Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 65, pp. 1–16. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu BOUNDARY REGULARITY FOR STRONGLY DEGENERATE OPERATORS OF GRUSHIN TYPE GIUSEPPE DI FAZIO, MARIA STELLA FANCIULLO, PIERO ZAMBONI Communicated by Giovanni Molica Bisci Abstract. We prove Harnack inequality and global regularity results for weak solutions of quasilinear degenerate equations driven by operators of Grushin type with natural growth. Degeneracy is a power of a strong A∞ weight. Regularity results are achieved under minimal assumptions on the lower order coefficients. 1. Introduction In recent decades regularity for elliptic PDEs became more and more important both in theoretical and in applied Mathematics. This paper contributes towards a complete regularity theory concerning solutions of degenerate elliptic equations under minimal assumptions on the coefficients. Concerning the study of the regularity properties of solutions of quasilinear el- liptic equations of the form divA(x, u,∇u) +B(x, u,∇u) = 0 (1.1) we recall the classical works [18, 23, 24], where regularity under Lp assumptions took its final form. The first paper where non-Lp-conditions appear, is [20], where (1.1) has been investigated under somewhat simplified structure although the right hand side is allowed to be a measure in some Morrey space. Following De Giorgi method, as adapted by Ladyzhenskaya and Ural’tseva in [18] to quasilinear equations, Hölder continuity of the weak solutions is proved. Improvements of [20] can be found in [17, 19] where some Lp assumptions are weakened or replaced by Morrey space assumptions. Equation (1.1) has been inves- tigated in [25] assuming Stummel-Kato type hypotheses on the lower order coeffi- cients. This is a kind of generalization of the [1, 4] to quasilinear elliptic equations. We now turn on degenerate elliptic equations. Many regularity results for weak solutions of elliptic equations have been generalized to the Carnot Caratheodory (CC) spaces. In such spaces the metric is generated by sub-unit curves associated to a system of non-commuting vector fields. In this direction we quote [3, 5, 6, 11, 12], 2020 Mathematics Subject Classification. 35B65. Key words and phrases. Harnack inequality; strong A∞ weights; stummel-Kato classes; Grushin operator. ©2022. This work is licensed under a CC BY 4.0 license. Submitted July 6, 2022. Published September 15, 2022. 1 2 G. DI FAZIO, M. S. FANCIULLO, P. ZAMBONI EJDE-2022/65 where the above mentioned results are obtained in the subelliptic setting through Morrey and Stummel-Kato type assumptions. Almost at the same time a parallel investigation has been performed in [13, 8], where degeneracy of operator is due to a suitable power of a strong A∞. There a weighted version of Stummel-Kato class has been defined to obtain the continuity of weak solutions. Strong A∞ weights have been introduced by David and Semmes in [7] for different purposes and it has been found useful in several problems related to geometric measure theory and quasiconformal mappings. In [8, 13] a strong A∞ weighted Stummel-Kato class has been defined and the continuity of weak solutions has been obtained. The first attempt to consider together the two types of degeneracy described above, has been exploited by Franchi, Gutierrez and Wheeden in [14, 15] where they proved Harnack inequality for positive weak solutions of equation div(w(z)∇λu(z)) = 0 where w is a power of a strong A∞ weight and ∇λu = (∇xu(z), λ(x)∇yu(z)). We remark that equations studied in [14, 15] do not contain lower order terms. In this article we extend results in [14, 15] to a more general equation. Namely, in Ω ⊂ RN = Rnx × Rmy , we consider the quasilinear elliptic equation in divergence form divA(z, u,∇λu) +B(z, u,∇λu) = 0 (1.2) where A and B are measurable functions satisfying suitable structure conditions of the form |A(z, u, ξ)| ≤ aw(z)[|ξx|2 + λ2(x)|ξy|2] p−1 2 + b(z)|u|p−1 + e(z) |B(z, u, ξ)| ≤ b0w(z)[|ξx|2 + λ2(x)|ξy|2]p/2 + b1(z)[|ξx|2 + λ2(x)|ξy|2] p−1 2 + d(z)|u|p−1 + f(z) ξ ·A(z, u, ξ) ≥ w(z)[|ξx|2 + λ2(x)|ξy|2]p/2 − d1(z)|u|p − g(z) (1.3) where w = v1− p N , 1 < p < N , v is a strong A∞ weight and λ is a suitable function we make precise later. We briefly describe the content of this paper. In Section 2 we recall the definition of strong A∞ weight and the related function spaces. In Section 3 we give the definitions of Stummel-Kato and Morrey classes introduced in [9, 10] (see e.g. [21, 22]) and state Fefferman type inequality which will allows us to control the integral arising from the coefficients in (1.3). In Section 4, following Trudinger pattern (see [24]), we obtain Harnack inequality for non negative weak solutions of equation (1.2) and, as consequence, the regularity of weak solutions. In particular, we prove continuity results under Stummel-Kato type assumptions, and Hölder continuity result under Morrey type assumptions. Finally, in Section 5 we use Trudinger technique to prove Harnack inequality up to the boundary. 2. Strong A∞ weights and function spaces We denote by z = (x, y) a point in RN , with x ∈ Rn and y ∈ Rm, n + m = N . We assume that there exists a function λ(x) defined on Rn such that (H1) λ = λ(x) is a continuous nonnegative function vanishing only at a finite number of points; EJDE-2022/65 BOUNDARY REGULARITY FOR STRONGLY DEGENERATE OPERATORS 3 (H2) λn is a strong A∞ weight (see Definition 2.2); (H3) λ satisfies an infinite order reverse Hölder inequality, i.e. for any x0 ∈ Rn, r > 0 we have − ∫ |x−x0| 1 and let v be a nonnegative locally integrable function in RN . We say that v is a weight of the Muckenhoupt class Aq if sup B ( 1 |B| ∫ B v(z) dz )( 1 |B| ∫ B [v(z)] −1 q−1 dz )q−1 ≡ C0 < +∞ , where the supremum is taken over all Carnot-Carathéodory metric balls B in RN . The number C0 is called the Aq constant of v. Definition 2.2. Let v be an Aq weight for some q > 1. If z1, z2 belong to RN , put δ(z1, z2) = inf (∫ B v(z)λ m N−1 (z)dz )1/N , where the infimum is taken over the balls B such that z1, z2 ∈ B. If γ : [0, T ]→ RN is a continuous curve, we define the v-lenght of γ as l(γ) = lim inf |σ|→0 p−1∑ i=0 δ(γ(ti+1), γ(ti)) , where σ = {t0, . . . , tp} is a partition of [0, T ], and we define a distance d(z1, z2) as the infimum of the v- lengths of sub-unit curves connecting z1 and z2. If there exist positive constants c1 and c2 such that c1δ(z1, z2) ≤ d(z1, z2) ≤ c2δ(z1, z2) , we say that v is a strong A∞ weight for the metric ρ. An example of strong A∞ weight is the function v(z) = ρ(z, z0)α with α ≥ 0 and z0 ∈ RN (see [15]). Using strong A∞ weights we define Lebesgue and Sobolev classes. 4 G. DI FAZIO, M. S. FANCIULLO, P. ZAMBONI EJDE-2022/65 Definition 2.3. Let v be a strong A∞ weight and w = v1−p/N , 1 ≤ p < N , Ω ⊂ RN . For any u ∈ C∞0 (Ω) we set ‖u‖Lpv(Ω) = (∫ Ω |u(z)|p w(z) dz )1/p . We define Lpv(Ω) to be the completion of C∞0 (Ω) with respect to the above norm. For u ∈ C∞(Ω) we set ‖u‖H1,p v (Ω) = (∫ Ω |u(z)|p w(z) dz )1/p + (∫ Ω |∇λu(z)|p w(z) dz )1/p . (2.1) We define H1,p 0,v (Ω) to be the completion of C∞0 (Ω) with respect to the norm (2.1) and H1,p v (Ω) to be the completion of C∞(Ω) with respect to the same norm. Now to recall the Sobolev embedding theorem and the representation formula proved in [15], Theorem I, we need another assumption on strong A∞ weights. A strong A∞ weight v satisfies the local boundedness condition near the zeros of λ if the following condition holds if λ(x1) = 0, then v(x, y) is bounded as x → x1 uniformly in y, for y in every bounded set. (2.2) Theorem 2.4. Let 1 < p < N and v be a strong A∞. If there exists a strong A∞ weight w satisfying (Zλ) such that v1−p/Nw−(N−1)/N belongs to Ap with respect to the (doubling) measure w(N−1)/Ndz, then there exists a constant q > p such that( − ∫ B(z0,r) |g − µ|qv1−p/Ndz )1/q ≤ Cr ( − ∫ B(z0,r) |∇λg|pv1−p/Ndz )1/p (2.3) for any Lipschitz continuous function g, where µ can be chosen to be the v1−p/N - average of g over B(z0, r). Remark 2.5. We stress that if we take the weights v = w = ρα(0, z) and the function λ = |x|σ, (α, σ > 0), the assumptions of Theorem 2.4 are satisfied (see also [14]). 3. Stummel-Kato type classes In this Section we recall a representation formula (see [15, Corollary 3.2]) to define Stummel - Kato type classes. Theorem 3.1. Let Ω be a bounded domain in RN , v a strong A∞ weight satisfying (2.2) and u a compactly supported smooth function in a metric ball B = BR ⊂ Ω. Then there exists c independent of u such that |u(z)| ≤ c ∫ B |∇λu(ξ)|v1− 1 N (ξ)k(z, ξ)dξ for almost all z ∈ B, where k(z, ξ) = (∫ Bρ(z,ξ)(z) v(ζ)λ m N−1 (ζ)dζ ) 1−N N . Now we give the definition of Stummel-Kato and Morrey classes. EJDE-2022/65 BOUNDARY REGULARITY FOR STRONGLY DEGENERATE OPERATORS 5 Definition 3.2. Let V be a locally integrable function in Ω, r > 0, and p ∈]1, N [. Let v be a strong A∞ weight. We set w(z ) ≡ v1− p N (z ) and φ(V ; r) ≡ sup z∈Ω (∫ B(z,r) k(z, ξ)v(ξ) (∫ Ω |V (ζ)|k(ζ, ξ)w(ζ)dζ ) 1 p−1 dξ )p−1 . We say that V belongs to the class S̃v(Ω) if φ(V ; r) is just a bounded function in a neighborhood of the origin. If, moreover limr→0 φ(V ; r) = 0 then we say that V belongs to the Stummel-Kato class Sv(Ω). If there exists ρ > 0 such that∫ ρ 0 φ(V ; t) t dt < +∞ , then we say that the function V belongs to the class S′v(Ω). If there exist c and σ > 0 such that φ(V ; r) ≤ crσ then we say that the function V belongs to the Morrey class L1,σ v (Ω). Remark 3.3. If v(z) = 1, λ(x) = 1 and p = 2 the previous definitions give back the classical Stummel-Kato class (see [1]) and Morrey space L1,n−2+σ for some σ > 0. Now we state a Fefferman type inequality related to Stummel-Kato classes and its corollary. Theorem 3.4. Let v be a strong A∞ weight satisfying (Zλ) and 1 < p < N . If V belongs to the class S̃v(Ω), then there exists a constant c such that ∀u ∈ C∞0 (Ω)(∫ B |V (z)||u(z)|p w(z) dz )1/p ≤ cφ1/p(V ; 2R) (∫ B |∇λu(z)|pw dz )1/p , where w(z ) ≡ v1− p N (z) and R is the radius of a metric ball B, containing the support of u. For a proof of the above theorem, see [9, Theorem 2.3]. Corollary 3.5. Let 1 < p < N and v be a strong A∞ weight satisfying (2.2). Let V belongs to the class Sv(Ω). For any ε > 0, there exists K(ε) such that∫ B |V (z)||u(z)|pw(z) dz ≤ ε ∫ B |∇λu(z)|pw(z) dz +K(ε) ∫ B |u(z)|pw(z) dz , for all u ∈ C∞0 (Ω), where w(z) = v(z)1− p N , K(ε) ∼ σ[ φ−1 ( V ; ε )]N+p and φ−1 denotes the inverse function of φ. For a proof of the above colollary, see [9, Corollary 2.1]. 4. Harnack inequality In this section we prove a weak Harnack inequality for non negative weak solu- tions of the equation divA(z, u,∇λu) +B(z, u,∇λu) = 0 . (4.1) First wee recall what we mean by weak solution of (4.1). 6 G. DI FAZIO, M. S. FANCIULLO, P. ZAMBONI EJDE-2022/65 Definition 4.1. A function u ∈ H1,p v (Ω) is a local weak subsolution (supersolution) of equation (4.1) in Ω if∫ Ω A(z, u(z),∇λu(z)) · ∇ϕdz − ∫ Ω B(z, u(z),∇λu(z))ϕdz ≤ 0 (≥ 0) (4.2) for every non negative ϕ ∈ H1,p 0,v (Ω). A function u is a weak solution if it is both super and sub solution. We require the functions A(z, u, ξ) and B(z, u, ξ) to be measurable functions satisfying the following structural conditions |A(z, u, ξ)| ≤ aw(z)[|ξx|2 + λ2(x)|ξy|2] p−1 2 + b(z)|u|p−1 + e(z) |B(z, u, ξ)| ≤ b0w(z)[|ξx|2 + λ2(x)|ξy|2]p/2 + b1(z)[|ξx|2 + λ2(x)|ξy|2] p−1 2 + d(z)|u|p−1 + f(z) ξ ·A(z, u, ξ) ≥ w(z)[|ξx|2 + λ2(x)|ξy|2]p/2 − d1(z)|u|p − g(z) . (4.3) where 1 < p < N , w = v1− p N and v is a strong A∞ weight. We show that locally bounded weak solutions satisfy a Harnack inequality and, as a consequence, some regularity properties. We shall make the following assumptions on the lower order terms to ensure the continuity of local weak solutions: a, b0 ∈ R, ( b w ) p p−1 , (b1 w )p , d w , d1 w , ( e w ) p p−1 , f w , g w ∈ S′v(Ω) . (4.4) Theorem 4.2. Let u be a non negative weak supersolution of equation (4.1) in Ω satisfying (4.3) and (4.4). Let Br be a ball such that B3r b Ω and let M be a constant such that u ≤M in B3r. Then there exists c depending on n, M , a, b0, p and the weight v such that w−1(B2r) ∫ B2r uwdz ≤ c { inf Br u+ h(r) } where h(r) = [ φ (( e w ) p p−1 ; r ) + φ ( g w ; r )]1/p + [ φ ( f w ; r )] 1 p−1 . Proof. We simplify the structure assumptions by setting uh = u+ h(r). We obtain |A(z, u, ξ)| ≤ aw(z)[|ξx|2 + λ2(x)|ξy|2] p−1 2 + b2(z)|uh|p−1 |B(z, u, ξ)| ≤ b0w(z)[|ξx|2 + λ2(x)|ξy|2]p/2 + b1(z)[|ξx|2 + λ2(x)|ξy|2] p−1 2 + d2(z)|uh|p−1 ξ ·A(z, u, ξ) ≥ w(z)[|ξx|2 + λ2(x)|ξy|2]p/2 − d3(z)|uh|p (4.5) where b2 = b+ h(r)1−pe, d2 = d+ h(r)1−pf , and d3 = d1 + h(r)−pg. It is easy to check that b2, d2, and d3 satisfy the same assumptions of b, d d1. We take ϕ = ηpuβhe −b0uh , β < 0, as test function in (4.2), (see [18]), where η ∈ C1 0 (B3r), η ≥ 0. We obtain∫ B3r ηpe−b0uh(b0u β h + |β|uβ−1 h )∇λuh ·Adz − p ∫ B3r uβhη p−1e−b0uh∇λη ·Adz + ∫ B3r ηpuβhe −b0uhBdz ≤ 0 . EJDE-2022/65 BOUNDARY REGULARITY FOR STRONGLY DEGENERATE OPERATORS 7 The previous inequality and the structure assumptions (4.5) yield∫ B3r e−b0uhηp(b0u β h + |β|uβ−1 h )|∇λuh|pwdz ≤ ∫ B3r e−b0uhηp(b0u β h + |β|uβ−1 h )(∇λuh ·A+ d3|uh|p)dz ≤ p ∫ B3r uβhη p−1e−b0uh∇λη ·Adz − ∫ B3r ηpuβhe −b0uhB dz + ∫ B3r e−b0uhηp(b0u β h + |β|uβ−1 h )d3|uh|p dz ≤ p ∫ B3r uβhη p−1e−b0uh∇λη ·Adz + ∫ B3r ηpuβhe −b0uh(b0|∇λuh|pw + b1|∇λuh|p−1 + d2|uh|p−1)dz + ∫ B3r e−b0uhηp(b0u β h + |β|uβ−1 h )d3|uh|p dz . By Young inequality and boundedness of uh in B3r we obtain |β| ∫ B3r ηpuβ−1 h |∇λuh|pwdz ≤ cp ∫ B3r uβhη p−1∇λη ·Adz + c ∫ B3r ηpuβh(b1|∇λuh|p−1 + d2|uh|p−1)dz + c ∫ B3r ηp(b0u β h + |β|uβ−1 h )d3|uh|p dz ≤ c ∫ B3r { puβhη p−1|∇λη|(aw|∇λuh|p−1 + b2|uh|p−1) + ηpuβhb1|∇λuh| p−1 + ηpuβ+p−1 h d2 + ηpb0u β+p h d3 + |β|ηpuβ+p−1 h d2 } dz ≤ c ∫ B3r { puβhη p−1|∇λη|aw|∇λuh|p−1 + puβ+p−1 h ηp−1|∇λη|b2 + ηpuβhb1|∇λuh| p−1 + (1 + |β|)ηpuβ+p−1 h d2 + ηpb0u β+p h d3 } dz . Then |β| ∫ B3r ηpuβ−1 h |∇λuh|pwdz ≤ c(b0,M, p) ∫ B3r { uβhη p−1|∇λη|a|∇λuh|p−1w dz + εηpuβ−1 h |∇λuh|pw + c(ε)ηp bp1 wp−1 uβ+p−1 h + ηp−1|∇λη|uβ+p−1 h b2 + (1 + |β|)ηpuβ+p−1 h d2 + ηpuβ+p h d3 } dz 8 G. DI FAZIO, M. S. FANCIULLO, P. ZAMBONI EJDE-2022/65 ≤ c(b0,M, a, p) ∫ B3r { uβhη p−1|∇λη||∇λuh|p−1wdz + εηpuβ−1 h |∇λuh|pw + c(ε)ηp bp1 wp−1 uβ+p−1 h + uβ+p−1 h |∇λη|pw + ηpuβ+p−1 h b p p−1 2 w 1 p−1 + (1 + |β|)ηpuβ+p−1 h d2 + ηpuβ+p−1 h d3 } dz . Setting V = b p p−1 2 w 1 p−1 + bp1 wp−1 + d2 + d3, we obtain∫ B3r ηpuβ−1 h |∇λuh|pwdz ≤ c(1 + |β|−1)p ∫ B3r { |∇λη|puβ+p−1 h w + V ηpuβ+p−1 h } dz . (4.6) Now we set U(x) = { uqh(x) where pq = p+ β − 1 if β 6= 1− p log uh(x) ifβ = 1− p by (4.6) we have∫ B3r ηp|∇λU|pw dz ≤ c|q|p(1 + |β|−1)p {∫ B3r |∇λη|pUpw dz + ∫ B3r V ηpUp dz } , β 6= 1− p , (4.7) while ∫ B3r ηp|∇λU|pw dz ≤ c {∫ B3r |∇λη|pw dz + ∫ B3r V ηp dz } (4.8) if β = 1− p. Let us start with the case β = 1− p. By Theorem 3.4 we have∫ B3r V ηp dz ≤ cφ (V w ; diam Ω )∫ B3r |∇λη|pw dz , and from (4.8), ∫ B3r ηp|∇λU|pw dz ≤ c ∫ B3r |∇λη|pw dz . Let Bh be a ball contained in B2r. Choosing η(x) so that η = 1 in Bh, 0 ≤ η ≤ 1 in B3r \Bh and |∇λη| ≤ 3 h , we obtain ‖∇λU‖Lpv(Bh) ≤ c w(Bh)1/p h . By Theorem 2.4 and John-Nirenberg lemma [2], there exist two positive constants p0 and c, such that( − ∫ B2r ep0Uw dz )1/p0( − ∫ B2r e−p0Uw dz )1/p0 ≤ c . (4.9) EJDE-2022/65 BOUNDARY REGULARITY FOR STRONGLY DEGENERATE OPERATORS 9 Let us consider the family of seminorms Φ(p, ρ) = (∫ Bρ |uh|pw dz )1/p , p 6= 0 . By (4.9) we have 1 w(B2r)1/p0 Φ(p0, 2r) ≤ cw(B2r) 1/p0Φ(−p0, 2r) . In the case (4.7) by Corollary 3.5 we obtain∫ B3r |∇λU|pηpw dz ≤ c { (|q|p + 1) ( 1 + 1 |β| )p ∫ B3r |∇λη|pUpwdz + [ 1 φ−1 ( V w ; |q|−p(1 + 1 |β| ) −p )]n+p ∫ B3r ηpUpw dz } . From (2.3), setting k = q/p, we have(∫ B3r |ηU|kpw dz )1/k ≤ cw(B) 1 k−1 { (|q|p + 2) ( 1 + 1 |β| )p ∫ B3r |∇λη|pUpw dz + [ 1 φ−1 ( V w ; |q|−p ( 1 + 1 |β| )−p)]n+p ∫ B3r ηpUpw dz } where c is a positive constant independent of w. Now we choose the function η. Let r1 and r2 be real numbers such that r ≤ r1 < r2 ≤ 2r and let the function η be chosen so that η(z) = 1 in Br1 , 0 ≤ η ≤ 1 in Br2 , η(z) = 0 outside Br2 , |∇λη| ≤ c r2−r1 for some fixed constant c. we have(∫ Br1 Ukpw dz )1/k ≤ cw(B) 1 k−1 1 (r2 − r1)p (|q|p + 2) ( 1 + 1 |β| )p × [ 1 φ−1 ( V w ; |q|−p ( 1 + 1 |β| )−p)]n+p ∫ Br2 Upw dz . Setting γ = pq = p+ β − 1 and recalling that U(z) = uqh(z), we obtain Φ(kγ, r1) ≥ c1/γw(B) 1 γ ( 1 k−1)(|q|p + 2)1/γ × [ 1 φ−1 ( V w ; |q|−p )]n+p γ 1 (r2 − r1)1/p Φ(γ, r2) , (4.10) for negative γ. This is the inequality we are going to iterate. If γi = kip0 and ri = r + r 2i , i = 1, 2, . . . iteration of (4.10) and use of [9, Lemma 2.4] yield Φ(−∞, r) ≥ c(p, a, φV w ,diam Ω)ω(Br) 1 p0 Φ(−p0, 2r) . Therefore by Hölder inequality, Φ(p′0, 2r) ≤ Φ(p0, 2r)w(Br) 1 p′0 − 1 p0 , p′0 ≤ p0 . So we obtain w−1(B2r)Φ(1, 2r) ≤ cΦ(−∞, r) where c ≡ c(p, a, φV w ,diam Ω) and the result follows. � We obtain a weak Harnack inequality for weak subsolutions in a similar way of Theorem 4.2. 10 G. DI FAZIO, M. S. FANCIULLO, P. ZAMBONI EJDE-2022/65 Theorem 4.3. Let u be a non negative weak subsolution of equation (4.1) in Ω satisfying (4.3) and (4.4). Let Br be a ball such that B3r b Ω and let M be a constant such that u ≤M in B3r. Then there exists c depending on n, M , a, b0, p and the weight v such that sup Br u ≤ c { w−1(B2r) ∫ B2r uwdz + h(r) } where h(r) = [ φ (( e w ) p p−1 ; 3r ) + φ ( g w ; 3r )]1/p + [ φ ( f w ; 3r )] 1 p−1 . If we take a non negative weak solution, we can put together the two previous results. Theorem 4.4. Let u be a non negative weak solution of equation (4.1) in Ω satis- fying (4.3) and (4.4). Let Br be a ball such that B3r b Ω and let M be a constant such that u ≤ M in B3r. Then there exists c depending on n, M , a, b0, p and the weight v such that sup Br u ≤ c { inf Br u+ h(r) } , where h(r) = [ φ (( e w ) p p−1 ; 3r ) + φ ( g w ; 3r )]1/p + [ φ ( f w ; 3r )] 1 p−1 . Now, as a simple consequence of Harnack inequality, we obtain some regularity results for weak solutions of (4.1). The proof is an immediate consequence of Harnack inequality so we omit it. Theorem 4.5. Let u be a locally bounded weak solution of equation (4.1) in Ω satisfying (4.3) and (4.4). Then u is continuous in Ω. If we assume more restrictive assumptions on the lower order terms we obtain the following refinement of the previous one. Theorem 4.6. Let u be a locally bounded weak solution of equation (4.1) in Ω satisfying (4.3) and a, b0 ∈ R, ( b w ) p p−1 , (b1 w )p , d w , d1 w , ( e w ) p p−1 , f w , g w ∈ L1,σ v (Ω), σ > 0 . Then u is locally Hölder continuous in Ω. 5. Boundary Harnack inequality Our next step is to show a Harnack inequality near the boundary of Ω for weak supersolutions and subsolutions to the equation divA(z, u,∇λu) +B(z, u,∇λu) = 0 , (5.1) with the structural conditions |A(z, u, ξ)| ≤ aw(z)[|ξx|2 + λ2(x)|ξy|2] p−1 2 + b(z)|u|p−1 + e(z) |B(z, u, ξ)| ≤ b0w(z)[|ξx|2 + λ2(x)|ξy|2]p/2 + b1(z)[|ξx|2 + λ2(x)|ξy|2] p−1 2 + d(z)|u|p−1 + f(z) ξ ·A(z, u, ξ) ≥ w(z)[|ξx|2 + λ2(x)|ξy|2]p/2 − d1(z)|u|p − g(z) (5.2) EJDE-2022/65 BOUNDARY REGULARITY FOR STRONGLY DEGENERATE OPERATORS 11 where 1 < p < N , w = v1− p N and v is a strong A∞ weight and a, b0 ∈ R, ( b w ) p p−1 , (b1 w )p , d w , d1 w , ( e w ) p p−1 , f w , g w ∈ S′v(Ω) . (5.3) Let S be a subset of ∂Ω and u be a function on Ω. We say that u ≤ M on S if for all ε > 0 there exists a neighborhood N of S such that u(x) ≤ M + ε for a.e. x ∈ N ∩ Ω. In this way we can define infS u, supS u and oscS u. Now, let Br be a ball centered at x0 ∈ ∂Ω and u ∈ H1,p v (Ω ∩B4r, w) we set ũ(x) = { min{u,m} if x ∈ Ω ∩B4r m if x ∈ Rn \ (Ω ∩B4r) where m = inf∂Ω∩B4r u. Moreover, we define b = 0, d = 0, d1 = 0, e = 0, f = 0, g = 0 outside Ω. Theorem 5.1. Let u ∈ H1,p v (Ω ∩ B4r) be a weak non negative supersolution of (5.1) in Ω ∩ B4r. Assume (5.2) and (5.3). Let M be a constant such that u ≤ M on Ω∩B4r. Then there exists c depending on n, M , a, b0, p and the weight v such that w−1(B2r) ∫ B2r ũ wdz ≤ c { inf Br ũ+ φ1/p [( e w ) p p−1 ; r ] + φ 1 p−1 ( f w ; r ) + φ1/p ( g w ; r )} . (5.4) Proof. Set h = φ1/p [( e w ) p p−1 ; r ] + φ 1 p−1 ( f w ; r ) + φ1/p ( g w ; r ) and ṽ = ũ + h. Let η ∈ C1 0 (B3r) and η ≥ 0. For β < 0 we define ϕ(z) = ηp[ṽβ − (m+ h)β ]e−|b0|ṽ ∈ H1,p 0,v (B3r). From (5.2), we obtain in the support of ϕ, |A(z, u,∇λu)| ≤ aw(z)|∇λu|p−1 + b2(z)|ṽ|p−1 |B(z, u,∇λu)| ≤ b0w(z)|∇λu|p + b1(z)|∇λu|p−1 + d2(z)|ṽ|p−1 ξ ·A(z, u,∇λu) ≥ w(z)|∇λu|p − d3(z)|ṽ|p (5.5) where b2(z) = b(z) + e(z) hp−1 , d2(z) = d+ f(z) hp−1 , and d3(z) = d1(z) + g(z) hp . Since, for any 0 < ρ < 3r, φ ((b2 w ) p p−1 ; ρ ) ≤ φ (( b w ) p p−1 ; ρ ) + 1 hp φ (( e w ) p p−1 ; ρ ) , φ (d2 w ; ρ ) ≤ φ ( d w ; ρ ) + 1 hp−1 φ ( f w ; ρ ) , φ (d3 w ; ρ ) ≤ φ (d1 w ; ρ ) + 1 hp φ ( g w ; ρ ) we obtain (b2 w ) p p−1 , d2 w , d3 w ∈ S′v(B3r). Since u is a supersolution of (5.1) we have∫ B3r ηpe−|b0|ṽ{|β|vβ−1 + b0[ṽβ − (m+ h)β ]}A(z, u,∇λu) · ∇λu dz + ∫ B3r B(z, u,∇λu)ϕdz 12 G. DI FAZIO, M. S. FANCIULLO, P. ZAMBONI EJDE-2022/65 ≤ p ∫ B3r ηp−1A(z, u,∇λu) · ∇λη[ṽβ − (m+ h)β ]e−|b0|ṽdz Using (5.5) we obtain∫ B3r (w|∇λu|pd3ṽ p)ηpe−|b0|ṽ{|β|ṽβ−1 + b0[ṽβ − (m+ h)β ]}dz ≤ p ∫ B3r (aw|∇λu|p−1 + b0ṽ p−1)ηp−1|∇λη|[ṽβ − (m+ h)β ]e−|b0|ṽdz + ∫ B3r (b0w|∇λu|p + b1|∇λu|p−1 + d2ṽ p−1)ηp[ṽβ − (m+ h)β ]e−|b0|ṽdz from which∫ B3r |β||∇λu|pηpe−|b0|ṽ ṽβ−1w dx ≤ |β| ∫ B3r d3η pṽp+β−1e−|b0|ṽdz + b0 ∫ B3r d3η pṽp[vβ − (m+ h)β ]e−|b0|ṽ + p ∫ B3r aηp−1|∇λη||∇λu|p−1[ṽβ − (m+ h)β ]e−|b0|ṽwdz + p ∫ B3r b2ṽ p−1ηp−1|∇λη|[ṽβ − (m+ h)β ]e−|b0|ṽdz + ∫ B3r b1η p|∇λu|p−1[ṽβ − (m+ h)β ]e−|b0|ṽdz + ∫ B3r d2ṽ p−1ηp[ṽβ − (m+ h)β ]e−|b0|ṽdz . Then ∫ B3r |β||∇λu|pηpṽβ−1w dx ≤ c|β| ∫ B3r d3η pṽp+β−1dz + c ∫ B3r d3η pṽp[ṽβ − (m+ h)β ] + c ∫ B3r aηp−1|∇λη||∇λu|p−1[ṽβ − (m+ h)β ]wdz + c ∫ B3r b2ṽ p−1ηp−1|∇λη|[ṽβ − (m+ h)β ]dz + c ∫ B3r b1η p|∇λu|p−1[ṽβ − (m+ h)β ]dz + c ∫ B3r d2ṽ p−1ηp[ṽβ − (m+ h)β ]dz . Since ṽβ − (m+ h)β ≤ ṽβ the proof follows as in the proof of Theorem 4.2. � In a similar way, let Br be a ball centered at x0 ∈ ∂Ω and u ∈ H1,p v (Ω ∩ B4r), we set u(x) = { max{u,M} if x ∈ Ω ∩B4r M if x ∈ Rn \ (Ω ∩B4r) where M = sup∂Ω∩B4r u. EJDE-2022/65 BOUNDARY REGULARITY FOR STRONGLY DEGENERATE OPERATORS 13 Theorem 5.2. Let u ∈ H1,p v (Ω∩B4r) be a weak non negative subsolution of (5.1) in Ω ∩ B4r. Assume (5.2) and (5.3). Let M be a constant such that u ≤ M on Ω ∩ B4r. Then there exists c depending on n, M , a, b0, p and the weight v such that sup Br u ≤ c { w−1(B2r) ∫ B2r uw dz + φ1/p [( e w ) p p−1 ; r ] + φ 1 p−1 ( f w ; r ) + φ1/p ( g w ; r )} . To obtain regularity up to the boundary of the domain we need some geometric assumptions. Definition 5.3. Let Ω be a domain in RN and z0 ∈ ∂Ω. Let v be a strong A∞ weight and w = v1− p N , 1 < p < N . We say that w satisfies the condition Av at z0 if there exist positive constants R0 and A such that w(Br(z0) \ Ω) w(Br(z0)) ≥ A, 0 < r < R0 . We say that Ω satisfies the condition Av if it satisfies the condition at any point. In the case v = 1 the Av condition gives back the outer sphere condition. Using the geometric assumption Av we give an estimate for the oscillation of solutions near the boundary. Theorem 5.4. Let Ω be a bounded open set satisfying the Av condition at z0 ∈ ∂Ω. Let u be a locally bounded weak solution of equation (5.1) in Ω satisfying (5.2) and (5.3). Then there exists R0 > 0 such that for any ball Br(z0), with 0 < r < R0 and µ ∈ (0, 1) we have oscBr∩Ω u ≤ c [( r R0 )α oscBR0 ∩Ω u+ oscB rµR 1−µ 0 ∩∂Ω u+ h(rµR1−µ 0 ) ] , where c and α are positive constants and h is an infinitesimal function. Proof. For ρ > 0 set M(ρ) = supBρ∩Ω u and m(ρ) = infBρ∩Ω u, with Bρ = Bρ(z0). Let 0 < r ≤ R0/4 the function M(4r)− u is solution of div Ã(z, u,∇λu) = B̃(z, u,∇λu) , where Ã(z, u, ξ) = A(z,M(4r)− u,−ξ), B̃(z, u, ξ) = B(z,M(4r)− u,−ξ) . Moreover à and B̃ satisfy |Ã(z, u, ξ)| ≤ aw(z)[|ξx|2 + λ2(x)|ξy|2] p−1 2 + b|u|p−1 + e, |B̃(z, u, ξ)| ≤ b0[|ξx|2 + λ2(x)|ξy|2] p−1 2 + b1[|ξx|2 + λ2(x)|ξy|2]p/2 + d|u|p−1 + f, ξ · Ã(z, u, ξ) ≥ w(z)[|ξx|2 + λ2(x)|ξy|2]p/2 − d1|u|p − g , where b(z) = 2pb(z), d(z) = 2pd(z), d1(z) = 2pd1(z), e(z) = 2pb(z)M(4r)p−1 + e(z), f(z) = 2pd(z)M(4r)p−1 + f(z), 14 G. DI FAZIO, M. S. FANCIULLO, P. ZAMBONI EJDE-2022/65 g(z) = 2pd1(z)M(4r)p−1 + g(z),( b w )p/p−1 , ( d w ) , (d1 w ) , ( e w )p/p−1 , f w , g w ∈ S′v(Ω) . Then by (5.4) and condition Av, we have M(4r)−M ≤ w(B2r \ Ω) Aw(B2r) [M(4r)−M ] = 1 Aw(B2r) ∫ B2r\Ω [M(4r)−M ]w dx ≤ 1 Aw(B2r) ∫ B2r\Ω [ ˜M(4r)− u]w dx ≤ c[ inf Br∩Ω (M(4r)− u) + h(r)] ≤ c[M(4r)−M(r) + h(r)] . (5.6) where M = supB4r∩∂Ω u, m = infB4r∩∂Ω u and h(r) = φ1/p [( e w ) p p−1 ; r ] + φ 1 p−1 ( f w ; r ) + φ1/p ( g w ; r ) . In a similar way, for u−m(4r), m−m(4r) ≤ c[m(r)−m(4r) + h(r)] . (5.7) From (5.6) and (5.7) we obtain, for θ < 1 M(r)−m(r) ≤ θ[M(4r)−m(4r)] +M −m+ ch(r) , from which applying [24, Lemma 8.23] (see also [16]) we obtain the result. � As consequences of the previous Theorem we obtain the following corollary. Corollary 5.5. Let Ω be a bounded open set satisfying condition Av in every x0 ∈ ∂Ω. Let u be a locally bounded weak solution of equation (5.1) in Ω satisfying (5.2) and (5.3). Let u = ϕ on ∂Ω. If ϕ is continuous in ∂Ω then u is continuous in Ω. Now we refine our assumptions on lower order terms. If we assume the coefficients in a suitable Morrey space we obtain Hölder continuity of the solution. Corollary 5.6. Let Ω be a bounded open set satisfying the Av condition in every z0 ∈ ∂Ω. Let u be a locally bounded weak solution of equation (5.1) in Ω satisfying (5.2) and a, b0 ∈ R, ( b w ) p p−1 , (b1 w )p , d w , d1 w , ( e w ) p p−1 , f w , g w ∈ L1,p−ε v (Ω) , (5.8) with 0 < ε < p. Let u = ϕ on ∂Ω. If ϕ is Hölder continuous in ∂Ω, then u is Hölder continuous in Ω. Acknowledgements. This work was supported by Universite.gli Studi di Catania, “Piano PIA.CE.RI”, upb 53722122154. EJDE-2022/65 BOUNDARY REGULARITY FOR STRONGLY DEGENERATE OPERATORS 15 References [1] M. Aizenman - B. Simon; Brownian motion and Harnack’s inequality for Schrödinger oper- ators, Comm. Pure Appl. Math., 35 (1982), 209–271. [2] S. M. Buckley; Inequalities of John–Nirenberg type in doubling spaces, J. Anal. Math. 79 (1999), 215–240. [3] L. Capogna, D. Danielli, N. Garofalo; An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. P.D.E. 18 (1993), 1765–1794. [4] F. Chiarenza, E. Fabes, N. Garofalo; Harnack’s inequality for Schrödinger operators and continuity of solutions, Proc. AMS 98 (1986), 415–425. [5] D. Danielli; A Fefferman-Phong type inequality and applications to quasilinear subelliptic equations, Potential Analysis 115 (1999), 387–413. [6] D. Danielli, N. Garofalo, D. Nhieu; Trace inequalities for Carnot-Caratheodory spaces and applications, Ann. Scuola Norm. Sup. Pisa, 4 (1998), 195–252. [7] G. David - S. Semmes; Strong A∞ weights, Sobolev inequalities and quasiconformal map- pings, Analysis and Partial Differential Equations, Lecture notes in Pure and Applied Math- ematics 122, Marcel Dekker, 1990. [8] G. Di Fazio, M. S. Fanciullo, P. Zamboni; Harnack inequality and regularity for degenerate quasilinear elliptic equations, Math. Z. 264 (2010), no. 3, 679–695. [9] G. Di Fazio, M. S. Fanciullo, P. Zamboni; Regularity for a class of strongly degenerate quasilinear operators, J. Differential Equations 255 (2013), no. 11, 3920–3939. [10] G. Di Fazio, M. S. Fanciullo, P. Zamboni; Harnack inequality and continuity of weak solutions for doubly degenerate elliptic equations, Mathematische Zeitschrift, 292 (2019), 1325–1336. [11] G. Di Fazio - P. Zamboni; Hölder continuity for quasilinear subelliptic equations in Carnot Caratheodory spaces, Math. Nachr., 272 (2004), 3–10. [12] G. Di Fazio, P. Zamboni; Local regularity of solutions to quasilinear subelliptic equations in Carnot Carathéodory spaces, Boll. Unione Mat. Ital., (8) 9 B (2006), no. 2, 485–504. [13] G. Di Fazio, P. Zamboni; Regularity for quasilinear degenerate elliptic equations, Math. Z., 253 (2006), 787–803. [14] B. Franchi, C. E. Gutierrez, R. Wheeden; Two-weight Sobolev-Poincaré inequalities and Harnack inequality for a class of degenerate elliptic operators, Rend. Mat. Acc. Lincei, 5 (1994), 167–175. [15] B. Franchi, C. E. Gutierrez, R. Wheeden; Weighted Sobolev-Poincaré inequalities for Grushin type operators, Commun. PDE 19 (1994), 523–604. [16] D. Gilbarg, N. S. Trudinger; Elliptic Partial Differential Equations of Second Order, Springer- Verlag, Berlin, 1983. [17] G. M. Lieberman; Sharp form of estimates for subsolutions and supersolutions of quasilinear elliptic equations involving measures, Comm. PDE 18 (1993), 1191–1212. [18] O. A. Ladyzhenskaya, N. N. Ural’tseva; Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968 xviii+495 pp. [19] J. M. Rakotoson; Quasilinear equations and Spaces of Campanato-Morrey type, Comm. P.D.E. 16 (1991), 1155–1182. [20] J. M. Rakotoson, W. P. Ziemer; Local behavior of solutions of quasilinear elliptic equations with general structure, Trans. AMS 319 (1990), 747–764. [21] Y. Sawano, G. Di Fazio, D. Hakim; Morrey Spaces: Introduction and Applications to Integral Operators and PDE’s, Volume I, Taylor and Francis,2020. [22] Y. Sawano, G. Di Fazio, D. Hakim; Morrey Spaces: Introduction and Applications to Integral Operators and PDE’s, Volume II , Taylor and Francis, 2020. [23] J. Serrin; Local behaviour of solutions of quasilinear equations, Acta Math., 111 (1964), 247–302. [24] N. S. Trudinger; On Harnack type inequalities and their application to quasilinear elliptic equations, CPAM XX (1967), 721–747. [25] P. Zamboni; The Harnack inequality for quasilinear elliptic equations under minimal as- sumptions, Manuscripta Math. 102 (2000), 311–323. 16 G. DI FAZIO, M. S. FANCIULLO, P. ZAMBONI EJDE-2022/65 Giuseppe Di Fazio Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125, Catania, Italy Email address: giuseppedifazio@unict.it Maria Stella Fanciullo Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125, Catania, Italy Email address: fanciullo@dmi.unict.it Piero Zamboni Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125, Catania, Italy Email address: zamboni@dmi.unict.it 1. Introduction 2. Strong A weights and function spaces 3. Stummel-Kato type classes 4. Harnack inequality 5. Boundary Harnack inequality Acknowledgements References