id	sid	tid	token	lemma	pos
ejde-92	1	1	electronic	electronic	ADJ
ejde-92	1	2	journal	journal	NOUN
ejde-92	1	3	of	of	ADP
ejde-92	1	4	differential	differential	ADJ
ejde-92	1	5	equations	equation	NOUN
ejde-92	1	6	,	,	PUNCT
ejde-92	1	7	vol	vol	NOUN
ejde-92	1	8	.	.	PUNCT
ejde-92	1	9	2022	2022	NUM
ejde-92	1	10	(	(	PUNCT
ejde-92	1	11	2022	2022	NUM
ejde-92	1	12	)	)	PUNCT
ejde-92	1	13	,	,	PUNCT
ejde-92	1	14	no	no	INTJ
ejde-92	1	15	.	.	NOUN
ejde-92	1	16	13	13	NUM
ejde-92	1	17	,	,	PUNCT
ejde-92	1	18	pp	pp	PROPN
ejde-92	1	19	.	.	PUNCT
ejde-92	2	1	1–12	1–12	NOUN
ejde-92	2	2	.	.	PUNCT
ejde-92	3	1	issn	issn	PROPN
ejde-92	3	2	:	:	PUNCT
ejde-92	3	3	1072	1072	NUM
ejde-92	3	4	-	-	SYM
ejde-92	3	5	6691	6691	NUM
ejde-92	3	6	.	.	PUNCT
ejde-92	4	1	url	url	PROPN
ejde-92	4	2	:	:	PUNCT
ejde-92	4	3	http://ejde.math.txstate.edu	http://ejde.math.txstate.edu	PROPN
ejde-92	4	4	or	or	CCONJ
ejde-92	4	5	http://ejde.math.unt.edu	http://ejde.math.unt.edu	PROPN
ejde-92	4	6	remarks	remark	NOUN
ejde-92	4	7	on	on	ADP
ejde-92	4	8	the	the	DET
ejde-92	4	9	second	second	ADJ
ejde-92	4	10	neumann	neumann	PROPN
ejde-92	4	11	eigenvalue	eigenvalue	PROPN
ejde-92	4	12	josé	josé	PROPN
ejde-92	4	13	c.	c.	PROPN
ejde-92	4	14	sabina	sabina	PROPN
ejde-92	4	15	de	de	PROPN
ejde-92	4	16	lis	lis	PROPN
ejde-92	4	17	abstract	abstract	NOUN
ejde-92	4	18	.	.	PUNCT
ejde-92	5	1	this	this	DET
ejde-92	5	2	work	work	NOUN
ejde-92	5	3	reviews	review	VERB
ejde-92	5	4	some	some	DET
ejde-92	5	5	basic	basic	ADJ
ejde-92	5	6	features	feature	NOUN
ejde-92	5	7	on	on	ADP
ejde-92	5	8	the	the	DET
ejde-92	5	9	second	second	ADJ
ejde-92	5	10	(	(	PUNCT
ejde-92	5	11	first	first	ADJ
ejde-92	5	12	nontrivial	nontrivial	NOUN
ejde-92	5	13	)	)	PUNCT
ejde-92	5	14	eigenvalue	eigenvalue	NOUN
ejde-92	5	15	λ2	λ2	NOUN
ejde-92	5	16	to	to	ADP
ejde-92	5	17	the	the	DET
ejde-92	5	18	neumann	neumann	PROPN
ejde-92	5	19	problem	problem	NOUN
ejde-92	5	20	−∆pu	−∆pu	X
ejde-92	6	1	=	=	SYM
ejde-92	6	2	λ|u|p−2u	λ|u|p−2u	PROPN
ejde-92	6	3	x	x	PUNCT
ejde-92	6	4	∈	∈	PROPN
ejde-92	6	5	ω	ω	NUM
ejde-92	6	6	|∇u|p−2	|∇u|p−2	PROPN
ejde-92	6	7	∂u	∂u	PROPN
ejde-92	6	8	∂ν	∂ν	X
ejde-92	7	1	=	=	PUNCT
ejde-92	7	2	0	0	PUNCT
ejde-92	7	3	x	x	SYM
ejde-92	7	4	∈	∈	PROPN
ejde-92	7	5	∂ω	∂ω	PROPN
ejde-92	7	6	,	,	PUNCT
ejde-92	7	7	where	where	SCONJ
ejde-92	7	8	ω	ω	PROPN
ejde-92	7	9	is	be	AUX
ejde-92	7	10	a	a	DET
ejde-92	7	11	bounded	bounded	ADJ
ejde-92	7	12	lipschitz	lipschitz	NOUN
ejde-92	7	13	domain	domain	NOUN
ejde-92	7	14	of	of	ADP
ejde-92	7	15	rn	rn	PROPN
ejde-92	7	16	,	,	PUNCT
ejde-92	7	17	ν	ν	PROPN
ejde-92	7	18	is	be	AUX
ejde-92	7	19	the	the	DET
ejde-92	7	20	outer	outer	ADJ
ejde-92	7	21	unit	unit	NOUN
ejde-92	7	22	normal	normal	ADJ
ejde-92	7	23	,	,	PUNCT
ejde-92	7	24	and	and	CCONJ
ejde-92	7	25	∆pu	∆pu	NOUN
ejde-92	7	26	=	=	SYM
ejde-92	7	27	div(|∇u|p−2∇u	div(|∇u|p−2∇u	PROPN
ejde-92	7	28	)	)	PUNCT
ejde-92	7	29	is	be	AUX
ejde-92	7	30	the	the	DET
ejde-92	7	31	p	p	PROPN
ejde-92	7	32	-	-	PUNCT
ejde-92	7	33	laplacian	laplacian	ADJ
ejde-92	7	34	operator	operator	NOUN
ejde-92	7	35	.	.	PUNCT
ejde-92	8	1	we	we	PRON
ejde-92	8	2	are	be	AUX
ejde-92	8	3	mainly	mainly	ADV
ejde-92	8	4	concerned	concerned	ADJ
ejde-92	8	5	with	with	ADP
ejde-92	8	6	the	the	DET
ejde-92	8	7	variational	variational	ADJ
ejde-92	8	8	characterization	characterization	NOUN
ejde-92	8	9	of	of	ADP
ejde-92	8	10	λ2	λ2	NOUN
ejde-92	8	11	and	and	CCONJ
ejde-92	8	12	place	place	NOUN
ejde-92	8	13	emphasis	emphasis	NOUN
ejde-92	8	14	on	on	ADP
ejde-92	8	15	the	the	DET
ejde-92	8	16	range	range	NOUN
ejde-92	8	17	1	1	NUM
ejde-92	8	18	<	<	X
ejde-92	8	19	p	p	X
ejde-92	8	20	<	<	X
ejde-92	8	21	2	2	NUM
ejde-92	8	22	,	,	PUNCT
ejde-92	8	23	where	where	SCONJ
ejde-92	8	24	the	the	DET
ejde-92	8	25	nonlinearity	nonlinearity	NOUN
ejde-92	8	26	|u|p−2u	|u|p−2u	NOUN
ejde-92	8	27	becomes	become	VERB
ejde-92	8	28	non	non	ADJ
ejde-92	8	29	smooth	smooth	ADJ
ejde-92	8	30	.	.	PUNCT
ejde-92	9	1	we	we	PRON
ejde-92	9	2	also	also	ADV
ejde-92	9	3	address	address	VERB
ejde-92	9	4	the	the	DET
ejde-92	9	5	corresponding	corresponding	ADJ
ejde-92	9	6	result	result	NOUN
ejde-92	9	7	for	for	ADP
ejde-92	9	8	the	the	DET
ejde-92	9	9	p	p	NOUN
ejde-92	9	10	-	-	PUNCT
ejde-92	9	11	laplacian	laplacian	NOUN
ejde-92	9	12	in	in	ADP
ejde-92	9	13	graphs	graph	NOUN
ejde-92	9	14	.	.	PUNCT
ejde-92	10	1	1	1	X
ejde-92	10	2	.	.	X
ejde-92	10	3	introduction	introduction	NOUN
ejde-92	10	4	the	the	DET
ejde-92	10	5	analysis	analysis	NOUN
ejde-92	10	6	of	of	ADP
ejde-92	10	7	the	the	DET
ejde-92	10	8	eigenvalues	eigenvalue	NOUN
ejde-92	10	9	of	of	ADP
ejde-92	10	10	the	the	DET
ejde-92	10	11	p	p	ADJ
ejde-92	10	12	-	-	PUNCT
ejde-92	10	13	laplacian	laplacian	ADJ
ejde-92	10	14	operator	operator	NOUN
ejde-92	10	15	under	under	ADP
ejde-92	10	16	different	different	ADJ
ejde-92	10	17	types	type	NOUN
ejde-92	10	18	of	of	ADP
ejde-92	10	19	boundary	boundary	ADJ
ejde-92	10	20	conditions	condition	NOUN
ejde-92	10	21	is	be	AUX
ejde-92	10	22	one	one	NUM
ejde-92	10	23	of	of	ADP
ejde-92	10	24	the	the	DET
ejde-92	10	25	most	most	ADV
ejde-92	10	26	interesting	interesting	ADJ
ejde-92	10	27	issues	issue	NOUN
ejde-92	10	28	in	in	ADP
ejde-92	10	29	nonlinear	nonlinear	ADJ
ejde-92	10	30	analysis	analysis	NOUN
ejde-92	10	31	[	[	X
ejde-92	10	32	13	13	NUM
ejde-92	10	33	,	,	PUNCT
ejde-92	10	34	14	14	NUM
ejde-92	10	35	,	,	PUNCT
ejde-92	10	36	17	17	NUM
ejde-92	10	37	]	]	PUNCT
ejde-92	10	38	.	.	PUNCT
ejde-92	11	1	here	here	ADV
ejde-92	11	2	,	,	PUNCT
ejde-92	11	3	we	we	PRON
ejde-92	11	4	focuss	focuss	ADJ
ejde-92	11	5	on	on	ADP
ejde-92	11	6	the	the	DET
ejde-92	11	7	neumann	neumann	PROPN
ejde-92	11	8	eigenvalue	eigenvalue	PROPN
ejde-92	11	9	problem	problem	NOUN
ejde-92	11	10	−∆pu	−∆pu	X
ejde-92	12	1	=	=	SYM
ejde-92	12	2	λ|u|p−2u	λ|u|p−2u	PROPN
ejde-92	12	3	x	x	PUNCT
ejde-92	12	4	∈	∈	PROPN
ejde-92	12	5	ω	ω	NUM
ejde-92	12	6	|∇u|p−2	|∇u|p−2	PROPN
ejde-92	12	7	∂u	∂u	PROPN
ejde-92	12	8	∂ν	∂ν	X
ejde-92	13	1	=	=	PUNCT
ejde-92	13	2	0	0	PUNCT
ejde-92	13	3	x	x	SYM
ejde-92	13	4	∈	∈	PROPN
ejde-92	13	5	∂ω	∂ω	PROPN
ejde-92	13	6	,	,	PUNCT
ejde-92	13	7	(	(	PUNCT
ejde-92	13	8	1.1	1.1	NUM
ejde-92	13	9	)	)	PUNCT
ejde-92	13	10	where	where	SCONJ
ejde-92	13	11	ω	ω	PROPN
ejde-92	13	12	⊂	⊂	PROPN
ejde-92	13	13	rn	rn	PROPN
ejde-92	13	14	is	be	AUX
ejde-92	13	15	a	a	DET
ejde-92	13	16	c0,1	c0,1	NOUN
ejde-92	13	17	bounded	bounded	ADJ
ejde-92	13	18	domain	domain	NOUN
ejde-92	13	19	,	,	PUNCT
ejde-92	13	20	ν	ν	PROPN
ejde-92	13	21	stands	stand	VERB
ejde-92	13	22	for	for	ADP
ejde-92	13	23	its	its	PRON
ejde-92	13	24	outer	outer	ADJ
ejde-92	13	25	unit	unit	NOUN
ejde-92	13	26	normal	normal	ADJ
ejde-92	13	27	on	on	ADP
ejde-92	13	28	∂ω	∂ω	PROPN
ejde-92	13	29	and	and	CCONJ
ejde-92	13	30	p	p	X
ejde-92	13	31	>	>	X
ejde-92	13	32	1	1	X
ejde-92	13	33	.	.	PUNCT
ejde-92	14	1	we	we	PRON
ejde-92	14	2	recall	recall	VERB
ejde-92	14	3	that	that	SCONJ
ejde-92	14	4	u	u	PROPN
ejde-92	14	5	∈	∈	PROPN
ejde-92	14	6	w	w	PROPN
ejde-92	14	7	1,p(ω	1,p(ω	NUM
ejde-92	14	8	)	)	PUNCT
ejde-92	14	9	\	\	NOUN
ejde-92	14	10	{	{	PUNCT
ejde-92	14	11	0	0	NUM
ejde-92	14	12	}	}	PUNCT
ejde-92	14	13	is	be	AUX
ejde-92	14	14	said	say	VERB
ejde-92	14	15	to	to	PART
ejde-92	14	16	be	be	AUX
ejde-92	14	17	a	a	DET
ejde-92	14	18	weak	weak	ADJ
ejde-92	14	19	eigenfunction	eigenfunction	NOUN
ejde-92	14	20	to	to	ADP
ejde-92	14	21	(	(	PUNCT
ejde-92	14	22	1.1	1.1	NUM
ejde-92	14	23	)	)	PUNCT
ejde-92	14	24	associated	associate	VERB
ejde-92	14	25	with	with	ADP
ejde-92	14	26	the	the	DET
ejde-92	14	27	eigenvalue	eigenvalue	PROPN
ejde-92	14	28	λ	λ	SYM
ejde-92	14	29	∈	∈	NOUN
ejde-92	14	30	r	r	NOUN
ejde-92	15	1	if	if	SCONJ
ejde-92	15	2	the	the	DET
ejde-92	15	3	equality∫	equality∫	NOUN
ejde-92	15	4	ω	ω	PROPN
ejde-92	15	5	|∇u|p−2∇u∇v	|∇u|p−2∇u∇v	NOUN
ejde-92	15	6	dx	dx	PROPN
ejde-92	16	1	=	=	SYM
ejde-92	16	2	λ	λ	PROPN
ejde-92	16	3	∫	∫	PROPN
ejde-92	16	4	ω	ω	PROPN
ejde-92	16	5	|u|p−2uv	|u|p−2uv	PROPN
ejde-92	16	6	dx	dx	PROPN
ejde-92	16	7	,	,	PUNCT
ejde-92	16	8	(	(	PUNCT
ejde-92	16	9	1.2	1.2	NUM
ejde-92	16	10	)	)	PUNCT
ejde-92	16	11	holds	hold	VERB
ejde-92	16	12	for	for	ADP
ejde-92	16	13	arbitrary	arbitrary	ADJ
ejde-92	16	14	test	test	NOUN
ejde-92	16	15	functions	function	NOUN
ejde-92	16	16	v	v	NOUN
ejde-92	16	17	in	in	ADP
ejde-92	16	18	w	w	PROPN
ejde-92	16	19	1,p(ω	1,p(ω	NUM
ejde-92	16	20	)	)	PUNCT
ejde-92	16	21	.	.	PUNCT
ejde-92	17	1	from	from	ADP
ejde-92	17	2	the	the	DET
ejde-92	17	3	definition	definition	NOUN
ejde-92	17	4	of	of	ADP
ejde-92	17	5	eigenvalue	eigenvalue	PROPN
ejde-92	17	6	it	it	PRON
ejde-92	17	7	follows	follow	VERB
ejde-92	17	8	by	by	ADP
ejde-92	17	9	choosing	choose	VERB
ejde-92	17	10	v	v	NUM
ejde-92	17	11	=	=	SYM
ejde-92	17	12	u	u	NOUN
ejde-92	17	13	in	in	ADP
ejde-92	17	14	(	(	PUNCT
ejde-92	17	15	1.2	1.2	NUM
ejde-92	17	16	)	)	PUNCT
ejde-92	17	17	that	that	PRON
ejde-92	17	18	eigenvalues	eigenvalue	VERB
ejde-92	17	19	λ	λ	NOUN
ejde-92	17	20	must	must	AUX
ejde-92	17	21	be	be	AUX
ejde-92	17	22	nonnegative	nonnegative	ADJ
ejde-92	17	23	.	.	PUNCT
ejde-92	18	1	thus	thus	ADV
ejde-92	18	2	λ1	λ1	VERB
ejde-92	18	3	=	=	SYM
ejde-92	18	4	0	0	NUM
ejde-92	18	5	becomes	become	VERB
ejde-92	18	6	the	the	DET
ejde-92	18	7	“	"	PUNCT
ejde-92	18	8	first	first	ADJ
ejde-92	18	9	”	"	PUNCT
ejde-92	18	10	(	(	PUNCT
ejde-92	18	11	lowest	low	ADJ
ejde-92	18	12	)	)	PUNCT
ejde-92	18	13	eigenvalue	eigenvalue	VERB
ejde-92	18	14	whose	whose	DET
ejde-92	18	15	eigenfunctions	eigenfunction	NOUN
ejde-92	18	16	are	be	AUX
ejde-92	18	17	constant	constant	ADJ
ejde-92	18	18	functions	function	NOUN
ejde-92	18	19	.	.	PUNCT
ejde-92	19	1	as	as	SCONJ
ejde-92	19	2	all	all	DET
ejde-92	19	3	those	those	DET
ejde-92	19	4	eigenfunctions	eigenfunction	NOUN
ejde-92	19	5	are	be	AUX
ejde-92	19	6	a	a	DET
ejde-92	19	7	multiple	multiple	NOUN
ejde-92	19	8	of	of	ADP
ejde-92	19	9	u	u	NOUN
ejde-92	19	10	=	=	PROPN
ejde-92	19	11	1	1	NUM
ejde-92	19	12	,	,	PUNCT
ejde-92	19	13	this	this	PRON
ejde-92	19	14	amounts	amount	VERB
ejde-92	19	15	to	to	PART
ejde-92	19	16	say	say	VERB
ejde-92	19	17	that	that	SCONJ
ejde-92	19	18	λ1	λ1	PROPN
ejde-92	19	19	is	be	AUX
ejde-92	19	20	simple	simple	ADJ
ejde-92	19	21	(	(	PUNCT
ejde-92	19	22	in	in	ADP
ejde-92	19	23	a	a	DET
ejde-92	19	24	proper	proper	ADJ
ejde-92	19	25	sense	sense	NOUN
ejde-92	19	26	)	)	PUNCT
ejde-92	19	27	.	.	PUNCT
ejde-92	20	1	moreover	moreover	ADV
ejde-92	20	2	,	,	PUNCT
ejde-92	20	3	λ1	λ1	PROPN
ejde-92	20	4	=	=	SYM
ejde-92	20	5	0	0	NUM
ejde-92	20	6	is	be	AUX
ejde-92	20	7	an	an	DET
ejde-92	20	8	isolated	isolate	VERB
ejde-92	20	9	eigenvalue	eigenvalue	NOUN
ejde-92	20	10	as	as	SCONJ
ejde-92	20	11	it	it	PRON
ejde-92	20	12	is	be	AUX
ejde-92	20	13	going	go	VERB
ejde-92	20	14	to	to	PART
ejde-92	20	15	be	be	AUX
ejde-92	20	16	checked	check	VERB
ejde-92	20	17	below	below	ADV
ejde-92	20	18	(	(	PUNCT
ejde-92	20	19	theorem	theorem	VERB
ejde-92	20	20	1.1	1.1	NUM
ejde-92	20	21	)	)	PUNCT
ejde-92	20	22	.	.	PUNCT
ejde-92	21	1	it	it	PRON
ejde-92	21	2	should	should	AUX
ejde-92	21	3	be	be	AUX
ejde-92	21	4	stressed	stress	VERB
ejde-92	21	5	that	that	SCONJ
ejde-92	21	6	for	for	ADP
ejde-92	21	7	the	the	DET
ejde-92	21	8	dirichlet	dirichlet	PROPN
ejde-92	21	9	boundary	boundary	PROPN
ejde-92	21	10	condition	condition	NOUN
ejde-92	21	11	u	u	NOUN
ejde-92	21	12	=	=	NOUN
ejde-92	21	13	0	0	NUM
ejde-92	21	14	on	on	ADP
ejde-92	21	15	∂ω	∂ω	PROPN
ejde-92	21	16	,	,	PUNCT
ejde-92	21	17	proving	prove	VERB
ejde-92	21	18	the	the	DET
ejde-92	21	19	simplicity	simplicity	NOUN
ejde-92	21	20	of	of	ADP
ejde-92	21	21	the	the	DET
ejde-92	21	22	first	first	ADJ
ejde-92	21	23	(	(	PUNCT
ejde-92	21	24	lowest	low	ADJ
ejde-92	21	25	)	)	PUNCT
ejde-92	21	26	eigenvalue	eigenvalue	NOUN
ejde-92	21	27	λd1	λd1	NOUN
ejde-92	21	28	turned	turn	VERB
ejde-92	21	29	out	out	ADP
ejde-92	21	30	to	to	ADP
ejde-92	21	31	2020	2020	NUM
ejde-92	21	32	mathematics	mathematic	NOUN
ejde-92	21	33	subject	subject	ADJ
ejde-92	21	34	classification	classification	NOUN
ejde-92	21	35	.	.	PUNCT
ejde-92	22	1	35j70	35j70	NUM
ejde-92	22	2	,	,	PUNCT
ejde-92	22	3	35j92	35j92	NUM
ejde-92	22	4	,	,	PUNCT
ejde-92	22	5	35p30	35p30	NUM
ejde-92	22	6	.	.	PUNCT
ejde-92	23	1	key	key	ADJ
ejde-92	23	2	words	word	NOUN
ejde-92	23	3	and	and	CCONJ
ejde-92	23	4	phrases	phrase	NOUN
ejde-92	23	5	.	.	PUNCT
ejde-92	24	1	p	p	X
ejde-92	24	2	-	-	PUNCT
ejde-92	24	3	laplacian	laplacian	ADJ
ejde-92	24	4	operator	operator	NOUN
ejde-92	24	5	;	;	PUNCT
ejde-92	24	6	eigenvalues	eigenvalue	NOUN
ejde-92	24	7	;	;	PUNCT
ejde-92	24	8	neumann	neumann	PROPN
ejde-92	24	9	conditions	condition	NOUN
ejde-92	24	10	.	.	PUNCT
ejde-92	25	1	©	©	ADP
ejde-92	25	2	2022	2022	NUM
ejde-92	25	3	.	.	PUNCT
ejde-92	26	1	this	this	DET
ejde-92	26	2	work	work	NOUN
ejde-92	26	3	is	be	AUX
ejde-92	26	4	licensed	license	VERB
ejde-92	26	5	under	under	ADP
ejde-92	26	6	a	a	DET
ejde-92	26	7	cc	cc	NOUN
ejde-92	26	8	by	by	ADP
ejde-92	26	9	4.0	4.0	NUM
ejde-92	26	10	license	license	NOUN
ejde-92	26	11	.	.	PUNCT
ejde-92	27	1	submitted	submit	VERB
ejde-92	27	2	august	august	PROPN
ejde-92	27	3	16	16	NUM
ejde-92	27	4	,	,	PUNCT
ejde-92	27	5	2021	2021	NUM
ejde-92	27	6	.	.	PUNCT
ejde-92	28	1	published	publish	VERB
ejde-92	28	2	february	february	PROPN
ejde-92	28	3	20	20	NUM
ejde-92	28	4	,	,	PUNCT
ejde-92	28	5	2022	2022	NUM
ejde-92	28	6	.	.	PUNCT
ejde-92	29	1	1	1	NUM
ejde-92	29	2	2	2	NUM
ejde-92	29	3	j.	j.	PROPN
ejde-92	29	4	c.	c.	PROPN
ejde-92	29	5	sabina	sabina	PROPN
ejde-92	29	6	de	de	PROPN
ejde-92	29	7	lis	lis	PROPN
ejde-92	29	8	ejde-2022/13	ejde-2022/13	ADV
ejde-92	29	9	be	be	AUX
ejde-92	29	10	a	a	DET
ejde-92	29	11	quite	quite	ADV
ejde-92	29	12	hard	hard	ADJ
ejde-92	29	13	question	question	NOUN
ejde-92	29	14	.	.	PUNCT
ejde-92	30	1	a	a	DET
ejde-92	30	2	successful	successful	ADJ
ejde-92	30	3	answer	answer	NOUN
ejde-92	30	4	was	be	AUX
ejde-92	30	5	given	give	VERB
ejde-92	30	6	in	in	ADP
ejde-92	30	7	[	[	X
ejde-92	30	8	3	3	NUM
ejde-92	30	9	]	]	PUNCT
ejde-92	30	10	,	,	PUNCT
ejde-92	30	11	where	where	SCONJ
ejde-92	30	12	the	the	DET
ejde-92	30	13	isolation	isolation	NOUN
ejde-92	30	14	of	of	ADP
ejde-92	30	15	λd1	λd1	NOUN
ejde-92	30	16	was	be	AUX
ejde-92	30	17	furthermore	furthermore	ADV
ejde-92	30	18	shown	show	VERB
ejde-92	30	19	(	(	PUNCT
ejde-92	30	20	a	a	DET
ejde-92	30	21	sharpened	sharpen	VERB
ejde-92	30	22	result	result	NOUN
ejde-92	30	23	was	be	AUX
ejde-92	30	24	later	later	ADV
ejde-92	30	25	proved	prove	VERB
ejde-92	30	26	in	in	ADP
ejde-92	30	27	[	[	X
ejde-92	30	28	15	15	NUM
ejde-92	30	29	,	,	PUNCT
ejde-92	30	30	16	16	NUM
ejde-92	30	31	]	]	PUNCT
ejde-92	30	32	)	)	PUNCT
ejde-92	30	33	.	.	PUNCT
ejde-92	31	1	moreover	moreover	ADV
ejde-92	31	2	,	,	PUNCT
ejde-92	31	3	the	the	DET
ejde-92	31	4	existence	existence	NOUN
ejde-92	31	5	of	of	ADP
ejde-92	31	6	a	a	DET
ejde-92	31	7	second	second	ADJ
ejde-92	31	8	dirichlet	dirichlet	PROPN
ejde-92	31	9	eigenvalue	eigenvalue	PROPN
ejde-92	31	10	λd2	λd2	PROPN
ejde-92	31	11	is	be	AUX
ejde-92	31	12	a	a	DET
ejde-92	31	13	consequence	consequence	NOUN
ejde-92	31	14	of	of	ADP
ejde-92	31	15	the	the	DET
ejde-92	31	16	latter	latter	ADJ
ejde-92	31	17	assertion	assertion	NOUN
ejde-92	31	18	.	.	PUNCT
ejde-92	32	1	in	in	ADP
ejde-92	32	2	addition	addition	NOUN
ejde-92	32	3	,	,	PUNCT
ejde-92	32	4	it	it	PRON
ejde-92	32	5	is	be	AUX
ejde-92	32	6	worth	worth	ADJ
ejde-92	32	7	remarking	remark	VERB
ejde-92	32	8	that	that	SCONJ
ejde-92	32	9	obtaining	obtain	VERB
ejde-92	32	10	a	a	DET
ejde-92	32	11	variational	variational	ADJ
ejde-92	32	12	characterization	characterization	NOUN
ejde-92	32	13	of	of	ADP
ejde-92	32	14	λd2	λd2	PROPN
ejde-92	32	15	is	be	AUX
ejde-92	32	16	by	by	ADP
ejde-92	32	17	no	no	DET
ejde-92	32	18	means	means	NOUN
ejde-92	32	19	an	an	DET
ejde-92	32	20	easy	easy	ADJ
ejde-92	32	21	task	task	NOUN
ejde-92	32	22	.	.	PUNCT
ejde-92	33	1	reader	reader	NOUN
ejde-92	33	2	is	be	AUX
ejde-92	33	3	referred	refer	VERB
ejde-92	33	4	to	to	ADP
ejde-92	33	5	[	[	X
ejde-92	33	6	4	4	NUM
ejde-92	33	7	,	,	PUNCT
ejde-92	33	8	11	11	NUM
ejde-92	33	9	,	,	PUNCT
ejde-92	33	10	12	12	NUM
ejde-92	33	11	]	]	PUNCT
ejde-92	33	12	for	for	ADP
ejde-92	33	13	different	different	ADJ
ejde-92	33	14	variational	variational	ADJ
ejde-92	33	15	expressions	expression	NOUN
ejde-92	33	16	of	of	ADP
ejde-92	33	17	λd2	λd2	PROPN
ejde-92	33	18	.	.	PUNCT
ejde-92	34	1	just	just	ADV
ejde-92	34	2	to	to	PART
ejde-92	34	3	grasp	grasp	VERB
ejde-92	34	4	an	an	DET
ejde-92	34	5	insight	insight	NOUN
ejde-92	34	6	,	,	PUNCT
ejde-92	34	7	the	the	DET
ejde-92	34	8	corresponding	correspond	VERB
ejde-92	34	9	one	one	NUM
ejde-92	34	10	in	in	ADP
ejde-92	34	11	[	[	X
ejde-92	34	12	11	11	NUM
ejde-92	34	13	]	]	PUNCT
ejde-92	34	14	is	be	AUX
ejde-92	34	15	λd2	λd2	X
ejde-92	34	16	=	=	PROPN
ejde-92	34	17	inf	inf	PROPN
ejde-92	34	18	γ	γ	PROPN
ejde-92	34	19	max	max	PROPN
ejde-92	34	20	u∈γ(i	u∈γ(i	PROPN
ejde-92	34	21	)	)	PUNCT
ejde-92	34	22	∫	∫	PROPN
ejde-92	35	1	ω	ω	PROPN
ejde-92	35	2	|∇u|p	|∇u|p	PROPN
ejde-92	35	3	dx∫	dx∫	PROPN
ejde-92	35	4	ω	ω	PROPN
ejde-92	35	5	|u|p	|u|p	PROPN
ejde-92	35	6	dx	dx	PROPN
ejde-92	35	7	,	,	PUNCT
ejde-92	35	8	(	(	PUNCT
ejde-92	35	9	1.3	1.3	NUM
ejde-92	35	10	)	)	PUNCT
ejde-92	35	11	where	where	SCONJ
ejde-92	35	12	γ	γ	PROPN
ejde-92	35	13	varies	vary	VERB
ejde-92	35	14	in	in	ADP
ejde-92	35	15	the	the	DET
ejde-92	35	16	set	set	NOUN
ejde-92	35	17	of	of	ADP
ejde-92	35	18	all	all	DET
ejde-92	35	19	continuous	continuous	ADJ
ejde-92	35	20	curves	curve	NOUN
ejde-92	35	21	γ	γ	NOUN
ejde-92	35	22	:	:	PUNCT
ejde-92	35	23	i	i	NOUN
ejde-92	35	24	=	=	PUNCT
ejde-92	36	1	[	[	X
ejde-92	36	2	0	0	NUM
ejde-92	36	3	,	,	PUNCT
ejde-92	36	4	1]→w	1]→w	NUM
ejde-92	36	5	1,p	1,p	NOUN
ejde-92	36	6	0	0	NUM
ejde-92	36	7	(	(	PUNCT
ejde-92	36	8	ω)\{0	ω)\{0	PROPN
ejde-92	36	9	}	}	PUNCT
ejde-92	36	10	such	such	ADJ
ejde-92	36	11	that	that	DET
ejde-92	36	12	γ(0	γ(0	PROPN
ejde-92	36	13	)	)	PUNCT
ejde-92	36	14	=	=	SYM
ejde-92	36	15	φ1	φ1	PROPN
ejde-92	36	16	,	,	PUNCT
ejde-92	36	17	γ(1	γ(1	PROPN
ejde-92	36	18	)	)	PUNCT
ejde-92	37	1	=	=	SYM
ejde-92	37	2	−φ1	−φ1	PROPN
ejde-92	37	3	,	,	PUNCT
ejde-92	37	4	φ1	φ1	PROPN
ejde-92	37	5	being	be	AUX
ejde-92	37	6	a	a	DET
ejde-92	37	7	fixed	fix	VERB
ejde-92	37	8	normalized	normalize	VERB
ejde-92	37	9	eigenfunction	eigenfunction	NOUN
ejde-92	37	10	associated	associate	VERB
ejde-92	37	11	with	with	ADP
ejde-92	37	12	λd1	λd1	PROPN
ejde-92	37	13	.	.	PUNCT
ejde-92	38	1	see	see	VERB
ejde-92	38	2	also	also	ADV
ejde-92	38	3	[	[	X
ejde-92	38	4	5	5	NUM
ejde-92	38	5	,	,	PUNCT
ejde-92	38	6	6	6	NUM
ejde-92	38	7	]	]	PUNCT
ejde-92	38	8	for	for	ADP
ejde-92	38	9	related	related	ADJ
ejde-92	38	10	results	result	NOUN
ejde-92	38	11	in	in	ADP
ejde-92	38	12	a	a	DET
ejde-92	38	13	“	"	PUNCT
ejde-92	38	14	nonsymmetric	nonsymmetric	ADJ
ejde-92	38	15	”	"	PUNCT
ejde-92	38	16	version	version	NOUN
ejde-92	38	17	of	of	ADP
ejde-92	38	18	the	the	DET
ejde-92	38	19	dirichlet	dirichlet	PROPN
ejde-92	38	20	and	and	CCONJ
ejde-92	38	21	neumann	neumann	PROPN
ejde-92	38	22	eigenvalue	eigenvalue	PROPN
ejde-92	38	23	problems	problem	NOUN
ejde-92	38	24	.	.	PUNCT
ejde-92	39	1	as	as	SCONJ
ejde-92	39	2	mentioned	mention	VERB
ejde-92	39	3	above	above	ADV
ejde-92	39	4	,	,	PUNCT
ejde-92	39	5	λ1	λ1	PROPN
ejde-92	39	6	=	=	SYM
ejde-92	39	7	0	0	NUM
ejde-92	39	8	is	be	AUX
ejde-92	39	9	isolated	isolate	VERB
ejde-92	39	10	and	and	CCONJ
ejde-92	39	11	so	so	ADV
ejde-92	39	12	the	the	DET
ejde-92	39	13	neumann	neumann	PROPN
ejde-92	39	14	problem	problem	NOUN
ejde-92	39	15	(	(	PUNCT
ejde-92	39	16	1.1	1.1	NUM
ejde-92	39	17	)	)	PUNCT
ejde-92	39	18	admits	admit	VERB
ejde-92	39	19	a	a	DET
ejde-92	39	20	second	second	ADJ
ejde-92	39	21	eigenvalue	eigenvalue	ADJ
ejde-92	39	22	λ2	λ2	NOUN
ejde-92	39	23	.	.	PUNCT
ejde-92	40	1	more	more	ADV
ejde-92	40	2	interestingly	interestingly	ADV
ejde-92	40	3	,	,	PUNCT
ejde-92	40	4	i	i	PRON
ejde-92	40	5	came	come	VERB
ejde-92	40	6	across	across	ADP
ejde-92	40	7	reference	reference	NOUN
ejde-92	40	8	[	[	X
ejde-92	40	9	7	7	X
ejde-92	40	10	]	]	PUNCT
ejde-92	40	11	when	when	SCONJ
ejde-92	40	12	searching	search	VERB
ejde-92	40	13	for	for	ADP
ejde-92	40	14	sharp	sharp	ADJ
ejde-92	40	15	lower	low	ADJ
ejde-92	40	16	estimates	estimate	NOUN
ejde-92	40	17	for	for	ADP
ejde-92	40	18	λ2	λ2	NOUN
ejde-92	40	19	in	in	ADP
ejde-92	40	20	a	a	DET
ejde-92	40	21	general	general	ADJ
ejde-92	40	22	domain	domain	NOUN
ejde-92	40	23	ω	ω	NOUN
ejde-92	40	24	.	.	PUNCT
ejde-92	41	1	i	i	PRON
ejde-92	41	2	was	be	AUX
ejde-92	41	3	amazed	amazed	ADJ
ejde-92	41	4	by	by	ADP
ejde-92	41	5	the	the	DET
ejde-92	41	6	existence	existence	NOUN
ejde-92	41	7	of	of	ADP
ejde-92	41	8	the	the	DET
ejde-92	41	9	elegant	elegant	ADJ
ejde-92	41	10	characterization	characterization	NOUN
ejde-92	41	11	of	of	ADP
ejde-92	41	12	this	this	DET
ejde-92	41	13	eigenvalue	eigenvalue	NOUN
ejde-92	41	14	,	,	PUNCT
ejde-92	41	15	freely	freely	ADV
ejde-92	41	16	used	use	VERB
ejde-92	41	17	there	there	ADV
ejde-92	41	18	.	.	PUNCT
ejde-92	42	1	the	the	DET
ejde-92	42	2	expression	expression	NOUN
ejde-92	42	3	is	be	AUX
ejde-92	42	4	considerably	considerably	ADV
ejde-92	42	5	much	much	ADV
ejde-92	42	6	simpler	simple	ADJ
ejde-92	42	7	than	than	ADP
ejde-92	42	8	its	its	PRON
ejde-92	42	9	dirichlet	dirichlet	NOUN
ejde-92	42	10	counterpart	counterpart	NOUN
ejde-92	42	11	(	(	PUNCT
ejde-92	42	12	1.3	1.3	NUM
ejde-92	42	13	)	)	PUNCT
ejde-92	42	14	,	,	PUNCT
ejde-92	42	15	and	and	CCONJ
ejde-92	42	16	seems	seem	VERB
ejde-92	42	17	indeed	indeed	ADV
ejde-92	42	18	closer	close	ADJ
ejde-92	42	19	to	to	ADP
ejde-92	42	20	the	the	DET
ejde-92	42	21	familiar	familiar	ADJ
ejde-92	42	22	‘	'	PUNCT
ejde-92	42	23	rayleigh	rayleigh	PROPN
ejde-92	42	24	quotient	quotient	NOUN
ejde-92	42	25	’	'	PUNCT
ejde-92	42	26	for	for	ADP
ejde-92	42	27	−∆.	−∆.	NOUN
ejde-92	42	28	namely	namely	ADV
ejde-92	43	1	,	,	PUNCT
ejde-92	43	2	λ2	λ2	PROPN
ejde-92	43	3	=	=	SYM
ejde-92	43	4	inf	inf	PROPN
ejde-92	43	5	∫	∫	PROPN
ejde-92	43	6	ω	ω	PROPN
ejde-92	43	7	|∇u|p	|∇u|p	PROPN
ejde-92	43	8	dx∫	dx∫	PROPN
ejde-92	43	9	ω	ω	PROPN
ejde-92	43	10	|u|p	|u|p	PROPN
ejde-92	43	11	dx	dx	PROPN
ejde-92	43	12	,	,	PUNCT
ejde-92	43	13	(	(	PUNCT
ejde-92	43	14	1.4	1.4	NUM
ejde-92	43	15	)	)	PUNCT
ejde-92	43	16	the	the	DET
ejde-92	43	17	infimum	infimum	ADJ
ejde-92	43	18	being	be	AUX
ejde-92	43	19	extended	extend	VERB
ejde-92	43	20	over	over	ADP
ejde-92	43	21	all	all	DET
ejde-92	43	22	those	those	DET
ejde-92	43	23	nonvanishing	nonvanishing	NOUN
ejde-92	43	24	functions	function	NOUN
ejde-92	43	25	u	u	NOUN
ejde-92	43	26	∈	∈	PROPN
ejde-92	43	27	w	w	PROPN
ejde-92	43	28	1,p(ω	1,p(ω	NUM
ejde-92	43	29	)	)	PUNCT
ejde-92	43	30	satisfying	satisfy	VERB
ejde-92	43	31	the	the	DET
ejde-92	43	32	“	"	PUNCT
ejde-92	43	33	null	null	ADJ
ejde-92	43	34	average	average	ADJ
ejde-92	43	35	”	"	PUNCT
ejde-92	43	36	type	type	NOUN
ejde-92	43	37	condition	condition	NOUN
ejde-92	43	38	(	(	PUNCT
ejde-92	43	39	see	see	VERB
ejde-92	43	40	section	section	NOUN
ejde-92	43	41	2),∫	2),∫	NUM
ejde-92	43	42	ω	ω	NUM
ejde-92	43	43	|u|p−2u	|u|p−2u	PROPN
ejde-92	43	44	dx	dx	PROPN
ejde-92	43	45	=	=	PROPN
ejde-92	43	46	0	0	PROPN
ejde-92	43	47	.	.	PUNCT
ejde-92	44	1	(	(	PUNCT
ejde-92	44	2	1.5	1.5	NUM
ejde-92	44	3	)	)	PUNCT
ejde-92	44	4	in	in	ADP
ejde-92	44	5	fact	fact	NOUN
ejde-92	44	6	,	,	PUNCT
ejde-92	44	7	by	by	ADP
ejde-92	44	8	setting	set	VERB
ejde-92	44	9	v	v	NOUN
ejde-92	44	10	=	=	SYM
ejde-92	44	11	1	1	NUM
ejde-92	44	12	as	as	ADP
ejde-92	44	13	a	a	DET
ejde-92	44	14	test	test	NOUN
ejde-92	44	15	function	function	NOUN
ejde-92	44	16	in	in	ADP
ejde-92	44	17	(	(	PUNCT
ejde-92	44	18	1.2	1.2	NUM
ejde-92	44	19	)	)	PUNCT
ejde-92	44	20	it	it	PRON
ejde-92	44	21	follows	follow	VERB
ejde-92	44	22	that	that	SCONJ
ejde-92	44	23	the	the	DET
ejde-92	44	24	eigenfunctions	eigenfunction	NOUN
ejde-92	44	25	u	u	PRON
ejde-92	44	26	∈w	∈w	NOUN
ejde-92	44	27	1,p(ω	1,p(ω	NUM
ejde-92	44	28	)	)	PUNCT
ejde-92	44	29	associated	associate	VERB
ejde-92	44	30	with	with	ADP
ejde-92	44	31	all	all	DET
ejde-92	44	32	possible	possible	ADJ
ejde-92	44	33	eigenvalues	eigenvalue	VERB
ejde-92	44	34	λ	λ	PROPN
ejde-92	44	35	6=	6=	NUM
ejde-92	44	36	0	0	NUM
ejde-92	44	37	must	must	AUX
ejde-92	44	38	satisfy	satisfy	VERB
ejde-92	44	39	(	(	PUNCT
ejde-92	44	40	1.5	1.5	NUM
ejde-92	44	41	)	)	PUNCT
ejde-92	44	42	.	.	PUNCT
ejde-92	45	1	on	on	ADP
ejde-92	45	2	the	the	DET
ejde-92	45	3	other	other	ADJ
ejde-92	45	4	hand	hand	NOUN
ejde-92	45	5	,	,	PUNCT
ejde-92	45	6	it	it	PRON
ejde-92	45	7	is	be	AUX
ejde-92	45	8	more	more	ADV
ejde-92	45	9	or	or	CCONJ
ejde-92	45	10	less	less	ADV
ejde-92	45	11	straightforward	straightforward	ADJ
ejde-92	45	12	for	for	SCONJ
ejde-92	45	13	the	the	DET
ejde-92	45	14	specialist	specialist	NOUN
ejde-92	45	15	to	to	PART
ejde-92	45	16	check	check	VERB
ejde-92	45	17	(	(	PUNCT
ejde-92	45	18	1.4	1.4	NUM
ejde-92	45	19	)	)	PUNCT
ejde-92	45	20	in	in	ADP
ejde-92	45	21	the	the	DET
ejde-92	45	22	case	case	NOUN
ejde-92	45	23	p	p	X
ejde-92	45	24	≥	≥	NUM
ejde-92	45	25	2	2	NUM
ejde-92	45	26	.	.	PUNCT
ejde-92	46	1	on	on	ADP
ejde-92	46	2	the	the	DET
ejde-92	46	3	contrary	contrary	NOUN
ejde-92	46	4	,	,	PUNCT
ejde-92	46	5	the	the	DET
ejde-92	46	6	proof	proof	NOUN
ejde-92	46	7	for	for	ADP
ejde-92	46	8	the	the	DET
ejde-92	46	9	complementary	complementary	ADJ
ejde-92	46	10	range	range	NOUN
ejde-92	46	11	1	1	NUM
ejde-92	46	12	<	<	X
ejde-92	46	13	p	p	X
ejde-92	46	14	<	<	X
ejde-92	46	15	2	2	NUM
ejde-92	46	16	is	be	AUX
ejde-92	46	17	far	far	ADV
ejde-92	46	18	from	from	ADP
ejde-92	46	19	obvious	obvious	ADJ
ejde-92	46	20	and	and	CCONJ
ejde-92	46	21	this	this	DET
ejde-92	46	22	brief	brief	ADJ
ejde-92	46	23	note	note	NOUN
ejde-92	46	24	is	be	AUX
ejde-92	46	25	just	just	ADV
ejde-92	46	26	devoted	devoted	ADJ
ejde-92	46	27	to	to	ADP
ejde-92	46	28	this	this	DET
ejde-92	46	29	goal	goal	NOUN
ejde-92	46	30	.	.	PUNCT
ejde-92	47	1	since	since	SCONJ
ejde-92	47	2	i	i	PRON
ejde-92	47	3	have	have	AUX
ejde-92	47	4	not	not	PART
ejde-92	47	5	been	be	AUX
ejde-92	47	6	able	able	ADJ
ejde-92	47	7	to	to	PART
ejde-92	47	8	find	find	VERB
ejde-92	47	9	a	a	DET
ejde-92	47	10	proof	proof	NOUN
ejde-92	47	11	,	,	PUNCT
ejde-92	47	12	i	i	PRON
ejde-92	47	13	decided	decide	VERB
ejde-92	47	14	to	to	PART
ejde-92	47	15	publish	publish	VERB
ejde-92	47	16	one	one	NUM
ejde-92	47	17	of	of	ADP
ejde-92	47	18	my	my	PRON
ejde-92	47	19	own	own	ADJ
ejde-92	47	20	.	.	PUNCT
ejde-92	48	1	it	it	PRON
ejde-92	48	2	should	should	AUX
ejde-92	48	3	be	be	AUX
ejde-92	48	4	mentioned	mention	VERB
ejde-92	48	5	that	that	SCONJ
ejde-92	48	6	the	the	DET
ejde-92	48	7	omission	omission	NOUN
ejde-92	48	8	of	of	ADP
ejde-92	48	9	the	the	DET
ejde-92	48	10	case	case	NOUN
ejde-92	48	11	1	1	NUM
ejde-92	48	12	<	<	X
ejde-92	48	13	p	p	X
ejde-92	48	14	<	<	X
ejde-92	48	15	2	2	NUM
ejde-92	48	16	is	be	AUX
ejde-92	48	17	striking	strike	VERB
ejde-92	48	18	in	in	ADP
ejde-92	48	19	some	some	DET
ejde-92	48	20	references	reference	NOUN
ejde-92	48	21	(	(	PUNCT
ejde-92	48	22	see	see	VERB
ejde-92	48	23	[	[	X
ejde-92	48	24	14	14	NUM
ejde-92	48	25	,	,	PUNCT
ejde-92	48	26	chapter	chapter	NOUN
ejde-92	48	27	iv	iv	NOUN
ejde-92	48	28	,	,	PUNCT
ejde-92	48	29	§	§	NOUN
ejde-92	48	30	2	2	NUM
ejde-92	48	31	]	]	PUNCT
ejde-92	48	32	)	)	PUNCT
ejde-92	48	33	.	.	PUNCT
ejde-92	49	1	it	it	PRON
ejde-92	49	2	should	should	AUX
ejde-92	49	3	be	be	AUX
ejde-92	49	4	also	also	ADV
ejde-92	49	5	remarked	remark	VERB
ejde-92	49	6	that	that	DET
ejde-92	49	7	c∗	c∗	NOUN
ejde-92	49	8	=	=	SYM
ejde-92	49	9	λ	λ	X
ejde-92	49	10	−1	−1	NOUN
ejde-92	49	11	/	/	SYM
ejde-92	49	12	p	p	NOUN
ejde-92	49	13	2	2	NUM
ejde-92	49	14	can	can	AUX
ejde-92	49	15	be	be	AUX
ejde-92	49	16	regarded	regard	VERB
ejde-92	49	17	as	as	ADP
ejde-92	49	18	the	the	DET
ejde-92	49	19	optimum	optimum	ADJ
ejde-92	49	20	constant	constant	ADJ
ejde-92	49	21	c	c	NOUN
ejde-92	49	22	in	in	ADP
ejde-92	49	23	poincaré	poincaré	PROPN
ejde-92	49	24	’s	’s	PART
ejde-92	49	25	inequality	inequality	PROPN
ejde-92	49	26	inf	inf	PROPN
ejde-92	49	27	t∈r	t∈r	NOUN
ejde-92	49	28	‖u−	‖u−	PROPN
ejde-92	49	29	t‖lp(ω	t‖lp(ω	PROPN
ejde-92	49	30	)	)	PUNCT
ejde-92	49	31	≤	≤	NOUN
ejde-92	49	32	c‖|∇u|‖lp(ω	c‖|∇u|‖lp(ω	PROPN
ejde-92	49	33	)	)	PUNCT
ejde-92	49	34	,	,	PUNCT
ejde-92	49	35	(	(	PUNCT
ejde-92	49	36	1.6	1.6	NUM
ejde-92	49	37	)	)	PUNCT
ejde-92	49	38	where	where	SCONJ
ejde-92	49	39	u	u	PROPN
ejde-92	49	40	∈	∈	PROPN
ejde-92	49	41	w	w	PROPN
ejde-92	49	42	1,p(ω	1,p(ω	NUM
ejde-92	49	43	)	)	PUNCT
ejde-92	49	44	.	.	PUNCT
ejde-92	50	1	in	in	ADP
ejde-92	50	2	spite	spite	NOUN
ejde-92	50	3	of	of	ADP
ejde-92	50	4	reference	reference	NOUN
ejde-92	50	5	[	[	X
ejde-92	50	6	18	18	NUM
ejde-92	50	7	]	]	PUNCT
ejde-92	50	8	containing	contain	VERB
ejde-92	50	9	general	general	ADJ
ejde-92	50	10	versions	version	NOUN
ejde-92	50	11	of	of	ADP
ejde-92	50	12	this	this	DET
ejde-92	50	13	inequality	inequality	NOUN
ejde-92	50	14	,	,	PUNCT
ejde-92	50	15	a	a	DET
ejde-92	50	16	connection	connection	NOUN
ejde-92	50	17	between	between	ADP
ejde-92	50	18	(	(	PUNCT
ejde-92	50	19	1.6	1.6	NUM
ejde-92	50	20	)	)	PUNCT
ejde-92	50	21	and	and	CCONJ
ejde-92	50	22	λ2	λ2	NOUN
ejde-92	50	23	is	be	AUX
ejde-92	50	24	not	not	PART
ejde-92	50	25	reported	report	VERB
ejde-92	50	26	there	there	ADV
ejde-92	50	27	.	.	PUNCT
ejde-92	51	1	our	our	PRON
ejde-92	51	2	main	main	ADJ
ejde-92	51	3	result	result	NOUN
ejde-92	51	4	reads	read	VERB
ejde-92	51	5	as	as	SCONJ
ejde-92	51	6	follows	follow	VERB
ejde-92	51	7	.	.	PUNCT
ejde-92	52	1	theorem	theorem	VERB
ejde-92	52	2	1.1	1.1	NUM
ejde-92	52	3	.	.	PUNCT
ejde-92	53	1	let	let	VERB
ejde-92	53	2	ω	ω	PROPN
ejde-92	53	3	⊂	⊂	PROPN
ejde-92	53	4	rn	rn	AUX
ejde-92	53	5	be	be	AUX
ejde-92	53	6	a	a	DET
ejde-92	53	7	bounded	bounded	ADJ
ejde-92	53	8	c0,1	c0,1	NOUN
ejde-92	53	9	domain	domain	NOUN
ejde-92	53	10	and	and	CCONJ
ejde-92	53	11	set	set	VERB
ejde-92	53	12	λ̂	λ̂	X
ejde-92	54	1	=	=	SYM
ejde-92	54	2	inf	inf	PROPN
ejde-92	54	3	u∈m0\{0	u∈m0\{0	PROPN
ejde-92	54	4	}	}	PUNCT
ejde-92	54	5	∫	∫	PROPN
ejde-92	54	6	ω	ω	PROPN
ejde-92	54	7	|∇u|p	|∇u|p	PROPN
ejde-92	54	8	dx∫	dx∫	PROPN
ejde-92	54	9	ω	ω	PROPN
ejde-92	54	10	|u|p	|u|p	PROPN
ejde-92	54	11	dx	dx	PROPN
ejde-92	54	12	,	,	PUNCT
ejde-92	54	13	(	(	PUNCT
ejde-92	54	14	1.7	1.7	NUM
ejde-92	54	15	)	)	PUNCT
ejde-92	54	16	ejde-2022/13	ejde-2022/13	ADJ
ejde-92	54	17	second	second	ADJ
ejde-92	54	18	neumann	neumann	PROPN
ejde-92	54	19	eigenvalue	eigenvalue	X
ejde-92	54	20	3	3	NUM
ejde-92	54	21	where	where	SCONJ
ejde-92	54	22	m0	m0	NOUN
ejde-92	54	23	=	=	PUNCT
ejde-92	54	24	{	{	PUNCT
ejde-92	54	25	u	u	NOUN
ejde-92	54	26	∈w	∈w	NOUN
ejde-92	54	27	1,p(ω	1,p(ω	NUM
ejde-92	54	28	)	)	PUNCT
ejde-92	54	29	:	:	PUNCT
ejde-92	55	1	∫	∫	PROPN
ejde-92	55	2	ω	ω	PROPN
ejde-92	55	3	|u|p−2u	|u|p−2u	PROPN
ejde-92	55	4	dx	dx	PROPN
ejde-92	55	5	=	=	PROPN
ejde-92	55	6	0	0	NUM
ejde-92	55	7	}	}	PUNCT
ejde-92	55	8	.	.	PUNCT
ejde-92	56	1	then	then	ADV
ejde-92	56	2	the	the	DET
ejde-92	56	3	following	follow	VERB
ejde-92	56	4	properties	property	NOUN
ejde-92	56	5	hold	hold	VERB
ejde-92	56	6	(	(	PUNCT
ejde-92	56	7	i	i	NOUN
ejde-92	56	8	)	)	PUNCT
ejde-92	56	9	the	the	DET
ejde-92	56	10	infimum	infimum	ADJ
ejde-92	56	11	in	in	ADP
ejde-92	56	12	(	(	PUNCT
ejde-92	56	13	1.7	1.7	NUM
ejde-92	56	14	)	)	PUNCT
ejde-92	56	15	is	be	AUX
ejde-92	56	16	achieved	achieve	VERB
ejde-92	56	17	at	at	ADP
ejde-92	56	18	some	some	DET
ejde-92	56	19	u	u	NOUN
ejde-92	56	20	∈m0	∈m0	ADJ
ejde-92	56	21	and	and	CCONJ
ejde-92	56	22	thus	thus	ADV
ejde-92	56	23	λ̂	λ̂	X
ejde-92	56	24	>	>	X
ejde-92	56	25	0	0	X
ejde-92	56	26	.	.	PUNCT
ejde-92	57	1	(	(	PUNCT
ejde-92	57	2	ii	ii	NOUN
ejde-92	57	3	)	)	PUNCT
ejde-92	57	4	every	every	DET
ejde-92	57	5	eigenvalue	eigenvalue	PROPN
ejde-92	57	6	λ	λ	PROPN
ejde-92	57	7	6=	6=	ADP
ejde-92	57	8	0	0	NUM
ejde-92	57	9	to	to	ADP
ejde-92	57	10	the	the	DET
ejde-92	57	11	neumann	neumann	PROPN
ejde-92	57	12	problem	problem	NOUN
ejde-92	57	13	(	(	PUNCT
ejde-92	57	14	1.1	1.1	NUM
ejde-92	57	15	)	)	PUNCT
ejde-92	57	16	satisfies	satisfie	NOUN
ejde-92	57	17	λ	λ	PROPN
ejde-92	57	18	≥	≥	NOUN
ejde-92	57	19	λ̂.	λ̂.	VERB
ejde-92	57	20	in	in	ADP
ejde-92	57	21	particular	particular	ADJ
ejde-92	57	22	,	,	PUNCT
ejde-92	57	23	λ	λ	X
ejde-92	57	24	=	=	SYM
ejde-92	57	25	0	0	NUM
ejde-92	57	26	is	be	AUX
ejde-92	57	27	an	an	DET
ejde-92	57	28	isolated	isolated	ADJ
ejde-92	57	29	eigenvalue	eigenvalue	NOUN
ejde-92	57	30	.	.	PUNCT
ejde-92	58	1	(	(	PUNCT
ejde-92	58	2	iii	iii	X
ejde-92	58	3	)	)	PUNCT
ejde-92	58	4	λ̂	λ̂	X
ejde-92	58	5	is	be	AUX
ejde-92	58	6	an	an	DET
ejde-92	58	7	eigenvalue	eigenvalue	NOUN
ejde-92	58	8	and	and	CCONJ
ejde-92	58	9	therefore	therefore	ADV
ejde-92	58	10	λ2	λ2	PROPN
ejde-92	58	11	=	=	SYM
ejde-92	58	12	λ̂.	λ̂.	PROPN
ejde-92	58	13	(	(	PUNCT
ejde-92	58	14	1.8	1.8	NUM
ejde-92	58	15	)	)	PUNCT
ejde-92	58	16	note	note	NOUN
ejde-92	58	17	that	that	SCONJ
ejde-92	58	18	(	(	PUNCT
ejde-92	58	19	i	i	NOUN
ejde-92	58	20	)	)	PUNCT
ejde-92	58	21	and	and	CCONJ
ejde-92	58	22	(	(	PUNCT
ejde-92	58	23	ii	ii	NOUN
ejde-92	58	24	)	)	PUNCT
ejde-92	58	25	are	be	AUX
ejde-92	58	26	well	well	ADV
ejde-92	58	27	-	-	PUNCT
ejde-92	58	28	known	know	VERB
ejde-92	58	29	and	and	CCONJ
ejde-92	58	30	that	that	SCONJ
ejde-92	58	31	(	(	PUNCT
ejde-92	58	32	iii	iii	NOUN
ejde-92	58	33	)	)	PUNCT
ejde-92	58	34	can	can	AUX
ejde-92	58	35	be	be	AUX
ejde-92	58	36	proved	prove	VERB
ejde-92	58	37	in	in	ADP
ejde-92	58	38	a	a	DET
ejde-92	58	39	standard	standard	ADJ
ejde-92	58	40	way	way	NOUN
ejde-92	58	41	when	when	SCONJ
ejde-92	58	42	p	p	NOUN
ejde-92	58	43	≥	≥	NUM
ejde-92	58	44	2	2	NUM
ejde-92	58	45	(	(	PUNCT
ejde-92	58	46	see	see	VERB
ejde-92	58	47	remark	remark	NOUN
ejde-92	58	48	3.2	3.2	NUM
ejde-92	58	49	)	)	PUNCT
ejde-92	58	50	.	.	PUNCT
ejde-92	59	1	therefore	therefore	ADV
ejde-92	59	2	we	we	PRON
ejde-92	59	3	focuss	focuss	ADJ
ejde-92	59	4	on	on	ADP
ejde-92	59	5	showing	show	VERB
ejde-92	59	6	(	(	PUNCT
ejde-92	59	7	1.8	1.8	NUM
ejde-92	59	8	)	)	PUNCT
ejde-92	59	9	when	when	SCONJ
ejde-92	59	10	p	p	NOUN
ejde-92	59	11	falls	fall	VERB
ejde-92	59	12	in	in	ADP
ejde-92	59	13	the	the	DET
ejde-92	59	14	‘	'	PUNCT
ejde-92	59	15	singular	singular	ADJ
ejde-92	59	16	’	'	PUNCT
ejde-92	59	17	range	range	NOUN
ejde-92	59	18	1	1	NUM
ejde-92	59	19	<	<	X
ejde-92	59	20	p	p	X
ejde-92	59	21	<	<	X
ejde-92	59	22	2	2	NUM
ejde-92	59	23	.	.	PUNCT
ejde-92	59	24	moving	move	VERB
ejde-92	59	25	now	now	ADV
ejde-92	59	26	to	to	ADP
ejde-92	59	27	a	a	DET
ejde-92	59	28	further	further	ADJ
ejde-92	59	29	scenario	scenario	NOUN
ejde-92	59	30	,	,	PUNCT
ejde-92	59	31	the	the	DET
ejde-92	59	32	version	version	NOUN
ejde-92	59	33	of	of	ADP
ejde-92	59	34	the	the	DET
ejde-92	59	35	p	p	ADJ
ejde-92	59	36	-	-	PUNCT
ejde-92	59	37	laplacian	laplacian	ADJ
ejde-92	59	38	operator	operator	NOUN
ejde-92	59	39	−∆p	−∆p	NOUN
ejde-92	59	40	for	for	ADP
ejde-92	59	41	graphs	graph	NOUN
ejde-92	59	42	g	g	PROPN
ejde-92	59	43	has	have	AUX
ejde-92	59	44	been	be	AUX
ejde-92	59	45	recently	recently	ADV
ejde-92	59	46	studied	study	VERB
ejde-92	59	47	in	in	ADP
ejde-92	59	48	[	[	X
ejde-92	59	49	2	2	NUM
ejde-92	59	50	]	]	PUNCT
ejde-92	59	51	,	,	PUNCT
ejde-92	59	52	where	where	SCONJ
ejde-92	59	53	its	its	PRON
ejde-92	59	54	second	second	ADJ
ejde-92	59	55	eigenvalue	eigenvalue	NOUN
ejde-92	59	56	λ2	λ2	PROPN
ejde-92	59	57	has	have	AUX
ejde-92	59	58	been	be	AUX
ejde-92	59	59	introduced	introduce	VERB
ejde-92	59	60	and	and	CCONJ
ejde-92	59	61	analyzed	analyze	VERB
ejde-92	59	62	in	in	ADP
ejde-92	59	63	detail	detail	NOUN
ejde-92	59	64	(	(	PUNCT
ejde-92	59	65	see	see	VERB
ejde-92	59	66	also	also	ADV
ejde-92	59	67	[	[	X
ejde-92	59	68	8	8	NUM
ejde-92	59	69	,	,	PUNCT
ejde-92	59	70	9	9	NUM
ejde-92	59	71	,	,	PUNCT
ejde-92	59	72	10	10	NUM
ejde-92	59	73	]	]	PUNCT
ejde-92	59	74	.	.	PUNCT
ejde-92	60	1	the	the	DET
ejde-92	60	2	latter	latter	ADJ
ejde-92	60	3	specifically	specifically	ADV
ejde-92	60	4	concerns	concern	VERB
ejde-92	60	5	a	a	DET
ejde-92	60	6	variety	variety	NOUN
ejde-92	60	7	of	of	ADP
ejde-92	60	8	features	feature	NOUN
ejde-92	60	9	on	on	ADP
ejde-92	60	10	the	the	DET
ejde-92	60	11	spectrum	spectrum	NOUN
ejde-92	60	12	of	of	ADP
ejde-92	60	13	−∆	−∆	NOUN
ejde-92	60	14	)	)	PUNCT
ejde-92	60	15	.	.	PUNCT
ejde-92	61	1	such	such	ADJ
ejde-92	61	2	eigenvalue	eigenvalue	NOUN
ejde-92	61	3	plays	play	VERB
ejde-92	61	4	an	an	DET
ejde-92	61	5	important	important	ADJ
ejde-92	61	6	rôle	rôle	NOUN
ejde-92	61	7	in	in	ADP
ejde-92	61	8	the	the	DET
ejde-92	61	9	so	so	ADV
ejde-92	61	10	-	-	PUNCT
ejde-92	61	11	called	call	VERB
ejde-92	61	12	‘	'	PUNCT
ejde-92	61	13	clustering	clustering	ADJ
ejde-92	61	14	problem	problem	NOUN
ejde-92	61	15	’	'	PUNCT
ejde-92	61	16	for	for	ADP
ejde-92	61	17	undirected	undirected	ADJ
ejde-92	61	18	graphs	graph	NOUN
ejde-92	61	19	[	[	X
ejde-92	61	20	8	8	NUM
ejde-92	61	21	]	]	PUNCT
ejde-92	61	22	.	.	PUNCT
ejde-92	62	1	our	our	PRON
ejde-92	62	2	next	next	ADJ
ejde-92	62	3	result	result	NOUN
ejde-92	62	4	extends	extend	VERB
ejde-92	62	5	[	[	X
ejde-92	62	6	2	2	NUM
ejde-92	62	7	,	,	PUNCT
ejde-92	62	8	theorem	theorem	VERB
ejde-92	62	9	1	1	NUM
ejde-92	62	10	]	]	PUNCT
ejde-92	62	11	to	to	ADP
ejde-92	62	12	the	the	DET
ejde-92	62	13	range	range	NOUN
ejde-92	62	14	p	p	X
ejde-92	62	15	>	>	X
ejde-92	62	16	1	1	NUM
ejde-92	62	17	.	.	PUNCT
ejde-92	63	1	in	in	ADP
ejde-92	63	2	fact	fact	NOUN
ejde-92	63	3	,	,	PUNCT
ejde-92	63	4	the	the	DET
ejde-92	63	5	proof	proof	NOUN
ejde-92	63	6	contained	contain	VERB
ejde-92	63	7	there	there	PRON
ejde-92	63	8	is	be	VERB
ejde-92	63	9	only	only	ADV
ejde-92	63	10	valid	valid	ADJ
ejde-92	63	11	in	in	ADP
ejde-92	63	12	the	the	DET
ejde-92	63	13	case	case	NOUN
ejde-92	63	14	p	p	X
ejde-92	63	15	≥	≥	NUM
ejde-92	63	16	2	2	NUM
ejde-92	63	17	,	,	PUNCT
ejde-92	63	18	and	and	CCONJ
ejde-92	63	19	the	the	DET
ejde-92	63	20	same	same	ADJ
ejde-92	63	21	remark	remark	NOUN
ejde-92	63	22	applies	apply	VERB
ejde-92	63	23	to	to	ADP
ejde-92	63	24	[	[	X
ejde-92	63	25	8	8	NUM
ejde-92	63	26	,	,	PUNCT
ejde-92	63	27	theorem	theorem	VERB
ejde-92	63	28	3.2	3.2	NUM
ejde-92	63	29	]	]	PUNCT
ejde-92	63	30	(	(	PUNCT
ejde-92	63	31	see	see	VERB
ejde-92	63	32	also	also	ADV
ejde-92	63	33	[	[	X
ejde-92	63	34	9	9	NUM
ejde-92	63	35	]	]	NUM
ejde-92	63	36	)	)	PUNCT
ejde-92	63	37	.	.	PUNCT
ejde-92	64	1	the	the	DET
ejde-92	64	2	reader	reader	NOUN
ejde-92	64	3	is	be	AUX
ejde-92	64	4	referred	refer	VERB
ejde-92	64	5	to	to	ADP
ejde-92	64	6	section	section	NOUN
ejde-92	64	7	4	4	NUM
ejde-92	64	8	for	for	ADP
ejde-92	64	9	the	the	DET
ejde-92	64	10	necessary	necessary	ADJ
ejde-92	64	11	background	background	NOUN
ejde-92	64	12	material	material	NOUN
ejde-92	64	13	concerning	concern	VERB
ejde-92	64	14	graph	graph	NOUN
ejde-92	64	15	theory	theory	NOUN
ejde-92	64	16	.	.	PUNCT
ejde-92	65	1	theorem	theorem	VERB
ejde-92	65	2	1.2	1.2	NUM
ejde-92	65	3	.	.	PUNCT
ejde-92	66	1	let	let	VERB
ejde-92	66	2	g	g	PROPN
ejde-92	66	3	=	=	SYM
ejde-92	66	4	(	(	PUNCT
ejde-92	66	5	v	v	NOUN
ejde-92	66	6	,	,	PUNCT
ejde-92	66	7	e	e	NOUN
ejde-92	66	8	)	)	PUNCT
ejde-92	66	9	,	,	PUNCT
ejde-92	66	10	v	v	X
ejde-92	66	11	=	=	SYM
ejde-92	66	12	{	{	PUNCT
ejde-92	66	13	v1	v1	NOUN
ejde-92	66	14	,	,	PUNCT
ejde-92	66	15	.	.	PUNCT
ejde-92	66	16	.	.	PUNCT
ejde-92	67	1	.	.	PUNCT
ejde-92	68	1	,	,	PUNCT
ejde-92	68	2	vn	vn	PROPN
ejde-92	68	3	}	}	PUNCT
ejde-92	68	4	,	,	PUNCT
ejde-92	68	5	be	be	AUX
ejde-92	68	6	a	a	DET
ejde-92	68	7	connected	connected	ADJ
ejde-92	68	8	graph	graph	NOUN
ejde-92	68	9	with	with	ADP
ejde-92	68	10	weights	weight	NOUN
ejde-92	68	11	matrix	matrix	NOUN
ejde-92	68	12	a	a	DET
ejde-92	68	13	=	=	X
ejde-92	68	14	(	(	PUNCT
ejde-92	68	15	ωij)1≤i≤n	ωij)1≤i≤n	PROPN
ejde-92	68	16	,	,	PUNCT
ejde-92	68	17	ωij	ωij	PROPN
ejde-92	68	18	=	=	NOUN
ejde-92	68	19	ωji	ωji	NOUN
ejde-92	68	20	,	,	PUNCT
ejde-92	68	21	ωij	ωij	X
ejde-92	68	22	≥	≥	NOUN
ejde-92	68	23	0	0	NUM
ejde-92	68	24	,	,	PUNCT
ejde-92	68	25	ωii	ωii	NOUN
ejde-92	68	26	=	=	SYM
ejde-92	68	27	0	0	NUM
ejde-92	68	28	,	,	PUNCT
ejde-92	68	29	a	a	DET
ejde-92	68	30	6=	6=	NUM
ejde-92	68	31	0	0	X
ejde-92	68	32	.	.	PUNCT
ejde-92	69	1	consider	consider	VERB
ejde-92	69	2	the	the	DET
ejde-92	69	3	eigenvalue	eigenvalue	PROPN
ejde-92	69	4	problem	problem	NOUN
ejde-92	69	5	−∆p(f	−∆p(f	PROPN
ejde-92	69	6	)	)	PUNCT
ejde-92	70	1	=	=	PUNCT
ejde-92	70	2	λνφp(f	λνφp(f	PROPN
ejde-92	70	3	)	)	PUNCT
ejde-92	70	4	,	,	PUNCT
ejde-92	70	5	(	(	PUNCT
ejde-92	70	6	1.9	1.9	NUM
ejde-92	70	7	)	)	PUNCT
ejde-92	70	8	where	where	SCONJ
ejde-92	70	9	f	f	NOUN
ejde-92	70	10	=	=	SYM
ejde-92	70	11	(	(	PUNCT
ejde-92	70	12	fi	fi	NOUN
ejde-92	70	13	)	)	PUNCT
ejde-92	70	14	∈	∈	PROPN
ejde-92	70	15	rn	rn	PROPN
ejde-92	70	16	,	,	PUNCT
ejde-92	70	17	ν	ν	X
ejde-92	70	18	=	=	SYM
ejde-92	70	19	(	(	PUNCT
ejde-92	70	20	νi	νi	NOUN
ejde-92	70	21	)	)	PUNCT
ejde-92	70	22	∈	∈	NOUN
ejde-92	70	23	rn+	rn+	NOUN
ejde-92	70	24	,	,	PUNCT
ejde-92	70	25	φp(f	φp(f	X
ejde-92	70	26	)	)	PUNCT
ejde-92	70	27	=	=	PRON
ejde-92	70	28	(	(	PUNCT
ejde-92	70	29	|pi|p−2pi	|pi|p−2pi	NOUN
ejde-92	70	30	)	)	PUNCT
ejde-92	70	31	and	and	CCONJ
ejde-92	70	32	−∆p	−∆p	NOUN
ejde-92	70	33	is	be	AUX
ejde-92	70	34	the	the	DET
ejde-92	70	35	plaplacian	plaplacian	PROPN
ejde-92	70	36	in	in	ADP
ejde-92	70	37	g.	g.	PROPN
ejde-92	70	38	then	then	ADV
ejde-92	70	39	the	the	DET
ejde-92	70	40	following	follow	VERB
ejde-92	70	41	properties	property	NOUN
ejde-92	70	42	hold	hold	VERB
ejde-92	70	43	(	(	PUNCT
ejde-92	70	44	i	i	NOUN
ejde-92	70	45	)	)	PUNCT
ejde-92	70	46	λ1	λ1	PROPN
ejde-92	71	1	=	=	SYM
ejde-92	71	2	0	0	NUM
ejde-92	71	3	is	be	AUX
ejde-92	71	4	the	the	DET
ejde-92	71	5	first	first	ADJ
ejde-92	71	6	eigenvalue	eigenvalue	NOUN
ejde-92	71	7	which	which	PRON
ejde-92	71	8	is	be	AUX
ejde-92	71	9	isolated	isolate	VERB
ejde-92	71	10	,	,	PUNCT
ejde-92	71	11	simple	simple	ADJ
ejde-92	71	12	and	and	CCONJ
ejde-92	71	13	has	have	VERB
ejde-92	71	14	span{1	span{1	NOUN
ejde-92	71	15	}	}	PUNCT
ejde-92	71	16	as	as	ADP
ejde-92	71	17	the	the	DET
ejde-92	71	18	set	set	NOUN
ejde-92	71	19	of	of	ADP
ejde-92	71	20	associated	associated	ADJ
ejde-92	71	21	eigenfunctions	eigenfunction	NOUN
ejde-92	71	22	.	.	PUNCT
ejde-92	72	1	(	(	PUNCT
ejde-92	72	2	ii	ii	X
ejde-92	72	3	)	)	PUNCT
ejde-92	72	4	the	the	DET
ejde-92	72	5	second	second	ADJ
ejde-92	72	6	eigenvalue	eigenvalue	PROPN
ejde-92	72	7	λ2	λ2	PROPN
ejde-92	72	8	is	be	AUX
ejde-92	72	9	expressed	express	VERB
ejde-92	72	10	as	as	ADP
ejde-92	72	11	λ2	λ2	NOUN
ejde-92	72	12	=	=	SYM
ejde-92	72	13	inf	inf	PROPN
ejde-92	72	14	f∈m0\{0	f∈m0\{0	PROPN
ejde-92	72	15	}	}	PUNCT
ejde-92	72	16	1	1	NUM
ejde-92	72	17	2	2	NUM
ejde-92	72	18	∑	∑	PROPN
ejde-92	72	19	i	i	PROPN
ejde-92	72	20	,	,	PUNCT
ejde-92	72	21	j	j	PROPN
ejde-92	72	22	ωij	ωij	VERB
ejde-92	72	23	|fi	|fi	ADP
ejde-92	72	24	−	−	PROPN
ejde-92	72	25	fj	fj	NUM
ejde-92	72	26	|p∑	|p∑	NOUN
ejde-92	72	27	i	i	PRON
ejde-92	72	28	νi|fi|p	νi|fi|p	VERB
ejde-92	72	29	,	,	PUNCT
ejde-92	72	30	(	(	PUNCT
ejde-92	72	31	1.10	1.10	NUM
ejde-92	72	32	)	)	PUNCT
ejde-92	72	33	where	where	SCONJ
ejde-92	72	34	m0	m0	NOUN
ejde-92	72	35	=	=	PUNCT
ejde-92	72	36	{	{	PUNCT
ejde-92	72	37	f	f	NOUN
ejde-92	72	38	:	:	PUNCT
ejde-92	72	39	∑	∑	PUNCT
ejde-92	72	40	i	i	PRON
ejde-92	72	41	νi|fi|p−2fi	νi|fi|p−2fi	VERB
ejde-92	72	42	=	=	NOUN
ejde-92	72	43	0	0	NUM
ejde-92	72	44	}	}	PUNCT
ejde-92	72	45	.	.	PUNCT
ejde-92	73	1	(	(	PUNCT
ejde-92	73	2	iii	iii	X
ejde-92	73	3	)	)	PUNCT
ejde-92	73	4	the	the	DET
ejde-92	73	5	maximum	maximum	PROPN
ejde-92	73	6	eigenvalue	eigenvalue	PROPN
ejde-92	73	7	λ∗	λ∗	PROPN
ejde-92	73	8	is	be	AUX
ejde-92	73	9	provided	provide	VERB
ejde-92	73	10	by	by	ADP
ejde-92	73	11	the	the	DET
ejde-92	73	12	expression	expression	NOUN
ejde-92	73	13	λ∗	λ∗	NOUN
ejde-92	73	14	=	=	SYM
ejde-92	73	15	sup	sup	NOUN
ejde-92	73	16	f∈m0\{0	f∈m0\{0	PROPN
ejde-92	73	17	}	}	PUNCT
ejde-92	73	18	1	1	NUM
ejde-92	73	19	2	2	NUM
ejde-92	73	20	∑	∑	PROPN
ejde-92	73	21	i	i	PROPN
ejde-92	73	22	,	,	PUNCT
ejde-92	73	23	j	j	PROPN
ejde-92	73	24	ωij	ωij	VERB
ejde-92	73	25	|fi	|fi	ADP
ejde-92	73	26	−	−	PROPN
ejde-92	73	27	fj	fj	NUM
ejde-92	73	28	|p∑	|p∑	NOUN
ejde-92	73	29	i	i	PRON
ejde-92	73	30	νi|fi|p	νi|fi|p	VERB
ejde-92	73	31	.	.	PUNCT
ejde-92	74	1	(	(	PUNCT
ejde-92	74	2	1.11	1.11	NUM
ejde-92	74	3	)	)	PUNCT
ejde-92	74	4	remark	remark	NOUN
ejde-92	74	5	1.3	1.3	NUM
ejde-92	74	6	.	.	PUNCT
ejde-92	75	1	in	in	ADP
ejde-92	75	2	this	this	DET
ejde-92	75	3	work	work	NOUN
ejde-92	75	4	we	we	PRON
ejde-92	75	5	denote	denote	VERB
ejde-92	75	6	by	by	ADP
ejde-92	75	7	−∆p	−∆p	NOUN
ejde-92	75	8	what	what	PRON
ejde-92	75	9	is	be	AUX
ejde-92	75	10	commonly	commonly	ADV
ejde-92	75	11	defined	define	VERB
ejde-92	75	12	as	as	ADP
ejde-92	75	13	the	the	DET
ejde-92	75	14	p	p	PROPN
ejde-92	75	15	-	-	PUNCT
ejde-92	75	16	laplacian	laplacian	ADJ
ejde-92	75	17	operator	operator	NOUN
ejde-92	75	18	in	in	ADP
ejde-92	75	19	graphs	graph	NOUN
ejde-92	75	20	,	,	PUNCT
ejde-92	75	21	usually	usually	ADV
ejde-92	75	22	designated	designate	VERB
ejde-92	75	23	as	as	ADP
ejde-92	75	24	∆p	∆p	PROPN
ejde-92	75	25	.	.	PUNCT
ejde-92	76	1	we	we	PRON
ejde-92	76	2	have	have	AUX
ejde-92	76	3	proceeded	proceed	VERB
ejde-92	76	4	in	in	ADP
ejde-92	76	5	this	this	DET
ejde-92	76	6	way	way	NOUN
ejde-92	76	7	to	to	PART
ejde-92	76	8	preserve	preserve	VERB
ejde-92	76	9	the	the	DET
ejde-92	76	10	analogies	analogy	NOUN
ejde-92	76	11	with	with	ADP
ejde-92	76	12	partial	partial	ADJ
ejde-92	76	13	differential	differential	ADJ
ejde-92	76	14	equations	equation	NOUN
ejde-92	76	15	(	(	PUNCT
ejde-92	76	16	see	see	VERB
ejde-92	76	17	section	section	NOUN
ejde-92	76	18	4	4	NUM
ejde-92	76	19	)	)	PUNCT
ejde-92	76	20	.	.	PUNCT
ejde-92	77	1	this	this	DET
ejde-92	77	2	work	work	NOUN
ejde-92	77	3	is	be	AUX
ejde-92	77	4	organized	organize	VERB
ejde-92	77	5	as	as	SCONJ
ejde-92	77	6	follows	follow	VERB
ejde-92	77	7	.	.	PUNCT
ejde-92	78	1	section	section	NOUN
ejde-92	78	2	2	2	NUM
ejde-92	78	3	analyzes	analyze	VERB
ejde-92	78	4	the	the	DET
ejde-92	78	5	differentiability	differentiability	NOUN
ejde-92	78	6	properties	property	NOUN
ejde-92	78	7	of	of	ADP
ejde-92	78	8	the	the	DET
ejde-92	78	9	variance	variance	NOUN
ejde-92	78	10	functional	functional	ADJ
ejde-92	78	11	in	in	ADP
ejde-92	78	12	lp(x,µ	lp(x,µ	NOUN
ejde-92	78	13	)	)	PUNCT
ejde-92	78	14	,	,	PUNCT
ejde-92	78	15	where	where	SCONJ
ejde-92	78	16	(	(	PUNCT
ejde-92	78	17	x,µ	x,µ	NOUN
ejde-92	78	18	)	)	PUNCT
ejde-92	78	19	is	be	AUX
ejde-92	78	20	a	a	DET
ejde-92	78	21	measurable	measurable	ADJ
ejde-92	78	22	space	space	NOUN
ejde-92	78	23	.	.	PUNCT
ejde-92	79	1	4	4	NUM
ejde-92	79	2	j.	j.	PROPN
ejde-92	79	3	c.	c.	PROPN
ejde-92	79	4	sabina	sabina	PROPN
ejde-92	79	5	de	de	PROPN
ejde-92	79	6	lis	lis	X
ejde-92	79	7	ejde-2022/13	ejde-2022/13	NOUN
ejde-92	79	8	the	the	DET
ejde-92	79	9	results	result	NOUN
ejde-92	79	10	attained	attain	VERB
ejde-92	79	11	seem	seem	VERB
ejde-92	79	12	to	to	PART
ejde-92	79	13	be	be	AUX
ejde-92	79	14	new	new	ADJ
ejde-92	79	15	and	and	CCONJ
ejde-92	79	16	are	be	AUX
ejde-92	79	17	instrumental	instrumental	ADJ
ejde-92	79	18	in	in	ADP
ejde-92	79	19	the	the	DET
ejde-92	79	20	subsequent	subsequent	ADJ
ejde-92	79	21	sections	section	NOUN
ejde-92	79	22	.	.	PUNCT
ejde-92	80	1	section	section	NOUN
ejde-92	80	2	3	3	NUM
ejde-92	80	3	is	be	AUX
ejde-92	80	4	devoted	devote	VERB
ejde-92	80	5	to	to	ADP
ejde-92	80	6	proving	prove	VERB
ejde-92	80	7	theorem	theorem	ADJ
ejde-92	80	8	1.1	1.1	NUM
ejde-92	80	9	.	.	PUNCT
ejde-92	81	1	there	there	ADV
ejde-92	81	2	,	,	PUNCT
ejde-92	81	3	we	we	PRON
ejde-92	81	4	are	be	AUX
ejde-92	81	5	dealing	deal	VERB
ejde-92	81	6	with	with	ADP
ejde-92	81	7	the	the	DET
ejde-92	81	8	slightly	slightly	ADV
ejde-92	81	9	more	more	ADV
ejde-92	81	10	general	general	ADJ
ejde-92	81	11	version	version	NOUN
ejde-92	81	12	of	of	ADP
ejde-92	81	13	(	(	PUNCT
ejde-92	81	14	1.1	1.1	NUM
ejde-92	81	15	)	)	PUNCT
ejde-92	81	16	,	,	PUNCT
ejde-92	81	17	−∆pu	−∆pu	NOUN
ejde-92	82	1	=	=	SYM
ejde-92	83	1	λm(x)|u|p−2u	λm(x)|u|p−2u	PUNCT
ejde-92	83	2	x	x	SYM
ejde-92	83	3	∈	∈	PROPN
ejde-92	83	4	ω	ω	NUM
ejde-92	83	5	|∇u|p−2	|∇u|p−2	PROPN
ejde-92	83	6	∂u	∂u	PROPN
ejde-92	84	1	∂ν	∂ν	X
ejde-92	84	2	=	=	PUNCT
ejde-92	84	3	0	0	PUNCT
ejde-92	84	4	x	x	SYM
ejde-92	84	5	∈	∈	PROPN
ejde-92	84	6	∂ω	∂ω	PROPN
ejde-92	84	7	,	,	PUNCT
ejde-92	84	8	(	(	PUNCT
ejde-92	84	9	1.12	1.12	NUM
ejde-92	84	10	)	)	PUNCT
ejde-92	84	11	where	where	SCONJ
ejde-92	84	12	m	m	VERB
ejde-92	84	13	∈	∈	NOUN
ejde-92	84	14	lr(ω	lr(ω	X
ejde-92	84	15	)	)	PUNCT
ejde-92	84	16	is	be	AUX
ejde-92	84	17	a	a	DET
ejde-92	84	18	weight	weight	NOUN
ejde-92	84	19	function	function	NOUN
ejde-92	84	20	such	such	ADJ
ejde-92	84	21	that	that	SCONJ
ejde-92	84	22	m(x	m(x	PROPN
ejde-92	84	23	)	)	PUNCT
ejde-92	84	24	>	>	X
ejde-92	84	25	0	0	NUM
ejde-92	85	1	a.e	a.e	PROPN
ejde-92	85	2	.	.	PROPN
ejde-92	86	1	in	in	ADP
ejde-92	86	2	ω	ω	NUM
ejde-92	86	3	,	,	PUNCT
ejde-92	86	4	while	while	SCONJ
ejde-92	86	5	exponent	exponent	ADJ
ejde-92	86	6	r	r	NOUN
ejde-92	86	7	≥	≥	NUM
ejde-92	86	8	1	1	NUM
ejde-92	86	9	is	be	AUX
ejde-92	86	10	suitably	suitably	ADV
ejde-92	86	11	chosen	choose	VERB
ejde-92	86	12	.	.	PUNCT
ejde-92	87	1	the	the	DET
ejde-92	87	2	tools	tool	NOUN
ejde-92	87	3	developed	develop	VERB
ejde-92	87	4	in	in	ADP
ejde-92	87	5	section	section	NOUN
ejde-92	87	6	2	2	NUM
ejde-92	87	7	turn	turn	NOUN
ejde-92	87	8	also	also	ADV
ejde-92	87	9	out	out	ADP
ejde-92	87	10	to	to	PART
ejde-92	87	11	be	be	AUX
ejde-92	87	12	useful	useful	ADJ
ejde-92	87	13	for	for	ADP
ejde-92	87	14	studying	study	VERB
ejde-92	87	15	the	the	DET
ejde-92	87	16	nonlinear	nonlinear	ADJ
ejde-92	87	17	diffusion	diffusion	NOUN
ejde-92	87	18	problem	problem	NOUN
ejde-92	87	19	−∆pu	−∆pu	X
ejde-92	87	20	=	=	SYM
ejde-92	87	21	λm(x)|u|q−2u	λm(x)|u|q−2u	PUNCT
ejde-92	87	22	x	x	SYM
ejde-92	87	23	∈	∈	PROPN
ejde-92	87	24	ω	ω	NUM
ejde-92	87	25	|∇u|p−2	|∇u|p−2	PROPN
ejde-92	87	26	∂u	∂u	PROPN
ejde-92	88	1	∂ν	∂ν	X
ejde-92	88	2	=	=	PUNCT
ejde-92	88	3	0	0	PUNCT
ejde-92	88	4	x	x	SYM
ejde-92	88	5	∈	∈	PROPN
ejde-92	88	6	∂ω	∂ω	PROPN
ejde-92	88	7	,	,	PUNCT
ejde-92	88	8	(	(	PUNCT
ejde-92	88	9	1.13	1.13	NUM
ejde-92	88	10	)	)	PUNCT
ejde-92	88	11	where	where	SCONJ
ejde-92	88	12	1	1	NUM
ejde-92	88	13	≤	≤	NOUN
ejde-92	88	14	q	q	NOUN
ejde-92	88	15	<	<	X
ejde-92	88	16	p∗	p∗	NOUN
ejde-92	88	17	with	with	ADP
ejde-92	88	18	p∗	p∗	PROPN
ejde-92	88	19	=	=	PUNCT
ejde-92	88	20	pn	pn	PROPN
ejde-92	88	21	n−p	n−p	PROPN
ejde-92	88	22	if	if	SCONJ
ejde-92	88	23	p	p	X
ejde-92	88	24	<	<	X
ejde-92	88	25	n	n	NOUN
ejde-92	88	26	,	,	PUNCT
ejde-92	88	27	p∗	p∗	ADJ
ejde-92	88	28	=	=	SYM
ejde-92	88	29	∞	∞	NUM
ejde-92	88	30	otherwise	otherwise	ADV
ejde-92	88	31	,	,	PUNCT
ejde-92	88	32	m	m	VERB
ejde-92	88	33	∈	∈	NOUN
ejde-92	88	34	lr(ω)+	lr(ω)+	NOUN
ejde-92	88	35	and	and	CCONJ
ejde-92	88	36	r	r	NOUN
ejde-92	88	37	varies	varie	NOUN
ejde-92	88	38	in	in	ADP
ejde-92	88	39	a	a	DET
ejde-92	88	40	convenient	convenient	ADJ
ejde-92	88	41	range	range	NOUN
ejde-92	88	42	.	.	PUNCT
ejde-92	89	1	it	it	PRON
ejde-92	89	2	should	should	AUX
ejde-92	89	3	be	be	AUX
ejde-92	89	4	pointed	point	VERB
ejde-92	89	5	out	out	ADP
ejde-92	89	6	that	that	SCONJ
ejde-92	89	7	the	the	DET
ejde-92	89	8	dirichlet	dirichlet	PROPN
ejde-92	89	9	counterpart	counterpart	NOUN
ejde-92	89	10	of	of	ADP
ejde-92	89	11	(	(	PUNCT
ejde-92	89	12	1.13	1.13	NUM
ejde-92	89	13	)	)	PUNCT
ejde-92	89	14	was	be	AUX
ejde-92	89	15	discussed	discuss	VERB
ejde-92	89	16	in	in	ADP
ejde-92	89	17	full	full	ADJ
ejde-92	89	18	detail	detail	NOUN
ejde-92	89	19	in	in	ADP
ejde-92	89	20	[	[	X
ejde-92	89	21	13	13	NUM
ejde-92	89	22	]	]	PUNCT
ejde-92	90	1	where	where	SCONJ
ejde-92	90	2	m	m	PROPN
ejde-92	90	3	∈	∈	PROPN
ejde-92	90	4	l∞(ω)+	l∞(ω)+	PROPN
ejde-92	90	5	.	.	PUNCT
ejde-92	91	1	in	in	ADP
ejde-92	91	2	section	section	NOUN
ejde-92	91	3	4	4	NUM
ejde-92	91	4	we	we	PRON
ejde-92	91	5	analyze	analyze	VERB
ejde-92	91	6	the	the	DET
ejde-92	91	7	main	main	ADJ
ejde-92	91	8	existence	existence	NOUN
ejde-92	91	9	issues	issue	NOUN
ejde-92	91	10	concerning	concern	VERB
ejde-92	91	11	(	(	PUNCT
ejde-92	91	12	1.13	1.13	NUM
ejde-92	91	13	)	)	PUNCT
ejde-92	91	14	.	.	PUNCT
ejde-92	92	1	finally	finally	ADV
ejde-92	92	2	section	section	VERB
ejde-92	92	3	5	5	NUM
ejde-92	92	4	contains	contain	VERB
ejde-92	92	5	the	the	DET
ejde-92	92	6	proof	proof	NOUN
ejde-92	92	7	of	of	ADP
ejde-92	92	8	theorem	theorem	ADJ
ejde-92	92	9	1.2	1.2	NUM
ejde-92	92	10	.	.	NOUN
ejde-92	92	11	2	2	NUM
ejde-92	92	12	.	.	X
ejde-92	92	13	variance	variance	NOUN
ejde-92	92	14	functional	functional	ADJ
ejde-92	92	15	in	in	ADP
ejde-92	92	16	lp(x,µ	lp(x,µ	NOUN
ejde-92	92	17	)	)	PUNCT
ejde-92	92	18	our	our	PRON
ejde-92	92	19	forthcoming	forthcoming	ADJ
ejde-92	92	20	results	result	NOUN
ejde-92	92	21	will	will	AUX
ejde-92	92	22	be	be	AUX
ejde-92	92	23	obtained	obtain	VERB
ejde-92	92	24	in	in	ADP
ejde-92	92	25	the	the	DET
ejde-92	92	26	general	general	ADJ
ejde-92	92	27	framework	framework	NOUN
ejde-92	92	28	of	of	ADP
ejde-92	92	29	a	a	DET
ejde-92	92	30	measurable	measurable	ADJ
ejde-92	92	31	space	space	NOUN
ejde-92	92	32	(	(	PUNCT
ejde-92	92	33	x	x	X
ejde-92	92	34	,	,	PUNCT
ejde-92	92	35	a	a	DET
ejde-92	92	36	,	,	PUNCT
ejde-92	92	37	µ	µ	NOUN
ejde-92	92	38	)	)	PUNCT
ejde-92	92	39	where	where	SCONJ
ejde-92	92	40	µ	µ	NOUN
ejde-92	92	41	is	be	AUX
ejde-92	92	42	a	a	DET
ejde-92	92	43	finite	finite	ADJ
ejde-92	92	44	measure	measure	NOUN
ejde-92	92	45	defined	define	VERB
ejde-92	92	46	in	in	ADP
ejde-92	92	47	some	some	DET
ejde-92	92	48	fixed	fix	VERB
ejde-92	92	49	σ	σ	PROPN
ejde-92	92	50	-	-	PUNCT
ejde-92	92	51	algebra	algebra	PROPN
ejde-92	92	52	a	a	PRON
ejde-92	92	53	in	in	ADP
ejde-92	92	54	x.	x.	NOUN
ejde-92	92	55	as	as	ADP
ejde-92	92	56	usual	usual	ADJ
ejde-92	92	57	,	,	PUNCT
ejde-92	92	58	lp(x,µ	lp(x,µ	NOUN
ejde-92	92	59	)	)	PUNCT
ejde-92	92	60	stands	stand	VERB
ejde-92	92	61	for	for	ADP
ejde-92	92	62	the	the	DET
ejde-92	92	63	space	space	NOUN
ejde-92	92	64	of	of	ADP
ejde-92	92	65	measurable	measurable	ADJ
ejde-92	92	66	functions	function	NOUN
ejde-92	92	67	f	f	PRON
ejde-92	92	68	such	such	ADJ
ejde-92	92	69	that∫	that∫	NOUN
ejde-92	92	70	x	x	X
ejde-92	92	71	|f	|f	PROPN
ejde-92	92	72	|p	|p	PROPN
ejde-92	92	73	dµ	dµ	VERB
ejde-92	92	74	<	<	X
ejde-92	92	75	∞.	∞.	PROPN
ejde-92	92	76	we	we	PRON
ejde-92	92	77	are	be	AUX
ejde-92	92	78	proceeding	proceed	VERB
ejde-92	92	79	in	in	ADP
ejde-92	92	80	this	this	DET
ejde-92	92	81	way	way	NOUN
ejde-92	92	82	for	for	ADP
ejde-92	92	83	the	the	DET
ejde-92	92	84	sake	sake	NOUN
ejde-92	92	85	of	of	ADP
ejde-92	92	86	completeness	completeness	NOUN
ejde-92	92	87	.	.	PUNCT
ejde-92	93	1	we	we	PRON
ejde-92	93	2	begin	begin	VERB
ejde-92	93	3	by	by	ADP
ejde-92	93	4	reviewing	review	VERB
ejde-92	93	5	some	some	DET
ejde-92	93	6	elementary	elementary	ADJ
ejde-92	93	7	features	feature	NOUN
ejde-92	93	8	which	which	PRON
ejde-92	93	9	permit	permit	VERB
ejde-92	93	10	us	we	PRON
ejde-92	93	11	refreshing	refresh	VERB
ejde-92	93	12	the	the	DET
ejde-92	93	13	notion	notion	NOUN
ejde-92	93	14	of	of	ADP
ejde-92	93	15	variance	variance	NOUN
ejde-92	93	16	.	.	PUNCT
ejde-92	94	1	lemma	lemma	PROPN
ejde-92	94	2	2.1	2.1	NUM
ejde-92	94	3	.	.	PUNCT
ejde-92	95	1	let	let	VERB
ejde-92	95	2	(	(	PUNCT
ejde-92	95	3	x	x	NOUN
ejde-92	95	4	,	,	PUNCT
ejde-92	95	5	a	a	PRON
ejde-92	95	6	)	)	PUNCT
ejde-92	95	7	be	be	AUX
ejde-92	95	8	a	a	DET
ejde-92	95	9	measurable	measurable	ADJ
ejde-92	95	10	space	space	NOUN
ejde-92	95	11	endowed	endow	VERB
ejde-92	95	12	with	with	ADP
ejde-92	95	13	a	a	DET
ejde-92	95	14	finite	finite	ADJ
ejde-92	95	15	measure	measure	NOUN
ejde-92	95	16	µ.	µ.	NOUN
ejde-92	95	17	then	then	ADV
ejde-92	95	18	,	,	PUNCT
ejde-92	95	19	for	for	ADP
ejde-92	95	20	every	every	DET
ejde-92	95	21	u	u	PROPN
ejde-92	95	22	∈	∈	PROPN
ejde-92	95	23	lp(x,µ	lp(x,µ	NOUN
ejde-92	95	24	)	)	PUNCT
ejde-92	95	25	there	there	PRON
ejde-92	95	26	exists	exist	VERB
ejde-92	95	27	a	a	DET
ejde-92	95	28	unique	unique	ADJ
ejde-92	95	29	ũ	ũ	PROPN
ejde-92	95	30	∈	∈	NOUN
ejde-92	95	31	r	r	NOUN
ejde-92	95	32	such	such	ADJ
ejde-92	95	33	that∫	that∫	NOUN
ejde-92	95	34	x	x	PUNCT
ejde-92	95	35	|u−	|u−	PROPN
ejde-92	95	36	ũ|p−2(u−	ũ|p−2(u−	PROPN
ejde-92	95	37	ũ	ũ	PROPN
ejde-92	95	38	)	)	PUNCT
ejde-92	95	39	dµ	dµ	PROPN
ejde-92	95	40	=	=	PUNCT
ejde-92	96	1	0	0	X
ejde-92	96	2	.	.	PUNCT
ejde-92	97	1	moreover	moreover	ADV
ejde-92	97	2	,	,	PUNCT
ejde-92	97	3	ũ	ũ	PROPN
ejde-92	97	4	is	be	AUX
ejde-92	97	5	variationally	variationally	ADV
ejde-92	97	6	characterized	characterize	VERB
ejde-92	97	7	by	by	ADP
ejde-92	97	8	the	the	DET
ejde-92	97	9	expression∫	expression∫	NOUN
ejde-92	97	10	x	x	PUNCT
ejde-92	97	11	|u−	|u−	PROPN
ejde-92	97	12	ũ|p	ũ|p	NOUN
ejde-92	97	13	dµ	dµ	PROPN
ejde-92	97	14	=	=	PUNCT
ejde-92	97	15	inf	inf	PROPN
ejde-92	97	16	t∈r	t∈r	ADJ
ejde-92	97	17	∫	∫	NOUN
ejde-92	97	18	x	x	PUNCT
ejde-92	97	19	|u−	|u−	NOUN
ejde-92	98	1	t|p	t|p	X
ejde-92	98	2	dµ.	dµ.	PROPN
ejde-92	98	3	(	(	PUNCT
ejde-92	98	4	2.1	2.1	NUM
ejde-92	98	5	)	)	PUNCT
ejde-92	98	6	furthermore	furthermore	ADV
ejde-92	98	7	,	,	PUNCT
ejde-92	98	8	ũ	ũ	PROPN
ejde-92	98	9	defines	define	VERB
ejde-92	98	10	a	a	DET
ejde-92	98	11	continuous	continuous	ADJ
ejde-92	98	12	functional	functional	NOUN
ejde-92	98	13	of	of	ADP
ejde-92	98	14	u	u	PROPN
ejde-92	98	15	∈	∈	PROPN
ejde-92	98	16	lp(x,µ	lp(x,µ	NOUN
ejde-92	98	17	)	)	PUNCT
ejde-92	98	18	.	.	PUNCT
ejde-92	99	1	definition	definition	NOUN
ejde-92	99	2	2.2	2.2	NUM
ejde-92	99	3	.	.	PUNCT
ejde-92	100	1	for	for	ADP
ejde-92	100	2	a	a	DET
ejde-92	100	3	function	function	NOUN
ejde-92	100	4	u	u	PROPN
ejde-92	100	5	∈	∈	PROPN
ejde-92	100	6	lp(x,µ	lp(x,µ	NOUN
ejde-92	100	7	)	)	PUNCT
ejde-92	100	8	its	its	PRON
ejde-92	100	9	variance	variance	NOUN
ejde-92	100	10	is	be	AUX
ejde-92	100	11	defined	define	VERB
ejde-92	100	12	as	as	ADP
ejde-92	100	13	vp(u	vp(u	NOUN
ejde-92	100	14	)	)	PUNCT
ejde-92	101	1	=	=	SYM
ejde-92	101	2	∫	∫	PROPN
ejde-92	101	3	x	x	PROPN
ejde-92	101	4	|u−	|u−	PROPN
ejde-92	101	5	ũ|p	ũ|p	PROPN
ejde-92	101	6	dµ	dµ	PROPN
ejde-92	101	7	,	,	PUNCT
ejde-92	101	8	while	while	SCONJ
ejde-92	101	9	ũ	ũ	PROPN
ejde-92	101	10	is	be	AUX
ejde-92	101	11	said	say	VERB
ejde-92	101	12	to	to	PART
ejde-92	101	13	be	be	AUX
ejde-92	101	14	the	the	DET
ejde-92	101	15	average	average	NOUN
ejde-92	101	16	of	of	ADP
ejde-92	101	17	u	u	NOUN
ejde-92	101	18	relative	relative	ADJ
ejde-92	101	19	to	to	ADP
ejde-92	101	20	lp(x,µ	lp(x,µ	NOUN
ejde-92	101	21	)	)	PUNCT
ejde-92	101	22	.	.	PUNCT
ejde-92	102	1	remark	remark	VERB
ejde-92	102	2	2.3	2.3	NUM
ejde-92	102	3	.	.	PUNCT
ejde-92	103	1	this	this	PRON
ejde-92	103	2	is	be	AUX
ejde-92	103	3	of	of	ADP
ejde-92	103	4	course	course	NOUN
ejde-92	103	5	the	the	DET
ejde-92	103	6	standard	standard	ADJ
ejde-92	103	7	definition	definition	NOUN
ejde-92	103	8	when	when	SCONJ
ejde-92	103	9	p	p	PROPN
ejde-92	103	10	=	=	NOUN
ejde-92	103	11	2	2	X
ejde-92	103	12	.	.	PUNCT
ejde-92	104	1	in	in	ADP
ejde-92	104	2	fact	fact	NOUN
ejde-92	104	3	,	,	PUNCT
ejde-92	104	4	ũ	ũ	PROPN
ejde-92	104	5	=	=	NOUN
ejde-92	104	6	1	1	NUM
ejde-92	104	7	|x|	|x|	PROPN
ejde-92	104	8	∫	∫	PROPN
ejde-92	104	9	x	x	SYM
ejde-92	104	10	u	u	NOUN
ejde-92	104	11	dµ	dµ	PROPN
ejde-92	104	12	,	,	PUNCT
ejde-92	104	13	|x|	|x|	PROPN
ejde-92	104	14	:	:	PUNCT
ejde-92	104	15	=	=	SYM
ejde-92	105	1	∫	∫	PROPN
ejde-92	105	2	x	x	X
ejde-92	105	3	dµ	dµ	PROPN
ejde-92	105	4	=	=	PUNCT
ejde-92	105	5	µ(x	µ(x	X
ejde-92	105	6	)	)	PUNCT
ejde-92	105	7	,	,	PUNCT
ejde-92	105	8	if	if	SCONJ
ejde-92	105	9	p	p	NOUN
ejde-92	105	10	=	=	SYM
ejde-92	105	11	2	2	NUM
ejde-92	105	12	and	and	CCONJ
ejde-92	105	13	ũ	ũ	PROPN
ejde-92	105	14	coincides	coincide	VERB
ejde-92	105	15	with	with	ADP
ejde-92	105	16	the	the	DET
ejde-92	105	17	standard	standard	ADJ
ejde-92	105	18	average	average	NOUN
ejde-92	105	19	of	of	ADP
ejde-92	105	20	u	u	NOUN
ejde-92	105	21	in	in	ADP
ejde-92	105	22	x.	x.	PROPN
ejde-92	105	23	ejde-2022/13	ejde-2022/13	PROPN
ejde-92	105	24	second	second	ADJ
ejde-92	105	25	neumann	neumann	PROPN
ejde-92	105	26	eigenvalue	eigenvalue	X
ejde-92	105	27	5	5	NUM
ejde-92	105	28	proof	proof	NOUN
ejde-92	105	29	of	of	ADP
ejde-92	105	30	lemma	lemma	PROPN
ejde-92	105	31	2.1	2.1	NUM
ejde-92	105	32	.	.	PUNCT
ejde-92	106	1	the	the	DET
ejde-92	106	2	function	function	PROPN
ejde-92	106	3	g(t	g(t	PROPN
ejde-92	106	4	)	)	PUNCT
ejde-92	107	1	=	=	SYM
ejde-92	107	2	∫	∫	PROPN
ejde-92	107	3	x	x	PUNCT
ejde-92	107	4	|u−	|u−	PROPN
ejde-92	107	5	t|p−2(u−	t|p−2(u−	PROPN
ejde-92	107	6	t	t	PROPN
ejde-92	107	7	)	)	PUNCT
ejde-92	107	8	dµ	dµ	PROPN
ejde-92	107	9	is	be	AUX
ejde-92	107	10	continuous	continuous	ADJ
ejde-92	107	11	and	and	CCONJ
ejde-92	107	12	decreasing	decrease	VERB
ejde-92	107	13	while	while	SCONJ
ejde-92	107	14	limt→±∞	limt→±∞	PROPN
ejde-92	107	15	g(t	g(t	PROPN
ejde-92	107	16	)	)	PUNCT
ejde-92	108	1	=	=	PUNCT
ejde-92	108	2	∓∞.	∓∞.	NOUN
ejde-92	108	3	hence	hence	ADV
ejde-92	108	4	,	,	PUNCT
ejde-92	108	5	the	the	DET
ejde-92	108	6	existence	existence	NOUN
ejde-92	108	7	and	and	CCONJ
ejde-92	108	8	uniqueness	uniqueness	ADJ
ejde-92	108	9	assertions	assertion	NOUN
ejde-92	108	10	follow	follow	VERB
ejde-92	108	11	.	.	PUNCT
ejde-92	109	1	on	on	ADP
ejde-92	109	2	the	the	DET
ejde-92	109	3	other	other	ADJ
ejde-92	109	4	hand	hand	NOUN
ejde-92	109	5	,	,	PUNCT
ejde-92	109	6	f(t	f(t	PROPN
ejde-92	109	7	)	)	PUNCT
ejde-92	109	8	=	=	SYM
ejde-92	110	1	∫	∫	PROPN
ejde-92	110	2	x	x	PUNCT
ejde-92	110	3	|u−	|u−	NOUN
ejde-92	110	4	t|p	t|p	PUNCT
ejde-92	111	1	dµ	dµ	PROPN
ejde-92	111	2	is	be	AUX
ejde-92	111	3	a	a	DET
ejde-92	111	4	convex	convex	ADJ
ejde-92	111	5	coercive	coercive	ADJ
ejde-92	111	6	function	function	NOUN
ejde-92	112	1	such	such	ADJ
ejde-92	112	2	that	that	SCONJ
ejde-92	112	3	f	f	PROPN
ejde-92	112	4	′	′	NUM
ejde-92	112	5	=	=	SYM
ejde-92	112	6	−pg	−pg	PROPN
ejde-92	112	7	.	.	PUNCT
ejde-92	112	8	this	this	PRON
ejde-92	112	9	implies	imply	VERB
ejde-92	112	10	(	(	PUNCT
ejde-92	112	11	2.1	2.1	NUM
ejde-92	112	12	)	)	PUNCT
ejde-92	112	13	.	.	PUNCT
ejde-92	113	1	as	as	ADP
ejde-92	113	2	for	for	ADP
ejde-92	113	3	the	the	DET
ejde-92	113	4	continuity	continuity	NOUN
ejde-92	113	5	assertion	assertion	NOUN
ejde-92	113	6	assume	assume	VERB
ejde-92	113	7	that	that	SCONJ
ejde-92	113	8	un	un	PROPN
ejde-92	113	9	→	→	SYM
ejde-92	113	10	u	u	PROPN
ejde-92	113	11	in	in	ADP
ejde-92	113	12	lp(x,µ	lp(x,µ	NOUN
ejde-92	113	13	)	)	PUNCT
ejde-92	113	14	.	.	PUNCT
ejde-92	114	1	the	the	DET
ejde-92	114	2	inequalities	inequality	NOUN
ejde-92	114	3	|x||ũ|p	|x||ũ|p	VERB
ejde-92	114	4	≤	≤	PROPN
ejde-92	114	5	2p−1	2p−1	NUM
ejde-92	114	6	∫	∫	NOUN
ejde-92	114	7	x	x	INTJ
ejde-92	114	8	{	{	PUNCT
ejde-92	114	9	|u−	|u−	ADJ
ejde-92	114	10	ũ|p	ũ|p	NOUN
ejde-92	114	11	+	+	CCONJ
ejde-92	114	12	|u|p	|u|p	PROPN
ejde-92	114	13	}	}	PUNCT
ejde-92	114	14	dµ	dµ	ADJ
ejde-92	114	15	≤	≤	NUM
ejde-92	114	16	2p	2p	NUM
ejde-92	114	17	∫	∫	NOUN
ejde-92	114	18	x	x	PUNCT
ejde-92	115	1	|u|p	|u|p	NOUN
ejde-92	115	2	dµ	dµ	PRON
ejde-92	115	3	imply	imply	VERB
ejde-92	115	4	|ũ|	|ũ|	ADJ
ejde-92	115	5	≤	≤	ADJ
ejde-92	115	6	2	2	NUM
ejde-92	115	7	|x|1	|x|1	NOUN
ejde-92	115	8	/	/	SYM
ejde-92	115	9	p	p	NOUN
ejde-92	115	10	‖u‖p	‖u‖p	NOUN
ejde-92	115	11	,	,	PUNCT
ejde-92	115	12	(	(	PUNCT
ejde-92	115	13	2.2	2.2	NUM
ejde-92	115	14	)	)	PUNCT
ejde-92	115	15	with	with	ADP
ejde-92	115	16	‖u‖p	‖u‖p	NOUN
ejde-92	115	17	=	=	SYM
ejde-92	115	18	‖u‖lp(x,µ	‖u‖lp(x,µ	NOUN
ejde-92	115	19	)	)	PUNCT
ejde-92	115	20	.	.	PUNCT
ejde-92	116	1	thus	thus	ADV
ejde-92	116	2	ũn	ũn	PROPN
ejde-92	116	3	is	be	AUX
ejde-92	116	4	bounded	bound	VERB
ejde-92	116	5	and	and	CCONJ
ejde-92	116	6	a	a	DET
ejde-92	116	7	convergent	convergent	NOUN
ejde-92	116	8	subsequence	subsequence	NOUN
ejde-92	116	9	ũn′	ũn′	PROPN
ejde-92	116	10	→	→	SYM
ejde-92	116	11	ũ′	ũ′	PROPN
ejde-92	116	12	can	can	AUX
ejde-92	116	13	be	be	AUX
ejde-92	116	14	extracted	extract	VERB
ejde-92	116	15	from	from	ADP
ejde-92	116	16	ũn	ũn	NOUN
ejde-92	116	17	.	.	PUNCT
ejde-92	117	1	since	since	SCONJ
ejde-92	117	2	|un′	|un′	PROPN
ejde-92	117	3	−	−	PROPN
ejde-92	117	4	ũn′	ũn′	PROPN
ejde-92	117	5	|p−2(un′	|p−2(un′	NOUN
ejde-92	117	6	−	−	PROPN
ejde-92	117	7	ũn′)→	ũn′)→	PROPN
ejde-92	117	8	|u−	|u−	PROPN
ejde-92	117	9	ũ′|p−2(u−	ũ′|p−2(u−	PROPN
ejde-92	117	10	ũ′	ũ′	PROPN
ejde-92	117	11	)	)	PUNCT
ejde-92	117	12	in	in	ADP
ejde-92	117	13	lp	lp	NOUN
ejde-92	117	14	′	′	NUM
ejde-92	117	15	(	(	PUNCT
ejde-92	117	16	x,µ	x,µ	NOUN
ejde-92	117	17	)	)	PUNCT
ejde-92	117	18	,	,	PUNCT
ejde-92	117	19	it	it	PRON
ejde-92	117	20	follows	follow	VERB
ejde-92	117	21	that∫	that∫	NOUN
ejde-92	117	22	x	x	SYM
ejde-92	117	23	|u−	|u−	PROPN
ejde-92	117	24	ũ′|p−2(u−	ũ′|p−2(u−	PROPN
ejde-92	117	25	ũ′	ũ′	PROPN
ejde-92	117	26	)	)	PUNCT
ejde-92	117	27	dµ	dµ	PROPN
ejde-92	117	28	=	=	PUNCT
ejde-92	118	1	0	0	X
ejde-92	118	2	.	.	PUNCT
ejde-92	119	1	this	this	PRON
ejde-92	119	2	means	mean	VERB
ejde-92	119	3	that	that	SCONJ
ejde-92	119	4	ũ′	ũ′	PROPN
ejde-92	119	5	=	=	SYM
ejde-92	119	6	ũ	ũ	PROPN
ejde-92	119	7	and	and	CCONJ
ejde-92	119	8	the	the	DET
ejde-92	119	9	continuity	continuity	NOUN
ejde-92	119	10	is	be	AUX
ejde-92	119	11	shown	show	VERB
ejde-92	119	12	.	.	PUNCT
ejde-92	120	1	�	�	PROPN
ejde-92	120	2	the	the	DET
ejde-92	120	3	main	main	ADJ
ejde-92	120	4	result	result	NOUN
ejde-92	120	5	of	of	ADP
ejde-92	120	6	this	this	DET
ejde-92	120	7	section	section	NOUN
ejde-92	120	8	is	be	AUX
ejde-92	120	9	the	the	DET
ejde-92	120	10	next	next	ADJ
ejde-92	120	11	one	one	NUM
ejde-92	120	12	.	.	PUNCT
ejde-92	121	1	it	it	PRON
ejde-92	121	2	states	state	VERB
ejde-92	121	3	the	the	DET
ejde-92	121	4	differentiability	differentiability	NOUN
ejde-92	121	5	of	of	ADP
ejde-92	121	6	the	the	DET
ejde-92	121	7	variance	variance	NOUN
ejde-92	121	8	functional	functional	ADJ
ejde-92	121	9	vp	vp	PROPN
ejde-92	121	10	and	and	CCONJ
ejde-92	121	11	seems	seem	VERB
ejde-92	121	12	new	new	ADJ
ejde-92	121	13	at	at	ADP
ejde-92	121	14	the	the	DET
ejde-92	121	15	best	good	ADJ
ejde-92	121	16	of	of	ADP
ejde-92	121	17	our	our	PRON
ejde-92	121	18	knowledge	knowledge	NOUN
ejde-92	121	19	.	.	PUNCT
ejde-92	122	1	in	in	ADP
ejde-92	122	2	this	this	DET
ejde-92	122	3	regard	regard	NOUN
ejde-92	122	4	,	,	PUNCT
ejde-92	122	5	it	it	PRON
ejde-92	122	6	should	should	AUX
ejde-92	122	7	be	be	AUX
ejde-92	122	8	remarked	remark	VERB
ejde-92	122	9	that	that	SCONJ
ejde-92	122	10	vp	vp	PROPN
ejde-92	122	11	is	be	AUX
ejde-92	122	12	defined	define	VERB
ejde-92	122	13	in	in	ADP
ejde-92	122	14	a	a	DET
ejde-92	122	15	variational	variational	ADJ
ejde-92	122	16	way	way	NOUN
ejde-92	123	1	and	and	CCONJ
ejde-92	123	2	so	so	ADV
ejde-92	123	3	this	this	PRON
ejde-92	123	4	is	be	AUX
ejde-92	123	5	not	not	PART
ejde-92	123	6	a	a	DET
ejde-92	123	7	straightforward	straightforward	ADJ
ejde-92	123	8	issue	issue	NOUN
ejde-92	123	9	.	.	PUNCT
ejde-92	124	1	theorem	theorem	VERB
ejde-92	124	2	2.4	2.4	NUM
ejde-92	124	3	.	.	PUNCT
ejde-92	125	1	under	under	ADP
ejde-92	125	2	the	the	DET
ejde-92	125	3	assumptions	assumption	NOUN
ejde-92	125	4	of	of	ADP
ejde-92	125	5	lemma	lemma	PROPN
ejde-92	125	6	2.1	2.1	NUM
ejde-92	125	7	,	,	PUNCT
ejde-92	125	8	the	the	DET
ejde-92	125	9	variance	variance	NOUN
ejde-92	125	10	functional	functional	ADJ
ejde-92	125	11	vp	vp	X
ejde-92	125	12	is	be	AUX
ejde-92	125	13	fréchet	fréchet	VERB
ejde-92	125	14	differentiable	differentiable	ADJ
ejde-92	125	15	in	in	ADP
ejde-92	125	16	lp(x,µ	lp(x,µ	NOUN
ejde-92	125	17	)	)	PUNCT
ejde-92	125	18	for	for	ADP
ejde-92	125	19	all	all	PRON
ejde-92	125	20	p	p	X
ejde-92	125	21	>	>	X
ejde-92	125	22	1	1	NUM
ejde-92	125	23	.	.	PUNCT
ejde-92	126	1	moreover	moreover	ADV
ejde-92	126	2	,	,	PUNCT
ejde-92	126	3	its	its	PRON
ejde-92	126	4	differential	differential	NOUN
ejde-92	126	5	dvp(u	dvp(u	PROPN
ejde-92	126	6	)	)	PUNCT
ejde-92	126	7	at	at	ADP
ejde-92	126	8	u	u	NOUN
ejde-92	126	9	is	be	AUX
ejde-92	126	10	represented	represent	VERB
ejde-92	126	11	as	as	ADP
ejde-92	126	12	〈	〈	PROPN
ejde-92	126	13	dvp(u	dvp(u	PROPN
ejde-92	126	14	)	)	PUNCT
ejde-92	126	15	,	,	PUNCT
ejde-92	126	16	v	v	NOUN
ejde-92	126	17	〉	〉	NOUN
ejde-92	126	18	=	=	SYM
ejde-92	127	1	p	p	NOUN
ejde-92	127	2	∫	∫	PROPN
ejde-92	127	3	x	x	PUNCT
ejde-92	127	4	|u−	|u−	PROPN
ejde-92	127	5	ũ|p−2(u−	ũ|p−2(u−	PROPN
ejde-92	127	6	ũ)v	ũ)v	PROPN
ejde-92	127	7	dµ	dµ	PROPN
ejde-92	127	8	,	,	PUNCT
ejde-92	127	9	v	v	PROPN
ejde-92	127	10	∈	∈	PROPN
ejde-92	127	11	lp(x,µ	lp(x,µ	NOUN
ejde-92	127	12	)	)	PUNCT
ejde-92	127	13	.	.	PUNCT
ejde-92	128	1	the	the	DET
ejde-92	128	2	proof	proof	NOUN
ejde-92	128	3	of	of	ADP
ejde-92	128	4	theorem	theorem	ADJ
ejde-92	128	5	2.4	2.4	NUM
ejde-92	128	6	relies	relie	NOUN
ejde-92	128	7	on	on	ADP
ejde-92	128	8	the	the	DET
ejde-92	128	9	next	next	ADJ
ejde-92	128	10	crucial	crucial	ADJ
ejde-92	128	11	lemma	lemma	PROPN
ejde-92	128	12	.	.	PUNCT
ejde-92	129	1	lemma	lemma	PROPN
ejde-92	129	2	2.5	2.5	NUM
ejde-92	129	3	.	.	PUNCT
ejde-92	130	1	let	let	VERB
ejde-92	130	2	u	u	PRON
ejde-92	130	3	∈	∈	PROPN
ejde-92	130	4	lp(x,µ	lp(x,µ	NOUN
ejde-92	130	5	)	)	PUNCT
ejde-92	130	6	be	be	AUX
ejde-92	130	7	fixed	fix	VERB
ejde-92	130	8	.	.	PUNCT
ejde-92	131	1	then	then	ADV
ejde-92	131	2	the	the	DET
ejde-92	131	3	gâteaux	gâteaux	ADJ
ejde-92	131	4	derivative	derivative	ADJ
ejde-92	131	5	dvp(u	dvp(u	PROPN
ejde-92	131	6	,	,	PUNCT
ejde-92	131	7	v	v	NOUN
ejde-92	131	8	)	)	PUNCT
ejde-92	131	9	of	of	ADP
ejde-92	131	10	the	the	DET
ejde-92	131	11	variance	variance	NOUN
ejde-92	131	12	vp	vp	NOUN
ejde-92	131	13	at	at	ADP
ejde-92	131	14	u	u	PROPN
ejde-92	131	15	in	in	ADP
ejde-92	131	16	the	the	DET
ejde-92	131	17	direction	direction	NOUN
ejde-92	131	18	v	v	ADP
ejde-92	131	19	∈	∈	PROPN
ejde-92	131	20	lp(x,µ	lp(x,µ	NOUN
ejde-92	131	21	)	)	PUNCT
ejde-92	131	22	exists	exist	VERB
ejde-92	131	23	and	and	CCONJ
ejde-92	131	24	is	be	AUX
ejde-92	131	25	given	give	VERB
ejde-92	131	26	by	by	ADP
ejde-92	131	27	dvp(u	dvp(u	PROPN
ejde-92	131	28	,	,	PUNCT
ejde-92	131	29	v	v	NOUN
ejde-92	131	30	)	)	PUNCT
ejde-92	131	31	=	=	PUNCT
ejde-92	132	1	p	p	X
ejde-92	132	2	∫	∫	PROPN
ejde-92	132	3	x	x	PUNCT
ejde-92	132	4	|u−	|u−	PROPN
ejde-92	132	5	ũ|p−2(u−	ũ|p−2(u−	PROPN
ejde-92	132	6	ũ)v	ũ)v	PROPN
ejde-92	132	7	dµ.	dµ.	PROPN
ejde-92	132	8	(	(	PUNCT
ejde-92	132	9	2.3	2.3	NUM
ejde-92	132	10	)	)	PUNCT
ejde-92	132	11	proof	proof	NOUN
ejde-92	132	12	.	.	PUNCT
ejde-92	133	1	fix	fix	VERB
ejde-92	133	2	u	u	NOUN
ejde-92	133	3	,	,	PUNCT
ejde-92	133	4	v	v	PROPN
ejde-92	133	5	∈	∈	PROPN
ejde-92	133	6	lp(x,µ	lp(x,µ	NOUN
ejde-92	133	7	)	)	PUNCT
ejde-92	133	8	.	.	PUNCT
ejde-92	134	1	for	for	ADP
ejde-92	134	2	t	t	PROPN
ejde-92	134	3	∈	∈	PROPN
ejde-92	134	4	r	r	NOUN
ejde-92	134	5	define	define	NOUN
ejde-92	134	6	ut	ut	PROPN
ejde-92	134	7	=	=	PUNCT
ejde-92	134	8	u+	u+	NUM
ejde-92	134	9	tv	tv	NOUN
ejde-92	134	10	,	,	PUNCT
ejde-92	134	11	ũt	ũt	PROPN
ejde-92	134	12	=	=	PUNCT
ejde-92	134	13	ũt	ũt	PROPN
ejde-92	134	14	=	=	SYM
ejde-92	134	15	ũ+	ũ+	PROPN
ejde-92	134	16	tv	tv	NOUN
ejde-92	134	17	,	,	PUNCT
ejde-92	134	18	together	together	ADV
ejde-92	134	19	with	with	ADP
ejde-92	134	20	v	v	PROPN
ejde-92	134	21	(	(	PUNCT
ejde-92	134	22	t	t	NOUN
ejde-92	134	23	)	)	PUNCT
ejde-92	134	24	=	=	PUNCT
ejde-92	134	25	vp(ut	vp(ut	PROPN
ejde-92	134	26	)	)	PUNCT
ejde-92	134	27	=	=	SYM
ejde-92	134	28	vp(u+	vp(u+	NOUN
ejde-92	134	29	tv	tv	NOUN
ejde-92	134	30	)	)	PUNCT
ejde-92	134	31	.	.	PUNCT
ejde-92	135	1	by	by	ADP
ejde-92	135	2	setting	set	VERB
ejde-92	135	3	ft(σ	ft(σ	PUNCT
ejde-92	135	4	)	)	PUNCT
ejde-92	135	5	=	=	SYM
ejde-92	135	6	∫	∫	PROPN
ejde-92	135	7	x	x	X
ejde-92	135	8	|u+	|u+	PROPN
ejde-92	135	9	tv	tv	PROPN
ejde-92	135	10	−	−	PROPN
ejde-92	135	11	σ|p	σ|p	NOUN
ejde-92	135	12	dµ	dµ	PROPN
ejde-92	136	1	=	=	SYM
ejde-92	136	2	∫	∫	PROPN
ejde-92	136	3	x	x	PROPN
ejde-92	136	4	|ut	|ut	PROPN
ejde-92	136	5	−	−	PROPN
ejde-92	136	6	σ|p	σ|p	PROPN
ejde-92	136	7	dµ	dµ	PROPN
ejde-92	136	8	,	,	PUNCT
ejde-92	136	9	it	it	PRON
ejde-92	136	10	holds	hold	VERB
ejde-92	136	11	that	that	SCONJ
ejde-92	136	12	v	v	NOUN
ejde-92	136	13	(	(	PUNCT
ejde-92	136	14	t	t	NOUN
ejde-92	136	15	)	)	PUNCT
ejde-92	136	16	=	=	SYM
ejde-92	136	17	ft(ũt	ft(ũt	X
ejde-92	136	18	)	)	PUNCT
ejde-92	136	19	=	=	PROPN
ejde-92	136	20	inf	inf	PROPN
ejde-92	136	21	σ∈r	σ∈r	PROPN
ejde-92	136	22	ft(σ	ft(σ	PROPN
ejde-92	136	23	)	)	PUNCT
ejde-92	136	24	,	,	PUNCT
ejde-92	136	25	v	v	X
ejde-92	136	26	(	(	PUNCT
ejde-92	136	27	0	0	NUM
ejde-92	136	28	)	)	PUNCT
ejde-92	136	29	=	=	SYM
ejde-92	136	30	f0(ũ	f0(ũ	PROPN
ejde-92	136	31	)	)	PUNCT
ejde-92	137	1	=	=	PROPN
ejde-92	138	1	inf	inf	PROPN
ejde-92	138	2	σ∈r	σ∈r	PROPN
ejde-92	138	3	f0(σ	f0(σ	PROPN
ejde-92	138	4	)	)	PUNCT
ejde-92	138	5	.	.	PUNCT
ejde-92	139	1	on	on	ADP
ejde-92	139	2	the	the	DET
ejde-92	139	3	other	other	ADJ
ejde-92	139	4	hand	hand	NOUN
ejde-92	139	5	,	,	PUNCT
ejde-92	139	6	ft(ũt)−	ft(ũt)−	PROPN
ejde-92	139	7	f0(ũt	f0(ũt	PROPN
ejde-92	139	8	)	)	PUNCT
ejde-92	139	9	≤	≤	NOUN
ejde-92	139	10	v	v	X
ejde-92	139	11	(	(	PUNCT
ejde-92	139	12	t)−	t)−	PROPN
ejde-92	139	13	v	v	NOUN
ejde-92	139	14	(	(	PUNCT
ejde-92	139	15	0	0	NUM
ejde-92	139	16	)	)	PUNCT
ejde-92	139	17	≤	≤	NUM
ejde-92	139	18	ft(ũ)−	ft(ũ)−	PROPN
ejde-92	139	19	f0(ũ	f0(ũ	PROPN
ejde-92	139	20	)	)	PUNCT
ejde-92	139	21	.	.	PUNCT
ejde-92	140	1	(	(	PUNCT
ejde-92	140	2	2.4	2.4	NUM
ejde-92	140	3	)	)	PUNCT
ejde-92	140	4	6	6	NUM
ejde-92	140	5	j.	j.	PROPN
ejde-92	140	6	c.	c.	PROPN
ejde-92	140	7	sabina	sabina	PROPN
ejde-92	140	8	de	de	PROPN
ejde-92	140	9	lis	lis	X
ejde-92	140	10	ejde-2022/13	ejde-2022/13	VERB
ejde-92	140	11	the	the	DET
ejde-92	140	12	first	first	ADJ
ejde-92	140	13	term	term	NOUN
ejde-92	140	14	in	in	ADP
ejde-92	140	15	inequality	inequality	NOUN
ejde-92	140	16	(	(	PUNCT
ejde-92	140	17	2.4	2.4	NUM
ejde-92	140	18	)	)	PUNCT
ejde-92	140	19	can	can	AUX
ejde-92	140	20	be	be	AUX
ejde-92	140	21	written	write	VERB
ejde-92	140	22	as	as	ADP
ejde-92	140	23	ft(ũt)−	ft(ũt)−	PROPN
ejde-92	140	24	f0(ũt	f0(ũt	NUM
ejde-92	140	25	)	)	PUNCT
ejde-92	140	26	=	=	SYM
ejde-92	141	1	pt	pt	NOUN
ejde-92	141	2	∫	∫	PROPN
ejde-92	141	3	x	x	X
ejde-92	141	4	{	{	PUNCT
ejde-92	141	5	∫	∫	PROPN
ejde-92	141	6	1	1	NUM
ejde-92	141	7	0	0	NUM
ejde-92	141	8	|u+	|u+	NOUN
ejde-92	141	9	tsv	tsv	VERB
ejde-92	141	10	−	−	NOUN
ejde-92	141	11	ũt|p−2(u+	ũt|p−2(u+	NOUN
ejde-92	141	12	tsv	tsv	NOUN
ejde-92	141	13	−	−	PROPN
ejde-92	141	14	ũt	ũt	SYM
ejde-92	141	15	)	)	PUNCT
ejde-92	141	16	ds	ds	ADJ
ejde-92	141	17	}	}	PUNCT
ejde-92	141	18	v	v	NOUN
ejde-92	141	19	dµ.	dµ.	NOUN
ejde-92	141	20	by	by	ADP
ejde-92	141	21	using	use	VERB
ejde-92	141	22	(	(	PUNCT
ejde-92	141	23	2.2	2.2	NUM
ejde-92	141	24	)	)	PUNCT
ejde-92	141	25	and	and	CCONJ
ejde-92	141	26	assuming	assume	VERB
ejde-92	141	27	that	that	PRON
ejde-92	141	28	|t|	|t|	VERB
ejde-92	141	29	≤	≤	NOUN
ejde-92	141	30	1	1	NUM
ejde-92	141	31	,	,	PUNCT
ejde-92	141	32	the	the	DET
ejde-92	141	33	integrand	integrand	NOUN
ejde-92	141	34	in	in	ADP
ejde-92	141	35	the	the	DET
ejde-92	141	36	last	last	ADJ
ejde-92	141	37	term	term	NOUN
ejde-92	141	38	can	can	AUX
ejde-92	141	39	be	be	AUX
ejde-92	141	40	estimated	estimate	VERB
ejde-92	141	41	as	as	ADP
ejde-92	141	42	∣∣∣	∣∣∣	ADJ
ejde-92	141	43	∫	∫	PROPN
ejde-92	141	44	1	1	NUM
ejde-92	141	45	0	0	NUM
ejde-92	141	46	|u+	|u+	NOUN
ejde-92	141	47	tsv	tsv	VERB
ejde-92	141	48	−	−	NOUN
ejde-92	141	49	ũt|p−2(u+	ũt|p−2(u+	NOUN
ejde-92	141	50	tsv	tsv	NOUN
ejde-92	141	51	−	−	PROPN
ejde-92	141	52	ũt	ũt	SYM
ejde-92	141	53	)	)	PUNCT
ejde-92	142	1	ds	ds	AUX
ejde-92	142	2	∣∣∣	∣∣∣	ADJ
ejde-92	142	3	≤	≤	NUM
ejde-92	142	4	∫	∫	PROPN
ejde-92	142	5	1	1	NUM
ejde-92	142	6	0	0	NUM
ejde-92	142	7	|u+	|u+	NOUN
ejde-92	142	8	tsv	tsv	VERB
ejde-92	142	9	−	−	X
ejde-92	142	10	ũt|p−1	ũt|p−1	PROPN
ejde-92	142	11	ds	ds	ADJ
ejde-92	142	12	≤	≤	NOUN
ejde-92	142	13	c{|u|p−1	c{|u|p−1	VERB
ejde-92	142	14	+	+	CCONJ
ejde-92	142	15	|v|p−1	|v|p−1	NUM
ejde-92	142	16	+	+	CCONJ
ejde-92	142	17	‖u‖p−1	‖u‖p−1	VERB
ejde-92	142	18	p	p	X
ejde-92	143	1	+	+	CCONJ
ejde-92	143	2	‖v‖p−1	‖v‖p−1	ADJ
ejde-92	143	3	p	p	X
ejde-92	143	4	}	}	PUNCT
ejde-92	143	5	,	,	PUNCT
ejde-92	143	6	where	where	SCONJ
ejde-92	143	7	c	c	PROPN
ejde-92	143	8	>	>	X
ejde-92	143	9	0	0	PUNCT
ejde-92	143	10	is	be	AUX
ejde-92	143	11	a	a	DET
ejde-92	143	12	constant	constant	ADJ
ejde-92	143	13	only	only	ADV
ejde-92	143	14	depending	depend	VERB
ejde-92	143	15	on	on	ADP
ejde-92	143	16	p	p	PROPN
ejde-92	143	17	and	and	CCONJ
ejde-92	143	18	|x|	|x|	PROPN
ejde-92	143	19	.	.	PROPN
ejde-92	143	20	since∫	since∫	VERB
ejde-92	143	21	1	1	NUM
ejde-92	143	22	0	0	NUM
ejde-92	143	23	|u+	|u+	NOUN
ejde-92	143	24	tsv	tsv	VERB
ejde-92	143	25	−	−	NOUN
ejde-92	143	26	ũt|p−2(u+	ũt|p−2(u+	NOUN
ejde-92	143	27	tsv	tsv	NOUN
ejde-92	143	28	−	−	PROPN
ejde-92	143	29	ũt	ũt	SYM
ejde-92	143	30	)	)	PUNCT
ejde-92	143	31	ds→	ds→	VERB
ejde-92	143	32	|u−	|u−	NOUN
ejde-92	143	33	ũ|p−2(u−	ũ|p−2(u−	PROPN
ejde-92	143	34	ũ	ũ	PROPN
ejde-92	143	35	)	)	PUNCT
ejde-92	143	36	,	,	PUNCT
ejde-92	143	37	a.e	a.e	PROPN
ejde-92	143	38	.	.	PROPN
ejde-92	144	1	in	in	ADP
ejde-92	144	2	x	x	PUNCT
ejde-92	144	3	as	as	ADP
ejde-92	144	4	t→	t→	X
ejde-92	144	5	0	0	NUM
ejde-92	144	6	,	,	PUNCT
ejde-92	144	7	the	the	DET
ejde-92	144	8	lebesgue	lebesgue	NOUN
ejde-92	144	9	dominated	dominate	VERB
ejde-92	144	10	convergence	convergence	NOUN
ejde-92	144	11	theorem	theorem	NOUN
ejde-92	144	12	implies	imply	VERB
ejde-92	144	13	that	that	SCONJ
ejde-92	144	14	lim	lim	PROPN
ejde-92	144	15	t→0	t→0	ADP
ejde-92	144	16	1	1	NUM
ejde-92	144	17	t	t	PROPN
ejde-92	144	18	(	(	PUNCT
ejde-92	144	19	ft(ũt)−	ft(ũt)−	PROPN
ejde-92	144	20	f0(ũt	f0(ũt	PROPN
ejde-92	144	21	)	)	PUNCT
ejde-92	144	22	)	)	PUNCT
ejde-92	145	1	=	=	PUNCT
ejde-92	146	1	p	p	X
ejde-92	146	2	∫	∫	PROPN
ejde-92	146	3	x	x	PUNCT
ejde-92	146	4	|u−	|u−	PROPN
ejde-92	146	5	ũ|p−2(u−	ũ|p−2(u−	PROPN
ejde-92	146	6	ũ)v	ũ)v	INTJ
ejde-92	146	7	dµ.	dµ.	VERB
ejde-92	146	8	an	an	DET
ejde-92	146	9	identical	identical	ADJ
ejde-92	146	10	argument	argument	NOUN
ejde-92	146	11	shows	show	VERB
ejde-92	146	12	that	that	SCONJ
ejde-92	146	13	lim	lim	PROPN
ejde-92	146	14	t→0	t→0	ADP
ejde-92	146	15	1	1	NUM
ejde-92	146	16	t	t	NOUN
ejde-92	146	17	(	(	PUNCT
ejde-92	146	18	ft(ũ)−	ft(ũ)−	PROPN
ejde-92	146	19	f0(ũ	f0(ũ	PROPN
ejde-92	146	20	)	)	PUNCT
ejde-92	146	21	)	)	PUNCT
ejde-92	147	1	=	=	PUNCT
ejde-92	148	1	p	p	X
ejde-92	148	2	∫	∫	PROPN
ejde-92	148	3	x	x	PUNCT
ejde-92	148	4	|u−	|u−	PROPN
ejde-92	148	5	ũ|p−2(u−	ũ|p−2(u−	PROPN
ejde-92	148	6	ũ)v	ũ)v	PROPN
ejde-92	148	7	dµ	dµ	PROPN
ejde-92	148	8	,	,	PUNCT
ejde-92	148	9	and	and	CCONJ
ejde-92	148	10	the	the	DET
ejde-92	148	11	desired	desire	VERB
ejde-92	148	12	result	result	NOUN
ejde-92	148	13	follows	follow	VERB
ejde-92	148	14	from	from	ADP
ejde-92	148	15	dividing	divide	VERB
ejde-92	148	16	the	the	DET
ejde-92	148	17	three	three	NUM
ejde-92	148	18	terms	term	NOUN
ejde-92	148	19	in	in	ADP
ejde-92	148	20	(	(	PUNCT
ejde-92	148	21	2.4	2.4	NUM
ejde-92	148	22	)	)	PUNCT
ejde-92	148	23	by	by	ADP
ejde-92	148	24	t	t	PROPN
ejde-92	148	25	6=	6=	ADP
ejde-92	148	26	0	0	NUM
ejde-92	148	27	and	and	CCONJ
ejde-92	148	28	passing	pass	VERB
ejde-92	148	29	to	to	ADP
ejde-92	148	30	the	the	DET
ejde-92	148	31	limit	limit	NOUN
ejde-92	148	32	as	as	ADP
ejde-92	148	33	t→	t→	X
ejde-92	148	34	0	0	X
ejde-92	148	35	.	.	PUNCT
ejde-92	148	36	�	�	PROPN
ejde-92	148	37	proof	proof	NOUN
ejde-92	148	38	of	of	ADP
ejde-92	148	39	theorem	theorem	ADJ
ejde-92	148	40	2.4	2.4	NUM
ejde-92	148	41	.	.	PUNCT
ejde-92	149	1	the	the	DET
ejde-92	149	2	gâteaux	gâteaux	ADJ
ejde-92	149	3	derivative	derivative	ADJ
ejde-92	149	4	dvp(u	dvp(u	PROPN
ejde-92	149	5	,	,	PUNCT
ejde-92	149	6	v	v	NOUN
ejde-92	149	7	)	)	PUNCT
ejde-92	149	8	is	be	AUX
ejde-92	149	9	linear	linear	ADJ
ejde-92	149	10	continuous	continuous	ADJ
ejde-92	149	11	in	in	ADP
ejde-92	149	12	v	v	NUM
ejde-92	149	13	∈	∈	PROPN
ejde-92	149	14	lp(x,µ	lp(x,µ	NOUN
ejde-92	149	15	)	)	PUNCT
ejde-92	149	16	for	for	ADP
ejde-92	149	17	u	u	PRON
ejde-92	149	18	fixed	fix	VERB
ejde-92	149	19	.	.	PUNCT
ejde-92	150	1	on	on	ADP
ejde-92	150	2	the	the	DET
ejde-92	150	3	other	other	ADJ
ejde-92	150	4	hand	hand	NOUN
ejde-92	150	5	,	,	PUNCT
ejde-92	150	6	mapping	map	VERB
ejde-92	150	7	u→	u→	PROPN
ejde-92	150	8	dvp(u	dvp(u	PROPN
ejde-92	150	9	,	,	PUNCT
ejde-92	150	10	·	·	PUNCT
ejde-92	150	11	)	)	PUNCT
ejde-92	150	12	regarded	regard	VERB
ejde-92	150	13	as	as	ADP
ejde-92	150	14	taking	take	VERB
ejde-92	150	15	values	value	NOUN
ejde-92	150	16	in	in	ADP
ejde-92	150	17	(	(	PUNCT
ejde-92	150	18	lp(x,µ))∗	lp(x,µ))∗	NOUN
ejde-92	150	19	=	=	SYM
ejde-92	150	20	lp	lp	NOUN
ejde-92	151	1	′	′	NUM
ejde-92	151	2	(	(	PUNCT
ejde-92	151	3	x,µ	x,µ	NOUN
ejde-92	151	4	)	)	PUNCT
ejde-92	151	5	is	be	AUX
ejde-92	151	6	continuous	continuous	ADJ
ejde-92	151	7	.	.	PUNCT
ejde-92	152	1	this	this	PRON
ejde-92	152	2	entails	entail	VERB
ejde-92	152	3	that	that	SCONJ
ejde-92	152	4	vp	vp	PROPN
ejde-92	152	5	is	be	AUX
ejde-92	152	6	fréchet	fréchet	VERB
ejde-92	152	7	differentiable	differentiable	ADJ
ejde-92	152	8	at	at	ADP
ejde-92	152	9	u	u	NOUN
ejde-92	152	10	(	(	PUNCT
ejde-92	152	11	see	see	VERB
ejde-92	152	12	for	for	ADP
ejde-92	152	13	instance	instance	NOUN
ejde-92	152	14	[	[	X
ejde-92	152	15	1	1	NUM
ejde-92	152	16	,	,	PUNCT
ejde-92	152	17	chapter	chapter	NOUN
ejde-92	152	18	1	1	NUM
ejde-92	152	19	]	]	PUNCT
ejde-92	152	20	)	)	PUNCT
ejde-92	152	21	.	.	PUNCT
ejde-92	153	1	�	�	PROPN
ejde-92	153	2	we	we	PRON
ejde-92	153	3	next	next	ADV
ejde-92	153	4	single	single	VERB
ejde-92	153	5	out	out	ADP
ejde-92	153	6	two	two	NUM
ejde-92	153	7	special	special	ADJ
ejde-92	153	8	cases	case	NOUN
ejde-92	153	9	where	where	SCONJ
ejde-92	153	10	theorem	theorem	ADJ
ejde-92	153	11	2.4	2.4	NUM
ejde-92	153	12	is	be	AUX
ejde-92	153	13	applied	apply	VERB
ejde-92	153	14	.	.	PUNCT
ejde-92	154	1	in	in	ADP
ejde-92	154	2	the	the	DET
ejde-92	154	3	first	first	ADJ
ejde-92	154	4	one	one	NUM
ejde-92	154	5	,	,	PUNCT
ejde-92	154	6	x	x	PUNCT
ejde-92	154	7	=	=	SYM
ejde-92	154	8	ω	ω	PROPN
ejde-92	154	9	is	be	AUX
ejde-92	154	10	a	a	DET
ejde-92	154	11	bounded	bounded	ADJ
ejde-92	154	12	set	set	NOUN
ejde-92	154	13	of	of	ADP
ejde-92	154	14	rn	rn	PROPN
ejde-92	154	15	,	,	PUNCT
ejde-92	154	16	endowed	endow	VERB
ejde-92	154	17	with	with	ADP
ejde-92	154	18	the	the	DET
ejde-92	154	19	measure	measure	NOUN
ejde-92	154	20	dµ	dµ	VERB
ejde-92	154	21	=	=	NOUN
ejde-92	154	22	m(x)dx	m(x)dx	NOUN
ejde-92	154	23	where	where	SCONJ
ejde-92	154	24	m	m	VERB
ejde-92	154	25	∈	∈	PROPN
ejde-92	154	26	l1(ω	l1(ω	PROPN
ejde-92	154	27	)	)	PUNCT
ejde-92	154	28	,	,	PUNCT
ejde-92	154	29	m(x	m(x	PROPN
ejde-92	154	30	)	)	PUNCT
ejde-92	154	31	>	>	X
ejde-92	154	32	0	0	NUM
ejde-92	155	1	a.e	a.e	PROPN
ejde-92	155	2	.	.	PROPN
ejde-92	155	3	in	in	ADP
ejde-92	155	4	ω	ω	PROPN
ejde-92	155	5	.	.	PUNCT
ejde-92	156	1	corollary	corollary	ADJ
ejde-92	156	2	2.6	2.6	NUM
ejde-92	156	3	.	.	PUNCT
ejde-92	157	1	let	let	VERB
ejde-92	157	2	vp	vp	PROPN
ejde-92	157	3	be	be	AUX
ejde-92	157	4	the	the	DET
ejde-92	157	5	variance	variance	NOUN
ejde-92	157	6	functional	functional	ADJ
ejde-92	157	7	defined	define	VERB
ejde-92	157	8	in	in	ADP
ejde-92	157	9	lp(ω	lp(ω	PROPN
ejde-92	157	10	,	,	PUNCT
ejde-92	157	11	mdx	mdx	NOUN
ejde-92	157	12	)	)	PUNCT
ejde-92	157	13	.	.	PUNCT
ejde-92	158	1	then	then	ADV
ejde-92	158	2	,	,	PUNCT
ejde-92	158	3	at	at	ADP
ejde-92	158	4	any	any	DET
ejde-92	158	5	u	u	NOUN
ejde-92	158	6	∈	∈	PROPN
ejde-92	158	7	lp(ω	lp(ω	PROPN
ejde-92	158	8	,	,	PUNCT
ejde-92	158	9	mdx	mdx	PROPN
ejde-92	158	10	)	)	PUNCT
ejde-92	158	11	we	we	PRON
ejde-92	158	12	have	have	VERB
ejde-92	158	13	〈	〈	PROPN
ejde-92	158	14	dvp(u	dvp(u	X
ejde-92	158	15	)	)	PUNCT
ejde-92	158	16	,	,	PUNCT
ejde-92	158	17	v	v	NOUN
ejde-92	158	18	〉	〉	NOUN
ejde-92	158	19	=	=	SYM
ejde-92	159	1	p	p	PROPN
ejde-92	159	2	∫	∫	PROPN
ejde-92	159	3	ω	ω	PROPN
ejde-92	159	4	|u−	|u−	PROPN
ejde-92	159	5	ũ|p−2(u−	ũ|p−2(u−	PROPN
ejde-92	159	6	ũ)vmdx	ũ)vmdx	NOUN
ejde-92	159	7	,	,	PUNCT
ejde-92	159	8	v	v	ADP
ejde-92	159	9	∈	∈	PROPN
ejde-92	159	10	lp(ω	lp(ω	X
ejde-92	159	11	,	,	PUNCT
ejde-92	159	12	mdx	mdx	NOUN
ejde-92	159	13	)	)	PUNCT
ejde-92	159	14	.	.	PUNCT
ejde-92	160	1	in	in	ADP
ejde-92	160	2	the	the	DET
ejde-92	160	3	second	second	ADJ
ejde-92	160	4	example	example	NOUN
ejde-92	160	5	x	x	PUNCT
ejde-92	160	6	is	be	AUX
ejde-92	160	7	a	a	DET
ejde-92	160	8	finite	finite	NOUN
ejde-92	160	9	set	set	VERB
ejde-92	160	10	v	v	NOUN
ejde-92	160	11	=	=	SYM
ejde-92	160	12	{	{	PUNCT
ejde-92	160	13	v1	v1	NOUN
ejde-92	160	14	,	,	PUNCT
ejde-92	160	15	.	.	PUNCT
ejde-92	160	16	.	.	PUNCT
ejde-92	161	1	.	.	PUNCT
ejde-92	162	1	,	,	PUNCT
ejde-92	162	2	vn	vn	PROPN
ejde-92	162	3	}	}	PUNCT
ejde-92	162	4	where	where	SCONJ
ejde-92	162	5	the	the	DET
ejde-92	162	6	measure	measure	NOUN
ejde-92	162	7	is	be	AUX
ejde-92	162	8	µ	µ	X
ejde-92	162	9	=	=	SYM
ejde-92	162	10	n∑	n∑	PROPN
ejde-92	162	11	i=1	i=1	PROPN
ejde-92	162	12	νiδ(x−	νiδ(x−	X
ejde-92	162	13	vi	vi	PROPN
ejde-92	162	14	)	)	PUNCT
ejde-92	162	15	,	,	PUNCT
ejde-92	162	16	while	while	SCONJ
ejde-92	162	17	δ	δ	PROPN
ejde-92	162	18	is	be	AUX
ejde-92	162	19	the	the	DET
ejde-92	162	20	dirac	dirac	NOUN
ejde-92	162	21	’s	’s	PART
ejde-92	162	22	delta	delta	NOUN
ejde-92	162	23	and	and	CCONJ
ejde-92	162	24	νi	νi	PRON
ejde-92	162	25	>	>	X
ejde-92	162	26	0	0	NUM
ejde-92	162	27	,	,	PUNCT
ejde-92	162	28	1	1	NUM
ejde-92	162	29	≤	≤	NUM
ejde-92	162	30	i	i	PRON
ejde-92	162	31	≤	≤	NUM
ejde-92	162	32	n.	n.	NOUN
ejde-92	162	33	functions	function	NOUN
ejde-92	163	1	f	f	X
ejde-92	163	2	:	:	PUNCT
ejde-92	163	3	v	v	X
ejde-92	163	4	→	→	SYM
ejde-92	163	5	r	r	NOUN
ejde-92	163	6	are	be	AUX
ejde-92	163	7	identified	identify	VERB
ejde-92	163	8	to	to	ADP
ejde-92	163	9	vectors	vector	NOUN
ejde-92	163	10	in	in	ADP
ejde-92	163	11	rn	rn	PROPN
ejde-92	163	12	by	by	ADP
ejde-92	163	13	means	mean	NOUN
ejde-92	163	14	of	of	ADP
ejde-92	163	15	the	the	DET
ejde-92	163	16	expression	expression	NOUN
ejde-92	163	17	f	f	NOUN
ejde-92	163	18	=	=	PUNCT
ejde-92	163	19	(	(	PUNCT
ejde-92	163	20	fi	fi	NOUN
ejde-92	163	21	)	)	PUNCT
ejde-92	163	22	with	with	ADP
ejde-92	163	23	fi	fi	NOUN
ejde-92	163	24	=	=	SYM
ejde-92	163	25	f(i	f(i	PROPN
ejde-92	163	26	)	)	PUNCT
ejde-92	163	27	.	.	PUNCT
ejde-92	164	1	then	then	ADV
ejde-92	164	2	vp(f	vp(f	PUNCT
ejde-92	164	3	)	)	PUNCT
ejde-92	164	4	=	=	PUNCT
ejde-92	164	5	∑	∑	PUNCT
ejde-92	164	6	i	i	PRON
ejde-92	164	7	|fi	|fi	X
ejde-92	164	8	−	−	PROPN
ejde-92	164	9	f̃	f̃	PROPN
ejde-92	164	10	|pνi	|pνi	PROPN
ejde-92	164	11	=	=	PROPN
ejde-92	164	12	inf	inf	NOUN
ejde-92	164	13	t∈r	t∈r	NOUN
ejde-92	164	14	∑	∑	PUNCT
ejde-92	164	15	i	i	PRON
ejde-92	164	16	|fi	|fi	X
ejde-92	164	17	−	−	X
ejde-92	164	18	t|pνi	t|pνi	NOUN
ejde-92	164	19	,	,	PUNCT
ejde-92	164	20	where	where	SCONJ
ejde-92	164	21	t	t	NOUN
ejde-92	164	22	=	=	SYM
ejde-92	164	23	f̃	f̃	PROPN
ejde-92	164	24	is	be	AUX
ejde-92	164	25	the	the	DET
ejde-92	164	26	unique	unique	ADJ
ejde-92	164	27	number	number	NOUN
ejde-92	164	28	so	so	SCONJ
ejde-92	164	29	that	that	SCONJ
ejde-92	164	30	∑	∑	ADP
ejde-92	164	31	i	i	PRON
ejde-92	164	32	|fi	|fi	ADP
ejde-92	164	33	−	−	PROPN
ejde-92	164	34	t|p−2(fi	t|p−2(fi	NOUN
ejde-92	164	35	−	−	PUNCT
ejde-92	165	1	t)νi	t)νi	PROPN
ejde-92	165	2	=	=	SYM
ejde-92	165	3	0	0	X
ejde-92	165	4	.	.	X
ejde-92	165	5	ejde-2022/13	ejde-2022/13	ADJ
ejde-92	165	6	second	second	ADJ
ejde-92	165	7	neumann	neumann	PROPN
ejde-92	165	8	eigenvalue	eigenvalue	PROPN
ejde-92	165	9	7	7	NUM
ejde-92	165	10	corollary	corollary	ADJ
ejde-92	165	11	2.7	2.7	NUM
ejde-92	165	12	.	.	PUNCT
ejde-92	166	1	function	function	NOUN
ejde-92	166	2	vp	vp	PROPN
ejde-92	166	3	is	be	AUX
ejde-92	166	4	differentiable	differentiable	ADJ
ejde-92	166	5	at	at	ADP
ejde-92	166	6	any	any	DET
ejde-92	166	7	f	f	PROPN
ejde-92	166	8	∈	∈	PROPN
ejde-92	166	9	rn	rn	PROPN
ejde-92	166	10	and	and	CCONJ
ejde-92	166	11	〈	〈	PROPN
ejde-92	166	12	dvp(f	dvp(f	PROPN
ejde-92	166	13	)	)	PUNCT
ejde-92	166	14	,	,	PUNCT
ejde-92	166	15	g	g	PROPN
ejde-92	166	16	〉	〉	NUM
ejde-92	166	17	=	=	SYM
ejde-92	167	1	p	p	NOUN
ejde-92	167	2	∑	∑	PUNCT
ejde-92	167	3	i	i	PRON
ejde-92	167	4	|fi	|fi	X
ejde-92	167	5	−	−	PROPN
ejde-92	167	6	f̃	f̃	PROPN
ejde-92	167	7	|p−2(fi	|p−2(fi	NUM
ejde-92	167	8	−	−	PROPN
ejde-92	167	9	f̃)giνi	f̃)giνi	NOUN
ejde-92	167	10	,	,	PUNCT
ejde-92	167	11	g	g	PROPN
ejde-92	167	12	∈	∈	PROPN
ejde-92	167	13	rn	rn	PROPN
ejde-92	167	14	.	.	PROPN
ejde-92	168	1	(	(	PUNCT
ejde-92	168	2	2.5	2.5	NUM
ejde-92	168	3	)	)	PUNCT
ejde-92	168	4	3	3	NUM
ejde-92	168	5	.	.	X
ejde-92	169	1	proof	proof	NOUN
ejde-92	169	2	of	of	ADP
ejde-92	169	3	theorem	theorem	ADJ
ejde-92	169	4	1.1	1.1	NUM
ejde-92	169	5	in	in	ADP
ejde-92	169	6	this	this	DET
ejde-92	169	7	section	section	NOUN
ejde-92	169	8	we	we	PRON
ejde-92	169	9	prove	prove	VERB
ejde-92	169	10	a	a	DET
ejde-92	169	11	slightly	slightly	ADV
ejde-92	169	12	more	more	ADJ
ejde-92	169	13	general	general	ADJ
ejde-92	169	14	version	version	NOUN
ejde-92	169	15	of	of	ADP
ejde-92	169	16	theorem	theorem	ADJ
ejde-92	169	17	1.1	1.1	NUM
ejde-92	169	18	that	that	PRON
ejde-92	169	19	can	can	AUX
ejde-92	169	20	be	be	AUX
ejde-92	169	21	stated	state	VERB
ejde-92	169	22	as	as	SCONJ
ejde-92	169	23	follows	follow	NOUN
ejde-92	169	24	.	.	PUNCT
ejde-92	170	1	theorem	theorem	ADJ
ejde-92	170	2	3.1	3.1	NUM
ejde-92	170	3	.	.	PUNCT
ejde-92	171	1	assume	assume	VERB
ejde-92	171	2	that	that	SCONJ
ejde-92	171	3	ω	ω	PROPN
ejde-92	171	4	⊂	⊂	PROPN
ejde-92	171	5	rn	rn	PROPN
ejde-92	171	6	is	be	AUX
ejde-92	171	7	class	class	NOUN
ejde-92	171	8	c0,1	c0,1	PROPN
ejde-92	171	9	bounded	bounded	ADJ
ejde-92	171	10	domain	domain	NOUN
ejde-92	171	11	,	,	PUNCT
ejde-92	171	12	m	m	NOUN
ejde-92	171	13	∈	∈	NOUN
ejde-92	171	14	lr(ω	lr(ω	NOUN
ejde-92	171	15	)	)	PUNCT
ejde-92	171	16	,	,	PUNCT
ejde-92	171	17	m(x	m(x	PROPN
ejde-92	171	18	)	)	PUNCT
ejde-92	171	19	>	>	X
ejde-92	171	20	0	0	NUM
ejde-92	172	1	a.e	a.e	PROPN
ejde-92	172	2	.	.	PROPN
ejde-92	173	1	in	in	ADP
ejde-92	173	2	ω	ω	PROPN
ejde-92	173	3	,	,	PUNCT
ejde-92	173	4	where	where	SCONJ
ejde-92	173	5	r	r	NOUN
ejde-92	173	6			PROPN
ejde-92	173	7	>	>	X
ejde-92	173	8	(	(	PUNCT
ejde-92	173	9	p∗	p∗	PROPN
ejde-92	173	10	p	p	NOUN
ejde-92	173	11	)	)	PUNCT
ejde-92	173	12	′	′	NUM
ejde-92	174	1	=	=	PUNCT
ejde-92	174	2	np	np	INTJ
ejde-92	174	3	if	if	SCONJ
ejde-92	174	4	1	1	NUM
ejde-92	174	5	<	<	X
ejde-92	174	6	p	p	X
ejde-92	174	7	<	<	X
ejde-92	174	8	n	n	CCONJ
ejde-92	174	9	,	,	PUNCT
ejde-92	174	10	>	>	X
ejde-92	174	11	1	1	NUM
ejde-92	174	12	if	if	SCONJ
ejde-92	174	13	p	p	NOUN
ejde-92	174	14	=	=	NOUN
ejde-92	174	15	n	n	CCONJ
ejde-92	174	16	,	,	PUNCT
ejde-92	174	17	=	=	NOUN
ejde-92	174	18	1	1	NUM
ejde-92	174	19	if	if	SCONJ
ejde-92	174	20	p	p	PROPN
ejde-92	174	21	>	>	X
ejde-92	174	22	n.	n.	NOUN
ejde-92	174	23	we	we	PRON
ejde-92	174	24	define	define	VERB
ejde-92	174	25	λ̂(m	λ̂(m	ADP
ejde-92	174	26	)	)	PUNCT
ejde-92	174	27	=	=	SYM
ejde-92	175	1	inf	inf	NOUN
ejde-92	175	2	u∈m0\{0	u∈m0\{0	PROPN
ejde-92	175	3	}	}	PUNCT
ejde-92	175	4	∫	∫	PROPN
ejde-92	175	5	ω	ω	PROPN
ejde-92	175	6	|∇u|p	|∇u|p	PROPN
ejde-92	175	7	dx∫	dx∫	PROPN
ejde-92	175	8	ω	ω	PROPN
ejde-92	175	9	|u|pmdx	|u|pmdx	NOUN
ejde-92	175	10	,	,	PUNCT
ejde-92	175	11	(	(	PUNCT
ejde-92	175	12	3.1	3.1	NUM
ejde-92	175	13	)	)	PUNCT
ejde-92	175	14	with	with	ADP
ejde-92	175	15	m0	m0	PROPN
ejde-92	175	16	=	=	PUNCT
ejde-92	175	17	{	{	PUNCT
ejde-92	175	18	u	u	NOUN
ejde-92	175	19	∈	∈	PROPN
ejde-92	175	20	w	w	PROPN
ejde-92	175	21	1,p(ω	1,p(ω	NUM
ejde-92	175	22	)	)	PUNCT
ejde-92	175	23	:	:	PUNCT
ejde-92	176	1	∫	∫	PROPN
ejde-92	176	2	ω	ω	PROPN
ejde-92	176	3	|u|p−2umdx	|u|p−2umdx	PUNCT
ejde-92	176	4	=	=	NOUN
ejde-92	176	5	0	0	NUM
ejde-92	176	6	}	}	PUNCT
ejde-92	176	7	.	.	PUNCT
ejde-92	177	1	then	then	ADV
ejde-92	177	2	eigenvalue	eigenvalue	VERB
ejde-92	177	3	problem	problem	NOUN
ejde-92	177	4	(	(	PUNCT
ejde-92	177	5	1.12	1.12	NUM
ejde-92	177	6	)	)	PUNCT
ejde-92	177	7	satisfies	satisfy	VERB
ejde-92	177	8	the	the	DET
ejde-92	177	9	assertions	assertion	NOUN
ejde-92	177	10	(	(	PUNCT
ejde-92	177	11	i)–(iii	i)–(iii	NOUN
ejde-92	177	12	)	)	PUNCT
ejde-92	177	13	in	in	ADP
ejde-92	177	14	theorem	theorem	ADJ
ejde-92	177	15	1.1	1.1	NUM
ejde-92	177	16	with	with	ADP
ejde-92	177	17	λ2	λ2	NOUN
ejde-92	177	18	replaced	replace	VERB
ejde-92	177	19	by	by	ADP
ejde-92	177	20	λ2(m	λ2(m	PROPN
ejde-92	177	21	)	)	PUNCT
ejde-92	177	22	,	,	PUNCT
ejde-92	177	23	the	the	DET
ejde-92	177	24	second	second	ADJ
ejde-92	177	25	eigenvalue	eigenvalue	NOUN
ejde-92	177	26	to	to	ADP
ejde-92	177	27	(	(	PUNCT
ejde-92	177	28	1.12	1.12	NUM
ejde-92	177	29	)	)	PUNCT
ejde-92	177	30	.	.	PUNCT
ejde-92	178	1	proof	proof	NOUN
ejde-92	178	2	.	.	PUNCT
ejde-92	179	1	to	to	PART
ejde-92	179	2	show	show	VERB
ejde-92	179	3	theorem	theorem	VERB
ejde-92	179	4	3.1	3.1	NUM
ejde-92	179	5	for	for	ADP
ejde-92	179	6	problem	problem	NOUN
ejde-92	179	7	(	(	PUNCT
ejde-92	179	8	1.12	1.12	NUM
ejde-92	179	9	)	)	PUNCT
ejde-92	179	10	we	we	PRON
ejde-92	179	11	first	first	ADV
ejde-92	179	12	notice	notice	VERB
ejde-92	179	13	that	that	SCONJ
ejde-92	179	14	λ̂(m	λ̂(m	ADP
ejde-92	179	15	)	)	PUNCT
ejde-92	179	16	=	=	SYM
ejde-92	179	17	inf	inf	NOUN
ejde-92	179	18	u∈m0\{0	u∈m0\{0	PROPN
ejde-92	179	19	}	}	PUNCT
ejde-92	179	20	∫	∫	PROPN
ejde-92	179	21	ω	ω	PROPN
ejde-92	179	22	|∇u|p	|∇u|p	PROPN
ejde-92	179	23	dx∫	dx∫	PROPN
ejde-92	179	24	ω	ω	PROPN
ejde-92	179	25	|u|pmdx	|u|pmdx	NOUN
ejde-92	179	26	=	=	PROPN
ejde-92	179	27	inf	inf	PROPN
ejde-92	179	28	u∈m1	u∈m1	PROPN
ejde-92	179	29	j	j	PROPN
ejde-92	179	30	(	(	PUNCT
ejde-92	179	31	u	u	NOUN
ejde-92	179	32	)	)	PUNCT
ejde-92	179	33	,	,	PUNCT
ejde-92	179	34	where	where	SCONJ
ejde-92	179	35	j	j	PROPN
ejde-92	179	36	(	(	PUNCT
ejde-92	179	37	u	u	NOUN
ejde-92	179	38	)	)	PUNCT
ejde-92	179	39	=	=	SYM
ejde-92	179	40	∫	∫	PROPN
ejde-92	179	41	ω	ω	PROPN
ejde-92	179	42	|∇u|p	|∇u|p	PROPN
ejde-92	179	43	dx	dx	PROPN
ejde-92	179	44	,	,	PUNCT
ejde-92	179	45	and	and	CCONJ
ejde-92	179	46	m0	m0	PROPN
ejde-92	179	47	=	=	PUNCT
ejde-92	179	48	{	{	PUNCT
ejde-92	179	49	u	u	NOUN
ejde-92	179	50	∈w	∈w	NOUN
ejde-92	179	51	1,p(ω	1,p(ω	NUM
ejde-92	179	52	)	)	PUNCT
ejde-92	179	53	:	:	PUNCT
ejde-92	180	1	∫	∫	PROPN
ejde-92	180	2	ω	ω	PROPN
ejde-92	180	3	|u|p−2u	|u|p−2u	PROPN
ejde-92	180	4	mdx	mdx	PROPN
ejde-92	180	5	=	=	PROPN
ejde-92	180	6	0	0	NUM
ejde-92	180	7	}	}	PUNCT
ejde-92	180	8	,	,	PUNCT
ejde-92	180	9	m1	m1	PROPN
ejde-92	180	10	=	=	SYM
ejde-92	180	11	m0	m0	PROPN
ejde-92	180	12	∩	∩	NOUN
ejde-92	180	13	{	{	PUNCT
ejde-92	180	14	∫	∫	PROPN
ejde-92	180	15	ω	ω	PROPN
ejde-92	180	16	|u|pmdx	|u|pmdx	NOUN
ejde-92	180	17	=	=	NOUN
ejde-92	180	18	1	1	NUM
ejde-92	180	19	}	}	PUNCT
ejde-92	180	20	.	.	PUNCT
ejde-92	181	1	functional	functional	ADJ
ejde-92	181	2	j	j	PROPN
ejde-92	181	3	is	be	AUX
ejde-92	181	4	coercive	coercive	ADJ
ejde-92	181	5	,	,	PUNCT
ejde-92	181	6	i.e.	i.e.	X
ejde-92	181	7	j	j	PROPN
ejde-92	181	8	(	(	PUNCT
ejde-92	181	9	u	u	NOUN
ejde-92	181	10	)	)	PUNCT
ejde-92	181	11	→	→	SYM
ejde-92	181	12	∞	∞	PROPN
ejde-92	181	13	as	as	ADP
ejde-92	181	14	‖u‖w	‖u‖w	NOUN
ejde-92	181	15	1,p(ω	1,p(ω	NUM
ejde-92	181	16	)	)	PUNCT
ejde-92	181	17	→	→	SYM
ejde-92	181	18	∞	∞	PROPN
ejde-92	181	19	,	,	PUNCT
ejde-92	181	20	and	and	CCONJ
ejde-92	181	21	weakly	weakly	ADV
ejde-92	181	22	lower	low	ADJ
ejde-92	181	23	semicontinuous	semicontinuous	ADJ
ejde-92	181	24	,	,	PUNCT
ejde-92	181	25	while	while	SCONJ
ejde-92	181	26	the	the	DET
ejde-92	181	27	election	election	NOUN
ejde-92	181	28	of	of	ADP
ejde-92	181	29	r	r	NOUN
ejde-92	181	30	entails	entail	VERB
ejde-92	181	31	thatm1	thatm1	NOUN
ejde-92	181	32	is	be	AUX
ejde-92	181	33	weakly	weakly	ADV
ejde-92	181	34	closed	closed	ADJ
ejde-92	181	35	in	in	ADP
ejde-92	181	36	w	w	PROPN
ejde-92	181	37	1,p(ω	1,p(ω	NUM
ejde-92	181	38	)	)	PUNCT
ejde-92	181	39	.	.	PUNCT
ejde-92	182	1	thus	thus	ADV
ejde-92	182	2	,	,	PUNCT
ejde-92	182	3	a	a	DET
ejde-92	182	4	well	well	ADV
ejde-92	182	5	-	-	PUNCT
ejde-92	182	6	known	know	VERB
ejde-92	182	7	result	result	NOUN
ejde-92	182	8	in	in	ADP
ejde-92	182	9	calculus	calculus	NOUN
ejde-92	182	10	of	of	ADP
ejde-92	182	11	variations	variation	NOUN
ejde-92	182	12	(	(	PUNCT
ejde-92	182	13	[	[	X
ejde-92	182	14	19	19	NUM
ejde-92	182	15	,	,	PUNCT
ejde-92	182	16	chapter	chapter	NOUN
ejde-92	182	17	i	i	PROPN
ejde-92	182	18	]	]	X
ejde-92	182	19	)	)	PUNCT
ejde-92	182	20	ensures	ensure	VERB
ejde-92	182	21	us	we	PRON
ejde-92	182	22	the	the	DET
ejde-92	182	23	existence	existence	NOUN
ejde-92	182	24	of	of	ADP
ejde-92	182	25	a	a	DET
ejde-92	182	26	global	global	ADJ
ejde-92	182	27	minimizer	minimizer	NOUN
ejde-92	182	28	u1	u1	NOUN
ejde-92	182	29	∈m1	∈m1	NOUN
ejde-92	182	30	.	.	PUNCT
ejde-92	183	1	hence	hence	ADV
ejde-92	183	2	0	0	NUM
ejde-92	183	3	<	<	X
ejde-92	183	4	λ̂(m	λ̂(m	NOUN
ejde-92	183	5	)	)	PUNCT
ejde-92	183	6	=	=	SYM
ejde-92	183	7	j	j	PROPN
ejde-92	183	8	(	(	PUNCT
ejde-92	183	9	u1	u1	PROPN
ejde-92	183	10	)	)	PUNCT
ejde-92	183	11	=	=	SYM
ejde-92	183	12	inf	inf	PROPN
ejde-92	183	13	u∈m0\{0	u∈m0\{0	PROPN
ejde-92	183	14	}	}	PUNCT
ejde-92	183	15	∫	∫	PROPN
ejde-92	183	16	ω	ω	PROPN
ejde-92	183	17	|∇u|p	|∇u|p	PROPN
ejde-92	183	18	dx∫	dx∫	PROPN
ejde-92	183	19	ω	ω	PROPN
ejde-92	183	20	|u|p	|u|p	PROPN
ejde-92	183	21	mdx	mdx	NOUN
ejde-92	183	22	.	.	PUNCT
ejde-92	184	1	since	since	SCONJ
ejde-92	184	2	every	every	DET
ejde-92	184	3	possible	possible	ADJ
ejde-92	184	4	eigenfunction	eigenfunction	NOUN
ejde-92	184	5	u	u	NOUN
ejde-92	184	6	associated	associate	VERB
ejde-92	184	7	with	with	ADP
ejde-92	184	8	a	a	DET
ejde-92	184	9	nonzero	nonzero	NOUN
ejde-92	184	10	eigenvalue	eigenvalue	ADJ
ejde-92	184	11	λ	λ	NOUN
ejde-92	184	12	to	to	ADP
ejde-92	184	13	(	(	PUNCT
ejde-92	184	14	1.12	1.12	NUM
ejde-92	184	15	)	)	PUNCT
ejde-92	184	16	lies	lie	NOUN
ejde-92	184	17	in	in	ADP
ejde-92	184	18	m0	m0	NOUN
ejde-92	184	19	then	then	ADV
ejde-92	184	20	λ	λ	X
ejde-92	184	21	≥	≥	NOUN
ejde-92	184	22	λ̂(m	λ̂(m	ADP
ejde-92	184	23	)	)	PUNCT
ejde-92	184	24	.	.	PUNCT
ejde-92	185	1	thus	thus	ADV
ejde-92	185	2	the	the	DET
ejde-92	185	3	main	main	ADJ
ejde-92	185	4	feature	feature	NOUN
ejde-92	185	5	remains	remain	VERB
ejde-92	185	6	to	to	PART
ejde-92	185	7	be	be	AUX
ejde-92	185	8	proved	prove	VERB
ejde-92	185	9	.	.	PUNCT
ejde-92	186	1	namely	namely	ADV
ejde-92	186	2	that	that	PRON
ejde-92	186	3	λ̂(m	λ̂(m	ADP
ejde-92	186	4	)	)	PUNCT
ejde-92	186	5	is	be	AUX
ejde-92	186	6	actually	actually	ADV
ejde-92	186	7	an	an	DET
ejde-92	186	8	eigenvalue	eigenvalue	NOUN
ejde-92	186	9	.	.	PUNCT
ejde-92	187	1	the	the	DET
ejde-92	187	2	next	next	ADJ
ejde-92	187	3	proof	proof	NOUN
ejde-92	187	4	,	,	PUNCT
ejde-92	187	5	relying	rely	VERB
ejde-92	187	6	on	on	ADP
ejde-92	187	7	the	the	DET
ejde-92	187	8	properties	property	NOUN
ejde-92	187	9	of	of	ADP
ejde-92	187	10	the	the	DET
ejde-92	187	11	variance	variance	NOUN
ejde-92	187	12	functional	functional	ADJ
ejde-92	187	13	vp	vp	PROPN
ejde-92	187	14	presented	present	VERB
ejde-92	187	15	in	in	ADP
ejde-92	187	16	section	section	NOUN
ejde-92	187	17	2	2	NUM
ejde-92	187	18	,	,	PUNCT
ejde-92	187	19	can	can	AUX
ejde-92	187	20	be	be	AUX
ejde-92	187	21	applied	apply	VERB
ejde-92	187	22	to	to	PART
ejde-92	187	23	cover	cover	VERB
ejde-92	187	24	both	both	DET
ejde-92	187	25	cases	case	NOUN
ejde-92	187	26	p	p	X
ejde-92	187	27	≥	≥	NUM
ejde-92	187	28	2	2	NUM
ejde-92	187	29	and	and	CCONJ
ejde-92	187	30	1	1	NUM
ejde-92	187	31	<	<	X
ejde-92	187	32	p	p	X
ejde-92	187	33	<	<	X
ejde-92	187	34	2	2	NUM
ejde-92	187	35	.	.	PUNCT
ejde-92	187	36	first	first	ADV
ejde-92	187	37	observe	observe	VERB
ejde-92	187	38	that	that	SCONJ
ejde-92	187	39	m0	m0	NOUN
ejde-92	187	40	=	=	PUNCT
ejde-92	187	41	{	{	PUNCT
ejde-92	187	42	u−	u−	PROPN
ejde-92	187	43	ũ	ũ	PROPN
ejde-92	187	44	:	:	PUNCT
ejde-92	187	45	u	u	NOUN
ejde-92	187	46	∈w	∈w	VERB
ejde-92	187	47	1,p(ω	1,p(ω	NUM
ejde-92	187	48	)	)	PUNCT
ejde-92	187	49	}	}	PUNCT
ejde-92	187	50	,	,	PUNCT
ejde-92	187	51	where	where	SCONJ
ejde-92	187	52	the	the	DET
ejde-92	187	53	notation	notation	NOUN
ejde-92	187	54	of	of	ADP
ejde-92	187	55	section	section	NOUN
ejde-92	187	56	2	2	NUM
ejde-92	187	57	has	have	AUX
ejde-92	187	58	been	be	AUX
ejde-92	187	59	used	use	VERB
ejde-92	187	60	.	.	PUNCT
ejde-92	188	1	hence	hence	ADV
ejde-92	188	2	,	,	PUNCT
ejde-92	188	3	λ̂(m	λ̂(m	ADP
ejde-92	188	4	)	)	PUNCT
ejde-92	188	5	=	=	SYM
ejde-92	188	6	inf	inf	PROPN
ejde-92	188	7	v∈m0\{0	v∈m0\{0	PROPN
ejde-92	188	8	}	}	PUNCT
ejde-92	188	9	∫	∫	PROPN
ejde-92	188	10	ω	ω	PROPN
ejde-92	188	11	|∇v|p	|∇v|p	PROPN
ejde-92	188	12	dx∫	dx∫	PROPN
ejde-92	188	13	ω	ω	PROPN
ejde-92	188	14	|v|p	|v|p	PROPN
ejde-92	188	15	mdx	mdx	NOUN
ejde-92	188	16	=	=	PROPN
ejde-92	188	17	inf	inf	PROPN
ejde-92	188	18	u/∈span{1	u/∈span{1	NOUN
ejde-92	188	19	}	}	PUNCT
ejde-92	188	20	∫	∫	PROPN
ejde-92	188	21	ω	ω	PROPN
ejde-92	188	22	|∇u|p	|∇u|p	PROPN
ejde-92	188	23	dx∫	dx∫	PROPN
ejde-92	188	24	ω	ω	PROPN
ejde-92	188	25	|u−	|u−	PROPN
ejde-92	188	26	ũ|p	ũ|p	PROPN
ejde-92	188	27	mdx	mdx	NOUN
ejde-92	188	28	,	,	PUNCT
ejde-92	188	29	8	8	NUM
ejde-92	188	30	j.	j.	PROPN
ejde-92	188	31	c.	c.	PROPN
ejde-92	188	32	sabina	sabina	PROPN
ejde-92	188	33	de	de	PROPN
ejde-92	188	34	lis	lis	X
ejde-92	188	35	ejde-2022/13	ejde-2022/13	ADJ
ejde-92	188	36	since	since	SCONJ
ejde-92	188	37	m0	m0	PROPN
ejde-92	188	38	\	\	PROPN
ejde-92	188	39	{	{	PUNCT
ejde-92	188	40	0	0	NUM
ejde-92	188	41	}	}	PUNCT
ejde-92	188	42	=	=	PRON
ejde-92	188	43	{	{	PUNCT
ejde-92	188	44	u	u	NOUN
ejde-92	188	45	−	−	PROPN
ejde-92	188	46	ũ	ũ	PROPN
ejde-92	188	47	:	:	PUNCT
ejde-92	188	48	u	u	PROPN
ejde-92	188	49	∈	∈	PROPN
ejde-92	188	50	w	w	PROPN
ejde-92	188	51	1,p(ω	1,p(ω	NUM
ejde-92	188	52	)	)	PUNCT
ejde-92	188	53	,	,	PUNCT
ejde-92	188	54	u	u	NOUN
ejde-92	188	55	/∈	/∈	NOUN
ejde-92	188	56	span{1	span{1	NOUN
ejde-92	188	57	}	}	PUNCT
ejde-92	188	58	}	}	PUNCT
ejde-92	188	59	.	.	PUNCT
ejde-92	189	1	thus	thus	ADV
ejde-92	189	2	,	,	PUNCT
ejde-92	189	3	an	an	DET
ejde-92	189	4	alternative	alternative	ADJ
ejde-92	189	5	expression	expression	NOUN
ejde-92	189	6	for	for	ADP
ejde-92	189	7	λ̂(m	λ̂(m	ADP
ejde-92	189	8	)	)	PUNCT
ejde-92	189	9	reads	read	NOUN
ejde-92	189	10	as	as	SCONJ
ejde-92	189	11	follows	follow	VERB
ejde-92	189	12	λ̂(m	λ̂(m	ADP
ejde-92	189	13	)	)	PUNCT
ejde-92	189	14	=	=	SYM
ejde-92	189	15	inf	inf	PROPN
ejde-92	189	16	u/∈span{1	u/∈span{1	NOUN
ejde-92	189	17	}	}	PUNCT
ejde-92	189	18	∫	∫	PROPN
ejde-92	189	19	ω	ω	PROPN
ejde-92	189	20	|∇u|p	|∇u|p	PROPN
ejde-92	189	21	dx	dx	PROPN
ejde-92	189	22	vp(u	vp(u	VERB
ejde-92	189	23	)	)	PUNCT
ejde-92	189	24	.	.	PUNCT
ejde-92	190	1	next	next	ADV
ejde-92	190	2	we	we	PRON
ejde-92	190	3	observe	observe	VERB
ejde-92	190	4	that	that	SCONJ
ejde-92	190	5	the	the	DET
ejde-92	190	6	quotient	quotient	NOUN
ejde-92	190	7	can	can	AUX
ejde-92	190	8	be	be	AUX
ejde-92	190	9	differentiated	differentiate	VERB
ejde-92	190	10	regardless	regardless	ADV
ejde-92	190	11	the	the	DET
ejde-92	190	12	value	value	NOUN
ejde-92	190	13	of	of	ADP
ejde-92	190	14	p	p	PROPN
ejde-92	190	15	>	>	X
ejde-92	190	16	1	1	X
ejde-92	190	17	.	.	PUNCT
ejde-92	190	18	accordingly	accordingly	ADV
ejde-92	190	19	,	,	PUNCT
ejde-92	190	20	by	by	ADP
ejde-92	190	21	using	use	VERB
ejde-92	190	22	the	the	DET
ejde-92	190	23	expression	expression	NOUN
ejde-92	190	24	for	for	ADP
ejde-92	190	25	the	the	DET
ejde-92	190	26	gâteaux	gâteaux	ADJ
ejde-92	190	27	derivative	derivative	NOUN
ejde-92	190	28	of	of	ADP
ejde-92	190	29	vp	vp	PROPN
ejde-92	190	30	stated	state	VERB
ejde-92	190	31	in	in	ADP
ejde-92	190	32	lemma	lemma	PROPN
ejde-92	190	33	2.5	2.5	NUM
ejde-92	190	34	(	(	PUNCT
ejde-92	190	35	corollary	corollary	ADJ
ejde-92	190	36	2.6	2.6	NUM
ejde-92	190	37	)	)	PUNCT
ejde-92	190	38	,	,	PUNCT
ejde-92	190	39	evaluated	evaluate	VERB
ejde-92	190	40	at	at	ADP
ejde-92	190	41	a	a	DET
ejde-92	190	42	minimizer	minimizer	NOUN
ejde-92	190	43	u1	u1	NOUN
ejde-92	190	44	and	and	CCONJ
ejde-92	190	45	in	in	ADP
ejde-92	190	46	the	the	DET
ejde-92	190	47	direction	direction	NOUN
ejde-92	190	48	v	v	ADP
ejde-92	190	49	∈w	∈w	NOUN
ejde-92	190	50	1,p(ω	1,p(ω	NUM
ejde-92	190	51	)	)	PUNCT
ejde-92	190	52	we	we	PRON
ejde-92	190	53	easily	easily	ADV
ejde-92	190	54	arrive	arrive	VERB
ejde-92	190	55	to	to	ADP
ejde-92	190	56	p	p	PROPN
ejde-92	190	57	∫	∫	PROPN
ejde-92	190	58	ω	ω	PROPN
ejde-92	190	59	|∇u1|p−2∇u1∇v	|∇u1|p−2∇u1∇v	PROPN
ejde-92	190	60	dx	dx	PROPN
ejde-92	190	61	=	=	SYM
ejde-92	190	62	pλ̂(m)〈dvp(u	pλ̂(m)〈dvp(u	PROPN
ejde-92	190	63	)	)	PUNCT
ejde-92	190	64	,	,	PUNCT
ejde-92	190	65	v	v	NOUN
ejde-92	190	66	〉	〉	NUM
ejde-92	190	67	,	,	PUNCT
ejde-92	190	68	equivalently,∫	equivalently,∫	NOUN
ejde-92	190	69	ω	ω	PROPN
ejde-92	190	70	|∇u1|p−2∇u1∇v	|∇u1|p−2∇u1∇v	PROPN
ejde-92	190	71	dx	dx	PROPN
ejde-92	190	72	=	=	PUNCT
ejde-92	191	1	λ̂(m	λ̂(m	PRON
ejde-92	191	2	)	)	PUNCT
ejde-92	192	1	∫	∫	PROPN
ejde-92	192	2	ω	ω	PROPN
ejde-92	192	3	|u1	|u1	PROPN
ejde-92	193	1	−	−	PROPN
ejde-92	193	2	ũ1|p−2(u1	ũ1|p−2(u1	NOUN
ejde-92	194	1	−	−	NOUN
ejde-92	194	2	ũ1)vmdx	ũ1)vmdx	NOUN
ejde-92	194	3	.	.	PUNCT
ejde-92	195	1	since	since	SCONJ
ejde-92	195	2	v	v	NUM
ejde-92	195	3	∈	∈	PROPN
ejde-92	195	4	w	w	PROPN
ejde-92	195	5	1,p(ω	1,p(ω	NUM
ejde-92	195	6	)	)	PUNCT
ejde-92	195	7	is	be	AUX
ejde-92	195	8	arbitrary	arbitrary	ADJ
ejde-92	195	9	,	,	PUNCT
ejde-92	195	10	this	this	PRON
ejde-92	195	11	means	mean	VERB
ejde-92	195	12	that	that	SCONJ
ejde-92	195	13	λ̂(m	λ̂(m	ADP
ejde-92	195	14	)	)	PUNCT
ejde-92	195	15	is	be	AUX
ejde-92	195	16	an	an	DET
ejde-92	195	17	eigenvalue	eigenvalue	NOUN
ejde-92	195	18	with	with	ADP
ejde-92	195	19	associated	associate	VERB
ejde-92	195	20	eigenfunction	eigenfunction	NOUN
ejde-92	195	21	u1	u1	NOUN
ejde-92	195	22	−	−	PROPN
ejde-92	195	23	ũ1	ũ1	PROPN
ejde-92	195	24	.	.	PUNCT
ejde-92	195	25	�	�	PROPN
ejde-92	195	26	remark	remark	VERB
ejde-92	195	27	3.2	3.2	NUM
ejde-92	195	28	.	.	PUNCT
ejde-92	196	1	an	an	DET
ejde-92	196	2	alternative	alternative	ADJ
ejde-92	196	3	proof	proof	NOUN
ejde-92	196	4	of	of	ADP
ejde-92	196	5	the	the	DET
ejde-92	196	6	fact	fact	NOUN
ejde-92	196	7	that	that	SCONJ
ejde-92	196	8	λ̂(m	λ̂(m	ADP
ejde-92	196	9	)	)	PUNCT
ejde-92	196	10	is	be	AUX
ejde-92	196	11	an	an	DET
ejde-92	196	12	eigenvalue	eigenvalue	NOUN
ejde-92	196	13	can	can	AUX
ejde-92	196	14	be	be	AUX
ejde-92	196	15	given	give	VERB
ejde-92	196	16	in	in	ADP
ejde-92	196	17	the	the	DET
ejde-92	196	18	case	case	NOUN
ejde-92	196	19	p	p	X
ejde-92	196	20	≥	≥	NUM
ejde-92	196	21	2	2	NUM
ejde-92	196	22	.	.	PUNCT
ejde-92	197	1	in	in	ADP
ejde-92	197	2	this	this	DET
ejde-92	197	3	case	case	NOUN
ejde-92	197	4	,	,	PUNCT
ejde-92	197	5	lagrange	lagrange	PROPN
ejde-92	197	6	’s	’s	PART
ejde-92	197	7	multiplier	multipli	ADJ
ejde-92	197	8	rule	rule	NOUN
ejde-92	197	9	can	can	AUX
ejde-92	197	10	be	be	AUX
ejde-92	197	11	employed	employ	VERB
ejde-92	197	12	in	in	ADP
ejde-92	197	13	a	a	DET
ejde-92	197	14	standard	standard	ADJ
ejde-92	197	15	way	way	NOUN
ejde-92	197	16	.	.	PUNCT
ejde-92	198	1	indeed	indeed	ADV
ejde-92	198	2	,	,	PUNCT
ejde-92	198	3	the	the	DET
ejde-92	198	4	constraint	constraint	NOUN
ejde-92	198	5	defining	define	VERB
ejde-92	198	6	m0	m0	NOUN
ejde-92	198	7	involves	involve	VERB
ejde-92	198	8	the	the	DET
ejde-92	198	9	functional	functional	ADJ
ejde-92	198	10	i(v	i(v	NOUN
ejde-92	198	11	)	)	PUNCT
ejde-92	198	12	=	=	SYM
ejde-92	199	1	∫	∫	PROPN
ejde-92	199	2	ω	ω	NUM
ejde-92	199	3	|v|p−2v	|v|p−2v	PROPN
ejde-92	199	4	which	which	PRON
ejde-92	199	5	is	be	AUX
ejde-92	199	6	c1	c1	PROPN
ejde-92	199	7	provided	provide	VERB
ejde-92	199	8	p	p	PRON
ejde-92	199	9	≥	≥	NUM
ejde-92	199	10	2	2	NUM
ejde-92	199	11	.	.	NOUN
ejde-92	199	12	4	4	NUM
ejde-92	199	13	.	.	X
ejde-92	199	14	a	a	DET
ejde-92	199	15	further	further	ADJ
ejde-92	199	16	subcritical	subcritical	ADJ
ejde-92	199	17	problem	problem	NOUN
ejde-92	199	18	the	the	DET
ejde-92	199	19	results	result	NOUN
ejde-92	199	20	of	of	ADP
ejde-92	199	21	section	section	NOUN
ejde-92	199	22	2	2	NUM
ejde-92	199	23	can	can	AUX
ejde-92	199	24	be	be	AUX
ejde-92	199	25	still	still	ADV
ejde-92	199	26	employed	employ	VERB
ejde-92	199	27	to	to	PART
ejde-92	199	28	study	study	VERB
ejde-92	199	29	the	the	DET
ejde-92	199	30	following	follow	VERB
ejde-92	199	31	nonlinear	nonlinear	ADJ
ejde-92	199	32	problems	problem	NOUN
ejde-92	199	33	−∆pu	−∆pu	X
ejde-92	200	1	=	=	SYM
ejde-92	200	2	λm(x)|u|q−2u	λm(x)|u|q−2u	PUNCT
ejde-92	200	3	x	x	SYM
ejde-92	200	4	∈	∈	PROPN
ejde-92	200	5	ω	ω	NUM
ejde-92	200	6	|∇u|p−2	|∇u|p−2	PROPN
ejde-92	200	7	∂u	∂u	PROPN
ejde-92	200	8	∂ν	∂ν	X
ejde-92	201	1	=	=	PUNCT
ejde-92	201	2	0	0	PUNCT
ejde-92	201	3	x	x	SYM
ejde-92	201	4	∈	∈	PROPN
ejde-92	201	5	∂ω	∂ω	PROPN
ejde-92	201	6	,	,	PUNCT
ejde-92	201	7	(	(	PUNCT
ejde-92	201	8	4.1	4.1	NUM
ejde-92	201	9	)	)	PUNCT
ejde-92	201	10	where	where	SCONJ
ejde-92	201	11	1	1	NUM
ejde-92	201	12	≤	≤	NOUN
ejde-92	201	13	q	q	X
ejde-92	201	14	<	<	X
ejde-92	201	15	p∗	p∗	PROPN
ejde-92	201	16	,	,	PUNCT
ejde-92	201	17	(	(	PUNCT
ejde-92	201	18	4.2	4.2	NUM
ejde-92	201	19	)	)	PUNCT
ejde-92	201	20	with	with	ADP
ejde-92	201	21	p∗	p∗	PROPN
ejde-92	201	22	=	=	PUNCT
ejde-92	201	23	pn	pn	PROPN
ejde-92	201	24	n−p	n−p	PROPN
ejde-92	201	25	if	if	SCONJ
ejde-92	201	26	p	p	X
ejde-92	201	27	<	<	X
ejde-92	201	28	n	n	NOUN
ejde-92	201	29	,	,	PUNCT
ejde-92	201	30	p∗	p∗	ADJ
ejde-92	201	31	=	=	NOUN
ejde-92	201	32	∞	∞	NOUN
ejde-92	201	33	otherwise	otherwise	ADV
ejde-92	201	34	.	.	PUNCT
ejde-92	202	1	in	in	ADP
ejde-92	202	2	addition	addition	NOUN
ejde-92	202	3	,	,	PUNCT
ejde-92	202	4	m	m	VERB
ejde-92	202	5	∈	∈	NOUN
ejde-92	202	6	lr(ω	lr(ω	X
ejde-92	202	7	)	)	PUNCT
ejde-92	202	8	is	be	AUX
ejde-92	202	9	positive	positive	ADJ
ejde-92	202	10	a.e	a.e	NOUN
ejde-92	202	11	.	.	PROPN
ejde-92	203	1	in	in	ADP
ejde-92	203	2	ω	ω	PROPN
ejde-92	203	3	and	and	CCONJ
ejde-92	203	4	r	r	NOUN
ejde-92	203	5			PROPN
ejde-92	203	6	>	>	X
ejde-92	203	7	(	(	PUNCT
ejde-92	203	8	p∗	p∗	PROPN
ejde-92	203	9	q	q	NOUN
ejde-92	203	10	)	)	PUNCT
ejde-92	203	11	′	′	NUM
ejde-92	204	1	=	=	PUNCT
ejde-92	204	2	np	np	INTJ
ejde-92	204	3	n(p−q)+pq	n(p−q)+pq	VERB
ejde-92	204	4	if	if	SCONJ
ejde-92	204	5	1	1	NUM
ejde-92	204	6	<	<	X
ejde-92	204	7	p	p	X
ejde-92	204	8	<	<	X
ejde-92	204	9	n	n	CCONJ
ejde-92	204	10	,	,	PUNCT
ejde-92	204	11	>	>	X
ejde-92	204	12	1	1	NUM
ejde-92	204	13	if	if	SCONJ
ejde-92	204	14	p	p	NOUN
ejde-92	204	15	=	=	NOUN
ejde-92	204	16	n	n	CCONJ
ejde-92	204	17	,	,	PUNCT
ejde-92	204	18	=	=	NOUN
ejde-92	204	19	1	1	NUM
ejde-92	204	20	if	if	SCONJ
ejde-92	204	21	p	p	PROPN
ejde-92	204	22	>	>	X
ejde-92	204	23	n.	n.	NOUN
ejde-92	204	24	(	(	PUNCT
ejde-92	204	25	4.3	4.3	NUM
ejde-92	204	26	)	)	PUNCT
ejde-92	204	27	theorem	theorem	VERB
ejde-92	204	28	4.1	4.1	NUM
ejde-92	204	29	.	.	PUNCT
ejde-92	205	1	assume	assume	VERB
ejde-92	205	2	that	that	SCONJ
ejde-92	205	3	the	the	DET
ejde-92	205	4	exponents	exponent	NOUN
ejde-92	205	5	q	q	PROPN
ejde-92	205	6	and	and	CCONJ
ejde-92	205	7	r	r	NOUN
ejde-92	205	8	satisfy	satisfy	NOUN
ejde-92	205	9	(	(	PUNCT
ejde-92	205	10	4.2	4.2	NUM
ejde-92	205	11	)	)	PUNCT
ejde-92	205	12	and	and	CCONJ
ejde-92	205	13	(	(	PUNCT
ejde-92	205	14	4.3	4.3	NUM
ejde-92	205	15	)	)	PUNCT
ejde-92	205	16	,	,	PUNCT
ejde-92	205	17	respectively	respectively	ADV
ejde-92	205	18	.	.	PUNCT
ejde-92	206	1	then	then	ADV
ejde-92	206	2	problem	problem	NOUN
ejde-92	206	3	(	(	PUNCT
ejde-92	206	4	4.1	4.1	NUM
ejde-92	206	5	)	)	PUNCT
ejde-92	206	6	admits	admit	VERB
ejde-92	206	7	a	a	DET
ejde-92	206	8	nontrivial	nontrivial	NOUN
ejde-92	206	9	(	(	PUNCT
ejde-92	206	10	nonconstant	nonconstant	ADJ
ejde-92	206	11	)	)	PUNCT
ejde-92	206	12	solution	solution	NOUN
ejde-92	206	13	u	u	NOUN
ejde-92	206	14	∈w	∈w	VERB
ejde-92	206	15	1,p(ω	1,p(ω	NUM
ejde-92	206	16	)	)	PUNCT
ejde-92	207	1	if	if	SCONJ
ejde-92	207	2	and	and	CCONJ
ejde-92	207	3	only	only	ADV
ejde-92	207	4	if	if	SCONJ
ejde-92	207	5	λ	λ	X
ejde-92	207	6	>	>	X
ejde-92	207	7	0	0	NUM
ejde-92	207	8	.	.	PUNCT
ejde-92	208	1	these	these	DET
ejde-92	208	2	nontrivial	nontrivial	ADJ
ejde-92	208	3	solutions	solution	NOUN
ejde-92	208	4	satisfy	satisfy	VERB
ejde-92	208	5	the	the	DET
ejde-92	208	6	average	average	ADJ
ejde-92	208	7	condition∫	condition∫	NOUN
ejde-92	208	8	ω	ω	NOUN
ejde-92	208	9	|u|q−2umdx	|u|q−2umdx	PROPN
ejde-92	209	1	=	=	NOUN
ejde-92	209	2	0	0	PROPN
ejde-92	209	3	.	.	PUNCT
ejde-92	210	1	(	(	PUNCT
ejde-92	210	2	4.4	4.4	NUM
ejde-92	210	3	)	)	PUNCT
ejde-92	210	4	proof	proof	NOUN
ejde-92	210	5	.	.	PUNCT
ejde-92	211	1	solutions	solution	NOUN
ejde-92	211	2	u	u	NOUN
ejde-92	211	3	∈w	∈w	NOUN
ejde-92	211	4	1,p(ω	1,p(ω	NUM
ejde-92	211	5	)	)	PUNCT
ejde-92	211	6	are	be	AUX
ejde-92	211	7	understood	understand	VERB
ejde-92	211	8	in	in	ADP
ejde-92	211	9	weak	weak	ADJ
ejde-92	211	10	sense	sense	NOUN
ejde-92	211	11	,	,	PUNCT
ejde-92	211	12	that	that	ADV
ejde-92	211	13	is	is	ADV
ejde-92	211	14	,	,	PUNCT
ejde-92	211	15	equality∫	equality∫	PROPN
ejde-92	211	16	ω	ω	PROPN
ejde-92	211	17	|∇u|p−2∇u∇v	|∇u|p−2∇u∇v	NOUN
ejde-92	211	18	dx	dx	PROPN
ejde-92	212	1	=	=	SYM
ejde-92	212	2	λ	λ	PROPN
ejde-92	212	3	∫	∫	PROPN
ejde-92	212	4	ω	ω	PROPN
ejde-92	212	5	|u|q−2uvmdx	|u|q−2uvmdx	PROPN
ejde-92	212	6	,	,	PUNCT
ejde-92	212	7	(	(	PUNCT
ejde-92	212	8	4.5	4.5	NUM
ejde-92	212	9	)	)	PUNCT
ejde-92	212	10	holds	hold	VERB
ejde-92	212	11	for	for	ADP
ejde-92	212	12	all	all	DET
ejde-92	212	13	v	v	NOUN
ejde-92	212	14	∈w	∈w	NOUN
ejde-92	212	15	1,p(ω	1,p(ω	NUM
ejde-92	212	16	)	)	PUNCT
ejde-92	212	17	.	.	PUNCT
ejde-92	213	1	observe	observe	VERB
ejde-92	213	2	that	that	SCONJ
ejde-92	213	3	the	the	DET
ejde-92	213	4	latter	latter	ADJ
ejde-92	213	5	integrand	integrand	NOUN
ejde-92	213	6	lays	lay	VERB
ejde-92	213	7	in	in	ADP
ejde-92	213	8	l1(ω	l1(ω	PROPN
ejde-92	213	9	)	)	PUNCT
ejde-92	213	10	because	because	SCONJ
ejde-92	213	11	of	of	ADP
ejde-92	213	12	conditions	condition	NOUN
ejde-92	213	13	(	(	PUNCT
ejde-92	213	14	4.3	4.3	NUM
ejde-92	213	15	)	)	PUNCT
ejde-92	213	16	satisfied	satisfy	VERB
ejde-92	213	17	by	by	ADP
ejde-92	213	18	m.	m.	NOUN
ejde-92	213	19	thus	thus	ADV
ejde-92	213	20	,	,	PUNCT
ejde-92	213	21	it	it	PRON
ejde-92	213	22	is	be	AUX
ejde-92	213	23	readily	readily	ADV
ejde-92	213	24	deduced	deduce	VERB
ejde-92	213	25	that	that	SCONJ
ejde-92	213	26	nonconstant	nonconstant	ADJ
ejde-92	213	27	ejde-2022/13	ejde-2022/13	ADJ
ejde-92	213	28	second	second	ADJ
ejde-92	213	29	neumann	neumann	PROPN
ejde-92	213	30	eigenvalue	eigenvalue	PROPN
ejde-92	213	31	9	9	NUM
ejde-92	213	32	nontrivial	nontrivial	NOUN
ejde-92	213	33	solutions	solution	NOUN
ejde-92	213	34	are	be	AUX
ejde-92	213	35	only	only	ADV
ejde-92	213	36	possible	possible	ADJ
ejde-92	213	37	when	when	SCONJ
ejde-92	213	38	λ	λ	X
ejde-92	213	39	>	>	X
ejde-92	213	40	0	0	PUNCT
ejde-92	214	1	while	while	SCONJ
ejde-92	214	2	in	in	ADP
ejde-92	214	3	addition	addition	NOUN
ejde-92	214	4	,	,	PUNCT
ejde-92	214	5	every	every	DET
ejde-92	214	6	solution	solution	NOUN
ejde-92	214	7	must	must	AUX
ejde-92	214	8	satisfy	satisfy	VERB
ejde-92	214	9	(	(	PUNCT
ejde-92	214	10	4.4	4.4	NUM
ejde-92	214	11	)	)	PUNCT
ejde-92	214	12	(	(	PUNCT
ejde-92	214	13	use	use	VERB
ejde-92	214	14	v	v	NOUN
ejde-92	214	15	=	=	SYM
ejde-92	214	16	1	1	NUM
ejde-92	214	17	as	as	ADP
ejde-92	214	18	a	a	DET
ejde-92	214	19	test	test	NOUN
ejde-92	214	20	function	function	NOUN
ejde-92	214	21	)	)	PUNCT
ejde-92	214	22	.	.	PUNCT
ejde-92	215	1	on	on	ADP
ejde-92	215	2	the	the	DET
ejde-92	215	3	other	other	ADJ
ejde-92	215	4	hand	hand	NOUN
ejde-92	215	5	,	,	PUNCT
ejde-92	215	6	solving	solve	VERB
ejde-92	215	7	(	(	PUNCT
ejde-92	215	8	4.1	4.1	NUM
ejde-92	215	9	)	)	PUNCT
ejde-92	215	10	for	for	ADP
ejde-92	215	11	a	a	DET
ejde-92	215	12	specific	specific	ADJ
ejde-92	215	13	λ	λ	NOUN
ejde-92	215	14	=	=	SYM
ejde-92	215	15	λ0	λ0	NOUN
ejde-92	215	16	>	>	X
ejde-92	215	17	0	0	NUM
ejde-92	215	18	amounts	amount	NOUN
ejde-92	215	19	to	to	ADP
ejde-92	215	20	solving	solve	VERB
ejde-92	215	21	it	it	PRON
ejde-92	215	22	for	for	ADP
ejde-92	215	23	every	every	DET
ejde-92	215	24	λ	λ	PROPN
ejde-92	215	25	>	>	X
ejde-92	215	26	0	0	NUM
ejde-92	215	27	.	.	PUNCT
ejde-92	216	1	in	in	ADP
ejde-92	216	2	fact	fact	NOUN
ejde-92	216	3	,	,	PUNCT
ejde-92	216	4	if	if	SCONJ
ejde-92	216	5	u0	u0	ADJ
ejde-92	216	6	is	be	AUX
ejde-92	216	7	a	a	DET
ejde-92	216	8	non	non	ADJ
ejde-92	216	9	trivial	trivial	ADJ
ejde-92	216	10	solution	solution	NOUN
ejde-92	216	11	for	for	ADP
ejde-92	216	12	λ	λ	NOUN
ejde-92	216	13	=	=	SYM
ejde-92	216	14	λ0	λ0	NOUN
ejde-92	216	15	then	then	ADV
ejde-92	216	16	uλ	uλ	ADV
ejde-92	216	17	=	=	PUNCT
ejde-92	216	18	(	(	PUNCT
ejde-92	216	19	λ	λ	X
ejde-92	216	20	λ0	λ0	NOUN
ejde-92	216	21	)	)	PUNCT
ejde-92	216	22	1	1	NUM
ejde-92	216	23	p−q	p−q	NOUN
ejde-92	216	24	u0	u0	NOUN
ejde-92	216	25	,	,	PUNCT
ejde-92	216	26	defines	define	VERB
ejde-92	216	27	a	a	DET
ejde-92	216	28	solution	solution	NOUN
ejde-92	216	29	to	to	ADP
ejde-92	216	30	(	(	PUNCT
ejde-92	216	31	4.1	4.1	NUM
ejde-92	216	32	)	)	PUNCT
ejde-92	216	33	corresponding	correspond	VERB
ejde-92	216	34	to	to	ADP
ejde-92	216	35	any	any	DET
ejde-92	216	36	λ	λ	PROPN
ejde-92	216	37	>	>	X
ejde-92	216	38	0	0	NUM
ejde-92	216	39	.	.	PUNCT
ejde-92	217	1	thus	thus	ADV
ejde-92	217	2	it	it	PRON
ejde-92	217	3	is	be	AUX
ejde-92	217	4	enough	enough	ADJ
ejde-92	217	5	with	with	ADP
ejde-92	217	6	finding	find	VERB
ejde-92	217	7	out	out	ADP
ejde-92	217	8	a	a	DET
ejde-92	217	9	solution	solution	NOUN
ejde-92	217	10	at	at	ADP
ejde-92	217	11	some	some	DET
ejde-92	217	12	fixed	fix	VERB
ejde-92	217	13	value	value	NOUN
ejde-92	217	14	of	of	ADP
ejde-92	217	15	λ0	λ0	NOUN
ejde-92	217	16	>	>	X
ejde-92	217	17	0	0	X
ejde-92	217	18	.	.	PUNCT
ejde-92	218	1	we	we	PRON
ejde-92	218	2	now	now	ADV
ejde-92	218	3	mimic	mimic	VERB
ejde-92	218	4	the	the	DET
ejde-92	218	5	proof	proof	NOUN
ejde-92	218	6	of	of	ADP
ejde-92	218	7	theorem	theorem	ADJ
ejde-92	218	8	1.1	1.1	NUM
ejde-92	218	9	and	and	CCONJ
ejde-92	218	10	minimize	minimize	VERB
ejde-92	218	11	j	j	PROPN
ejde-92	218	12	(	(	PUNCT
ejde-92	218	13	u	u	NOUN
ejde-92	218	14	)	)	PUNCT
ejde-92	218	15	=	=	SYM
ejde-92	218	16	∫	∫	PROPN
ejde-92	218	17	ω	ω	NUM
ejde-92	218	18	|∇u|p	|∇u|p	NOUN
ejde-92	218	19	on	on	ADP
ejde-92	218	20	m1,q	m1,q	PROPN
ejde-92	218	21	=	=	SYM
ejde-92	218	22	m0,q	m0,q	NOUN
ejde-92	218	23	∩	∩	NOUN
ejde-92	218	24	{	{	PUNCT
ejde-92	218	25	u	u	NOUN
ejde-92	218	26	∈	∈	PROPN
ejde-92	218	27	w	w	PROPN
ejde-92	218	28	1,p(ω	1,p(ω	NUM
ejde-92	218	29	)	)	PUNCT
ejde-92	218	30	:	:	PUNCT
ejde-92	219	1	∫	∫	PROPN
ejde-92	219	2	ω	ω	PROPN
ejde-92	219	3	|u|q	|u|q	PROPN
ejde-92	219	4	mdx	mdx	NOUN
ejde-92	219	5	=	=	NOUN
ejde-92	219	6	1	1	NUM
ejde-92	219	7	}	}	PUNCT
ejde-92	219	8	,	,	PUNCT
ejde-92	219	9	m0,q	m0,q	PROPN
ejde-92	219	10	being	be	AUX
ejde-92	219	11	the	the	DET
ejde-92	219	12	functions	function	NOUN
ejde-92	219	13	of	of	ADP
ejde-92	219	14	w	w	PROPN
ejde-92	219	15	1,p(ω	1,p(ω	NUM
ejde-92	219	16	)	)	PUNCT
ejde-92	219	17	satisfying	satisfy	VERB
ejde-92	219	18	∫	∫	PROPN
ejde-92	219	19	ω	ω	PROPN
ejde-92	219	20	|u|q−2umdx	|u|q−2umdx	PROPN
ejde-92	220	1	=	=	NOUN
ejde-92	220	2	0	0	PROPN
ejde-92	220	3	.	.	PUNCT
ejde-92	221	1	by	by	ADP
ejde-92	221	2	similar	similar	ADJ
ejde-92	221	3	reasons	reason	NOUN
ejde-92	221	4	as	as	ADP
ejde-92	221	5	in	in	ADP
ejde-92	221	6	section	section	NOUN
ejde-92	221	7	3	3	NUM
ejde-92	221	8	,	,	PUNCT
ejde-92	221	9	j	j	PROPN
ejde-92	221	10	achieves	achieve	VERB
ejde-92	221	11	a	a	DET
ejde-92	221	12	minimum	minimum	NOUN
ejde-92	221	13	at	at	ADP
ejde-92	221	14	some	some	DET
ejde-92	221	15	u1	u1	NOUN
ejde-92	221	16	∈	∈	PROPN
ejde-92	221	17	m1,q	m1,q	PROPN
ejde-92	221	18	and	and	CCONJ
ejde-92	221	19	so	so	ADV
ejde-92	221	20	0	0	NUM
ejde-92	221	21	<	<	X
ejde-92	221	22	µ1	µ1	PROPN
ejde-92	221	23	:	:	PUNCT
ejde-92	221	24	=	=	SYM
ejde-92	221	25	j	j	PROPN
ejde-92	221	26	(	(	PUNCT
ejde-92	221	27	u1	u1	PROPN
ejde-92	221	28	)	)	PUNCT
ejde-92	221	29	=	=	SYM
ejde-92	221	30	inf	inf	PROPN
ejde-92	221	31	m1,q	m1,q	PROPN
ejde-92	221	32	j	j	PROPN
ejde-92	221	33	=	=	PROPN
ejde-92	221	34	inf	inf	PROPN
ejde-92	221	35	u∈m0,q\{0	u∈m0,q\{0	PROPN
ejde-92	221	36	}	}	PUNCT
ejde-92	221	37	∫	∫	PROPN
ejde-92	221	38	ω	ω	PROPN
ejde-92	221	39	|∇u|p	|∇u|p	PROPN
ejde-92	221	40	dx	dx	PROPN
ejde-92	221	41	(	(	PUNCT
ejde-92	221	42	∫	∫	PROPN
ejde-92	221	43	ω	ω	PROPN
ejde-92	221	44	|u|q	|u|q	PROPN
ejde-92	221	45	mdx	mdx	NOUN
ejde-92	221	46	)	)	PUNCT
ejde-92	221	47	p	p	X
ejde-92	221	48	/	/	SYM
ejde-92	221	49	q	q	NOUN
ejde-92	221	50	=	=	X
ejde-92	221	51	inf	inf	PROPN
ejde-92	221	52	u/∈span{1	u/∈span{1	NOUN
ejde-92	221	53	}	}	PUNCT
ejde-92	221	54	∫	∫	PROPN
ejde-92	221	55	ω	ω	PROPN
ejde-92	221	56	|∇u|p	|∇u|p	PROPN
ejde-92	221	57	dx	dx	PROPN
ejde-92	221	58	vq(u)p	vq(u)p	PROPN
ejde-92	221	59	/	/	SYM
ejde-92	221	60	q	q	NOUN
ejde-92	221	61	.	.	PUNCT
ejde-92	222	1	(	(	PUNCT
ejde-92	222	2	4.6	4.6	NUM
ejde-92	222	3	)	)	PUNCT
ejde-92	222	4	in	in	ADP
ejde-92	222	5	the	the	DET
ejde-92	222	6	above	above	ADJ
ejde-92	222	7	expression	expression	NOUN
ejde-92	222	8	we	we	PRON
ejde-92	222	9	used	use	VERB
ejde-92	222	10	that	that	SCONJ
ejde-92	222	11	m0,q	m0,q	PROPN
ejde-92	222	12	\	\	PROPN
ejde-92	222	13	{	{	PUNCT
ejde-92	222	14	0	0	NUM
ejde-92	222	15	}	}	PUNCT
ejde-92	222	16	=	=	PRON
ejde-92	222	17	{	{	PUNCT
ejde-92	222	18	u	u	NOUN
ejde-92	222	19	−	−	PROPN
ejde-92	222	20	ũq	ũq	NOUN
ejde-92	222	21	:	:	PUNCT
ejde-92	223	1	u	u	PROPN
ejde-92	223	2	∈	∈	PROPN
ejde-92	223	3	w	w	PROPN
ejde-92	223	4	1,p(ω	1,p(ω	NUM
ejde-92	223	5	)	)	PUNCT
ejde-92	223	6	,	,	PUNCT
ejde-92	223	7	u	u	NOUN
ejde-92	223	8	/∈	/∈	NOUN
ejde-92	223	9	span{1	span{1	NOUN
ejde-92	223	10	}	}	PUNCT
ejde-92	223	11	}	}	PUNCT
ejde-92	223	12	where	where	SCONJ
ejde-92	223	13	ũq	ũq	PROPN
ejde-92	223	14	means	mean	VERB
ejde-92	223	15	the	the	DET
ejde-92	223	16	average	average	NOUN
ejde-92	223	17	of	of	ADP
ejde-92	223	18	u	u	NOUN
ejde-92	223	19	relative	relative	ADJ
ejde-92	223	20	to	to	ADP
ejde-92	223	21	lq(ω	lq(ω	PROPN
ejde-92	223	22	,	,	PUNCT
ejde-92	223	23	mdx	mdx	NOUN
ejde-92	223	24	)	)	PUNCT
ejde-92	223	25	(	(	PUNCT
ejde-92	223	26	section	section	NOUN
ejde-92	223	27	2	2	NUM
ejde-92	223	28	)	)	PUNCT
ejde-92	223	29	.	.	PUNCT
ejde-92	224	1	by	by	ADP
ejde-92	224	2	taking	take	VERB
ejde-92	224	3	the	the	DET
ejde-92	224	4	directional	directional	ADJ
ejde-92	224	5	derivative	derivative	NOUN
ejde-92	224	6	of	of	ADP
ejde-92	224	7	the	the	DET
ejde-92	224	8	latter	latter	ADJ
ejde-92	224	9	quotient	quotient	NOUN
ejde-92	224	10	at	at	ADP
ejde-92	224	11	u	u	NOUN
ejde-92	224	12	=	=	NOUN
ejde-92	224	13	u1	u1	NOUN
ejde-92	224	14	in	in	ADP
ejde-92	224	15	the	the	DET
ejde-92	224	16	direction	direction	NOUN
ejde-92	224	17	v	v	ADP
ejde-92	224	18	∈w	∈w	NOUN
ejde-92	224	19	1,p(ω	1,p(ω	NUM
ejde-92	224	20	)	)	PUNCT
ejde-92	224	21	,	,	PUNCT
ejde-92	224	22	and	and	CCONJ
ejde-92	224	23	then	then	ADV
ejde-92	224	24	equaling	equal	VERB
ejde-92	224	25	to	to	ADP
ejde-92	224	26	zero	zero	NUM
ejde-92	224	27	we	we	PRON
ejde-92	224	28	obtain	obtain	VERB
ejde-92	224	29	〈	〈	PROPN
ejde-92	224	30	−∆pu1	−∆pu1	NOUN
ejde-92	224	31	,	,	PUNCT
ejde-92	224	32	v	v	NOUN
ejde-92	224	33	〉	〉	NOUN
ejde-92	224	34	=	=	NOUN
ejde-92	224	35	vq(u1	vq(u1	NOUN
ejde-92	224	36	)	)	PUNCT
ejde-92	224	37	p	p	NOUN
ejde-92	224	38	q−1µ1	q−1µ1	NOUN
ejde-92	224	39	〈	〈	PROPN
ejde-92	224	40	1	1	NUM
ejde-92	224	41	q	q	NOUN
ejde-92	224	42	dvq(u1	dvq(u1	NOUN
ejde-92	224	43	)	)	PUNCT
ejde-92	224	44	,	,	PUNCT
ejde-92	224	45	v	v	X
ejde-92	224	46	〉	〉	NOUN
ejde-92	224	47	,	,	PUNCT
ejde-92	224	48	for	for	ADP
ejde-92	224	49	all	all	PRON
ejde-92	224	50	v	v	ADP
ejde-92	224	51	∈	∈	NOUN
ejde-92	224	52	w	w	NOUN
ejde-92	224	53	1,p(ω	1,p(ω	NUM
ejde-92	224	54	)	)	PUNCT
ejde-92	224	55	.	.	PUNCT
ejde-92	225	1	by	by	ADP
ejde-92	225	2	using	use	VERB
ejde-92	225	3	lemma	lemma	PROPN
ejde-92	225	4	2.5	2.5	NUM
ejde-92	225	5	this	this	PRON
ejde-92	225	6	implies	imply	VERB
ejde-92	225	7	that	that	SCONJ
ejde-92	225	8	u0	u0	ADJ
ejde-92	225	9	=	=	NOUN
ejde-92	225	10	u1	u1	NOUN
ejde-92	225	11	−	−	PROPN
ejde-92	225	12	ũ1	ũ1	PROPN
ejde-92	225	13	solves	solve	NOUN
ejde-92	225	14	(	(	PUNCT
ejde-92	225	15	4.1	4.1	NUM
ejde-92	225	16	)	)	PUNCT
ejde-92	225	17	for	for	ADP
ejde-92	225	18	the	the	DET
ejde-92	225	19	special	special	ADJ
ejde-92	225	20	value	value	NOUN
ejde-92	225	21	:	:	PUNCT
ejde-92	225	22	λ0	λ0	NOUN
ejde-92	225	23	=	=	SYM
ejde-92	225	24	vq(u1	vq(u1	NOUN
ejde-92	225	25	)	)	PUNCT
ejde-92	225	26	p	p	NOUN
ejde-92	225	27	q−1µ1	q−1µ1	NOUN
ejde-92	225	28	.	.	PUNCT
ejde-92	226	1	as	as	SCONJ
ejde-92	226	2	already	already	ADV
ejde-92	226	3	pointed	point	VERB
ejde-92	226	4	out	out	ADP
ejde-92	226	5	this	this	PRON
ejde-92	226	6	permits	permit	VERB
ejde-92	226	7	us	we	PRON
ejde-92	226	8	obtaining	obtain	VERB
ejde-92	226	9	a	a	DET
ejde-92	226	10	solution	solution	NOUN
ejde-92	226	11	for	for	ADP
ejde-92	226	12	any	any	DET
ejde-92	226	13	positive	positive	ADJ
ejde-92	226	14	value	value	NOUN
ejde-92	227	1	λ	λ	X
ejde-92	227	2	>	>	X
ejde-92	227	3	0	0	NUM
ejde-92	227	4	.	.	PUNCT
ejde-92	227	5	�	�	PROPN
ejde-92	227	6	remark	remark	VERB
ejde-92	227	7	4.2	4.2	NUM
ejde-92	227	8	.	.	PUNCT
ejde-92	228	1	as	as	ADP
ejde-92	228	2	in	in	ADP
ejde-92	228	3	the	the	DET
ejde-92	228	4	case	case	NOUN
ejde-92	228	5	of	of	ADP
ejde-92	228	6	poincaré	poincaré	PROPN
ejde-92	228	7	’s	’s	PART
ejde-92	228	8	inequality	inequality	NOUN
ejde-92	228	9	(	(	PUNCT
ejde-92	228	10	1.6	1.6	NUM
ejde-92	228	11	)	)	PUNCT
ejde-92	228	12	,	,	PUNCT
ejde-92	228	13	µ	µ	X
ejde-92	228	14	−	−	PROPN
ejde-92	228	15	1	1	NUM
ejde-92	228	16	q	q	NOUN
ejde-92	228	17	1	1	NUM
ejde-92	228	18	defines	define	VERB
ejde-92	228	19	the	the	DET
ejde-92	228	20	optimum	optimum	ADJ
ejde-92	228	21	constant	constant	ADJ
ejde-92	228	22	c	c	NOUN
ejde-92	228	23	in	in	ADP
ejde-92	228	24	the	the	DET
ejde-92	228	25	inequality	inequality	NOUN
ejde-92	228	26	:	:	PUNCT
ejde-92	228	27	inf	inf	NOUN
ejde-92	228	28	t∈r	t∈r	NOUN
ejde-92	228	29	‖u−	‖u−	PROPN
ejde-92	228	30	t‖lq(ω	t‖lq(ω	PROPN
ejde-92	228	31	,	,	PUNCT
ejde-92	228	32	mdx	mdx	NOUN
ejde-92	228	33	)	)	PUNCT
ejde-92	228	34	≤	≤	NOUN
ejde-92	228	35	c‖|∇u|‖lp(ω	c‖|∇u|‖lp(ω	PROPN
ejde-92	228	36	)	)	PUNCT
ejde-92	228	37	,	,	PUNCT
ejde-92	228	38	(	(	PUNCT
ejde-92	228	39	4.7	4.7	NUM
ejde-92	228	40	)	)	PUNCT
ejde-92	228	41	associated	associate	VERB
ejde-92	228	42	with	with	ADP
ejde-92	228	43	the	the	DET
ejde-92	228	44	embedding	embed	VERB
ejde-92	228	45	w	w	PROPN
ejde-92	228	46	1,p(ω	1,p(ω	NUM
ejde-92	228	47	)	)	PUNCT
ejde-92	229	1	⊂	⊂	PROPN
ejde-92	229	2	lq(ω	lq(ω	PROPN
ejde-92	229	3	,	,	PUNCT
ejde-92	229	4	mdx	mdx	NOUN
ejde-92	229	5	)	)	PUNCT
ejde-92	229	6	.	.	PUNCT
ejde-92	230	1	see	see	VERB
ejde-92	230	2	[	[	X
ejde-92	230	3	18	18	NUM
ejde-92	230	4	,	,	PUNCT
ejde-92	230	5	lemma	lemma	PROPN
ejde-92	230	6	v.2.3–2	v.2.3–2	PROPN
ejde-92	230	7	]	]	X
ejde-92	230	8	for	for	ADP
ejde-92	230	9	the	the	DET
ejde-92	230	10	case	case	NOUN
ejde-92	230	11	m	m	NOUN
ejde-92	230	12	=	=	NOUN
ejde-92	230	13	1	1	NUM
ejde-92	230	14	.	.	NOUN
ejde-92	230	15	5	5	NUM
ejde-92	230	16	.	.	X
ejde-92	231	1	p	p	X
ejde-92	231	2	-	-	PUNCT
ejde-92	231	3	laplacian	laplacian	NOUN
ejde-92	231	4	on	on	ADP
ejde-92	231	5	graphs	graph	NOUN
ejde-92	231	6	an	an	DET
ejde-92	231	7	order	order	NOUN
ejde-92	231	8	n	n	PRON
ejde-92	231	9	graph	graph	NOUN
ejde-92	231	10	g	g	NOUN
ejde-92	231	11	is	be	AUX
ejde-92	231	12	defined	define	VERB
ejde-92	231	13	through	through	ADP
ejde-92	231	14	a	a	DET
ejde-92	231	15	couple	couple	NOUN
ejde-92	231	16	(	(	PUNCT
ejde-92	231	17	v	v	NOUN
ejde-92	231	18	,	,	PUNCT
ejde-92	231	19	e	e	NOUN
ejde-92	231	20	)	)	PUNCT
ejde-92	231	21	where	where	SCONJ
ejde-92	231	22	v	v	NOUN
ejde-92	231	23	=	=	SYM
ejde-92	231	24	{	{	PUNCT
ejde-92	231	25	v1	v1	NOUN
ejde-92	231	26	,	,	PUNCT
ejde-92	231	27	.	.	PUNCT
ejde-92	231	28	.	.	PUNCT
ejde-92	232	1	.	.	PUNCT
ejde-92	233	1	,	,	PUNCT
ejde-92	233	2	vn	vn	PROPN
ejde-92	233	3	}	}	PUNCT
ejde-92	233	4	is	be	AUX
ejde-92	233	5	a	a	DET
ejde-92	233	6	set	set	NOUN
ejde-92	233	7	with	with	ADP
ejde-92	233	8	n	n	NUM
ejde-92	233	9	≥	≥	NUM
ejde-92	233	10	2	2	NUM
ejde-92	233	11	elements	element	NOUN
ejde-92	233	12	,	,	PUNCT
ejde-92	233	13	the	the	DET
ejde-92	233	14	vertices	vertex	NOUN
ejde-92	233	15	of	of	ADP
ejde-92	233	16	the	the	DET
ejde-92	233	17	graph	graph	NOUN
ejde-92	233	18	,	,	PUNCT
ejde-92	233	19	together	together	ADV
ejde-92	233	20	with	with	ADP
ejde-92	233	21	a	a	DET
ejde-92	233	22	family	family	NOUN
ejde-92	233	23	e	e	NOUN
ejde-92	233	24	of	of	ADP
ejde-92	233	25	two	two	NUM
ejde-92	233	26	–	–	PUNCT
ejde-92	233	27	elements	element	NOUN
ejde-92	233	28	subsets	subset	NOUN
ejde-92	233	29	e	e	NOUN
ejde-92	233	30	=	=	PRON
ejde-92	233	31	{	{	PUNCT
ejde-92	233	32	u	u	NOUN
ejde-92	233	33	,	,	PUNCT
ejde-92	233	34	v	v	NOUN
ejde-92	233	35	}	}	PUNCT
ejde-92	233	36	of	of	ADP
ejde-92	233	37	v.	v.	ADP
ejde-92	233	38	members	member	NOUN
ejde-92	233	39	of	of	ADP
ejde-92	233	40	e	e	PRON
ejde-92	233	41	define	define	VERB
ejde-92	233	42	the	the	DET
ejde-92	233	43	edges	edge	NOUN
ejde-92	233	44	of	of	ADP
ejde-92	233	45	g	g	NOUN
ejde-92	234	1	and	and	CCONJ
ejde-92	234	2	it	it	PRON
ejde-92	234	3	is	be	AUX
ejde-92	234	4	said	say	VERB
ejde-92	234	5	that	that	SCONJ
ejde-92	234	6	u	u	NOUN
ejde-92	234	7	is	be	AUX
ejde-92	234	8	adjacent	adjacent	ADJ
ejde-92	234	9	(	(	PUNCT
ejde-92	234	10	or	or	CCONJ
ejde-92	234	11	‘	'	PUNCT
ejde-92	234	12	connected	connect	VERB
ejde-92	234	13	’	'	PUNCT
ejde-92	234	14	)	)	PUNCT
ejde-92	234	15	to	to	ADP
ejde-92	234	16	v	v	NOUN
ejde-92	234	17	when	when	SCONJ
ejde-92	234	18	{	{	PUNCT
ejde-92	234	19	u	u	NOUN
ejde-92	234	20	,	,	PUNCT
ejde-92	234	21	v	v	NOUN
ejde-92	234	22	}	}	PUNCT
ejde-92	234	23	∈	∈	PROPN
ejde-92	234	24	e.	e.	NOUN
ejde-92	235	1	it	it	PRON
ejde-92	235	2	is	be	AUX
ejde-92	235	3	often	often	ADV
ejde-92	235	4	convenient	convenient	ADJ
ejde-92	235	5	to	to	PART
ejde-92	235	6	associate	associate	VERB
ejde-92	235	7	a	a	DET
ejde-92	235	8	weight	weight	NOUN
ejde-92	235	9	ω	ω	NOUN
ejde-92	235	10	>	>	X
ejde-92	235	11	0	0	PUNCT
ejde-92	235	12	to	to	ADP
ejde-92	235	13	every	every	DET
ejde-92	235	14	edge	edge	NOUN
ejde-92	235	15	e	e	NOUN
ejde-92	235	16	=	=	SYM
ejde-92	235	17	{	{	PUNCT
ejde-92	235	18	u	u	NOUN
ejde-92	235	19	,	,	PUNCT
ejde-92	235	20	v	v	NOUN
ejde-92	235	21	}	}	PUNCT
ejde-92	235	22	∈	∈	PROPN
ejde-92	235	23	e	e	NOUN
ejde-92	235	24	,	,	PUNCT
ejde-92	235	25	which	which	PRON
ejde-92	235	26	could	could	AUX
ejde-92	235	27	be	be	AUX
ejde-92	235	28	understood	understand	VERB
ejde-92	235	29	as	as	ADP
ejde-92	235	30	the	the	DET
ejde-92	235	31	‘	'	PUNCT
ejde-92	235	32	connection	connection	NOUN
ejde-92	235	33	intensity	intensity	NOUN
ejde-92	235	34	’	'	PUNCT
ejde-92	235	35	between	between	ADP
ejde-92	235	36	the	the	DET
ejde-92	235	37	vertices	vertex	NOUN
ejde-92	235	38	u	u	NOUN
ejde-92	235	39	and	and	CCONJ
ejde-92	235	40	v.	v.	ADP
ejde-92	235	41	observe	observe	VERB
ejde-92	235	42	that	that	SCONJ
ejde-92	235	43	no	no	DET
ejde-92	235	44	order	order	NOUN
ejde-92	235	45	is	be	AUX
ejde-92	235	46	prescribed	prescribe	VERB
ejde-92	235	47	in	in	ADP
ejde-92	235	48	the	the	DET
ejde-92	235	49	edges	edge	NOUN
ejde-92	235	50	.	.	PUNCT
ejde-92	236	1	a	a	DET
ejde-92	236	2	possible	possible	ADJ
ejde-92	236	3	way	way	NOUN
ejde-92	236	4	of	of	ADP
ejde-92	236	5	simultaneously	simultaneously	ADV
ejde-92	236	6	defining	define	VERB
ejde-92	236	7	both	both	DET
ejde-92	236	8	10	10	NUM
ejde-92	236	9	j.	j.	PROPN
ejde-92	236	10	c.	c.	PROPN
ejde-92	236	11	sabina	sabina	PROPN
ejde-92	236	12	de	de	PROPN
ejde-92	236	13	lis	lis	X
ejde-92	236	14	ejde-2022/13	ejde-2022/13	VERB
ejde-92	236	15	the	the	DET
ejde-92	236	16	edges	edge	NOUN
ejde-92	236	17	and	and	CCONJ
ejde-92	236	18	their	their	PRON
ejde-92	236	19	weights	weight	NOUN
ejde-92	236	20	consists	consist	VERB
ejde-92	236	21	in	in	ADP
ejde-92	236	22	introducing	introduce	VERB
ejde-92	236	23	the	the	DET
ejde-92	236	24	weights	weight	NOUN
ejde-92	236	25	matrix	matrix	NOUN
ejde-92	236	26	a	a	DET
ejde-92	236	27	=	=	X
ejde-92	236	28	(	(	PUNCT
ejde-92	236	29	ωij	ωij	PROPN
ejde-92	236	30	)	)	PUNCT
ejde-92	236	31	of	of	ADP
ejde-92	236	32	g.	g.	PROPN
ejde-92	236	33	such	such	DET
ejde-92	236	34	a	a	DET
ejde-92	236	35	matrix	matrix	NOUN
ejde-92	236	36	a	a	PRON
ejde-92	236	37	is	be	AUX
ejde-92	236	38	always	always	ADV
ejde-92	236	39	chosen	choose	VERB
ejde-92	236	40	nonnegative	nonnegative	ADJ
ejde-92	236	41	and	and	CCONJ
ejde-92	236	42	symmetric	symmetric	ADJ
ejde-92	236	43	,	,	PUNCT
ejde-92	236	44	ωij	ωij	PROPN
ejde-92	236	45	=	=	SYM
ejde-92	236	46	ωji	ωji	NOUN
ejde-92	236	47	,	,	PUNCT
ejde-92	236	48	ωij	ωij	X
ejde-92	236	49	≥	≥	NOUN
ejde-92	236	50	0	0	NUM
ejde-92	236	51	,	,	PUNCT
ejde-92	236	52	while	while	SCONJ
ejde-92	236	53	ωij	ωij	VERB
ejde-92	236	54	>	>	X
ejde-92	236	55	0	0	PUNCT
ejde-92	237	1	both	both	PRON
ejde-92	237	2	means	mean	VERB
ejde-92	237	3	that	that	SCONJ
ejde-92	237	4	{	{	PUNCT
ejde-92	237	5	vi	vi	NOUN
ejde-92	237	6	,	,	PUNCT
ejde-92	237	7	vj	vj	ADJ
ejde-92	237	8	}	}	PUNCT
ejde-92	237	9	∈	∈	PROPN
ejde-92	237	10	e	e	NOUN
ejde-92	237	11	and	and	CCONJ
ejde-92	237	12	that	that	PRON
ejde-92	237	13	has	have	AUX
ejde-92	237	14	weight	weight	NOUN
ejde-92	237	15	ωij	ωij	NOUN
ejde-92	237	16	as	as	ADP
ejde-92	237	17	an	an	DET
ejde-92	237	18	edge	edge	NOUN
ejde-92	237	19	of	of	ADP
ejde-92	237	20	e.	e.	PROPN
ejde-92	237	21	observe	observe	VERB
ejde-92	237	22	that	that	SCONJ
ejde-92	237	23	,	,	PUNCT
ejde-92	237	24	from	from	ADP
ejde-92	237	25	the	the	DET
ejde-92	237	26	definition	definition	NOUN
ejde-92	237	27	of	of	ADP
ejde-92	237	28	e	e	NOUN
ejde-92	237	29	,	,	PUNCT
ejde-92	237	30	ωii	ωii	ADJ
ejde-92	237	31	=	=	NOUN
ejde-92	237	32	0	0	NUM
ejde-92	237	33	for	for	ADP
ejde-92	237	34	every	every	DET
ejde-92	237	35	1	1	NUM
ejde-92	237	36	≤	≤	NUM
ejde-92	237	37	i	i	PRON
ejde-92	237	38	≤	≤	ADJ
ejde-92	237	39	n.	n.	NOUN
ejde-92	237	40	matrix	matrix	NOUN
ejde-92	237	41	a	a	PRON
ejde-92	237	42	is	be	AUX
ejde-92	237	43	termed	term	VERB
ejde-92	237	44	as	as	ADP
ejde-92	237	45	the	the	DET
ejde-92	237	46	‘	'	PUNCT
ejde-92	237	47	adjacency	adjacency	NOUN
ejde-92	237	48	’	'	PUNCT
ejde-92	237	49	matrix	matrix	NOUN
ejde-92	237	50	when	when	SCONJ
ejde-92	237	51	the	the	DET
ejde-92	237	52	weights	weight	NOUN
ejde-92	237	53	ωij	ωij	VERB
ejde-92	237	54	∈	∈	PROPN
ejde-92	237	55	{	{	PUNCT
ejde-92	237	56	0	0	NUM
ejde-92	237	57	,	,	PUNCT
ejde-92	237	58	1	1	NUM
ejde-92	237	59	}	}	PUNCT
ejde-92	237	60	.	.	PUNCT
ejde-92	238	1	a	a	DET
ejde-92	238	2	subset	subset	NOUN
ejde-92	238	3	{	{	PUNCT
ejde-92	238	4	vi0	vi0	NOUN
ejde-92	238	5	,	,	PUNCT
ejde-92	238	6	vi1	vi1	INTJ
ejde-92	238	7	,	,	PUNCT
ejde-92	238	8	.	.	PUNCT
ejde-92	238	9	.	.	PUNCT
ejde-92	238	10	.	.	PUNCT
ejde-92	239	1	,	,	PUNCT
ejde-92	239	2	vim	vim	NOUN
ejde-92	239	3	}	}	PUNCT
ejde-92	239	4	of	of	ADP
ejde-92	239	5	m	m	PROPN
ejde-92	239	6	+	+	NOUN
ejde-92	239	7	1	1	NUM
ejde-92	239	8	vertices	vertex	NOUN
ejde-92	239	9	so	so	SCONJ
ejde-92	239	10	that	that	SCONJ
ejde-92	239	11	{	{	PUNCT
ejde-92	239	12	vik−1	vik−1	PROPN
ejde-92	239	13	,	,	PUNCT
ejde-92	239	14	vik	vik	PROPN
ejde-92	239	15	}	}	PUNCT
ejde-92	239	16	∈	∈	PROPN
ejde-92	239	17	e	e	NOUN
ejde-92	239	18	for	for	ADP
ejde-92	239	19	k	k	PROPN
ejde-92	239	20	=	=	SYM
ejde-92	239	21	1	1	NUM
ejde-92	239	22	,	,	PUNCT
ejde-92	239	23	.	.	PUNCT
ejde-92	239	24	.	.	PUNCT
ejde-92	240	1	.	.	PUNCT
ejde-92	241	1	,	,	PUNCT
ejde-92	241	2	m	m	NOUN
ejde-92	241	3	is	be	AUX
ejde-92	241	4	defined	define	VERB
ejde-92	241	5	to	to	PART
ejde-92	241	6	be	be	AUX
ejde-92	241	7	a	a	DET
ejde-92	241	8	path	path	NOUN
ejde-92	241	9	of	of	ADP
ejde-92	241	10	length	length	NOUN
ejde-92	241	11	m	m	AUX
ejde-92	241	12	connecting	connect	VERB
ejde-92	241	13	vi0	vi0	NOUN
ejde-92	241	14	to	to	ADP
ejde-92	241	15	vim	vim	PROPN
ejde-92	241	16	.	.	PUNCT
ejde-92	242	1	accordingly	accordingly	ADV
ejde-92	242	2	,	,	PUNCT
ejde-92	242	3	a	a	DET
ejde-92	242	4	graph	graph	NOUN
ejde-92	242	5	g	g	NOUN
ejde-92	242	6	is	be	AUX
ejde-92	242	7	said	say	VERB
ejde-92	242	8	to	to	PART
ejde-92	242	9	be	be	AUX
ejde-92	242	10	connected	connect	VERB
ejde-92	242	11	if	if	SCONJ
ejde-92	242	12	every	every	DET
ejde-92	242	13	couple	couple	NOUN
ejde-92	242	14	of	of	ADP
ejde-92	242	15	distinct	distinct	ADJ
ejde-92	242	16	vertices	vertex	NOUN
ejde-92	242	17	x	x	X
ejde-92	242	18	,	,	PUNCT
ejde-92	242	19	y	y	PROPN
ejde-92	242	20	∈	∈	PROPN
ejde-92	242	21	v	v	NOUN
ejde-92	242	22	can	can	AUX
ejde-92	242	23	be	be	AUX
ejde-92	242	24	joined	join	VERB
ejde-92	242	25	through	through	ADP
ejde-92	242	26	a	a	DET
ejde-92	242	27	path	path	NOUN
ejde-92	242	28	.	.	PUNCT
ejde-92	243	1	let	let	VERB
ejde-92	243	2	us	we	PRON
ejde-92	243	3	next	next	ADV
ejde-92	243	4	define	define	VERB
ejde-92	243	5	the	the	DET
ejde-92	243	6	p	p	ADJ
ejde-92	243	7	-	-	PUNCT
ejde-92	243	8	laplacian	laplacian	ADJ
ejde-92	243	9	operator	operator	NOUN
ejde-92	243	10	−∆p	−∆p	NOUN
ejde-92	243	11	on	on	ADP
ejde-92	243	12	g.	g.	PROPN
ejde-92	243	13	as	as	SCONJ
ejde-92	243	14	pointed	point	VERB
ejde-92	243	15	out	out	ADP
ejde-92	243	16	at	at	ADP
ejde-92	243	17	the	the	DET
ejde-92	243	18	end	end	NOUN
ejde-92	243	19	of	of	ADP
ejde-92	243	20	section	section	NOUN
ejde-92	243	21	2	2	NUM
ejde-92	243	22	,	,	PUNCT
ejde-92	244	1	the	the	DET
ejde-92	244	2	set	set	ADJ
ejde-92	244	3	h	h	NOUN
ejde-92	244	4	of	of	ADP
ejde-92	244	5	real	real	ADJ
ejde-92	244	6	functions	function	NOUN
ejde-92	244	7	f	f	NOUN
ejde-92	244	8	:	:	PUNCT
ejde-92	244	9	v	v	X
ejde-92	244	10	→	→	SYM
ejde-92	244	11	r	r	NOUN
ejde-92	244	12	can	can	AUX
ejde-92	244	13	be	be	AUX
ejde-92	244	14	identified	identify	VERB
ejde-92	244	15	with	with	ADP
ejde-92	244	16	rn	rn	PROPN
ejde-92	244	17	(	(	PUNCT
ejde-92	244	18	f	f	PROPN
ejde-92	244	19	=	=	SYM
ejde-92	244	20	(	(	PUNCT
ejde-92	244	21	fi	fi	NOUN
ejde-92	244	22	)	)	PUNCT
ejde-92	244	23	,	,	PUNCT
ejde-92	244	24	fi	fi	NOUN
ejde-92	244	25	=	=	SYM
ejde-92	244	26	f(i	f(i	PROPN
ejde-92	244	27	)	)	PUNCT
ejde-92	244	28	)	)	PUNCT
ejde-92	244	29	.	.	PUNCT
ejde-92	245	1	the	the	DET
ejde-92	245	2	p	p	NOUN
ejde-92	245	3	-	-	PUNCT
ejde-92	245	4	laplacian	laplacian	NOUN
ejde-92	245	5	in	in	ADP
ejde-92	245	6	g	g	PROPN
ejde-92	245	7	is	be	AUX
ejde-92	245	8	defined	define	VERB
ejde-92	245	9	as	as	ADP
ejde-92	245	10	a	a	DET
ejde-92	245	11	mapping	mapping	NOUN
ejde-92	245	12	from	from	ADP
ejde-92	245	13	h	h	NOUN
ejde-92	245	14	into	into	ADP
ejde-92	245	15	itself	itself	PRON
ejde-92	245	16	according	accord	VERB
ejde-92	245	17	the	the	DET
ejde-92	245	18	next	next	ADJ
ejde-92	245	19	definition	definition	NOUN
ejde-92	245	20	.	.	PUNCT
ejde-92	246	1	the	the	DET
ejde-92	246	2	eigenvalue	eigenvalue	PROPN
ejde-92	246	3	problem	problem	NOUN
ejde-92	246	4	for	for	ADP
ejde-92	246	5	−∆p	−∆p	NOUN
ejde-92	246	6	is	be	AUX
ejde-92	246	7	also	also	ADV
ejde-92	246	8	introduced	introduce	VERB
ejde-92	246	9	there	there	ADV
ejde-92	246	10	.	.	PUNCT
ejde-92	247	1	warning	warn	VERB
ejde-92	247	2	:	:	PUNCT
ejde-92	247	3	to	to	PART
ejde-92	247	4	keep	keep	VERB
ejde-92	247	5	the	the	DET
ejde-92	247	6	similarities	similarity	NOUN
ejde-92	247	7	with	with	ADP
ejde-92	247	8	the	the	DET
ejde-92	247	9	partial	partial	ADJ
ejde-92	247	10	differential	differential	ADJ
ejde-92	247	11	equations	equation	NOUN
ejde-92	247	12	setting	set	VERB
ejde-92	247	13	,	,	PUNCT
ejde-92	247	14	a	a	DET
ejde-92	247	15	minus	minus	NOUN
ejde-92	247	16	sign	sign	NOUN
ejde-92	247	17	preceding	precede	VERB
ejde-92	247	18	the	the	DET
ejde-92	247	19	operator	operator	NOUN
ejde-92	247	20	is	be	AUX
ejde-92	247	21	employed	employ	VERB
ejde-92	247	22	.	.	PUNCT
ejde-92	248	1	definition	definition	NOUN
ejde-92	248	2	5.1	5.1	NUM
ejde-92	248	3	.	.	PUNCT
ejde-92	249	1	let	let	VERB
ejde-92	249	2	g	g	PROPN
ejde-92	249	3	=	=	SYM
ejde-92	249	4	(	(	PUNCT
ejde-92	249	5	v	v	NOUN
ejde-92	249	6	,	,	PUNCT
ejde-92	249	7	e	e	NOUN
ejde-92	249	8	)	)	PUNCT
ejde-92	249	9	be	be	AUX
ejde-92	249	10	a	a	DET
ejde-92	249	11	graph	graph	NOUN
ejde-92	249	12	with	with	ADP
ejde-92	249	13	weights	weight	NOUN
ejde-92	249	14	matrix	matrix	NOUN
ejde-92	249	15	a	a	DET
ejde-92	249	16	=	=	X
ejde-92	249	17	(	(	PUNCT
ejde-92	249	18	ωij	ωij	PROPN
ejde-92	249	19	)	)	PUNCT
ejde-92	249	20	.	.	PUNCT
ejde-92	250	1	the	the	DET
ejde-92	250	2	p	p	PROPN
ejde-92	250	3	-	-	PUNCT
ejde-92	250	4	laplacian	laplacian	ADJ
ejde-92	250	5	operator	operator	NOUN
ejde-92	250	6	−∆p	−∆p	NOUN
ejde-92	250	7	:	:	PUNCT
ejde-92	250	8	h	h	NOUN
ejde-92	250	9	→	→	PUNCT
ejde-92	250	10	h	h	NOUN
ejde-92	250	11	is	be	AUX
ejde-92	250	12	defined	define	VERB
ejde-92	250	13	as	as	ADP
ejde-92	250	14	−∆p(f)(i	−∆p(f)(i	NOUN
ejde-92	250	15	)	)	PUNCT
ejde-92	251	1	=	=	PUNCT
ejde-92	252	1	∑	∑	PUNCT
ejde-92	252	2	j	j	PROPN
ejde-92	252	3	ωij	ωij	VERB
ejde-92	252	4	|fi	|fi	ADP
ejde-92	252	5	−	−	PROPN
ejde-92	252	6	fj	fj	INTJ
ejde-92	252	7	|p−2(fi	|p−2(fi	NUM
ejde-92	252	8	−	−	PROPN
ejde-92	252	9	fj	fj	PROPN
ejde-92	252	10	)	)	PUNCT
ejde-92	252	11	,	,	PUNCT
ejde-92	252	12	1	1	NUM
ejde-92	252	13	≤	≤	NUM
ejde-92	252	14	i	i	PRON
ejde-92	252	15	≤	≤	ADJ
ejde-92	252	16	n.	n.	NOUN
ejde-92	252	17	(	(	PUNCT
ejde-92	252	18	5.1	5.1	NUM
ejde-92	252	19	)	)	PUNCT
ejde-92	252	20	it	it	PRON
ejde-92	252	21	is	be	AUX
ejde-92	252	22	said	say	VERB
ejde-92	252	23	that	that	SCONJ
ejde-92	252	24	λ	λ	PROPN
ejde-92	252	25	∈	∈	NOUN
ejde-92	252	26	r	r	NOUN
ejde-92	252	27	is	be	AUX
ejde-92	252	28	an	an	DET
ejde-92	252	29	eigenvalue	eigenvalue	NOUN
ejde-92	252	30	to	to	ADP
ejde-92	252	31	−∆p	−∆p	NOUN
ejde-92	252	32	with	with	ADP
ejde-92	252	33	associated	associated	ADJ
ejde-92	252	34	eigenvector	eigenvector	PROPN
ejde-92	252	35	f	f	PROPN
ejde-92	252	36	∈	∈	PROPN
ejde-92	252	37	h\{0	h\{0	PROPN
ejde-92	252	38	}	}	PUNCT
ejde-92	252	39	provided	provide	VERB
ejde-92	252	40	that	that	SCONJ
ejde-92	252	41	−∆p(f	−∆p(f	PROPN
ejde-92	252	42	)	)	PUNCT
ejde-92	253	1	=	=	SYM
ejde-92	253	2	λνφp(f	λνφp(f	PROPN
ejde-92	253	3	)	)	PUNCT
ejde-92	253	4	,	,	PUNCT
ejde-92	253	5	(	(	PUNCT
ejde-92	253	6	5.2	5.2	NUM
ejde-92	253	7	)	)	PUNCT
ejde-92	253	8	where	where	SCONJ
ejde-92	253	9	ν	ν	NOUN
ejde-92	253	10	=	=	SYM
ejde-92	253	11	(	(	PUNCT
ejde-92	253	12	νi	νi	NOUN
ejde-92	253	13	)	)	PUNCT
ejde-92	253	14	∈	∈	NOUN
ejde-92	253	15	rn+	rn+	NOUN
ejde-92	253	16	is	be	AUX
ejde-92	253	17	a	a	DET
ejde-92	253	18	given	give	VERB
ejde-92	253	19	weight	weight	NOUN
ejde-92	253	20	function	function	NOUN
ejde-92	253	21	,	,	PUNCT
ejde-92	253	22	φp(f)(i	φp(f)(i	NOUN
ejde-92	253	23	)	)	PUNCT
ejde-92	253	24	=	=	SYM
ejde-92	254	1	|fi|p−2fi	|fi|p−2fi	PROPN
ejde-92	254	2	,	,	PUNCT
ejde-92	254	3	1	1	NUM
ejde-92	254	4	≤	≤	NUM
ejde-92	254	5	i	i	PRON
ejde-92	254	6	≤	≤	ADJ
ejde-92	254	7	n.	n.	NOUN
ejde-92	254	8	remark	remark	NOUN
ejde-92	254	9	5.2	5.2	NUM
ejde-92	254	10	.	.	PUNCT
ejde-92	255	1	when	when	SCONJ
ejde-92	255	2	p	p	NOUN
ejde-92	255	3	=	=	SYM
ejde-92	255	4	2	2	NUM
ejde-92	255	5	,	,	PUNCT
ejde-92	255	6	−∆p	−∆p	NOUN
ejde-92	255	7	becomes	become	VERB
ejde-92	255	8	the	the	DET
ejde-92	255	9	linear	linear	ADJ
ejde-92	255	10	operator	operator	NOUN
ejde-92	255	11	(	(	PUNCT
ejde-92	255	12	the	the	DET
ejde-92	255	13	laplacian	laplacian	NOUN
ejde-92	255	14	in	in	ADP
ejde-92	255	15	g	g	NOUN
ejde-92	255	16	):	):	PUNCT
ejde-92	255	17	−∆2f	−∆2f	X
ejde-92	255	18	=	=	PUNCT
ejde-92	255	19	(	(	PUNCT
ejde-92	255	20	d	d	ADP
ejde-92	255	21	−a)f	−a)f	ADJ
ejde-92	255	22	,	,	PUNCT
ejde-92	255	23	a	a	DET
ejde-92	255	24	being	be	AUX
ejde-92	255	25	the	the	DET
ejde-92	255	26	weights	weight	NOUN
ejde-92	255	27	matrix	matrix	NOUN
ejde-92	255	28	of	of	ADP
ejde-92	255	29	g	g	PROPN
ejde-92	255	30	and	and	CCONJ
ejde-92	255	31	d	d	NOUN
ejde-92	255	32	=	=	PUNCT
ejde-92	255	33	diag(d1	diag(d1	NOUN
ejde-92	255	34	,	,	PUNCT
ejde-92	255	35	.	.	PUNCT
ejde-92	255	36	.	.	PUNCT
ejde-92	256	1	.	.	PUNCT
ejde-92	257	1	,	,	PUNCT
ejde-92	257	2	dn	dn	PROPN
ejde-92	257	3	)	)	PUNCT
ejde-92	257	4	where	where	SCONJ
ejde-92	257	5	di	di	X
ejde-92	257	6	=	=	PUNCT
ejde-92	257	7	∑	∑	PUNCT
ejde-92	257	8	j	j	PROPN
ejde-92	257	9	ωij	ωij	VERB
ejde-92	257	10	.	.	PUNCT
ejde-92	258	1	in	in	ADP
ejde-92	258	2	this	this	DET
ejde-92	258	3	case	case	NOUN
ejde-92	258	4	the	the	DET
ejde-92	258	5	spectrum	spectrum	NOUN
ejde-92	258	6	of	of	ADP
ejde-92	258	7	−∆2	−∆2	NOUN
ejde-92	258	8	corresponding	correspond	VERB
ejde-92	258	9	to	to	ADP
ejde-92	258	10	the	the	DET
ejde-92	258	11	weight	weight	NOUN
ejde-92	258	12	ν	ν	NOUN
ejde-92	258	13	=	=	SYM
ejde-92	258	14	1	1	NUM
ejde-92	258	15	consists	consist	VERB
ejde-92	258	16	of	of	ADP
ejde-92	258	17	the	the	DET
ejde-92	258	18	eigenvalues	eigenvalue	NOUN
ejde-92	258	19	of	of	ADP
ejde-92	258	20	d	d	NOUN
ejde-92	258	21	−	−	PROPN
ejde-92	258	22	a.	a.	NOUN
ejde-92	258	23	in	in	ADP
ejde-92	258	24	some	some	DET
ejde-92	258	25	cases	case	NOUN
ejde-92	258	26	the	the	DET
ejde-92	258	27	interest	interest	NOUN
ejde-92	258	28	is	be	AUX
ejde-92	258	29	focussed	focusse	VERB
ejde-92	258	30	on	on	ADP
ejde-92	258	31	the	the	DET
ejde-92	258	32	normalized	normalize	VERB
ejde-92	258	33	eigenvalues	eigenvalue	NOUN
ejde-92	258	34	of	of	ADP
ejde-92	258	35	−∆2	−∆2	NOUN
ejde-92	258	36	(	(	PUNCT
ejde-92	258	37	respectively	respectively	ADV
ejde-92	258	38	,	,	PUNCT
ejde-92	258	39	−∆p	−∆p	NOUN
ejde-92	258	40	)	)	PUNCT
ejde-92	258	41	.	.	PUNCT
ejde-92	259	1	these	these	PRON
ejde-92	259	2	are	be	AUX
ejde-92	259	3	the	the	DET
ejde-92	259	4	eigenvalues	eigenvalue	NOUN
ejde-92	259	5	corresponding	correspond	VERB
ejde-92	259	6	to	to	ADP
ejde-92	259	7	the	the	DET
ejde-92	259	8	choice	choice	NOUN
ejde-92	259	9	ν	ν	X
ejde-92	259	10	=	=	SYM
ejde-92	259	11	(	(	PUNCT
ejde-92	259	12	di	di	NOUN
ejde-92	259	13	)	)	PUNCT
ejde-92	259	14	as	as	ADP
ejde-92	259	15	a	a	DET
ejde-92	259	16	weight	weight	NOUN
ejde-92	259	17	function	function	NOUN
ejde-92	259	18	in	in	ADP
ejde-92	259	19	(	(	PUNCT
ejde-92	259	20	5.2	5.2	NUM
ejde-92	259	21	)	)	PUNCT
ejde-92	259	22	(	(	PUNCT
ejde-92	259	23	see	see	VERB
ejde-92	259	24	[	[	X
ejde-92	259	25	10	10	NUM
ejde-92	259	26	,	,	PUNCT
ejde-92	259	27	8	8	NUM
ejde-92	259	28	]	]	NUM
ejde-92	259	29	)	)	PUNCT
ejde-92	259	30	.	.	PUNCT
ejde-92	260	1	proof	proof	NOUN
ejde-92	260	2	of	of	ADP
ejde-92	260	3	theorem	theorem	ADJ
ejde-92	260	4	1.2	1.2	NUM
ejde-92	260	5	.	.	PUNCT
ejde-92	261	1	we	we	PRON
ejde-92	261	2	first	first	ADV
ejde-92	261	3	review	review	VERB
ejde-92	261	4	some	some	DET
ejde-92	261	5	preliminary	preliminary	ADJ
ejde-92	261	6	well	well	ADV
ejde-92	261	7	known	know	VERB
ejde-92	261	8	features	feature	NOUN
ejde-92	261	9	[	[	X
ejde-92	261	10	2	2	NUM
ejde-92	261	11	,	,	PUNCT
ejde-92	261	12	8	8	NUM
ejde-92	261	13	]	]	PUNCT
ejde-92	261	14	.	.	PUNCT
ejde-92	262	1	by	by	ADP
ejde-92	262	2	using	use	VERB
ejde-92	262	3	the	the	DET
ejde-92	262	4	euclidean	euclidean	ADJ
ejde-92	262	5	scalar	scalar	ADJ
ejde-92	262	6	product	product	NOUN
ejde-92	262	7	〈	〈	PROPN
ejde-92	262	8	·	·	SYM
ejde-92	262	9	,	,	PUNCT
ejde-92	262	10	·	·	SYM
ejde-92	262	11	〉	〉	NOUN
ejde-92	262	12	2	2	NUM
ejde-92	262	13	of	of	ADP
ejde-92	262	14	rn	rn	PROPN
ejde-92	262	15	it	it	PRON
ejde-92	262	16	is	be	AUX
ejde-92	262	17	found	find	VERB
ejde-92	262	18	that	that	SCONJ
ejde-92	262	19	〈	〈	PROPN
ejde-92	262	20	−∆p(f	−∆p(f	PROPN
ejde-92	262	21	)	)	PUNCT
ejde-92	262	22	,	,	PUNCT
ejde-92	262	23	f〉2	f〉2	NOUN
ejde-92	262	24	=	=	PUNCT
ejde-92	262	25	∑	∑	PUNCT
ejde-92	262	26	i	i	X
ejde-92	262	27	∑	∑	PUNCT
ejde-92	262	28	j	j	PROPN
ejde-92	262	29	ωij	ωij	VERB
ejde-92	262	30	|fi	|fi	ADP
ejde-92	262	31	−	−	PROPN
ejde-92	262	32	fj	fj	INTJ
ejde-92	262	33	|p−2(fi	|p−2(fi	NUM
ejde-92	262	34	−	−	NOUN
ejde-92	262	35	fj)fi	fj)fi	X
ejde-92	262	36	=	=	PUNCT
ejde-92	262	37	−	−	PROPN
ejde-92	262	38	∑	∑	PUNCT
ejde-92	262	39	i	i	PRON
ejde-92	262	40	∑	∑	PUNCT
ejde-92	262	41	j	j	PROPN
ejde-92	262	42	ωij	ωij	VERB
ejde-92	262	43	|fi	|fi	ADP
ejde-92	262	44	−	−	PROPN
ejde-92	262	45	fj	fj	INTJ
ejde-92	262	46	|p−2(fi	|p−2(fi	NUM
ejde-92	262	47	−	−	PROPN
ejde-92	263	1	fj)fj	fj)fj	PROPN
ejde-92	263	2	,	,	PUNCT
ejde-92	263	3	which	which	PRON
ejde-92	263	4	implies	imply	VERB
ejde-92	263	5	〈	〈	PROPN
ejde-92	263	6	−∆p(f	−∆p(f	PROPN
ejde-92	263	7	)	)	PUNCT
ejde-92	263	8	,	,	PUNCT
ejde-92	263	9	f〉2	f〉2	NOUN
ejde-92	263	10	=	=	PUNCT
ejde-92	263	11	1	1	NUM
ejde-92	263	12	2	2	NUM
ejde-92	263	13	∑	∑	PROPN
ejde-92	263	14	i	i	PROPN
ejde-92	263	15	,	,	PUNCT
ejde-92	263	16	j	j	PROPN
ejde-92	263	17	ωij	ωij	VERB
ejde-92	263	18	|fi	|fi	ADP
ejde-92	263	19	−	−	PROPN
ejde-92	263	20	fj	fj	INTJ
ejde-92	263	21	|p	|p	NOUN
ejde-92	263	22	=	=	NOUN
ejde-92	263	23	:	:	PUNCT
ejde-92	263	24	d(f	d(f	NOUN
ejde-92	263	25	)	)	PUNCT
ejde-92	263	26	.	.	PUNCT
ejde-92	264	1	(	(	PUNCT
ejde-92	264	2	5.3	5.3	NUM
ejde-92	264	3	)	)	PUNCT
ejde-92	264	4	as	as	ADP
ejde-92	264	5	a	a	DET
ejde-92	264	6	consequence	consequence	NOUN
ejde-92	264	7	,	,	PUNCT
ejde-92	264	8	all	all	DET
ejde-92	264	9	possible	possible	ADJ
ejde-92	264	10	eigenvalues	eigenvalue	VERB
ejde-92	264	11	λ	λ	NOUN
ejde-92	264	12	of	of	ADP
ejde-92	264	13	−∆p	−∆p	NOUN
ejde-92	264	14	must	must	AUX
ejde-92	264	15	be	be	AUX
ejde-92	264	16	nonnegative	nonnegative	ADJ
ejde-92	264	17	since	since	SCONJ
ejde-92	264	18	the	the	DET
ejde-92	264	19	weight	weight	NOUN
ejde-92	264	20	function	function	NOUN
ejde-92	264	21	ν	ν	NOUN
ejde-92	264	22	is	be	AUX
ejde-92	264	23	positive	positive	ADJ
ejde-92	264	24	.	.	PUNCT
ejde-92	265	1	another	another	DET
ejde-92	265	2	implication	implication	NOUN
ejde-92	265	3	of	of	ADP
ejde-92	265	4	(	(	PUNCT
ejde-92	265	5	5.3	5.3	NUM
ejde-92	265	6	)	)	PUNCT
ejde-92	265	7	is	be	AUX
ejde-92	265	8	the	the	DET
ejde-92	265	9	fact	fact	NOUN
ejde-92	265	10	that	that	SCONJ
ejde-92	265	11	λ	λ	NOUN
ejde-92	265	12	=	=	SYM
ejde-92	265	13	0	0	NUM
ejde-92	265	14	is	be	AUX
ejde-92	265	15	a	a	DET
ejde-92	265	16	simple	simple	ADJ
ejde-92	265	17	eigenvalue	eigenvalue	NOUN
ejde-92	265	18	since	since	SCONJ
ejde-92	265	19	its	its	PRON
ejde-92	265	20	set	set	NOUN
ejde-92	265	21	of	of	ADP
ejde-92	265	22	associated	associated	ADJ
ejde-92	265	23	eigenfunctions	eigenfunction	NOUN
ejde-92	265	24	is	be	AUX
ejde-92	265	25	span{1	span{1	NOUN
ejde-92	265	26	}	}	PUNCT
ejde-92	265	27	,	,	PUNCT
ejde-92	265	28	1	1	NUM
ejde-92	265	29	standing	stand	VERB
ejde-92	265	30	for	for	ADP
ejde-92	265	31	the	the	DET
ejde-92	265	32	function	function	NOUN
ejde-92	265	33	f	f	NOUN
ejde-92	265	34	=	=	SYM
ejde-92	265	35	1	1	X
ejde-92	265	36	.	.	PUNCT
ejde-92	266	1	in	in	ADP
ejde-92	266	2	this	this	DET
ejde-92	266	3	regard	regard	NOUN
ejde-92	266	4	,	,	PUNCT
ejde-92	266	5	the	the	DET
ejde-92	266	6	connectedness	connectedness	NOUN
ejde-92	266	7	of	of	ADP
ejde-92	266	8	g	g	PROPN
ejde-92	266	9	is	be	AUX
ejde-92	266	10	employed	employ	VERB
ejde-92	266	11	to	to	PART
ejde-92	266	12	show	show	VERB
ejde-92	266	13	that	that	SCONJ
ejde-92	266	14	for	for	ADP
ejde-92	266	15	every	every	DET
ejde-92	266	16	associated	associated	ADJ
ejde-92	266	17	eigenfunction	eigenfunction	NOUN
ejde-92	266	18	f	f	NOUN
ejde-92	266	19	it	it	PRON
ejde-92	266	20	holds	hold	VERB
ejde-92	266	21	fi	fi	NOUN
ejde-92	266	22	=	=	PROPN
ejde-92	266	23	fj	fj	PROPN
ejde-92	266	24	for	for	ADP
ejde-92	266	25	every	every	DET
ejde-92	266	26	couple	couple	NOUN
ejde-92	266	27	of	of	ADP
ejde-92	266	28	indices	index	NOUN
ejde-92	266	29	ejde-2022/13	ejde-2022/13	ADJ
ejde-92	266	30	second	second	ADJ
ejde-92	266	31	neumann	neumann	PROPN
ejde-92	266	32	eigenvalue	eigenvalue	X
ejde-92	266	33	11	11	NUM
ejde-92	266	34	1	1	NUM
ejde-92	266	35	≤	≤	NOUN
ejde-92	266	36	i	i	PRON
ejde-92	266	37	,	,	PUNCT
ejde-92	266	38	j	j	PROPN
ejde-92	266	39	≤	≤	PROPN
ejde-92	266	40	n.	n.	NOUN
ejde-92	266	41	in	in	ADP
ejde-92	266	42	fact	fact	NOUN
ejde-92	266	43	,	,	PUNCT
ejde-92	266	44	the	the	DET
ejde-92	266	45	constant	constant	ADJ
ejde-92	266	46	value	value	NOUN
ejde-92	266	47	f	f	PROPN
ejde-92	266	48	=	=	NOUN
ejde-92	266	49	fi	fi	NOUN
ejde-92	266	50	is	be	AUX
ejde-92	266	51	‘	'	PUNCT
ejde-92	266	52	propagated	propagate	VERB
ejde-92	266	53	’	'	PUNCT
ejde-92	266	54	through	through	ADP
ejde-92	266	55	the	the	DET
ejde-92	266	56	path	path	NOUN
ejde-92	266	57	connecting	connect	VERB
ejde-92	266	58	vi	vi	PROPN
ejde-92	266	59	to	to	ADP
ejde-92	266	60	vj	vj	PROPN
ejde-92	266	61	.	.	PUNCT
ejde-92	267	1	next	next	ADJ
ejde-92	267	2	observe	observe	VERB
ejde-92	267	3	that	that	SCONJ
ejde-92	267	4	〈	〈	PROPN
ejde-92	267	5	−∆p(f	−∆p(f	PROPN
ejde-92	267	6	)	)	PUNCT
ejde-92	267	7	,	,	PUNCT
ejde-92	267	8	1〉2	1〉2	NUM
ejde-92	268	1	=	=	PUNCT
ejde-92	268	2	∑	∑	PUNCT
ejde-92	268	3	i	i	PRON
ejde-92	268	4	(	(	PUNCT
ejde-92	268	5	−∆p(f))i	−∆p(f))i	PROPN
ejde-92	268	6	=	=	PUNCT
ejde-92	268	7	∑	∑	PUNCT
ejde-92	268	8	i	i	X
ejde-92	268	9	∑	∑	PUNCT
ejde-92	268	10	j	j	PROPN
ejde-92	268	11	ωij	ωij	VERB
ejde-92	268	12	|fi	|fi	ADP
ejde-92	268	13	−	−	PROPN
ejde-92	268	14	fj	fj	INTJ
ejde-92	268	15	|p−2(fi	|p−2(fi	NUM
ejde-92	268	16	−	−	PROPN
ejde-92	268	17	fj	fj	PROPN
ejde-92	268	18	)	)	PUNCT
ejde-92	268	19	=	=	NOUN
ejde-92	268	20	0	0	X
ejde-92	268	21	.	.	PUNCT
ejde-92	269	1	that	that	PRON
ejde-92	269	2	is	be	AUX
ejde-92	269	3	why	why	SCONJ
ejde-92	269	4	any	any	DET
ejde-92	269	5	eigenfunction	eigenfunction	NOUN
ejde-92	269	6	f	f	PROPN
ejde-92	269	7	associated	associate	VERB
ejde-92	269	8	with	with	ADP
ejde-92	269	9	an	an	DET
ejde-92	269	10	eigenvalue	eigenvalue	PROPN
ejde-92	269	11	λ	λ	PROPN
ejde-92	269	12	6=	6=	ADP
ejde-92	269	13	0	0	NUM
ejde-92	269	14	must	must	AUX
ejde-92	269	15	satisfy	satisfy	VERB
ejde-92	269	16	the	the	DET
ejde-92	269	17	condition	condition	NOUN
ejde-92	269	18	equivalent	equivalent	ADJ
ejde-92	269	19	to	to	ADP
ejde-92	269	20	(	(	PUNCT
ejde-92	269	21	1.5	1.5	NUM
ejde-92	269	22	)	)	PUNCT
ejde-92	269	23	,	,	PUNCT
ejde-92	269	24	i.	i.	PROPN
ejde-92	269	25	e.	e.	PROPN
ejde-92	269	26	,∑	,∑	PUNCT
ejde-92	269	27	i	i	PRON
ejde-92	269	28	|fi|p−2fiνi	|fi|p−2fiνi	X
ejde-92	269	29	=	=	SYM
ejde-92	269	30	0	0	PROPN
ejde-92	269	31	.	.	PUNCT
ejde-92	270	1	(	(	PUNCT
ejde-92	270	2	5.4	5.4	NUM
ejde-92	270	3	)	)	PUNCT
ejde-92	270	4	set	set	VERB
ejde-92	270	5	now	now	ADV
ejde-92	270	6	m1	m1	PROPN
ejde-92	270	7	=	=	PUNCT
ejde-92	271	1	{	{	PUNCT
ejde-92	271	2	f	f	PROPN
ejde-92	271	3	∈	∈	PROPN
ejde-92	271	4	rn	rn	PROPN
ejde-92	271	5	:	:	PUNCT
ejde-92	271	6	f	f	PROPN
ejde-92	271	7	satisfies	satisfie	NOUN
ejde-92	271	8	(	(	PUNCT
ejde-92	271	9	5.4	5.4	NUM
ejde-92	271	10	)	)	PUNCT
ejde-92	271	11	and	and	CCONJ
ejde-92	271	12	∑	∑	PUNCT
ejde-92	271	13	i	i	PRON
ejde-92	271	14	|fi|pνi	|fi|pνi	ADJ
ejde-92	271	15	=	=	NOUN
ejde-92	271	16	1	1	X
ejde-92	271	17	}	}	PUNCT
ejde-92	271	18	.	.	PUNCT
ejde-92	272	1	by	by	ADP
ejde-92	272	2	compactness	compactness	NOUN
ejde-92	272	3	,	,	PUNCT
ejde-92	272	4	function	function	NOUN
ejde-92	272	5	d	d	AUX
ejde-92	272	6	achieves	achieve	VERB
ejde-92	272	7	its	its	PRON
ejde-92	272	8	minimum	minimum	NOUN
ejde-92	272	9	at	at	ADP
ejde-92	272	10	some	some	DET
ejde-92	272	11	f1	f1	NOUN
ejde-92	272	12	∈m1	∈m1	NOUN
ejde-92	272	13	.	.	PUNCT
ejde-92	273	1	in	in	ADP
ejde-92	273	2	addition	addition	NOUN
ejde-92	273	3	,	,	PUNCT
ejde-92	273	4	0	0	PUNCT
ejde-92	273	5	<	<	X
ejde-92	273	6	λ̂	λ̂	X
ejde-92	273	7	:	:	PUNCT
ejde-92	273	8	=	=	PUNCT
ejde-92	273	9	d(f1	d(f1	NOUN
ejde-92	273	10	)	)	PUNCT
ejde-92	274	1	=	=	SYM
ejde-92	274	2	min	min	PROPN
ejde-92	274	3	f	f	PROPN
ejde-92	274	4	/∈span{1	/∈span{1	PROPN
ejde-92	274	5	}	}	PUNCT
ejde-92	274	6	d(f	d(f	PROPN
ejde-92	274	7	)	)	PUNCT
ejde-92	274	8	vp(f	vp(f	NUM
ejde-92	274	9	)	)	PUNCT
ejde-92	274	10	,	,	PUNCT
ejde-92	274	11	since	since	SCONJ
ejde-92	274	12	{	{	PUNCT
ejde-92	274	13	f	f	PROPN
ejde-92	274	14	∈	∈	PROPN
ejde-92	274	15	rn	rn	PROPN
ejde-92	274	16	:	:	PUNCT
ejde-92	274	17	∑	∑	PUNCT
ejde-92	274	18	|fi|p−2fiνi	|fi|p−2fiνi	X
ejde-92	274	19	=	=	SYM
ejde-92	274	20	0	0	NUM
ejde-92	274	21	}	}	PUNCT
ejde-92	274	22	\	\	NOUN
ejde-92	274	23	{	{	PUNCT
ejde-92	274	24	0	0	NUM
ejde-92	274	25	}	}	PUNCT
ejde-92	274	26	=	=	PRON
ejde-92	274	27	{	{	PUNCT
ejde-92	274	28	f	f	X
ejde-92	274	29	−	−	PROPN
ejde-92	274	30	f̃	f̃	PROPN
ejde-92	274	31	:	:	PUNCT
ejde-92	274	32	f	f	PROPN
ejde-92	274	33	∈	∈	PROPN
ejde-92	274	34	rn	rn	PROPN
ejde-92	274	35	\	\	PROPN
ejde-92	274	36	span{1	span{1	NOUN
ejde-92	274	37	}	}	PUNCT
ejde-92	274	38	}	}	PUNCT
ejde-92	274	39	.	.	PUNCT
ejde-92	275	1	next	next	ADJ
ejde-92	275	2	observe	observe	VERB
ejde-92	275	3	that	that	SCONJ
ejde-92	275	4	∇d(f	∇d(f	NOUN
ejde-92	275	5	)	)	PUNCT
ejde-92	275	6	=	=	SYM
ejde-92	275	7	p(−∆p(f	p(−∆p(f	PROPN
ejde-92	275	8	)	)	PUNCT
ejde-92	275	9	)	)	PUNCT
ejde-92	275	10	.	.	PUNCT
ejde-92	276	1	hence	hence	ADV
ejde-92	276	2	,	,	PUNCT
ejde-92	276	3	it	it	PRON
ejde-92	276	4	readily	readily	ADV
ejde-92	276	5	follows	follow	VERB
ejde-92	276	6	by	by	ADP
ejde-92	276	7	differentiating	differentiate	VERB
ejde-92	276	8	d(u	d(u	PROPN
ejde-92	276	9	)	)	PUNCT
ejde-92	276	10	vp(u	vp(u	VERB
ejde-92	276	11	)	)	PUNCT
ejde-92	276	12	and	and	CCONJ
ejde-92	276	13	employing	employ	VERB
ejde-92	276	14	(	(	PUNCT
ejde-92	276	15	2.5	2.5	NUM
ejde-92	276	16	)	)	PUNCT
ejde-92	276	17	that	that	SCONJ
ejde-92	276	18	−∆p(f1	−∆p(f1	VERB
ejde-92	276	19	−	−	PUNCT
ejde-92	276	20	f̃1	f̃1	NOUN
ejde-92	276	21	)	)	PUNCT
ejde-92	276	22	=	=	PUNCT
ejde-92	277	1	λ̂νφp(f1	λ̂νφp(f1	ADP
ejde-92	277	2	−	−	PROPN
ejde-92	277	3	f̃1	f̃1	NOUN
ejde-92	277	4	)	)	PUNCT
ejde-92	277	5	.	.	PUNCT
ejde-92	278	1	this	this	PRON
ejde-92	278	2	means	mean	VERB
ejde-92	278	3	that	that	SCONJ
ejde-92	278	4	λ̂	λ̂	X
ejde-92	278	5	defines	define	VERB
ejde-92	278	6	the	the	DET
ejde-92	278	7	second	second	ADJ
ejde-92	278	8	eigenvalue	eigenvalue	PROPN
ejde-92	278	9	λ2	λ2	PROPN
ejde-92	278	10	.	.	PUNCT
ejde-92	279	1	observe	observe	VERB
ejde-92	279	2	that	that	PRON
ejde-92	279	3	corollary	corollary	ADJ
ejde-92	279	4	2.7	2.7	NUM
ejde-92	279	5	has	have	AUX
ejde-92	279	6	permitted	permit	VERB
ejde-92	279	7	us	we	PRON
ejde-92	279	8	handling	handle	VERB
ejde-92	279	9	both	both	DET
ejde-92	279	10	the	the	DET
ejde-92	279	11	case	case	NOUN
ejde-92	279	12	p	p	X
ejde-92	279	13	≥	≥	NUM
ejde-92	279	14	2	2	NUM
ejde-92	279	15	and	and	CCONJ
ejde-92	279	16	the	the	DET
ejde-92	279	17	singular	singular	NOUN
ejde-92	279	18	one	one	NUM
ejde-92	279	19	1	1	NUM
ejde-92	279	20	<	<	X
ejde-92	280	1	p	p	X
ejde-92	281	1	<	<	X
ejde-92	281	2	2	2	NUM
ejde-92	281	3	.	.	PUNCT
ejde-92	281	4	by	by	ADP
ejde-92	281	5	arguing	argue	VERB
ejde-92	281	6	in	in	ADP
ejde-92	281	7	the	the	DET
ejde-92	281	8	same	same	ADJ
ejde-92	281	9	way	way	NOUN
ejde-92	281	10	one	one	PRON
ejde-92	281	11	obtains	obtain	VERB
ejde-92	281	12	that	that	DET
ejde-92	281	13	λ∗	λ∗	NOUN
ejde-92	281	14	:	:	PUNCT
ejde-92	281	15	=	=	SYM
ejde-92	281	16	max	max	PROPN
ejde-92	281	17	f	f	PROPN
ejde-92	281	18	/∈span{1	/∈span{1	PROPN
ejde-92	281	19	}	}	PUNCT
ejde-92	281	20	d(f	d(f	PROPN
ejde-92	281	21	)	)	PUNCT
ejde-92	281	22	vp(f	vp(f	NUM
ejde-92	281	23	)	)	PUNCT
ejde-92	281	24	,	,	PUNCT
ejde-92	281	25	constitutes	constitute	VERB
ejde-92	281	26	the	the	DET
ejde-92	281	27	maximum	maximum	PROPN
ejde-92	281	28	eigenvalue	eigenvalue	NOUN
ejde-92	281	29	of	of	ADP
ejde-92	281	30	−∆p	−∆p	PRON
ejde-92	281	31	in	in	ADP
ejde-92	281	32	g.	g.	PROPN
ejde-92	281	33	�	�	PROPN
ejde-92	281	34	acknowledgments	acknowledgment	NOUN
ejde-92	281	35	.	.	PUNCT
ejde-92	282	1	this	this	DET
ejde-92	282	2	research	research	NOUN
ejde-92	282	3	was	be	AUX
ejde-92	282	4	partially	partially	ADV
ejde-92	282	5	supported	support	VERB
ejde-92	282	6	by	by	ADP
ejde-92	282	7	cece	cece	PROPN
ejde-92	282	8	(	(	PUNCT
ejde-92	282	9	generalitat	generalitat	PROPN
ejde-92	282	10	valenciana	valenciana	PROPN
ejde-92	282	11	)	)	PUNCT
ejde-92	282	12	under	under	ADP
ejde-92	282	13	project	project	NOUN
ejde-92	282	14	aico/2021/223	aico/2021/223	PROPN
ejde-92	282	15	.	.	PUNCT
ejde-92	283	1	references	reference	NOUN
ejde-92	283	2	[	[	X
ejde-92	283	3	1	1	NUM
ejde-92	283	4	]	]	PUNCT
ejde-92	283	5	a.	a.	NOUN
ejde-92	283	6	ambrosetti	ambrosetti	PROPN
ejde-92	283	7	,	,	PUNCT
ejde-92	283	8	g.	g.	PROPN
ejde-92	283	9	prodi	prodi	PROPN
ejde-92	283	10	;	;	PUNCT
ejde-92	283	11	a	a	DET
ejde-92	283	12	primer	primer	NOUN
ejde-92	283	13	of	of	ADP
ejde-92	283	14	nonlinear	nonlinear	ADJ
ejde-92	283	15	analysis	analysis	NOUN
ejde-92	283	16	.	.	PUNCT
ejde-92	284	1	cambridge	cambridge	PROPN
ejde-92	284	2	university	university	PROPN
ejde-92	284	3	press	press	PROPN
ejde-92	284	4	,	,	PUNCT
ejde-92	284	5	cambridge	cambridge	PROPN
ejde-92	284	6	,	,	PUNCT
ejde-92	284	7	1993	1993	NUM
ejde-92	284	8	.	.	PUNCT
ejde-92	285	1	[	[	X
ejde-92	285	2	2	2	X
ejde-92	285	3	]	]	PUNCT
ejde-92	285	4	s.	s.	PROPN
ejde-92	285	5	amghibech	amghibech	PROPN
ejde-92	285	6	;	;	PUNCT
ejde-92	285	7	eigenvalues	eigenvalue	NOUN
ejde-92	285	8	of	of	ADP
ejde-92	285	9	the	the	DET
ejde-92	285	10	discrete	discrete	ADJ
ejde-92	285	11	p	p	NOUN
ejde-92	285	12	-	-	PUNCT
ejde-92	285	13	laplacian	laplacian	NOUN
ejde-92	285	14	for	for	ADP
ejde-92	285	15	graphs	graph	NOUN
ejde-92	285	16	,	,	PUNCT
ejde-92	285	17	ars	ar	NOUN
ejde-92	285	18	combin	combin	NOUN
ejde-92	285	19	.	.	PUNCT
ejde-92	286	1	,	,	PUNCT
ejde-92	286	2	67	67	NUM
ejde-92	286	3	(	(	PUNCT
ejde-92	286	4	2003	2003	NUM
ejde-92	286	5	)	)	PUNCT
ejde-92	286	6	,	,	PUNCT
ejde-92	286	7	283–302	283–302	NUM
ejde-92	286	8	.	.	PUNCT
ejde-92	287	1	[	[	X
ejde-92	287	2	3	3	NUM
ejde-92	287	3	]	]	PUNCT
ejde-92	287	4	a.	a.	NOUN
ejde-92	287	5	anane	anane	PROPN
ejde-92	287	6	;	;	PUNCT
ejde-92	287	7	simplicité	simplicité	NOUN
ejde-92	287	8	et	et	NOUN
ejde-92	287	9	isolation	isolation	NOUN
ejde-92	287	10	de	de	X
ejde-92	287	11	la	la	X
ejde-92	287	12	première	première	PROPN
ejde-92	287	13	valeur	valeur	PROPN
ejde-92	287	14	propre	propre	PROPN
ejde-92	287	15	du	du	PROPN
ejde-92	287	16	p	p	PROPN
ejde-92	287	17	-	-	PUNCT
ejde-92	287	18	laplacien	laplacien	NOUN
ejde-92	287	19	avec	avec	X
ejde-92	287	20	poids	poids	PROPN
ejde-92	287	21	,	,	PUNCT
ejde-92	287	22	c.	c.	PROPN
ejde-92	287	23	r.	r.	PROPN
ejde-92	287	24	acad	acad	PROPN
ejde-92	287	25	.	.	PUNCT
ejde-92	288	1	sci	sci	PROPN
ejde-92	288	2	.	.	PROPN
ejde-92	288	3	paris	paris	PROPN
ejde-92	288	4	sér	sér	PROPN
ejde-92	288	5	.	.	PUNCT
ejde-92	289	1	i	i	PRON
ejde-92	289	2	math	math	PROPN
ejde-92	289	3	.	.	PUNCT
ejde-92	290	1	,	,	PUNCT
ejde-92	290	2	305	305	NUM
ejde-92	290	3	(	(	PUNCT
ejde-92	290	4	1987	1987	NUM
ejde-92	290	5	)	)	PUNCT
ejde-92	290	6	,	,	PUNCT
ejde-92	290	7	725–728	725–728	NUM
ejde-92	290	8	.	.	PUNCT
ejde-92	291	1	[	[	X
ejde-92	291	2	4	4	NUM
ejde-92	291	3	]	]	PUNCT
ejde-92	291	4	a.	a.	NOUN
ejde-92	291	5	anane	anane	PROPN
ejde-92	291	6	,	,	PUNCT
ejde-92	291	7	n.	n.	PROPN
ejde-92	291	8	tsouli	tsouli	PROPN
ejde-92	291	9	;	;	PUNCT
ejde-92	291	10	on	on	ADP
ejde-92	291	11	the	the	DET
ejde-92	291	12	second	second	ADJ
ejde-92	291	13	eigenvalue	eigenvalue	NOUN
ejde-92	291	14	of	of	ADP
ejde-92	291	15	the	the	DET
ejde-92	291	16	p	p	NOUN
ejde-92	291	17	-	-	PUNCT
ejde-92	291	18	laplacian	laplacian	NOUN
ejde-92	291	19	.	.	PUNCT
ejde-92	292	1	in	in	ADP
ejde-92	292	2	nonlinear	nonlinear	ADJ
ejde-92	292	3	partial	partial	ADJ
ejde-92	292	4	differential	differential	NOUN
ejde-92	292	5	equations	equation	NOUN
ejde-92	292	6	(	(	PUNCT
ejde-92	292	7	fès	fès	PROPN
ejde-92	292	8	,	,	PUNCT
ejde-92	292	9	1994	1994	NUM
ejde-92	292	10	)	)	PUNCT
ejde-92	292	11	,	,	PUNCT
ejde-92	292	12	pitman	pitman	NOUN
ejde-92	292	13	res	res	PROPN
ejde-92	292	14	.	.	PROPN
ejde-92	292	15	notes	note	VERB
ejde-92	292	16	math	math	PROPN
ejde-92	292	17	.	.	PUNCT
ejde-92	293	1	ser	ser	PROPN
ejde-92	293	2	.	.	PROPN
ejde-92	294	1	vol	vol	NOUN
ejde-92	294	2	.	.	PROPN
ejde-92	295	1	343	343	NUM
ejde-92	295	2	,	,	PUNCT
ejde-92	295	3	pages	page	NOUN
ejde-92	295	4	1–9	1–9	NUM
ejde-92	295	5	,	,	PUNCT
ejde-92	295	6	longman	longman	NOUN
ejde-92	295	7	,	,	PUNCT
ejde-92	295	8	harlow	harlow	NOUN
ejde-92	295	9	,	,	PUNCT
ejde-92	295	10	1996	1996	NUM
ejde-92	295	11	.	.	PUNCT
ejde-92	296	1	[	[	X
ejde-92	296	2	5	5	NUM
ejde-92	296	3	]	]	PUNCT
ejde-92	296	4	m.	m.	NOUN
ejde-92	296	5	arias	arias	PROPN
ejde-92	296	6	,	,	PUNCT
ejde-92	296	7	j.	j.	PROPN
ejde-92	296	8	campos	campos	PROPN
ejde-92	296	9	,	,	PUNCT
ejde-92	296	10	m.	m.	NOUN
ejde-92	296	11	cuesta	cuesta	PROPN
ejde-92	296	12	,	,	PUNCT
ejde-92	296	13	j.	j.	PROPN
ejde-92	296	14	p.	p.	PROPN
ejde-92	296	15	gossez	gossez	NOUN
ejde-92	296	16	;	;	PUNCT
ejde-92	296	17	asymmetric	asymmetric	ADJ
ejde-92	296	18	elliptic	elliptic	ADJ
ejde-92	296	19	problems	problem	NOUN
ejde-92	296	20	with	with	ADP
ejde-92	296	21	indefinite	indefinite	ADJ
ejde-92	296	22	weights	weight	NOUN
ejde-92	296	23	,	,	PUNCT
ejde-92	296	24	ann	ann	PROPN
ejde-92	296	25	.	.	PROPN
ejde-92	296	26	inst	inst	PROPN
ejde-92	296	27	.	.	PUNCT
ejde-92	297	1	h.	h.	PROPN
ejde-92	297	2	poincaré	poincaré	PROPN
ejde-92	297	3	anal	anal	PROPN
ejde-92	297	4	.	.	PUNCT
ejde-92	298	1	non	non	PROPN
ejde-92	298	2	linéaire	linéaire	PROPN
ejde-92	298	3	,	,	PUNCT
ejde-92	298	4	19	19	NUM
ejde-92	298	5	(	(	PUNCT
ejde-92	298	6	2002	2002	NUM
ejde-92	298	7	)	)	PUNCT
ejde-92	298	8	,	,	PUNCT
ejde-92	298	9	581–616	581–616	NUM
ejde-92	298	10	.	.	PUNCT
ejde-92	299	1	[	[	X
ejde-92	299	2	6	6	NUM
ejde-92	299	3	]	]	PUNCT
ejde-92	299	4	m.	m.	NOUN
ejde-92	299	5	arias	arias	PROPN
ejde-92	299	6	,	,	PUNCT
ejde-92	299	7	j.	j.	PROPN
ejde-92	299	8	campos	campos	PROPN
ejde-92	299	9	,	,	PUNCT
ejde-92	299	10	m.	m.	NOUN
ejde-92	299	11	cuesta	cuesta	PROPN
ejde-92	299	12	,	,	PUNCT
ejde-92	299	13	j.	j.	PROPN
ejde-92	299	14	p.	p.	PROPN
ejde-92	299	15	gossez	gossez	NOUN
ejde-92	299	16	;	;	PUNCT
ejde-92	299	17	an	an	DET
ejde-92	299	18	asymmetric	asymmetric	ADJ
ejde-92	299	19	neumann	neumann	PROPN
ejde-92	299	20	problem	problem	NOUN
ejde-92	299	21	with	with	ADP
ejde-92	299	22	weights	weight	NOUN
ejde-92	299	23	,	,	PUNCT
ejde-92	299	24	ann	ann	PROPN
ejde-92	299	25	.	.	PROPN
ejde-92	299	26	inst	inst	PROPN
ejde-92	299	27	.	.	PUNCT
ejde-92	300	1	h.	h.	PROPN
ejde-92	300	2	poincaré	poincaré	PROPN
ejde-92	300	3	anal	anal	PROPN
ejde-92	300	4	.	.	PUNCT
ejde-92	301	1	non	non	PROPN
ejde-92	301	2	linéaire	linéaire	PROPN
ejde-92	301	3	,	,	PUNCT
ejde-92	301	4	25	25	NUM
ejde-92	301	5	(	(	PUNCT
ejde-92	301	6	2008	2008	NUM
ejde-92	301	7	)	)	PUNCT
ejde-92	301	8	,	,	PUNCT
ejde-92	301	9	267–280	267–280	NUM
ejde-92	301	10	.	.	PUNCT
ejde-92	302	1	[	[	X
ejde-92	302	2	7	7	X
ejde-92	302	3	]	]	X
ejde-92	302	4	b.	b.	PROPN
ejde-92	302	5	brandolini	brandolini	PROPN
ejde-92	302	6	,	,	PUNCT
ejde-92	302	7	f.	f.	PROPN
ejde-92	302	8	chiacchio	chiacchio	PROPN
ejde-92	302	9	,	,	PUNCT
ejde-92	302	10	c.	c.	PROPN
ejde-92	302	11	trombetti	trombetti	PROPN
ejde-92	302	12	;	;	PUNCT
ejde-92	302	13	optimal	optimal	ADJ
ejde-92	302	14	lower	low	ADJ
ejde-92	302	15	bounds	bound	NOUN
ejde-92	302	16	for	for	ADP
ejde-92	302	17	eigenvalues	eigenvalue	NOUN
ejde-92	302	18	of	of	ADP
ejde-92	302	19	linear	linear	PROPN
ejde-92	302	20	and	and	CCONJ
ejde-92	302	21	nonlinear	nonlinear	PROPN
ejde-92	302	22	neumann	neumann	PROPN
ejde-92	302	23	problems	problems	PROPN
ejde-92	302	24	,	,	PUNCT
ejde-92	302	25	proc	proc	PROPN
ejde-92	302	26	.	.	PUNCT
ejde-92	302	27	roy	roy	PROPN
ejde-92	302	28	.	.	PROPN
ejde-92	302	29	soc	soc	PROPN
ejde-92	302	30	.	.	PUNCT
ejde-92	302	31	edinburgh	edinburgh	PROPN
ejde-92	302	32	sect	sect	PROPN
ejde-92	302	33	.	.	PUNCT
ejde-92	303	1	a	a	DET
ejde-92	303	2	,	,	PUNCT
ejde-92	303	3	145	145	NUM
ejde-92	303	4	(	(	PUNCT
ejde-92	303	5	2015	2015	NUM
ejde-92	303	6	)	)	PUNCT
ejde-92	303	7	,	,	PUNCT
ejde-92	303	8	31–45	31–45	NUM
ejde-92	303	9	.	.	PUNCT
ejde-92	304	1	12	12	NUM
ejde-92	304	2	j.	j.	PROPN
ejde-92	304	3	c.	c.	PROPN
ejde-92	304	4	sabina	sabina	PROPN
ejde-92	304	5	de	de	PROPN
ejde-92	304	6	lis	lis	X
ejde-92	304	7	ejde-2022/13	ejde-2022/13	NOUN
ejde-92	304	8	[	[	X
ejde-92	304	9	8	8	NUM
ejde-92	304	10	]	]	PUNCT
ejde-92	304	11	t.	t.	NOUN
ejde-92	304	12	bühler	bühler	NOUN
ejde-92	304	13	,	,	PUNCT
ejde-92	304	14	m.	m.	NOUN
ejde-92	304	15	hein	hein	PROPN
ejde-92	304	16	;	;	PUNCT
ejde-92	304	17	spectral	spectral	ADJ
ejde-92	304	18	clustering	clustering	NOUN
ejde-92	304	19	based	base	VERB
ejde-92	304	20	on	on	ADP
ejde-92	304	21	the	the	DET
ejde-92	304	22	graph	graph	NOUN
ejde-92	304	23	p	p	NOUN
ejde-92	304	24	-	-	PUNCT
ejde-92	304	25	laplacian	laplacian	NOUN
ejde-92	304	26	.	.	PUNCT
ejde-92	305	1	in	in	ADP
ejde-92	305	2	l.	l.	PROPN
ejde-92	305	3	bottou	bottou	PROPN
ejde-92	305	4	and	and	CCONJ
ejde-92	305	5	m.	m.	PROPN
ejde-92	305	6	littman	littman	PROPN
ejde-92	305	7	(	(	PUNCT
ejde-92	305	8	eds	eds	PROPN
ejde-92	305	9	.	.	PUNCT
ejde-92	305	10	)	)	PUNCT
ejde-92	305	11	,	,	PUNCT
ejde-92	305	12	proc	proc	NOUN
ejde-92	305	13	.	.	PUNCT
ejde-92	306	1	of	of	ADP
ejde-92	306	2	the	the	DET
ejde-92	306	3	26th	26th	ADJ
ejde-92	306	4	int	int	NOUN
ejde-92	306	5	.	.	PUNCT
ejde-92	307	1	conf	conf	NOUN
ejde-92	307	2	.	.	PUNCT
ejde-92	308	1	mach	mach	PROPN
ejde-92	308	2	.	.	PUNCT
ejde-92	309	1	learn	learn	VERB
ejde-92	309	2	.	.	PUNCT
ejde-92	310	1	(	(	PUNCT
ejde-92	310	2	icml	icml	PROPN
ejde-92	310	3	)	)	PUNCT
ejde-92	310	4	,	,	PUNCT
ejde-92	310	5	canada	canada	PROPN
ejde-92	310	6	,	,	PUNCT
ejde-92	310	7	pages	page	NOUN
ejde-92	310	8	81–88	81–88	NUM
ejde-92	310	9	,	,	PUNCT
ejde-92	310	10	association	association	NOUN
ejde-92	310	11	for	for	ADP
ejde-92	310	12	computing	computing	NOUN
ejde-92	310	13	machinery	machinery	NOUN
ejde-92	310	14	,	,	PUNCT
ejde-92	310	15	new	new	PROPN
ejde-92	310	16	york	york	PROPN
ejde-92	310	17	,	,	PUNCT
ejde-92	310	18	united	united	PROPN
ejde-92	310	19	states	states	PROPN
ejde-92	310	20	,	,	PUNCT
ejde-92	310	21	2009	2009	NUM
ejde-92	310	22	.	.	PUNCT
ejde-92	311	1	[	[	X
ejde-92	311	2	9	9	NUM
ejde-92	311	3	]	]	PUNCT
ejde-92	311	4	t.	t.	NOUN
ejde-92	311	5	bühler	bühler	NOUN
ejde-92	311	6	,	,	PUNCT
ejde-92	311	7	m.	m.	PROPN
ejde-92	311	8	hein	hein	PROPN
ejde-92	311	9	;	;	PUNCT
ejde-92	311	10	supplementary	supplementary	ADJ
ejde-92	311	11	material	material	NOUN
ejde-92	311	12	.	.	PUNCT
ejde-92	312	1	http://www.ml.uni-saarland.de/publications/buehei09tech.pdf	http://www.ml.uni-saarland.de/publications/buehei09tech.pdf	PROPN
ejde-92	312	2	,	,	PUNCT
ejde-92	312	3	2009	2009	NUM
ejde-92	312	4	.	.	PUNCT
ejde-92	313	1	[	[	X
ejde-92	313	2	10	10	NUM
ejde-92	313	3	]	]	X
ejde-92	313	4	f.	f.	PROPN
ejde-92	313	5	r.	r.	PROPN
ejde-92	313	6	k.	k.	PROPN
ejde-92	313	7	chung	chung	PROPN
ejde-92	313	8	;	;	PUNCT
ejde-92	313	9	spectral	spectral	ADJ
ejde-92	313	10	graph	graph	NOUN
ejde-92	313	11	theory	theory	NOUN
ejde-92	313	12	.	.	PUNCT
ejde-92	314	1	cbms	cbms	PROPN
ejde-92	314	2	regional	regional	PROPN
ejde-92	314	3	conference	conference	NOUN
ejde-92	314	4	series	series	NOUN
ejde-92	314	5	in	in	ADP
ejde-92	314	6	mathematics	mathematics	PROPN
ejde-92	314	7	vol	vol	NOUN
ejde-92	314	8	.	.	PROPN
ejde-92	315	1	92	92	NUM
ejde-92	315	2	,	,	PUNCT
ejde-92	315	3	american	american	PROPN
ejde-92	315	4	mathematical	mathematical	ADJ
ejde-92	315	5	society	society	NOUN
ejde-92	315	6	,	,	PUNCT
ejde-92	315	7	providence	providence	NOUN
ejde-92	315	8	,	,	PUNCT
ejde-92	315	9	ri	ri	NOUN
ejde-92	315	10	,	,	PUNCT
ejde-92	315	11	1997	1997	NUM
ejde-92	315	12	.	.	PUNCT
ejde-92	316	1	[	[	X
ejde-92	316	2	11	11	NUM
ejde-92	316	3	]	]	PUNCT
ejde-92	316	4	m.	m.	NOUN
ejde-92	316	5	cuesta	cuesta	PROPN
ejde-92	316	6	,	,	PUNCT
ejde-92	316	7	d.	d.	PROPN
ejde-92	316	8	de	de	PROPN
ejde-92	316	9	figueiredo	figueiredo	PROPN
ejde-92	316	10	,	,	PUNCT
ejde-92	316	11	j.	j.	PROPN
ejde-92	316	12	p.	p.	PROPN
ejde-92	316	13	gossez	gossez	NOUN
ejde-92	316	14	;	;	PUNCT
ejde-92	316	15	the	the	DET
ejde-92	316	16	beginning	beginning	NOUN
ejde-92	316	17	of	of	ADP
ejde-92	316	18	the	the	DET
ejde-92	316	19	fučik	fučik	PROPN
ejde-92	316	20	spectrum	spectrum	NOUN
ejde-92	316	21	for	for	ADP
ejde-92	316	22	the	the	DET
ejde-92	316	23	p	p	NOUN
ejde-92	316	24	-	-	PUNCT
ejde-92	316	25	laplacian	laplacian	NOUN
ejde-92	316	26	.	.	PUNCT
ejde-92	317	1	j.	j.	PROPN
ejde-92	317	2	differential	differential	PROPN
ejde-92	317	3	equations	equations	PROPN
ejde-92	317	4	,	,	PUNCT
ejde-92	317	5	159	159	NUM
ejde-92	317	6	(	(	PUNCT
ejde-92	317	7	1999	1999	NUM
ejde-92	317	8	)	)	PUNCT
ejde-92	317	9	,	,	PUNCT
ejde-92	317	10	212–238	212–238	NUM
ejde-92	317	11	.	.	PUNCT
ejde-92	318	1	[	[	X
ejde-92	318	2	12	12	NUM
ejde-92	318	3	]	]	PUNCT
ejde-92	318	4	p.	p.	NOUN
ejde-92	318	5	drábek	drábek	PROPN
ejde-92	318	6	,	,	PUNCT
ejde-92	319	1	s.	s.	PROPN
ejde-92	319	2	b.	b.	PROPN
ejde-92	319	3	robinson	robinson	PROPN
ejde-92	319	4	;	;	PUNCT
ejde-92	319	5	resonance	resonance	NOUN
ejde-92	319	6	problems	problem	NOUN
ejde-92	319	7	for	for	ADP
ejde-92	319	8	the	the	DET
ejde-92	319	9	p	p	NOUN
ejde-92	319	10	-	-	PUNCT
ejde-92	319	11	laplacian	laplacian	NOUN
ejde-92	319	12	,	,	PUNCT
ejde-92	319	13	j.	j.	PROPN
ejde-92	319	14	funct	funct	PROPN
ejde-92	319	15	.	.	PUNCT
ejde-92	320	1	anal	anal	PROPN
ejde-92	320	2	.	.	PROPN
ejde-92	320	3	,	,	PUNCT
ejde-92	320	4	169	169	NUM
ejde-92	320	5	(	(	PUNCT
ejde-92	320	6	1999	1999	NUM
ejde-92	320	7	)	)	PUNCT
ejde-92	320	8	,	,	PUNCT
ejde-92	320	9	189–200	189–200	NUM
ejde-92	320	10	.	.	PUNCT
ejde-92	321	1	[	[	X
ejde-92	321	2	13	13	NUM
ejde-92	321	3	]	]	X
ejde-92	321	4	j.	j.	PROPN
ejde-92	321	5	garćıa	garćıa	PROPN
ejde-92	321	6	azorero	azorero	PROPN
ejde-92	321	7	,	,	PUNCT
ejde-92	321	8	i.	i.	PROPN
ejde-92	321	9	peral	peral	PROPN
ejde-92	321	10	alonso	alonso	PROPN
ejde-92	321	11	;	;	PUNCT
ejde-92	321	12	existence	existence	NOUN
ejde-92	321	13	and	and	CCONJ
ejde-92	321	14	nonuniqueness	nonuniqueness	NOUN
ejde-92	321	15	for	for	ADP
ejde-92	321	16	the	the	DET
ejde-92	321	17	p	p	NOUN
ejde-92	321	18	-	-	PUNCT
ejde-92	321	19	laplacian	laplacian	NOUN
ejde-92	321	20	:	:	PUNCT
ejde-92	321	21	nonlinear	nonlinear	ADJ
ejde-92	321	22	eigenvalues	eigenvalue	NOUN
ejde-92	321	23	.	.	PUNCT
ejde-92	322	1	comm	comm	NOUN
ejde-92	322	2	.	.	PUNCT
ejde-92	323	1	partial	partial	ADJ
ejde-92	323	2	differential	differential	NOUN
ejde-92	323	3	equations	equation	NOUN
ejde-92	323	4	,	,	PUNCT
ejde-92	323	5	12	12	NUM
ejde-92	323	6	(	(	PUNCT
ejde-92	323	7	1987	1987	NUM
ejde-92	323	8	)	)	PUNCT
ejde-92	323	9	,	,	PUNCT
ejde-92	323	10	1389–1430	1389–1430	NUM
ejde-92	323	11	.	.	PUNCT
ejde-92	324	1	[	[	X
ejde-92	324	2	14	14	NUM
ejde-92	324	3	]	]	X
ejde-92	324	4	l.	l.	PROPN
ejde-92	324	5	gasiński	gasiński	PROPN
ejde-92	324	6	,	,	PUNCT
ejde-92	324	7	n.	n.	PROPN
ejde-92	324	8	s.	s.	PROPN
ejde-92	324	9	papageorgiou	papageorgiou	PROPN
ejde-92	324	10	;	;	PUNCT
ejde-92	324	11	nonlinear	nonlinear	ADJ
ejde-92	324	12	analysis	analysis	NOUN
ejde-92	324	13	.	.	PUNCT
ejde-92	324	14	series	series	PROPN
ejde-92	324	15	in	in	ADP
ejde-92	324	16	mathematical	mathematical	ADJ
ejde-92	324	17	analysis	analysis	NOUN
ejde-92	324	18	and	and	CCONJ
ejde-92	324	19	applications	application	NOUN
ejde-92	324	20	vol	vol	NOUN
ejde-92	324	21	.	.	PUNCT
ejde-92	324	22	9	9	NUM
ejde-92	324	23	,	,	PUNCT
ejde-92	324	24	chapman	chapman	NOUN
ejde-92	324	25	&	&	CCONJ
ejde-92	324	26	hall	hall	PROPN
ejde-92	324	27	/	/	SYM
ejde-92	324	28	crc	crc	PROPN
ejde-92	324	29	,	,	PUNCT
ejde-92	324	30	boca	boca	PROPN
ejde-92	324	31	raton	raton	PROPN
ejde-92	324	32	,	,	PUNCT
ejde-92	324	33	fl	fl	PROPN
ejde-92	324	34	,	,	PUNCT
ejde-92	324	35	2006	2006	NUM
ejde-92	324	36	.	.	PUNCT
ejde-92	325	1	[	[	X
ejde-92	325	2	15	15	NUM
ejde-92	325	3	]	]	X
ejde-92	325	4	p.	p.	NOUN
ejde-92	325	5	lindqvist	lindqvist	NOUN
ejde-92	325	6	;	;	PUNCT
ejde-92	325	7	on	on	ADP
ejde-92	325	8	the	the	DET
ejde-92	325	9	equation	equation	NOUN
ejde-92	325	10	div	div	X
ejde-92	325	11	(	(	PUNCT
ejde-92	325	12	|∇u|p−2∇u	|∇u|p−2∇u	NUM
ejde-92	325	13	)	)	PUNCT
ejde-92	326	1	+	+	CCONJ
ejde-92	326	2	λ|u|p−2u	λ|u|p−2u	PROPN
ejde-92	326	3	=	=	SYM
ejde-92	326	4	0	0	NUM
ejde-92	326	5	,	,	PUNCT
ejde-92	326	6	proc	proc	NOUN
ejde-92	326	7	.	.	PUNCT
ejde-92	327	1	amer	amer	PROPN
ejde-92	327	2	.	.	PUNCT
ejde-92	327	3	math	math	PROPN
ejde-92	327	4	.	.	PUNCT
ejde-92	328	1	soc	soc	PROPN
ejde-92	328	2	.	.	PUNCT
ejde-92	328	3	,	,	PUNCT
ejde-92	328	4	109	109	NUM
ejde-92	328	5	(	(	PUNCT
ejde-92	328	6	1990	1990	NUM
ejde-92	328	7	)	)	PUNCT
ejde-92	328	8	,	,	PUNCT
ejde-92	328	9	157–164	157–164	NUM
ejde-92	328	10	.	.	PUNCT
ejde-92	329	1	[	[	X
ejde-92	329	2	16	16	NUM
ejde-92	329	3	]	]	PUNCT
ejde-92	329	4	p.	p.	NOUN
ejde-92	329	5	lindqvist	lindqvist	NOUN
ejde-92	329	6	;	;	PUNCT
ejde-92	329	7	addendum	addendum	PROPN
ejde-92	329	8	:	:	PUNCT
ejde-92	329	9	“	"	PUNCT
ejde-92	329	10	on	on	ADP
ejde-92	329	11	the	the	DET
ejde-92	329	12	equation	equation	NOUN
ejde-92	329	13	div	div	X
ejde-92	329	14	(	(	PUNCT
ejde-92	329	15	|∇u|p−2∇u	|∇u|p−2∇u	NUM
ejde-92	329	16	)	)	PUNCT
ejde-92	329	17	+	+	CCONJ
ejde-92	330	1	λ|u|p−2u	λ|u|p−2u	PROPN
ejde-92	330	2	=	=	SYM
ejde-92	330	3	0	0	NUM
ejde-92	330	4	”	"	PUNCT
ejde-92	330	5	,	,	PUNCT
ejde-92	330	6	proc	proc	PROPN
ejde-92	330	7	.	.	PUNCT
ejde-92	331	1	amer	amer	PROPN
ejde-92	331	2	.	.	PUNCT
ejde-92	331	3	math	math	PROPN
ejde-92	331	4	.	.	PUNCT
ejde-92	332	1	soc	soc	PROPN
ejde-92	332	2	.	.	PUNCT
ejde-92	333	1	,	,	PUNCT
ejde-92	333	2	116	116	NUM
ejde-92	333	3	(	(	PUNCT
ejde-92	333	4	1992	1992	NUM
ejde-92	333	5	)	)	PUNCT
ejde-92	333	6	,	,	PUNCT
ejde-92	333	7	583–584	583–584	NUM
ejde-92	333	8	.	.	PUNCT
ejde-92	334	1	[	[	X
ejde-92	334	2	17	17	NUM
ejde-92	334	3	]	]	PUNCT
ejde-92	334	4	p.	p.	NOUN
ejde-92	334	5	lindqvist	lindqvist	NOUN
ejde-92	334	6	;	;	PUNCT
ejde-92	334	7	a	a	DET
ejde-92	334	8	nonlinear	nonlinear	ADJ
ejde-92	334	9	eigenvalue	eigenvalue	PROPN
ejde-92	334	10	problem	problem	NOUN
ejde-92	334	11	.	.	PUNCT
ejde-92	335	1	in	in	ADP
ejde-92	335	2	topics	topic	NOUN
ejde-92	335	3	in	in	ADP
ejde-92	335	4	mathematical	mathematical	ADJ
ejde-92	335	5	analysis	analysis	NOUN
ejde-92	335	6	,	,	PUNCT
ejde-92	335	7	ser	ser	NOUN
ejde-92	335	8	.	.	PROPN
ejde-92	335	9	anal	anal	PROPN
ejde-92	335	10	.	.	PUNCT
ejde-92	335	11	appl	appl	PROPN
ejde-92	335	12	.	.	PUNCT
ejde-92	336	1	comput	comput	PROPN
ejde-92	336	2	.	.	PUNCT
ejde-92	337	1	vol	vol	NOUN
ejde-92	337	2	.	.	PROPN
ejde-92	338	1	3	3	NUM
ejde-92	338	2	,	,	PUNCT
ejde-92	338	3	pages	page	NOUN
ejde-92	338	4	175–203	175–203	NUM
ejde-92	338	5	,	,	PUNCT
ejde-92	338	6	world	world	PROPN
ejde-92	338	7	sci	sci	PROPN
ejde-92	338	8	.	.	PROPN
ejde-92	338	9	publ	publ	PROPN
ejde-92	338	10	.	.	PROPN
ejde-92	338	11	,	,	PUNCT
ejde-92	338	12	hackensack	hackensack	PROPN
ejde-92	338	13	,	,	PUNCT
ejde-92	338	14	nj	nj	PROPN
ejde-92	338	15	,	,	PUNCT
ejde-92	338	16	2008	2008	NUM
ejde-92	338	17	.	.	PUNCT
ejde-92	339	1	[	[	X
ejde-92	339	2	18	18	NUM
ejde-92	339	3	]	]	X
ejde-92	339	4	v.	v.	ADP
ejde-92	339	5	maz’ya	maz’ya	PROPN
ejde-92	339	6	;	;	PUNCT
ejde-92	339	7	sobolev	sobolev	NOUN
ejde-92	339	8	spaces	space	VERB
ejde-92	339	9	with	with	ADP
ejde-92	339	10	applications	application	NOUN
ejde-92	339	11	to	to	ADP
ejde-92	339	12	elliptic	elliptic	ADJ
ejde-92	339	13	partial	partial	ADJ
ejde-92	339	14	differential	differential	NOUN
ejde-92	339	15	equations	equation	NOUN
ejde-92	339	16	.	.	PUNCT
ejde-92	340	1	springer	springer	NOUN
ejde-92	340	2	,	,	PUNCT
ejde-92	340	3	heidelberg	heidelberg	PROPN
ejde-92	340	4	,	,	PUNCT
ejde-92	340	5	2011	2011	NUM
ejde-92	340	6	.	.	PUNCT
ejde-92	341	1	[	[	X
ejde-92	341	2	19	19	NUM
ejde-92	341	3	]	]	PUNCT
ejde-92	341	4	m.	m.	NOUN
ejde-92	341	5	struwe	struwe	NOUN
ejde-92	341	6	;	;	PUNCT
ejde-92	341	7	variational	variational	ADJ
ejde-92	341	8	methods	method	NOUN
ejde-92	341	9	.	.	PUNCT
ejde-92	342	1	springer	springer	NOUN
ejde-92	342	2	-	-	PUNCT
ejde-92	342	3	verlag	verlag	PROPN
ejde-92	342	4	,	,	PUNCT
ejde-92	342	5	berlin	berlin	PROPN
ejde-92	342	6	,	,	PUNCT
ejde-92	342	7	2008	2008	NUM
ejde-92	342	8	.	.	PUNCT
ejde-92	343	1	josé	josé	PROPN
ejde-92	343	2	c.	c.	PROPN
ejde-92	343	3	sabina	sabina	PROPN
ejde-92	343	4	de	de	PROPN
ejde-92	343	5	lis	lis	PROPN
ejde-92	343	6	departamento	departamento	NOUN
ejde-92	343	7	de	de	PROPN
ejde-92	343	8	análisis	análisis	PROPN
ejde-92	343	9	matemático	matemático	PROPN
ejde-92	343	10	and	and	CCONJ
ejde-92	343	11	iuea	iuea	NOUN
ejde-92	343	12	,	,	PUNCT
ejde-92	343	13	universidad	universidad	PROPN
ejde-92	343	14	de	de	PROPN
ejde-92	343	15	la	la	PROPN
ejde-92	343	16	laguna	laguna	PROPN
ejde-92	343	17	,	,	PUNCT
ejde-92	343	18	c.	c.	PROPN
ejde-92	343	19	astrof́ısico	astrof́ısico	PROPN
ejde-92	343	20	francisco	francisco	PROPN
ejde-92	344	1	sánchez	sánchez	PROPN
ejde-92	344	2	s	s	PROPN
ejde-92	344	3	/	/	SYM
ejde-92	344	4	n	n	CCONJ
ejde-92	344	5	,	,	PUNCT
ejde-92	344	6	38203	38203	NUM
ejde-92	344	7	–	–	PUNCT
ejde-92	344	8	la	la	PROPN
ejde-92	344	9	laguna	laguna	PROPN
ejde-92	344	10	,	,	PUNCT
ejde-92	344	11	spain	spain	PROPN
ejde-92	344	12	email	email	NOUN
ejde-92	344	13	address	address	NOUN
ejde-92	344	14	:	:	PUNCT
ejde-92	344	15	josabina@ull.edu.es	josabina@ull.edu.es	PROPN
ejde-92	344	16	1	1	NUM
ejde-92	344	17	.	.	PUNCT
ejde-92	344	18	introduction	introduction	NOUN
ejde-92	344	19	2	2	NUM
ejde-92	344	20	.	.	PUNCT
ejde-92	344	21	variance	variance	NOUN
ejde-92	344	22	functional	functional	ADJ
ejde-92	344	23	in	in	ADP
ejde-92	344	24	lp(x	lp(x	PROPN
ejde-92	344	25	,	,	PUNCT
ejde-92	344	26	)	)	PUNCT
ejde-92	344	27	3	3	X
ejde-92	344	28	.	.	X
ejde-92	344	29	proof	proof	NOUN
ejde-92	344	30	of	of	ADP
ejde-92	344	31	theorem	theorem	NOUN
ejde-92	344	32	?	?	PUNCT
ejde-92	344	33	?	?	PUNCT
ejde-92	345	1	4	4	X
ejde-92	345	2	.	.	X
ejde-92	345	3	a	a	DET
ejde-92	345	4	further	further	ADJ
ejde-92	345	5	subcritical	subcritical	ADJ
ejde-92	345	6	problem	problem	NOUN
ejde-92	345	7	5	5	NUM
ejde-92	345	8	.	.	PUNCT
ejde-92	346	1	p	p	X
ejde-92	346	2	-	-	PUNCT
ejde-92	346	3	laplacian	laplacian	NOUN
ejde-92	346	4	on	on	ADP
ejde-92	346	5	graphs	graph	NOUN
ejde-92	346	6	acknowledgments	acknowledgment	NOUN
ejde-92	346	7	references	reference	NOUN