Holder regularity for bounded solutions to a class of anisotropic operators

Testo completo

(1)manuscripta math. 158, 421–439 (2019). © Springer-Verlag GmbH Germany, part of Springer Nature 2018. Stella Piro-Vernier · Francesco Ragnedda · Vincenzo Vespri. Hölder regularity for bounded solutions to a class of anisotropic operators Received: 25 November 2017 / Accepted: 22 April 2018 / Published online: 2 May 2018 Abstract. In this note we show the Hölder regularity for bounded solutions to a class of anisotropic elliptic operators. This result is the dual of the one proved by Liskevich and Skrypnik (Nonlinear Anal 71:1699–1708, 2009).. 1. Introduction In this paper we consider the following class of anisotropic equations N −1 i=1. ∂ ∂ Aq,i (x, u, Du) + A p (x, u, Du) = 0 in , ∂ xi ∂ xN. (1). with a regular domain in R N , p > q > 1. The functions Aq,i (x, u, Du) and A p (x, u, Du) : × R N +1 → R N are assumed to be measurable and satisfying the structure conditions ⎧ N −1 ⎪ ⎪ ⎪ ⎪ i) Aq,i (x, u, η) · ηi ≥ C0 | η |q ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎨ ii) A p (x, u, η) · η N ≥ C0 | η N | p (2) N −1 ⎪ ⎪ ⎪ q−1 ⎪ iii) | Aq,i (x, u, η) |≤ C1 | η | ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎩ iv) | A p (x, u, η) |≤ C1 | η N | p−1 , where C0 , C1 are positive constants and for any vector η = (η1 , . . . , η N ) in RN , η = (η1 , . . . , η N −1 ). (3) S. Piro-Vernier (B)· F. Ragnedda: Dipartimento di Matematica e Informatica, Università di Cagliari, Viale Merello 92, 09123 Cagliari, Italy. e-mail: svernier@unica.it; ragnedda@unica.it V. Vespri: Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134 Florence, Italy. e-mail: vespri@math.unifi.it Mathematics Subject Classification: 35J70 · 35J92 · 35B65; Secondary 35B45. https://doi.org/10.1007/s00229-018-1034-z.

(2) 422. S. Piro-Vernier et al.. The prototype of this class of operators is the anisotropic p-Laplacean: N −1 i=1. ∂ ∂u q−2 ∂u ∂ ∂u p−2 ∂u + = 0 in . ∂ xi ∂ xi ∂ xi ∂ xN ∂ xN ∂ xN. Let us introduce two functional spaces:. W 1,[q, p] () = u ∈ L 1 () : Dxi u ∈ L q (), i = 1, . . . , N − 1; Dx N u ∈ L p () 1,[q, p]. W0. () = W 1,[q, p] () ∩ W01,1 (). 1,[q, p]. Definition. A function u ∈ Wloc compact set K ⊂

(3) N −1 K i=1. () is a local, weak solution to (1) if for every. ∂ϕ Aq,i (x, u, Du) dx + ∂ xi 1,[q, p]. for all test functions ϕ ∈ W0.

(4) A p (x, u, Du) K. ∂ϕ d x = 0, ∂ xN. (4). (), where ϕ has compact support in K .. In the last decades many papers were devoted to the study on the anisotropic operators (see, for instance [1–3,17,18]) but the question of the Hölder regularity of solutions to equations with measurable coefficients was left open (see, for instance, the survey paper [19]). A first step to face this problem was made by Liskevich and Skrypnik [15] in 2009. They considered the following class of anisotropic quasilinear equations ∂ 2u ∂ + Ai (x, u, Du) + b(x, u, Du) = 0 ∂ x12 i=2 ∂ xi N. in .. (5). They assumed p > 2, a suitable condition on the lower terms, the following structure conditions ⎧ N ⎪ ⎪ p ⎨ i) Ai (x, u, η) · ηi ≥ C0 | η¯ | N −1 ⎪ ⎪ ⎩. i=2. ii). p−1. | Ai (x, u, η) |≤ C1 | η¯ | N −1 ,. where η¯ = (η2 , . . . , η N ) and the following nearness between the exponents holds: 1 1 1 N − 1 N p¯ , = + , p¯ < N . (6) 2< p≤ N − p¯ p¯ N 2 p We recall that this condition implies the local boundedness of the solutions (see [13,16]). More precisely they realized that the parabolic techniques (the so called time expansion of positivity techniques) developed in [7] to study the degenerate equation u t = p u,. p > 2,.

(5) Hölder regularity for bounded solutions. 423. can be adapted to study the elliptic case too, by proving a suitable expansion of positivity in space variables. In [11], the result proved in [15] was extended to the following class of anisotropic equations ∂ ∂ A p (x, u, Du) + Aq,i (x, u, Du) = 0 in . ∂ x1 ∂ xi N. (7). i=2. The authors assumed q > p > 1 , the structure conditions ⎧ i) A p (x, u, η) · η1 ≥ C0 | η1 | p ⎪ ⎪ ⎪ ⎪ ⎪ ii) | A p (x, u, η) |≤ C1 | η1 | p−1 ⎪ ⎪ ⎨ N ⎪ iii) Aq,i (x, u, η) · ηi ≥ C0 | η¯ |q ⎪ ⎪ ⎪ ⎪ i=2 ⎪ ⎪ ⎩ iv) |Aq,i (x, u, η)| ≤ C1 | η¯ |q−1 and the local boundness of u. The last assumption is alternative to the assumption of the nearness of p and q. The approach of the space expansion of positivity seems very promising and it works in different context (see [10,12]). More recently, in [9], a stability result was proved. Let u be a locally bounded weak solution u of the equation N . Dxi (A pi (x, u, Du)) = 0 in .. i=1. Assume pi > 1, i = 1, . . . , N , their harmonic mean p¯ less than the dimension of the space N and the following structure conditions: ⎧ N N ⎪ ⎪ ⎨ i) A p (x, u, Du) · Dx u ≥ C0 | Dx u | pi , i. ⎪ ⎪ ⎩. i. i=1. ii). i. i=1 pi −1. | A pi (x, u, Du) |≤ C1 | Dxi u |. , ∀i = 1, . . . , N .. There exists a q0 > 1 such that if pmax − pmin ≤ 1/q0 , with pmin = min{ p1 , . . . , p N } and pmax = max{ p1 , . . . , p N }, then the solution is locally Hölder continuous in . The aim of the present paper is to give a new insight of this difficult, challenging and still open problem. Here we prove, in a certain sense, a dual result with respect the one proved in [11,15]. This should be an important step to prove the regularity in the physical case of 3 spatial dimensions having each direction a different diffusion exponent..

(6) 424. S. Piro-Vernier et al.. As in [11], the condition on the nearness of the exponents does not appear. Perhaps, such a condition is necessary for the boundedness of the solutions but it does not play any role for the regularity of already bounded solutions. As in [11,15], the proof is based on a suitable space expansion of positivity, but, differently from the quoted papers, here one can avoid to work with the transformed equation and this makes simpler the calculations. In the sequel we say that a constant depends only upon the data if it depends only upon N , C0 , C1 , p, q and u L ∞ () . Theorem 1.1. Let u(x) be a locally bounded weak solution to (1)–(2) with p > q > 1, then u(x) is locally Hölder continuous. More precisely, in analogy with the parabolic p-Laplacean, it is possible to prove that there are two positive constants β < 1 and C depending only upon the data such that for any compact K strictly contained in and for any P1 = (x , x N ), P2 = (y , y N ) ∈ K ⎞β ⎛ p−q q p p ⎜ |x N − y N | + u L ∞ () |x − y | ⎟ |u(P1 ) − u(P2 )| ≤ C u L ∞ () ⎝ ⎠ , (8) ( p, q) − dist (K , ∂) p−q. q. p p where ( p, q) − dist (K , ∂) is the infimum of |x N − y N | + u L ∞ () |x − y | ranging (x , x N ) ∈ K and (y , y N ) ∈ ∂. Such a proof is quite technical and, nowadays, standard. For these reason we omit it and we refer the reader to Chapter 3 of [5]. In our forthcoming researches, as already written, we intend to combine this new approach with the ideas developed in [11,15] to prove the regularity of the solutions under more general assumptions. The scheme of this paper is the following: in Sect. 2 we state some preliminary results. In Sect. 3 we begin to prove the main Theorem starting an alternative argument that we conclude in Sect. 4 . The proof of Theorem 1.1 will result a direct consequence of the alternative argument.. 2. Preliminary results In this section we collect some results we will use in the sequel. Let r > 0 and define a ball in the last N − 1 variables N xi2 ≤ r 2 , B¯ r = (x2 , . . . , x N ) ∈ R N −1 : i=2. and define the following cylinder in , Q L ,r = {x ∈ R N : |x1 | ≤ L , (x2 , . . . , x N ) ∈ B¯ r }. Assume Q 1 := Q L ,r contained in . Consider another cylinder Q 2 := Q L 1 ,r1 with L 1 < L and r1 < r ..

(7) Hölder regularity for bounded solutions. 425. Denote with ω the oscillation of u in and let μ− be the ess inf of u in . Define 1 1 G(u) := − , (u − μ− + aωH )q−1 (ωH )q−1 + where [·]+ denotes the positive part, thus G(u) = 0 if u > μ− + (1 − a)ωH and G(u) is positive a.e. if u ≤ μ− + (1 − a)ωH. Here 0 < a < 1, 0 < H < 1. Lemma 2.1. Let u be a locally bounded weak solution of equation (1) and assume that the structure conditions (2) are satisfied. Then there exists a constant C > 0 (depending only upon the data) such that

(8) Q2. . A. q

(9) Hω dx ≤ C Dx ln+ |Dx ξ |q d x 1 u − μ− + aωH Q1.

(10) +C Q1. . (9). A. Dx ξ p (u − μ− + aωH ) p−q d x N A. where Dx is the gradient in the first N-1 variables, A = {x ∈ : u < μ− + (1 − a)ωH } and ξ ∈ C ∞ is a function such that ξ = 1 in Q 2 , ξ = 0 in \Q 1 . Proof. Let us take G(u)ξ p as a test function. Then using equation (1), we have

(11) N −1.

(12) Aq,i (x, u, Du)Dxi (G(u)ξ p )d x +. i=1. A p (x, u, Du)Dx N (G(u)ξ p )d x = 0. . If we use the definition of G(u) and condition (2), then

(13) (q − 1)C0 Q1. . |Dx u|q A.

(14). + (q − 1)C0 Q1.

(15) ≤ pC1 Q1. .

(16). + pC1 Q1. . 1 ξ pd x (u − μ− + aωH )q. Dx u p N A. |Dx u|q−1 A. . 1 ξ p−1 |Dx ξ | d x + (u − μ− + aωH )q−1. Dx u p−1 N A. 1 ξ pd x (u − μ− + aωH )q. 1 ξ p−1 Dx N ξ d x q−1 (u − μ− + aωH ). The estimate (9) follows by using Young inequality in the right hand side of the previous estimate..

(17).

(18) 426. S. Piro-Vernier et al.. Lemma 2.2. (Sobolev–Troisi Inequality [21], see also [20] and [14]) Let ⊂ R N 1,[ p ,..., p N ] be a bounded open set and consider u ∈ W0 1 (), pi ≥ 1 for all i = 1, . . . , N . Let 1 11 = , p¯ N pi N. p¯ ∗ =. i=1. N p¯ . N − p¯. Then there exists c depending on N , p1 , . . . , p N if p¯ < N such that. u LN p∗ () ≤ c. N Dx u i L i=1. pi (). .. Lemma 2.3. (Algebraical lemma [4], see also [8]) Let {Ym }, m = 0, 1, 2, . . . , be a sequence of positive numbers, satisfying the recursive inequalities Ym+1 ≤ Cbm Ym1+λ where C, b > 1 and λ > 0 are given numbers. If Y0 ≤ C. −1 λ. −1. b λ2 ,. then {Ym } converges to zero as m → ∞. Lemma 2.4. (Generalised Caccioppoli’s Inequality) Let u be a locally bounded weak solution of (1) and assume that the structure conditions in (2) are satisfied. Then there exists a constant C > 0 (depending only upon the data) such that for every test function θ ∈ C01 (), we have

(19). p θ p (|Dx (u − k)− |q + Dx N (u − k)− )d x. .

(20). p (θ p−q |Dx θ |q |(u − k)− |q + Dx N θ |(u − k)− | p )d x,. ≤C . where Dx is the gradient in the first N − 1 variables and k any positive constant. Note that an analogous result also holds if we deal with (u − k)+ . Proof. Take ψ = ψ(x) = −θ (x) p (u − k)− , θ ∈ C01 (). as a test function. Note that Dx ψ = pθ p−1 (u − k)− Dx θ + θ p Dx (u − k)− Dx N ψ = pθ p−1 (u − k)− Dx N θ + θ p Dx N (u − k)− ..

(21) Hölder regularity for bounded solutions. 427. Then using Eq. (1), we have

(22) pθ p−1 (u − k)−. N −1 . Aq,i (x, u, Du)Dxi θ d x. i=1. .

(23) N −1 Aq,i (x, u, Du)Dxi (u − k)− d x + θp . i=1.

(24) + pθ p−1 (u − k)− A p (x, u, Du)Dx N θ d x .

(25) + θ p A p (x, u, Du)Dx N (u − k)− d x = 0. . If we take into account (2), we obtain the following inequality

(26) p C0 θ p (|Dx (u − k)− |q + Dx N (u − k)− )d x .

(27) (N − 1)θ p−1 |(u − k)− | |Dx θ | |Dx (u − k)− |q−1 d x. ≤ pC1 .

(28). + pC1. p−1 θ p−1 |(u − k)− | Dx N θ Dx N (u − k)− d x.. .

(29). Again, the statement follows by applying Young inequality.. In the next lemma we will consider a DeGiorgi type lemma. It is necessary to introduce the intrinsic geometry induced by the anisotropy of the operator itself (for more details about the intrinsic geometry see [5,22]). For a Lebesgue measurable set E ⊂ R N denote by |E| its measure. q. p−q. Let R0 , k0 > 0 be given numbers and R N = R0p k0 p . Define for j = 0, 1, 2, . . . Q R0 , j := x : |xi | < R j , ∀i = 1, . . . , N − 1, |x N | < r j , N 0 where R j = R20 + 2Rj+1 and r j = R2N + 2Rj+1 . Let c j = |Q R0 , j | the measure of the parallelepiped Q R0 , j . Clearly c j is a decreasing sequence converging to a strictly positive constant c∞ . Let u be a bounded weak solution of (1) in Q R0 ,0 , assume that the structure conditions in (2) are satisfied and let μ+ = sup Q R ,0 u and μ− = inf Q R0 ,0 u. 0 Define k0 k0 As, j := x ∈ Q R0 ,s : u(x) ≤ k j , k j = + j+1 + μ− . 2 2. and Z j :=. |A2 j, j | |Q R0 ,2 j |.

(30) 428. S. Piro-Vernier et al.. Let denote A∞ as the intersection of the sets A2 j, j , i.e, A∞ = ∩ A2 j, j = {x ∈ Q R∞ : u(x) ≤ j. k0 + μ− }. 2. Lemma 2.5. (DeGiorgi type lemma) Let u be a bounded weak solution of (1) in Q R0 ,0 , assume that the structures conditions in (2) are satisfied . There is a number ν > 0 depending only upon the data (but not depending on u, R0 , and k0 ) such that if Z 0 < ν then {Z j } converges to zero as j goes to infinity. Proof. Let p¯ and p¯ ∗ be defined as in Sobolev-Troisi Lemma 2.2. If we consider the set A2 j+2, j+1 , we can write

(31) u − k j p¯ d x ≥ (k j − k j+1 ) p¯ |A2 j+2, j+1 |. A2 j+2, j+1. Then 1 |A2 j+2, j+1 | ≤ (k j − k j+1 ) p¯ 1 ≤ (k j − k j+1 ) p¯.

(32). (u − k j )− p¯ d x. A2 j+2, j+1.

(33). p¯ p¯ θ2 j+1 (u − k j )− d x.. A2 j+1, j. Here θ2 j+1 ∈ C∞ is a function such that θ2 j+1 = 1 in Q R0 ,2 j+2 and θ2 j+1 = 0 j out of Q R0 ,2 j+1 satisfying Dx θ2 j+1 (z) ≤ c˜ R4 0 , (where we recall once again that j Dx is the gradient with respect the first N − 1 variables), Dx θ2 j+1 (z) ≤ c˜ 4 for a positive constant c˜ and for all z ∈ Q R0 ,2 j+1 . If we use Hölder inequality, we obtain ⎡ |A2 j+2, j+1 | ≤. 1 ⎢ ⎣ (k j − k j+1 ) p¯.

(34). N. ⎤ p¯∗ p¯ p¯ ∗ ⎥ 1− p¯ (θ2 j+1 (u − k j )− ) d x ⎦ |A2 j+1, j | p¯ ∗ ,. A2 j+1, j. Now by Lemma 2.2, we have 1 1− p¯p¯∗ |A | 2 j+1, j (k j − k j+1 ) p¯ ⎡ ⎤ p¯ qN " #

(35) N −1 q ⎢ ⎥ Dx θ2 j+1 (u − k j )− d x ⎦ ⎣ i. |A2 j+2, j+1 | ≤ c. i=1. ⎡ ⎢ ⎣. A2 j+1, j.

(36) A2 j+1, j. RN. ⎤ p¯ pN " # p Dx θ2 j+1 (u − k j )− d x ⎥ ⎦ N.

(37) Hölder regularity for bounded solutions. 429. 1 1− p¯ |A2 j+1, j | p¯ ∗ p ¯ (k j − k j+1 ) ⎡ ⎤ (N −1) p¯ Nq #q

(38) " ⎢ ⎥ Dx θ2 j+1 (u − k j )− d x ⎦ ⎣ . ≤c. A2 j+1, j. ⎡ ⎢ ⎣.

(39). ⎤ p¯ pN " # p ⎥ Dx θ2 j+1 (u − k j )− d x ⎦ N. (10). A2 j+1, j.

(40) Let us estimate. " #q Dx θ2 j+1 (u − k j )− d x. We have . A2 j+1, j. " #q Dx θ2 j+1 (u − k j )− d x .

(41) A2 j+1, j.

(42). ≤C. Dx (u − k j )− q d x. Dx (θ2 j+1 )q (u − k j )− q + θ q 2 j+1. A2 j+1, j. Considering the properties of the function θ2 j+1 , we have

(43) A2 j+1, j.

(44). jq Dx (θ2 j+1 (u − k j )− )q d x ≤ C 4 q R0. (u − k j )− q d x. A2 j+1, j.

(45). Dx (u − k j )− q d x.. +C A2 j+1, j. If we pass to the larger set A2 j, j and use the function θ2 j with exponent p, we obtain

(46) A2 j+1, j. jq Dx (θ2 j+1 (u − k j )− )q d x ≤ C 4 q R0.

(47). (u − k j )− q d x. A2 j, j.

(48) +C. q p θ2 j Dx (u − k j )− d x.. A2 j, j. Here we can apply Caccioppoli’s Inequality of Lemma 2.4 to the second integral on the right hand side and we get,.

(49) 430. S. Piro-Vernier et al..

(50) A2 j+1, j. jq Dx (θ2 j+1 (u − k j )− )q d x ≤ C 4 q R0.

(51). (u − k j )− q d x. A2 j, j.

(52). Dx θ2 j q (u − k j )− q d x. +C A2 j, j.

(53). Dx θ2 j p (u − k j )− p d x. N. +C A2 j, j. If we use the properties of θ2 j ,

(54) A2 j+1, j. jq Dx (θ2 j+1 (u − k j )− ) p d x ≤ C 4 q R0. +C.

(55). (u − k j )− q d x. A2 j, j.

(56). 4 jp p RN. (u − k j )− p d x.. A2 j, j. Noting that u − k j ≤ 2k0 and by the definition of R N we have

(57) A2 j+1, j. jp Dx (θ2 j+1 (u − k j )− ) p d x ≤ C 4 k q |A2 j, j | q R0 0. Arguing as above, one obtains a similar estimate also for

(58) Dx (θ2 j+1 (u − k j )− ) p d x, N A2 j+1, j. i.e.

(59) A2 j+1, j. jq Dx (θ2 j+1 (u − k j )− ) p d x ≤ C 4 q N R0. +C. ≤C Then since. p(N ¯ −1) qN. +. p¯ pN.

(60) A2 j, j. 4 jp p RN. 4 jp q R0. (u − k j )− q d x

(61). (u − k j )− p d x. A2 j, j. q. k0 |A2 j, j |.. = 1 we obtain from (10),. 1 1− p¯p¯∗ |A2 j+2, j+1 | ≤ C |A | 2 j+1, j (k j − k j+1 ) p¯. $. % 4 jp q q k |A2 j, j |. . R0 0. (11).

(62) Hölder regularity for bounded solutions. 431. Noting that |A2 j+1, j | ≤ |A2 j, j | we get from (11) that, |A2 j+2, j+1 | ≤. 2( j+2) p¯ p¯. k0. |A2 j, j |. 1+(1− p¯p¯∗ ) ( j+1) p. 4. q. k0. q.. R0. Divide both sides by |Q R0 ,2 j | to have the following inequality Z j+1 ≤ C 2( j+2) p¯ 4( j+1) p Z 1+λ j where λ := (1 − p¯p¯∗ ). Then the required statement follows by Lemma 2.3..

(63). In the sequel we need to apply the DeGiorgi type lemma in a very special case that allows us to get rid of the anisotropy. More specifically: consider a cube of side 2R0 , i.e Q R0 := {x : |xi | ≤ R0 , ∀i = 1, . . . , N }. Let u be a bounded weak solution of (1) in Q R0 , assume that the structure conditions in (2) are satisfied and let μ+ = sup Q R u and μ− = inf Q R0 u. 0 Assume that u(x1 , . . . , x N ) ≥ μ− + k0 ,. (12). when |xi | ≤ R0 for i = 1, . . . , N − 1 and x N = ±R0 . Lemma 2.6. (When the anisotropic DeGiorgi Lemma becomes isotropic) Let u be a bounded weak solution of (1) in Q R0 , assume that conditions (2) and (12) are satisfied. Then there is a number ν > 0 depending only upon the data (but not depending on u, R0 , and k0 ) and a constant γ depending only upon the data and k0 such that if Z 0 < ν either μ+ − μ− ≤ γ R0N or {Z j } converges to zero as j goes to infinity. Proof. Define Q ∗R0 , j := x : |xi | < R j , ∀i = 1, . . . , N − 1, |x N | < R0 , 0 where R j = R20 + 2Rj+1 . ∗ Let c˜ j = |Q R0 , j |. Clearly c˜ j is a decreasing sequence converging to a strictly positive constant c∞ . Define. As, j := {x ∈ Q ∗R0,s : u(x) ≤ k j }, k j = and Z j :=. |A2 j, j | |Q ∗R |. 0,2 j. k0 k0 + j+1 + μ− . 2 2.

(64) 432. S. Piro-Vernier et al.. In the sets Q ∗R0 , j , choosing as test function −θ q (u − ki )− = −θ q (ki − u)+ where θ does not depend on x N , the Caccioppoli generalised inequality becomes

(65) p θ q (|Dx (u − ki )− |q + Dx N (u − ki )− )d x Q ∗R. 0, j.

(66) |Dx θ |q |(u − ki )− |q d x,. ≤C Q ∗R. 0, j. where with Dx we denoted the gradient with respect the first N − 1 variables. On the other hand

(67) q θ q (|Dx (u − ki )− |q + Dx N (u − ki )− )d x Q ∗R. 0, j.

(68). p θ q (|Dx (u − ki )− |q + Dx N (u − ki )− )d x + |A j,i |. ≤ Q ∗R. Hence. 0, j.

(69) Q ∗R. q θ q (|Dx (u − ki )− |q + Dx N (u − ki )− )d x 0, j.

(70) |Dx θ |q |(u − ki )− |q d x + |A j,i |. ≤C Q ∗R. 0, j. The statement, now, follows by the classical properties of non homogeneous DeGiorgi classes, by repeating in a straightforward way the proof of Proposition 4.1, Chapter 10 of [6]..

(71) Before stating the last preliminary result, let us introduce some notation. For a function v defined in a set E and real numbers k < l, put [v > l] = {x ∈ E v(x) > ł} [v < k] = {x ∈ E v(x) < k} [k < v < ] = {x ∈ E k < v(x) < l}. For ρ > 0 and y ∈ R N , denote by Bρ (y) the ball of radius ρ centered at y, and by K ρ (y) the cube of edge ρ, centered at y and with faces parallel to the coordinate planes. If y is the origin, let Bρ (0) = Bρ , and K ρ (0) = K ρ . Lemma 2.7. (DeGiorgi [4]) Let v ∈ W 1,1 (K ρ (y)), and let k < l be real numbers. There exists a constant γ depending only on N , p and independent of k, l, v, y, ρ, such that

(72) ρ N +1 (l − k)|[v < k]| ≤ γ |Dv|d x. (13) |[v > l]| [k<v<l].

(73) Hölder regularity for bounded solutions. 433. Remark 2.8. The conclusion of the lemma continues to hold for functions v ∈ W 1,1 (E) provided E is convex; in such a case, instead of ρ N +1 , we have d N +1 , where d is the diameter of the set E.. 3. First alternative Without loss of generality (modulo suitable standard homothetical transformations), we may assume to work in the unitary cube Q 1 ; we may also assume that 0 < 2 u < 1 in Q 1 . Consider now a slide S N of this cube, S N = {(x1 , . . . , x N −1 , x N ) ∈ 2. R N : − 21 < xi < 21 , ∀ 1 ≤ i ≤ N − 1, −ε < x N < ε} with ε a small positive number to be quantified later. By using a suitable homothetical transformation, we can transform S N into TN where the set TN = {(x1 , . . . , x N −1 , x N ) ∈ R N : − 21 θ < xi < 21 θ, ∀ 1 ≤ i ≤ N − 1, − 21 < x N < 21 } where θ is a very large positive parameter to be chosen. Note that, when θ is chosen, ε too is quantified. Without loss of generality, we may assume θ ∈ N. Now, partition TN in sub-cubes Q ( j) of unitary wedge. Assume that for each Q ( j) 1 [x ∈ Q ( j) : u(x) ≤ ] > ν|Q ( j) | 2. (14). where ν is the quantity claimed in Lemma 2.5. Lemma 3.1. (The first alternative) Let u be a bounded weak solution of (1) in Q 1 . 2 Assume that the structure conditions (2) are satisfied and that (14) holds for any 1 Q ( j) . Then there is a n 0 ∈ N such that u ≤ 1 − 2no1+1 for a.e. x ∈ TN . 2 Proof. Apply Lemma 2.7 to any Q ( j) with l =1−. 1 , 2n. k =1−. 1 , 2n−1. h =l −k =. 1 , 2n. n = 2, . . . , n o ,. where n o is to be chosen. We obtain & '

(74) 1 1 ( j) u >1− n ∩ Q ≤γ |Du| d x. 2n 2 [k<v<l]∩Q ( j) If we sum over all the Q ( j) , we get & '

(75) 1 1 u > 1 − n ∩ TN ≤ γ |Du| d x. 2n 2 [k<v<l]∩TN Since

(76) [k<v<l]∩TN.

(77) |Du| d x ≤.

(78) [k<v<l]∩TN. |Dx u| d x +. [k<v<l]∩TN. |Dx N u| d x,.

(79) 434. S. Piro-Vernier et al.. (where with Dx we have denoted the gradient with respect the first N −1 variables), 1 denoting [u > 1 − n ] ∩ TN by An , we have 2

(80) 1 |A | ≤ γ |Du| d x n 2n [k<v<l]∩Tn #1 "

(81) q q−1 q |Dx u| d x |An−1 \An | q ≤γ [k<v<l]∩TN. "

(82) +γ. #1 |Dx N u| d x p. [k<v<l]∩TN. p. |An−1 \An |. p−1 p. .. By generalised Caccioppoli inequality applied to (u − k)+ it follows 1 |An | 2n &

(83) ≤γ 2TN. & ζ p−q |Dx ζ |q |(u − k)+ |q. ' '1 q q−1 p p + Dx N ζ |(u − k)+ | d x · |An−1 \An | q &

(84) & ζ p−q |Dx ζ |q |(u − k)+ |q +γ 2TN. ' '1 p p−1 p p + Dx N ζ |(u − k)+ | d x · |An−1 \An | p . By choosing a suitable ζ 1 |An | 2n & " ≤ γ hp. # '1 q q−1 1 q + 1 |T | · |A \A | N n−1 n θ q h p−q & " # '1 p p−1 1 p p , +γ h + 1 |T | · |A \A | N n−1 n θ q h p−q. 1 since by the choice of k and h we have (u − k)+ < 1 − (1 − 2n−1 ) < 2h. Moreover. 1 |An | 2n " # p &" # '1 q q−1 1 1 q ≤γ + 1 |T | · |An−1 \An | q N n q p−q 2 θ h " # &" # '1 p p−1 1 1 +γ + 1 |T | · |An−1 \An | p . N n q p−q 2 θ h.

(85) Hölder regularity for bounded solutions. 435. Suppose that n o has already been chosen, and let θ =2. no. p−q q. As mentioned above, provided n o is sufficiently large, we can always assume that n o p−q q is an integer. Due to the previous choice for θ we derive # " p−1 q−1 1 1 p p q q |An | ≤ γ |TN | |An−1 \An | . + |TN | |An−1 \An | At each step,we have that one of these two inequalities holds: ⎧ p 1 ⎪ ⎨ |An | p−1 ≤ γ |Q| p−1 |An−1 \An | or ⎪ q 1 ⎩ |An | q−1 ≤ γ |Q| q−1 |An−1 \An |.. (15). Once more, suppose that n o has already been chosen: then, or for p or for q the previous inequality holds at least n2o times. Therefore, summing An in (15) when it holds for p or q, we conclude that 1 no |[x ∈ TN : u(x) ≥ 1 − n ]| < γ |TN |. 2 2 o The value of n o is determined in such a way that 2γ ≤ ν, no where ν is the same parameter as in Lemma 2.5. Notice that TN is scaled correctly according to the intrinsic geometry considered in Lemma 2.5. Therefore, we can apply such a Lemma, and conclude that u ≤1− for a.e. x ∈. 1 2n o +1. 1 TN . 2.

(86). Hence, if the first alternative occurs, we have reduced the oscillation of the solution u in a quantitative way in S N . 4. Second alternative and Proof of Theorem 1.1 If the first alternative does not occur, there exists at least one Q ( j) ⊂ TN , centered ( j) ( j) at x0 = (x1 , . . . , x N −1 , 0), such that 1 x ∈ Q ( j) : u(x) ≤ ≤ ν|Q ( j) |. 2 ( j). Without loss of generality we may assume x0 = (x1 , 0, 0, . . . , 0). Define Q 0 the cube centered in the origin, so Q ( j) = x0 + Q 0 ..

(87) 436. S. Piro-Vernier et al.. In the next lemma, we apply the logarithmic estimates to expand the positivity of Q ( j) till the origin. Apply Lemma 2.5 to Q ( j) to get, for a.e. x ∈ 21 Q ( j) 1 . 4 Lemma 4.1. (The second alternative) Let u be a bounded weak solution of (1) in Q 1 . Assume that conditions (2) are satisfied and that (14) does not hold for a Q ( j) . 2 Then for any positive constant ν0 ∈ (0, 1), there exists a positive integer s0 such that 1 1 (16) {x ∈ Q 0 : u(x) ≤ e−s0 ≤ ν0 | Q 0 |. 2 2 Proof. If we perform the change of variable y1 ( j) x1 − x1 = 2 yi xi = , ∀i = 2, . . . , N − 1 2 1 p−q y p N xN = 4 2 and calling v = 4u, you have that v satisfies an equation of the type (1) satisfying (2). Moreover v ≥ 1 in the unitary cube centered in the origin. As already noticed, the Lemma is proved if we are able to transport some positivity till to the point ( j) z = (−2x1 , 0, . . . , 0). Let us denote the solution v with u and the coordinates (y1 , . . . , y N ) with ( j) (x1 , . . . , x N ). Without loss of generality assume −2x1 > 0, denote with L = ( j) −2x1 and assume L > 1. Define S the slide of the half unitary cube centered in the origin in the x = (x2 , . . . , x N ) variables , i.e {x ∈ RN : x = (0, x2 , . . . , x N ) : − 41 < xi < 14 , ∀i = 2, . . . , N }. Let s0 > 1 be an integer, to be fixed later, and define a set A in S so defined: u(x) ≥. A = {(0, x ) ∈ S : ∃ t ∈ [0, 2L], u(t, x ) ≤ e−s0 } We want to prove that for any positive constant ν0 ∈ (0, 1), there exists a positive integer s0 such that |A| ≤ ν0 . (17) |S| Let x ∈ A. Then ∃ t ∈ [0, 2L] such that u(t, x ) ≤ e−s0 . Therefore, u(0, x ) + e−s0 u(t, x ) + e−s0 1 1 = ln+ − ln+ −s 0 u(t, x ) + e u(0, x ) + e−s0

(88) t

(89) 2L 1 1 = Dx1 (ln+ )ds ≤ Dx1 (ln+ ) ds u(s, x ) + e−s0 u(s, x ) + e−s0 . s0 − 1 ≤ ln+. 0. 0.

(90) Hölder regularity for bounded solutions. 437. Now if we integrate the inequality above over the set A, we obtain

(91)

(92) 2L 1 Dx (ln+ dx ) (s0 − 1)|A| ≤ 1 −s 0 u(x) + e S 0. Using Hölder Inequality, we have ⎡. ⎤ q1 q

(93)

(94) 2L q−1 1 Dx ln+ ( (s0 − 1)|A| ≤ ⎣ ) d x ⎦ [2L|S|] q . 1 −s 0 u(x) + e S 0. Here if we consider Lemma 2.1 choosing ξ = 1 in [0, 2L] × S and ξ = 0 out of [0, 3L] × 2S, and putting H ω = 1 and a = e−s0 we get ⎡ 3L

(95)

(96) N −1 Dx ξ q d x (s0 − 1)|A| ≤ C ⎣ i 2S 0.

(97)

(98) 3L +. i=1. ⎤ q1. q−1 Dx ξ p e−s0 ( p−q) d x ⎦ [2L|S|] q . N. 2S 0. (s0 − 1)|A| ≤ C L|S| Dividing by |S| we have L |A| ≤C |S| s0 − 1. and (17) follows choosing s0 large enough, and (17), in turn, implies (16).

(99) Thanks to the previous result, we have proved that, around the origin, the solution is far away from zero in most of the cube. However the geometry is not the intrinsic one, so we cannot apply directly the De Giorgi- type lemma. To overcome this difficulty we find two slides, one up and the other one down the origin in the x N variable, where the solution is far away from zero everywhere. Now we are in the condition to apply Lemma 2.6 and get the positivity everywhere inside the two slides. Let us prove the main result of this note. Proof of Theorem 1.1. By using the same notation of Lemma 4.1, choose ν0 = q. 1 ν , where ν 4 N 10. p−q. is the small constant defined in Lemma 2.5, where R N = R0p k0 p . Let P be the parallelepiped having the first N − 1 sides long 21 and the last one q. −s. p−q. ( 21 ) p e 0 p . N ∈ [ 1 , 1 ] such that in the set With this choice there exists a x0+ 8 4 R L := (L , 0, . . . , 0, x N0+ ) + P.

(100).

(101) 438. S. Piro-Vernier et al.. the measure where u is smaller than e−s0 is smaller than ν|P|. If such thing was not occurring, then, in the parallelepiped P = [L − 41 , L + 41 ] × [− 41 , 41 ] N −2 × [ 18 , 41 ], the measure where u is smaller than e−s0 would be larger than ν|P | and this ν . Therefore by Lemma 2.5. is in contradiction with (16) and the choice ν0 = 41N 10 1 1 N ) + P we have u ≥ e−s0 . in the set (L , 0, . . . , 0, x0+ 2 2 N ∈ [− 1 , − 1 ] such that in the set (L , 0, . . . , 0, x N )+ Analogously there exists x0− 0− 4 8 1 1 −s0 . 2 P we have u ≥ 2 e Therefore in the parallelepiped P1 = {x ∈ R N : L − 41 < x1 < L + 41 , − 41 < N < x N x2 < 41 , . . . , − 41 < x N −1 < 41 , x0− N < x 0+ } we have that on the faces N and x = x N the solution u is bigger or equal than 1 e−s0 . Moreover, x N = x0− N 0+ 2 by the choice of ν0 , analogously as above, one can prove that the measure where u < e−s0 is smaller than ν|P1 |. Therefore, the assumptions of Lemma 2.6 are fulfilled and we have either u ≥ 14 e−s0 in 21 P1 or the oscillation is quantitatively small. Hence, also in the case of the second alternative, we reduced the oscillation of the solution u in a quantitative way and this implies the Hölder continuity of the solution. Acknowledgements The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). We want also to thank the anonymous referee for the careful reading of the paper and for the valuable suggestions.. References [1] Boccardo, L., Marcellini, P., Sbordone, C.: L ∞ -regularity for variational problems with sharp nonstandard growth conditions. Boll. Un. Mat. Ital. 4–A, 219–226 (1990) [2] Cupini, G., Marcellini, P., Mascolo, E.: Regularity under sharp anisotropic general growth conditions. Discrete Contin. Dyn. Syst. Ser. B 11, 66–86 (2009) [3] Cupini, G., Marcellini, P., Mascolo, E.: Local boundedness of solutions to quasilinear elliptic systems. Manuscr. Math. 137, 287–315 (2012) [4] De Giorgi, E.: Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3, 25–43 (1957) [5] DiBenedetto, E.: Degenerate Parabolic Equations. Springer, New York (1993) [6] DiBenedetto, E.: Partial Differential Equations, Second Edition (Cornerstones). Birkhuser Boston Inc., Boston (2010) [7] DiBenedetto, E., Gianazza, U., Vespri, V.: Harnack estimates for quasi-linear degenerate parabolic differential equations. Acta Math. 200, 181–209 (2008) [8] DiBenedetto, E., Gianazza, U., Vespri, V.: Harnack’s Inequality for Degenerate and Singular Parabolic Equations (Springer Monographs in Mathematics). Springer, New York (2012) [9] DiBenedetto, E., Gianazza, U., Vespri, V.: Remarks on local boundedness and local Hölder continuity of local weak solutions to anisotropic p-Laplacian type equations. J. Elliptic Parabol. Equ. 2(1–2), 157–169 (2016) [10] Düzgün, F.G., Gianazza, U., Vespri, V.: 1-dimensional Harnack estimates. Discrete Contin. Dyn. Syst. Ser. S 9, 675–685 (2016).

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