Gradient estimates for the strong $p(x)$-Laplace equation

We study nonlinear elliptic equations of strong $p(x)$-Laplacian type to obtain an interior Calderon-Zygmund type estimates by finding a correct regularity assumption on the variable exponent $p(x)$. Our proof is based on the maximal function technique and the appropriate localization method.


Introduction.
Let Ω be a bounded domain in R N .In this paper we study the gradient estimates of weak solutions for the following strong p(x)-Laplacian type equation: in Ω, (1) where |f | p(x) belongs to L q(x) loc (Ω) with q(x) > 1.The strong p(•)-Laplace operator was first introduced by Adamowicz and Hästö in [2,3] In order to state our problem in a weak form, we note that ∆p(•) u = div(|∇u| p(x)−2 ∇u) − |∇u| p(x)−2 log(|∇u|)∇u • ∇p (2) when u ∈ C 2 (Ω), which is scaling invariant.The corresponding weak formulation of problem (1) then requires that u ∈ W 1,p(x) loc (Ω) satisfies for all ϕ ∈ W 1,p(x) 0 (Ω).Obviously, this equation reduces to the ordinary p-Laplace equation when p is constant.Another point worth mentioning is that we must require that ∇p has some integrability.To do this, we assume that p(x) is Lipschitz continuous, which ensures that the second term on the left-hand side of (3) makes sense.On the other hand, it is crucial in the proof of Lemma 3.3.
The strong p(•)-Laplace equation is different from the ususl p(•)-Laplace equation, because the term with the logarithm in (1) has supercritical growth: it is of order t p(•)−1 log t while the main term has only order t p(•)−1 .At first sight the strong p(•)-Laplacian seems to have a distinct disadvantage over the p(•)-Laplacian, but Adamowicz and Hästö discovered in [2,3] that solutions of the following equation − div(|∇u| p(x)−2 ∇u) + |∇u| p(x)−2 log(|∇u|)∇u • ∇p = 0 (4) possess some advantages over solutions of the usual p(•)-Laplacian, such as the scalability and the fine geometric regularity [2], and the scale-invariant Harnack inequality [3].Zhang and Zhou [18] further obtained the Hölder regularity for the gradients of solutions of (4).However, owing to the incompleteness of the theory of weak solutions of (4), the full comparison principle, the equivalence of weak and viscosity solutions and the Calderón-Zygmund theory remain open.Throughout this paper, we assume that the variable exponent p(x) satisfies that which implies that the strong log-Hölder condition holds, i.e., for any Ω 0 ⊂⊂ Ω, there exists a nondecreasing continuous function w : Furthermore, we assume that q(x) satisfies and is log-Hölder continuous in Ω 0 : for some constant L > 0. We remark that q(•) is log-Hölder continuous in Ω 0 if and only if q(•) is modulus continuous in Ω 0 , i.e., there exists a nondecreasing continuous function ρ : R + → R + satisfying lim r→0 ρ(r) = 0, and a constant L q(•) > 0 such that As usual, solutions of Eq. ( 1) are understood in a weak sense.
Definition 1.1.We say that u ∈ W 1,p(x) loc (Ω) is a weak solution of Eq. ( 1) if In this paper, we are mainly interested in establishing an interior Calderón-Zygmund theory for the strong p(x)-Laplacian equation ( 1) with a datum of divergence type on the right-hand side under minimal assumptions on the variable exponent p(x), which can be seen as a further step to the results in [2,3,18].In fact, there have been many results in the literature.We here mention some nonlinear results for the usual p(x)-Laplacian equation.Especially, Acerbi and Mingione in [1] first obtained a local gradient estimates for the equation by proving that |f | p(•) ∈ L q loc (Ω) ⇒ |∇u| p(•) ∈ L q loc (Ω) for every 1 < q < ∞, assuming that the variable exponent p(•) satisfies strong log-Hölder condition (6).In [6,17] Byun, Zhang and their collaborators obtained a global Calderón-Zygmund estimate for p(x)-laplacian type equation provided that the nonlinearity satisfies a small BMO (bounded mean oscillation) condition with respect to x and Ω is sufficiently flat in the Reifenberg sense.Moreover, Byun and Ok in [5] proved the W 1,q(•) regularity theory for every variable exponent q(•) strictly bigger that p(•).Inspired by the above mentioned works, we will combine the maximal function method, the modified Vitali covering lemma and the appropriate localization technique to find a version of L q(•) -type Calderón-Zygmund estimates for Eq.(1).More precisely, we will show that for each q(•) > 1 provided q(•) satisfies the log-Hölder condition (8).Although the outline of our proof is similar to the previous works about the usual p(x)-Laplacian equation, there are several significant differences in the details.Perhaps the most important of them is the fact that the left-hand side of Eq. ( 1) is not a divergence form and the term with the logarithm in (3) has supercritical growth: it is of order t p(•)−1 log t while the main term has only order t p(•)−1 .We have to carefully exploit the information coming from the the logarithm term and fit together the estimates by combining all of these terms in a suitable way by virtue of logarithm type inequalities.In the case of constant p, this problem never occurs.The core of our argument is based on a proper reverse Hölder inequality and an appropriate approximation argument combining with a new decay estimates for the super-level sets {x : M(|∇u| The main result of this paper is stated as follows. Theorem 1.2.Assume that p(•) and q(•) satisfy the conditions (5), (7) and (8).Let u ∈ W 1,p(•) (Ω) be a local weak solution to (1) loc (Ω).This paper is organized as follows.In Section 2, we first recall some properties for generalized Lebesgue-Sobolev spaces and then state some preliminary tools and known results which will be used later.We will finish the proof of Theorem 1.2 in Section 3.

Preliminaries.
2.1.Generalized Lebesgue-Sobolev spaces.The variable exponent Lebesgue space L p(•) (Ω) consists of all measurable functions satisfying with the following Luxemburg norm and the variable exponent Sobolev space is equipped with the norm We also denote the Sobolev space with zero boundary values, W 1,p(•) 0 (Ω), as the completion of C ∞ 0 (Ω) with respect to the norm u W 1,p(•) (Ω) .Then, they are separable reflexive Banach spaces.We refer to [8,9,13] for more details.

Maximal function.
For a locally integrable function f defined on R N , we define its maximal function M(f )(x) as If f is not defined outside a bounded domain Ω, then we define for the standard characteristic function χ on Ω.
The basic properties for the Hardy-Littlewood maximal function are the followings.
Remark 1.In the same way, if s ≥ 1 we define . By means of Theorem 7.1 in [10], we have the following estimate 2.3.The spaces L log β L. The Orlicz space L p (Ω; R N ) can embed in this space, that is, for any p > 1, we have that where the constant C is depending only on p, and blows up when where Basically due to T. Iwaniec [4,10,11], we recall that there exists a constant C = C(β) ≥ 1 such that for every positive constants a and b, and for every β and every 0 < σ < 1.

Technical lemma.
In this paper, we use the following version of the Vitali covering lemma, which will be a crucial ingredient in obtaining our main result.

Lemma 2.2 ([7]
). Assume that C and D are measurable sets, C ⊂ D ⊂ B 1 , and that there exists an ε > 0 such that |C| < ε|B 1 | and that for all x ∈ B 1 and for all r ∈ (0, 1] with 3. Proof of the main result.Hereafter we set where σ is a fixed constant defined in Lemma 3.2 and select 0 < 2r ≤ R 0 ≤ min {1, 1/K} with B 4R0 ⊂⊂ Ω which will be specified later.We consider the following reference problems: where p 2 = sup x∈B2r p(x).

Lemma 3.1 ([12]
).There exists a constant m 1 > 1 such that for all 1 ≤ µ ≤ m 1 the following holds.If u ∈ W µp2 (B 2r ) and v is the weak solution of Eq. ( 17), then there exists a constant C depending on p 2 such that We start with the following reverse Hölder's estimate by applying the classical Gehring's lemma to the quantity |∇u| p(•) to derive higher integrability for small exponents.As usual, we denote B r (x) as a ball in R N , centered at the point x, with radius r > 0 and B r := B r (0) is a ball with center 0. The integral average of an integral function f ∈ L 1 (E) on a measurable subset E of R N is defined by (Ω) be a local weak solution of Eq. (1) and |f | p(•) ∈ L γ3 loc (Ω), where γ 3 is given by (7).Assume R 0 > 0 satisfies that Then there exists a positive constant σ 0 = σ 0 (N, K 0 , w(•), R 0 , γ 1 , γ 2 , γ 3 ) ≤ γ 3 − 1 such that the following holds: for any σ ∈ (0, σ 0 ] and any B R ⊂ B 4R0 , we have R , we derive the following equality Recalling the logarithm type inequality: for any γ ∈ (0, 1), there exists C = C(γ) such that for any t > 0, For the second term on the left-hand side in (20), the inequality (21) and Young's inequality yield that for any ε > 0, Therefore, following the proof of Theorem 5 in [1] we deduce the Caccioppoli type inequality 1−γ dx.
On the other hand, if p− s ≥ N , u is of class of Hölder continuous or BMO spaces.We can also obtain the above estimate.
Note that Then we have Using Gehring's lemma, see Theorem 4 in [1] and references therein, we get the conclusion.
We will use the following approximation lemma which plays an important role in proving our main result.Lemma 3.3.Let p(x) satisfy the assumption (5).For any ε > 0, there exists a small δ = δ(ε) > 0 such that if u ∈ W 1,p(x) (Ω) is a local weak solution of (1), with and where σ is a fixed constant defined in Lemma 3.2 and λ ≥ 1 is a positive constant.Then there exists a corresponding weak solution v ∈ W 1,p2 (B 2r ) of (17) such that where p 2 = sup x∈B2r p(x) and Proof.Using the definition of weak solutions, we have where Next we estimate I 1 -I 4 , one by one.
Estimate of I 1 .We divide it into two cases.Case 1. p 2 ≥ 2. Using the elementary inequality for every s, t ∈ R N , we have Using the elementary inequality for every s, t ∈ R N and for every θ ∈ (0, 1], we have that is, Selecting τ = θ 2 p 2 /C, we observe that Estimate of I 2 .From the mean-value theorem and Young's inequality with τ , we can deduce that Denoting p 1 = inf x∈B2r p(x) and noting that for all x ∈ B 2r , where w(4r) ≤ σ 8 and σ is defined in Lemma 3.2.And for all x ∈ B 2r , Then from Lemma 3.2 and (23) we obtain where we used the fact that r −N w(4r) stays bounded as 0 < 4r < 2R 0 , δ < 1, w(4r) ≤ σ/8 and the Hölder's inequality: p2−1 .We now estimate the term from the above estimates and inequalities ( 13)-( 16) about the properties of the space L log β L(Ω) as follows: Following the arguments as the Step 3 of Lemma 2 in [1] by some necessary changes, we can further deduce Thus we obtain that Estimate of I 3 .Hölder's and Young's inequalities imply that Here we used the fact that ∇p L ∞ (B2r) ≤ C, r ≤ δ and the Poincaré's type inequality Thus this term can be estimated as the ones of I 2 similarly.Estimate of I 4 .Note that Therefore, from Young's inequality with τ we observe Combining all the estimates I 1 − I 4 , by choosing the constant τ so small with 0 < τ δ, we get Thus we have by choosing δ satisfying the last identity above.This finishes the proof. and where M, M 1+σ are the Hardy-Littlewood maximal operators defined as before.
Lemma 3.4.There is a constant N 1 > 0 so that for any ε > 0, there exists a small Proof.From (28), there exists a point x 0 ∈ B r such that q(x) and q 2 := sup Then it is clear that for any x ∈ B 5r (x 0 ) ⊂ B 6r , Since B 4r ⊂ B 5r (x 0 ), we observe that Here, in the last inequality, we used by assuming r ≤ min{1, 1/K} and the fact that ρ(r) satisfies the condition (8).Similarly, Now applying Lemma 3.3 to problem (1), we obtain that for any η > 0 there exist a small δ = δ(η) > 0 and a weak solution v ∈ W 1,p2 (B 2r ) of ( 17) such that By Remark 2 and (27), we know that sup It follows that ≤ CK σ λ.
Using the above results we obtain that which can be derived as follows.By Hölder's inequality and the fact that p(x) p2 ≥ p1 p2 > 1 2 , for all x ∈ B 2r , we have .
The following lemma can be obtained by the same arguments as in [17].So we omit the proof.for every integer n ≥ 0. Now we can prove our main result.