Gradient estimate of a variable power for nonlinear elliptic equations with Orlicz growth

Abstract: In this paper,weprove a global Calderón-Zygmund type estimate in the frameworkof Lorentz spaces for a variable power of the gradients to the zero-Dirichlet problemof general nonlinear elliptic equationswith the nonlinearities satisfying Orlicz growth. It is mainly assumed that the variable exponents p(x) satisfy the log-Hölder continuity, while the nonlinearity and underlying domain (A, Ω) is (δ, R0)-vanishing in x ∈ Ω.


Introduction
Throughout this paper, let Ω ⊂ R n for n ≥ be a given bounded domain with its rough boundary speci ed later. Given a vectorial valued function f = (f , f , · · · , f n ) : Ω → R n . The aim of this present article is to study a global Calderón-Zygmund type estimate in the framework of Lorentz spaces for a variable power of the gradients of weak solutions to the zero-Dirichlet problem of general nonlinear elliptic equations div A(x, Du) =div G(x, f ) in Ω, u = on ∂Ω, (1.1) where the nonlinearity A = A(x, ξ ) : R n × R n → R n is measurable for each ξ ∈ R n and di erentiable for almost every x ∈ R n , and there exist constants < ν ≤ ≤ Λ < ∞ such that for all x, ξ , η ∈ R n , |A(x, ξ )| + |ξ ||D ξ A(x, ξ )| ≤ Λφ(|ξ |). Moreover, the nonhomogeneous term G(x, ξ ) : R n × R n → R n satis es |G(x, ξ )| ≤ Λφ(|ξ |) for any x, ξ ∈ R n . (1.4) We also de ne Φ = Φ(s) =ˆs φ(r) dr for s ≥ . (1.5) The weak solution of (1.1) is understood in the following usual sense, if for u ∈ W ,Φ (Ω) it holdŝ where W ,Φ is a Orlicz-Sobolev space de ned in the following. For more details about Orlicz spaces, we refer to [14,15,28,29]. The equation of generalized p-Laplacian type is arising in the elds of uid dynamics, magnetism and mechanics. Particularly, if φ(s) = s p− for < p < ∞, our problem becomes the p-Laplace equation. For this case, there is a great deal of literature concerning the regularity of weak solutions, for instance, see [4,9,11,13,24]. While φ(s) = s p− + as q− , where < p < q and a is a positive constant, Filippis and Mingione [18] obtained a global estimate in the setting of Lebesgue spaces, which is covered by our problem (1.1). We also refer to [8] for the existence of solutions to the (p, q)-Laplace equations.
In recent years, there have been signi cant advances of this type of problem in the regularity theory, see for example [12,14,34,35] since starting with a seminal paper of Lieberman [23]. In 2011, Verde [34] (1.3). He proved the global Calderón-Zygmund estimates over whole domain R n for the weak solutions. Very recently, Yao and Zhou [35] obtained local L qestimates for weak solutions of (1.1), which implies the fact that Also, Byun and Cho [12] further extended it to a global gradient estimate in the setting of Orlicz spaces under the assumption that the boundary of underlying domain is Reifenberg at. We would like to point out that Cho [14] also considered the zero-Dirichlet problem of (1.1) for A de ned as in (1.2) and G(x, ξ ) = φ(|ξ |) |ξ | ξ , who obtained a global Calderón-Zygmund estimate in the setting of Orlicz spaces. We also refer to [6,7,17,36] for a further study of these problems with Orlicz growth. In particular, Baroni and Lindfors [7], and Yao [36] studied parabolic problems with Orlicz growth.
In this paper we are to prove a global Calderón-Zygmund type estimate in the Lorentz spaces for general nonlinear elliptic equations (1.1). As we know, there are mainly three kinds of di erent arguments to handle the Calderón-Zygmund theory for elliptic and parabolic problems with VMO or small BMO discontinuous coe cients except for a classical technique by using singular integral operators and their commutators. The rst one is the so-called geometric method originally traced from Byun and Wang's work in [11], which is based on the weak compactness, the Hardy-Littlewood maximal operators and the modi ed Vitali covering. Here, the so-called modi ed Vitali covering actually refers to the argument as "crawling of ink spots" as in the early papers by Safonov and Krylov [20,30]. Indeed, this is also a development from Ca arelli and Peral's paper in [13]. Secondly, Kim and Krylov [19,21] gave a uni ed approach of studying L p solvability for elliptic and parabolic problems due to the Fe erman-Stein theorem, which is mainly based on the sharp functions. In this present paper, we have to highlight the third technique being called large-M-inequality principle originating from Acerbi and Mingione's work [1,2], which is directly based on arguing on certain Calderón-Zygmund-type covering instead of the boundedness of a maximal function operator and the so-called good-λ-inequality used by Byun and Wang [11] and Kim and Krylov's papers [19,21].
Here we are revising the so-called large-M-inequality principle and geometric method to get an estimate in the Lorentz spaces for the variable power of the gradients to (1.1). Regarding what we consider, we would like to point out that Byun, Ok and Wang [10] studied the zero-Dirichlet problem of linear elliptic system in divergence form under the assumptions that their coe cients are partially BMO and the variable exponent p(x) is log-Hölder continuous. They rst obtained a global Calderón-Zygmund estimate in the variable exponent Lebesgue spaces: Also, Tian and Zheng in [32] further extended the above result to the global Calderón-Zygmund type estimate in the Lorentz spaces for a variable power of the gradients of weak solutions. Note that the Lorentz space is a two-parameter scale of the Lebesgue space obtained by re ning it in the fashion of a second index, and there are a lot of research activities on Lorentz regularity for partial di erential equations. For examples, the rst estimates in the Lorentz spaces are obtained by Mingione [27]. Later, Mengesha and Phuc [24] derived the weighted Lorentz estimate to quasilinear p-Laplacian type equations based on the geometric approach. In 2014, Zhang and Zhou [39] extended the above result in [24] to that for a general equation of p(x)-Laplacian also using a geometrical argument. Adimurthil and Phuc [3] proved the global Lorentz and Lorentz-Morrey estimates for quasilinear equations below the natural exponent. Meanwhile, Baroni [4,5] obtained interior Lorentz estimates to evolutionary p-Laplacian systems and obstacle parabolic p-Laplacian with the given obstacle function Dψ ∈ L γ,q locally in Ω T , respectively, which means that for γ > p and q ∈ ( , ∞], where he just used the so-called large-M-inequality principle. Very recently, Tian and Zheng [31] showed a global weighted Lorentz estimate to linear elliptic equations with lower order items under assumptions of partially BMO coe cients in Reifenberg at domain. Zhang and Zheng [37,38] also proved Hessian Lorentz estimates for fully nonlinear parabolic and elliptic equations with small BMO nonlinearities, and weighted Hessian Lorentz estimates of strong solutions for nondivergence linear elliptic equations with partially BMO coe cients, respectively. To this end, we introduce some related notation and basic facts being useful in the article. The Lorentz space L t,q (U) for open subset U ⊂ R n with parameters ≤ t < ∞ and < q < ∞, is the set of all measurable functions g : U → R requiring while the Lorentz space L t,∞ for ≤ t < ∞ and q = ∞ is de ned by the Marcinkiewicz space M t (U) as usual, which is the set of all measurable functions g with The local variant of such spaces is de ned as usual way. If t = q, then the Lorentz space L t,t (U) is nothing but a classical Lebesgue space. Indeed, by Fubini's theorem it yields g t L t (U) = tˆ∞ µ t |{ξ ∈ U : |g(ξ )| > µ}| which implies L t (U) = L t,t (U), see also [4,5,24,37]. Let us denote Br(y) = x ∈ R n : |x − y| < r for y ∈ Ω and radius r > , and with brie y Br = Br( ), Ωr = Ωr( ) and Tr = Tr( ). For a bounded open set U ⊂ R n , we write the integral average of g(x) over U of a locally integrable function g in R n by As usual, it is a necessary assumption that the variable exponent p(·) is log-Hölder continuous, which ensures that the molli cation, the singular integrals and the Hardy-Littlewood maximal operator are all bounded within the framework of generalized Lebesgue space. For this, we are recalling the de nition that p(x) is log-Hölder continuous denoted it by p(x) ∈ LH(Ω), if there exist positive constants c and δ such that In the following context, we assume that p(x) : Ω → R is a log-Hölder continuous function and there exist positive constants γ and γ such that Without loss of generality, let where ω : [ , ∞) → [ , ∞) is a modulus of continuity of p(x) such that ω is a nondecreasing continuous function with ω( ) = and lim sup r→ ω(r) log r < ∞. With the above assumptions in hand, it is clear that To obtain a global Calderón-Zygmund type estimate for general nonlinear elliptic problem (1.1), it is also necessary to impose some regular assumptions on the nonlinearity A = A(x, ξ ) and the rough boundary of underlying domain Ω. Let us set ensuring some natural properties in geometric analysis to hold, such as Sobolev embedding theorem and Sobolev extension theorem, see [11,22,25,33].
Finally, we state the main result of this article.
This article is devoted to a global Calderón-Zygmund type estimate in the framework of Lorentz spaces for a variable power of the gradients of weak solutions to the zero-Dirichlet problem of general nonlinear elliptic equations (1.1) over Reifenberg domains. As already mentioned, our problem and proof are inspired by work of Acerbi and Mingione [1,2] and Baroni [4,5], and recent work from Cho [14]. The key ingredient is to make use of the so-called large-M-inequality principle, Calderón-Zygmund-type covering, approximate estimate and an iteration argument to attain the variable Lorentz estimate (1.12). The rest of the paper is organized as follows. In Section 2, we introduce notation and some useful lemmas. In Section 3, we focus on proving the main theorem.

Technical tools
In this section, we introduce some useful lemmas. From now on, we denote c to mean a universal constant that can be computed in terms of given data such as n, ν, Λ, γ , γ , t, q, σφ , τφ , R , ω(·) and |Ω|. First of all, we recall the existence and energy estimate of weak solution to general nonlinear elliptic problem (1.1).
Proof. The existence and uniqueness of weak solution to (1.1) has been proved in [14]. Next, we take a test function ϕ = u ∈ W ,Φ (Ω) and use (1.2) where ξ ∈ R n and t ≥ . It leads to the following conclusions: (II)Ã andG satisfy assumptions (1.2) and (1.4), respectively, with the same constants ν and L.
Proof. By a similar procedure to Theorem 9 in [16] and using the measure density property (1 .11) for Ω, and a zero-extension of u in B r , the conclusion is clearly true.
Note that Φ(s) is an N-function from the de nition of Φ(s) in (1.5). Besides, we deduce that min α σφ+ , α τφ+ Φ(s) ≤ Φ(αs) ≤ max α σφ+ , α τφ+ Φ(s), for any s, α ≥ , and We de ne an auxiliary vector eld V : R n → R n given by Then there is the following relation between V and Φ, see [14]: for all ξ , η ∈ R n and m > depending only on n, ν, Λ, σφ and τφ. Now, we give the comparison estimates with a limiting problem, and show its Lipschitz regularity by following from Proposition 5.5 and 5.11 in [14]. We state the required comparison estimates in the following lemmas.
For the boundary case, let Ω satisfy that where c = c (n, ν, Λ, σφ , τφ) > andv is zero extension of v from B + to Ω .
Let us now collect some preliminary results concerning the so-called embedding relations involved in the Lorentz spaces, which will be used in the sequel.
Proposition 2.7. Let U be a bounded measurable subset of R n , then the following relations hold: (I) If < q ≤ ∞, and ≤ t < t < ∞, then L t ,q (U) ⊂ L t ,q (U) with the estimate The following two lemmas will play important roles in our main proof, which are the variants of classical Hardy's inequality and a reverse Hölder inequality, respectively, see Lemma 3.4 and 3.5 in [4]. where the constant c depends only on α , α and r; except in the case α = ∞ with c ≡ c(α , r).

Proof of Theorem 1.4
This section is mainly devoted to proving Theorem 1.4. Our proof consists of 6 steps. In step 1, for given λ in (3.7), we show the Calderón-Zygmund type covering on the super-level set E λ, Ωr (x ) , and establish a decay estimate of Ωr y (y). In step 2, we give various comparison estimates to the reference problems. In step 3, we employ the so-called "crawling of ink spots" approach to show an estimate for the super-level set. In steps 4, we get the estimate of Φ(|Du|) p(x) L t,q (Ωr (x )) for q < ∞. In steps 5 and 6, we deduce our conclusions in the cases of q < ∞ and q = ∞, respectively, under a priori assumption Φ(|Du|) p(x) L t,q (Ωr (x )) < ∞ which is proved in step 5. Proof. Let σ be the same as in Lemma 2.4, and let σ = min{σ , γ − } > due to γ > . For a xed point y ∈ Ω, we take ry < R with where c * = c * (n, γ , γ , ν, Λ, σφ , τφ , ω(·), |Ω|) ≥ |Ω| + determined later.
Here, we only consider the boundary case, saying B ry (y) ⊄ Ω. For this, we can nd a boundary point y ∈ B ry (y) ∩ ∂Ω and for ry ∈ ( , R ), there exists a coordinate system depending only onȳ and ry, such that in this new coordinate system it holds z = y,ȳ + δry( , · · · , , ) is the origin, We select < δ < such that Ω ry (z) ⊂ Ω ry ( ), which implies the fact that Then we also have For a xed point x ∈ Ω, we set For the weak solution u of original problem, by a scaling argument to the nonhomogeneous terms f and the N-functions φ, Φ, then we writē Hereafter, for the sake of simplicity, we still use u, f , φ and Φ replacingū,f ,φ andΦ in the following.
Step 1. In this step, we give a Calderón-Zygmund type covering on the super-level set E λ, Ωr (x ) as below. Let u be the weak solution of (1.1), we de ne the quantity for any r and r with R ≤ r ≤ r ≤ R λ := where δ > and η > will be speci ed later. We treat the super-level set For y ∈ E λ, Ωr (x ) and radii < r ≤ r − r , we let CZ(Ωr(y)) := (3.8) If r −r ≤ r ≤ r − r , then we discover that which means that while r −r ≤ r ≤ r − r one has CZ(Ωr(y)) < λ. At the same time, by Lebesgue's di erentiation theorem we nd that CZ(Ωr(y)) > λ for < r . Therefore, by absolute continuity of the integral with respect to the domain we can pick the maximal radius ry such that CZ(Ωr y (y)) = Ωr y (y) for each point y ∈ E λ, Ωr (x ) . Moreover, one has CZ(Ωr(y)) < λ, for any r ∈ (ry , r − r ]. (3.10) From (3.9), we conclude the following alternatives: First, we suppose that the rst case of (3.11) is valid to have where σ > is determined later.

which implies
Ωr y (y) ≤ c Ωr y (y) ∩ E( λ , Ωr (x )) (3.14) with the positive constant c depending only on n, ν, Λ, γ , γ , σφ , τφ , R , K , and |Ω|. If the case of the second estimate in (3.11) is valid, by taking ζ = δ and Fubini's theorem, we get Let δ = ζ , we derive that Now we put (3.14) and (3.15) together to get that Step 2. This step is devoted to various comparison estimates with the reference problems and the limiting one. Note that Ω is (δ, R )-Reifenberg at, then it follows from (3.3) and (3.10) that where we still use variable x for simplicity. Therefore, it su ces to show that for a constant c ≥ . We rst claim that with c ≥ a universal constant. In fact, since Φ(|f |) p(x) ∈ L t,q (Ω) for t > and < q ≤ ∞, it deduces that where we have used (3.6 On the other hand, we use (3.21) and where we have used the so-called log-Hölder continuity (1.8) for p(x) in the last inequality, which yields (3.20).
Recalling γ ≤ p + y and (3.20) with λ > , we obtain Note that Φ(|f |) p(x) ∈ L t,q (Ω) for < γ ≤ p(x) ≤ γ < ∞, and < η ≤ + σ with σ as the same of Lemma Similarly, recalling δλ ≥ and λ ≥ Mλ we nd where ϵ > is small and c = max{c , c } ≥ . For the case of interior, by Lemma 2.5 similarly we have Step 3. We are here to estimate the super level set E λ, Ωr (x ) . For any xed point x ∈ Ω, we select a universal constant R with < R ≤ min R , R |Ω|+ , , and there exists a constant δ = δ(ϵ) > such that Lemma 2.5 and 2.6 hold. Let A = ( τφ+ c (m + )) γ γ , for any x ∈ E(Aλ, Ωr (x )) we consider the collection B λ of all subset Ωr y (y). By "crawling of ink spots" argument, we extract a countable subcollection {Ωr i (y i )} ∈ B λ , such that For the boundary case, we recall (2.2), (2.3) and (2.4) to nd that which implies that By a simple change of variable and (3.26) it leads to that We are now to estimate I . For this we part it in two cases.
Therefore, we deduce that Similarly, we also show that ˆΩ |Φ(|f |) For the case of q < ∞, from (3.33) we then infer the following relations L t,q (Ω) ≤ c, where c depends only on n, ν, Λ, γ , γ , t, q, σφ , τφ , R , K , ω(·), and |Ω|. And recalling the de nition in (3.5), we get the desired result (1.12) Step 6. Finally, for the case of q = ∞, we come back to the second inequality in (3.11) and split it into two parts with a small ι > determined later: If we take ϵ > so small that it ensures cϵ t ≤ , it follows that In the remainder we use a similar way of the argument in Step 5, and it leads to the desired result for the case q = ∞.