN
ejde-673	599	15	)	)	PUNCT
ejde-673	599	16	,	,	PUNCT
ejde-673	599	17	then	then	ADV
ejde-673	599	18	γ(a	γ(a	NOUN
ejde-673	599	19	)	)	PUNCT
ejde-673	599	20	≤	≤	NUM
ejde-673	599	21	γ(b	γ(b	NOUN
ejde-673	599	22	)	)	PUNCT
ejde-673	599	23	.	.	PUNCT
ejde-673	600	1	(	(	PUNCT
ejde-673	600	2	ii	ii	NOUN
ejde-673	600	3	)	)	PUNCT
ejde-673	600	4	if	if	SCONJ
ejde-673	600	5	a	a	DET
ejde-673	600	6	⊂	⊂	PROPN
ejde-673	600	7	b	b	PROPN
ejde-673	600	8	,	,	PUNCT
ejde-673	600	9	then	then	ADV
ejde-673	600	10	γ(a	γ(a	NOUN
ejde-673	600	11	)	)	PUNCT
ejde-673	600	12	≤	≤	NUM
ejde-673	600	13	γ(b	γ(b	NOUN
ejde-673	600	14	)	)	PUNCT
ejde-673	600	15	.	.	PUNCT
ejde-673	601	1	(	(	PUNCT
ejde-673	601	2	iii	iii	X
ejde-673	601	3	)	)	PUNCT
ejde-673	601	4	γ(a	γ(a	NOUN
ejde-673	601	5	∪b	∪b	NOUN
ejde-673	601	6	)	)	PUNCT
ejde-673	601	7	≤	≤	NUM
ejde-673	601	8	γ(a	γ(a	NOUN
ejde-673	601	9	)	)	PUNCT
ejde-673	602	1	+	+	NUM
ejde-673	602	2	γ(b	γ(b	NOUN
ejde-673	602	3	)	)	PUNCT
ejde-673	602	4	.	.	PUNCT
ejde-673	603	1	(	(	PUNCT
ejde-673	603	2	iv	iv	X
ejde-673	603	3	)	)	PUNCT
ejde-673	603	4	if	if	SCONJ
ejde-673	603	5	a	a	PRON
ejde-673	603	6	is	be	AUX
ejde-673	603	7	compact	compact	ADJ
ejde-673	603	8	,	,	PUNCT
ejde-673	603	9	then	then	ADV
ejde-673	603	10	γ(a	γ(a	NOUN
ejde-673	603	11	)	)	PUNCT
ejde-673	603	12	<	<	X
ejde-673	603	13	∞	∞	PROPN
ejde-673	603	14	and	and	CCONJ
ejde-673	603	15	there	there	PRON
ejde-673	603	16	exists	exist	VERB
ejde-673	603	17	δ	δ	PROPN
ejde-673	603	18	>	>	X
ejde-673	603	19	0	0	NUM
ejde-673	604	1	such	such	ADJ
ejde-673	604	2	that	that	SCONJ
ejde-673	604	3	if	if	SCONJ
ejde-673	604	4	we	we	PRON
ejde-673	604	5	put	put	VERB
ejde-673	604	6	nδ(a	nδ(a	PRON
ejde-673	604	7	)	)	PUNCT
ejde-673	604	8	=	=	PRON
ejde-673	604	9	{	{	PUNCT
ejde-673	604	10	x	x	PUNCT
ejde-673	604	11	∈	∈	PROPN
ejde-673	604	12	e	e	NOUN
ejde-673	604	13	;	;	PUNCT
ejde-673	604	14	∥x	∥x	PROPN
ejde-673	604	15	−	−	PROPN
ejde-673	604	16	a∥	a∥	PROPN
ejde-673	604	17	:	:	PUNCT
ejde-673	604	18	=	=	PUNCT
ejde-673	604	19	inf{∥x	inf{∥x	PROPN
ejde-673	604	20	−	−	NOUN
ejde-673	604	21	y∥	y∥	NOUN
ejde-673	604	22	;	;	PUNCT
ejde-673	604	23	y	y	PROPN
ejde-673	604	24	∈	∈	PROPN
ejde-673	604	25	a	a	DET
ejde-673	604	26	}	}	PUNCT
ejde-673	604	27	≤	≤	NUM
ejde-673	604	28	δ	δ	PROPN
ejde-673	604	29	}	}	PUNCT
ejde-673	604	30	,	,	PUNCT
ejde-673	604	31	then	then	ADV
ejde-673	604	32	nδ(a	nδ(a	NUM
ejde-673	604	33	)	)	PUNCT
ejde-673	604	34	∈	∈	PROPN
ejde-673	604	35	e	e	NOUN
ejde-673	604	36	and	and	CCONJ
ejde-673	604	37	γ(nδ(a	γ(nδ(a	PROPN
ejde-673	604	38	)	)	PUNCT
ejde-673	604	39	)	)	PUNCT
ejde-673	605	1	=	=	SYM
ejde-673	605	2	γ(a	γ(a	PROPN
ejde-673	605	3	)	)	PUNCT
ejde-673	605	4	.	.	PUNCT
ejde-673	606	1	(	(	PUNCT
ejde-673	606	2	v	v	NOUN
ejde-673	606	3	)	)	PUNCT
ejde-673	606	4	if	if	SCONJ
ejde-673	606	5	ω	ω	PROPN
ejde-673	606	6	is	be	AUX
ejde-673	606	7	a	a	DET
ejde-673	606	8	bounded	bounded	ADJ
ejde-673	606	9	neighborhood	neighborhood	NOUN
ejde-673	606	10	of	of	ADP
ejde-673	606	11	0	0	NUM
ejde-673	606	12	in	in	ADP
ejde-673	606	13	rn	rn	PROPN
ejde-673	606	14	,	,	PUNCT
ejde-673	606	15	and	and	CCONJ
ejde-673	606	16	there	there	PRON
ejde-673	606	17	exists	exist	VERB
ejde-673	606	18	a	a	DET
ejde-673	606	19	mapping	mapping	NOUN
ejde-673	606	20	h	h	NOUN
ejde-673	606	21	:	:	PUNCT
ejde-673	606	22	a	a	DET
ejde-673	606	23	→	→	SYM
ejde-673	606	24	∂ω	∂ω	ADJ
ejde-673	606	25	with	with	ADP
ejde-673	606	26	h	h	DET
ejde-673	606	27	an	an	DET
ejde-673	606	28	odd	odd	ADJ
ejde-673	606	29	homeomorphism	homeomorphism	NOUN
ejde-673	606	30	,	,	PUNCT
ejde-673	606	31	then	then	ADV
ejde-673	606	32	γ(a	γ(a	PROPN
ejde-673	606	33	)	)	PUNCT
ejde-673	606	34	=	=	VERB
ejde-673	606	35	n.	n.	NOUN
ejde-673	606	36	for	for	ADP
ejde-673	606	37	a	a	DET
ejde-673	606	38	proof	proof	NOUN
ejde-673	606	39	of	of	ADP
ejde-673	606	40	the	the	DET
ejde-673	606	41	above	above	ADJ
ejde-673	606	42	proposition	proposition	NOUN
ejde-673	606	43	,	,	PUNCT
ejde-673	606	44	see	see	VERB
ejde-673	606	45	[	[	X
ejde-673	606	46	30	30	NUM
ejde-673	606	47	,	,	PUNCT
ejde-673	606	48	lemma	lemma	PROPN
ejde-673	606	49	7.5	7.5	NUM
ejde-673	606	50	and	and	CCONJ
ejde-673	606	51	proposition	proposition	NOUN
ejde-673	606	52	7.7	7.7	NUM
ejde-673	606	53	]	]	PUNCT
ejde-673	606	54	or	or	CCONJ
ejde-673	606	55	[	[	X
ejde-673	606	56	32	32	NUM
ejde-673	606	57	,	,	PUNCT
ejde-673	606	58	proposition	proposition	NOUN
ejde-673	606	59	2.3	2.3	NUM
ejde-673	606	60	]	]	PUNCT
ejde-673	606	61	.	.	PUNCT
ejde-673	607	1	we	we	PRON
ejde-673	607	2	note	note	VERB
ejde-673	607	3	that	that	SCONJ
ejde-673	607	4	it	it	PRON
ejde-673	607	5	can	can	AUX
ejde-673	607	6	be	be	AUX
ejde-673	607	7	easily	easily	ADV
ejde-673	607	8	seen	see	VERB
ejde-673	607	9	that	that	SCONJ
ejde-673	607	10	when	when	SCONJ
ejde-673	607	11	a	a	DET
ejde-673	607	12	∈	∈	PROPN
ejde-673	607	13	e	e	NOUN
ejde-673	607	14	,	,	PUNCT
ejde-673	607	15	a	a	DET
ejde-673	607	16	̸=	̸=	PROPN
ejde-673	607	17	∅	∅	NOUN
ejde-673	607	18	if	if	SCONJ
ejde-673	607	19	and	and	CCONJ
ejde-673	607	20	only	only	ADV
ejde-673	607	21	if	if	SCONJ
ejde-673	607	22	γ(a	γ(a	NOUN
ejde-673	607	23	)	)	PUNCT
ejde-673	607	24	≥	≥	NOUN
ejde-673	607	25	1	1	NUM
ejde-673	607	26	.	.	PUNCT
ejde-673	608	1	we	we	PRON
ejde-673	608	2	apply	apply	VERB
ejde-673	608	3	the	the	DET
ejde-673	608	4	notion	notion	NOUN
ejde-673	608	5	with	with	ADP
ejde-673	608	6	e	e	NOUN
ejde-673	608	7	=	=	SYM
ejde-673	608	8	y	y	PROPN
ejde-673	608	9	.	.	PUNCT
ejde-673	609	1	let	let	VERB
ejde-673	609	2	σα	σα	PRON
ejde-673	609	3	=	=	PUNCT
ejde-673	609	4	{	{	PUNCT
ejde-673	609	5	h	h	NOUN
ejde-673	609	6	⊂	⊂	PROPN
ejde-673	609	7	mα	mα	PROPN
ejde-673	609	8	:	:	PUNCT
ejde-673	609	9	h	h	NOUN
ejde-673	609	10	is	be	AUX
ejde-673	609	11	compact	compact	ADJ
ejde-673	609	12	and	and	CCONJ
ejde-673	609	13	symmetric	symmetric	ADJ
ejde-673	609	14	}	}	PUNCT
ejde-673	609	15	,	,	PUNCT
ejde-673	609	16	γ(h	γ(h	NOUN
ejde-673	609	17	)	)	PUNCT
ejde-673	609	18	be	be	VERB
ejde-673	609	19	the	the	DET
ejde-673	609	20	genus	genus	NOUN
ejde-673	609	21	of	of	ADP
ejde-673	609	22	h	h	NOUN
ejde-673	609	23	∈	∈	PROPN
ejde-673	609	24	σα	σα	PROPN
ejde-673	609	25	,	,	PUNCT
ejde-673	609	26	and	and	CCONJ
ejde-673	609	27	define	define	VERB
ejde-673	609	28	c(n	c(n	PROPN
ejde-673	609	29	,	,	PUNCT
ejde-673	609	30	α	α	NOUN
ejde-673	609	31	)	)	PUNCT
ejde-673	609	32	=	=	SYM
ejde-673	609	33	inf	inf	ADJ
ejde-673	609	34	h∈σα	h∈σα	NOUN
ejde-673	609	35	,	,	PUNCT
ejde-673	609	36	γ(h)≥n	γ(h)≥n	NUM
ejde-673	609	37	sup	sup	NOUN
ejde-673	609	38	u∈h	u∈h	ADJ
ejde-673	609	39	ψ̃(u	ψ̃(u	NOUN
ejde-673	609	40	)	)	PUNCT
ejde-673	609	41	(	(	PUNCT
ejde-673	609	42	n	n	NOUN
ejde-673	609	43	=	=	SYM
ejde-673	609	44	1	1	NUM
ejde-673	609	45	,	,	PUNCT
ejde-673	609	46	2	2	NUM
ejde-673	609	47	,	,	PUNCT
ejde-673	609	48	.	.	PUNCT
ejde-673	609	49	.	.	PUNCT
ejde-673	609	50	.	.	PUNCT
ejde-673	609	51	)	)	PUNCT
ejde-673	609	52	.	.	PUNCT
ejde-673	610	1	(	(	PUNCT
ejde-673	610	2	3.19	3.19	NUM
ejde-673	610	3	)	)	PUNCT
ejde-673	610	4	the	the	DET
ejde-673	610	5	following	follow	VERB
ejde-673	610	6	proposition	proposition	NOUN
ejde-673	610	7	is	be	AUX
ejde-673	610	8	due	due	ADJ
ejde-673	610	9	to	to	ADP
ejde-673	610	10	[	[	X
ejde-673	610	11	32	32	NUM
ejde-673	610	12	,	,	PUNCT
ejde-673	610	13	corollary	corollary	NOUN
ejde-673	610	14	4.3	4.3	NUM
ejde-673	610	15	]	]	PUNCT
ejde-673	610	16	.	.	PUNCT
ejde-673	611	1	proposition	proposition	NOUN
ejde-673	611	2	3.15	3.15	NUM
ejde-673	611	3	(	(	PUNCT
ejde-673	611	4	ljusternik	ljusternik	X
ejde-673	611	5	-	-	PUNCT
ejde-673	611	6	schnirelmann	schnirelmann	ADJ
ejde-673	611	7	principle	principle	NOUN
ejde-673	611	8	)	)	PUNCT
ejde-673	611	9	.	.	PUNCT
ejde-673	612	1	assume	assume	VERB
ejde-673	612	2	that	that	SCONJ
ejde-673	612	3	m	m	PROPN
ejde-673	612	4	is	be	AUX
ejde-673	612	5	a	a	DET
ejde-673	612	6	closed	closed	ADJ
ejde-673	612	7	symmetric	symmetric	ADJ
ejde-673	612	8	c1	c1	NOUN
ejde-673	612	9	-	-	PUNCT
ejde-673	612	10	submanifold	submanifold	NOUN
ejde-673	612	11	of	of	ADP
ejde-673	612	12	a	a	DET
ejde-673	612	13	real	real	ADJ
ejde-673	612	14	banach	banach	NOUN
ejde-673	612	15	space	space	NOUN
ejde-673	612	16	b	b	PROPN
ejde-673	612	17	and	and	CCONJ
ejde-673	613	1	0	0	NUM
ejde-673	613	2	̸∈	̸∈	PROPN
ejde-673	613	3	m	m	PROPN
ejde-673	613	4	.	.	PUNCT
ejde-673	614	1	let	let	VERB
ejde-673	614	2	f	f	PROPN
ejde-673	614	3	∈	∈	PROPN
ejde-673	614	4	c1(m	c1(m	PROPN
ejde-673	614	5	,	,	PUNCT
ejde-673	614	6	r	r	NOUN
ejde-673	614	7	)	)	PUNCT
ejde-673	614	8	be	be	AUX
ejde-673	614	9	an	an	DET
ejde-673	614	10	even	even	ADV
ejde-673	614	11	functional	functional	ADJ
ejde-673	614	12	and	and	CCONJ
ejde-673	614	13	bounded	bound	VERB
ejde-673	614	14	from	from	ADP
ejde-673	614	15	below	below	ADV
ejde-673	614	16	.	.	PUNCT
ejde-673	615	1	define	define	VERB
ejde-673	615	2	cj	cj	NOUN
ejde-673	615	3	=	=	PROPN
ejde-673	615	4	inf	inf	PROPN
ejde-673	615	5	h∈γj	h∈γj	ADJ
ejde-673	615	6	sup	sup	PROPN
ejde-673	615	7	u∈h	u∈h	X
ejde-673	615	8	f(u	f(u	PROPN
ejde-673	615	9	)	)	PUNCT
ejde-673	615	10	for	for	ADP
ejde-673	615	11	j	j	PROPN
ejde-673	615	12	=	=	SYM
ejde-673	615	13	1	1	NUM
ejde-673	615	14	,	,	PUNCT
ejde-673	615	15	2	2	NUM
ejde-673	615	16	,	,	PUNCT
ejde-673	615	17	.	.	PUNCT
ejde-673	615	18	.	.	PUNCT
ejde-673	616	1	.	.	PUNCT
ejde-673	617	1	,	,	PUNCT
ejde-673	617	2	where	where	SCONJ
ejde-673	617	3	γj	γj	SCONJ
ejde-673	617	4	=	=	PRON
ejde-673	617	5	{	{	PUNCT
ejde-673	617	6	h	h	NOUN
ejde-673	617	7	⊂	⊂	X
ejde-673	617	8	m	m	VERB
ejde-673	617	9	:	:	PUNCT
ejde-673	617	10	h	h	NOUN
ejde-673	617	11	is	be	AUX
ejde-673	617	12	compact	compact	ADJ
ejde-673	617	13	,	,	PUNCT
ejde-673	617	14	symmetric	symmetric	ADJ
ejde-673	617	15	and	and	CCONJ
ejde-673	617	16	γ(h	γ(h	PROPN
ejde-673	617	17	)	)	PUNCT
ejde-673	617	18	≥	≥	NOUN
ejde-673	617	19	j	j	NOUN
ejde-673	617	20	}	}	PUNCT
ejde-673	617	21	.	.	PUNCT
ejde-673	618	1	if	if	SCONJ
ejde-673	618	2	γk	γk	PROPN
ejde-673	618	3	̸=	̸=	PROPN
ejde-673	618	4	∅	∅	NOUN
ejde-673	618	5	for	for	ADP
ejde-673	618	6	some	some	DET
ejde-673	618	7	k	k	PROPN
ejde-673	618	8	≥	≥	NUM
ejde-673	618	9	1	1	NUM
ejde-673	618	10	and	and	CCONJ
ejde-673	618	11	f	f	PROPN
ejde-673	618	12	satisfies	satisfie	NOUN
ejde-673	618	13	(	(	PUNCT
ejde-673	618	14	ps)c	ps)c	NOUN
ejde-673	618	15	-	-	PUNCT
ejde-673	618	16	condition	condition	NOUN
ejde-673	618	17	for	for	ADP
ejde-673	618	18	c	c	NOUN
ejde-673	618	19	:	:	PUNCT
ejde-673	618	20	=	=	SYM
ejde-673	618	21	cm	cm	NOUN
ejde-673	618	22	=	=	SYM
ejde-673	618	23	cm+1	cm+1	PROPN
ejde-673	618	24	=	=	PUNCT
ejde-673	618	25	·	·	PUNCT
ejde-673	618	26	·	·	PUNCT
ejde-673	618	27	·	·	PUNCT
ejde-673	619	1	=	=	PUNCT
ejde-673	619	2	ck	ck	ADJ
ejde-673	619	3	with	with	ADP
ejde-673	619	4	1	1	NUM
ejde-673	619	5	≤	≤	NUM
ejde-673	619	6	m	m	VERB
ejde-673	619	7	≤	≤	NOUN
ejde-673	620	1	k	k	ADP
ejde-673	620	2	,	,	PUNCT
ejde-673	620	3	then	then	ADV
ejde-673	620	4	f	f	PROPN
ejde-673	620	5	has	have	VERB
ejde-673	621	1	at	at	ADV
ejde-673	621	2	least	least	ADJ
ejde-673	621	3	k	k	NOUN
ejde-673	622	1	−	−	PROPN
ejde-673	623	1	m	m	VERB
ejde-673	623	2	+	+	ADJ
ejde-673	623	3	1	1	NUM
ejde-673	623	4	distinct	distinct	ADJ
ejde-673	623	5	pairs	pair	NOUN
ejde-673	623	6	of	of	ADP
ejde-673	623	7	critical	critical	ADJ
ejde-673	623	8	points	point	NOUN
ejde-673	623	9	.	.	PUNCT
ejde-673	624	1	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	624	2	eigenvalue	eigenvalue	VERB
ejde-673	624	3	problems	problem	NOUN
ejde-673	624	4	for	for	ADP
ejde-673	624	5	kirchhoff	kirchhoff	NOUN
ejde-673	624	6	-	-	PUNCT
ejde-673	624	7	type	type	NOUN
ejde-673	624	8	equations	equation	NOUN
ejde-673	624	9	21	21	NUM
ejde-673	624	10	since	since	SCONJ
ejde-673	624	11	y	y	PROPN
ejde-673	624	12	is	be	AUX
ejde-673	624	13	a	a	DET
ejde-673	624	14	separable	separable	ADJ
ejde-673	624	15	reflexive	reflexive	ADJ
ejde-673	624	16	banach	banach	NOUN
ejde-673	624	17	space	space	NOUN
ejde-673	624	18	,	,	PUNCT
ejde-673	624	19	it	it	PRON
ejde-673	624	20	is	be	AUX
ejde-673	624	21	well	well	ADV
ejde-673	624	22	known	know	VERB
ejde-673	624	23	that	that	SCONJ
ejde-673	624	24	there	there	PRON
ejde-673	624	25	exist	exist	VERB
ejde-673	624	26	{	{	PUNCT
ejde-673	624	27	en}∞n=1	en}∞n=1	X
ejde-673	624	28	⊂	⊂	PROPN
ejde-673	624	29	y	y	PROPN
ejde-673	624	30	and	and	CCONJ
ejde-673	624	31	{	{	PUNCT
ejde-673	624	32	fn}∞n=1	fn}∞n=1	PROPN
ejde-673	624	33	⊂	⊂	PROPN
ejde-673	624	34	y	y	PROPN
ejde-673	624	35	∗	∗	VERB
ejde-673	624	36	such	such	ADJ
ejde-673	624	37	that	that	DET
ejde-673	624	38	⟨fn	⟨fn	PROPN
ejde-673	624	39	,	,	PUNCT
ejde-673	624	40	em⟩y	em⟩y	ADP
ejde-673	624	41	∗,y	∗,y	NOUN
ejde-673	624	42	=	=	SYM
ejde-673	624	43	δnm	δnm	NOUN
ejde-673	624	44	,	,	PUNCT
ejde-673	624	45	where	where	SCONJ
ejde-673	624	46	δnm	δnm	NOUN
ejde-673	624	47	is	be	AUX
ejde-673	624	48	the	the	DET
ejde-673	624	49	kronecker	kronecker	NOUN
ejde-673	624	50	delta	delta	NOUN
ejde-673	624	51	and	and	CCONJ
ejde-673	624	52	y	y	PROPN
ejde-673	624	53	=	=	PUNCT
ejde-673	624	54	span{e1	span{e1	PROPN
ejde-673	624	55	,	,	PUNCT
ejde-673	624	56	e2	e2	PROPN
ejde-673	624	57	,	,	PUNCT
ejde-673	624	58	.	.	PUNCT
ejde-673	624	59	.	.	PUNCT
ejde-673	625	1	.	.	PUNCT
ejde-673	625	2	}	}	PUNCT
ejde-673	626	1	and	and	CCONJ
ejde-673	626	2	y	y	PROPN
ejde-673	626	3	∗	∗	NOUN
ejde-673	626	4	=	=	SYM
ejde-673	626	5	span{f1	span{f1	X
ejde-673	626	6	,	,	PUNCT
ejde-673	626	7	f2	f2	PROPN
ejde-673	626	8	,	,	PUNCT
ejde-673	626	9	.	.	PUNCT
ejde-673	626	10	.	.	PUNCT
ejde-673	626	11	.	.	PUNCT
ejde-673	626	12	}	}	PUNCT
ejde-673	626	13	.	.	PUNCT
ejde-673	627	1	we	we	PRON
ejde-673	627	2	define	define	VERB
ejde-673	627	3	the	the	DET
ejde-673	627	4	spaces	space	NOUN
ejde-673	627	5	yj	yj	NOUN
ejde-673	627	6	=	=	SYM
ejde-673	627	7	span{ej	span{ej	PROPN
ejde-673	627	8	}	}	PUNCT
ejde-673	627	9	,	,	PUNCT
ejde-673	627	10	zn	zn	NOUN
ejde-673	627	11	=	=	SYM
ejde-673	627	12	⊕n	⊕n	NOUN
ejde-673	627	13	j=1yj	j=1yj	NOUN
ejde-673	627	14	,	,	PUNCT
ejde-673	627	15	wn	wn	PROPN
ejde-673	627	16	=	=	SYM
ejde-673	627	17	⊕∞	⊕∞	PROPN
ejde-673	627	18	j	j	PROPN
ejde-673	628	1	=	=	NOUN
ejde-673	628	2	nyj	nyj	X
ejde-673	628	3	.	.	PUNCT
ejde-673	629	1	if	if	SCONJ
ejde-673	629	2	we	we	PRON
ejde-673	629	3	apply	apply	VERB
ejde-673	629	4	proposition	proposition	NOUN
ejde-673	629	5	3.15	3.15	NUM
ejde-673	629	6	with	with	ADP
ejde-673	629	7	b	b	PROPN
ejde-673	629	8	=	=	SYM
ejde-673	629	9	y	y	PROPN
ejde-673	629	10	,	,	PUNCT
ejde-673	629	11	m	m	VERB
ejde-673	629	12	=	=	SYM
ejde-673	629	13	mα	mα	PROPN
ejde-673	629	14	and	and	CCONJ
ejde-673	629	15	f	f	PROPN
ejde-673	629	16	=	=	SYM
ejde-673	629	17	ψ̃	ψ̃	PROPN
ejde-673	629	18	,	,	PUNCT
ejde-673	629	19	then	then	ADV
ejde-673	629	20	we	we	PRON
ejde-673	629	21	obtain	obtain	VERB
ejde-673	629	22	the	the	DET
ejde-673	629	23	following	follow	VERB
ejde-673	629	24	lemma	lemma	PROPN
ejde-673	629	25	.	.	PUNCT
ejde-673	630	1	we	we	PRON
ejde-673	630	2	note	note	VERB
ejde-673	630	3	that	that	SCONJ
ejde-673	630	4	ψ̃	ψ̃	PROPN
ejde-673	630	5	is	be	AUX
ejde-673	630	6	bounded	bound	VERB
ejde-673	630	7	from	from	ADP
ejde-673	630	8	below	below	ADV
ejde-673	630	9	on	on	ADP
ejde-673	630	10	mα	mα	PROPN
ejde-673	630	11	and	and	CCONJ
ejde-673	630	12	satisfies	satisfie	NOUN
ejde-673	630	13	(	(	PUNCT
ejde-673	630	14	ps)c	ps)c	NOUN
ejde-673	630	15	-	-	PUNCT
ejde-673	630	16	condition	condition	NOUN
ejde-673	630	17	with	with	ADP
ejde-673	630	18	respect	respect	NOUN
ejde-673	630	19	ot	ot	INTJ
ejde-673	630	20	mα	mα	NOUN
ejde-673	630	21	for	for	ADP
ejde-673	630	22	any	any	DET
ejde-673	630	23	c	c	NOUN
ejde-673	630	24	∈	∈	NOUN
ejde-673	630	25	r	r	NOUN
ejde-673	630	26	by	by	ADP
ejde-673	630	27	proposition	proposition	NOUN
ejde-673	630	28	3.13	3.13	NUM
ejde-673	630	29	.	.	PUNCT
ejde-673	631	1	lemma	lemma	PROPN
ejde-673	631	2	3.16	3.16	NUM
ejde-673	631	3	.	.	PUNCT
ejde-673	632	1	for	for	ADP
ejde-673	632	2	any	any	DET
ejde-673	632	3	m	m	PROPN
ejde-673	632	4	∈	∈	NOUN
ejde-673	632	5	n	n	CCONJ
ejde-673	632	6	,	,	PUNCT
ejde-673	632	7	we	we	PRON
ejde-673	632	8	have	have	VERB
ejde-673	632	9	γm	γm	PRON
ejde-673	632	10	̸=	̸=	PROPN
ejde-673	632	11	∅.	∅.	ADV
ejde-673	632	12	thus	thus	ADV
ejde-673	632	13	we	we	PRON
ejde-673	632	14	see	see	VERB
ejde-673	632	15	that	that	SCONJ
ejde-673	632	16	all	all	DET
ejde-673	632	17	c(m	c(m	PROPN
ejde-673	632	18	,	,	PUNCT
ejde-673	632	19	α	α	NOUN
ejde-673	632	20	)	)	PUNCT
ejde-673	632	21	defined	define	VERB
ejde-673	632	22	by	by	ADP
ejde-673	632	23	(	(	PUNCT
ejde-673	632	24	3.19	3.19	NUM
ejde-673	632	25	)	)	PUNCT
ejde-673	632	26	are	be	AUX
ejde-673	632	27	critical	critical	ADJ
ejde-673	632	28	values	value	NOUN
ejde-673	632	29	of	of	ADP
ejde-673	632	30	ψ̃	ψ̃	PROPN
ejde-673	632	31	with	with	ADP
ejde-673	632	32	respect	respect	NOUN
ejde-673	632	33	to	to	ADP
ejde-673	632	34	mα	mα	PROPN
ejde-673	632	35	and	and	CCONJ
ejde-673	632	36	−∞	−∞	X
ejde-673	632	37	<	<	X
ejde-673	632	38	c(m	c(m	PROPN
ejde-673	632	39	,	,	PUNCT
ejde-673	632	40	α	α	NOUN
ejde-673	632	41	)	)	PUNCT
ejde-673	632	42	≤	≤	NOUN
ejde-673	632	43	c(m+1,α	c(m+1,α	PROPN
ejde-673	632	44	)	)	PUNCT
ejde-673	633	1	<	<	X
ejde-673	633	2	∞	∞	PROPN
ejde-673	633	3	for	for	ADP
ejde-673	633	4	every	every	DET
ejde-673	633	5	m	m	PROPN
ejde-673	633	6	∈	∈	PROPN
ejde-673	633	7	n.	n.	NOUN
ejde-673	633	8	proof	proof	NOUN
ejde-673	633	9	.	.	PUNCT
ejde-673	634	1	for	for	ADP
ejde-673	634	2	each	each	DET
ejde-673	634	3	fixed	fix	VERB
ejde-673	634	4	m	m	PROPN
ejde-673	634	5	∈	∈	PROPN
ejde-673	634	6	n	n	CCONJ
ejde-673	634	7	,	,	PUNCT
ejde-673	634	8	we	we	PRON
ejde-673	634	9	claim	claim	VERB
ejde-673	634	10	that	that	SCONJ
ejde-673	634	11	c(m	c(m	PROPN
ejde-673	634	12	)	)	PUNCT
ejde-673	634	13	:	:	PUNCT
ejde-673	635	1	=	=	PUNCT
ejde-673	635	2	inf{k(u	inf{k(u	NOUN
ejde-673	635	3	)	)	PUNCT
ejde-673	635	4	:	:	PUNCT
ejde-673	635	5	u	u	PROPN
ejde-673	635	6	∈	∈	PROPN
ejde-673	635	7	zm	zm	PROPN
ejde-673	635	8	,	,	PUNCT
ejde-673	635	9	∥u∥y	∥u∥y	X
ejde-673	635	10	=	=	SYM
ejde-673	635	11	1	1	NUM
ejde-673	635	12	}	}	PUNCT
ejde-673	635	13	>	>	X
ejde-673	635	14	0	0	X
ejde-673	635	15	.	.	PUNCT
ejde-673	635	16	(	(	PUNCT
ejde-673	635	17	3.20	3.20	NUM
ejde-673	635	18	)	)	PUNCT
ejde-673	635	19	indeed	indeed	ADV
ejde-673	635	20	,	,	PUNCT
ejde-673	635	21	assume	assume	VERB
ejde-673	635	22	that	that	SCONJ
ejde-673	635	23	c(m	c(m	NOUN
ejde-673	635	24	)	)	PUNCT
ejde-673	636	1	=	=	PUNCT
ejde-673	636	2	0	0	X
ejde-673	636	3	.	.	PUNCT
ejde-673	637	1	then	then	ADV
ejde-673	637	2	there	there	PRON
ejde-673	637	3	exists	exist	VERB
ejde-673	637	4	a	a	DET
ejde-673	637	5	sequence	sequence	NOUN
ejde-673	637	6	{	{	PUNCT
ejde-673	637	7	uj	uj	PROPN
ejde-673	637	8	}	}	PUNCT
ejde-673	637	9	⊂	⊂	PROPN
ejde-673	637	10	zm	zm	PROPN
ejde-673	637	11	such	such	ADJ
ejde-673	637	12	that	that	DET
ejde-673	637	13	∥uj∥y	∥uj∥y	NOUN
ejde-673	637	14	=	=	SYM
ejde-673	637	15	1	1	NUM
ejde-673	637	16	and	and	CCONJ
ejde-673	637	17	0	0	NUM
ejde-673	637	18	≤	≤	NOUN
ejde-673	637	19	k(uj	k(uj	NOUN
ejde-673	637	20	)	)	PUNCT
ejde-673	637	21	≤	≤	NUM
ejde-673	637	22	1	1	NUM
ejde-673	637	23	j	j	NOUN
ejde-673	637	24	.	.	PUNCT
ejde-673	638	1	(	(	PUNCT
ejde-673	638	2	3.21	3.21	NUM
ejde-673	638	3	)	)	PUNCT
ejde-673	638	4	since	since	SCONJ
ejde-673	638	5	the	the	DET
ejde-673	638	6	sequence	sequence	NOUN
ejde-673	638	7	{	{	PUNCT
ejde-673	638	8	uj	uj	PROPN
ejde-673	638	9	}	}	PUNCT
ejde-673	638	10	is	be	AUX
ejde-673	638	11	bounded	bound	VERB
ejde-673	638	12	in	in	ADP
ejde-673	638	13	y	y	PROPN
ejde-673	638	14	,	,	PUNCT
ejde-673	638	15	there	there	PRON
ejde-673	638	16	exist	exist	VERB
ejde-673	638	17	a	a	DET
ejde-673	638	18	subsequence	subsequence	NOUN
ejde-673	638	19	{	{	PUNCT
ejde-673	638	20	uj′	uj′	NOUN
ejde-673	638	21	}	}	PUNCT
ejde-673	638	22	of	of	ADP
ejde-673	638	23	{	{	PUNCT
ejde-673	638	24	uj	uj	PROPN
ejde-673	638	25	}	}	PUNCT
ejde-673	638	26	and	and	CCONJ
ejde-673	638	27	u0	u0	PROPN
ejde-673	638	28	∈	∈	PROPN
ejde-673	638	29	y	y	PROPN
ejde-673	638	30	such	such	ADJ
ejde-673	638	31	that	that	SCONJ
ejde-673	638	32	uj′	uj′	ADP
ejde-673	638	33	→	→	SYM
ejde-673	638	34	u0	u0	ADJ
ejde-673	638	35	weakly	weakly	ADJ
ejde-673	638	36	in	in	ADP
ejde-673	638	37	y	y	PROPN
ejde-673	638	38	as	as	ADP
ejde-673	638	39	j′	j′	PROPN
ejde-673	638	40	→	→	SYM
ejde-673	638	41	∞.	∞.	PROPN
ejde-673	638	42	since	since	SCONJ
ejde-673	638	43	⟨fk	⟨fk	PROPN
ejde-673	638	44	,	,	PUNCT
ejde-673	638	45	uj′⟩y	uj′⟩y	PROPN
ejde-673	638	46	∗,y	∗,y	NOUN
ejde-673	638	47	=	=	SYM
ejde-673	638	48	0	0	NUM
ejde-673	638	49	for	for	ADP
ejde-673	638	50	any	any	DET
ejde-673	638	51	k	k	PROPN
ejde-673	638	52	>	>	X
ejde-673	638	53	m	m	PROPN
ejde-673	638	54	,	,	PUNCT
ejde-673	638	55	we	we	PRON
ejde-673	638	56	have	have	VERB
ejde-673	638	57	⟨fk	⟨fk	NOUN
ejde-673	638	58	,	,	PUNCT
ejde-673	638	59	u0⟩y	u0⟩y	NOUN
ejde-673	638	60	∗,y	∗,y	NOUN
ejde-673	638	61	=	=	SYM
ejde-673	638	62	0	0	NUM
ejde-673	638	63	for	for	ADP
ejde-673	638	64	all	all	PRON
ejde-673	638	65	k	k	PROPN
ejde-673	638	66	>	>	X
ejde-673	638	67	m	m	PROPN
ejde-673	638	68	,	,	PUNCT
ejde-673	638	69	so	so	SCONJ
ejde-673	638	70	we	we	PRON
ejde-673	638	71	see	see	VERB
ejde-673	638	72	that	that	DET
ejde-673	638	73	u0	u0	PROPN
ejde-673	638	74	∈	∈	PROPN
ejde-673	638	75	zm	zm	PROPN
ejde-673	638	76	.	.	PUNCT
ejde-673	639	1	since	since	SCONJ
ejde-673	639	2	dimzm	dimzm	NOUN
ejde-673	639	3	=	=	VERB
ejde-673	639	4	m	m	VERB
ejde-673	639	5	<	<	X
ejde-673	639	6	∞	∞	PROPN
ejde-673	639	7	,	,	PUNCT
ejde-673	639	8	uj′	uj′	ADP
ejde-673	639	9	→	→	SYM
ejde-673	639	10	u0	u0	ADJ
ejde-673	639	11	strongly	strongly	ADV
ejde-673	639	12	in	in	ADP
ejde-673	639	13	zm	zm	PROPN
ejde-673	639	14	,	,	PUNCT
ejde-673	639	15	so	so	ADV
ejde-673	639	16	in	in	ADP
ejde-673	639	17	y	y	PROPN
ejde-673	639	18	.	.	PUNCT
ejde-673	640	1	thereby	thereby	ADV
ejde-673	640	2	∥u0∥y	∥u0∥y	PROPN
ejde-673	640	3	=	=	SYM
ejde-673	640	4	1	1	NUM
ejde-673	640	5	,	,	PUNCT
ejde-673	640	6	so	so	SCONJ
ejde-673	640	7	we	we	PRON
ejde-673	640	8	can	can	AUX
ejde-673	640	9	see	see	VERB
ejde-673	640	10	that	that	DET
ejde-673	640	11	k(u0	k(u0	PROPN
ejde-673	640	12	)	)	PUNCT
ejde-673	640	13	>	>	X
ejde-673	641	1	0	0	X
ejde-673	641	2	.	.	PUNCT
ejde-673	642	1	on	on	ADP
ejde-673	642	2	the	the	DET
ejde-673	642	3	other	other	ADJ
ejde-673	642	4	hand	hand	NOUN
ejde-673	642	5	,	,	PUNCT
ejde-673	642	6	letting	let	VERB
ejde-673	642	7	j′	j′	NOUN
ejde-673	642	8	→	→	SYM
ejde-673	642	9	∞	∞	NUM
ejde-673	642	10	in	in	ADP
ejde-673	642	11	(	(	PUNCT
ejde-673	642	12	3.21	3.21	NUM
ejde-673	642	13	)	)	PUNCT
ejde-673	642	14	,	,	PUNCT
ejde-673	642	15	we	we	PRON
ejde-673	642	16	see	see	VERB
ejde-673	642	17	that	that	SCONJ
ejde-673	642	18	k(u0	k(u0	PROPN
ejde-673	642	19	)	)	PUNCT
ejde-673	643	1	=	=	PUNCT
ejde-673	643	2	0	0	X
ejde-673	643	3	.	.	PUNCT
ejde-673	644	1	this	this	PRON
ejde-673	644	2	is	be	AUX
ejde-673	644	3	a	a	DET
ejde-673	644	4	contradiction	contradiction	NOUN
ejde-673	644	5	.	.	PUNCT
ejde-673	645	1	for	for	ADP
ejde-673	645	2	0	0	NUM
ejde-673	645	3	̸=	̸=	PROPN
ejde-673	645	4	u	u	PROPN
ejde-673	645	5	∈	∈	PROPN
ejde-673	645	6	zm	zm	PROPN
ejde-673	645	7	,	,	PUNCT
ejde-673	645	8	since	since	SCONJ
ejde-673	645	9	∥u/∥u∥y	∥u/∥u∥y	NUM
ejde-673	645	10	∥y	∥y	PROPN
ejde-673	645	11	=	=	SYM
ejde-673	645	12	1	1	NUM
ejde-673	645	13	,	,	PUNCT
ejde-673	645	14	it	it	PRON
ejde-673	645	15	follows	follow	VERB
ejde-673	645	16	from	from	ADP
ejde-673	645	17	(	(	PUNCT
ejde-673	645	18	3.20	3.20	NUM
ejde-673	645	19	)	)	PUNCT
ejde-673	645	20	and	and	CCONJ
ejde-673	645	21	(	(	PUNCT
ejde-673	645	22	3.4	3.4	NUM
ejde-673	645	23	)	)	PUNCT
ejde-673	646	1	that	that	PRON
ejde-673	646	2	c(m	c(m	NOUN
ejde-673	646	3	)	)	PUNCT
ejde-673	646	4	≤	≤	PUNCT
ejde-673	647	1	k	k	X
ejde-673	647	2	(	(	PUNCT
ejde-673	647	3	u	u	NOUN
ejde-673	647	4	∥u∥y	∥u∥y	NOUN
ejde-673	647	5	)	)	PUNCT
ejde-673	647	6	=	=	SYM
ejde-673	648	1	∫	∫	PROPN
ejde-673	648	2	γ2	γ2	NOUN
ejde-673	648	3	1	1	NUM
ejde-673	648	4	∥u∥r(x)y	∥u∥r(x)y	PUNCT
ejde-673	648	5	g(x	g(x	NOUN
ejde-673	648	6	,	,	PUNCT
ejde-673	648	7	u(x))dσx	u(x))dσx	NOUN
ejde-673	648	8	≤	≤	NUM
ejde-673	648	9	1	1	NUM
ejde-673	648	10	∥u∥r+y	∥u∥r+y	PROPN
ejde-673	648	11	∧	∧	PROPN
ejde-673	648	12	∥u∥r−y	∥u∥r−y	PUNCT
ejde-673	648	13	k(u	k(u	NOUN
ejde-673	648	14	)	)	PUNCT
ejde-673	648	15	.	.	PUNCT
ejde-673	649	1	thus	thus	ADV
ejde-673	649	2	we	we	PRON
ejde-673	649	3	have	have	VERB
ejde-673	649	4	k(u	k(u	X
ejde-673	649	5	)	)	PUNCT
ejde-673	649	6	≥	≥	PROPN
ejde-673	649	7	c(m)∥u∥r+y	c(m)∥u∥r+y	NOUN
ejde-673	649	8	∧	∧	PROPN
ejde-673	649	9	∥u∥r−y	∥u∥r−y	PUNCT
ejde-673	649	10	for	for	ADP
ejde-673	649	11	all	all	DET
ejde-673	649	12	u	u	PROPN
ejde-673	649	13	∈	∈	PROPN
ejde-673	649	14	zm	zm	PROPN
ejde-673	649	15	.	.	PUNCT
ejde-673	650	1	therefore	therefore	ADV
ejde-673	650	2	,	,	PUNCT
ejde-673	650	3	zm	zm	PROPN
ejde-673	650	4	∩mα	∩mα	PROPN
ejde-673	650	5	is	be	AUX
ejde-673	650	6	a	a	DET
ejde-673	650	7	bounded	bounded	ADJ
ejde-673	650	8	and	and	CCONJ
ejde-673	650	9	closed	closed	ADJ
ejde-673	650	10	subset	subset	NOUN
ejde-673	650	11	of	of	ADP
ejde-673	650	12	zm	zm	PROPN
ejde-673	650	13	,	,	PUNCT
ejde-673	650	14	so	so	ADV
ejde-673	650	15	is	be	AUX
ejde-673	650	16	compact	compact	ADJ
ejde-673	650	17	by	by	ADP
ejde-673	650	18	dimzm	dimzm	NOUN
ejde-673	650	19	<	<	X
ejde-673	650	20	∞.	∞.	PROPN
ejde-673	650	21	since	since	SCONJ
ejde-673	650	22	k	k	PROPN
ejde-673	650	23	is	be	AUX
ejde-673	650	24	an	an	DET
ejde-673	650	25	even	even	ADV
ejde-673	650	26	functional	functional	ADJ
ejde-673	650	27	,	,	PUNCT
ejde-673	650	28	zm	zm	PROPN
ejde-673	650	29	∩mα	∩mα	PROPN
ejde-673	650	30	is	be	AUX
ejde-673	650	31	clearly	clearly	ADV
ejde-673	650	32	symmetric	symmetric	ADJ
ejde-673	650	33	.	.	PUNCT
ejde-673	651	1	let	let	VERB
ejde-673	651	2	g	g	NOUN
ejde-673	651	3	=	=	PUNCT
ejde-673	651	4	{	{	PUNCT
ejde-673	651	5	u	u	NOUN
ejde-673	651	6	=	=	NOUN
ejde-673	651	7	u1e1	u1e1	PROPN
ejde-673	651	8	+	+	X
ejde-673	651	9	·	·	PUNCT
ejde-673	651	10	·	·	PUNCT
ejde-673	651	11	·	·	PUNCT
ejde-673	651	12	+	+	NUM
ejde-673	651	13	umem	umem	ADJ
ejde-673	651	14	∈	∈	PROPN
ejde-673	651	15	zm;k(u	zm;k(u	NOUN
ejde-673	651	16	)	)	PUNCT
ejde-673	651	17	<	<	X
ejde-673	651	18	α	α	X
ejde-673	651	19	}	}	PUNCT
ejde-673	651	20	.	.	PUNCT
ejde-673	652	1	then	then	ADV
ejde-673	652	2	g	g	PROPN
ejde-673	652	3	can	can	AUX
ejde-673	652	4	be	be	AUX
ejde-673	652	5	identified	identify	VERB
ejde-673	652	6	with	with	ADP
ejde-673	652	7	an	an	DET
ejde-673	652	8	open	open	ADJ
ejde-673	652	9	neighborhood	neighborhood	NOUN
ejde-673	652	10	of	of	ADP
ejde-673	652	11	0	0	NUM
ejde-673	652	12	in	in	ADP
ejde-673	652	13	rm	rm	NOUN
ejde-673	652	14	by	by	ADP
ejde-673	652	15	a	a	DET
ejde-673	652	16	trivial	trivial	ADJ
ejde-673	652	17	odd	odd	ADJ
ejde-673	652	18	homeomorphism	homeomorphism	NOUN
ejde-673	652	19	.	.	PUNCT
ejde-673	653	1	since	since	SCONJ
ejde-673	653	2	the	the	DET
ejde-673	653	3	identity	identity	NOUN
ejde-673	653	4	map	map	NOUN
ejde-673	653	5	:	:	PUNCT
ejde-673	653	6	zm	zm	PROPN
ejde-673	653	7	∩	∩	PROPN
ejde-673	653	8	mα	mα	PROPN
ejde-673	653	9	→	→	SYM
ejde-673	653	10	∂g	∂g	PROPN
ejde-673	653	11	is	be	AUX
ejde-673	653	12	an	an	DET
ejde-673	653	13	odd	odd	ADJ
ejde-673	653	14	homeomorphism	homeomorphism	NOUN
ejde-673	653	15	,	,	PUNCT
ejde-673	653	16	using	use	VERB
ejde-673	653	17	proposition	proposition	NOUN
ejde-673	653	18	3.14	3.14	NUM
ejde-673	653	19	(	(	PUNCT
ejde-673	653	20	v	v	NOUN
ejde-673	653	21	)	)	PUNCT
ejde-673	653	22	,	,	PUNCT
ejde-673	653	23	we	we	PRON
ejde-673	653	24	have	have	VERB
ejde-673	653	25	γ(zm	γ(zm	NUM
ejde-673	653	26	∩	∩	ADJ
ejde-673	653	27	mα	mα	NOUN
ejde-673	653	28	)	)	PUNCT
ejde-673	653	29	=	=	SYM
ejde-673	653	30	m	m	PROPN
ejde-673	653	31	,	,	PUNCT
ejde-673	653	32	so	so	ADV
ejde-673	653	33	γm	γm	ADJ
ejde-673	653	34	̸=	̸=	PROPN
ejde-673	653	35	∅.	∅.	NOUN
ejde-673	653	36	since	since	SCONJ
ejde-673	653	37	γm+1	γm+1	PROPN
ejde-673	653	38	⊂	⊂	PROPN
ejde-673	653	39	γm	γm	X
ejde-673	653	40	,	,	PUNCT
ejde-673	653	41	we	we	PRON
ejde-673	653	42	can	can	AUX
ejde-673	653	43	see	see	VERB
ejde-673	653	44	that	that	PRON
ejde-673	653	45	−∞	−∞	ADP
ejde-673	653	46	<	<	X
ejde-673	653	47	c(m	c(m	PROPN
ejde-673	653	48	,	,	PUNCT
ejde-673	653	49	α	α	NOUN
ejde-673	653	50	)	)	PUNCT
ejde-673	653	51	≤	≤	NOUN
ejde-673	653	52	c(m+1,α	c(m+1,α	PROPN
ejde-673	653	53	)	)	PUNCT
ejde-673	654	1	<	<	X
ejde-673	654	2	∞.	∞.	PROPN
ejde-673	654	3	□	□	PUNCT
ejde-673	654	4	lemma	lemma	PROPN
ejde-673	654	5	3.17	3.17	NUM
ejde-673	654	6	.	.	PUNCT
ejde-673	654	7	assume	assume	VERB
ejde-673	654	8	that	that	SCONJ
ejde-673	654	9	a	a	DET
ejde-673	654	10	functional	functional	ADJ
ejde-673	654	11	χ	χ	X
ejde-673	654	12	:	:	PUNCT
ejde-673	654	13	y	y	PROPN
ejde-673	654	14	→	→	PUNCT
ejde-673	654	15	r	r	NOUN
ejde-673	654	16	is	be	AUX
ejde-673	654	17	sequentially	sequentially	ADV
ejde-673	654	18	weakly	weakly	ADV
ejde-673	654	19	continuous	continuous	ADJ
ejde-673	654	20	and	and	CCONJ
ejde-673	654	21	satisfies	satisfie	NOUN
ejde-673	654	22	χ(0	χ(0	NOUN
ejde-673	654	23	)	)	PUNCT
ejde-673	654	24	=	=	SYM
ejde-673	655	1	0	0	X
ejde-673	655	2	.	.	PUNCT
ejde-673	656	1	then	then	ADV
ejde-673	656	2	for	for	ADP
ejde-673	656	3	any	any	DET
ejde-673	656	4	fixed	fixed	ADJ
ejde-673	656	5	r	r	NOUN
ejde-673	656	6	>	>	X
ejde-673	656	7	0	0	PROPN
ejde-673	656	8	,	,	PUNCT
ejde-673	656	9	lim	lim	PROPN
ejde-673	656	10	n→∞	n→∞	NUM
ejde-673	656	11	sup	sup	NOUN
ejde-673	656	12	u∈wn,∥u∥y	u∈wn,∥u∥y	PRON
ejde-673	656	13	≤r	≤r	ADJ
ejde-673	656	14	|χ(u)|	|χ(u)|	NOUN
ejde-673	656	15	=	=	SYM
ejde-673	656	16	0	0	X
ejde-673	656	17	.	.	PUNCT
ejde-673	657	1	(	(	PUNCT
ejde-673	657	2	3.22	3.22	NUM
ejde-673	657	3	)	)	PUNCT
ejde-673	657	4	proof	proof	NOUN
ejde-673	657	5	.	.	PUNCT
ejde-673	658	1	put	put	VERB
ejde-673	658	2	dn	dn	NOUN
ejde-673	658	3	=	=	PUNCT
ejde-673	658	4	supu∈wn,∥u∥y	supu∈wn,∥u∥y	NOUN
ejde-673	658	5	≤r	≤r	PROPN
ejde-673	658	6	|χ(u)|	|χ(u)|	PROPN
ejde-673	658	7	.	.	PUNCT
ejde-673	659	1	then	then	ADV
ejde-673	659	2	there	there	PRON
ejde-673	659	3	exists	exist	VERB
ejde-673	659	4	uj	uj	PROPN
ejde-673	659	5	∈	∈	PROPN
ejde-673	659	6	wn	wn	PROPN
ejde-673	659	7	with	with	ADP
ejde-673	659	8	∥uj∥y	∥uj∥y	NOUN
ejde-673	659	9	≤	≤	NUM
ejde-673	659	10	r	r	NOUN
ejde-673	660	1	such	such	ADJ
ejde-673	660	2	that	that	DET
ejde-673	660	3	limj→∞	limj→∞	PROPN
ejde-673	660	4	|χ(uj)|	|χ(uj)|	X
ejde-673	660	5	=	=	SYM
ejde-673	660	6	dn	dn	PROPN
ejde-673	660	7	.	.	PROPN
ejde-673	661	1	since	since	SCONJ
ejde-673	661	2	y	y	PROPN
ejde-673	661	3	is	be	AUX
ejde-673	661	4	a	a	DET
ejde-673	661	5	reflexive	reflexive	ADJ
ejde-673	661	6	banach	banach	NOUN
ejde-673	661	7	space	space	NOUN
ejde-673	661	8	,	,	PUNCT
ejde-673	661	9	there	there	PRON
ejde-673	661	10	exist	exist	VERB
ejde-673	661	11	a	a	DET
ejde-673	661	12	subsequence	subsequence	NOUN
ejde-673	661	13	{	{	PUNCT
ejde-673	661	14	uj′	uj′	NOUN
ejde-673	661	15	}	}	PUNCT
ejde-673	661	16	of	of	ADP
ejde-673	661	17	{	{	PUNCT
ejde-673	661	18	uj	uj	PROPN
ejde-673	661	19	}	}	PUNCT
ejde-673	661	20	and	and	CCONJ
ejde-673	661	21	u(n	u(n	PROPN
ejde-673	661	22	)	)	PUNCT
ejde-673	661	23	∈	∈	PROPN
ejde-673	661	24	y	y	PROPN
ejde-673	661	25	such	such	ADJ
ejde-673	661	26	that	that	SCONJ
ejde-673	661	27	uj′	uj′	PROPN
ejde-673	661	28	→	→	SYM
ejde-673	661	29	u(n	u(n	PROPN
ejde-673	661	30	)	)	PUNCT
ejde-673	661	31	weakly	weakly	ADV
ejde-673	661	32	in	in	ADP
ejde-673	661	33	y	y	PROPN
ejde-673	661	34	.	.	PUNCT
ejde-673	662	1	hence	hence	ADV
ejde-673	662	2	∥u(n)∥y	∥u(n)∥y	ADJ
ejde-673	662	3	≤	≤	NUM
ejde-673	662	4	lim	lim	PROPN
ejde-673	662	5	infj′→∞	infj′→∞	PROPN
ejde-673	662	6	∥uj′∥y	∥uj′∥y	PROPN
ejde-673	662	7	≤	≤	PROPN
ejde-673	662	8	r.	r.	PROPN
ejde-673	662	9	since	since	SCONJ
ejde-673	662	10	wn	wn	PROPN
ejde-673	662	11	is	be	AUX
ejde-673	662	12	a	a	DET
ejde-673	662	13	closed	closed	ADJ
ejde-673	662	14	subspace	subspace	NOUN
ejde-673	662	15	of	of	ADP
ejde-673	662	16	y	y	PROPN
ejde-673	662	17	,	,	PUNCT
ejde-673	662	18	we	we	PRON
ejde-673	662	19	see	see	VERB
ejde-673	662	20	22	22	NUM
ejde-673	662	21	j.	j.	PROPN
ejde-673	662	22	aramaki	aramaki	PROPN
ejde-673	662	23	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	662	24	that	that	SCONJ
ejde-673	662	25	wn	wn	PROPN
ejde-673	662	26	is	be	AUX
ejde-673	662	27	weakly	weakly	ADV
ejde-673	662	28	closed	closed	ADJ
ejde-673	662	29	,	,	PUNCT
ejde-673	662	30	so	so	CCONJ
ejde-673	663	1	u(n	u(n	PROPN
ejde-673	663	2	)	)	PUNCT
ejde-673	663	3	∈	∈	PROPN
ejde-673	663	4	wn	wn	PROPN
ejde-673	663	5	.	.	PUNCT
ejde-673	664	1	since	since	SCONJ
ejde-673	664	2	χ	χ	PROPN
ejde-673	664	3	is	be	AUX
ejde-673	664	4	sequentially	sequentially	ADV
ejde-673	664	5	weakly	weakly	ADV
ejde-673	664	6	continuous	continuous	ADJ
ejde-673	664	7	,	,	PUNCT
ejde-673	664	8	|χ(uj′)|	|χ(uj′)|	ADJ
ejde-673	664	9	→	→	SYM
ejde-673	664	10	|χ(u(n))|	|χ(u(n))|	PROPN
ejde-673	664	11	as	as	ADP
ejde-673	664	12	j′	j′	PROPN
ejde-673	664	13	→	→	SYM
ejde-673	664	14	∞.	∞.	PROPN
ejde-673	664	15	thereby	thereby	ADV
ejde-673	664	16	|χ(u(n))|	|χ(u(n))|	PROPN
ejde-673	664	17	=	=	SYM
ejde-673	664	18	dn	dn	PROPN
ejde-673	664	19	.	.	NOUN
ejde-673	664	20	since	since	SCONJ
ejde-673	664	21	dn+1	dn+1	NOUN
ejde-673	664	22	≤	≤	X
ejde-673	664	23	dn	dn	NOUN
ejde-673	664	24	for	for	ADP
ejde-673	664	25	all	all	PRON
ejde-673	664	26	n	n	PRON
ejde-673	664	27	∈	∈	PROPN
ejde-673	664	28	n	n	CCONJ
ejde-673	664	29	,	,	PUNCT
ejde-673	664	30	limn→∞	limn→∞	X
ejde-673	664	31	dn	dn	PROPN
ejde-673	664	32	=	=	PUNCT
ejde-673	664	33	d0	d0	PROPN
ejde-673	664	34	≥	≥	NOUN
ejde-673	664	35	0	0	NUM
ejde-673	664	36	exists	exist	VERB
ejde-673	664	37	.	.	PUNCT
ejde-673	665	1	since	since	SCONJ
ejde-673	665	2	{	{	PUNCT
ejde-673	665	3	u(n	u(n	PROPN
ejde-673	665	4	)	)	PUNCT
ejde-673	665	5	}	}	PUNCT
ejde-673	665	6	satisfies	satisfy	VERB
ejde-673	665	7	∥u(n)∥y	∥u(n)∥y	NOUN
ejde-673	665	8	≤	≤	ADJ
ejde-673	665	9	r	r	NOUN
ejde-673	665	10	,	,	PUNCT
ejde-673	665	11	there	there	PRON
ejde-673	665	12	exists	exist	VERB
ejde-673	665	13	a	a	DET
ejde-673	665	14	subsequence	subsequence	NOUN
ejde-673	665	15	{	{	PUNCT
ejde-673	665	16	u(n′	u(n′	PROPN
ejde-673	665	17	)	)	PUNCT
ejde-673	665	18	}	}	PUNCT
ejde-673	665	19	of	of	ADP
ejde-673	665	20	{	{	PUNCT
ejde-673	665	21	u(n	u(n	PROPN
ejde-673	665	22	)	)	PUNCT
ejde-673	665	23	}	}	PUNCT
ejde-673	665	24	and	and	CCONJ
ejde-673	665	25	u0	u0	PROPN
ejde-673	665	26	∈	∈	PROPN
ejde-673	665	27	y	y	PROPN
ejde-673	665	28	such	such	ADJ
ejde-673	665	29	that	that	SCONJ
ejde-673	665	30	u(n′	u(n′	PROPN
ejde-673	665	31	)	)	PUNCT
ejde-673	665	32	→	→	SYM
ejde-673	665	33	u0	u0	ADJ
ejde-673	665	34	weakly	weakly	ADV
ejde-673	665	35	in	in	ADP
ejde-673	665	36	y	y	PROPN
ejde-673	665	37	,	,	PUNCT
ejde-673	665	38	so	so	ADV
ejde-673	665	39	∥u0∥y	∥u0∥y	PROPN
ejde-673	665	40	≤	≤	PROPN
ejde-673	665	41	r.	r.	NOUN
ejde-673	665	42	since	since	SCONJ
ejde-673	665	43	again	again	ADV
ejde-673	665	44	χ	χ	X
ejde-673	665	45	is	be	AUX
ejde-673	665	46	sequentially	sequentially	ADV
ejde-673	665	47	weakly	weakly	ADV
ejde-673	665	48	continuous	continuous	ADJ
ejde-673	665	49	,	,	PUNCT
ejde-673	665	50	|χ(u(n′))|	|χ(u(n′))|	PROPN
ejde-673	665	51	=	=	SYM
ejde-673	665	52	dn′	dn′	NOUN
ejde-673	665	53	→	→	SYM
ejde-673	665	54	|χ(u0)|	|χ(u0)|	NOUN
ejde-673	665	55	=	=	SYM
ejde-673	665	56	d0	d0	NOUN
ejde-673	665	57	.	.	PUNCT
ejde-673	666	1	since	since	SCONJ
ejde-673	666	2	y	y	PROPN
ejde-673	666	3	is	be	AUX
ejde-673	666	4	reflexive	reflexive	ADJ
ejde-673	666	5	,	,	PUNCT
ejde-673	666	6	we	we	PRON
ejde-673	666	7	can	can	AUX
ejde-673	666	8	look	look	VERB
ejde-673	666	9	upon	upon	SCONJ
ejde-673	666	10	u0	u0	ADJ
ejde-673	666	11	∈	∈	PROPN
ejde-673	666	12	y	y	PROPN
ejde-673	666	13	∗∗	∗∗	PROPN
ejde-673	666	14	=	=	SYM
ejde-673	666	15	y	y	PROPN
ejde-673	666	16	.	.	PUNCT
ejde-673	667	1	therefore	therefore	ADV
ejde-673	667	2	,	,	PUNCT
ejde-673	667	3	for	for	ADP
ejde-673	667	4	any	any	DET
ejde-673	667	5	fj	fj	PROPN
ejde-673	667	6	∈	∈	PROPN
ejde-673	667	7	y	y	PROPN
ejde-673	667	8	∗	∗	NOUN
ejde-673	667	9	,	,	PUNCT
ejde-673	667	10	since	since	SCONJ
ejde-673	667	11	u(n′	u(n′	PROPN
ejde-673	667	12	)	)	PUNCT
ejde-673	667	13	∈	∈	PROPN
ejde-673	667	14	wn′	wn′	NOUN
ejde-673	667	15	,	,	PUNCT
ejde-673	667	16	we	we	PRON
ejde-673	667	17	have	have	VERB
ejde-673	667	18	⟨u0	⟨u0	PROPN
ejde-673	667	19	,	,	PUNCT
ejde-673	667	20	fj⟩y	fj⟩y	PROPN
ejde-673	667	21	∗∗,y	∗∗,y	ADV
ejde-673	667	22	∗	∗	NOUN
ejde-673	667	23	=	=	SYM
ejde-673	667	24	⟨fj	⟨fj	PRON
ejde-673	667	25	,	,	PUNCT
ejde-673	667	26	u0⟩y	u0⟩y	NOUN
ejde-673	668	1	∗,y	∗,y	NOUN
ejde-673	668	2	=	=	SYM
ejde-673	668	3	lim	lim	PROPN
ejde-673	668	4	n′→∞	n′→∞	PROPN
ejde-673	668	5	⟨fj	⟨fj	NUM
ejde-673	668	6	,	,	PUNCT
ejde-673	668	7	u(n′)⟩y	u(n′)⟩y	PROPN
ejde-673	668	8	∗,y	∗,y	PROPN
ejde-673	668	9	=	=	SYM
ejde-673	668	10	0	0	NUM
ejde-673	668	11	.	.	PUNCT
ejde-673	669	1	thus	thus	ADV
ejde-673	669	2	we	we	PRON
ejde-673	669	3	have	have	VERB
ejde-673	669	4	u0	u0	ADJ
ejde-673	669	5	=	=	ADJ
ejde-673	669	6	0	0	NUM
ejde-673	669	7	.	.	PUNCT
ejde-673	670	1	since	since	SCONJ
ejde-673	670	2	χ(0	χ(0	NOUN
ejde-673	670	3	)	)	PUNCT
ejde-673	670	4	=	=	SYM
ejde-673	670	5	0	0	NUM
ejde-673	670	6	,	,	PUNCT
ejde-673	670	7	we	we	PRON
ejde-673	670	8	have	have	VERB
ejde-673	670	9	d0	d0	NOUN
ejde-673	670	10	=	=	SYM
ejde-673	670	11	0	0	NUM
ejde-673	670	12	,	,	PUNCT
ejde-673	670	13	that	that	ADV
ejde-673	670	14	is	is	ADV
ejde-673	670	15	,	,	PUNCT
ejde-673	670	16	(	(	PUNCT
ejde-673	670	17	3.22	3.22	NUM
ejde-673	670	18	)	)	PUNCT
ejde-673	670	19	holds	hold	VERB
ejde-673	670	20	.	.	PUNCT
ejde-673	671	1	□	□	PUNCT
ejde-673	671	2	proposition	proposition	NOUN
ejde-673	671	3	3.18	3.18	NUM
ejde-673	671	4	.	.	PUNCT
ejde-673	672	1	we	we	PRON
ejde-673	672	2	have	have	VERB
ejde-673	672	3	limn→∞	limn→∞	PROPN
ejde-673	672	4	infu∈wn∩mα	infu∈wn∩mα	NOUN
ejde-673	672	5	∥u∥y	∥u∥y	X
ejde-673	672	6	=	=	SYM
ejde-673	672	7	∞.	∞.	PROPN
ejde-673	672	8	proof	proof	NOUN
ejde-673	672	9	.	.	PUNCT
ejde-673	673	1	suppose	suppose	VERB
ejde-673	673	2	that	that	SCONJ
ejde-673	673	3	the	the	DET
ejde-673	673	4	conclusion	conclusion	NOUN
ejde-673	673	5	is	be	AUX
ejde-673	673	6	false	false	ADJ
ejde-673	673	7	.	.	PUNCT
ejde-673	674	1	then	then	ADV
ejde-673	674	2	there	there	PRON
ejde-673	674	3	exist	exist	VERB
ejde-673	674	4	c1	c1	PROPN
ejde-673	674	5	>	>	X
ejde-673	674	6	0	0	PUNCT
ejde-673	675	1	and	and	CCONJ
ejde-673	675	2	un	un	PROPN
ejde-673	675	3	⊂	⊂	PROPN
ejde-673	675	4	wn	wn	PROPN
ejde-673	675	5	∩mα	∩mα	PROPN
ejde-673	675	6	such	such	ADJ
ejde-673	675	7	that	that	SCONJ
ejde-673	675	8	∥un∥y	∥un∥y	SCONJ
ejde-673	675	9	≤	≤	ADJ
ejde-673	675	10	c1	c1	NOUN
ejde-673	675	11	for	for	ADP
ejde-673	675	12	large	large	ADJ
ejde-673	675	13	n	n	CCONJ
ejde-673	675	14	∈	∈	PROPN
ejde-673	675	15	n.	n.	NOUN
ejde-673	675	16	then	then	ADV
ejde-673	675	17	sup	sup	VERB
ejde-673	675	18	u∈wn,∥u∥y	u∈wn,∥u∥y	PRON
ejde-673	675	19	≤c1	≤c1	ADJ
ejde-673	675	20	|k(u)|	|k(u)|	ADP
ejde-673	675	21	≥	≥	NOUN
ejde-673	675	22	|k(un)|	|k(un)|	NOUN
ejde-673	675	23	=	=	SYM
ejde-673	675	24	α	α	NOUN
ejde-673	675	25	.	.	PUNCT
ejde-673	676	1	therefore	therefore	ADV
ejde-673	676	2	,	,	PUNCT
ejde-673	676	3	lim	lim	PROPN
ejde-673	676	4	n→∞	n→∞	NUM
ejde-673	676	5	sup	sup	NOUN
ejde-673	676	6	u∈wn,∥u∥y	u∈wn,∥u∥y	PRON
ejde-673	676	7	≤c1	≤c1	ADJ
ejde-673	676	8	|k(u)|	|k(u)|	DET
ejde-673	676	9	≥	≥	NOUN
ejde-673	676	10	lim	lim	PROPN
ejde-673	676	11	n→∞	n→∞	PRON
ejde-673	676	12	|k(un)|	|k(un)|	NOUN
ejde-673	676	13	=	=	SYM
ejde-673	676	14	α	α	X
ejde-673	676	15	>	>	X
ejde-673	676	16	0	0	PROPN
ejde-673	676	17	.	.	PUNCT
ejde-673	677	1	if	if	SCONJ
ejde-673	677	2	we	we	PRON
ejde-673	677	3	apply	apply	VERB
ejde-673	677	4	lemma	lemma	PROPN
ejde-673	677	5	3.17	3.17	NUM
ejde-673	677	6	with	with	ADP
ejde-673	677	7	χ	χ	PROPN
ejde-673	677	8	=	=	SYM
ejde-673	677	9	k	k	PROPN
ejde-673	677	10	,	,	PUNCT
ejde-673	677	11	this	this	PRON
ejde-673	677	12	is	be	AUX
ejde-673	677	13	a	a	DET
ejde-673	677	14	contradiction	contradiction	NOUN
ejde-673	677	15	.	.	PUNCT
ejde-673	678	1	□	□	PUNCT
ejde-673	678	2	proposition	proposition	NOUN
ejde-673	678	3	3.19	3.19	NUM
ejde-673	678	4	.	.	PUNCT
ejde-673	679	1	we	we	PRON
ejde-673	679	2	have	have	VERB
ejde-673	679	3	lim	lim	PROPN
ejde-673	679	4	n→∞	n→∞	NUM
ejde-673	679	5	c(n	c(n	PROPN
ejde-673	679	6	,	,	PUNCT
ejde-673	679	7	α	α	X
ejde-673	679	8	)	)	PUNCT
ejde-673	679	9	=	=	SYM
ejde-673	679	10	∞.	∞.	PROPN
ejde-673	679	11	(	(	PUNCT
ejde-673	679	12	3.23	3.23	NUM
ejde-673	679	13	)	)	PUNCT
ejde-673	679	14	proof	proof	NOUN
ejde-673	679	15	.	.	PUNCT
ejde-673	680	1	by	by	ADP
ejde-673	680	2	proposition	proposition	NOUN
ejde-673	680	3	3.18	3.18	NUM
ejde-673	680	4	,	,	PUNCT
ejde-673	680	5	for	for	ADP
ejde-673	680	6	any	any	DET
ejde-673	680	7	c	c	PROPN
ejde-673	680	8	>	>	X
ejde-673	680	9	1	1	NUM
ejde-673	680	10	,	,	PUNCT
ejde-673	680	11	there	there	PRON
ejde-673	680	12	exists	exist	VERB
ejde-673	680	13	n0	n0	PROPN
ejde-673	680	14	∈	∈	PROPN
ejde-673	680	15	n	n	PRON
ejde-673	680	16	such	such	ADJ
ejde-673	680	17	that	that	PRON
ejde-673	680	18	for	for	ADP
ejde-673	680	19	any	any	DET
ejde-673	680	20	n	n	DET
ejde-673	680	21	≥	≥	NOUN
ejde-673	680	22	n0	n0	NOUN
ejde-673	680	23	and	and	CCONJ
ejde-673	680	24	u	u	PROPN
ejde-673	680	25	∈	∈	PROPN
ejde-673	680	26	wn	wn	PROPN
ejde-673	680	27	∩	∩	PROPN
ejde-673	680	28	mα	mα	PROPN
ejde-673	680	29	,	,	PUNCT
ejde-673	680	30	we	we	PRON
ejde-673	680	31	have	have	VERB
ejde-673	680	32	∥u∥y	∥u∥y	ADJ
ejde-673	680	33	>	>	X
ejde-673	680	34	c.	c.	NOUN
ejde-673	680	35	for	for	ADP
ejde-673	680	36	any	any	DET
ejde-673	680	37	h	h	NOUN
ejde-673	680	38	∈	∈	NOUN
ejde-673	681	1	σα	σα	INTJ
ejde-673	681	2	,	,	PUNCT
ejde-673	681	3	we	we	PRON
ejde-673	681	4	have	have	VERB
ejde-673	681	5	γ(h	γ(h	PROPN
ejde-673	681	6	∩	∩	ADJ
ejde-673	681	7	zn−1	zn−1	ADJ
ejde-673	681	8	)	)	PUNCT
ejde-673	681	9	≤	≤	NOUN
ejde-673	681	10	n	n	CCONJ
ejde-673	681	11	−	−	PROPN
ejde-673	681	12	1	1	NUM
ejde-673	681	13	.	.	PUNCT
ejde-673	682	1	on	on	ADP
ejde-673	682	2	the	the	DET
ejde-673	682	3	other	other	ADJ
ejde-673	682	4	hand	hand	NOUN
ejde-673	682	5	,	,	PUNCT
ejde-673	682	6	we	we	PRON
ejde-673	682	7	have	have	VERB
ejde-673	682	8	codimwn	codimwn	NOUN
ejde-673	682	9	=	=	SYM
ejde-673	682	10	n	n	CCONJ
ejde-673	682	11	−	−	PROPN
ejde-673	682	12	1	1	NUM
ejde-673	682	13	.	.	PUNCT
ejde-673	683	1	hence	hence	ADV
ejde-673	683	2	for	for	ADP
ejde-673	683	3	any	any	DET
ejde-673	683	4	h	h	NOUN
ejde-673	683	5	∈	∈	NOUN
ejde-673	683	6	σα	σα	PROPN
ejde-673	683	7	with	with	ADP
ejde-673	683	8	γ(h	γ(h	PROPN
ejde-673	683	9	)	)	PUNCT
ejde-673	683	10	≥	≥	NOUN
ejde-673	683	11	n	n	CCONJ
ejde-673	683	12	,	,	PUNCT
ejde-673	683	13	h	h	PROPN
ejde-673	683	14	∩	∩	NOUN
ejde-673	683	15	wn	wn	PROPN
ejde-673	683	16	is	be	AUX
ejde-673	683	17	non	non	ADJ
ejde-673	683	18	-	-	ADJ
ejde-673	683	19	empty	empty	ADJ
ejde-673	683	20	.	.	PUNCT
ejde-673	684	1	indeed	indeed	ADV
ejde-673	684	2	,	,	PUNCT
ejde-673	684	3	since	since	SCONJ
ejde-673	684	4	h	h	NOUN
ejde-673	684	5	=	=	PUNCT
ejde-673	684	6	(	(	PUNCT
ejde-673	684	7	h	h	PROPN
ejde-673	684	8	∩	∩	X
ejde-673	684	9	zn−1	zn−1	ADJ
ejde-673	684	10	)	)	PUNCT
ejde-673	684	11	∪	∪	NOUN
ejde-673	684	12	(	(	PUNCT
ejde-673	684	13	h	h	NOUN
ejde-673	684	14	∩wn	∩wn	NOUN
ejde-673	684	15	)	)	PUNCT
ejde-673	684	16	,	,	PUNCT
ejde-673	684	17	it	it	PRON
ejde-673	684	18	follows	follow	VERB
ejde-673	684	19	from	from	ADP
ejde-673	684	20	proposition	proposition	NOUN
ejde-673	684	21	3.14	3.14	NUM
ejde-673	684	22	(	(	PUNCT
ejde-673	684	23	iii	iii	NOUN
ejde-673	684	24	)	)	PUNCT
ejde-673	685	1	that	that	PRON
ejde-673	685	2	n	n	CCONJ
ejde-673	685	3	≤	≤	PROPN
ejde-673	685	4	γ(h	γ(h	NOUN
ejde-673	685	5	)	)	PUNCT
ejde-673	685	6	≤	≤	NUM
ejde-673	685	7	γ(h	γ(h	PROPN
ejde-673	685	8	∩	∩	X
ejde-673	685	9	zn−1	zn−1	PROPN
ejde-673	685	10	)	)	PUNCT
ejde-673	685	11	+	+	NUM
ejde-673	685	12	γ(h	γ(h	PROPN
ejde-673	685	13	∩wn	∩wn	NOUN
ejde-673	685	14	)	)	PUNCT
ejde-673	685	15	≤	≤	NUM
ejde-673	686	1	n−	n−	NOUN
ejde-673	686	2	1	1	NUM
ejde-673	686	3	+	+	CCONJ
ejde-673	686	4	γ(h	γ(h	PROPN
ejde-673	686	5	∩wn	∩wn	NOUN
ejde-673	686	6	)	)	PUNCT
ejde-673	686	7	,	,	PUNCT
ejde-673	686	8	so	so	SCONJ
ejde-673	686	9	γ(h	γ(h	PROPN
ejde-673	686	10	∩wn	∩wn	PROPN
ejde-673	686	11	)	)	PUNCT
ejde-673	686	12	≥	≥	NOUN
ejde-673	686	13	1	1	NUM
ejde-673	686	14	.	.	PUNCT
ejde-673	687	1	hence	hence	ADV
ejde-673	687	2	h	h	NOUN
ejde-673	687	3	∩wn	∩wn	NOUN
ejde-673	687	4	̸=	̸=	PROPN
ejde-673	687	5	∅.	∅.	ADV
ejde-673	687	6	for	for	ADP
ejde-673	687	7	n	n	PRON
ejde-673	687	8	≥	≥	NOUN
ejde-673	687	9	n0	n0	NUM
ejde-673	687	10	,	,	PUNCT
ejde-673	687	11	using	use	VERB
ejde-673	687	12	(	(	PUNCT
ejde-673	687	13	3.19	3.19	NUM
ejde-673	687	14	)	)	PUNCT
ejde-673	687	15	,	,	PUNCT
ejde-673	687	16	we	we	PRON
ejde-673	687	17	have	have	VERB
ejde-673	687	18	c(n	c(n	PROPN
ejde-673	687	19	,	,	PUNCT
ejde-673	687	20	α	α	NOUN
ejde-673	687	21	)	)	PUNCT
ejde-673	687	22	=	=	SYM
ejde-673	687	23	inf	inf	ADJ
ejde-673	687	24	h∈σα	h∈σα	NOUN
ejde-673	687	25	,	,	PUNCT
ejde-673	687	26	γ(h)≥n	γ(h)≥n	NUM
ejde-673	687	27	sup	sup	NOUN
ejde-673	687	28	u∈h	u∈h	ADJ
ejde-673	687	29	ψ̃(u	ψ̃(u	NOUN
ejde-673	687	30	)	)	PUNCT
ejde-673	687	31	=	=	SYM
ejde-673	687	32	inf	inf	ADJ
ejde-673	687	33	h∈σα	h∈σα	NOUN
ejde-673	687	34	,	,	PUNCT
ejde-673	687	35	γ(h)≥n	γ(h)≥n	SYM
ejde-673	687	36	max	max	PROPN
ejde-673	687	37	{	{	PUNCT
ejde-673	687	38	sup	sup	NOUN
ejde-673	687	39	u∈h∩(y	u∈h∩(y	PROPN
ejde-673	687	40	\zn−1	\zn−1	PROPN
ejde-673	687	41	)	)	PUNCT
ejde-673	687	42	ψ̃(u	ψ̃(u	NOUN
ejde-673	687	43	)	)	PUNCT
ejde-673	687	44	,	,	PUNCT
ejde-673	687	45	sup	sup	NOUN
ejde-673	687	46	u∈h∩zn−1	u∈h∩zn−1	PROPN
ejde-673	687	47	ψ̃(u	ψ̃(u	NOUN
ejde-673	687	48	)	)	PUNCT
ejde-673	687	49	}	}	PUNCT
ejde-673	687	50	≥	≥	PROPN
ejde-673	687	51	inf	inf	VERB
ejde-673	687	52	h∈σα	h∈σα	NOUN
ejde-673	687	53	,	,	PUNCT
ejde-673	687	54	γ(h)≥n	γ(h)≥n	NUM
ejde-673	687	55	sup	sup	NOUN
ejde-673	687	56	u∈h∩(y	u∈h∩(y	PROPN
ejde-673	687	57	\zn−1	\zn−1	PROPN
ejde-673	687	58	)	)	PUNCT
ejde-673	687	59	ψ̃(u	ψ̃(u	ADJ
ejde-673	687	60	)	)	PUNCT
ejde-673	688	1	=	=	SYM
ejde-673	688	2	inf	inf	ADJ
ejde-673	688	3	h∈σα	h∈σα	NOUN
ejde-673	688	4	,	,	PUNCT
ejde-673	688	5	γ(h)≥n	γ(h)≥n	SYM
ejde-673	688	6	max	max	PROPN
ejde-673	688	7	{	{	PUNCT
ejde-673	688	8	sup	sup	NOUN
ejde-673	688	9	u∈h∩((y	u∈h∩((y	PUNCT
ejde-673	688	10	\zn−1)\wn	\zn−1)\wn	PROPN
ejde-673	688	11	)	)	PUNCT
ejde-673	688	12	ψ̃(u	ψ̃(u	NOUN
ejde-673	688	13	)	)	PUNCT
ejde-673	688	14	,	,	PUNCT
ejde-673	688	15	sup	sup	NOUN
ejde-673	688	16	u∈h∩wn	u∈h∩wn	NOUN
ejde-673	688	17	ψ̃(u	ψ̃(u	NOUN
ejde-673	688	18	)	)	PUNCT
ejde-673	688	19	}	}	PUNCT
ejde-673	688	20	≥	≥	PROPN
ejde-673	688	21	inf	inf	VERB
ejde-673	688	22	h∈σα	h∈σα	NOUN
ejde-673	688	23	,	,	PUNCT
ejde-673	688	24	γ(h)≥n	γ(h)≥n	NUM
ejde-673	688	25	sup	sup	NOUN
ejde-673	688	26	u∈h∩wn	u∈h∩wn	NOUN
ejde-673	688	27	ψ̃(u	ψ̃(u	NOUN
ejde-673	688	28	)	)	PUNCT
ejde-673	688	29	≥	≥	NOUN
ejde-673	688	30	inf	inf	VERB
ejde-673	688	31	h∈σα	h∈σα	NOUN
ejde-673	688	32	,	,	PUNCT
ejde-673	688	33	γ(h)≥n	γ(h)≥n	NUM
ejde-673	688	34	sup	sup	NOUN
ejde-673	688	35	u∈h∩wn	u∈h∩wn	NOUN
ejde-673	688	36	m0	m0	PROPN
ejde-673	688	37	l	l	PROPN
ejde-673	689	1	(	(	PUNCT
ejde-673	689	2	k0	k0	PROPN
ejde-673	689	3	p+	p+	PROPN
ejde-673	689	4	∥u∥p	∥u∥p	NOUN
ejde-673	689	5	−	−	PROPN
ejde-673	689	6	y	y	PROPN
ejde-673	689	7	)	)	PUNCT
ejde-673	689	8	l	l	NOUN
ejde-673	689	9	≥	≥	NOUN
ejde-673	689	10	m0	m0	PROPN
ejde-673	689	11	l	l	PROPN
ejde-673	689	12	(	(	PUNCT
ejde-673	689	13	k0	k0	PROPN
ejde-673	689	14	p+	p+	PROPN
ejde-673	689	15	cp	cp	PROPN
ejde-673	689	16	−	−	PROPN
ejde-673	689	17	)	)	PUNCT
ejde-673	689	18	l	l	NOUN
ejde-673	689	19	.	.	PUNCT
ejde-673	690	1	since	since	SCONJ
ejde-673	690	2	c	c	PROPN
ejde-673	690	3	>	>	X
ejde-673	690	4	1	1	NUM
ejde-673	690	5	is	be	AUX
ejde-673	690	6	arbitrary	arbitrary	ADJ
ejde-673	690	7	,	,	PUNCT
ejde-673	690	8	we	we	PRON
ejde-673	690	9	thus	thus	ADV
ejde-673	690	10	get	get	VERB
ejde-673	690	11	(	(	PUNCT
ejde-673	690	12	3.23	3.23	NUM
ejde-673	690	13	)	)	PUNCT
ejde-673	690	14	.	.	PUNCT
ejde-673	691	1	□	□	PUNCT
ejde-673	691	2	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	691	3	eigenvalue	eigenvalue	NOUN
ejde-673	691	4	problems	problem	NOUN
ejde-673	691	5	for	for	ADP
ejde-673	691	6	kirchhoff	kirchhoff	NOUN
ejde-673	691	7	-	-	PUNCT
ejde-673	691	8	type	type	NOUN
ejde-673	691	9	equations	equation	NOUN
ejde-673	691	10	23	23	NUM
ejde-673	691	11	theorem	theorem	VERB
ejde-673	691	12	3.20	3.20	NUM
ejde-673	691	13	.	.	PUNCT
ejde-673	692	1	assume	assume	VERB
ejde-673	692	2	that	that	SCONJ
ejde-673	692	3	(	(	PUNCT
ejde-673	692	4	a2)–(a8	a2)–(a8	ADJ
ejde-673	692	5	)	)	PUNCT
ejde-673	692	6	hold	hold	VERB
ejde-673	692	7	and	and	CCONJ
ejde-673	692	8	fix	fix	VERB
ejde-673	692	9	α	α	PRON
ejde-673	692	10	>	>	X
ejde-673	692	11	0	0	NUM
ejde-673	692	12	.	.	PUNCT
ejde-673	693	1	then	then	ADV
ejde-673	693	2	for	for	ADP
ejde-673	693	3	every	every	DET
ejde-673	693	4	n	n	PRON
ejde-673	693	5	∈	∈	PROPN
ejde-673	693	6	n	n	CCONJ
ejde-673	693	7	,	,	PUNCT
ejde-673	693	8	c(n	c(n	PROPN
ejde-673	693	9	,	,	PUNCT
ejde-673	693	10	α	α	NOUN
ejde-673	693	11	)	)	PUNCT
ejde-673	693	12	defined	define	VERB
ejde-673	693	13	by	by	ADP
ejde-673	693	14	(	(	PUNCT
ejde-673	693	15	3.19	3.19	NUM
ejde-673	693	16	)	)	PUNCT
ejde-673	693	17	is	be	AUX
ejde-673	693	18	a	a	DET
ejde-673	693	19	critical	critical	ADJ
ejde-673	693	20	value	value	NOUN
ejde-673	693	21	of	of	ADP
ejde-673	693	22	ψ̃	ψ̃	PROPN
ejde-673	693	23	with	with	ADP
ejde-673	693	24	respect	respect	NOUN
ejde-673	693	25	to	to	ADP
ejde-673	693	26	the	the	DET
ejde-673	693	27	submanifold	submanifold	NOUN
ejde-673	693	28	mα	mα	ADP
ejde-673	693	29	such	such	ADJ
ejde-673	693	30	that	that	SCONJ
ejde-673	693	31	0	0	NUM
ejde-673	693	32	<	<	X
ejde-673	693	33	c(n	c(n	PROPN
ejde-673	693	34	,	,	PUNCT
ejde-673	693	35	α	α	NOUN
ejde-673	693	36	)	)	PUNCT
ejde-673	693	37	≤	≤	NOUN
ejde-673	693	38	c(n+1,α	c(n+1,α	NUM
ejde-673	693	39	)	)	PUNCT
ejde-673	693	40	<	<	X
ejde-673	694	1	∞	∞	PROPN
ejde-673	694	2	and	and	CCONJ
ejde-673	694	3	c(n	c(n	PROPN
ejde-673	694	4	,	,	PUNCT
ejde-673	694	5	α	α	NOUN
ejde-673	694	6	)	)	PUNCT
ejde-673	694	7	→	→	SYM
ejde-673	694	8	∞	∞	PROPN
ejde-673	694	9	as	as	ADP
ejde-673	694	10	n	n	PROPN
ejde-673	694	11	→	→	SYM
ejde-673	694	12	∞.	∞.	PROPN
ejde-673	694	13	moreover	moreover	ADV
ejde-673	694	14	,	,	PUNCT
ejde-673	694	15	(	(	PUNCT
ejde-673	694	16	1.1	1.1	NUM
ejde-673	694	17	)	)	PUNCT
ejde-673	694	18	has	have	VERB
ejde-673	694	19	infinitely	infinitely	ADV
ejde-673	694	20	many	many	ADJ
ejde-673	694	21	eigenpair	eigenpair	NOUN
ejde-673	694	22	sequence	sequence	NOUN
ejde-673	694	23	{	{	PUNCT
ejde-673	694	24	(	(	PUNCT
ejde-673	694	25	u(n	u(n	PROPN
ejde-673	694	26	,	,	PUNCT
ejde-673	694	27	α	α	NOUN
ejde-673	694	28	)	)	PUNCT
ejde-673	694	29	,	,	PUNCT
ejde-673	694	30	λ(n	λ(n	PROPN
ejde-673	694	31	,	,	PUNCT
ejde-673	694	32	α	α	NOUN
ejde-673	694	33	)	)	PUNCT
ejde-673	694	34	)	)	PUNCT
ejde-673	694	35	}	}	PUNCT
ejde-673	694	36	such	such	ADJ
ejde-673	694	37	that	that	SCONJ
ejde-673	694	38	k(±u(n	k(±u(n	PROPN
ejde-673	694	39	,	,	PUNCT
ejde-673	694	40	α	α	NOUN
ejde-673	694	41	)	)	PUNCT
ejde-673	694	42	)	)	PUNCT
ejde-673	695	1	=	=	SYM
ejde-673	695	2	α	α	X
ejde-673	695	3	,	,	PUNCT
ejde-673	695	4	ψ(±u(n	ψ(±u(n	NOUN
ejde-673	695	5	,	,	PUNCT
ejde-673	695	6	α	α	NOUN
ejde-673	695	7	)	)	PUNCT
ejde-673	695	8	)	)	PUNCT
ejde-673	696	1	=	=	SYM
ejde-673	696	2	c(n	c(n	PROPN
ejde-673	696	3	,	,	PUNCT
ejde-673	696	4	α	α	NOUN
ejde-673	696	5	)	)	PUNCT
ejde-673	696	6	and	and	CCONJ
ejde-673	696	7	0	0	NUM
ejde-673	696	8	<	<	X
ejde-673	696	9	λ(n	λ(n	PROPN
ejde-673	696	10	,	,	PUNCT
ejde-673	696	11	α	α	NOUN
ejde-673	696	12	)	)	PUNCT
ejde-673	696	13	→	→	SYM
ejde-673	696	14	∞	∞	PROPN
ejde-673	696	15	as	as	ADP
ejde-673	696	16	n	n	PROPN
ejde-673	696	17	→	→	SYM
ejde-673	696	18	∞.	∞.	PROPN
ejde-673	696	19	proof	proof	NOUN
ejde-673	696	20	.	.	PUNCT
ejde-673	697	1	taking	take	VERB
ejde-673	697	2	proposition	proposition	NOUN
ejde-673	697	3	3.15	3.15	NUM
ejde-673	697	4	,	,	PUNCT
ejde-673	697	5	(	(	PUNCT
ejde-673	697	6	3.12	3.12	NUM
ejde-673	697	7	)	)	PUNCT
ejde-673	697	8	,	,	PUNCT
ejde-673	697	9	(	(	PUNCT
ejde-673	697	10	3.13	3.13	NUM
ejde-673	697	11	)	)	PUNCT
ejde-673	697	12	,	,	PUNCT
ejde-673	697	13	lemma	lemma	PROPN
ejde-673	697	14	3.16	3.16	NUM
ejde-673	697	15	,	,	PUNCT
ejde-673	697	16	and	and	CCONJ
ejde-673	697	17	proposition	proposition	NOUN
ejde-673	697	18	3.19	3.19	NUM
ejde-673	697	19	into	into	ADP
ejde-673	697	20	consideration	consideration	NOUN
ejde-673	697	21	,	,	PUNCT
ejde-673	697	22	it	it	PRON
ejde-673	697	23	suffices	suffice	VERB
ejde-673	697	24	to	to	PART
ejde-673	697	25	show	show	VERB
ejde-673	697	26	that	that	SCONJ
ejde-673	697	27	λ(n	λ(n	PROPN
ejde-673	697	28	,	,	PUNCT
ejde-673	697	29	α	α	NOUN
ejde-673	697	30	)	)	PUNCT
ejde-673	697	31	→	→	SYM
ejde-673	697	32	∞	∞	PROPN
ejde-673	697	33	as	as	ADP
ejde-673	697	34	n	n	PROPN
ejde-673	697	35	→	→	SYM
ejde-673	697	36	∞.	∞.	PROPN
ejde-673	697	37	it	it	PRON
ejde-673	697	38	follows	follow	VERB
ejde-673	697	39	from	from	ADP
ejde-673	697	40	(	(	PUNCT
ejde-673	697	41	a8	a8	PROPN
ejde-673	697	42	)	)	PUNCT
ejde-673	698	1	that	that	PRON
ejde-673	698	2	⟨k	⟨k	PROPN
ejde-673	698	3	′(u(n	′(u(n	NOUN
ejde-673	698	4	,	,	PUNCT
ejde-673	698	5	α	α	NOUN
ejde-673	698	6	)	)	PUNCT
ejde-673	698	7	)	)	PUNCT
ejde-673	698	8	,	,	PUNCT
ejde-673	698	9	u(n	u(n	PROPN
ejde-673	698	10	,	,	PUNCT
ejde-673	698	11	α)⟩y	α)⟩y	PROPN
ejde-673	698	12	∗,y	∗,y	PROPN
ejde-673	698	13	≤	≤	NUM
ejde-673	698	14	r+k(u(n	r+k(u(n	NOUN
ejde-673	698	15	,	,	PUNCT
ejde-673	698	16	α	α	NOUN
ejde-673	698	17	)	)	PUNCT
ejde-673	698	18	)	)	PUNCT
ejde-673	698	19	=	=	PUNCT
ejde-673	698	20	r+α	r+α	NUM
ejde-673	698	21	.	.	PUNCT
ejde-673	698	22	hence	hence	ADV
ejde-673	698	23	λ(n	λ(n	PROPN
ejde-673	698	24	,	,	PUNCT
ejde-673	698	25	α	α	NOUN
ejde-673	698	26	)	)	PUNCT
ejde-673	698	27	=	=	SYM
ejde-673	698	28	⟨ψ′(u(n	⟨ψ′(u(n	NOUN
ejde-673	698	29	,	,	PUNCT
ejde-673	698	30	α	α	NOUN
ejde-673	698	31	)	)	PUNCT
ejde-673	698	32	)	)	PUNCT
ejde-673	698	33	,	,	PUNCT
ejde-673	698	34	u(n	u(n	PROPN
ejde-673	698	35	,	,	PUNCT
ejde-673	698	36	α)⟩y	α)⟩y	PROPN
ejde-673	698	37	∗,y	∗,y	PROPN
ejde-673	698	38	⟨k	⟨k	NOUN
ejde-673	698	39	′(u(n	′(u(n	NOUN
ejde-673	698	40	,	,	PUNCT
ejde-673	698	41	α	α	NOUN
ejde-673	698	42	)	)	PUNCT
ejde-673	698	43	)	)	PUNCT
ejde-673	698	44	,	,	PUNCT
ejde-673	698	45	u(n	u(n	PROPN
ejde-673	698	46	,	,	PUNCT
ejde-673	698	47	α)⟩y	α)⟩y	PROPN
ejde-673	698	48	∗,y	∗,y	PROPN
ejde-673	698	49	≥	≥	NUM
ejde-673	698	50	⟨ψ′(u(n	⟨ψ′(u(n	NOUN
ejde-673	698	51	,	,	PUNCT
ejde-673	698	52	α	α	NOUN
ejde-673	698	53	)	)	PUNCT
ejde-673	698	54	)	)	PUNCT
ejde-673	698	55	,	,	PUNCT
ejde-673	698	56	u(n	u(n	PROPN
ejde-673	698	57	,	,	PUNCT
ejde-673	698	58	α)⟩y	α)⟩y	PROPN
ejde-673	698	59	∗,y	∗,y	PROPN
ejde-673	698	60	r+α	r+α	X
ejde-673	698	61	.	.	PUNCT
ejde-673	699	1	(	(	PUNCT
ejde-673	699	2	3.24	3.24	NUM
ejde-673	699	3	)	)	PUNCT
ejde-673	699	4	assume	assume	VERB
ejde-673	699	5	that	that	SCONJ
ejde-673	699	6	λ(n	λ(n	PROPN
ejde-673	699	7	,	,	PUNCT
ejde-673	699	8	α	α	NOUN
ejde-673	699	9	)	)	PUNCT
ejde-673	699	10	≤	≤	NUM
ejde-673	699	11	m	m	VERB
ejde-673	699	12	for	for	ADP
ejde-673	699	13	all	all	DET
ejde-673	699	14	n	n	PRON
ejde-673	699	15	∈	∈	PROPN
ejde-673	699	16	n.	n.	NOUN
ejde-673	699	17	then	then	ADV
ejde-673	699	18	by	by	ADP
ejde-673	699	19	(	(	PUNCT
ejde-673	699	20	3.24	3.24	NUM
ejde-673	699	21	)	)	PUNCT
ejde-673	699	22	,	,	PUNCT
ejde-673	699	23	⟨ψ′(u(n	⟨ψ′(u(n	PROPN
ejde-673	699	24	,	,	PUNCT
ejde-673	699	25	α	α	NOUN
ejde-673	699	26	)	)	PUNCT
ejde-673	699	27	)	)	PUNCT
ejde-673	699	28	,	,	PUNCT
ejde-673	699	29	u(n	u(n	PROPN
ejde-673	699	30	,	,	PUNCT
ejde-673	699	31	α)⟩y	α)⟩y	PROPN
ejde-673	699	32	∗,y	∗,y	PROPN
ejde-673	699	33	≤	≤	NOUN
ejde-673	699	34	mr+α	mr+α	PROPN
ejde-673	699	35	=	=	NOUN
ejde-673	699	36	:	:	PUNCT
ejde-673	699	37	c2	c2	PROPN
ejde-673	699	38	.	.	PUNCT
ejde-673	700	1	on	on	ADP
ejde-673	700	2	the	the	DET
ejde-673	700	3	other	other	ADJ
ejde-673	700	4	hand	hand	NOUN
ejde-673	700	5	,	,	PUNCT
ejde-673	700	6	from	from	ADP
ejde-673	700	7	(	(	PUNCT
ejde-673	700	8	a5	a5	PROPN
ejde-673	700	9	)	)	PUNCT
ejde-673	700	10	,	,	PUNCT
ejde-673	700	11	we	we	PRON
ejde-673	700	12	have	have	VERB
ejde-673	700	13	⟨ψ′(u(n	⟨ψ′(u(n	NOUN
ejde-673	700	14	,	,	PUNCT
ejde-673	700	15	α	α	NOUN
ejde-673	700	16	)	)	PUNCT
ejde-673	700	17	)	)	PUNCT
ejde-673	700	18	,	,	PUNCT
ejde-673	701	1	u(n	u(n	PROPN
ejde-673	701	2	,	,	PUNCT
ejde-673	701	3	α)⟩y	α)⟩y	PROPN
ejde-673	701	4	∗,y	∗,y	NOUN
ejde-673	701	5	=	=	SYM
ejde-673	701	6	m(φ(u(n	m(φ(u(n	PROPN
ejde-673	701	7	,	,	PUNCT
ejde-673	701	8	α)))⟨φ′(u(n	α)))⟨φ′(u(n	NOUN
ejde-673	701	9	,	,	PUNCT
ejde-673	701	10	α	α	NOUN
ejde-673	701	11	)	)	PUNCT
ejde-673	701	12	)	)	PUNCT
ejde-673	701	13	,	,	PUNCT
ejde-673	701	14	u(n	u(n	PROPN
ejde-673	701	15	,	,	PUNCT
ejde-673	701	16	α)⟩y	α)⟩y	PROPN
ejde-673	701	17	∗,y	∗,y	PROPN
ejde-673	701	18	≥	≥	NOUN
ejde-673	701	19	m0	m0	NOUN
ejde-673	701	20	(	(	PUNCT
ejde-673	701	21	φ(u(n	φ(u(n	PROPN
ejde-673	701	22	,	,	PUNCT
ejde-673	701	23	α	α	NOUN
ejde-673	701	24	)	)	PUNCT
ejde-673	701	25	)	)	PUNCT
ejde-673	701	26	)	)	PUNCT
ejde-673	702	1	l−1	l−1	PROPN
ejde-673	702	2	∫	∫	PROPN
ejde-673	702	3	ω	ω	PROPN
ejde-673	702	4	a(x,∇u(n	a(x,∇u(n	PROPN
ejde-673	702	5	,	,	PUNCT
ejde-673	702	6	α)(x	α)(x	PROPN
ejde-673	702	7	)	)	PUNCT
ejde-673	702	8	)	)	PUNCT
ejde-673	702	9	·	·	PUNCT
ejde-673	702	10	∇u(n	∇u(n	PROPN
ejde-673	702	11	,	,	PUNCT
ejde-673	702	12	α)(x	α)(x	NOUN
ejde-673	702	13	)	)	PUNCT
ejde-673	702	14	dx	dx	PROPN
ejde-673	702	15	≥	≥	PROPN
ejde-673	702	16	m0	m0	PROPN
ejde-673	702	17	(	(	PUNCT
ejde-673	702	18	1	1	NUM
ejde-673	702	19	p+	p+	NOUN
ejde-673	702	20	∫	∫	PROPN
ejde-673	702	21	ω	ω	PROPN
ejde-673	702	22	a(x,∇u(n	a(x,∇u(n	PROPN
ejde-673	702	23	,	,	PUNCT
ejde-673	702	24	α)(x	α)(x	PROPN
ejde-673	702	25	)	)	PUNCT
ejde-673	702	26	)	)	PUNCT
ejde-673	702	27	·	·	PUNCT
ejde-673	702	28	∇u(n	∇u(n	PROPN
ejde-673	702	29	,	,	PUNCT
ejde-673	702	30	α)(x	α)(x	NOUN
ejde-673	702	31	)	)	PUNCT
ejde-673	702	32	dx	dx	PROPN
ejde-673	702	33	)	)	PUNCT
ejde-673	703	1	l−1	l−1	PROPN
ejde-673	703	2	×	×	NOUN
ejde-673	703	3	∫	∫	PROPN
ejde-673	703	4	ω	ω	PROPN
ejde-673	703	5	a(x,∇u(n	a(x,∇u(n	PROPN
ejde-673	703	6	,	,	PUNCT
ejde-673	703	7	α)(x	α)(x	PROPN
ejde-673	703	8	)	)	PUNCT
ejde-673	703	9	)	)	PUNCT
ejde-673	703	10	·	·	PUNCT
ejde-673	703	11	∇u(n	∇u(n	PROPN
ejde-673	703	12	,	,	PUNCT
ejde-673	703	13	α)(x	α)(x	NOUN
ejde-673	703	14	)	)	PUNCT
ejde-673	703	15	dx	dx	PROPN
ejde-673	704	1	=	=	PROPN
ejde-673	704	2	m0	m0	PROPN
ejde-673	704	3	(	(	PUNCT
ejde-673	704	4	p+)l−1	p+)l−1	NUM
ejde-673	704	5	(	(	PUNCT
ejde-673	704	6	∫	∫	PROPN
ejde-673	704	7	ω	ω	PROPN
ejde-673	704	8	a(x,∇u(n	a(x,∇u(n	PROPN
ejde-673	704	9	,	,	PUNCT
ejde-673	704	10	α)(x	α)(x	PROPN
ejde-673	704	11	)	)	PUNCT
ejde-673	704	12	)	)	PUNCT
ejde-673	704	13	·	·	PUNCT
ejde-673	704	14	∇u(n	∇u(n	PROPN
ejde-673	704	15	,	,	PUNCT
ejde-673	704	16	α)(x	α)(x	NOUN
ejde-673	704	17	)	)	PUNCT
ejde-673	704	18	dx	dx	PROPN
ejde-673	704	19	)	)	PUNCT
ejde-673	704	20	l	l	NOUN
ejde-673	704	21	≥	≥	NOUN
ejde-673	704	22	m0	m0	NOUN
ejde-673	704	23	(	(	PUNCT
ejde-673	704	24	p+)l−1	p+)l−1	NUM
ejde-673	704	25	kl0	kl0	NOUN
ejde-673	704	26	(	(	PUNCT
ejde-673	704	27	∫	∫	PROPN
ejde-673	704	28	ω	ω	NUM
ejde-673	704	29	h1(x)|∇u(n	h1(x)|∇u(n	PROPN
ejde-673	704	30	,	,	PUNCT
ejde-673	704	31	α)(x)|p(x)dx	α)(x)|p(x)dx	NOUN
ejde-673	704	32	)	)	PUNCT
ejde-673	705	1	l	l	NOUN
ejde-673	705	2	=	=	SYM
ejde-673	705	3	m0	m0	PROPN
ejde-673	705	4	(	(	PUNCT
ejde-673	705	5	p+)l−1	p+)l−1	NUM
ejde-673	705	6	kl0(ρ̃(p(·),h1(·))(u(n	kl0(ρ̃(p(·),h1(·))(u(n	PROPN
ejde-673	705	7	,	,	PUNCT
ejde-673	705	8	α	α	NOUN
ejde-673	705	9	)	)	PUNCT
ejde-673	705	10	)	)	PUNCT
ejde-673	705	11	)	)	PUNCT
ejde-673	706	1	l.	l.	PROPN
ejde-673	706	2	therefore	therefore	ADV
ejde-673	706	3	,	,	PUNCT
ejde-673	706	4	we	we	PRON
ejde-673	706	5	have	have	VERB
ejde-673	706	6	ρ̃(p(·),h1(·))(u(n	ρ̃(p(·),h1(·))(u(n	NOUN
ejde-673	706	7	,	,	PUNCT
ejde-673	706	8	α	α	NOUN
ejde-673	706	9	)	)	PUNCT
ejde-673	706	10	)	)	PUNCT
ejde-673	706	11	≤	≤	NUM
ejde-673	706	12	c3	c3	NOUN
ejde-673	706	13	for	for	ADP
ejde-673	706	14	some	some	DET
ejde-673	706	15	constant	constant	ADJ
ejde-673	706	16	c3	c3	NOUN
ejde-673	706	17	.	.	PUNCT
ejde-673	707	1	in	in	ADP
ejde-673	707	2	particular	particular	ADJ
ejde-673	707	3	,	,	PUNCT
ejde-673	707	4	∥u(n	∥u(n	PROPN
ejde-673	707	5	,	,	PUNCT
ejde-673	707	6	α)∥y	α)∥y	X
ejde-673	707	7	≤	≤	NUM
ejde-673	707	8	c4	c4	NOUN
ejde-673	707	9	for	for	ADP
ejde-673	707	10	all	all	PRON
ejde-673	707	11	n	n	PRON
ejde-673	707	12	∈	∈	NOUN
ejde-673	707	13	n	n	NOUN
ejde-673	707	14	with	with	ADP
ejde-673	707	15	some	some	DET
ejde-673	707	16	constant	constant	ADJ
ejde-673	707	17	c4	c4	NOUN
ejde-673	707	18	.	.	PUNCT
ejde-673	708	1	then	then	ADV
ejde-673	708	2	from	from	ADP
ejde-673	708	3	lemma	lemma	PROPN
ejde-673	708	4	3.7	3.7	NUM
ejde-673	708	5	(	(	PUNCT
ejde-673	708	6	i	i	NOUN
ejde-673	708	7	)	)	PUNCT
ejde-673	708	8	,	,	PUNCT
ejde-673	708	9	φ(u(n	φ(u(n	PROPN
ejde-673	708	10	,	,	PUNCT
ejde-673	708	11	α	α	NOUN
ejde-673	708	12	)	)	PUNCT
ejde-673	708	13	)	)	PUNCT
ejde-673	708	14	≤	≤	NUM
ejde-673	708	15	c(2∥h0∥lp′(·)(ω)∥u(n	c(2∥h0∥lp′(·)(ω)∥u(n	NOUN
ejde-673	708	16	,	,	PUNCT
ejde-673	708	17	α)∥y	α)∥y	X
ejde-673	708	18	+	+	CCONJ
ejde-673	708	19	ρ̃(p(·),h1(·))(u(n	ρ̃(p(·),h1(·))(u(n	PROPN
ejde-673	708	20	,	,	PUNCT
ejde-673	708	21	α	α	NOUN
ejde-673	708	22	)	)	PUNCT
ejde-673	708	23	)	)	PUNCT
ejde-673	708	24	)	)	PUNCT
ejde-673	708	25	≤	≤	PUNCT
ejde-673	708	26	c5	c5	PROPN
ejde-673	708	27	for	for	ADP
ejde-673	708	28	some	some	DET
ejde-673	708	29	constant	constant	ADJ
ejde-673	708	30	c5	c5	PROPN
ejde-673	708	31	>	>	X
ejde-673	708	32	0	0	X
ejde-673	708	33	.	.	PUNCT
ejde-673	709	1	since	since	SCONJ
ejde-673	709	2	m̂	m̂	PROPN
ejde-673	709	3	is	be	AUX
ejde-673	709	4	bounded	bound	VERB
ejde-673	709	5	for	for	ADP
ejde-673	709	6	every	every	DET
ejde-673	709	7	bounded	bounded	NOUN
ejde-673	709	8	subset	subset	NOUN
ejde-673	709	9	from	from	ADP
ejde-673	709	10	(	(	PUNCT
ejde-673	709	11	a1	a1	NOUN
ejde-673	709	12	)	)	PUNCT
ejde-673	709	13	,	,	PUNCT
ejde-673	709	14	we	we	PRON
ejde-673	709	15	see	see	VERB
ejde-673	709	16	that	that	SCONJ
ejde-673	709	17	c(n	c(n	PROPN
ejde-673	709	18	,	,	PUNCT
ejde-673	709	19	α	α	NOUN
ejde-673	709	20	)	)	PUNCT
ejde-673	709	21	=	=	SYM
ejde-673	709	22	ψ(u(n	ψ(u(n	PROPN
ejde-673	709	23	,	,	PUNCT
ejde-673	709	24	α	α	NOUN
ejde-673	709	25	)	)	PUNCT
ejde-673	709	26	)	)	PUNCT
ejde-673	709	27	=	=	PUNCT
ejde-673	710	1	m̂(φ(u(n	m̂(φ(u(n	NOUN
ejde-673	710	2	,	,	PUNCT
ejde-673	710	3	α	α	NOUN
ejde-673	710	4	)	)	PUNCT
ejde-673	710	5	)	)	PUNCT
ejde-673	710	6	)	)	PUNCT
ejde-673	710	7	is	be	AUX
ejde-673	710	8	bounded	bound	VERB
ejde-673	710	9	from	from	ADP
ejde-673	710	10	above	above	ADV
ejde-673	710	11	.	.	PUNCT
ejde-673	711	1	this	this	PRON
ejde-673	711	2	contradicts	contradict	VERB
ejde-673	711	3	proposition	proposition	NOUN
ejde-673	711	4	3.19	3.19	NUM
ejde-673	711	5	.	.	PUNCT
ejde-673	712	1	□	□	PUNCT
ejde-673	712	2	remark	remark	NOUN
ejde-673	712	3	3.21	3.21	NUM
ejde-673	712	4	.	.	PUNCT
ejde-673	713	1	we	we	PRON
ejde-673	713	2	do	do	AUX
ejde-673	713	3	not	not	PART
ejde-673	713	4	know	know	VERB
ejde-673	713	5	whether	whether	SCONJ
ejde-673	713	6	problem	problem	NOUN
ejde-673	713	7	(	(	PUNCT
ejde-673	713	8	1.1	1.1	NUM
ejde-673	713	9	)	)	PUNCT
ejde-673	713	10	only	only	ADV
ejde-673	713	11	has	have	AUX
ejde-673	713	12	eigenvalue	eigenvalue	VERB
ejde-673	713	13	sequences	sequence	NOUN
ejde-673	713	14	of	of	ADP
ejde-673	713	15	the	the	DET
ejde-673	713	16	form	form	NOUN
ejde-673	713	17	{	{	PUNCT
ejde-673	713	18	λ(n	λ(n	PROPN
ejde-673	713	19	,	,	PUNCT
ejde-673	713	20	α	α	NOUN
ejde-673	713	21	)	)	PUNCT
ejde-673	713	22	}	}	PUNCT
ejde-673	713	23	.	.	PUNCT
ejde-673	714	1	remark	remark	PROPN
ejde-673	714	2	3.22	3.22	NUM
ejde-673	714	3	.	.	PUNCT
ejde-673	715	1	we	we	PRON
ejde-673	715	2	assume	assume	VERB
ejde-673	715	3	the	the	DET
ejde-673	715	4	following	follow	VERB
ejde-673	715	5	more	more	ADV
ejde-673	715	6	restrictive	restrictive	ADJ
ejde-673	715	7	conditions	condition	NOUN
ejde-673	715	8	instead	instead	ADV
ejde-673	715	9	of	of	ADP
ejde-673	715	10	(	(	PUNCT
ejde-673	715	11	a1	a1	NOUN
ejde-673	715	12	)	)	PUNCT
ejde-673	715	13	and	and	CCONJ
ejde-673	715	14	(	(	PUNCT
ejde-673	715	15	a5	a5	PROPN
ejde-673	715	16	):	):	PUNCT
ejde-673	715	17	(	(	PUNCT
ejde-673	715	18	a1	a1	PROPN
ejde-673	715	19	’	'	PUNCT
ejde-673	715	20	)	)	PUNCT
ejde-673	716	1	m	m	VERB
ejde-673	716	2	:	:	PUNCT
ejde-673	717	1	[	[	X
ejde-673	717	2	0,∞	0,∞	NOUN
ejde-673	717	3	)	)	PUNCT
ejde-673	717	4	→	→	PUNCT
ejde-673	718	1	[	[	X
ejde-673	718	2	0,∞	0,∞	NUM
ejde-673	718	3	)	)	PUNCT
ejde-673	718	4	is	be	AUX
ejde-673	718	5	continuous	continuous	ADJ
ejde-673	718	6	and	and	CCONJ
ejde-673	718	7	monotone	monotone	ADJ
ejde-673	718	8	non	non	ADJ
ejde-673	718	9	-	-	ADJ
ejde-673	718	10	decreasing	decrease	VERB
ejde-673	718	11	,	,	PUNCT
ejde-673	718	12	and	and	CCONJ
ejde-673	718	13	there	there	PRON
ejde-673	718	14	exist	exist	VERB
ejde-673	718	15	0	0	NUM
ejde-673	718	16	<	<	X
ejde-673	718	17	m0	m0	PROPN
ejde-673	718	18	≤	≤	PROPN
ejde-673	718	19	m1	m1	PROPN
ejde-673	718	20	<	<	X
ejde-673	718	21	∞	∞	PROPN
ejde-673	718	22	and	and	CCONJ
ejde-673	718	23	l	l	NOUN
ejde-673	718	24	≥	≥	NUM
ejde-673	718	25	1	1	NUM
ejde-673	718	26	such	such	ADJ
ejde-673	718	27	that	that	DET
ejde-673	718	28	m0s	m0s	PROPN
ejde-673	718	29	l−1	l−1	PROPN
ejde-673	718	30	≤	≤	NUM
ejde-673	718	31	m(s	m(s	PROPN
ejde-673	718	32	)	)	PUNCT
ejde-673	718	33	≤	≤	NOUN
ejde-673	718	34	m1s	m1s	PROPN
ejde-673	718	35	l−1	l−1	PROPN
ejde-673	718	36	for	for	ADP
ejde-673	718	37	s	s	PRON
ejde-673	718	38	≥	≥	NOUN
ejde-673	718	39	0	0	NUM
ejde-673	718	40	.	.	X
ejde-673	718	41	24	24	NUM
ejde-673	718	42	j.	j.	PROPN
ejde-673	718	43	aramaki	aramaki	PROPN
ejde-673	718	44	ejde-2025/17	ejde-2025/17	X
ejde-673	718	45	(	(	PUNCT
ejde-673	718	46	a5	a5	PROPN
ejde-673	718	47	’	'	PUNCT
ejde-673	718	48	)	)	PUNCT
ejde-673	719	1	k0h1(x)|ξ|p(x	k0h1(x)|ξ|p(x	PROPN
ejde-673	719	2	)	)	PUNCT
ejde-673	719	3	≤	≤	PUNCT
ejde-673	719	4	a(x	a(x	PROPN
ejde-673	719	5	,	,	PUNCT
ejde-673	719	6	ξ	ξ	NOUN
ejde-673	719	7	)	)	PUNCT
ejde-673	719	8	·	·	PUNCT
ejde-673	720	1	ξ	ξ	X
ejde-673	720	2	=	=	SYM
ejde-673	720	3	p(x)a(x	p(x)a(x	NOUN
ejde-673	720	4	,	,	PUNCT
ejde-673	720	5	ξ	ξ	NOUN
ejde-673	720	6	)	)	PUNCT
ejde-673	720	7	for	for	ADP
ejde-673	720	8	a.e	a.e	PROPN
ejde-673	720	9	.	.	PUNCT
ejde-673	720	10	x	x	PUNCT
ejde-673	720	11	∈	∈	PROPN
ejde-673	720	12	ω	ω	NOUN
ejde-673	720	13	and	and	CCONJ
ejde-673	720	14	all	all	DET
ejde-673	720	15	ξ	ξ	PROPN
ejde-673	720	16	∈	∈	PROPN
ejde-673	720	17	rn	rn	PROPN
ejde-673	720	18	.	.	PUNCT
ejde-673	721	1	we	we	PRON
ejde-673	721	2	note	note	VERB
ejde-673	721	3	that	that	SCONJ
ejde-673	721	4	(	(	PUNCT
ejde-673	721	5	i	i	NOUN
ejde-673	721	6	)	)	PUNCT
ejde-673	721	7	in	in	ADP
ejde-673	721	8	example	example	NOUN
ejde-673	721	9	3.2	3.2	NUM
ejde-673	721	10	satisfies	satisfie	NOUN
ejde-673	721	11	(	(	PUNCT
ejde-673	721	12	a5	a5	PROPN
ejde-673	721	13	’	'	PUNCT
ejde-673	721	14	)	)	PUNCT
ejde-673	721	15	,	,	PUNCT
ejde-673	721	16	but	but	CCONJ
ejde-673	721	17	(	(	PUNCT
ejde-673	721	18	ii	ii	NOUN
ejde-673	721	19	)	)	PUNCT
ejde-673	721	20	does	do	AUX
ejde-673	721	21	not	not	PART
ejde-673	721	22	satisfy	satisfy	VERB
ejde-673	721	23	this	this	DET
ejde-673	721	24	condition	condition	NOUN
ejde-673	721	25	.	.	PUNCT
ejde-673	722	1	under	under	ADP
ejde-673	722	2	assumptions	assumption	NOUN
ejde-673	722	3	(	(	PUNCT
ejde-673	722	4	a1)–(a4	a1)–(a4	NOUN
ejde-673	722	5	)	)	PUNCT
ejde-673	722	6	,	,	PUNCT
ejde-673	722	7	(	(	PUNCT
ejde-673	722	8	a6)–(a8	a6)–(a8	ADV
ejde-673	722	9	)	)	PUNCT
ejde-673	722	10	,	,	PUNCT
ejde-673	722	11	(	(	PUNCT
ejde-673	722	12	a1	a1	PROPN
ejde-673	722	13	’	'	PUNCT
ejde-673	722	14	)	)	PUNCT
ejde-673	722	15	,	,	PUNCT
ejde-673	722	16	and	and	CCONJ
ejde-673	722	17	(	(	PUNCT
ejde-673	722	18	a5	a5	PROPN
ejde-673	722	19	’	'	PUNCT
ejde-673	722	20	)	)	PUNCT
ejde-673	722	21	,	,	PUNCT
ejde-673	722	22	we	we	PRON
ejde-673	722	23	have	have	VERB
ejde-673	722	24	λ(n+1,α	λ(n+1,α	PROPN
ejde-673	722	25	)	)	PUNCT
ejde-673	722	26	≥	≥	PROPN
ejde-673	722	27	p−r−m2	p−r−m2	NOUN
ejde-673	722	28	0	0	NUM
ejde-673	722	29	p+r+m2	p+r+m2	NOUN
ejde-673	722	30	1	1	NUM
ejde-673	722	31	λ(n	λ(n	PROPN
ejde-673	722	32	,	,	PUNCT
ejde-673	722	33	α	α	NOUN
ejde-673	722	34	)	)	PUNCT
ejde-673	722	35	.	.	PUNCT
ejde-673	723	1	(	(	PUNCT
ejde-673	723	2	3.25	3.25	NUM
ejde-673	723	3	)	)	PUNCT
ejde-673	723	4	in	in	ADP
ejde-673	723	5	particular	particular	ADJ
ejde-673	723	6	,	,	PUNCT
ejde-673	723	7	if	if	SCONJ
ejde-673	723	8	p(x	p(x	VERB
ejde-673	723	9	)	)	PUNCT
ejde-673	723	10	=	=	SYM
ejde-673	724	1	p	p	X
ejde-673	724	2	(	(	PUNCT
ejde-673	724	3	a	a	DET
ejde-673	724	4	constant	constant	ADJ
ejde-673	724	5	)	)	PUNCT
ejde-673	724	6	,	,	PUNCT
ejde-673	724	7	r(x	r(x	PROPN
ejde-673	724	8	)	)	PUNCT
ejde-673	725	1	=	=	SYM
ejde-673	725	2	r	r	NOUN
ejde-673	725	3	(	(	PUNCT
ejde-673	725	4	a	a	DET
ejde-673	725	5	constant	constant	ADJ
ejde-673	725	6	)	)	PUNCT
ejde-673	725	7	and	and	CCONJ
ejde-673	725	8	m0	m0	PROPN
ejde-673	725	9	=	=	SYM
ejde-673	725	10	m1	m1	PROPN
ejde-673	725	11	,	,	PUNCT
ejde-673	725	12	then	then	ADV
ejde-673	725	13	we	we	PRON
ejde-673	725	14	have	have	VERB
ejde-673	725	15	λ(n+1,α	λ(n+1,α	PROPN
ejde-673	725	16	)	)	PUNCT
ejde-673	725	17	≥	≥	NOUN
ejde-673	725	18	λ(n	λ(n	NOUN
ejde-673	725	19	,	,	PUNCT
ejde-673	725	20	α	α	NOUN
ejde-673	725	21	)	)	PUNCT
ejde-673	725	22	.	.	PUNCT
ejde-673	726	1	proof	proof	NOUN
ejde-673	726	2	.	.	PUNCT
ejde-673	727	1	let	let	VERB
ejde-673	727	2	un	un	PROPN
ejde-673	727	3	be	be	AUX
ejde-673	727	4	the	the	DET
ejde-673	727	5	eigenfuntion	eigenfuntion	NOUN
ejde-673	727	6	associated	associate	VERB
ejde-673	727	7	with	with	ADP
ejde-673	727	8	the	the	DET
ejde-673	727	9	eigenvalue	eigenvalue	ADJ
ejde-673	727	10	λ(n	λ(n	PROPN
ejde-673	727	11	,	,	PUNCT
ejde-673	727	12	α	α	NOUN
ejde-673	727	13	)	)	PUNCT
ejde-673	727	14	for	for	ADP
ejde-673	727	15	n	n	NOUN
ejde-673	727	16	=	=	SYM
ejde-673	727	17	1	1	NUM
ejde-673	727	18	,	,	PUNCT
ejde-673	727	19	2	2	NUM
ejde-673	727	20	,	,	PUNCT
ejde-673	727	21	.	.	PUNCT
ejde-673	727	22	.	.	PUNCT
ejde-673	728	1	..	..	PUNCT
ejde-673	728	2	from	from	ADP
ejde-673	728	3	assumption(a5	assumption(a5	ADV
ejde-673	728	4	’	'	PUNCT
ejde-673	728	5	)	)	PUNCT
ejde-673	728	6	,	,	PUNCT
ejde-673	728	7	(	(	PUNCT
ejde-673	728	8	a8	a8	PROPN
ejde-673	728	9	)	)	PUNCT
ejde-673	728	10	and	and	CCONJ
ejde-673	728	11	theorem	theorem	VERB
ejde-673	728	12	3.20	3.20	NUM
ejde-673	728	13	,	,	PUNCT
ejde-673	728	14	we	we	PRON
ejde-673	728	15	have	have	VERB
ejde-673	728	16	λ(n+1,α	λ(n+1,α	PRON
ejde-673	728	17	)	)	PUNCT
ejde-673	728	18	=	=	PUNCT
ejde-673	729	1	⟨ψ′(un+1	⟨ψ′(un+1	X
ejde-673	729	2	)	)	PUNCT
ejde-673	729	3	,	,	PUNCT
ejde-673	729	4	un+1⟩y	un+1⟩y	VERB
ejde-673	729	5	∗,y	∗,y	PROPN
ejde-673	729	6	⟨k	⟨k	PROPN
ejde-673	729	7	′(un+1	′(un+1	PROPN
ejde-673	729	8	)	)	PUNCT
ejde-673	729	9	,	,	PUNCT
ejde-673	729	10	un+1⟩y	un+1⟩y	VERB
ejde-673	729	11	∗,y	∗,y	PROPN
ejde-673	729	12	=	=	SYM
ejde-673	729	13	m(φ(un+1))⟨φ′(un+1	m(φ(un+1))⟨φ′(un+1	PROPN
ejde-673	729	14	)	)	PUNCT
ejde-673	729	15	,	,	PUNCT
ejde-673	729	16	un+1⟩y	un+1⟩y	VERB
ejde-673	729	17	∗,y∫	∗,y∫	PROPN
ejde-673	729	18	γ2	γ2	PROPN
ejde-673	729	19	k	k	PROPN
ejde-673	729	20	′(un+1	′(un+1	PROPN
ejde-673	729	21	)	)	PUNCT
ejde-673	729	22	,	,	PUNCT
ejde-673	729	23	un+1⟩y	un+1⟩y	VERB
ejde-673	729	24	∗,y	∗,y	PROPN
ejde-673	729	25	=	=	SYM
ejde-673	729	26	m(φ(un+1	m(φ(un+1	PROPN
ejde-673	729	27	)	)	PUNCT
ejde-673	729	28	)	)	PUNCT
ejde-673	730	1	∫	∫	PROPN
ejde-673	730	2	ω	ω	PROPN
ejde-673	730	3	a(x,∇un+1(x	a(x,∇un+1(x	PROPN
ejde-673	730	4	)	)	PUNCT
ejde-673	730	5	)	)	PUNCT
ejde-673	730	6	·	·	PUNCT
ejde-673	731	1	∇un+1(x	∇un+1(x	NOUN
ejde-673	731	2	)	)	PUNCT
ejde-673	731	3	dx∫	dx∫	PROPN
ejde-673	731	4	γ2	γ2	PROPN
ejde-673	731	5	g(x	g(x	PROPN
ejde-673	731	6	,	,	PUNCT
ejde-673	731	7	un+1(x))un+1(x)dσx	un+1(x))un+1(x)dσx	PROPN
ejde-673	731	8	≥	≥	NOUN
ejde-673	731	9	m0φ(un+1	m0φ(un+1	NOUN
ejde-673	731	10	)	)	PUNCT
ejde-673	732	1	l−1	l−1	PROPN
ejde-673	732	2	∫	∫	PROPN
ejde-673	732	3	ω	ω	PROPN
ejde-673	732	4	p(x)a(x,∇un+1(x	p(x)a(x,∇un+1(x	NOUN
ejde-673	732	5	)	)	PUNCT
ejde-673	732	6	)	)	PUNCT
ejde-673	733	1	dx∫	dx∫	PROPN
ejde-673	733	2	γ2	γ2	PROPN
ejde-673	733	3	r(x)g(x	r(x)g(x	ADJ
ejde-673	733	4	,	,	PUNCT
ejde-673	733	5	un+1(x))dσx	un+1(x))dσx	VERB
ejde-673	733	6	≥	≥	PRON
ejde-673	733	7	m0p	m0p	PROPN
ejde-673	733	8	−	−	PROPN
ejde-673	733	9	r+α	r+α	NUM
ejde-673	733	10	φ(un+1	φ(un+1	X
ejde-673	733	11	)	)	PUNCT
ejde-673	733	12	l	l	NOUN
ejde-673	733	13	≥	≥	NOUN
ejde-673	733	14	m0p	m0p	PROPN
ejde-673	733	15	−	−	PROPN
ejde-673	733	16	r+α	r+α	NUM
ejde-673	733	17	l	l	PROPN
ejde-673	733	18	m1	m1	PROPN
ejde-673	733	19	m̂(φ(un+1	m̂(φ(un+1	PROPN
ejde-673	733	20	)	)	PUNCT
ejde-673	733	21	)	)	PUNCT
ejde-673	734	1	=	=	PUNCT
ejde-673	734	2	m0p	m0p	PROPN
ejde-673	734	3	−l	−l	PROPN
ejde-673	734	4	r+αm1	r+αm1	NUM
ejde-673	734	5	c(n+1,α	c(n+1,α	NOUN
ejde-673	734	6	)	)	PUNCT
ejde-673	734	7	≥	≥	X
ejde-673	734	8	m0lp	m0lp	X
ejde-673	734	9	−	−	PROPN
ejde-673	734	10	r+αm1	r+αm1	NUM
ejde-673	734	11	c(n	c(n	PROPN
ejde-673	734	12	,	,	PUNCT
ejde-673	734	13	α	α	NOUN
ejde-673	734	14	)	)	PUNCT
ejde-673	734	15	.	.	PUNCT
ejde-673	735	1	the	the	DET
ejde-673	735	2	last	last	ADJ
ejde-673	735	3	inequality	inequality	NOUN
ejde-673	735	4	follows	follow	VERB
ejde-673	735	5	from	from	ADP
ejde-673	735	6	theorem	theorem	ADJ
ejde-673	735	7	3.20	3.20	NUM
ejde-673	735	8	.	.	PUNCT
ejde-673	736	1	on	on	ADP
ejde-673	736	2	the	the	DET
ejde-673	736	3	other	other	ADJ
ejde-673	736	4	hand	hand	NOUN
ejde-673	736	5	,	,	PUNCT
ejde-673	736	6	from	from	ADP
ejde-673	736	7	(	(	PUNCT
ejde-673	736	8	a1	a1	PROPN
ejde-673	736	9	’	'	PUNCT
ejde-673	736	10	)	)	PUNCT
ejde-673	736	11	,	,	PUNCT
ejde-673	736	12	(	(	PUNCT
ejde-673	736	13	a5	a5	PROPN
ejde-673	736	14	’	'	PUNCT
ejde-673	736	15	)	)	PUNCT
ejde-673	736	16	and	and	CCONJ
ejde-673	736	17	(	(	PUNCT
ejde-673	736	18	a8	a8	PROPN
ejde-673	736	19	)	)	PUNCT
ejde-673	736	20	,	,	PUNCT
ejde-673	736	21	we	we	PRON
ejde-673	736	22	have	have	VERB
ejde-673	736	23	c(n	c(n	PROPN
ejde-673	736	24	,	,	PUNCT
ejde-673	736	25	α	α	NOUN
ejde-673	736	26	)	)	PUNCT
ejde-673	736	27	=	=	SYM
ejde-673	736	28	α	α	PROPN
ejde-673	736	29	ψ(un	ψ(un	PROPN
ejde-673	736	30	)	)	PUNCT
ejde-673	736	31	k(un	k(un	PROPN
ejde-673	736	32	)	)	PUNCT
ejde-673	736	33	=	=	PUNCT
ejde-673	737	1	α	α	NUM
ejde-673	737	2	m̂(φ(un))∫	m̂(φ(un))∫	NOUN
ejde-673	737	3	γ2	γ2	PROPN
ejde-673	737	4	g(x	g(x	PROPN
ejde-673	737	5	,	,	PUNCT
ejde-673	737	6	un(x))dσx	un(x))dσx	PROPN
ejde-673	737	7	≥	≥	NOUN
ejde-673	737	8	α	α	NUM
ejde-673	737	9	m0	m0	PROPN
ejde-673	737	10	l	l	X
ejde-673	737	11	φ(un	φ(un	PROPN
ejde-673	737	12	)	)	PUNCT
ejde-673	737	13	l∫	l∫	ADJ
ejde-673	737	14	γ2	γ2	NOUN
ejde-673	737	15	1	1	NUM
ejde-673	737	16	r(x)g(x	r(x)g(x	X
ejde-673	737	17	,	,	PUNCT
ejde-673	737	18	un(x))un(x)dσx	un(x))un(x)dσx	VERB
ejde-673	737	19	≥	≥	NOUN
ejde-673	737	20	αm0	αm0	NOUN
ejde-673	737	21	l	l	NOUN
ejde-673	737	22	φ(un	φ(un	PROPN
ejde-673	737	23	)	)	PUNCT
ejde-673	737	24	l−1φ(un	l−1φ(un	NOUN
ejde-673	737	25	)	)	PUNCT
ejde-673	737	26	1	1	NUM
ejde-673	737	27	r−	r−	PROPN
ejde-673	737	28	⟨k	⟨k	PROPN
ejde-673	737	29	′(un	′(un	PROPN
ejde-673	737	30	)	)	PUNCT
ejde-673	737	31	,	,	PUNCT
ejde-673	737	32	un⟩y	un⟩y	PROPN
ejde-673	737	33	∗,y	∗,y	PROPN
ejde-673	737	34	≥	≥	NUM
ejde-673	737	35	αm0	αm0	NOUN
ejde-673	737	36	l	l	NOUN
ejde-673	737	37	1	1	NUM
ejde-673	737	38	m1	m1	PROPN
ejde-673	737	39	m(φ(un	m(φ(un	NOUN
ejde-673	737	40	)	)	PUNCT
ejde-673	737	41	)	)	PUNCT
ejde-673	737	42	∫	∫	PROPN
ejde-673	738	1	ω	ω	NUM
ejde-673	738	2	1	1	NUM
ejde-673	738	3	p(x)a(x,∇un(x	p(x)a(x,∇un(x	PROPN
ejde-673	738	4	)	)	PUNCT
ejde-673	738	5	)	)	PUNCT
ejde-673	738	6	·	·	PUNCT
ejde-673	738	7	∇un(x	∇un(x	X
ejde-673	738	8	)	)	PUNCT
ejde-673	738	9	dx	dx	PROPN
ejde-673	738	10	1	1	NUM
ejde-673	738	11	r−	r−	PROPN
ejde-673	738	12	⟨k	⟨k	PROPN
ejde-673	738	13	′(un	′(un	PROPN
ejde-673	738	14	)	)	PUNCT
ejde-673	738	15	,	,	PUNCT
ejde-673	738	16	un⟩y	un⟩y	PROPN
ejde-673	738	17	∗,y	∗,y	PROPN
ejde-673	738	18	≥	≥	NUM
ejde-673	738	19	αm0	αm0	NOUN
ejde-673	738	20	l	l	NOUN
ejde-673	738	21	1	1	NUM
ejde-673	738	22	m1	m1	PROPN
ejde-673	738	23	1	1	NUM
ejde-673	738	24	p+m(φ(un))⟨φ′(un	p+m(φ(un))⟨φ′(un	NOUN
ejde-673	738	25	)	)	PUNCT
ejde-673	738	26	,	,	PUNCT
ejde-673	739	1	un⟩y	un⟩y	PROPN
ejde-673	739	2	∗,y	∗,y	NUM
ejde-673	739	3	1	1	NUM
ejde-673	739	4	r−	r−	PROPN
ejde-673	739	5	⟨k	⟨k	PROPN
ejde-673	739	6	′(un	′(un	PROPN
ejde-673	739	7	)	)	PUNCT
ejde-673	739	8	,	,	PUNCT
ejde-673	739	9	un⟩y	un⟩y	PROPN
ejde-673	739	10	∗,y	∗,y	PROPN
ejde-673	739	11	=	=	SYM
ejde-673	739	12	αm0r	αm0r	PROPN
ejde-673	739	13	−	−	PROPN
ejde-673	739	14	lm1p+	lm1p+	NUM
ejde-673	739	15	⟨ψ′(un	⟨ψ′(un	PROPN
ejde-673	739	16	)	)	PUNCT
ejde-673	739	17	,	,	PUNCT
ejde-673	739	18	un⟩y	un⟩y	PROPN
ejde-673	739	19	∗,y	∗,y	PROPN
ejde-673	739	20	⟨k	⟨k	PROPN
ejde-673	739	21	′(un	′(un	PROPN
ejde-673	739	22	)	)	PUNCT
ejde-673	739	23	,	,	PUNCT
ejde-673	739	24	un⟩y	un⟩y	PROPN
ejde-673	739	25	∗,y	∗,y	PROPN
ejde-673	739	26	=	=	SYM
ejde-673	739	27	αm0r	αm0r	PROPN
ejde-673	739	28	−	−	NOUN
ejde-673	739	29	lm1p+	lm1p+	X
ejde-673	739	30	λ(n	λ(n	PROPN
ejde-673	739	31	,	,	PUNCT
ejde-673	739	32	α	α	NOUN
ejde-673	739	33	)	)	PUNCT
ejde-673	739	34	.	.	PUNCT
ejde-673	740	1	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	740	2	eigenvalue	eigenvalue	VERB
ejde-673	740	3	problems	problem	NOUN
ejde-673	740	4	for	for	ADP
ejde-673	740	5	kirchhoff	kirchhoff	NOUN
ejde-673	740	6	-	-	PUNCT
ejde-673	740	7	type	type	NOUN
ejde-673	740	8	equations	equation	NOUN
ejde-673	740	9	25	25	NUM
ejde-673	740	10	thus	thus	ADV
ejde-673	740	11	we	we	PRON
ejde-673	740	12	obtain	obtain	VERB
ejde-673	740	13	the	the	DET
ejde-673	740	14	estimate	estimate	NOUN
ejde-673	740	15	(	(	PUNCT
ejde-673	740	16	3.25	3.25	NUM
ejde-673	740	17	)	)	PUNCT
ejde-673	740	18	.	.	PUNCT
ejde-673	741	1	□	□	PUNCT
ejde-673	741	2	4	4	X
ejde-673	741	3	.	.	PUNCT
ejde-673	742	1	the	the	DET
ejde-673	742	2	infimum	infimum	NOUN
ejde-673	742	3	of	of	ADP
ejde-673	742	4	all	all	DET
ejde-673	742	5	the	the	DET
ejde-673	742	6	eigenvalues	eigenvalue	NOUN
ejde-673	742	7	in	in	ADP
ejde-673	742	8	this	this	DET
ejde-673	742	9	section	section	NOUN
ejde-673	742	10	,	,	PUNCT
ejde-673	742	11	we	we	PRON
ejde-673	742	12	consider	consider	VERB
ejde-673	742	13	the	the	DET
ejde-673	742	14	infimum	infimum	NOUN
ejde-673	742	15	of	of	ADP
ejde-673	742	16	all	all	DET
ejde-673	742	17	the	the	DET
ejde-673	742	18	eigenvalues	eigenvalue	NOUN
ejde-673	742	19	of	of	ADP
ejde-673	742	20	the	the	DET
ejde-673	742	21	problem	problem	NOUN
ejde-673	742	22	(	(	PUNCT
ejde-673	742	23	1.1	1.1	NUM
ejde-673	742	24	)	)	PUNCT
ejde-673	742	25	.	.	PUNCT
ejde-673	743	1	we	we	PRON
ejde-673	743	2	show	show	VERB
ejde-673	743	3	that	that	SCONJ
ejde-673	743	4	there	there	PRON
ejde-673	743	5	exist	exist	VERB
ejde-673	743	6	two	two	NUM
ejde-673	743	7	cases	case	NOUN
ejde-673	743	8	where	where	SCONJ
ejde-673	743	9	the	the	DET
ejde-673	743	10	infimum	infimum	NOUN
ejde-673	743	11	is	be	AUX
ejde-673	743	12	equal	equal	ADJ
ejde-673	743	13	to	to	ADP
ejde-673	743	14	zero	zero	NUM
ejde-673	743	15	,	,	PUNCT
ejde-673	743	16	and	and	CCONJ
ejde-673	743	17	positive	positive	ADJ
ejde-673	743	18	according	accord	VERB
ejde-673	743	19	to	to	ADP
ejde-673	743	20	the	the	DET
ejde-673	743	21	hypoetheses	hypoethese	NOUN
ejde-673	743	22	on	on	ADP
ejde-673	743	23	the	the	DET
ejde-673	743	24	variable	variable	ADJ
ejde-673	743	25	exponent	exponent	NOUN
ejde-673	743	26	.	.	PUNCT
ejde-673	744	1	put	put	VERB
ejde-673	744	2	λ	λ	NOUN
ejde-673	744	3	=	=	PRON
ejde-673	744	4	{	{	PUNCT
ejde-673	744	5	λ	λ	PROPN
ejde-673	744	6	is	be	AUX
ejde-673	744	7	an	an	DET
ejde-673	744	8	eigenvalue	eigenvalue	NOUN
ejde-673	744	9	of	of	ADP
ejde-673	744	10	problem	problem	NOUN
ejde-673	744	11	(	(	PUNCT
ejde-673	744	12	1.1	1.1	NUM
ejde-673	744	13	)	)	PUNCT
ejde-673	744	14	}	}	PUNCT
ejde-673	744	15	and	and	CCONJ
ejde-673	744	16	λ∗	λ∗	X
ejde-673	744	17	=	=	SYM
ejde-673	744	18	inf	inf	PROPN
ejde-673	744	19	λ	λ	PROPN
ejde-673	744	20	.	.	PROPN
ejde-673	744	21	for	for	ADP
ejde-673	744	22	a	a	DET
ejde-673	744	23	subset	subset	NOUN
ejde-673	744	24	a	a	DET
ejde-673	744	25	⊂	⊂	PROPN
ejde-673	744	26	ω	ω	PROPN
ejde-673	744	27	and	and	CCONJ
ejde-673	744	28	δ	δ	PROPN
ejde-673	744	29	>	>	X
ejde-673	744	30	0	0	PROPN
ejde-673	744	31	,	,	PUNCT
ejde-673	744	32	put	put	VERB
ejde-673	744	33	b(a	b(a	NOUN
ejde-673	744	34	,	,	PUNCT
ejde-673	744	35	δ	δ	PROPN
ejde-673	744	36	)	)	PUNCT
ejde-673	744	37	=	=	PRON
ejde-673	745	1	{	{	PUNCT
ejde-673	745	2	x	x	PUNCT
ejde-673	745	3	∈	∈	PROPN
ejde-673	745	4	rn	rn	PROPN
ejde-673	745	5	;	;	PUNCT
ejde-673	746	1	dist(x	dist(x	PROPN
ejde-673	746	2	,	,	PUNCT
ejde-673	746	3	a	a	PRON
ejde-673	746	4	)	)	PUNCT
ejde-673	746	5	<	<	X
ejde-673	746	6	δ	δ	X
ejde-673	746	7	}	}	PUNCT
ejde-673	746	8	,	,	PUNCT
ejde-673	746	9	bω(a	bω(a	X
ejde-673	746	10	,	,	PUNCT
ejde-673	746	11	δ	δ	PROPN
ejde-673	746	12	)	)	PUNCT
ejde-673	746	13	=	=	SYM
ejde-673	747	1	b(a	b(a	X
ejde-673	747	2	,	,	PUNCT
ejde-673	747	3	δ	δ	PROPN
ejde-673	747	4	)	)	PUNCT
ejde-673	747	5	∩	∩	PROPN
ejde-673	747	6	ω	ω	PROPN
ejde-673	747	7	,	,	PUNCT
ejde-673	747	8	bγ2	bγ2	PROPN
ejde-673	747	9	(	(	PUNCT
ejde-673	747	10	a	a	PRON
ejde-673	747	11	,	,	PUNCT
ejde-673	747	12	δ	δ	PROPN
ejde-673	747	13	)	)	PUNCT
ejde-673	747	14	=	=	SYM
ejde-673	747	15	b(a	b(a	X
ejde-673	747	16	,	,	PUNCT
ejde-673	747	17	δ	δ	PROPN
ejde-673	747	18	)	)	PUNCT
ejde-673	747	19	∩	∩	ADJ
ejde-673	747	20	γ2	γ2	PROPN
ejde-673	747	21	.	.	PUNCT
ejde-673	748	1	here	here	ADV
ejde-673	748	2	,	,	PUNCT
ejde-673	748	3	for	for	ADP
ejde-673	748	4	x0	x0	PROPN
ejde-673	748	5	∈	∈	PROPN
ejde-673	748	6	ω	ω	PROPN
ejde-673	748	7	,	,	PUNCT
ejde-673	748	8	if	if	SCONJ
ejde-673	748	9	a	a	PRON
ejde-673	748	10	=	=	X
ejde-673	748	11	{	{	PUNCT
ejde-673	748	12	x0	x0	PROPN
ejde-673	748	13	}	}	PUNCT
ejde-673	748	14	,	,	PUNCT
ejde-673	748	15	then	then	ADV
ejde-673	748	16	we	we	PRON
ejde-673	748	17	simply	simply	ADV
ejde-673	748	18	write	write	VERB
ejde-673	748	19	b({x0	b({x0	NOUN
ejde-673	748	20	}	}	PUNCT
ejde-673	748	21	,	,	PUNCT
ejde-673	748	22	δ	δ	PROPN
ejde-673	748	23	)	)	PUNCT
ejde-673	748	24	,	,	PUNCT
ejde-673	748	25	bω({x0	bω({x0	PROPN
ejde-673	748	26	}	}	PUNCT
ejde-673	748	27	,	,	PUNCT
ejde-673	748	28	δ	δ	PROPN
ejde-673	748	29	)	)	PUNCT
ejde-673	748	30	and	and	CCONJ
ejde-673	748	31	bγ2({x0	bγ2({x0	NOUN
ejde-673	748	32	}	}	PUNCT
ejde-673	748	33	,	,	PUNCT
ejde-673	748	34	δ	δ	PROPN
ejde-673	748	35	)	)	PUNCT
ejde-673	748	36	by	by	ADP
ejde-673	748	37	b(x0	b(x0	NOUN
ejde-673	748	38	,	,	PUNCT
ejde-673	748	39	δ	δ	PROPN
ejde-673	748	40	)	)	PUNCT
ejde-673	748	41	,	,	PUNCT
ejde-673	748	42	bω(x0	bω(x0	PROPN
ejde-673	748	43	,	,	PUNCT
ejde-673	748	44	δ	δ	PROPN
ejde-673	748	45	)	)	PUNCT
ejde-673	748	46	and	and	CCONJ
ejde-673	748	47	bγ2(x0	bγ2(x0	PROPN
ejde-673	748	48	,	,	PUNCT
ejde-673	748	49	δ	δ	PROPN
ejde-673	748	50	)	)	PUNCT
ejde-673	748	51	,	,	PUNCT
ejde-673	748	52	respectively	respectively	ADV
ejde-673	748	53	.	.	PUNCT
ejde-673	749	1	assume	assume	VERB
ejde-673	749	2	that	that	SCONJ
ejde-673	749	3	(	(	PUNCT
ejde-673	749	4	a1)–(a8	a1)–(a8	ADJ
ejde-673	749	5	)	)	PUNCT
ejde-673	749	6	,	,	PUNCT
ejde-673	749	7	hold	hold	VERB
ejde-673	749	8	.	.	PUNCT
ejde-673	750	1	lemma	lemma	PROPN
ejde-673	750	2	4.1	4.1	NUM
ejde-673	750	3	.	.	PUNCT
ejde-673	751	1	for	for	ADP
ejde-673	751	2	δ	δ	PROPN
ejde-673	751	3	,	,	PUNCT
ejde-673	751	4	α	α	PROPN
ejde-673	751	5	>	>	X
ejde-673	751	6	0	0	NUM
ejde-673	751	7	,	,	PUNCT
ejde-673	751	8	if	if	SCONJ
ejde-673	751	9	we	we	PRON
ejde-673	751	10	define	define	VERB
ejde-673	751	11	βδ(u	βδ(u	PUNCT
ejde-673	751	12	)	)	PUNCT
ejde-673	751	13	=	=	SYM
ejde-673	752	1	∫	∫	PROPN
ejde-673	752	2	bω(γ2,δ	bω(γ2,δ	PROPN
ejde-673	752	3	)	)	PUNCT
ejde-673	752	4	h1(x)|∇u(x)|p(x	h1(x)|∇u(x)|p(x	PROPN
ejde-673	752	5	)	)	PUNCT
ejde-673	752	6	dx	dx	PROPN
ejde-673	752	7	for	for	ADP
ejde-673	752	8	u	u	PROPN
ejde-673	752	9	∈	∈	PROPN
ejde-673	752	10	y	y	PROPN
ejde-673	752	11	,	,	PUNCT
ejde-673	752	12	then	then	ADV
ejde-673	752	13	we	we	PRON
ejde-673	752	14	have	have	VERB
ejde-673	752	15	β(δ	β(δ	PROPN
ejde-673	752	16	,	,	PUNCT
ejde-673	752	17	α	α	NOUN
ejde-673	752	18	)	)	PUNCT
ejde-673	752	19	:	:	PUNCT
ejde-673	752	20	=	=	NUM
ejde-673	752	21	inf	inf	ADJ
ejde-673	752	22	u∈mα	u∈mα	NOUN
ejde-673	752	23	βδ(u	βδ(u	PUNCT
ejde-673	752	24	)	)	PUNCT
ejde-673	752	25	>	>	X
ejde-673	752	26	0	0	X
ejde-673	752	27	.	.	PUNCT
ejde-673	752	28	proof	proof	NOUN
ejde-673	752	29	.	.	PUNCT
ejde-673	753	1	first	first	ADV
ejde-673	753	2	we	we	PRON
ejde-673	753	3	consider	consider	VERB
ejde-673	753	4	y	y	PRON
ejde-673	753	5	(	(	PUNCT
ejde-673	753	6	bω(γ2	bω(γ2	NOUN
ejde-673	753	7	,	,	PUNCT
ejde-673	753	8	δ	δ	PROPN
ejde-673	753	9	)	)	PUNCT
ejde-673	753	10	)	)	PUNCT
ejde-673	753	11	.	.	PUNCT
ejde-673	754	1	we	we	PRON
ejde-673	754	2	extend	extend	VERB
ejde-673	754	3	the	the	DET
ejde-673	754	4	function	function	NOUN
ejde-673	754	5	b(x	b(x	VERB
ejde-673	754	6	)	)	PUNCT
ejde-673	754	7	on	on	ADP
ejde-673	754	8	γ2	γ2	PROPN
ejde-673	754	9	in	in	ADP
ejde-673	754	10	(	(	PUNCT
ejde-673	754	11	a8	a8	PROPN
ejde-673	754	12	)	)	PUNCT
ejde-673	754	13	to	to	ADP
ejde-673	754	14	a	a	DET
ejde-673	754	15	function	function	NOUN
ejde-673	754	16	b̃(x	b̃(x	NOUN
ejde-673	754	17	)	)	PUNCT
ejde-673	754	18	on	on	ADP
ejde-673	754	19	γ̃2	γ̃2	PROPN
ejde-673	754	20	,	,	PUNCT
ejde-673	754	21	where	where	SCONJ
ejde-673	754	22	γ̃2	γ̃2	PROPN
ejde-673	754	23	:	:	PUNCT
ejde-673	754	24	=	=	NOUN
ejde-673	754	25	γ2	γ2	ADJ
ejde-673	754	26	∪	∪	X
ejde-673	754	27	(	(	PUNCT
ejde-673	754	28	∂bω(γ2	∂bω(γ2	NOUN
ejde-673	754	29	,	,	PUNCT
ejde-673	754	30	δ	δ	PROPN
ejde-673	754	31	)	)	PUNCT
ejde-673	754	32	\	\	PROPN
ejde-673	754	33	γ	γ	PROPN
ejde-673	754	34	)	)	PUNCT
ejde-673	754	35	by	by	ADP
ejde-673	754	36	a	a	DET
ejde-673	754	37	positive	positive	ADJ
ejde-673	754	38	constant	constant	ADJ
ejde-673	754	39	outside	outside	ADP
ejde-673	754	40	∂bω(γ2	∂bω(γ2	NOUN
ejde-673	754	41	,	,	PUNCT
ejde-673	754	42	δ	δ	PROPN
ejde-673	754	43	)	)	PUNCT
ejde-673	754	44	\	\	PROPN
ejde-673	754	45	γ	γ	NOUN
ejde-673	754	46	,	,	PUNCT
ejde-673	754	47	and	and	CCONJ
ejde-673	754	48	define	define	VERB
ejde-673	754	49	g̃	g̃	PROPN
ejde-673	754	50	and	and	CCONJ
ejde-673	754	51	k̃	k̃	PROPN
ejde-673	754	52	as	as	ADP
ejde-673	754	53	in	in	ADP
ejde-673	754	54	(	(	PUNCT
ejde-673	754	55	3.10	3.10	NUM
ejde-673	754	56	)	)	PUNCT
ejde-673	754	57	and	and	CCONJ
ejde-673	754	58	(	(	PUNCT
ejde-673	754	59	3.12	3.12	NUM
ejde-673	754	60	)	)	PUNCT
ejde-673	754	61	,	,	PUNCT
ejde-673	754	62	respectively	respectively	ADV
ejde-673	754	63	.	.	PUNCT
ejde-673	755	1	since	since	SCONJ
ejde-673	755	2	δ	δ	PROPN
ejde-673	755	3	>	>	X
ejde-673	755	4	0	0	PROPN
ejde-673	755	5	,	,	PUNCT
ejde-673	755	6	we	we	PRON
ejde-673	755	7	have	have	VERB
ejde-673	755	8	γ̃1	γ̃1	NOUN
ejde-673	755	9	:	:	PUNCT
ejde-673	755	10	=	=	SYM
ejde-673	755	11	∂bω(γ2	∂bω(γ2	X
ejde-673	755	12	,	,	PUNCT
ejde-673	755	13	δ	δ	NOUN
ejde-673	755	14	)	)	PUNCT
ejde-673	755	15	∩	∩	PROPN
ejde-673	755	16	γ1	γ1	PROPN
ejde-673	755	17	̸=	̸=	PROPN
ejde-673	755	18	∅	∅	NOUN
ejde-673	755	19	,	,	PUNCT
ejde-673	755	20	so	so	SCONJ
ejde-673	755	21	y	y	PROPN
ejde-673	755	22	(	(	PUNCT
ejde-673	755	23	bω(γ2	bω(γ2	NOUN
ejde-673	755	24	,	,	PUNCT
ejde-673	755	25	δ	δ	NOUN
ejde-673	755	26	)	)	PUNCT
ejde-673	755	27	)	)	PUNCT
ejde-673	755	28	is	be	AUX
ejde-673	755	29	the	the	DET
ejde-673	755	30	same	same	ADJ
ejde-673	755	31	properties	property	NOUN
ejde-673	755	32	as	as	ADP
ejde-673	755	33	y	y	PROPN
ejde-673	755	34	,	,	PUNCT
ejde-673	755	35	if	if	SCONJ
ejde-673	755	36	we	we	PRON
ejde-673	755	37	replace	replace	VERB
ejde-673	755	38	γ1	γ1	NOUN
ejde-673	755	39	in	in	ADP
ejde-673	755	40	y	y	PROPN
ejde-673	755	41	with	with	ADP
ejde-673	755	42	γ̃1	γ̃1	PROPN
ejde-673	755	43	.	.	PUNCT
ejde-673	756	1	thus	thus	ADV
ejde-673	756	2	y	y	PROPN
ejde-673	756	3	↪	↪	PROPN
ejde-673	756	4	→	→	SYM
ejde-673	756	5	y	y	PROPN
ejde-673	756	6	(	(	PUNCT
ejde-673	756	7	bω(γ2	bω(γ2	NOUN
ejde-673	756	8	,	,	PUNCT
ejde-673	756	9	δ	δ	NOUN
ejde-673	756	10	)	)	PUNCT
ejde-673	756	11	)	)	PUNCT
ejde-673	756	12	and	and	CCONJ
ejde-673	756	13	βδ	βδ	PRON
ejde-673	756	14	is	be	AUX
ejde-673	756	15	a	a	DET
ejde-673	756	16	modular	modular	NOUN
ejde-673	756	17	on	on	ADP
ejde-673	756	18	y	y	PROPN
ejde-673	756	19	(	(	PUNCT
ejde-673	756	20	bω(γ2	bω(γ2	NOUN
ejde-673	756	21	,	,	PUNCT
ejde-673	756	22	δ	δ	PROPN
ejde-673	756	23	)	)	PUNCT
ejde-673	756	24	)	)	PUNCT
ejde-673	756	25	.	.	PUNCT
ejde-673	757	1	assume	assume	VERB
ejde-673	757	2	that	that	SCONJ
ejde-673	757	3	β(δ	β(δ	PROPN
ejde-673	757	4	,	,	PUNCT
ejde-673	757	5	α	α	X
ejde-673	757	6	)	)	PUNCT
ejde-673	757	7	=	=	SYM
ejde-673	757	8	0	0	X
ejde-673	757	9	.	.	PUNCT
ejde-673	758	1	then	then	ADV
ejde-673	758	2	there	there	PRON
ejde-673	758	3	exist	exist	VERB
ejde-673	758	4	{	{	PUNCT
ejde-673	758	5	un	un	PROPN
ejde-673	758	6	}	}	PUNCT
ejde-673	758	7	⊂	⊂	NOUN
ejde-673	758	8	mα	mα	ADP
ejde-673	758	9	such	such	ADJ
ejde-673	758	10	that	that	DET
ejde-673	758	11	βδ(un	βδ(un	PROPN
ejde-673	758	12	)	)	PUNCT
ejde-673	758	13	→	→	SYM
ejde-673	758	14	0	0	NUM
ejde-673	758	15	as	as	ADP
ejde-673	758	16	n	n	NOUN
ejde-673	758	17	→	→	SYM
ejde-673	758	18	∞.	∞.	PROPN
ejde-673	758	19	hence	hence	ADV
ejde-673	758	20	∥un∥y	∥un∥y	PROPN
ejde-673	758	21	(	(	PUNCT
ejde-673	758	22	bω(γ2,δ	bω(γ2,δ	PROPN
ejde-673	758	23	)	)	PUNCT
ejde-673	758	24	)	)	PUNCT
ejde-673	759	1	→	→	SYM
ejde-673	759	2	0	0	NUM
ejde-673	759	3	as	as	ADP
ejde-673	759	4	n	n	NUM
ejde-673	759	5	→	→	SYM
ejde-673	759	6	∞	∞	PROPN
ejde-673	759	7	,	,	PUNCT
ejde-673	759	8	where	where	SCONJ
ejde-673	759	9	∥u∥y	∥u∥y	ADJ
ejde-673	759	10	(	(	PUNCT
ejde-673	759	11	bω(γ2,δ	bω(γ2,δ	PROPN
ejde-673	759	12	)	)	PUNCT
ejde-673	759	13	)	)	PUNCT
ejde-673	760	1	=	=	PRON
ejde-673	760	2	inf	inf	NOUN
ejde-673	760	3	{	{	PUNCT
ejde-673	760	4	τ	τ	PROPN
ejde-673	760	5	>	>	X
ejde-673	760	6	0;βδ	0;βδ	PUNCT
ejde-673	761	1	(	(	PUNCT
ejde-673	761	2	u	u	NOUN
ejde-673	761	3	τ	τ	PROPN
ejde-673	761	4	)	)	PUNCT
ejde-673	761	5	≤	≤	NUM
ejde-673	761	6	1	1	NUM
ejde-673	761	7	}	}	PUNCT
ejde-673	761	8	.	.	PUNCT
ejde-673	762	1	on	on	ADP
ejde-673	762	2	the	the	DET
ejde-673	762	3	other	other	ADJ
ejde-673	762	4	hand	hand	NOUN
ejde-673	762	5	,	,	PUNCT
ejde-673	762	6	we	we	PRON
ejde-673	762	7	have	have	VERB
ejde-673	762	8	k̃(un	k̃(un	VERB
ejde-673	762	9	)	)	PUNCT
ejde-673	763	1	=	=	SYM
ejde-673	763	2	∫	∫	PROPN
ejde-673	763	3	∂bω(γ2,δ	∂bω(γ2,δ	PROPN
ejde-673	763	4	)	)	PUNCT
ejde-673	763	5	)	)	PUNCT
ejde-673	764	1	g̃(x	g̃(x	PROPN
ejde-673	764	2	,	,	PUNCT
ejde-673	764	3	un(x))dσx	un(x))dσx	PROPN
ejde-673	764	4	≥	≥	NOUN
ejde-673	764	5	∫	∫	PROPN
ejde-673	764	6	γ2	γ2	PROPN
ejde-673	764	7	g(x	g(x	PROPN
ejde-673	764	8	,	,	PUNCT
ejde-673	764	9	un(x))dσx	un(x))dσx	X
ejde-673	764	10	=	=	SYM
ejde-673	764	11	k(un	k(un	PROPN
ejde-673	764	12	)	)	PUNCT
ejde-673	764	13	=	=	SYM
ejde-673	764	14	α	α	X
ejde-673	764	15	>	>	X
ejde-673	764	16	0	0	PROPN
ejde-673	764	17	.	.	PUNCT
ejde-673	765	1	since	since	SCONJ
ejde-673	765	2	k̃	k̃	PROPN
ejde-673	765	3	is	be	AUX
ejde-673	765	4	continuous	continuous	ADJ
ejde-673	765	5	on	on	ADP
ejde-673	765	6	y	y	PROPN
ejde-673	765	7	(	(	PUNCT
ejde-673	765	8	bω(γ2	bω(γ2	NOUN
ejde-673	765	9	,	,	PUNCT
ejde-673	765	10	δ	δ	PROPN
ejde-673	765	11	)	)	PUNCT
ejde-673	765	12	)	)	PUNCT
ejde-673	765	13	,	,	PUNCT
ejde-673	765	14	we	we	PRON
ejde-673	765	15	can	can	AUX
ejde-673	765	16	see	see	VERB
ejde-673	765	17	that	that	DET
ejde-673	765	18	k̃(un	k̃(un	PROPN
ejde-673	765	19	)	)	PUNCT
ejde-673	765	20	→	→	SYM
ejde-673	765	21	k̃(0	k̃(0	X
ejde-673	765	22	)	)	PUNCT
ejde-673	765	23	=	=	SYM
ejde-673	766	1	0	0	X
ejde-673	766	2	.	.	PUNCT
ejde-673	767	1	this	this	PRON
ejde-673	767	2	is	be	AUX
ejde-673	767	3	a	a	DET
ejde-673	767	4	contradiction	contradiction	NOUN
ejde-673	767	5	.	.	PUNCT
ejde-673	768	1	□	□	PUNCT
ejde-673	768	2	lemma	lemma	PROPN
ejde-673	768	3	4.2	4.2	NUM
ejde-673	768	4	.	.	PUNCT
ejde-673	769	1	for	for	ADP
ejde-673	769	2	α	α	PROPN
ejde-673	769	3	>	>	X
ejde-673	769	4	0	0	NUM
ejde-673	769	5	,	,	PUNCT
ejde-673	769	6	let	let	VERB
ejde-673	769	7	u0	u0	ADJ
ejde-673	769	8	be	be	AUX
ejde-673	769	9	an	an	DET
ejde-673	769	10	eigenfunction	eigenfunction	NOUN
ejde-673	769	11	associated	associate	VERB
ejde-673	769	12	with	with	ADP
ejde-673	769	13	λ(1,α	λ(1,α	PROPN
ejde-673	769	14	)	)	PUNCT
ejde-673	769	15	.	.	PUNCT
ejde-673	770	1	then	then	ADV
ejde-673	770	2	ψ(u0	ψ(u0	NUM
ejde-673	770	3	)	)	PUNCT
ejde-673	770	4	=	=	SYM
ejde-673	770	5	c(1,α	c(1,α	PROPN
ejde-673	770	6	)	)	PUNCT
ejde-673	771	1	=	=	PUNCT
ejde-673	771	2	inf{ψ(u);u	inf{ψ(u);u	PROPN
ejde-673	771	3	∈	∈	PROPN
ejde-673	771	4	mα	mα	PROPN
ejde-673	771	5	}	}	PUNCT
ejde-673	771	6	.	.	PUNCT
ejde-673	772	1	proof	proof	NOUN
ejde-673	772	2	.	.	PUNCT
ejde-673	773	1	put	put	VERB
ejde-673	773	2	bα	bα	NOUN
ejde-673	773	3	=	=	PUNCT
ejde-673	773	4	inf{ψ(u);u	inf{ψ(u);u	PROPN
ejde-673	773	5	∈	∈	PROPN
ejde-673	773	6	mα	mα	PROPN
ejde-673	773	7	}	}	PUNCT
ejde-673	773	8	.	.	PUNCT
ejde-673	774	1	since	since	SCONJ
ejde-673	774	2	c(1,α	c(1,α	PROPN
ejde-673	774	3	)	)	PUNCT
ejde-673	774	4	=	=	SYM
ejde-673	774	5	infh∈σα	infh∈σα	PROPN
ejde-673	774	6	,	,	PUNCT
ejde-673	774	7	γ(h)≥1	γ(h)≥1	PROPN
ejde-673	774	8	supu∈h	supu∈h	PROPN
ejde-673	774	9	ψ̃(u	ψ̃(u	PROPN
ejde-673	774	10	)	)	PUNCT
ejde-673	774	11	,	,	PUNCT
ejde-673	774	12	if	if	SCONJ
ejde-673	774	13	u	u	PROPN
ejde-673	774	14	∈	∈	PROPN
ejde-673	774	15	h	h	NOUN
ejde-673	774	16	and	and	CCONJ
ejde-673	774	17	h	h	NOUN
ejde-673	774	18	∈	∈	PROPN
ejde-673	774	19	σα	σα	INTJ
ejde-673	774	20	⊂	⊂	X
ejde-673	774	21	mα	mα	PROPN
ejde-673	774	22	with	with	ADP
ejde-673	774	23	γ(h	γ(h	NOUN
ejde-673	774	24	)	)	PUNCT
ejde-673	774	25	≥	≥	NOUN
ejde-673	774	26	1	1	NUM
ejde-673	774	27	,	,	PUNCT
ejde-673	774	28	then	then	ADV
ejde-673	774	29	ψ̃(u	ψ̃(u	VERB
ejde-673	774	30	)	)	PUNCT
ejde-673	774	31	=	=	SYM
ejde-673	774	32	ψ(u	ψ(u	PROPN
ejde-673	774	33	)	)	PUNCT
ejde-673	774	34	≥	≥	NUM
ejde-673	774	35	bα	bα	NOUN
ejde-673	774	36	.	.	PUNCT
ejde-673	775	1	thus	thus	ADV
ejde-673	775	2	c(1,α	c(1,α	PROPN
ejde-673	775	3	)	)	PUNCT
ejde-673	775	4	≥	≥	AUX
ejde-673	775	5	bα	bα	PROPN
ejde-673	775	6	.	.	PUNCT
ejde-673	776	1	by	by	ADP
ejde-673	776	2	the	the	DET
ejde-673	776	3	definition	definition	NOUN
ejde-673	776	4	of	of	ADP
ejde-673	776	5	bα	bα	NOUN
ejde-673	776	6	,	,	PUNCT
ejde-673	776	7	there	there	PRON
ejde-673	776	8	exists	exist	VERB
ejde-673	776	9	a	a	DET
ejde-673	776	10	sequence	sequence	NOUN
ejde-673	776	11	{	{	PUNCT
ejde-673	776	12	un	un	PROPN
ejde-673	776	13	}	}	PUNCT
ejde-673	776	14	⊂	⊂	NOUN
ejde-673	776	15	mα	mα	ADP
ejde-673	776	16	such	such	ADJ
ejde-673	776	17	that	that	DET
ejde-673	776	18	bα	bα	NOUN
ejde-673	776	19	=	=	PUNCT
ejde-673	776	20	limn→∞	limn→∞	X
ejde-673	776	21	ψ(un	ψ(un	NOUN
ejde-673	776	22	)	)	PUNCT
ejde-673	776	23	.	.	PUNCT
ejde-673	777	1	for	for	ADP
ejde-673	777	2	large	large	ADJ
ejde-673	777	3	n	n	CCONJ
ejde-673	777	4	,	,	PUNCT
ejde-673	777	5	bα	bα	PROPN
ejde-673	777	6	+	+	NOUN
ejde-673	777	7	1	1	NUM
ejde-673	777	8	≥	≥	NOUN
ejde-673	777	9	ψ(un	ψ(un	PROPN
ejde-673	777	10	)	)	PUNCT
ejde-673	777	11	=	=	SYM
ejde-673	777	12	m̂(φ(un	m̂(φ(un	NOUN
ejde-673	777	13	)	)	PUNCT
ejde-673	777	14	)	)	PUNCT
ejde-673	777	15	≥	≥	PROPN
ejde-673	777	16	m0	m0	PROPN
ejde-673	777	17	l	l	PROPN
ejde-673	777	18	(	(	PUNCT
ejde-673	777	19	k0	k0	PROPN
ejde-673	777	20	p+	p+	PROPN
ejde-673	777	21	∥un∥p	∥un∥p	PROPN
ejde-673	778	1	+	+	CCONJ
ejde-673	778	2	y	y	PROPN
ejde-673	778	3	∧	∧	PROPN
ejde-673	778	4	∥un∥p	∥un∥p	NUM
ejde-673	779	1	−	−	PROPN
ejde-673	779	2	y	y	PROPN
ejde-673	779	3	)	)	PUNCT
ejde-673	779	4	l.	l.	PROPN
ejde-673	779	5	thus	thus	ADV
ejde-673	779	6	{	{	PUNCT
ejde-673	779	7	un	un	PROPN
ejde-673	779	8	}	}	PUNCT
ejde-673	779	9	is	be	AUX
ejde-673	779	10	bounded	bound	VERB
ejde-673	779	11	in	in	ADP
ejde-673	779	12	y	y	PROPN
ejde-673	779	13	.	.	PUNCT
ejde-673	780	1	so	so	ADV
ejde-673	780	2	there	there	PRON
ejde-673	780	3	exist	exist	VERB
ejde-673	780	4	a	a	DET
ejde-673	780	5	subsequence	subsequence	NOUN
ejde-673	780	6	{	{	PUNCT
ejde-673	780	7	un′	un′	NOUN
ejde-673	780	8	}	}	PUNCT
ejde-673	780	9	of	of	ADP
ejde-673	780	10	{	{	PUNCT
ejde-673	780	11	un	un	PROPN
ejde-673	780	12	}	}	PUNCT
ejde-673	780	13	and	and	CCONJ
ejde-673	780	14	u∗	u∗	PROPN
ejde-673	780	15	∈	∈	PROPN
ejde-673	780	16	y	y	NOUN
ejde-673	780	17	such	such	ADJ
ejde-673	780	18	that	that	SCONJ
ejde-673	780	19	un′	un′	PROPN
ejde-673	780	20	→	→	SYM
ejde-673	780	21	u∗	u∗	ADJ
ejde-673	780	22	weakly	weakly	ADJ
ejde-673	780	23	in	in	ADP
ejde-673	780	24	y	y	PROPN
ejde-673	780	25	.	.	PUNCT
ejde-673	781	1	then	then	ADV
ejde-673	781	2	ψ(u∗	ψ(u∗	NOUN
ejde-673	781	3	)	)	PUNCT
ejde-673	781	4	≤	≤	PROPN
ejde-673	781	5	lim	lim	PROPN
ejde-673	781	6	infn′→∞	infn′→∞	PROPN
ejde-673	781	7	ψ(un′	ψ(un′	PROPN
ejde-673	781	8	)	)	PUNCT
ejde-673	782	1	=	=	SYM
ejde-673	782	2	26	26	NUM
ejde-673	782	3	j.	j.	PROPN
ejde-673	782	4	aramaki	aramaki	PROPN
ejde-673	782	5	ejde-2025/17	ejde-2025/17	PROPN
ejde-673	782	6	bα	bα	PROPN
ejde-673	782	7	.	.	PUNCT
ejde-673	783	1	since	since	SCONJ
ejde-673	783	2	mα	mα	PROPN
ejde-673	783	3	is	be	AUX
ejde-673	783	4	a	a	DET
ejde-673	783	5	weakly	weakly	ADJ
ejde-673	783	6	closed	closed	ADJ
ejde-673	783	7	subset	subset	NOUN
ejde-673	783	8	of	of	ADP
ejde-673	783	9	y	y	PROPN
ejde-673	783	10	,	,	PUNCT
ejde-673	783	11	u∗	u∗	PROPN
ejde-673	783	12	∈	∈	PROPN
ejde-673	783	13	mα	mα	PROPN
ejde-673	783	14	,	,	PUNCT
ejde-673	783	15	so	so	SCONJ
ejde-673	783	16	ψ(u∗	ψ(u∗	PRON
ejde-673	783	17	)	)	PUNCT
ejde-673	783	18	≥	≥	NUM
ejde-673	783	19	bα	bα	PROPN
ejde-673	783	20	.	.	PUNCT
ejde-673	784	1	thus	thus	ADV
ejde-673	784	2	we	we	PRON
ejde-673	784	3	have	have	VERB
ejde-673	784	4	ψ(u∗	ψ(u∗	NUM
ejde-673	784	5	)	)	PUNCT
ejde-673	785	1	=	=	SYM
ejde-673	785	2	bα	bα	PROPN
ejde-673	785	3	.	.	PUNCT
ejde-673	786	1	by	by	ADP
ejde-673	786	2	(	(	PUNCT
ejde-673	786	3	a7	a7	PROPN
ejde-673	786	4	)	)	PUNCT
ejde-673	786	5	,	,	PUNCT
ejde-673	786	6	ψ(±u∗	ψ(±u∗	NOUN
ejde-673	786	7	)	)	PUNCT
ejde-673	786	8	=	=	SYM
ejde-673	786	9	bα	bα	PROPN
ejde-673	786	10	.	.	PUNCT
ejde-673	786	11	leth0	leth0	X
ejde-673	787	1	=	=	PRON
ejde-673	787	2	{	{	PUNCT
ejde-673	787	3	±u∗	±u∗	NOUN
ejde-673	787	4	}	}	PUNCT
ejde-673	787	5	,	,	PUNCT
ejde-673	787	6	then	then	ADV
ejde-673	787	7	clearly	clearly	ADV
ejde-673	787	8	γ(h0	γ(h0	X
ejde-673	787	9	)	)	PUNCT
ejde-673	787	10	=	=	SYM
ejde-673	788	1	1	1	X
ejde-673	788	2	.	.	PUNCT
ejde-673	788	3	therefore	therefore	ADV
ejde-673	788	4	,	,	PUNCT
ejde-673	788	5	c(1,α	c(1,α	PROPN
ejde-673	788	6	)	)	PUNCT
ejde-673	788	7	≤	≤	NUM
ejde-673	788	8	supu∈h0	supu∈h0	PROPN
ejde-673	788	9	ψ(u	ψ(u	PROPN
ejde-673	788	10	)	)	PUNCT
ejde-673	789	1	=	=	SYM
ejde-673	789	2	bα	bα	PROPN
ejde-673	789	3	.	.	PUNCT
ejde-673	790	1	thus	thus	ADV
ejde-673	790	2	we	we	PRON
ejde-673	790	3	have	have	VERB
ejde-673	790	4	c(1,α	c(1,α	PROPN
ejde-673	790	5	)	)	PUNCT
ejde-673	791	1	=	=	SYM
ejde-673	791	2	bα	bα	PROPN
ejde-673	791	3	.	.	PUNCT
ejde-673	791	4	□	□	PUNCT
ejde-673	791	5	from	from	ADP
ejde-673	791	6	now	now	ADV
ejde-673	791	7	on	on	ADV
ejde-673	791	8	,	,	PUNCT
ejde-673	791	9	we	we	PRON
ejde-673	791	10	suppose	suppose	VERB
ejde-673	791	11	that	that	SCONJ
ejde-673	791	12	the	the	DET
ejde-673	791	13	following	follow	VERB
ejde-673	791	14	more	more	ADV
ejde-673	791	15	restrictive	restrictive	ADJ
ejde-673	791	16	assumption	assumption	NOUN
ejde-673	791	17	than	than	ADP
ejde-673	791	18	(	(	PUNCT
ejde-673	791	19	a8	a8	PROPN
ejde-673	791	20	)	)	PUNCT
ejde-673	791	21	on	on	ADP
ejde-673	791	22	the	the	DET
ejde-673	791	23	given	give	VERB
ejde-673	791	24	function	function	NOUN
ejde-673	791	25	g	g	PROPN
ejde-673	791	26	hold	hold	NOUN
ejde-673	791	27	.	.	PUNCT
ejde-673	792	1	(	(	PUNCT
ejde-673	792	2	a8	a8	PROPN
ejde-673	792	3	’	'	PUNCT
ejde-673	792	4	)	)	PUNCT
ejde-673	792	5	(	(	PUNCT
ejde-673	792	6	a8	a8	PROPN
ejde-673	792	7	)	)	PUNCT
ejde-673	792	8	holds	hold	VERB
ejde-673	792	9	with	with	ADP
ejde-673	792	10	r(x	r(x	NOUN
ejde-673	792	11	)	)	PUNCT
ejde-673	792	12	=	=	SYM
ejde-673	792	13	lp(x	lp(x	PROPN
ejde-673	792	14	)	)	PUNCT
ejde-673	792	15	,	,	PUNCT
ejde-673	792	16	where	where	SCONJ
ejde-673	792	17	l	l	NOUN
ejde-673	792	18	is	be	AUX
ejde-673	792	19	a	a	DET
ejde-673	792	20	constant	constant	ADJ
ejde-673	792	21	in	in	ADP
ejde-673	792	22	(	(	PUNCT
ejde-673	792	23	a1	a1	NOUN
ejde-673	792	24	)	)	PUNCT
ejde-673	792	25	,	,	PUNCT
ejde-673	792	26	that	that	ADV
ejde-673	792	27	is	is	ADV
ejde-673	792	28	,	,	PUNCT
ejde-673	792	29	g(x	g(x	PROPN
ejde-673	792	30	,	,	PUNCT
ejde-673	792	31	t	t	PROPN
ejde-673	792	32	)	)	PUNCT
ejde-673	792	33	=	=	SYM
ejde-673	792	34	b(x)|t|lp(x)−2	b(x)|t|lp(x)−2	NOUN
ejde-673	792	35	t	t	NOUN
ejde-673	792	36	with	with	ADP
ejde-673	792	37	a	a	DET
ejde-673	792	38	function	function	NOUN
ejde-673	792	39	b(x	b(x	NOUN
ejde-673	792	40	)	)	PUNCT
ejde-673	792	41	satisfying	satisfy	VERB
ejde-673	792	42	the	the	DET
ejde-673	792	43	condition	condition	NOUN
ejde-673	792	44	in	in	ADP
ejde-673	792	45	(	(	PUNCT
ejde-673	792	46	a8	a8	PROPN
ejde-673	792	47	)	)	PUNCT
ejde-673	792	48	with	with	ADP
ejde-673	792	49	r(x	r(x	NOUN
ejde-673	792	50	)	)	PUNCT
ejde-673	792	51	=	=	SYM
ejde-673	792	52	lp(x	lp(x	PROPN
ejde-673	792	53	)	)	PUNCT
ejde-673	792	54	.	.	PUNCT
ejde-673	793	1	theorem	theorem	VERB
ejde-673	793	2	4.3	4.3	NUM
ejde-673	793	3	.	.	PUNCT
ejde-673	794	1	assume	assume	VERB
ejde-673	794	2	that	that	SCONJ
ejde-673	794	3	(	(	PUNCT
ejde-673	794	4	a1)–(a7	a1)–(a7	PROPN
ejde-673	794	5	)	)	PUNCT
ejde-673	794	6	,	,	PUNCT
ejde-673	794	7	(	(	PUNCT
ejde-673	794	8	a8	a8	PROPN
ejde-673	794	9	’	'	PUNCT
ejde-673	794	10	)	)	PUNCT
ejde-673	794	11	hold	hold	VERB
ejde-673	794	12	,	,	PUNCT
ejde-673	794	13	moreover	moreover	ADV
ejde-673	794	14	,	,	PUNCT
ejde-673	794	15	suppose	suppose	VERB
ejde-673	794	16	that	that	SCONJ
ejde-673	794	17	there	there	PRON
ejde-673	794	18	exists	exist	VERB
ejde-673	794	19	δ	δ	PROPN
ejde-673	794	20	>	>	X
ejde-673	794	21	0	0	NUM
ejde-673	794	22	such	such	ADJ
ejde-673	794	23	that	that	SCONJ
ejde-673	794	24	p(x	p(x	NOUN
ejde-673	794	25	)	)	PUNCT
ejde-673	794	26	=	=	SYM
ejde-673	795	1	p	p	X
ejde-673	795	2	(	(	PUNCT
ejde-673	795	3	a	a	DET
ejde-673	795	4	constant	constant	ADJ
ejde-673	795	5	)	)	PUNCT
ejde-673	795	6	for	for	ADP
ejde-673	795	7	all	all	PRON
ejde-673	795	8	x	x	SYM
ejde-673	795	9	∈	∈	PROPN
ejde-673	795	10	bω(γ2	bω(γ2	NOUN
ejde-673	795	11	,	,	PUNCT
ejde-673	795	12	δ	δ	PROPN
ejde-673	795	13	)	)	PUNCT
ejde-673	795	14	.	.	PUNCT
ejde-673	796	1	then	then	ADV
ejde-673	796	2	we	we	PRON
ejde-673	796	3	have	have	VERB
ejde-673	796	4	λ∗	λ∗	PROPN
ejde-673	796	5	>	>	X
ejde-673	796	6	0	0	X
ejde-673	796	7	.	.	PUNCT
ejde-673	797	1	proof	proof	NOUN
ejde-673	797	2	.	.	PUNCT
ejde-673	798	1	let	let	VERB
ejde-673	798	2	u	u	PRON
ejde-673	798	3	be	be	AUX
ejde-673	798	4	the	the	DET
ejde-673	798	5	eigenfunction	eigenfunction	NOUN
ejde-673	798	6	of	of	ADP
ejde-673	798	7	problem	problem	NOUN
ejde-673	798	8	(	(	PUNCT
ejde-673	798	9	1.1	1.1	NUM
ejde-673	798	10	)	)	PUNCT
ejde-673	798	11	,	,	PUNCT
ejde-673	798	12	associated	associate	VERB
ejde-673	798	13	with	with	ADP
ejde-673	798	14	λ	λ	PROPN
ejde-673	798	15	.	.	PROPN
ejde-673	798	16	thenk(u	thenk(u	PROPN
ejde-673	798	17	)	)	PUNCT
ejde-673	798	18	>	>	X
ejde-673	799	1	0	0	X
ejde-673	799	2	.	.	PUNCT
ejde-673	800	1	in	in	ADP
ejde-673	800	2	fact	fact	NOUN
ejde-673	800	3	,	,	PUNCT
ejde-673	800	4	let	let	VERB
ejde-673	800	5	k(u	k(u	PRON
ejde-673	800	6	)	)	PUNCT
ejde-673	800	7	=	=	SYM
ejde-673	801	1	0	0	X
ejde-673	801	2	.	.	PUNCT
ejde-673	802	1	since	since	SCONJ
ejde-673	802	2	⟨ψ′(u	⟨ψ′(u	PROPN
ejde-673	802	3	)	)	PUNCT
ejde-673	802	4	,	,	PUNCT
ejde-673	802	5	u⟩y	u⟩y	NUM
ejde-673	802	6	∗,y	∗,y	PROPN
ejde-673	802	7	=	=	SYM
ejde-673	802	8	λ⟨k	λ⟨k	NUM
ejde-673	802	9	′(u	′(u	NOUN
ejde-673	802	10	)	)	PUNCT
ejde-673	802	11	,	,	PUNCT
ejde-673	802	12	u⟩y	u⟩y	NUM
ejde-673	802	13	∗,y	∗,y	PROPN
ejde-673	802	14	,	,	PUNCT
ejde-673	802	15	it	it	PRON
ejde-673	802	16	follows	follow	VERB
ejde-673	802	17	from	from	ADP
ejde-673	802	18	(	(	PUNCT
ejde-673	802	19	a8	a8	PROPN
ejde-673	802	20	’	'	PUNCT
ejde-673	802	21	)	)	PUNCT
ejde-673	802	22	that	that	SCONJ
ejde-673	802	23	m(φ(u	m(φ(u	NOUN
ejde-673	802	24	)	)	PUNCT
ejde-673	802	25	)	)	PUNCT
ejde-673	803	1	∫	∫	PROPN
ejde-673	803	2	ω	ω	NUM
ejde-673	803	3	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	803	4	)	)	PUNCT
ejde-673	803	5	)	)	PUNCT
ejde-673	803	6	·	·	PUNCT
ejde-673	803	7	∇u(x	∇u(x	NOUN
ejde-673	803	8	)	)	PUNCT
ejde-673	803	9	dx	dx	PROPN
ejde-673	804	1	=	=	SYM
ejde-673	804	2	λ	λ	PROPN
ejde-673	804	3	∫	∫	PROPN
ejde-673	804	4	γ2	γ2	PROPN
ejde-673	804	5	g(x	g(x	PROPN
ejde-673	804	6	,	,	PUNCT
ejde-673	804	7	u(x))u(x)dσx	u(x))u(x)dσx	NOUN
ejde-673	804	8	=	=	SYM
ejde-673	804	9	λlp	λlp	X
ejde-673	804	10	∫	∫	PROPN
ejde-673	804	11	γ2	γ2	PROPN
ejde-673	804	12	g(x	g(x	PROPN
ejde-673	804	13	,	,	PUNCT
ejde-673	804	14	u(x))dσx	u(x))dσx	X
ejde-673	804	15	=	=	SYM
ejde-673	804	16	λlpk(u	λlpk(u	NOUN
ejde-673	804	17	)	)	PUNCT
ejde-673	804	18	=	=	SYM
ejde-673	805	1	0	0	X
ejde-673	805	2	.	.	PUNCT
ejde-673	806	1	hence	hence	ADV
ejde-673	806	2	from	from	ADP
ejde-673	806	3	(	(	PUNCT
ejde-673	806	4	a5	a5	PROPN
ejde-673	806	5	)	)	PUNCT
ejde-673	806	6	and	and	CCONJ
ejde-673	806	7	m(φ(u	m(φ(u	NOUN
ejde-673	806	8	)	)	PUNCT
ejde-673	806	9	)	)	PUNCT
ejde-673	806	10	>	>	X
ejde-673	806	11	0	0	NUM
ejde-673	806	12	,	,	PUNCT
ejde-673	806	13	we	we	PRON
ejde-673	806	14	have	have	VERB
ejde-673	806	15	0	0	NUM
ejde-673	806	16	=	=	SYM
ejde-673	806	17	∫	∫	PROPN
ejde-673	806	18	ω	ω	NUM
ejde-673	806	19	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	806	20	)	)	PUNCT
ejde-673	806	21	)	)	PUNCT
ejde-673	806	22	·	·	PUNCT
ejde-673	806	23	∇u(x	∇u(x	NOUN
ejde-673	806	24	)	)	PUNCT
ejde-673	806	25	dx	dx	PROPN
ejde-673	806	26	≥	≥	PROPN
ejde-673	806	27	k0	k0	PROPN
ejde-673	806	28	∫	∫	PROPN
ejde-673	806	29	ω	ω	PROPN
ejde-673	806	30	h1(x)|∇u(x)|p(x	h1(x)|∇u(x)|p(x	PROPN
ejde-673	806	31	)	)	PUNCT
ejde-673	806	32	dx	dx	PROPN
ejde-673	806	33	.	.	PUNCT
ejde-673	807	1	thus	thus	ADV
ejde-673	807	2	we	we	PRON
ejde-673	807	3	have	have	VERB
ejde-673	807	4	∇u(x	∇u(x	NOUN
ejde-673	807	5	)	)	PUNCT
ejde-673	807	6	=	=	SYM
ejde-673	807	7	0	0	NUM
ejde-673	808	1	a.e	a.e	PROPN
ejde-673	808	2	.	.	PUNCT
ejde-673	808	3	x	x	SYM
ejde-673	808	4	∈	∈	PROPN
ejde-673	808	5	ω	ω	PROPN
ejde-673	808	6	.	.	PUNCT
ejde-673	809	1	from	from	ADP
ejde-673	809	2	proposition	proposition	NOUN
ejde-673	809	3	2.14	2.14	NUM
ejde-673	809	4	,	,	PUNCT
ejde-673	809	5	we	we	PRON
ejde-673	809	6	have	have	VERB
ejde-673	809	7	u	u	NOUN
ejde-673	809	8	=	=	NOUN
ejde-673	809	9	0	0	NUM
ejde-673	809	10	a.e	a.e	PROPN
ejde-673	809	11	.	.	PROPN
ejde-673	810	1	in	in	ADP
ejde-673	810	2	ω	ω	PROPN
ejde-673	810	3	.	.	PUNCT
ejde-673	811	1	this	this	PRON
ejde-673	811	2	is	be	AUX
ejde-673	811	3	a	a	DET
ejde-673	811	4	contradiction	contradiction	NOUN
ejde-673	811	5	.	.	PUNCT
ejde-673	812	1	we	we	PRON
ejde-673	812	2	show	show	VERB
ejde-673	812	3	that	that	SCONJ
ejde-673	812	4	there	there	PRON
ejde-673	812	5	exists	exist	VERB
ejde-673	812	6	t0	t0	PROPN
ejde-673	812	7	>	>	X
ejde-673	812	8	0	0	NUM
ejde-673	812	9	such	such	ADJ
ejde-673	812	10	that	that	DET
ejde-673	812	11	u1	u1	NOUN
ejde-673	812	12	:	:	PUNCT
ejde-673	812	13	=	=	SYM
ejde-673	812	14	1	1	NUM
ejde-673	812	15	t0	t0	NUM
ejde-673	812	16	u	u	NOUN
ejde-673	812	17	∈	∈	PROPN
ejde-673	812	18	m1	m1	NOUN
ejde-673	812	19	.	.	PUNCT
ejde-673	813	1	indeed	indeed	ADV
ejde-673	813	2	,	,	PUNCT
ejde-673	813	3	since	since	SCONJ
ejde-673	813	4	g(x	g(x	NOUN
ejde-673	813	5	,	,	PUNCT
ejde-673	813	6	t)t	t)t	PUNCT
ejde-673	813	7	=	=	SYM
ejde-673	813	8	lpg(x	lpg(x	NUM
ejde-673	813	9	,	,	PUNCT
ejde-673	813	10	t	t	PROPN
ejde-673	813	11	)	)	PUNCT
ejde-673	813	12	for	for	ADP
ejde-673	813	13	σ	σ	PROPN
ejde-673	813	14	-	-	PROPN
ejde-673	813	15	a.e	a.e	PROPN
ejde-673	813	16	.	.	PUNCT
ejde-673	813	17	x	x	SYM
ejde-673	813	18	∈	∈	PROPN
ejde-673	813	19	γ2	γ2	NOUN
ejde-673	813	20	and	and	CCONJ
ejde-673	813	21	all	all	DET
ejde-673	813	22	t	t	NOUN
ejde-673	813	23	∈	∈	NOUN
ejde-673	813	24	r	r	NOUN
ejde-673	813	25	,	,	PUNCT
ejde-673	813	26	it	it	PRON
ejde-673	813	27	follows	follow	VERB
ejde-673	813	28	from	from	ADP
ejde-673	813	29	(	(	PUNCT
ejde-673	813	30	3.4	3.4	NUM
ejde-673	813	31	)	)	PUNCT
ejde-673	813	32	that	that	PRON
ejde-673	813	33	k(ut	k(ut	ADV
ejde-673	813	34	)	)	PUNCT
ejde-673	814	1	=	=	SYM
ejde-673	814	2	t−lpk(u	t−lpk(u	NUM
ejde-673	814	3	)	)	PUNCT
ejde-673	814	4	for	for	ADP
ejde-673	814	5	t	t	PROPN
ejde-673	814	6	>	>	X
ejde-673	814	7	0	0	PROPN
ejde-673	814	8	.	.	PUNCT
ejde-673	815	1	here	here	ADV
ejde-673	815	2	we	we	PRON
ejde-673	815	3	can	can	AUX
ejde-673	815	4	see	see	VERB
ejde-673	815	5	that	that	SCONJ
ejde-673	815	6	k(ut	k(ut	PROPN
ejde-673	815	7	)	)	PUNCT
ejde-673	815	8	→	→	SYM
ejde-673	815	9	0	0	NUM
ejde-673	815	10	as	as	ADP
ejde-673	815	11	t	t	PROPN
ejde-673	815	12	→	→	SYM
ejde-673	815	13	∞	∞	PROPN
ejde-673	815	14	and	and	CCONJ
ejde-673	815	15	k(ut	k(ut	PROPN
ejde-673	815	16	)	)	PUNCT
ejde-673	816	1	→	→	SYM
ejde-673	816	2	∞	∞	PROPN
ejde-673	816	3	as	as	ADP
ejde-673	816	4	t	t	PROPN
ejde-673	816	5	→	→	SYM
ejde-673	816	6	+0	+0	PROPN
ejde-673	816	7	.	.	PROPN
ejde-673	816	8	since	since	SCONJ
ejde-673	816	9	k(ut	k(ut	PROPN
ejde-673	816	10	)	)	PUNCT
ejde-673	816	11	is	be	AUX
ejde-673	816	12	continuous	continuous	ADJ
ejde-673	816	13	with	with	ADP
ejde-673	816	14	respect	respect	NOUN
ejde-673	816	15	to	to	ADP
ejde-673	816	16	t	t	PROPN
ejde-673	816	17	∈	∈	PROPN
ejde-673	816	18	(	(	PUNCT
ejde-673	816	19	0,∞	0,∞	NUM
ejde-673	816	20	)	)	PUNCT
ejde-673	816	21	,	,	PUNCT
ejde-673	816	22	it	it	PRON
ejde-673	816	23	follows	follow	VERB
ejde-673	816	24	from	from	ADP
ejde-673	816	25	the	the	DET
ejde-673	816	26	intermediate	intermediate	ADJ
ejde-673	816	27	value	value	NOUN
ejde-673	816	28	theorem	theorem	VERB
ejde-673	816	29	that	that	SCONJ
ejde-673	816	30	there	there	PRON
ejde-673	816	31	exists	exist	VERB
ejde-673	816	32	t0	t0	PROPN
ejde-673	816	33	>	>	X
ejde-673	816	34	0	0	NUM
ejde-673	817	1	such	such	ADJ
ejde-673	817	2	that	that	SCONJ
ejde-673	817	3	k	k	NOUN
ejde-673	817	4	(	(	PUNCT
ejde-673	817	5	u	u	NOUN
ejde-673	817	6	t0	t0	PROPN
ejde-673	817	7	)	)	PUNCT
ejde-673	817	8	=	=	PUNCT
ejde-673	817	9	1	1	NUM
ejde-673	817	10	,	,	PUNCT
ejde-673	817	11	so	so	ADV
ejde-673	817	12	u1	u1	VERB
ejde-673	817	13	:	:	PUNCT
ejde-673	817	14	=	=	SYM
ejde-673	817	15	u	u	X
ejde-673	817	16	t0	t0	PROPN
ejde-673	817	17	∈	∈	PROPN
ejde-673	817	18	m1	m1	PROPN
ejde-673	817	19	.	.	PUNCT
ejde-673	818	1	now	now	ADV
ejde-673	818	2	since	since	SCONJ
ejde-673	818	3	k(u1	k(u1	NOUN
ejde-673	818	4	)	)	PUNCT
ejde-673	818	5	=	=	SYM
ejde-673	818	6	1	1	X
ejde-673	818	7	,	,	PUNCT
ejde-673	818	8	it	it	PRON
ejde-673	818	9	follows	follow	VERB
ejde-673	818	10	from	from	ADP
ejde-673	818	11	(	(	PUNCT
ejde-673	818	12	a1	a1	NOUN
ejde-673	818	13	)	)	PUNCT
ejde-673	818	14	,	,	PUNCT
ejde-673	818	15	(	(	PUNCT
ejde-673	818	16	a5	a5	PROPN
ejde-673	818	17	)	)	PUNCT
ejde-673	818	18	,	,	PUNCT
ejde-673	818	19	(	(	PUNCT
ejde-673	818	20	a8	a8	PROPN
ejde-673	818	21	’	'	PUNCT
ejde-673	818	22	)	)	PUNCT
ejde-673	818	23	,	,	PUNCT
ejde-673	818	24	and	and	CCONJ
ejde-673	818	25	lemma	lemma	PROPN
ejde-673	818	26	4.1	4.1	NUM
ejde-673	818	27	that	that	PRON
ejde-673	818	28	λ	λ	PROPN
ejde-673	818	29	=	=	SYM
ejde-673	818	30	⟨ψ′(u	⟨ψ′(u	PROPN
ejde-673	818	31	)	)	PUNCT
ejde-673	818	32	,	,	PUNCT
ejde-673	818	33	u⟩y	u⟩y	NUM
ejde-673	818	34	∗,y	∗,y	PROPN
ejde-673	818	35	⟨k	⟨k	NOUN
ejde-673	818	36	′(u	′(u	NOUN
ejde-673	818	37	)	)	PUNCT
ejde-673	818	38	,	,	PUNCT
ejde-673	818	39	u⟩y	u⟩y	NUM
ejde-673	818	40	∗,y	∗,y	NOUN
ejde-673	818	41	=	=	SYM
ejde-673	818	42	m(φ(u	m(φ(u	NOUN
ejde-673	818	43	)	)	PUNCT
ejde-673	818	44	)	)	PUNCT
ejde-673	819	1	∫	∫	PROPN
ejde-673	819	2	ω	ω	NUM
ejde-673	819	3	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	819	4	)	)	PUNCT
ejde-673	819	5	)	)	PUNCT
ejde-673	819	6	·	·	PUNCT
ejde-673	819	7	∇u(x	∇u(x	NOUN
ejde-673	819	8	)	)	PUNCT
ejde-673	819	9	dx∫	dx∫	PROPN
ejde-673	819	10	γ2	γ2	PROPN
ejde-673	819	11	g(x	g(x	PROPN
ejde-673	819	12	,	,	PUNCT
ejde-673	819	13	u(x))u(x)dσx	u(x))u(x)dσx	VERB
ejde-673	819	14	≥	≥	NOUN
ejde-673	819	15	m0	m0	PROPN
ejde-673	819	16	(	(	PUNCT
ejde-673	819	17	∫	∫	PROPN
ejde-673	819	18	ω	ω	PROPN
ejde-673	819	19	1	1	NUM
ejde-673	819	20	p(x)a(x,∇u(x	p(x)a(x,∇u(x	NOUN
ejde-673	819	21	)	)	PUNCT
ejde-673	819	22	)	)	PUNCT
ejde-673	819	23	·	·	PUNCT
ejde-673	819	24	∇u(x	∇u(x	NOUN
ejde-673	819	25	)	)	PUNCT
ejde-673	819	26	dx	dx	PROPN
ejde-673	819	27	)	)	PUNCT
ejde-673	820	1	l−1	l−1	PROPN
ejde-673	820	2	∫	∫	PROPN
ejde-673	820	3	ω	ω	NUM
ejde-673	820	4	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	820	5	)	)	PUNCT
ejde-673	820	6	)	)	PUNCT
ejde-673	820	7	·	·	PUNCT
ejde-673	820	8	∇u(x	∇u(x	NOUN
ejde-673	820	9	)	)	PUNCT
ejde-673	820	10	dx∫	dx∫	PROPN
ejde-673	820	11	γ2	γ2	PROPN
ejde-673	820	12	g(x	g(x	PROPN
ejde-673	820	13	,	,	PUNCT
ejde-673	820	14	u(x))u(x)dσx	u(x))u(x)dσx	VERB
ejde-673	820	15	≥	≥	NOUN
ejde-673	820	16	m0	m0	PROPN
ejde-673	820	17	(	(	PUNCT
ejde-673	820	18	p+)l−1	p+)l−1	NUM
ejde-673	820	19	(	(	PUNCT
ejde-673	820	20	∫	∫	PROPN
ejde-673	820	21	bω(γ2,δ	bω(γ2,δ	PROPN
ejde-673	820	22	)	)	PUNCT
ejde-673	820	23	a(x,∇u(x	a(x,∇u(x	NOUN
ejde-673	820	24	)	)	PUNCT
ejde-673	820	25	)	)	PUNCT
ejde-673	820	26	·	·	PUNCT
ejde-673	820	27	∇u(x	∇u(x	NOUN
ejde-673	820	28	)	)	PUNCT
ejde-673	820	29	dx	dx	PROPN
ejde-673	820	30	)	)	PUNCT
ejde-673	821	1	l	l	NOUN
ejde-673	821	2	lp	lp	ADJ
ejde-673	821	3	∫	∫	PROPN
ejde-673	821	4	γ2	γ2	PROPN
ejde-673	821	5	g(x	g(x	PROPN
ejde-673	821	6	,	,	PUNCT
ejde-673	821	7	u(x))dσx	u(x))dσx	NOUN
ejde-673	821	8	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	821	9	eigenvalue	eigenvalue	VERB
ejde-673	821	10	problems	problem	NOUN
ejde-673	821	11	for	for	ADP
ejde-673	821	12	kirchhoff	kirchhoff	NOUN
ejde-673	821	13	-	-	PUNCT
ejde-673	821	14	type	type	NOUN
ejde-673	821	15	equations	equation	NOUN
ejde-673	821	16	27	27	NUM
ejde-673	821	17	≥	≥	NOUN
ejde-673	821	18	m0k	m0k	PROPN
ejde-673	821	19	l	l	NOUN
ejde-673	821	20	0	0	NUM
ejde-673	821	21	lp(p+)l−1	lp(p+)l−1	NOUN
ejde-673	821	22	(	(	PUNCT
ejde-673	821	23	∫	∫	PROPN
ejde-673	821	24	bω(γ2,δ	bω(γ2,δ	PROPN
ejde-673	821	25	)	)	PUNCT
ejde-673	821	26	h1(x)|∇u(x)|pdx	h1(x)|∇u(x)|pdx	PROPN
ejde-673	821	27	)	)	PUNCT
ejde-673	821	28	l∫	l∫	ADJ
ejde-673	821	29	γ2	γ2	ADJ
ejde-673	821	30	g(x	g(x	NOUN
ejde-673	821	31	,	,	PUNCT
ejde-673	821	32	u(x))dσx	u(x))dσx	X
ejde-673	821	33	=	=	PUNCT
ejde-673	821	34	m0k	m0k	NOUN
ejde-673	821	35	l	l	NOUN
ejde-673	821	36	0	0	NUM
ejde-673	821	37	lp(p+)l−1	lp(p+)l−1	NOUN
ejde-673	821	38	(	(	PUNCT
ejde-673	821	39	∫	∫	PROPN
ejde-673	821	40	bω(γ2,δ	bω(γ2,δ	PROPN
ejde-673	821	41	)	)	PUNCT
ejde-673	821	42	h1(x)|t0∇u1(x)|pdx	h1(x)|t0∇u1(x)|pdx	PROPN
ejde-673	821	43	)	)	PUNCT
ejde-673	821	44	l∫	l∫	ADJ
ejde-673	821	45	γ2	γ2	ADJ
ejde-673	821	46	g(x	g(x	NOUN
ejde-673	821	47	,	,	PUNCT
ejde-673	822	1	t0u1(x))dσx	t0u1(x))dσx	PROPN
ejde-673	822	2	=	=	SYM
ejde-673	822	3	m0k	m0k	NOUN
ejde-673	822	4	l	l	NOUN
ejde-673	822	5	0	0	PUNCT
ejde-673	822	6	lp(p+)l−1	lp(p+)l−1	ADJ
ejde-673	822	7	tlp0	tlp0	NOUN
ejde-673	822	8	(	(	PUNCT
ejde-673	822	9	∫	∫	PROPN
ejde-673	822	10	bω(γ2,δ	bω(γ2,δ	PROPN
ejde-673	822	11	)	)	PUNCT
ejde-673	823	1	h1(x)|∇u1(x)|pdx	h1(x)|∇u1(x)|pdx	ADJ
ejde-673	823	2	)	)	PUNCT
ejde-673	823	3	l	l	NOUN
ejde-673	823	4	tlp0	tlp0	NOUN
ejde-673	823	5	∫	∫	PROPN
ejde-673	823	6	γ2	γ2	PROPN
ejde-673	823	7	g(x	g(x	PROPN
ejde-673	823	8	,	,	PUNCT
ejde-673	823	9	u1(x))dσx	u1(x))dσx	NUM
ejde-673	823	10	≥	≥	NOUN
ejde-673	823	11	m0k	m0k	PROPN
ejde-673	823	12	l	l	X
ejde-673	823	13	0	0	PUNCT
ejde-673	823	14	lp(p+)l−1	lp(p+)l−1	ADJ
ejde-673	823	15	βl	βl	NOUN
ejde-673	823	16	(	(	PUNCT
ejde-673	823	17	δ,1	δ,1	NOUN
ejde-673	823	18	)	)	PUNCT
ejde-673	823	19	>	>	X
ejde-673	823	20	0	0	X
ejde-673	823	21	.	.	PUNCT
ejde-673	824	1	thus	thus	ADV
ejde-673	824	2	we	we	PRON
ejde-673	824	3	have	have	VERB
ejde-673	824	4	λ∗	λ∗	NOUN
ejde-673	824	5	=	=	SYM
ejde-673	824	6	inf	inf	PROPN
ejde-673	824	7	λ	λ	PROPN
ejde-673	824	8	≥	≥	NOUN
ejde-673	824	9	m0k	m0k	PROPN
ejde-673	824	10	l	l	X
ejde-673	824	11	0	0	PUNCT
ejde-673	824	12	lp(p+)l−1	lp(p+)l−1	NOUN
ejde-673	824	13	β	β	X
ejde-673	824	14	l	l	X
ejde-673	824	15	(	(	PUNCT
ejde-673	824	16	δ,1	δ,1	NOUN
ejde-673	824	17	)	)	PUNCT
ejde-673	824	18	>	>	X
ejde-673	824	19	0	0	X
ejde-673	824	20	.	.	PUNCT
ejde-673	825	1	□	□	PUNCT
ejde-673	825	2	next	next	ADV
ejde-673	825	3	we	we	PRON
ejde-673	825	4	will	will	AUX
ejde-673	825	5	treat	treat	VERB
ejde-673	825	6	the	the	DET
ejde-673	825	7	case	case	NOUN
ejde-673	825	8	λ∗	λ∗	NOUN
ejde-673	825	9	=	=	X
ejde-673	825	10	0	0	X
ejde-673	825	11	.	.	PUNCT
ejde-673	825	12	from	from	ADP
ejde-673	825	13	the	the	DET
ejde-673	825	14	absolute	absolute	ADJ
ejde-673	825	15	continuity	continuity	NOUN
ejde-673	825	16	of	of	ADP
ejde-673	825	17	integral	integral	ADJ
ejde-673	825	18	,	,	PUNCT
ejde-673	825	19	we	we	PRON
ejde-673	825	20	obtain	obtain	VERB
ejde-673	825	21	the	the	DET
ejde-673	825	22	following	follow	VERB
ejde-673	825	23	lemma	lemma	PROPN
ejde-673	825	24	which	which	PRON
ejde-673	825	25	is	be	AUX
ejde-673	825	26	needed	need	VERB
ejde-673	825	27	later	later	ADV
ejde-673	825	28	.	.	PUNCT
ejde-673	826	1	lemma	lemma	PROPN
ejde-673	826	2	4.4	4.4	NUM
ejde-673	826	3	.	.	PUNCT
ejde-673	827	1	let	let	VERB
ejde-673	827	2	u	u	PRON
ejde-673	827	3	∈	∈	PROPN
ejde-673	827	4	y	y	PROPN
ejde-673	827	5	be	be	AUX
ejde-673	827	6	given	give	VERB
ejde-673	827	7	.	.	PUNCT
ejde-673	828	1	then	then	ADV
ejde-673	828	2	for	for	ADP
ejde-673	828	3	any	any	DET
ejde-673	828	4	ε	ε	PROPN
ejde-673	828	5	>	>	X
ejde-673	828	6	0	0	PROPN
ejde-673	828	7	,	,	PUNCT
ejde-673	828	8	there	there	PRON
ejde-673	828	9	exists	exist	VERB
ejde-673	828	10	δ0	δ0	NOUN
ejde-673	828	11	>	>	X
ejde-673	828	12	0	0	NUM
ejde-673	829	1	such	such	ADJ
ejde-673	829	2	that	that	PRON
ejde-673	829	3	for	for	ADP
ejde-673	829	4	any	any	DET
ejde-673	829	5	0	0	PUNCT
ejde-673	829	6	<	<	X
ejde-673	829	7	δ	δ	X
ejde-673	829	8	<	<	X
ejde-673	829	9	δ0	δ0	NOUN
ejde-673	829	10	,	,	PUNCT
ejde-673	829	11	βu(δ	βu(δ	PUNCT
ejde-673	829	12	)	)	PUNCT
ejde-673	829	13	:	:	PUNCT
ejde-673	830	1	=	=	SYM
ejde-673	830	2	∫	∫	PROPN
ejde-673	830	3	bω(γ2,δ	bω(γ2,δ	PROPN
ejde-673	830	4	)	)	PUNCT
ejde-673	830	5	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	830	6	)	)	PUNCT
ejde-673	830	7	)	)	PUNCT
ejde-673	830	8	dx	dx	PROPN
ejde-673	830	9	<	<	X
ejde-673	830	10	ε	ε	PROPN
ejde-673	830	11	.	.	PUNCT
ejde-673	830	12	theorem	theorem	VERB
ejde-673	830	13	4.5	4.5	NUM
ejde-673	830	14	.	.	PUNCT
ejde-673	831	1	assume	assume	VERB
ejde-673	831	2	that	that	SCONJ
ejde-673	831	3	(	(	PUNCT
ejde-673	831	4	a2)–(a7	a2)–(a7	NOUN
ejde-673	831	5	)	)	PUNCT
ejde-673	831	6	,	,	PUNCT
ejde-673	831	7	(	(	PUNCT
ejde-673	831	8	a1	a1	PROPN
ejde-673	831	9	’	'	PUNCT
ejde-673	831	10	)	)	PUNCT
ejde-673	831	11	,	,	PUNCT
ejde-673	831	12	(	(	PUNCT
ejde-673	831	13	a8	a8	PROPN
ejde-673	831	14	’	'	PUNCT
ejde-673	831	15	)	)	PUNCT
ejde-673	831	16	hold	hold	VERB
ejde-673	831	17	.	.	PUNCT
ejde-673	832	1	moreover	moreover	ADV
ejde-673	832	2	,	,	PUNCT
ejde-673	832	3	suppose	suppose	VERB
ejde-673	832	4	that	that	SCONJ
ejde-673	832	5	there	there	PRON
ejde-673	832	6	exist	exist	VERB
ejde-673	832	7	δ	δ	PROPN
ejde-673	832	8	>	>	X
ejde-673	832	9	0	0	PUNCT
ejde-673	832	10	and	and	CCONJ
ejde-673	832	11	x0	x0	PROPN
ejde-673	832	12	∈	∈	PROPN
ejde-673	832	13	γ2	γ2	NOUN
ejde-673	832	14	such	such	ADJ
ejde-673	832	15	that	that	SCONJ
ejde-673	832	16	the	the	DET
ejde-673	832	17	following	follow	VERB
ejde-673	832	18	hold	hold	NOUN
ejde-673	832	19	:	:	PUNCT
ejde-673	832	20	(	(	PUNCT
ejde-673	832	21	i	i	NOUN
ejde-673	832	22	)	)	PUNCT
ejde-673	832	23	p(x	p(x	PROPN
ejde-673	832	24	)	)	PUNCT
ejde-673	832	25	=	=	SYM
ejde-673	833	1	p	p	X
ejde-673	833	2	(	(	PUNCT
ejde-673	833	3	a	a	DET
ejde-673	833	4	constant	constant	ADJ
ejde-673	833	5	)	)	PUNCT
ejde-673	833	6	for	for	ADP
ejde-673	833	7	all	all	DET
ejde-673	833	8	x	x	SYM
ejde-673	833	9	∈	∈	NUM
ejde-673	833	10	bγ2	bγ2	NOUN
ejde-673	833	11	(	(	PUNCT
ejde-673	833	12	x0	x0	PROPN
ejde-673	833	13	,	,	PUNCT
ejde-673	833	14	δ	δ	PROPN
ejde-673	833	15	)	)	PUNCT
ejde-673	833	16	.	.	PUNCT
ejde-673	834	1	(	(	PUNCT
ejde-673	834	2	ii	ii	NOUN
ejde-673	834	3	)	)	PUNCT
ejde-673	834	4	p(x	p(x	PROPN
ejde-673	834	5	)	)	PUNCT
ejde-673	834	6	<	<	X
ejde-673	835	1	p	p	X
ejde-673	835	2	for	for	ADP
ejde-673	835	3	all	all	DET
ejde-673	835	4	x	x	PROPN
ejde-673	835	5	∈	∈	PROPN
ejde-673	835	6	bω(x0	bω(x0	PROPN
ejde-673	835	7	,	,	PUNCT
ejde-673	835	8	δ	δ	PROPN
ejde-673	835	9	)	)	PUNCT
ejde-673	835	10	.	.	PUNCT
ejde-673	836	1	(	(	PUNCT
ejde-673	836	2	iii	iii	X
ejde-673	836	3	)	)	PUNCT
ejde-673	836	4	h1	h1	PROPN
ejde-673	836	5	∈	∈	PROPN
ejde-673	836	6	l1(bω(x0	l1(bω(x0	NOUN
ejde-673	836	7	,	,	PUNCT
ejde-673	836	8	δ	δ	PROPN
ejde-673	836	9	)	)	PUNCT
ejde-673	836	10	)	)	PUNCT
ejde-673	836	11	,	,	PUNCT
ejde-673	836	12	where	where	SCONJ
ejde-673	836	13	h1	h1	PROPN
ejde-673	836	14	is	be	AUX
ejde-673	836	15	the	the	DET
ejde-673	836	16	function	function	NOUN
ejde-673	836	17	of	of	ADP
ejde-673	836	18	(	(	PUNCT
ejde-673	836	19	a3)–(a5	a3)–(a5	ADJ
ejde-673	836	20	)	)	PUNCT
ejde-673	836	21	.	.	PUNCT
ejde-673	837	1	then	then	ADV
ejde-673	837	2	we	we	PRON
ejde-673	837	3	have	have	VERB
ejde-673	837	4	limα→∞	limα→∞	PROPN
ejde-673	837	5	λ(1,α	λ(1,α	NOUN
ejde-673	837	6	)	)	PUNCT
ejde-673	838	1	=	=	SYM
ejde-673	838	2	0	0	NUM
ejde-673	838	3	,	,	PUNCT
ejde-673	838	4	so	so	ADV
ejde-673	838	5	λ∗	λ∗	NOUN
ejde-673	838	6	=	=	SYM
ejde-673	838	7	0	0	X
ejde-673	838	8	.	.	PUNCT
ejde-673	839	1	proof	proof	NOUN
ejde-673	839	2	.	.	PUNCT
ejde-673	840	1	replacing	replace	VERB
ejde-673	840	2	δ	δ	PROPN
ejde-673	840	3	>	>	X
ejde-673	840	4	0	0	PUNCT
ejde-673	840	5	with	with	ADP
ejde-673	840	6	smaller	small	ADJ
ejde-673	840	7	one	one	NOUN
ejde-673	840	8	,	,	PUNCT
ejde-673	840	9	if	if	SCONJ
ejde-673	840	10	necessary	necessary	ADJ
ejde-673	840	11	,	,	PUNCT
ejde-673	840	12	we	we	PRON
ejde-673	840	13	may	may	AUX
ejde-673	840	14	assume	assume	VERB
ejde-673	840	15	thatb(x0	thatb(x0	PROPN
ejde-673	840	16	,	,	PUNCT
ejde-673	840	17	δ)∩	δ)∩	PROPN
ejde-673	840	18	γ	γ	PROPN
ejde-673	840	19	⊂	⊂	PROPN
ejde-673	840	20	γ2	γ2	PROPN
ejde-673	840	21	.	.	PUNCT
ejde-673	841	1	choose	choose	VERB
ejde-673	841	2	0	0	NUM
ejde-673	841	3	≤	≤	NUM
ejde-673	841	4	u	u	PROPN
ejde-673	841	5	∈	∈	PROPN
ejde-673	841	6	c∞(ω	c∞(ω	NOUN
ejde-673	841	7	)	)	PUNCT
ejde-673	841	8	such	such	ADJ
ejde-673	841	9	that	that	SCONJ
ejde-673	841	10	u(x	u(x	VERB
ejde-673	841	11	)	)	PUNCT
ejde-673	841	12	=	=	SYM
ejde-673	841	13	1	1	NUM
ejde-673	841	14	for	for	ADP
ejde-673	841	15	x	x	PROPN
ejde-673	841	16	∈	∈	PROPN
ejde-673	841	17	bω(x0	bω(x0	NOUN
ejde-673	841	18	,	,	PUNCT
ejde-673	841	19	δ/4	δ/4	NUM
ejde-673	841	20	)	)	PUNCT
ejde-673	841	21	and	and	CCONJ
ejde-673	841	22	u(x	u(x	NOUN
ejde-673	841	23	)	)	PUNCT
ejde-673	841	24	=	=	SYM
ejde-673	841	25	0	0	NUM
ejde-673	842	1	for	for	ADP
ejde-673	842	2	x	x	PROPN
ejde-673	842	3	∈	∈	PROPN
ejde-673	842	4	ω	ω	NUM
ejde-673	842	5	\bω(x0	\bω(x0	PROPN
ejde-673	842	6	,	,	PUNCT
ejde-673	842	7	δ/2	δ/2	NUM
ejde-673	842	8	)	)	PUNCT
ejde-673	842	9	.	.	PUNCT
ejde-673	843	1	we	we	PRON
ejde-673	843	2	note	note	VERB
ejde-673	843	3	that	that	SCONJ
ejde-673	843	4	from	from	ADP
ejde-673	843	5	(	(	PUNCT
ejde-673	843	6	iii	iii	X
ejde-673	843	7	)	)	PUNCT
ejde-673	843	8	it	it	PRON
ejde-673	843	9	follows	follow	VERB
ejde-673	843	10	that	that	SCONJ
ejde-673	843	11	u	u	PROPN
ejde-673	843	12	∈	∈	PROPN
ejde-673	843	13	y	y	PROPN
ejde-673	843	14	.	.	PUNCT
ejde-673	844	1	by	by	ADP
ejde-673	844	2	lemma	lemma	PROPN
ejde-673	844	3	4.4	4.4	NUM
ejde-673	844	4	,	,	PUNCT
ejde-673	844	5	for	for	ADP
ejde-673	844	6	any	any	DET
ejde-673	844	7	ε	ε	PROPN
ejde-673	844	8	>	>	X
ejde-673	844	9	0	0	PROPN
ejde-673	844	10	,	,	PUNCT
ejde-673	844	11	there	there	PRON
ejde-673	844	12	exists	exist	VERB
ejde-673	844	13	δ0	δ0	NOUN
ejde-673	844	14	∈	∈	PROPN
ejde-673	844	15	(	(	PUNCT
ejde-673	844	16	0	0	NUM
ejde-673	844	17	,	,	PUNCT
ejde-673	844	18	δ/4	δ/4	NUM
ejde-673	844	19	)	)	PUNCT
ejde-673	844	20	such	such	ADJ
ejde-673	844	21	that	that	PRON
ejde-673	844	22	for	for	ADP
ejde-673	844	23	each	each	DET
ejde-673	844	24	δ1	δ1	NOUN
ejde-673	844	25	∈	∈	PROPN
ejde-673	844	26	(	(	PUNCT
ejde-673	844	27	0	0	NUM
ejde-673	844	28	,	,	PUNCT
ejde-673	844	29	δ0	δ0	NOUN
ejde-673	844	30	)	)	PUNCT
ejde-673	844	31	,	,	PUNCT
ejde-673	844	32	(	(	PUNCT
ejde-673	844	33	∫	∫	PROPN
ejde-673	844	34	bω(γ2,δ1	bω(γ2,δ1	PROPN
ejde-673	844	35	)	)	PUNCT
ejde-673	844	36	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	844	37	)	)	PUNCT
ejde-673	844	38	)	)	PUNCT
ejde-673	844	39	dx	dx	PROPN
ejde-673	844	40	)	)	PUNCT
ejde-673	845	1	l	l	NOUN
ejde-673	845	2	k(u	k(u	X
ejde-673	845	3	)	)	PUNCT
ejde-673	845	4	<	<	X
ejde-673	845	5	ε/(2c	ε/(2c	X
ejde-673	845	6	)	)	PUNCT
ejde-673	845	7	,	,	PUNCT
ejde-673	845	8	where	where	SCONJ
ejde-673	845	9	c	c	NOUN
ejde-673	845	10	=	=	SYM
ejde-673	845	11	m2	m2	PROPN
ejde-673	845	12	1p	1p	PROPN
ejde-673	845	13	+	+	CCONJ
ejde-673	845	14	lm0p−	lm0p−	NOUN
ejde-673	845	15	2l−1	2l−1	NUM
ejde-673	845	16	.	.	PUNCT
ejde-673	846	1	since	since	SCONJ
ejde-673	846	2	p	p	PROPN
ejde-673	846	3	∈	∈	PROPN
ejde-673	846	4	c(ω	c(ω	PROPN
ejde-673	846	5	)	)	PUNCT
ejde-673	846	6	,	,	PUNCT
ejde-673	846	7	it	it	PRON
ejde-673	846	8	follows	follow	VERB
ejde-673	846	9	from	from	ADP
ejde-673	846	10	(	(	PUNCT
ejde-673	846	11	ii	ii	NOUN
ejde-673	846	12	)	)	PUNCT
ejde-673	846	13	that	that	SCONJ
ejde-673	846	14	for	for	ADP
ejde-673	846	15	any	any	DET
ejde-673	846	16	x	x	SYM
ejde-673	846	17	∈	∈	PROPN
ejde-673	846	18	bω(x0	bω(x0	NOUN
ejde-673	846	19	,	,	PUNCT
ejde-673	846	20	δ/2	δ/2	NUM
ejde-673	846	21	)	)	PUNCT
ejde-673	846	22	\	\	NOUN
ejde-673	846	23	b(γ2	b(γ2	NOUN
ejde-673	846	24	,	,	PUNCT
ejde-673	846	25	δ0	δ0	NOUN
ejde-673	846	26	)	)	PUNCT
ejde-673	846	27	,	,	PUNCT
ejde-673	846	28	we	we	PRON
ejde-673	846	29	have	have	VERB
ejde-673	846	30	p(x)−	p(x)−	PROPN
ejde-673	846	31	p	p	PROPN
ejde-673	846	32	≤	≤	ADJ
ejde-673	846	33	p+(bω(x0	p+(bω(x0	NOUN
ejde-673	846	34	,	,	PUNCT
ejde-673	846	35	δ/2	δ/2	NUM
ejde-673	846	36	)	)	PUNCT
ejde-673	847	1	\b(γ2	\b(γ2	PROPN
ejde-673	847	2	,	,	PUNCT
ejde-673	847	3	δ0))−	δ0))−	ADP
ejde-673	847	4	p	p	NOUN
ejde-673	847	5	:	:	PUNCT
ejde-673	847	6	=	=	SYM
ejde-673	847	7	−ε0	−ε0	X
ejde-673	847	8	<	<	X
ejde-673	847	9	0	0	NUM
ejde-673	847	10	.	.	PUNCT
ejde-673	848	1	we	we	PRON
ejde-673	848	2	note	note	VERB
ejde-673	848	3	that	that	SCONJ
ejde-673	848	4	p(x	p(x	NOUN
ejde-673	848	5	)	)	PUNCT
ejde-673	848	6	=	=	PUNCT
ejde-673	849	1	p	p	NOUN
ejde-673	849	2	on	on	ADP
ejde-673	849	3	suppu	suppu	NOUN
ejde-673	849	4	∩	∩	ADJ
ejde-673	849	5	γ2	γ2	PROPN
ejde-673	849	6	.	.	PUNCT
ejde-673	850	1	if	if	SCONJ
ejde-673	850	2	we	we	PRON
ejde-673	850	3	define	define	VERB
ejde-673	850	4	h(t	h(t	PRON
ejde-673	850	5	)	)	PUNCT
ejde-673	850	6	=	=	SYM
ejde-673	850	7	k(tu	k(tu	X
ejde-673	850	8	)	)	PUNCT
ejde-673	850	9	=	=	SYM
ejde-673	850	10	tlpk(u	tlpk(u	NOUN
ejde-673	850	11	)	)	PUNCT
ejde-673	850	12	,	,	PUNCT
ejde-673	850	13	then	then	ADV
ejde-673	850	14	h	h	NOUN
ejde-673	850	15	is	be	AUX
ejde-673	850	16	differentiable	differentiable	ADJ
ejde-673	850	17	in	in	ADP
ejde-673	850	18	(	(	PUNCT
ejde-673	850	19	0,∞	0,∞	NOUN
ejde-673	850	20	)	)	PUNCT
ejde-673	850	21	and	and	CCONJ
ejde-673	850	22	h′(t	h′(t	NOUN
ejde-673	850	23	)	)	PUNCT
ejde-673	850	24	=	=	SYM
ejde-673	851	1	lptlp−1k(u	lptlp−1k(u	PROPN
ejde-673	851	2	)	)	PUNCT
ejde-673	851	3	>	>	X
ejde-673	851	4	0	0	NUM
ejde-673	851	5	,	,	PUNCT
ejde-673	851	6	so	so	ADV
ejde-673	851	7	h	h	NOUN
ejde-673	851	8	is	be	AUX
ejde-673	851	9	strictly	strictly	ADV
ejde-673	851	10	monotone	monotone	ADJ
ejde-673	851	11	increasing	increase	VERB
ejde-673	851	12	and	and	CCONJ
ejde-673	851	13	clearly	clearly	ADV
ejde-673	851	14	h(t	h(t	ADJ
ejde-673	851	15	)	)	PUNCT
ejde-673	852	1	→	→	SYM
ejde-673	852	2	0	0	NUM
ejde-673	852	3	as	as	ADP
ejde-673	852	4	t	t	PROPN
ejde-673	852	5	→	→	SYM
ejde-673	852	6	+0	+0	ADP
ejde-673	852	7	and	and	CCONJ
ejde-673	852	8	h(t	h(t	NUM
ejde-673	852	9	)	)	PUNCT
ejde-673	852	10	→	→	SYM
ejde-673	852	11	∞	∞	PROPN
ejde-673	852	12	as	as	ADP
ejde-673	852	13	t	t	PROPN
ejde-673	852	14	→	→	SYM
ejde-673	852	15	∞.	∞.	PROPN
ejde-673	852	16	hence	hence	ADV
ejde-673	852	17	for	for	ADP
ejde-673	852	18	any	any	DET
ejde-673	852	19	α	α	NOUN
ejde-673	852	20	>	>	X
ejde-673	852	21	0	0	NUM
ejde-673	852	22	,	,	PUNCT
ejde-673	852	23	there	there	PRON
ejde-673	852	24	exists	exist	VERB
ejde-673	852	25	unique	unique	ADJ
ejde-673	852	26	t(α	t(α	NOUN
ejde-673	852	27	)	)	PUNCT
ejde-673	852	28	>	>	X
ejde-673	852	29	0	0	NUM
ejde-673	853	1	such	such	ADJ
ejde-673	853	2	that	that	SCONJ
ejde-673	853	3	t(α)u	t(α)u	DET
ejde-673	853	4	∈	∈	PROPN
ejde-673	853	5	mα	mα	PROPN
ejde-673	853	6	.	.	PUNCT
ejde-673	853	7	clearly	clearly	ADV
ejde-673	853	8	t(α	t(α	NOUN
ejde-673	853	9	)	)	PUNCT
ejde-673	853	10	→	→	SYM
ejde-673	853	11	0	0	NUM
ejde-673	853	12	as	as	ADP
ejde-673	853	13	α	α	NOUN
ejde-673	853	14	→	→	SYM
ejde-673	853	15	+0	+0	ADP
ejde-673	853	16	and	and	CCONJ
ejde-673	853	17	t(α	t(α	NOUN
ejde-673	853	18	)	)	PUNCT
ejde-673	853	19	→	→	SYM
ejde-673	853	20	∞	∞	PROPN
ejde-673	853	21	as	as	ADP
ejde-673	853	22	α	α	PROPN
ejde-673	853	23	→	→	SYM
ejde-673	853	24	∞.	∞.	PROPN
ejde-673	853	25	so	so	ADV
ejde-673	853	26	there	there	PRON
ejde-673	853	27	exists	exist	VERB
ejde-673	853	28	α0	α0	ADJ
ejde-673	853	29	>	>	X
ejde-673	853	30	1	1	NUM
ejde-673	853	31	such	such	ADJ
ejde-673	853	32	that	that	PRON
ejde-673	853	33	for	for	ADP
ejde-673	853	34	any	any	DET
ejde-673	853	35	α	α	NOUN
ejde-673	853	36	∈	∈	PROPN
ejde-673	853	37	(	(	PUNCT
ejde-673	853	38	α0,∞	α0,∞	NOUN
ejde-673	853	39	)	)	PUNCT
ejde-673	853	40	,	,	PUNCT
ejde-673	853	41	max{1	max{1	NOUN
ejde-673	853	42	,	,	PUNCT
ejde-673	853	43	(	(	PUNCT
ejde-673	853	44	2ε−1cφ(u)l	2ε−1cφ(u)l	NUM
ejde-673	853	45	k(u	k(u	NOUN
ejde-673	853	46	)	)	PUNCT
ejde-673	853	47	)	)	PUNCT
ejde-673	853	48	1/(lε0	1/(lε0	NUM
ejde-673	853	49	)	)	PUNCT
ejde-673	853	50	}	}	PUNCT
ejde-673	853	51	<	<	X
ejde-673	853	52	t(α	t(α	NOUN
ejde-673	853	53	)	)	PUNCT
ejde-673	853	54	.	.	PUNCT
ejde-673	854	1	let	let	VERB
ejde-673	854	2	u0	u0	ADJ
ejde-673	854	3	be	be	AUX
ejde-673	854	4	the	the	DET
ejde-673	854	5	eigenfunction	eigenfunction	NOUN
ejde-673	854	6	associated	associate	VERB
ejde-673	854	7	with	with	ADP
ejde-673	854	8	λ(1,α	λ(1,α	PROPN
ejde-673	854	9	)	)	PUNCT
ejde-673	854	10	.	.	PUNCT
ejde-673	855	1	then	then	ADV
ejde-673	855	2	from	from	ADP
ejde-673	855	3	(	(	PUNCT
ejde-673	855	4	a1	a1	PROPN
ejde-673	855	5	’	'	PUNCT
ejde-673	855	6	)	)	PUNCT
ejde-673	855	7	we	we	PRON
ejde-673	855	8	have	have	VERB
ejde-673	855	9	λ(1,α	λ(1,α	NUM
ejde-673	855	10	)	)	PUNCT
ejde-673	856	1	=	=	SYM
ejde-673	856	2	m(φ(u0	m(φ(u0	NOUN
ejde-673	856	3	)	)	PUNCT
ejde-673	856	4	)	)	PUNCT
ejde-673	857	1	∫	∫	PROPN
ejde-673	857	2	ω	ω	NUM
ejde-673	857	3	a(x,∇u0(x	a(x,∇u0(x	PROPN
ejde-673	857	4	)	)	PUNCT
ejde-673	857	5	)	)	PUNCT
ejde-673	857	6	·	·	PUNCT
ejde-673	857	7	∇u0(x	∇u0(x	NOUN
ejde-673	857	8	)	)	PUNCT
ejde-673	857	9	dx∫	dx∫	PROPN
ejde-673	857	10	γ2	γ2	PROPN
ejde-673	857	11	g(x	g(x	PROPN
ejde-673	857	12	,	,	PUNCT
ejde-673	857	13	u0(x))u0(x	u0(x))u0(x	NOUN
ejde-673	857	14	)	)	PUNCT
ejde-673	857	15	dσx	dσx	NOUN
ejde-673	857	16	28	28	NUM
ejde-673	857	17	j.	j.	PROPN
ejde-673	857	18	aramaki	aramaki	PROPN
ejde-673	857	19	ejde-2025/17	ejde-2025/17	PROPN
ejde-673	857	20	≤	≤	PROPN
ejde-673	857	21	m1φ(u0	m1φ(u0	PROPN
ejde-673	857	22	)	)	PUNCT
ejde-673	858	1	l−1	l−1	PROPN
ejde-673	858	2	∫	∫	PROPN
ejde-673	858	3	ω	ω	PROPN
ejde-673	858	4	p(x)a(x,∇u0(x	p(x)a(x,∇u0(x	PROPN
ejde-673	858	5	)	)	PUNCT
ejde-673	858	6	)	)	PUNCT
ejde-673	859	1	dx	dx	PROPN
ejde-673	859	2	lp−	lp−	PROPN
ejde-673	859	3	∫	∫	PROPN
ejde-673	859	4	γ2	γ2	PROPN
ejde-673	859	5	g(x	g(x	PROPN
ejde-673	859	6	,	,	PUNCT
ejde-673	859	7	u0(x))dσx	u0(x))dσx	NUM
ejde-673	859	8	≤	≤	NOUN
ejde-673	859	9	m1p	m1p	X
ejde-673	860	1	+	+	NOUN
ejde-673	860	2	φ(u0	φ(u0	NOUN
ejde-673	860	3	)	)	PUNCT
ejde-673	860	4	l	l	NOUN
ejde-673	860	5	lp−α	lp−α	PROPN
ejde-673	860	6	≤	≤	NUM
ejde-673	860	7	m1p	m1p	PRON
ejde-673	860	8	+	+	NOUN
ejde-673	860	9	ψ(u0	ψ(u0	X
ejde-673	860	10	)	)	PUNCT
ejde-673	860	11	m0p−α	m0p−α	X
ejde-673	860	12	.	.	PUNCT
ejde-673	861	1	by	by	ADP
ejde-673	861	2	lemma	lemma	PROPN
ejde-673	861	3	4.2	4.2	NUM
ejde-673	861	4	,	,	PUNCT
ejde-673	861	5	since	since	SCONJ
ejde-673	861	6	ψ(u0	ψ(u0	NUM
ejde-673	861	7	)	)	PUNCT
ejde-673	861	8	=	=	SYM
ejde-673	861	9	c(1,α	c(1,α	PROPN
ejde-673	861	10	)	)	PUNCT
ejde-673	861	11	=	=	PUNCT
ejde-673	861	12	inf{ψ(u);u	inf{ψ(u);u	PROPN
ejde-673	861	13	∈	∈	PROPN
ejde-673	861	14	mα	mα	PROPN
ejde-673	861	15	}	}	PUNCT
ejde-673	861	16	,	,	PUNCT
ejde-673	861	17	we	we	PRON
ejde-673	861	18	have	have	VERB
ejde-673	861	19	ψ(u0	ψ(u0	NUM
ejde-673	861	20	)	)	PUNCT
ejde-673	861	21	≤	≤	NOUN
ejde-673	861	22	ψ(t(α)u	ψ(t(α)u	NOUN
ejde-673	861	23	)	)	PUNCT
ejde-673	861	24	.	.	PUNCT
ejde-673	862	1	hence	hence	ADV
ejde-673	862	2	λ(1,α	λ(1,α	NUM
ejde-673	862	3	)	)	PUNCT
ejde-673	863	1	≤	≤	PUNCT
ejde-673	863	2	m1p	m1p	PRON
ejde-673	863	3	+	+	NOUN
ejde-673	863	4	ψ(t(α)u	ψ(t(α)u	NOUN
ejde-673	863	5	)	)	PUNCT
ejde-673	863	6	m0p−α	m0p−α	X
ejde-673	864	1	=	=	SYM
ejde-673	864	2	m1p	m1p	NOUN
ejde-673	864	3	+	+	CCONJ
ejde-673	864	4	m0p−	m0p−	NOUN
ejde-673	864	5	ψ(t(α)u	ψ(t(α)u	NOUN
ejde-673	864	6	)	)	PUNCT
ejde-673	864	7	k(t(α)u	k(t(α)u	PROPN
ejde-673	864	8	)	)	PUNCT
ejde-673	864	9	.	.	PUNCT
ejde-673	865	1	thus	thus	ADV
ejde-673	865	2	using	use	VERB
ejde-673	865	3	(	(	PUNCT
ejde-673	865	4	3.1	3.1	NUM
ejde-673	865	5	)	)	PUNCT
ejde-673	865	6	,	,	PUNCT
ejde-673	865	7	we	we	PRON
ejde-673	865	8	have	have	VERB
ejde-673	865	9	λ(1,α	λ(1,α	NOUN
ejde-673	865	10	)	)	PUNCT
ejde-673	865	11	≤	≤	NUM
ejde-673	866	1	m1p	m1p	NOUN
ejde-673	866	2	+	+	CCONJ
ejde-673	866	3	m0p−	m0p−	NOUN
ejde-673	866	4	ψ(t(α)u	ψ(t(α)u	NOUN
ejde-673	866	5	)	)	PUNCT
ejde-673	866	6	k(t(α)u	k(t(α)u	PROPN
ejde-673	866	7	)	)	PUNCT
ejde-673	866	8	≤	≤	PUNCT
ejde-673	866	9	m1p	m1p	NOUN
ejde-673	867	1	+	+	CCONJ
ejde-673	867	2	m0p−	m0p−	NOUN
ejde-673	867	3	m1	m1	PROPN
ejde-673	867	4	l	l	PROPN
ejde-673	867	5	(	(	PUNCT
ejde-673	867	6	∫	∫	PROPN
ejde-673	867	7	ω	ω	PROPN
ejde-673	867	8	a(x	a(x	PROPN
ejde-673	867	9	,	,	PUNCT
ejde-673	867	10	t(α)∇u(x	t(α)∇u(x	PROPN
ejde-673	867	11	)	)	PUNCT
ejde-673	867	12	)	)	PUNCT
ejde-673	867	13	dx	dx	PROPN
ejde-673	867	14	)	)	PUNCT
ejde-673	867	15	l∫	l∫	ADJ
ejde-673	867	16	γ2	γ2	ADJ
ejde-673	867	17	g(x	g(x	NOUN
ejde-673	867	18	,	,	PUNCT
ejde-673	867	19	t(α)u(x))dσx	t(α)u(x))dσx	NOUN
ejde-673	867	20	≤	≤	PROPN
ejde-673	867	21	c	c	X
ejde-673	867	22	(	(	PUNCT
ejde-673	867	23	∫	∫	PROPN
ejde-673	867	24	bω(x0,δ/2)\bω(γ2,δ1	bω(x0,δ/2)\bω(γ2,δ1	PROPN
ejde-673	867	25	)	)	PUNCT
ejde-673	867	26	a(x	a(x	PROPN
ejde-673	867	27	,	,	PUNCT
ejde-673	867	28	t(α)∇u(x	t(α)∇u(x	NOUN
ejde-673	867	29	)	)	PUNCT
ejde-673	867	30	)	)	PUNCT
ejde-673	867	31	dx	dx	PROPN
ejde-673	867	32	)	)	PUNCT
ejde-673	867	33	l∫	l∫	ADJ
ejde-673	867	34	bγ2	bγ2	NOUN
ejde-673	867	35	(	(	PUNCT
ejde-673	867	36	x0,δ/2	x0,δ/2	PROPN
ejde-673	867	37	)	)	PUNCT
ejde-673	867	38	g(x	g(x	NOUN
ejde-673	867	39	,	,	PUNCT
ejde-673	867	40	t(α)u(x))dσx	t(α)u(x))dσx	PROPN
ejde-673	867	41	+	+	CCONJ
ejde-673	867	42	c	c	PROPN
ejde-673	867	43	(	(	PUNCT
ejde-673	867	44	∫	∫	PROPN
ejde-673	867	45	bω(x0,δ/2)∩bω(γ2,δ1	bω(x0,δ/2)∩bω(γ2,δ1	NOUN
ejde-673	867	46	)	)	PUNCT
ejde-673	867	47	a(x	a(x	PROPN
ejde-673	867	48	,	,	PUNCT
ejde-673	867	49	t(α)∇u(x	t(α)∇u(x	NOUN
ejde-673	867	50	)	)	PUNCT
ejde-673	867	51	)	)	PUNCT
ejde-673	867	52	dx	dx	PROPN
ejde-673	867	53	)	)	PUNCT
ejde-673	867	54	l∫	l∫	ADJ
ejde-673	867	55	bγ2	bγ2	NOUN
ejde-673	867	56	(	(	PUNCT
ejde-673	867	57	x0,δ/2	x0,δ/2	PROPN
ejde-673	867	58	)	)	PUNCT
ejde-673	867	59	g(x	g(x	NOUN
ejde-673	867	60	,	,	PUNCT
ejde-673	867	61	t(α)u(x))dσx	t(α)u(x))dσx	NOUN
ejde-673	867	62	≤	≤	PROPN
ejde-673	867	63	c	c	PROPN
ejde-673	867	64	(	(	PUNCT
ejde-673	867	65	∫	∫	PROPN
ejde-673	867	66	bω(x0,δ/2)\bω(γ2,δ1	bω(x0,δ/2)\bω(γ2,δ1	PROPN
ejde-673	867	67	)	)	PUNCT
ejde-673	867	68	t(α)p(x)a(x,∇u(x	t(α)p(x)a(x,∇u(x	PROPN
ejde-673	867	69	)	)	PUNCT
ejde-673	867	70	)	)	PUNCT
ejde-673	867	71	dx	dx	PROPN
ejde-673	867	72	)	)	PUNCT
ejde-673	867	73	l	l	NOUN
ejde-673	867	74	t(α)lp	t(α)lp	NUM
ejde-673	867	75	∫	∫	PROPN
ejde-673	867	76	bγ2	bγ2	NOUN
ejde-673	867	77	(	(	PUNCT
ejde-673	867	78	x0,δ/2	x0,δ/2	PROPN
ejde-673	867	79	)	)	PUNCT
ejde-673	867	80	g(x	g(x	NOUN
ejde-673	867	81	,	,	PUNCT
ejde-673	867	82	u(x))dσx	u(x))dσx	X
ejde-673	867	83	+	+	CCONJ
ejde-673	867	84	c	c	X
ejde-673	867	85	(	(	PUNCT
ejde-673	867	86	∫	∫	PROPN
ejde-673	867	87	bω(x0,δ/2)∩bω(γ2,δ1	bω(x0,δ/2)∩bω(γ2,δ1	PROPN
ejde-673	867	88	)	)	PUNCT
ejde-673	867	89	t(α)p(x)a(x,∇u(x	t(α)p(x)a(x,∇u(x	PROPN
ejde-673	867	90	)	)	PUNCT
ejde-673	867	91	)	)	PUNCT
ejde-673	867	92	dx	dx	PROPN
ejde-673	867	93	)	)	PUNCT
ejde-673	867	94	l	l	NOUN
ejde-673	867	95	t(α)lp	t(α)lp	NUM
ejde-673	867	96	∫	∫	PROPN
ejde-673	867	97	bγ2	bγ2	NOUN
ejde-673	867	98	(	(	PUNCT
ejde-673	867	99	x0,δ/2	x0,δ/2	PROPN
ejde-673	867	100	)	)	PUNCT
ejde-673	867	101	g(x	g(x	NOUN
ejde-673	867	102	,	,	PUNCT
ejde-673	867	103	u(x))dσx	u(x))dσx	X
ejde-673	867	104	=	=	PUNCT
ejde-673	867	105	c	c	X
ejde-673	867	106	(	(	PUNCT
ejde-673	867	107	∫	∫	PROPN
ejde-673	867	108	bω(x0,δ/2)\bω(γ2,δ1	bω(x0,δ/2)\bω(γ2,δ1	PROPN
ejde-673	867	109	)	)	PUNCT
ejde-673	867	110	t(α)p(x)−pa(x,∇u(x	t(α)p(x)−pa(x,∇u(x	NOUN
ejde-673	867	111	)	)	PUNCT
ejde-673	867	112	)	)	PUNCT
ejde-673	867	113	dx	dx	PROPN
ejde-673	867	114	)	)	PUNCT
ejde-673	867	115	l∫	l∫	ADJ
ejde-673	867	116	bγ2	bγ2	NOUN
ejde-673	867	117	(	(	PUNCT
ejde-673	867	118	x0,δ/2	x0,δ/2	PROPN
ejde-673	867	119	)	)	PUNCT
ejde-673	867	120	g(x	g(x	NOUN
ejde-673	867	121	,	,	PUNCT
ejde-673	867	122	u(x))dσx	u(x))dσx	X
ejde-673	867	123	+	+	CCONJ
ejde-673	867	124	c	c	X
ejde-673	867	125	(	(	PUNCT
ejde-673	867	126	∫	∫	PROPN
ejde-673	867	127	bω(x0,δ/2)∩bω(γ2,δ1	bω(x0,δ/2)∩bω(γ2,δ1	PROPN
ejde-673	867	128	)	)	PUNCT
ejde-673	867	129	t(α)p(x)−pa(x,∇u(x	t(α)p(x)−pa(x,∇u(x	NOUN
ejde-673	867	130	)	)	PUNCT
ejde-673	867	131	)	)	PUNCT
ejde-673	867	132	dx	dx	PROPN
ejde-673	867	133	)	)	PUNCT
ejde-673	867	134	l∫	l∫	ADJ
ejde-673	867	135	bγ2	bγ2	NOUN
ejde-673	867	136	(	(	PUNCT
ejde-673	867	137	x0,δ/2	x0,δ/2	PROPN
ejde-673	867	138	)	)	PUNCT
ejde-673	867	139	g(x	g(x	NOUN
ejde-673	867	140	,	,	PUNCT
ejde-673	867	141	u(x))dσx	u(x))dσx	NOUN
ejde-673	867	142	≤	≤	PUNCT
ejde-673	867	143	ct(α)−lε0	ct(α)−lε0	PROPN
ejde-673	867	144	φ(u)l	φ(u)l	PROPN
ejde-673	867	145	k(u	k(u	NOUN
ejde-673	867	146	)	)	PUNCT
ejde-673	868	1	+	+	CCONJ
ejde-673	868	2	c	c	X
ejde-673	868	3	(	(	PUNCT
ejde-673	868	4	∫	∫	PROPN
ejde-673	868	5	bω(γ2,δ1	bω(γ2,δ1	PROPN
ejde-673	868	6	)	)	PUNCT
ejde-673	868	7	a(x,∇u(x	a(x,∇u(x	PROPN
ejde-673	868	8	)	)	PUNCT
ejde-673	868	9	)	)	PUNCT
ejde-673	868	10	dx	dx	PROPN
ejde-673	868	11	)	)	PUNCT
ejde-673	868	12	l	l	NOUN
ejde-673	868	13	k(u	k(u	X
ejde-673	868	14	)	)	PUNCT
ejde-673	868	15	<	<	X
ejde-673	868	16	ε	ε	PROPN
ejde-673	868	17	2	2	NUM
ejde-673	868	18	+	+	CCONJ
ejde-673	868	19	ε	ε	PROPN
ejde-673	868	20	2	2	NUM
ejde-673	868	21	=	=	SYM
ejde-673	868	22	ε	ε	PROPN
ejde-673	868	23	.	.	PUNCT
ejde-673	868	24	therefore	therefore	ADV
ejde-673	868	25	,	,	PUNCT
ejde-673	868	26	0	0	PUNCT
ejde-673	868	27	<	<	X
ejde-673	868	28	λ(1,α	λ(1,α	NOUN
ejde-673	868	29	)	)	PUNCT
ejde-673	868	30	<	<	X
ejde-673	868	31	ε	ε	PROPN
ejde-673	868	32	for	for	ADP
ejde-673	868	33	all	all	DET
ejde-673	868	34	α	α	PROPN
ejde-673	868	35	>	>	X
ejde-673	868	36	α0	α0	PROPN
ejde-673	868	37	.	.	PUNCT
ejde-673	869	1	since	since	SCONJ
ejde-673	869	2	ε	ε	PROPN
ejde-673	869	3	>	>	X
ejde-673	869	4	0	0	NUM
ejde-673	869	5	is	be	AUX
ejde-673	869	6	arbitrary	arbitrary	ADJ
ejde-673	869	7	,	,	PUNCT
ejde-673	869	8	we	we	PRON
ejde-673	869	9	have	have	VERB
ejde-673	869	10	limα→∞	limα→∞	PROPN
ejde-673	869	11	λ(1,α	λ(1,α	NOUN
ejde-673	869	12	)	)	PUNCT
ejde-673	870	1	=	=	SYM
ejde-673	870	2	0	0	X
ejde-673	870	3	.	.	PUNCT
ejde-673	870	4	□	□	PUNCT
ejde-673	870	5	remark	remark	NOUN
ejde-673	870	6	4.6	4.6	NUM
ejde-673	870	7	.	.	PUNCT
ejde-673	871	1	(	(	PUNCT
ejde-673	871	2	1	1	X
ejde-673	871	3	)	)	PUNCT
ejde-673	871	4	if	if	SCONJ
ejde-673	871	5	p(x	p(x	VERB
ejde-673	871	6	)	)	PUNCT
ejde-673	871	7	=	=	SYM
ejde-673	871	8	p	p	X
ejde-673	871	9	(	(	PUNCT
ejde-673	871	10	a	a	DET
ejde-673	871	11	constant	constant	ADJ
ejde-673	871	12	)	)	PUNCT
ejde-673	871	13	in	in	ADP
ejde-673	871	14	ω	ω	PROPN
ejde-673	871	15	,	,	PUNCT
ejde-673	871	16	then	then	ADV
ejde-673	871	17	it	it	PRON
ejde-673	871	18	is	be	AUX
ejde-673	871	19	well	well	ADV
ejde-673	871	20	known	know	VERB
ejde-673	871	21	that	that	SCONJ
ejde-673	871	22	λ∗	λ∗	NOUN
ejde-673	872	1	=	=	PUNCT
ejde-673	872	2	λ(1,α	λ(1,α	NOUN
ejde-673	872	3	)	)	PUNCT
ejde-673	873	1	=	=	SYM
ejde-673	874	1	λ1	λ1	ADJ
ejde-673	874	2	and	and	CCONJ
ejde-673	874	3	so	so	ADV
ejde-673	874	4	λ∗	λ∗	PROPN
ejde-673	874	5	is	be	AUX
ejde-673	874	6	a	a	DET
ejde-673	874	7	principal	principal	ADJ
ejde-673	874	8	eigenvalue	eigenvalue	NOUN
ejde-673	874	9	.	.	PUNCT
ejde-673	875	1	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	875	2	eigenvalue	eigenvalue	VERB
ejde-673	875	3	problems	problem	NOUN
ejde-673	875	4	for	for	ADP
ejde-673	875	5	kirchhoff	kirchhoff	NOUN
ejde-673	875	6	-	-	PUNCT
ejde-673	875	7	type	type	NOUN
ejde-673	875	8	equations	equation	NOUN
ejde-673	875	9	29	29	NUM
ejde-673	875	10	(	(	PUNCT
ejde-673	875	11	2	2	NUM
ejde-673	875	12	)	)	PUNCT
ejde-673	875	13	for	for	ADP
ejde-673	875	14	a	a	DET
ejde-673	875	15	variable	variable	ADJ
ejde-673	875	16	exponent	exponent	NOUN
ejde-673	875	17	p(x	p(x	PROPN
ejde-673	875	18	)	)	PUNCT
ejde-673	875	19	,	,	PUNCT
ejde-673	875	20	under	under	ADP
ejde-673	875	21	some	some	DET
ejde-673	875	22	assumptions	assumption	NOUN
ejde-673	875	23	,	,	PUNCT
ejde-673	875	24	λ∗	λ∗	PROPN
ejde-673	875	25	=	=	PUNCT
ejde-673	875	26	0	0	X
ejde-673	875	27	.	.	PUNCT
ejde-673	876	1	this	this	PRON
ejde-673	876	2	means	mean	VERB
ejde-673	876	3	that	that	SCONJ
ejde-673	876	4	under	under	ADP
ejde-673	876	5	some	some	DET
ejde-673	876	6	assumptions	assumption	NOUN
ejde-673	876	7	,	,	PUNCT
ejde-673	876	8	there	there	PRON
ejde-673	876	9	does	do	AUX
ejde-673	876	10	not	not	PART
ejde-673	876	11	exist	exist	VERB
ejde-673	876	12	a	a	DET
ejde-673	876	13	principal	principal	ADJ
ejde-673	876	14	eigenvalue	eigenvalue	NOUN
ejde-673	876	15	and	and	CCONJ
ejde-673	876	16	the	the	DET
ejde-673	876	17	set	set	NOUN
ejde-673	876	18	of	of	ADP
ejde-673	876	19	eigenvalues	eigenvalue	NOUN
ejde-673	876	20	is	be	AUX
ejde-673	876	21	not	not	PART
ejde-673	876	22	closed	closed	ADJ
ejde-673	876	23	.	.	PUNCT
ejde-673	877	1	(	(	PUNCT
ejde-673	877	2	3	3	X
ejde-673	877	3	)	)	PUNCT
ejde-673	877	4	for	for	ADP
ejde-673	877	5	a	a	DET
ejde-673	877	6	variable	variable	ADJ
ejde-673	877	7	exponent	exponent	NOUN
ejde-673	877	8	p(x	p(x	PROPN
ejde-673	877	9	)	)	PUNCT
ejde-673	877	10	,	,	PUNCT
ejde-673	877	11	under	under	ADP
ejde-673	877	12	some	some	DET
ejde-673	877	13	assumptions	assumption	NOUN
ejde-673	877	14	,	,	PUNCT
ejde-673	877	15	we	we	PRON
ejde-673	877	16	have	have	VERB
ejde-673	877	17	λ∗	λ∗	PROPN
ejde-673	877	18	>	>	X
ejde-673	877	19	0	0	X
ejde-673	877	20	.	.	PUNCT
ejde-673	878	1	references	reference	NOUN
ejde-673	878	2	[	[	X
ejde-673	878	3	1	1	NUM
ejde-673	878	4	]	]	PUNCT
ejde-673	878	5	g.	g.	PROPN
ejde-673	878	6	a.	a.	PROPN
ejde-673	878	7	afrouzi	afrouzi	PROPN
ejde-673	878	8	,	,	PUNCT
ejde-673	878	9	m.	m.	NOUN
ejde-673	878	10	mirzapour	mirzapour	PROPN
ejde-673	878	11	;	;	PUNCT
ejde-673	878	12	eigenvalue	eigenvalue	NOUN
ejde-673	878	13	probelms	probelm	NOUN
ejde-673	878	14	for	for	ADP
ejde-673	878	15	p(x)-kirchhoff	p(x)-kirchhoff	PROPN
ejde-673	878	16	type	type	NOUN
ejde-673	878	17	equations	equation	NOUN
ejde-673	878	18	,	,	PUNCT
ejde-673	878	19	electr	electr	PROPN
ejde-673	878	20	.	.	PUNCT
ejde-673	879	1	j.	j.	PROPN
ejde-673	879	2	differ	differ	VERB
ejde-673	879	3	.	.	PUNCT
ejde-673	880	1	equa	equa	NOUN
ejde-673	880	2	.	.	PUNCT
ejde-673	881	1	,	,	PUNCT
ejde-673	881	2	vol	vol	NOUN
ejde-673	881	3	.	.	PUNCT
ejde-673	882	1	2013(253	2013(253	NUM
ejde-673	882	2	)	)	PUNCT
ejde-673	883	1	,	,	PUNCT
ejde-673	883	2	(	(	PUNCT
ejde-673	883	3	2013	2013	NUM
ejde-673	883	4	)	)	PUNCT
ejde-673	883	5	,	,	PUNCT
ejde-673	883	6	1	1	NUM
ejde-673	883	7	-	-	SYM
ejde-673	883	8	10	10	NUM
ejde-673	883	9	.	.	PUNCT
ejde-673	884	1	[	[	X
ejde-673	884	2	2	2	X
ejde-673	884	3	]	]	PUNCT
ejde-673	884	4	c.	c.	PROPN
ejde-673	884	5	alves	alves	PROPN
ejde-673	884	6	,	,	PUNCT
ejde-673	884	7	a.	a.	PROPN
ejde-673	884	8	moussoui	moussoui	PROPN
ejde-673	884	9	,	,	PUNCT
ejde-673	884	10	l.	l.	PROPN
ejde-673	884	11	tavares	tavares	PROPN
ejde-673	884	12	;	;	PUNCT
ejde-673	884	13	an	an	DET
ejde-673	884	14	elliptic	elliptic	ADJ
ejde-673	884	15	system	system	NOUN
ejde-673	884	16	with	with	ADP
ejde-673	884	17	logarithmic	logarithmic	ADJ
ejde-673	884	18	nonlinearity	nonlinearity	NOUN
ejde-673	884	19	,	,	PUNCT
ejde-673	884	20	advances	advance	NOUN
ejde-673	884	21	in	in	ADP
ejde-673	884	22	nonl	nonl	NOUN
ejde-673	884	23	.	.	PUNCT
ejde-673	885	1	anal	anal	PROPN
ejde-673	885	2	.	.	PUNCT
ejde-673	885	3	,	,	PUNCT
ejde-673	885	4	vol	vol	NOUN
ejde-673	885	5	.	.	PROPN
ejde-673	885	6	8	8	NUM
ejde-673	885	7	,	,	PUNCT
ejde-673	885	8	(	(	PUNCT
ejde-673	885	9	2019	2019	NUM
ejde-673	885	10	)	)	PUNCT
ejde-673	885	11	,	,	PUNCT
ejde-673	885	12	928	928	NUM
ejde-673	885	13	-	-	SYM
ejde-673	885	14	945	945	NUM
ejde-673	885	15	.	.	PUNCT
ejde-673	886	1	[	[	X
ejde-673	886	2	3	3	X
ejde-673	886	3	]	]	X
ejde-673	886	4	c.	c.	PROPN
ejde-673	886	5	alves	alves	PROPN
ejde-673	886	6	,	,	PUNCT
ejde-673	886	7	l.	l.	PROPN
ejde-673	886	8	s.	s.	PROPN
ejde-673	886	9	tavares	tavares	PROPN
ejde-673	886	10	;	;	PUNCT
ejde-673	886	11	a	a	DET
ejde-673	886	12	hardy	hardy	ADJ
ejde-673	886	13	-	-	PUNCT
ejde-673	886	14	littlewood	littlewood	NOUN
ejde-673	886	15	-	-	PUNCT
ejde-673	886	16	sobolev	sobolev	NOUN
ejde-673	886	17	-	-	PUNCT
ejde-673	886	18	type	type	NOUN
ejde-673	886	19	inequality	inequality	NOUN
ejde-673	886	20	for	for	ADP
ejde-673	886	21	variable	variable	ADJ
ejde-673	886	22	exponents	exponent	NOUN
ejde-673	886	23	and	and	CCONJ
ejde-673	886	24	applications	application	NOUN
ejde-673	886	25	to	to	PART
ejde-673	886	26	quasilinear	quasilinear	VERB
ejde-673	886	27	choquard	choquard	NOUN
ejde-673	886	28	equations	equation	NOUN
ejde-673	886	29	involving	involve	VERB
ejde-673	886	30	variable	variable	ADJ
ejde-673	886	31	exponent	exponent	NOUN
ejde-673	886	32	,	,	PUNCT
ejde-673	886	33	mediterranean	mediterranean	PROPN
ejde-673	886	34	j.	j.	PROPN
ejde-673	886	35	math	math	PROPN
ejde-673	886	36	.	.	PUNCT
ejde-673	886	37	,	,	PUNCT
ejde-673	886	38	vol	vol	NOUN
ejde-673	886	39	.	.	PROPN
ejde-673	886	40	16(2	16(2	NUM
ejde-673	886	41	)	)	PUNCT
ejde-673	886	42	,	,	PUNCT
ejde-673	886	43	(	(	PUNCT
ejde-673	886	44	2019	2019	NUM
ejde-673	886	45	)	)	PUNCT
ejde-673	886	46	,	,	PUNCT
ejde-673	886	47	paper	paper	NOUN
ejde-673	886	48	no	no	NOUN
ejde-673	886	49	.	.	PROPN
ejde-673	887	1	55	55	NUM
ejde-673	887	2	:	:	PUNCT
ejde-673	887	3	1	1	NUM
ejde-673	887	4	-	-	SYM
ejde-673	887	5	27	27	NUM
ejde-673	887	6	.	.	PUNCT
ejde-673	888	1	[	[	X
ejde-673	888	2	4	4	NUM
ejde-673	888	3	]	]	PUNCT
ejde-673	888	4	a.	a.	NOUN
ejde-673	888	5	anane	anane	PROPN
ejde-673	888	6	;	;	PUNCT
ejde-673	888	7	simplicité	simplicité	NOUN
ejde-673	888	8	et	et	NOUN
ejde-673	888	9	isolation	isolation	NOUN
ejde-673	888	10	de	de	X
ejde-673	888	11	la	la	X
ejde-673	888	12	premiére	premiére	PROPN
ejde-673	888	13	valeur	valeur	PROPN
ejde-673	888	14	propre	propre	PROPN
ejde-673	888	15	de	de	PROPN
ejde-673	888	16	p	p	PROPN
ejde-673	888	17	-	-	PUNCT
ejde-673	888	18	laplacian	laplacian	ADJ
ejde-673	888	19	avec	avec	X
ejde-673	888	20	poids	poids	PROPN
ejde-673	888	21	,	,	PUNCT
ejde-673	888	22	c.	c.	PROPN
ejde-673	888	23	r.	r.	PROPN
ejde-673	888	24	acad	acad	PROPN
ejde-673	888	25	.	.	PUNCT
ejde-673	889	1	sci	sci	PROPN
ejde-673	889	2	.	.	PROPN
ejde-673	889	3	paris	paris	PROPN
ejde-673	889	4	,	,	PUNCT
ejde-673	889	5	sér	sér	PROPN
ejde-673	889	6	.	.	PUNCT
ejde-673	890	1	i.	i.	PROPN
ejde-673	890	2	math	math	PROPN
ejde-673	890	3	.	.	PUNCT
ejde-673	891	1	,	,	PUNCT
ejde-673	891	2	vol	vol	NOUN
ejde-673	891	3	.	.	PROPN
ejde-673	891	4	305	305	NUM
ejde-673	891	5	,	,	PUNCT
ejde-673	891	6	(	(	PUNCT
ejde-673	891	7	1987	1987	NUM
ejde-673	891	8	)	)	PUNCT
ejde-673	891	9	,	,	PUNCT
ejde-673	891	10	725	725	NUM
ejde-673	891	11	-	-	SYM
ejde-673	891	12	728	728	NUM
ejde-673	891	13	.	.	PUNCT
ejde-673	892	1	[	[	X
ejde-673	892	2	5	5	X
ejde-673	892	3	]	]	PUNCT
ejde-673	892	4	j.	j.	PROPN
ejde-673	892	5	aramaki	aramaki	PROPN
ejde-673	892	6	;	;	PUNCT
ejde-673	892	7	existence	existence	NOUN
ejde-673	892	8	of	of	ADP
ejde-673	892	9	three	three	NUM
ejde-673	892	10	weak	weak	ADJ
ejde-673	892	11	solutions	solution	NOUN
ejde-673	892	12	for	for	ADP
ejde-673	892	13	a	a	DET
ejde-673	892	14	class	class	NOUN
ejde-673	892	15	of	of	ADP
ejde-673	892	16	nonlinear	nonlinear	ADJ
ejde-673	892	17	operators	operator	NOUN
ejde-673	892	18	involving	involve	VERB
ejde-673	892	19	p(x)-laplacian	p(x)-laplacian	PROPN
ejde-673	892	20	with	with	ADP
ejde-673	892	21	mixed	mixed	ADJ
ejde-673	892	22	boundary	boundary	ADJ
ejde-673	892	23	conditions	condition	NOUN
ejde-673	892	24	.	.	PUNCT
ejde-673	893	1	nonlinear	nonlinear	ADJ
ejde-673	893	2	funct	funct	NOUN
ejde-673	893	3	.	.	PUNCT
ejde-673	894	1	anal	anal	PROPN
ejde-673	894	2	.	.	PUNCT
ejde-673	894	3	appl	appl	PROPN
ejde-673	894	4	.	.	PROPN
ejde-673	894	5	,	,	PUNCT
ejde-673	894	6	vol	vol	NOUN
ejde-673	894	7	.	.	PUNCT
ejde-673	894	8	26(3	26(3	NUM
ejde-673	894	9	)	)	PUNCT
ejde-673	894	10	,	,	PUNCT
ejde-673	894	11	(	(	PUNCT
ejde-673	894	12	2021	2021	NUM
ejde-673	894	13	)	)	PUNCT
ejde-673	894	14	,	,	PUNCT
ejde-673	894	15	531	531	NUM
ejde-673	894	16	-	-	SYM
ejde-673	894	17	551	551	NUM
ejde-673	894	18	.	.	PUNCT
ejde-673	895	1	[	[	X
ejde-673	895	2	6	6	NUM
ejde-673	895	3	]	]	PUNCT
ejde-673	895	4	j.	j.	PROPN
ejde-673	895	5	aramaki	aramaki	PROPN
ejde-673	895	6	;	;	PUNCT
ejde-673	895	7	mixed	mixed	ADJ
ejde-673	895	8	boundary	boundary	ADJ
ejde-673	895	9	value	value	NOUN
ejde-673	895	10	problem	problem	NOUN
ejde-673	895	11	for	for	ADP
ejde-673	895	12	a	a	DET
ejde-673	895	13	class	class	NOUN
ejde-673	895	14	of	of	ADP
ejde-673	895	15	quasi	quasi	ADJ
ejde-673	895	16	-	-	ADJ
ejde-673	895	17	linear	linear	ADJ
ejde-673	895	18	elliptic	elliptic	ADJ
ejde-673	895	19	operators	operator	NOUN
ejde-673	895	20	containing	contain	VERB
ejde-673	895	21	p(·)-laplacian	p(·)-laplacian	NOUN
ejde-673	895	22	in	in	ADP
ejde-673	895	23	a	a	DET
ejde-673	895	24	variable	variable	ADJ
ejde-673	895	25	exponent	exponent	NOUN
ejde-673	895	26	sobolev	sobolev	NOUN
ejde-673	895	27	space	space	NOUN
ejde-673	895	28	,	,	PUNCT
ejde-673	895	29	adv	adv	PROPN
ejde-673	895	30	.	.	PUNCT
ejde-673	895	31	math	math	PROPN
ejde-673	895	32	.	.	PUNCT
ejde-673	896	1	sci	sci	PROPN
ejde-673	896	2	.	.	PUNCT
ejde-673	896	3	appl	appl	PROPN
ejde-673	896	4	.	.	PROPN
ejde-673	896	5	,	,	PUNCT
ejde-673	896	6	vol	vol	NOUN
ejde-673	896	7	.	.	PUNCT
ejde-673	897	1	31(2	31(2	NUM
ejde-673	897	2	)	)	PUNCT
ejde-673	897	3	,	,	PUNCT
ejde-673	897	4	(	(	PUNCT
ejde-673	897	5	2022	2022	NUM
ejde-673	897	6	)	)	PUNCT
ejde-673	897	7	,	,	PUNCT
ejde-673	897	8	207	207	NUM
ejde-673	897	9	-	-	SYM
ejde-673	897	10	239	239	NUM
ejde-673	897	11	.	.	PUNCT
ejde-673	898	1	[	[	X
ejde-673	898	2	7	7	X
ejde-673	898	3	]	]	X
ejde-673	898	4	j.	j.	PROPN
ejde-673	898	5	aramaki	aramaki	PROPN
ejde-673	898	6	;	;	PUNCT
ejde-673	898	7	existence	existence	NOUN
ejde-673	898	8	of	of	ADP
ejde-673	898	9	nontrivial	nontrivial	ADJ
ejde-673	898	10	weak	weak	ADJ
ejde-673	898	11	solutions	solution	NOUN
ejde-673	898	12	for	for	ADP
ejde-673	898	13	nonuniformly	nonuniformly	ADJ
ejde-673	898	14	elliptic	elliptic	ADJ
ejde-673	898	15	equation	equation	NOUN
ejde-673	898	16	with	with	ADP
ejde-673	898	17	mixed	mixed	ADJ
ejde-673	898	18	boundary	boundary	ADJ
ejde-673	898	19	condition	condition	NOUN
ejde-673	898	20	in	in	ADP
ejde-673	898	21	a	a	DET
ejde-673	898	22	variable	variable	ADJ
ejde-673	898	23	exponent	exponent	NOUN
ejde-673	898	24	sobolev	sobolev	NOUN
ejde-673	898	25	space	space	NOUN
ejde-673	898	26	,	,	PUNCT
ejde-673	898	27	electronic	electronic	ADJ
ejde-673	898	28	j.	j.	PROPN
ejde-673	898	29	qualitative	qualitative	PROPN
ejde-673	898	30	theory	theory	NOUN
ejde-673	898	31	differ	differ	VERB
ejde-673	898	32	.	.	PUNCT
ejde-673	899	1	eq	eq	ADP
ejde-673	899	2	.	.	PROPN
ejde-673	899	3	,	,	PUNCT
ejde-673	899	4	vol	vol	NOUN
ejde-673	899	5	.	.	PUNCT
ejde-673	899	6	2023(12	2023(12	NUM
ejde-673	899	7	)	)	PUNCT
ejde-673	899	8	,	,	PUNCT
ejde-673	899	9	(	(	PUNCT
ejde-673	899	10	2023	2023	NUM
ejde-673	899	11	)	)	PUNCT
ejde-673	899	12	,	,	PUNCT
ejde-673	899	13	1	1	NUM
ejde-673	899	14	-	-	SYM
ejde-673	899	15	22	22	NUM
ejde-673	899	16	.	.	PUNCT
ejde-673	900	1	[	[	X
ejde-673	900	2	8	8	X
ejde-673	900	3	]	]	X
ejde-673	900	4	j.	j.	PROPN
ejde-673	900	5	aramaki	aramaki	PROPN
ejde-673	900	6	;	;	PUNCT
ejde-673	900	7	existence	existence	NOUN
ejde-673	900	8	of	of	ADP
ejde-673	900	9	three	three	NUM
ejde-673	900	10	weak	weak	ADJ
ejde-673	900	11	solutions	solution	NOUN
ejde-673	900	12	for	for	ADP
ejde-673	900	13	the	the	DET
ejde-673	900	14	kirchhoff	kirchhoff	NOUN
ejde-673	900	15	-	-	PUNCT
ejde-673	900	16	type	type	NOUN
ejde-673	900	17	problem	problem	NOUN
ejde-673	900	18	with	with	ADP
ejde-673	900	19	mixed	mixed	ADJ
ejde-673	900	20	boundary	boundary	ADJ
ejde-673	900	21	condition	condition	NOUN
ejde-673	900	22	in	in	ADP
ejde-673	900	23	a	a	DET
ejde-673	900	24	variable	variable	ADJ
ejde-673	900	25	exponent	exponent	NOUN
ejde-673	900	26	sobolev	sobolev	NOUN
ejde-673	900	27	space	space	NOUN
ejde-673	900	28	,	,	PUNCT
ejde-673	900	29	east	east	PROPN
ejde-673	900	30	-	-	PUNCT
ejde-673	900	31	west	west	PROPN
ejde-673	900	32	j.	j.	PROPN
ejde-673	900	33	math	math	PROPN
ejde-673	900	34	.	.	PUNCT
ejde-673	900	35	,	,	PUNCT
ejde-673	900	36	vol	vol	NOUN
ejde-673	900	37	.	.	PUNCT
ejde-673	901	1	24(2	24(2	NUM
ejde-673	901	2	)	)	PUNCT
ejde-673	901	3	,	,	PUNCT
ejde-673	901	4	(	(	PUNCT
ejde-673	901	5	2023	2023	NUM
ejde-673	901	6	)	)	PUNCT
ejde-673	901	7	,	,	PUNCT
ejde-673	901	8	89	89	NUM
ejde-673	901	9	-	-	SYM
ejde-673	901	10	117	117	NUM
ejde-673	901	11	.	.	PUNCT
ejde-673	902	1	[	[	X
ejde-673	902	2	9	9	X
ejde-673	902	3	]	]	X
ejde-673	902	4	j.	j.	PROPN
ejde-673	902	5	aramaki	aramaki	PROPN
ejde-673	902	6	;	;	PUNCT
ejde-673	902	7	existence	existence	NOUN
ejde-673	902	8	of	of	ADP
ejde-673	902	9	three	three	NUM
ejde-673	902	10	weak	weak	ADJ
ejde-673	902	11	solutions	solution	NOUN
ejde-673	902	12	for	for	ADP
ejde-673	902	13	a	a	DET
ejde-673	902	14	nonlinear	nonlinear	ADJ
ejde-673	902	15	problem	problem	NOUN
ejde-673	902	16	with	with	ADP
ejde-673	902	17	mixed	mixed	ADJ
ejde-673	902	18	boundary	boundary	ADJ
ejde-673	902	19	condition	condition	NOUN
ejde-673	902	20	in	in	ADP
ejde-673	902	21	a	a	DET
ejde-673	902	22	variable	variable	ADJ
ejde-673	902	23	exponent	exponent	NOUN
ejde-673	902	24	soboev	soboev	NOUN
ejde-673	902	25	space	space	NOUN
ejde-673	902	26	,	,	PUNCT
ejde-673	902	27	j.	j.	PROPN
ejde-673	902	28	analysis	analysis	PROPN
ejde-673	902	29	,	,	PUNCT
ejde-673	902	30	vol	vol	NOUN
ejde-673	902	31	.	.	PUNCT
ejde-673	902	32	32(2	32(2	NUM
ejde-673	902	33	)	)	PUNCT
ejde-673	902	34	,	,	PUNCT
ejde-673	902	35	(	(	PUNCT
ejde-673	902	36	2024	2024	NUM
ejde-673	902	37	)	)	PUNCT
ejde-673	902	38	,	,	PUNCT
ejde-673	902	39	733	733	NUM
ejde-673	902	40	-	-	SYM
ejde-673	902	41	755	755	NUM
ejde-673	902	42	.	.	PUNCT
ejde-673	903	1	[	[	X
ejde-673	903	2	10	10	NUM
ejde-673	903	3	]	]	X
ejde-673	903	4	p.	p.	NOUN
ejde-673	903	5	g.	g.	PROPN
ejde-673	903	6	ciarlet	ciarlet	PROPN
ejde-673	903	7	,	,	PUNCT
ejde-673	903	8	g.	g.	PROPN
ejde-673	903	9	dinca	dinca	PROPN
ejde-673	903	10	;	;	PUNCT
ejde-673	903	11	a	a	DET
ejde-673	903	12	poincaré	poincaré	ADJ
ejde-673	903	13	inequality	inequality	NOUN
ejde-673	903	14	in	in	ADP
ejde-673	903	15	a	a	DET
ejde-673	903	16	sobolev	sobolev	ADJ
ejde-673	903	17	space	space	NOUN
ejde-673	903	18	with	with	ADP
ejde-673	903	19	a	a	DET
ejde-673	903	20	variable	variable	ADJ
ejde-673	903	21	exponent	exponent	NOUN
ejde-673	903	22	,	,	PUNCT
ejde-673	903	23	chin	chin	PROPN
ejde-673	903	24	.	.	PUNCT
ejde-673	904	1	ann	ann	PROPN
ejde-673	904	2	.	.	PUNCT
ejde-673	904	3	math	math	PROPN
ejde-673	904	4	.	.	PUNCT
ejde-673	905	1	,	,	PUNCT
ejde-673	905	2	vol	vol	NOUN
ejde-673	905	3	.	.	PUNCT
ejde-673	906	1	32b(3	32b(3	NUM
ejde-673	906	2	)	)	PUNCT
ejde-673	906	3	,	,	PUNCT
ejde-673	906	4	(	(	PUNCT
ejde-673	906	5	2011	2011	NUM
ejde-673	906	6	)	)	PUNCT
ejde-673	906	7	,	,	PUNCT
ejde-673	907	1	333	333	NUM
ejde-673	907	2	-	-	SYM
ejde-673	907	3	342	342	NUM
ejde-673	907	4	.	.	PUNCT
ejde-673	908	1	[	[	X
ejde-673	908	2	11	11	NUM
ejde-673	908	3	]	]	PUNCT
ejde-673	908	4	s.	s.	PROPN
ejde-673	908	5	g.	g.	PROPN
ejde-673	908	6	deng	deng	PROPN
ejde-673	908	7	;	;	PUNCT
ejde-673	908	8	eigenvalues	eigenvalue	NOUN
ejde-673	908	9	of	of	ADP
ejde-673	908	10	the	the	DET
ejde-673	908	11	p(x)-laplacian	p(x)-laplacian	ADJ
ejde-673	908	12	steklov	steklov	NOUN
ejde-673	908	13	problem	problem	NOUN
ejde-673	908	14	,	,	PUNCT
ejde-673	908	15	j.	j.	PROPN
ejde-673	908	16	math	math	PROPN
ejde-673	908	17	.	.	PUNCT
ejde-673	909	1	anal	anal	PROPN
ejde-673	909	2	.	.	PUNCT
ejde-673	910	1	appl	appl	PROPN
ejde-673	910	2	.	.	PROPN
ejde-673	910	3	,	,	PUNCT
ejde-673	910	4	vol	vol	NOUN
ejde-673	910	5	.	.	PROPN
ejde-673	910	6	339	339	NUM
ejde-673	910	7	,	,	PUNCT
ejde-673	910	8	(	(	PUNCT
ejde-673	910	9	2008	2008	NUM
ejde-673	910	10	)	)	PUNCT
ejde-673	910	11	,	,	PUNCT
ejde-673	910	12	925	925	NUM
ejde-673	910	13	-	-	SYM
ejde-673	910	14	937	937	NUM
ejde-673	910	15	.	.	PUNCT
ejde-673	911	1	[	[	X
ejde-673	911	2	12	12	NUM
ejde-673	911	3	]	]	X
ejde-673	911	4	l.	l.	PROPN
ejde-673	911	5	diening	diening	PROPN
ejde-673	911	6	;	;	PUNCT
ejde-673	911	7	theoretical	theoretical	ADJ
ejde-673	911	8	and	and	CCONJ
ejde-673	911	9	numerical	numerical	ADJ
ejde-673	911	10	results	result	NOUN
ejde-673	911	11	for	for	ADP
ejde-673	911	12	electrorheological	electrorheological	ADJ
ejde-673	911	13	fluids	fluid	NOUN
ejde-673	911	14	,	,	PUNCT
ejde-673	911	15	ph	ph	PROPN
ejde-673	911	16	.	.	PROPN
ejde-673	911	17	d.	d.	PROPN
ejde-673	911	18	thesis	thesis	PROPN
ejde-673	911	19	,	,	PUNCT
ejde-673	911	20	university	university	NOUN
ejde-673	911	21	of	of	ADP
ejde-673	911	22	frieburg	frieburg	PROPN
ejde-673	911	23	,	,	PUNCT
ejde-673	911	24	germany	germany	PROPN
ejde-673	911	25	2002	2002	NUM
ejde-673	911	26	.	.	PUNCT
ejde-673	912	1	[	[	X
ejde-673	912	2	13	13	NUM
ejde-673	912	3	]	]	X
ejde-673	912	4	l.	l.	PROPN
ejde-673	912	5	diening	diening	PROPN
ejde-673	912	6	,	,	PUNCT
ejde-673	912	7	p.	p.	PROPN
ejde-673	912	8	harjulehto	harjulehto	PROPN
ejde-673	912	9	,	,	PUNCT
ejde-673	912	10	p.	p.	PROPN
ejde-673	912	11	hästö	hästö	PROPN
ejde-673	912	12	,	,	PUNCT
ejde-673	912	13	m.	m.	NOUN
ejde-673	912	14	růz̆ic̆ka	růz̆ic̆ka	PROPN
ejde-673	912	15	;	;	PUNCT
ejde-673	912	16	lebesgue	lebesgue	NOUN
ejde-673	912	17	and	and	CCONJ
ejde-673	912	18	sobolev	sobolev	NOUN
ejde-673	912	19	spaces	space	NOUN
ejde-673	912	20	with	with	ADP
ejde-673	912	21	variable	variable	ADJ
ejde-673	912	22	exponent	exponent	NOUN
ejde-673	912	23	,	,	PUNCT
ejde-673	912	24	lecture	lecture	NOUN
ejde-673	912	25	notes	note	NOUN
ejde-673	912	26	in	in	ADP
ejde-673	912	27	math	math	NOUN
ejde-673	912	28	.	.	PUNCT
ejde-673	913	1	springer	springer	NOUN
ejde-673	913	2	,	,	PUNCT
ejde-673	913	3	2017	2017	NUM
ejde-673	913	4	.	.	PUNCT
ejde-673	914	1	[	[	X
ejde-673	914	2	14	14	NUM
ejde-673	914	3	]	]	PUNCT
ejde-673	914	4	x.	x.	PROPN
ejde-673	914	5	l.	l.	PROPN
ejde-673	914	6	fan	fan	PROPN
ejde-673	914	7	;	;	PUNCT
ejde-673	914	8	solutions	solution	NOUN
ejde-673	914	9	for	for	ADP
ejde-673	914	10	p(x)-laplacian	p(x)-laplacian	ADJ
ejde-673	914	11	dirichlet	dirichlet	NOUN
ejde-673	914	12	problems	problem	NOUN
ejde-673	914	13	with	with	ADP
ejde-673	914	14	singular	singular	ADJ
ejde-673	914	15	coefficients	coefficient	NOUN
ejde-673	914	16	,	,	PUNCT
ejde-673	914	17	j.	j.	PROPN
ejde-673	914	18	math	math	PROPN
ejde-673	914	19	.	.	PUNCT
ejde-673	915	1	anal	anal	PROPN
ejde-673	915	2	.	.	PUNCT
ejde-673	916	1	appl	appl	PROPN
ejde-673	916	2	.	.	PROPN
ejde-673	916	3	,	,	PUNCT
ejde-673	916	4	vol	vol	NOUN
ejde-673	916	5	.	.	PROPN
ejde-673	916	6	312	312	NUM
ejde-673	916	7	,	,	PUNCT
ejde-673	916	8	(	(	PUNCT
ejde-673	916	9	2005	2005	NUM
ejde-673	916	10	)	)	PUNCT
ejde-673	916	11	,	,	PUNCT
ejde-673	916	12	464	464	NUM
ejde-673	916	13	-	-	SYM
ejde-673	916	14	477	477	NUM
ejde-673	916	15	.	.	PUNCT
ejde-673	917	1	[	[	X
ejde-673	917	2	15	15	NUM
ejde-673	917	3	]	]	X
ejde-673	917	4	x.	x.	PROPN
ejde-673	917	5	l.	l.	PROPN
ejde-673	917	6	fan	fan	PROPN
ejde-673	917	7	;	;	PUNCT
ejde-673	917	8	eigenvalues	eigenvalue	NOUN
ejde-673	917	9	of	of	ADP
ejde-673	917	10	the	the	DET
ejde-673	917	11	p(x)-laplacian	p(x)-laplacian	PROPN
ejde-673	917	12	neumann	neumann	PROPN
ejde-673	917	13	problem	problem	NOUN
ejde-673	917	14	,	,	PUNCT
ejde-673	917	15	nonlinear	nonlinear	ADJ
ejde-673	917	16	anal	anal	NOUN
ejde-673	917	17	.	.	PUNCT
ejde-673	917	18	,	,	PUNCT
ejde-673	917	19	vol	vol	NOUN
ejde-673	917	20	.	.	PROPN
ejde-673	917	21	67	67	NUM
ejde-673	917	22	,	,	PUNCT
ejde-673	917	23	(	(	PUNCT
ejde-673	917	24	2007	2007	NUM
ejde-673	917	25	)	)	PUNCT
ejde-673	917	26	,	,	PUNCT
ejde-673	917	27	2982	2982	NUM
ejde-673	917	28	-	-	SYM
ejde-673	917	29	2992	2992	NUM
ejde-673	917	30	.	.	PUNCT
ejde-673	918	1	[	[	X
ejde-673	918	2	16	16	NUM
ejde-673	918	3	]	]	PUNCT
ejde-673	918	4	x.	x.	PROPN
ejde-673	918	5	l.	l.	PROPN
ejde-673	918	6	fan	fan	PROPN
ejde-673	918	7	;	;	PUNCT
ejde-673	918	8	boundary	boundary	ADJ
ejde-673	918	9	trace	trace	NOUN
ejde-673	918	10	embedding	embed	VERB
ejde-673	918	11	theorems	theorem	NOUN
ejde-673	918	12	for	for	ADP
ejde-673	918	13	variable	variable	ADJ
ejde-673	918	14	exponent	exponent	NOUN
ejde-673	918	15	sobolev	sobolev	NOUN
ejde-673	918	16	spaces	space	VERB
ejde-673	918	17	,	,	PUNCT
ejde-673	918	18	j.	j.	PROPN
ejde-673	918	19	math	math	PROPN
ejde-673	918	20	.	.	PUNCT
ejde-673	919	1	anal	anal	PROPN
ejde-673	919	2	.	.	PUNCT
ejde-673	920	1	appl	appl	PROPN
ejde-673	920	2	.	.	PROPN
ejde-673	920	3	,	,	PUNCT
ejde-673	920	4	vol	vol	NOUN
ejde-673	920	5	.	.	PROPN
ejde-673	920	6	339	339	NUM
ejde-673	920	7	,	,	PUNCT
ejde-673	920	8	(	(	PUNCT
ejde-673	920	9	2008	2008	NUM
ejde-673	920	10	)	)	PUNCT
ejde-673	920	11	,	,	PUNCT
ejde-673	920	12	1395	1395	NUM
ejde-673	920	13	-	-	SYM
ejde-673	920	14	1412	1412	NUM
ejde-673	920	15	.	.	PUNCT
ejde-673	921	1	[	[	X
ejde-673	921	2	17	17	NUM
ejde-673	921	3	]	]	PUNCT
ejde-673	921	4	x.	x.	PROPN
ejde-673	921	5	l.	l.	PROPN
ejde-673	921	6	fan	fan	PROPN
ejde-673	921	7	,	,	PUNCT
ejde-673	921	8	q.	q.	PROPN
ejde-673	921	9	h	h	PROPN
ejde-673	921	10	zhang	zhang	PROPN
ejde-673	921	11	;	;	PUNCT
ejde-673	921	12	existence	existence	NOUN
ejde-673	921	13	of	of	ADP
ejde-673	921	14	solutions	solution	NOUN
ejde-673	921	15	for	for	ADP
ejde-673	921	16	p(x)-laplacian	p(x)-laplacian	ADJ
ejde-673	921	17	dirichlet	dirichlet	PROPN
ejde-673	921	18	problem	problem	NOUN
ejde-673	921	19	,	,	PUNCT
ejde-673	921	20	nonlinear	nonlinear	ADJ
ejde-673	921	21	anal	anal	NOUN
ejde-673	921	22	.	.	PUNCT
ejde-673	921	23	,	,	PUNCT
ejde-673	921	24	vol	vol	NOUN
ejde-673	921	25	.	.	PROPN
ejde-673	921	26	52	52	NUM
ejde-673	921	27	,	,	PUNCT
ejde-673	921	28	(	(	PUNCT
ejde-673	921	29	2003	2003	NUM
ejde-673	921	30	)	)	PUNCT
ejde-673	921	31	,	,	PUNCT
ejde-673	921	32	1843	1843	NUM
ejde-673	921	33	-	-	SYM
ejde-673	921	34	1852	1852	NUM
ejde-673	921	35	.	.	PUNCT
ejde-673	922	1	[	[	X
ejde-673	922	2	18	18	NUM
ejde-673	922	3	]	]	PUNCT
ejde-673	922	4	x.	x.	PROPN
ejde-673	922	5	l.	l.	PROPN
ejde-673	922	6	fan	fan	PROPN
ejde-673	922	7	,	,	PUNCT
ejde-673	922	8	d.	d.	PROPN
ejde-673	922	9	zhao	zhao	PROPN
ejde-673	922	10	;	;	PUNCT
ejde-673	922	11	on	on	ADP
ejde-673	922	12	the	the	DET
ejde-673	922	13	spaces	space	NOUN
ejde-673	922	14	lp(x)(ω	lp(x)(ω	NOUN
ejde-673	922	15	)	)	PUNCT
ejde-673	922	16	and	and	CCONJ
ejde-673	922	17	wm	wm	PROPN
ejde-673	922	18	,	,	PUNCT
ejde-673	922	19	p(x)(ω	p(x)(ω	NUM
ejde-673	922	20	)	)	PUNCT
ejde-673	922	21	,	,	PUNCT
ejde-673	922	22	j.	j.	PROPN
ejde-673	922	23	math	math	PROPN
ejde-673	922	24	.	.	PUNCT
ejde-673	923	1	anal	anal	PROPN
ejde-673	923	2	.	.	PUNCT
ejde-673	924	1	appl	appl	PROPN
ejde-673	924	2	.	.	PROPN
ejde-673	924	3	,	,	PUNCT
ejde-673	924	4	vol	vol	NOUN
ejde-673	924	5	.	.	PROPN
ejde-673	924	6	263	263	NUM
ejde-673	924	7	,	,	PUNCT
ejde-673	924	8	(	(	PUNCT
ejde-673	924	9	2001	2001	NUM
ejde-673	924	10	)	)	PUNCT
ejde-673	924	11	,	,	PUNCT
ejde-673	924	12	424–446	424–446	NUM
ejde-673	924	13	.	.	PUNCT
ejde-673	925	1	[	[	X
ejde-673	925	2	19	19	NUM
ejde-673	925	3	]	]	PUNCT
ejde-673	925	4	x.	x.	PROPN
ejde-673	925	5	l.	l.	PROPN
ejde-673	925	6	fan	fan	PROPN
ejde-673	925	7	,	,	PUNCT
ejde-673	925	8	q.	q.	PROPN
ejde-673	925	9	zhang	zhang	PROPN
ejde-673	925	10	,	,	PUNCT
ejde-673	925	11	d.	d.	PROPN
ejde-673	925	12	zhao	zhao	PROPN
ejde-673	925	13	;	;	PUNCT
ejde-673	925	14	eigenvalues	eigenvalue	NOUN
ejde-673	925	15	of	of	ADP
ejde-673	925	16	p(x)-laplacian	p(x)-laplacian	ADJ
ejde-673	925	17	dirichlet	dirichlet	NOUN
ejde-673	925	18	problem	problem	NOUN
ejde-673	925	19	,	,	PUNCT
ejde-673	925	20	j.	j.	PROPN
ejde-673	925	21	math	math	PROPN
ejde-673	925	22	.	.	PUNCT
ejde-673	926	1	anal	anal	PROPN
ejde-673	926	2	.	.	PUNCT
ejde-673	927	1	appl	appl	PROPN
ejde-673	927	2	.	.	PROPN
ejde-673	927	3	,	,	PUNCT
ejde-673	927	4	vol	vol	NOUN
ejde-673	927	5	.	.	PROPN
ejde-673	927	6	302	302	NUM
ejde-673	927	7	,	,	PUNCT
ejde-673	927	8	(	(	PUNCT
ejde-673	927	9	2015	2015	NUM
ejde-673	927	10	)	)	PUNCT
ejde-673	927	11	,	,	PUNCT
ejde-673	927	12	306–317	306–317	NUM
ejde-673	927	13	.	.	PUNCT
ejde-673	928	1	[	[	X
ejde-673	928	2	20	20	NUM
ejde-673	928	3	]	]	PUNCT
ejde-673	928	4	l.	l.	PROPN
ejde-673	928	5	friedlander	friedlander	PROPN
ejde-673	928	6	;	;	PUNCT
ejde-673	928	7	asymptotic	asymptotic	ADJ
ejde-673	928	8	behavior	behavior	NOUN
ejde-673	928	9	of	of	ADP
ejde-673	928	10	the	the	DET
ejde-673	928	11	eigenvalues	eigenvalue	NOUN
ejde-673	928	12	of	of	ADP
ejde-673	928	13	the	the	DET
ejde-673	928	14	p	p	NOUN
ejde-673	928	15	-	-	PUNCT
ejde-673	928	16	laplacian	laplacian	ADJ
ejde-673	928	17	,	,	PUNCT
ejde-673	928	18	comm	comm	NOUN
ejde-673	928	19	.	.	PUNCT
ejde-673	929	1	partial	partial	ADJ
ejde-673	929	2	differential	differential	NOUN
ejde-673	929	3	equations	equation	NOUN
ejde-673	929	4	,	,	PUNCT
ejde-673	929	5	vol	vol	NOUN
ejde-673	929	6	.	.	PROPN
ejde-673	929	7	14	14	NUM
ejde-673	929	8	,	,	PUNCT
ejde-673	929	9	(	(	PUNCT
ejde-673	929	10	1989	1989	NUM
ejde-673	929	11	)	)	PUNCT
ejde-673	929	12	,	,	PUNCT
ejde-673	929	13	1059	1059	NUM
ejde-673	929	14	-	-	SYM
ejde-673	929	15	1069	1069	NUM
ejde-673	929	16	.	.	PUNCT
ejde-673	930	1	[	[	X
ejde-673	930	2	21	21	NUM
ejde-673	930	3	]	]	X
ejde-673	930	4	t.	t.	PROPN
ejde-673	930	5	c.	c.	PROPN
ejde-673	930	6	halsey	halsey	PROPN
ejde-673	930	7	;	;	PUNCT
ejde-673	930	8	electrorheological	electrorheological	ADJ
ejde-673	930	9	fluids	fluid	NOUN
ejde-673	930	10	,	,	PUNCT
ejde-673	930	11	science	science	NOUN
ejde-673	930	12	,	,	PUNCT
ejde-673	930	13	vol	vol	NOUN
ejde-673	930	14	.	.	PROPN
ejde-673	930	15	258	258	NUM
ejde-673	930	16	,	,	PUNCT
ejde-673	930	17	(	(	PUNCT
ejde-673	930	18	1992	1992	NUM
ejde-673	930	19	)	)	PUNCT
ejde-673	930	20	,	,	PUNCT
ejde-673	930	21	761–766	761–766	NUM
ejde-673	930	22	.	.	PUNCT
ejde-673	931	1	[	[	X
ejde-673	931	2	22	22	NUM
ejde-673	931	3	]	]	X
ejde-673	931	4	g.	g.	PROPN
ejde-673	931	5	kirchhoff	kirchhoff	PROPN
ejde-673	931	6	;	;	PUNCT
ejde-673	931	7	mechanik	mechanik	PROPN
ejde-673	931	8	,	,	PUNCT
ejde-673	931	9	teubner	teubner	NOUN
ejde-673	931	10	,	,	PUNCT
ejde-673	931	11	leipzig	leipzig	NOUN
ejde-673	931	12	,	,	PUNCT
ejde-673	931	13	1883	1883	NUM
ejde-673	931	14	.	.	PUNCT
ejde-673	932	1	[	[	X
ejde-673	932	2	23	23	NUM
ejde-673	932	3	]	]	PUNCT
ejde-673	932	4	o.	o.	PROPN
ejde-673	932	5	kovăc̆ik	kovăc̆ik	PROPN
ejde-673	932	6	,	,	PUNCT
ejde-673	932	7	j.	j.	PROPN
ejde-673	932	8	rákosńık	rákosńık	PROPN
ejde-673	932	9	;	;	PUNCT
ejde-673	932	10	on	on	ADP
ejde-673	932	11	spaces	space	NOUN
ejde-673	932	12	lp(x)(ω	lp(x)(ω	PROPN
ejde-673	932	13	)	)	PUNCT
ejde-673	932	14	and	and	CCONJ
ejde-673	932	15	wk	wk	INTJ
ejde-673	932	16	,	,	PUNCT
ejde-673	932	17	p(x)(ω	p(x)(ω	NUM
ejde-673	932	18	)	)	PUNCT
ejde-673	932	19	,	,	PUNCT
ejde-673	932	20	czechoslovak	czechoslovak	ADJ
ejde-673	932	21	math	math	NOUN
ejde-673	932	22	.	.	PUNCT
ejde-673	933	1	j.	j.	PROPN
ejde-673	933	2	,	,	PUNCT
ejde-673	933	3	vol	vol	NOUN
ejde-673	933	4	.	.	PUNCT
ejde-673	933	5	41(116	41(116	PROPN
ejde-673	933	6	)	)	PUNCT
ejde-673	933	7	,	,	PUNCT
ejde-673	933	8	(	(	PUNCT
ejde-673	933	9	1991	1991	NUM
ejde-673	933	10	)	)	PUNCT
ejde-673	933	11	,	,	PUNCT
ejde-673	933	12	592–618	592–618	NUM
ejde-673	933	13	.	.	PUNCT
ejde-673	934	1	[	[	X
ejde-673	934	2	24	24	NUM
ejde-673	934	3	]	]	PUNCT
ejde-673	934	4	a.	a.	NOUN
ejde-673	934	5	lê	lê	PROPN
ejde-673	934	6	;	;	PUNCT
ejde-673	934	7	eigenvalue	eigenvalue	NOUN
ejde-673	934	8	problems	problem	NOUN
ejde-673	934	9	for	for	ADP
ejde-673	934	10	the	the	DET
ejde-673	934	11	p	p	NOUN
ejde-673	934	12	-	-	PUNCT
ejde-673	934	13	laplacian	laplacian	ADJ
ejde-673	934	14	,	,	PUNCT
ejde-673	934	15	nonlinear	nonlinear	ADJ
ejde-673	934	16	anal	anal	NOUN
ejde-673	934	17	.	.	PUNCT
ejde-673	934	18	,	,	PUNCT
ejde-673	934	19	vol	vol	NOUN
ejde-673	934	20	.	.	PROPN
ejde-673	934	21	64	64	NUM
ejde-673	934	22	(	(	PUNCT
ejde-673	934	23	2006	2006	NUM
ejde-673	934	24	)	)	PUNCT
ejde-673	934	25	,	,	PUNCT
ejde-673	934	26	1057	1057	NUM
ejde-673	934	27	-	-	SYM
ejde-673	934	28	1099	1099	NUM
ejde-673	934	29	.	.	PUNCT
ejde-673	935	1	30	30	NUM
ejde-673	935	2	j.	j.	PROPN
ejde-673	935	3	aramaki	aramaki	PROPN
ejde-673	935	4	ejde-2025/17	ejde-2025/17	NOUN
ejde-673	935	5	[	[	X
ejde-673	935	6	25	25	NUM
ejde-673	935	7	]	]	PUNCT
ejde-673	935	8	l.	l.	PROPN
ejde-673	935	9	ljusternik	ljusternik	PROPN
ejde-673	935	10	,	,	PUNCT
ejde-673	935	11	l.	l.	PROPN
ejde-673	935	12	schnirelmann	schnirelmann	PROPN
ejde-673	935	13	;	;	PUNCT
ejde-673	935	14	méthodes	méthodes	PROPN
ejde-673	935	15	topologiques	topologique	NOUN
ejde-673	935	16	dans	dan	NOUN
ejde-673	935	17	les	le	NOUN
ejde-673	935	18	problémes	probléme	NOUN
ejde-673	935	19	variationels	variationel	NOUN
ejde-673	935	20	,	,	PUNCT
ejde-673	935	21	hermann	hermann	PROPN
ejde-673	935	22	,	,	PUNCT
ejde-673	935	23	paris	paris	PROPN
ejde-673	935	24	,	,	PUNCT
ejde-673	935	25	1934	1934	NUM
ejde-673	935	26	.	.	PUNCT
ejde-673	936	1	[	[	X
ejde-673	936	2	26	26	NUM
ejde-673	936	3	]	]	X
ejde-673	936	4	r.	r.	PROPN
ejde-673	936	5	a.	a.	PROPN
ejde-673	936	6	mashiyev	mashiyev	PROPN
ejde-673	936	7	,	,	PUNCT
ejde-673	936	8	b.	b.	PROPN
ejde-673	936	9	cekic	cekic	PROPN
ejde-673	936	10	,	,	PUNCT
ejde-673	936	11	m.	m.	NOUN
ejde-673	936	12	avci	avci	PROPN
ejde-673	936	13	,	,	PUNCT
ejde-673	936	14	z.	z.	PROPN
ejde-673	936	15	yucedag	yucedag	PROPN
ejde-673	936	16	;	;	PUNCT
ejde-673	936	17	existence	existence	NOUN
ejde-673	936	18	and	and	CCONJ
ejde-673	936	19	multiplicity	multiplicity	NOUN
ejde-673	936	20	of	of	ADP
ejde-673	936	21	weak	weak	ADJ
ejde-673	936	22	solutions	solution	NOUN
ejde-673	936	23	for	for	ADP
ejde-673	936	24	nonuniformly	nonuniformly	ADJ
ejde-673	936	25	elliptic	elliptic	ADJ
ejde-673	936	26	equations	equation	NOUN
ejde-673	936	27	with	with	ADP
ejde-673	936	28	nonstandard	nonstandard	ADJ
ejde-673	936	29	growth	growth	NOUN
ejde-673	936	30	condition	condition	NOUN
ejde-673	936	31	,	,	PUNCT
ejde-673	936	32	complex	complex	ADJ
ejde-673	936	33	variables	variable	NOUN
ejde-673	936	34	elliptic	elliptic	ADJ
ejde-673	936	35	equa	equa	NOUN
ejde-673	936	36	.	.	PUNCT
ejde-673	936	37	,	,	PUNCT
ejde-673	936	38	vol	vol	NOUN
ejde-673	936	39	.	.	PUNCT
ejde-673	937	1	57(5	57(5	NOUN
ejde-673	937	2	)	)	PUNCT
ejde-673	937	3	,	,	PUNCT
ejde-673	937	4	(	(	PUNCT
ejde-673	937	5	2012	2012	NUM
ejde-673	937	6	)	)	PUNCT
ejde-673	937	7	,	,	PUNCT
ejde-673	937	8	579	579	NUM
ejde-673	937	9	-	-	SYM
ejde-673	937	10	595	595	NUM
ejde-673	937	11	.	.	PUNCT
ejde-673	938	1	[	[	X
ejde-673	938	2	27	27	NUM
ejde-673	938	3	]	]	X
ejde-673	938	4	o.	o.	NOUN
ejde-673	938	5	méndez	méndez	PROPN
ejde-673	938	6	;	;	PUNCT
ejde-673	938	7	on	on	ADP
ejde-673	938	8	the	the	DET
ejde-673	938	9	eigenvalue	eigenvalue	PROPN
ejde-673	938	10	problem	problem	NOUN
ejde-673	938	11	for	for	ADP
ejde-673	938	12	a	a	DET
ejde-673	938	13	class	class	NOUN
ejde-673	938	14	of	of	ADP
ejde-673	938	15	kirchhoff	kirchhoff	NOUN
ejde-673	938	16	-	-	PUNCT
ejde-673	938	17	type	type	NOUN
ejde-673	938	18	equations	equation	NOUN
ejde-673	938	19	,	,	PUNCT
ejde-673	938	20	j.	j.	PROPN
ejde-673	938	21	math	math	PROPN
ejde-673	938	22	.	.	PUNCT
ejde-673	939	1	anal	anal	PROPN
ejde-673	939	2	.	.	PUNCT
ejde-673	940	1	appl	appl	PROPN
ejde-673	940	2	.	.	PROPN
ejde-673	940	3	,	,	PUNCT
ejde-673	940	4	vol	vol	NOUN
ejde-673	940	5	.	.	PROPN
ejde-673	940	6	494	494	NUM
ejde-673	940	7	,	,	PUNCT
ejde-673	940	8	(	(	PUNCT
ejde-673	940	9	2021	2021	NUM
ejde-673	940	10	)	)	PUNCT
ejde-673	940	11	,	,	PUNCT
ejde-673	940	12	124671	124671	NUM
ejde-673	940	13	.	.	PUNCT
ejde-673	941	1	[	[	X
ejde-673	941	2	28	28	NUM
ejde-673	941	3	]	]	X
ejde-673	941	4	m.	m.	NOUN
ejde-673	941	5	mihăilescu	mihăilescu	PROPN
ejde-673	941	6	,	,	PUNCT
ejde-673	941	7	v.	v.	ADP
ejde-673	941	8	rădulescu	rădulescu	PROPN
ejde-673	941	9	;	;	PUNCT
ejde-673	941	10	a	a	DET
ejde-673	941	11	multiplicity	multiplicity	NOUN
ejde-673	941	12	result	result	NOUN
ejde-673	941	13	for	for	ADP
ejde-673	941	14	a	a	DET
ejde-673	941	15	nonlinear	nonlinear	ADJ
ejde-673	941	16	degenerate	degenerate	ADJ
ejde-673	941	17	problem	problem	NOUN
ejde-673	941	18	arising	arise	VERB
ejde-673	941	19	in	in	ADP
ejde-673	941	20	the	the	DET
ejde-673	941	21	theory	theory	NOUN
ejde-673	941	22	of	of	ADP
ejde-673	941	23	electrorheological	electrorheological	ADJ
ejde-673	941	24	fluids	fluid	NOUN
ejde-673	941	25	,	,	PUNCT
ejde-673	941	26	proceeding	proceeding	NOUN
ejde-673	941	27	of	of	ADP
ejde-673	941	28	the	the	DET
ejde-673	941	29	royal	royal	PROPN
ejde-673	941	30	society	society	PROPN
ejde-673	941	31	a.	a.	NOUN
ejde-673	941	32	,	,	PUNCT
ejde-673	941	33	vol	vol	NOUN
ejde-673	941	34	.	.	PROPN
ejde-673	941	35	462	462	NUM
ejde-673	941	36	,	,	PUNCT
ejde-673	941	37	(	(	PUNCT
ejde-673	941	38	2006	2006	NUM
ejde-673	941	39	)	)	PUNCT
ejde-673	941	40	,	,	PUNCT
ejde-673	941	41	2625–2641	2625–2641	NUM
ejde-673	941	42	.	.	PUNCT
ejde-673	942	1	[	[	X
ejde-673	942	2	29	29	NUM
ejde-673	942	3	]	]	X
ejde-673	942	4	m.	m.	NOUN
ejde-673	942	5	mihăilescu	mihăilescu	PROPN
ejde-673	942	6	,	,	PUNCT
ejde-673	942	7	v.	v.	ADP
ejde-673	942	8	rădulescu	rădulescu	PROPN
ejde-673	942	9	;	;	PUNCT
ejde-673	942	10	on	on	ADP
ejde-673	942	11	a	a	DET
ejde-673	942	12	nonhomogenuous	nonhomogenuous	ADJ
ejde-673	942	13	quasilinear	quasilinear	NOUN
ejde-673	942	14	eigenvalue	eigenvalue	NOUN
ejde-673	942	15	problem	problem	NOUN
ejde-673	942	16	in	in	ADP
ejde-673	942	17	sobolev	sobolev	PROPN
ejde-673	942	18	cpaces	cpace	NOUN
ejde-673	942	19	with	with	ADP
ejde-673	942	20	variable	variable	ADJ
ejde-673	942	21	exponent	exponent	NOUN
ejde-673	942	22	,	,	PUNCT
ejde-673	942	23	proc	proc	PROPN
ejde-673	942	24	.	.	PUNCT
ejde-673	943	1	amer	amer	PROPN
ejde-673	943	2	.	.	PUNCT
ejde-673	943	3	math	math	PROPN
ejde-673	943	4	.	.	PUNCT
ejde-673	944	1	soc	soc	PROPN
ejde-673	944	2	.	.	PUNCT
ejde-673	944	3	,	,	PUNCT
ejde-673	944	4	vol	vol	NOUN
ejde-673	944	5	.	.	PROPN
ejde-673	944	6	135	135	NUM
ejde-673	944	7	,	,	PUNCT
ejde-673	944	8	(	(	PUNCT
ejde-673	944	9	2007	2007	NUM
ejde-673	944	10	)	)	PUNCT
ejde-673	944	11	,	,	PUNCT
ejde-673	944	12	2929	2929	NUM
ejde-673	944	13	-	-	SYM
ejde-673	944	14	2937	2937	NUM
ejde-673	944	15	.	.	PUNCT
ejde-673	945	1	[	[	X
ejde-673	945	2	30	30	NUM
ejde-673	945	3	]	]	PUNCT
ejde-673	945	4	p.	p.	NOUN
ejde-673	945	5	h.	h.	PROPN
ejde-673	945	6	rabinowitz	rabinowitz	PROPN
ejde-673	945	7	;	;	PUNCT
ejde-673	945	8	minimax	minimax	NOUN
ejde-673	945	9	methods	method	NOUN
ejde-673	945	10	in	in	ADP
ejde-673	945	11	critical	critical	ADJ
ejde-673	945	12	point	point	NOUN
ejde-673	945	13	theory	theory	NOUN
ejde-673	945	14	with	with	ADP
ejde-673	945	15	application	application	NOUN
ejde-673	945	16	to	to	ADP
ejde-673	945	17	differential	differential	ADJ
ejde-673	945	18	equations	equation	NOUN
ejde-673	945	19	,	,	PUNCT
ejde-673	945	20	cbms	cbms	PROPN
ejde-673	945	21	reg	reg	PROPN
ejde-673	945	22	.	.	PUNCT
ejde-673	945	23	conf	conf	PROPN
ejde-673	945	24	.	.	PUNCT
ejde-673	946	1	ser	ser	PROPN
ejde-673	946	2	.	.	PUNCT
ejde-673	947	1	in	in	ADP
ejde-673	947	2	math	math	NOUN
ejde-673	947	3	.	.	PUNCT
ejde-673	948	1	,	,	PUNCT
ejde-673	948	2	vol	vol	NOUN
ejde-673	948	3	.	.	PROPN
ejde-673	948	4	65	65	NUM
ejde-673	948	5	,	,	PUNCT
ejde-673	948	6	am	be	AUX
ejde-673	948	7	.	.	PUNCT
ejde-673	949	1	math	math	NOUN
ejde-673	949	2	.	.	PUNCT
ejde-673	950	1	soc	soc	PROPN
ejde-673	950	2	.	.	PUNCT
ejde-673	950	3	,	,	PUNCT
ejde-673	950	4	providence	providence	NOUN
ejde-673	950	5	,	,	PUNCT
ejde-673	950	6	1986	1986	NUM
ejde-673	950	7	.	.	PUNCT
ejde-673	951	1	[	[	X
ejde-673	951	2	31	31	NUM
ejde-673	951	3	]	]	PUNCT
ejde-673	951	4	m.	m.	NOUN
ejde-673	951	5	růz̆ic̆ka	růz̆ic̆ka	PROPN
ejde-673	951	6	;	;	PUNCT
ejde-673	951	7	electrotheological	electrotheological	ADJ
ejde-673	951	8	fluids	fluid	NOUN
ejde-673	951	9	:	:	PUNCT
ejde-673	951	10	modeling	modeling	NOUN
ejde-673	951	11	and	and	CCONJ
ejde-673	951	12	mathematical	mathematical	ADJ
ejde-673	951	13	theory	theory	NOUN
ejde-673	951	14	,	,	PUNCT
ejde-673	951	15	lecture	lecture	NOUN
ejde-673	951	16	notes	note	NOUN
ejde-673	951	17	in	in	ADP
ejde-673	951	18	mathematics	mathematic	NOUN
ejde-673	951	19	,	,	PUNCT
ejde-673	951	20	vol	vol	NOUN
ejde-673	951	21	.	.	PUNCT
ejde-673	951	22	1784	1784	NUM
ejde-673	951	23	,	,	PUNCT
ejde-673	951	24	berlin	berlin	PROPN
ejde-673	951	25	,	,	PUNCT
ejde-673	951	26	springer	springer	NOUN
ejde-673	951	27	,	,	PUNCT
ejde-673	951	28	2000	2000	NUM
ejde-673	951	29	.	.	PUNCT
ejde-673	952	1	[	[	X
ejde-673	952	2	32	32	NUM
ejde-673	952	3	]	]	PUNCT
ejde-673	952	4	a.	a.	NOUN
ejde-673	952	5	szulkin	szulkin	PROPN
ejde-673	952	6	;	;	PUNCT
ejde-673	952	7	ljusternik	ljusternik	X
ejde-673	952	8	-	-	PUNCT
ejde-673	952	9	schnirelmann	schnirelmann	PROPN
ejde-673	952	10	theory	theory	NOUN
ejde-673	952	11	on	on	ADP
ejde-673	952	12	c1	c1	NOUN
ejde-673	952	13	-	-	PUNCT
ejde-673	952	14	manifolds	manifolds	PROPN
ejde-673	952	15	,	,	PUNCT
ejde-673	952	16	ann	ann	PROPN
ejde-673	952	17	.	.	PROPN
ejde-673	952	18	inst	inst	PROPN
ejde-673	952	19	.	.	PUNCT
ejde-673	953	1	henri	henri	PROPN
ejde-673	953	2	poincaré	poincaré	ADJ
ejde-673	953	3	,	,	PUNCT
ejde-673	953	4	vol	vol	NOUN
ejde-673	953	5	.	.	PUNCT
ejde-673	953	6	5(2	5(2	NUM
ejde-673	953	7	)	)	PUNCT
ejde-673	953	8	,	,	PUNCT
ejde-673	953	9	(	(	PUNCT
ejde-673	953	10	1988	1988	NUM
ejde-673	953	11	)	)	PUNCT
ejde-673	953	12	,	,	PUNCT
ejde-673	953	13	119	119	NUM
ejde-673	953	14	-	-	SYM
ejde-673	953	15	139	139	NUM
ejde-673	953	16	.	.	PUNCT
ejde-673	954	1	[	[	X
ejde-673	954	2	33	33	NUM
ejde-673	954	3	]	]	PUNCT
ejde-673	954	4	j.	j.	PROPN
ejde-673	954	5	yao	yao	PROPN
ejde-673	954	6	;	;	PUNCT
ejde-673	954	7	solutions	solution	NOUN
ejde-673	954	8	for	for	ADP
ejde-673	954	9	neumann	neumann	PROPN
ejde-673	954	10	boundary	boundary	PROPN
ejde-673	954	11	value	value	NOUN
ejde-673	954	12	problem	problem	NOUN
ejde-673	954	13	involving	involve	VERB
ejde-673	954	14	p(x)-laplace	p(x)-laplace	PROPN
ejde-673	954	15	operators	operator	NOUN
ejde-673	954	16	,	,	PUNCT
ejde-673	954	17	nonlinear	nonlinear	ADJ
ejde-673	954	18	anal	anal	NOUN
ejde-673	954	19	.	.	PUNCT
ejde-673	954	20	,	,	PUNCT
ejde-673	954	21	vol	vol	NOUN
ejde-673	954	22	.	.	PROPN
ejde-673	954	23	68	68	NUM
ejde-673	954	24	,	,	PUNCT
ejde-673	954	25	(	(	PUNCT
ejde-673	954	26	2008	2008	NUM
ejde-673	954	27	)	)	PUNCT
ejde-673	954	28	,	,	PUNCT
ejde-673	954	29	1271	1271	NUM
ejde-673	954	30	-	-	SYM
ejde-673	954	31	1283	1283	NUM
ejde-673	954	32	.	.	PUNCT
ejde-673	955	1	[	[	X
ejde-673	955	2	34	34	NUM
ejde-673	955	3	]	]	X
ejde-673	955	4	e.	e.	PROPN
ejde-673	955	5	zeidler	zeidler	PROPN
ejde-673	955	6	;	;	PUNCT
ejde-673	955	7	nonlinear	nonlinear	ADJ
ejde-673	955	8	functional	functional	ADJ
ejde-673	955	9	analysis	analysis	NOUN
ejde-673	955	10	and	and	CCONJ
ejde-673	955	11	its	its	PRON
ejde-673	955	12	applications	application	NOUN
ejde-673	955	13	i	i	PRON
ejde-673	955	14	:	:	PUNCT
ejde-673	955	15	fixed	fix	VERB
ejde-673	955	16	-	-	PUNCT
ejde-673	955	17	point	point	NOUN
ejde-673	955	18	theorems	theorem	NOUN
ejde-673	955	19	,	,	PUNCT
ejde-673	955	20	iii	iii	NOUN
ejde-673	955	21	:	:	PUNCT
ejde-673	955	22	variational	variational	ADJ
ejde-673	955	23	methods	method	NOUN
ejde-673	955	24	and	and	CCONJ
ejde-673	955	25	optimization	optimization	NOUN
ejde-673	955	26	,	,	PUNCT
ejde-673	955	27	springer	springer	NOUN
ejde-673	955	28	-	-	PUNCT
ejde-673	955	29	verlag	verlag	PROPN
ejde-673	955	30	,	,	PUNCT
ejde-673	955	31	now	now	ADV
ejde-673	955	32	york	york	PROPN
ejde-673	955	33	,	,	PUNCT
ejde-673	955	34	berlin	berlin	PROPN
ejde-673	955	35	,	,	PUNCT
ejde-673	955	36	heidelberg	heidelberg	PROPN
ejde-673	955	37	,	,	PUNCT
ejde-673	955	38	london	london	PROPN
ejde-673	955	39	,	,	PUNCT
ejde-673	955	40	paris	paris	PROPN
ejde-673	955	41	,	,	PUNCT
ejde-673	955	42	tokyo	tokyo	PROPN
ejde-673	955	43	,	,	PUNCT
ejde-673	955	44	1990	1990	NUM
ejde-673	955	45	.	.	PUNCT
ejde-673	956	1	[	[	X
ejde-673	956	2	35	35	NUM
ejde-673	956	3	]	]	X
ejde-673	956	4	d.	d.	PROPN
ejde-673	956	5	zhao	zhao	PROPN
ejde-673	956	6	,	,	PUNCT
ejde-673	956	7	wj	wj	PROPN
ejde-673	956	8	.	.	PUNCT
ejde-673	957	1	qing	qing	PROPN
ejde-673	957	2	,	,	PUNCT
ejde-673	957	3	x.	x.	PROPN
ejde-673	957	4	l.	l.	PROPN
ejde-673	957	5	fan	fan	PROPN
ejde-673	957	6	;	;	PUNCT
ejde-673	957	7	on	on	ADP
ejde-673	957	8	generalized	generalize	VERB
ejde-673	957	9	orlicz	orlicz	ADJ
ejde-673	957	10	space	space	NOUN
ejde-673	957	11	lp(x)(ω	lp(x)(ω	NOUN
ejde-673	957	12	)	)	PUNCT
ejde-673	957	13	,	,	PUNCT
ejde-673	957	14	j.	j.	PROPN
ejde-673	957	15	gansu	gansu	PROPN
ejde-673	957	16	sci	sci	PROPN
ejde-673	957	17	.	.	PROPN
ejde-673	957	18	,	,	PUNCT
ejde-673	957	19	vol	vol	NOUN
ejde-673	957	20	.	.	PUNCT
ejde-673	957	21	9(2	9(2	NUM
ejde-673	957	22	)	)	PUNCT
ejde-673	957	23	,	,	PUNCT
ejde-673	957	24	(	(	PUNCT
ejde-673	957	25	1996	1996	NUM
ejde-673	957	26	)	)	PUNCT
ejde-673	957	27	,	,	PUNCT
ejde-673	957	28	1–7	1–7	X
ejde-673	957	29	.	.	PUNCT
ejde-673	957	30	(	(	PUNCT
ejde-673	957	31	in	in	ADP
ejde-673	957	32	chinese	chinese	PROPN
ejde-673	957	33	)	)	PUNCT
ejde-673	957	34	.	.	PUNCT
ejde-673	958	1	[	[	X
ejde-673	958	2	36	36	NUM
ejde-673	958	3	]	]	SYM
ejde-673	958	4	vv	vv	PROPN
ejde-673	958	5	.	.	PUNCT
ejde-673	958	6	zhikov	zhikov	PROPN
ejde-673	958	7	;	;	PUNCT
ejde-673	958	8	averaging	average	VERB
ejde-673	958	9	of	of	ADP
ejde-673	958	10	functionals	functional	NOUN
ejde-673	958	11	of	of	ADP
ejde-673	958	12	the	the	DET
ejde-673	958	13	calculus	calculus	NOUN
ejde-673	958	14	of	of	ADP
ejde-673	958	15	variation	variation	NOUN
ejde-673	958	16	and	and	CCONJ
ejde-673	958	17	elasticity	elasticity	NOUN
ejde-673	958	18	theory	theory	NOUN
ejde-673	958	19	,	,	PUNCT
ejde-673	958	20	math	math	NOUN
ejde-673	958	21	.	.	PUNCT
ejde-673	959	1	ussr	ussr	PROPN
ejde-673	959	2	,	,	PUNCT
ejde-673	959	3	izv	izv	PROPN
ejde-673	959	4	.	.	PROPN
ejde-673	959	5	,	,	PUNCT
ejde-673	959	6	vol	vol	NOUN
ejde-673	959	7	.	.	PROPN
ejde-673	959	8	29	29	NUM
ejde-673	959	9	,	,	PUNCT
ejde-673	959	10	(	(	PUNCT
ejde-673	959	11	1987	1987	NUM
ejde-673	959	12	)	)	PUNCT
ejde-673	959	13	,	,	PUNCT
ejde-673	959	14	33–66	33–66	NUM
ejde-673	959	15	.	.	PUNCT
ejde-673	960	1	junichi	junichi	PROPN
ejde-673	960	2	aramaki	aramaki	PROPN
ejde-673	960	3	division	division	PROPN
ejde-673	960	4	of	of	ADP
ejde-673	960	5	science	science	NOUN
ejde-673	960	6	,	,	PUNCT
ejde-673	960	7	faculty	faculty	NOUN
ejde-673	960	8	of	of	ADP
ejde-673	960	9	science	science	NOUN
ejde-673	960	10	and	and	CCONJ
ejde-673	960	11	engineering	engineering	NOUN
ejde-673	960	12	,	,	PUNCT
ejde-673	960	13	tokyo	tokyo	PROPN
ejde-673	960	14	denki	denki	PROPN
ejde-673	960	15	university	university	PROPN
ejde-673	960	16	,	,	PUNCT
ejde-673	960	17	hatoyama	hatoyama	NOUN
ejde-673	960	18	-	-	PUNCT
ejde-673	960	19	machi	machi	NOUN
ejde-673	960	20	,	,	PUNCT
ejde-673	960	21	saitama	saitama	PROPN
ejde-673	960	22	350	350	NUM
ejde-673	960	23	-	-	SYM
ejde-673	960	24	0394	0394	NUM
ejde-673	960	25	,	,	PUNCT
ejde-673	960	26	japan	japan	PROPN
ejde-673	960	27	email	email	PROPN
ejde-673	960	28	address	address	NOUN
ejde-673	960	29	:	:	PUNCT
ejde-673	960	30	aramaki@hctv.ne.jp	aramaki@hctv.ne.jp	PROPN
ejde-673	960	31	1	1	X
ejde-673	960	32	.	.	PUNCT
ejde-673	961	1	introduction	introduction	NOUN
ejde-673	961	2	2	2	NUM
ejde-673	961	3	.	.	PUNCT
ejde-673	961	4	preliminaries	preliminary	NOUN
ejde-673	961	5	3	3	NUM
ejde-673	961	6	.	.	PUNCT
ejde-673	961	7	assumptions	assumption	NOUN
ejde-673	961	8	and	and	CCONJ
ejde-673	961	9	main	main	ADJ
ejde-673	961	10	theorem	theorem	NOUN
ejde-673	961	11	4	4	NUM
ejde-673	961	12	.	.	PUNCT
ejde-673	962	1	the	the	DET
ejde-673	962	2	infimum	infimum	NOUN
ejde-673	962	3	of	of	ADP
ejde-673	962	4	all	all	DET
ejde-673	962	5	the	the	DET
ejde-673	962	6	eigenvalues	eigenvalue	NOUN
ejde-673	962	7	references	reference	NOUN