Nonlinear gradient estimates for elliptic double obstacle problems with measure data

We study quasilinear elliptic double obstacle problems with a variable exponent growth when the right-hand side is a measure. A global Calder\'{o}n-Zygmund estimate for the gradient of an approximable solution is obtained in terms of the associated double obstacles and a given measure, identifying minimal requirements for the regularity estimate.


Introduction and main results
We consider a double obstacle problem involving the following elliptic equation with measure data: where µ is a signed Radon measure on Ω with finite total mass |µ|(Ω) < ∞.
Under possibly optimal conditions on p(·), a and Ω, we show that such a solution u satisfies the following gradient bound: |Ψ i (y)| dy (1.4) for x ∈ R n , where Ψ i (x) := div a(Dψ i , x) for i = 1, 2. Here we have assumed that µ and Ψ i are defined in R n by considering the zero extension to R n \ Ω (see [2,29] for a further discussion on the fractional maximal function).
. This property is necessary to prove the existence of a limit of approximating solution (see Section 3 below). There have been many existence results for obstacle problems with measure data. For instance, see [22,23] for linear problems, [9,10] for semilinear problems, and [5,31,32,37,38,41] for p-Laplace type problems. We also refer to [41,Section 1.1] for further discussions in the literature. On the other hand, the uniqueness of a limit of approximating solution remains open except for the linear problems, even in cases without obstacles (see [21]).

(iii)
If Ω is (δ, R)-Reifenberg flat, then we have the following measure density properties: We refer to [17,30,43] for further discussions on Reifenberg flat domains.
For simplicity of notation, we employ the word "data" to denote any structure constant that depends only on n, Λ 1 , Λ 2 , p − and p + . Moreover we assume that g ∈ W 1,p(·) 0 (Ω) satisfies in Ω and div a(Dg, ·) ∈ L 1 (Ω). Now, we are ready to state our main result.
In [40,41], Scheven proved some potential estimates for measure data problems with a one-sided obstacle and a constant p-growth. On the other hand, we are dealing with two-sided obstacles and a variable exponent p(·)-growth, which poses many difficulties (see for example Remark 1.7 above) in obtaining the desired estimate (1.16). To handle these difficulties, we revisit the maximal function approach in [1,19,34] alongside with L 1 -comparison estimates (see Section 4 below), a standard energy L 1 -estimate (see Section 5.2 below) and a Vitali type covering lemma (see Lemma 2.2 and Section 5.1 below). In addition, we obtain (1.16) on a nonsmooth domain beyond the Lipschitz category.
We organize this paper as follow. In Section 2, we introduce standard notation, some function spaces and auxiliary results. Section 3 provides the existence of a limit of approximating solutions of the obstacle problem OP (ψ 1 , ψ 2 ; µ), previously introduced in Definition 1.1. In Section 4, we deduce comparison estimates between our obstacle problem and its reference problems. In Section 5, we develop a standard energy L 1 -estimate, and then we finally complete the proof of Theorem 1.5 by using a Vitali type covering lemma.

Preliminaries
We start with notation which is used throughout the paper. Let us denote by  We denote by M b (Ω) a collection of signed Radon measures on Ω with |µ|(Ω) < ∞ and define the truncation operator with k > 0 (Ω) for all k > 0.
Next, we briefly review Lebesgue and Sobolev spaces with variable exponents. Let p(·) : R n → (1, ∞) be a continuous function with (1.5). We define the variable exponent Lebesgue space L p(·) (Ω) as the collection of all measurable functions f on Ω with Then there is the following relation between the Luxemburg norm and the integral version: (Ω) whose first order derivatives belong to L p(·) (Ω), and the norm is defined by (Ω), and also the dual Note that they are all separable reflexive Banach spaces.
We now give a crucial condition on variable exponents p(·) to have some important properties for L p(·) (Ω) and W 1,p(·) (Ω). We say that p(·) is log-Hölder continuous in Ω if there is a constant c l > 0 such that 2 . This is equivalent to the existence of a nondecreasing concave function ω : for all x, y ∈ Ω, and sup 0<r≤ 1 2 ω(r) log 1 r ≤c l for some constantc l > 0. If p(·) is log-Hölder continuous, then the variable exponent function spaces introduced above have important properties such as the Sobolev embedding theorem and Poincaré's inequality. For further details on the variable exponent function spaces, we refer to the monographs [20,24]. We next introduce the Hardy-Littlewood maximal function as an important tool for the proof of our main result (see Section 5 below), defined by If f is defined on a bounded open set D ⊂ R n , then we write where χ D is the characteristic function over D. For the sake of simplicity, we drop the index D when D = Ω. We shall use the following well-known estimates: The following standard measure theoretical property will be used in Section 5.3 later: Lemma 7.3]). Let f be a measurable function on a bounded open set Ω ⊂ R n . Let λ > 0 and m > 1. Then, for any 0 < q < ∞, for some c = c(m, q) > 0.
We end this section with a Vitali type covering lemma as follows: For measurable sets C and D with C ⊂ D ⊂ Ω and for 0 < R 0 ≤ R, the two followings hold:

Existence for a limit of approximating solutions
In this section we derive the existence of a limit of approximating solutions (introduced in Definition 1.1) of the obstacle problem OP (ψ 1 , ψ 2 ; µ). We first recall the truncation function T k in (2.1) and introduce another truncation function we consider the weak solution u i ∈ A g ψ1,ψ2 to the variational inequality

. in Ω and
We shall show that u i → u in the sense of (1.12) for some u ∈ T 1,p(·) g (Ω) with ψ 1 ≤ u ≤ ψ 2 a.e. in Ω. To see this, we first introduce the following technical lemma (see [ Then there exists a constant c = c(c o ) > 0 such that (Ω) ≤ c for every continuous function r(·) on Ω satisfying Now we derive some uniform bounds for the variational inequality (3.3) in the variable exponent setting. For the constant exponent case p(·) ≡ p, we refer to [41,Lemma 3.3] (see also [4,6]).
be the weak solution to the variational inequality (3.3) with (3.2). Then there exists a constant c 1 depending only on data such that for any k > 0. Moreover, we have (Ω) ≤ c 2 for some c 2 = c 2 (data, K, Dg L p(·) (Ω) ) > 0 and for any continuous function r(·) on Ω with (3.4).
Next, by taking Proceeding analogously to the proof of (3.5), we deducê for all k > 0, where c is a positive constant depending only on data. Applying Lemma 3.1, we obtain (3.6) as desired.
Proof. The idea of the proof follows from [41,Lemma 3.4]. For the sake of completeness, we give the proof. In view of (3.5) and (2.2), for any fixed k > 0, the sequence {T k (u i − g)} i∈N is uniformly bounded in W 1,p(·) 0 (Ω). Since W 1,p(·) 0 (Ω) ֒→ L p(·) (Ω) is continuous compact embedding, we can assume that (3.10) {T k (u i − g)} i∈N is a Cauchy sequence in L p(·) (Ω) for all k > 0. Next, according to (3.6), there are a subsequence of {u i } i∈N , still denoted by {u i } i∈N , and a function u ∈ g + W 1,r(·) 0 (Ω) such that (3.11) u i → u a.e. in Ω, This implies that ψ 1 ≤ u ≤ ψ 2 a.e. in Ω, and that T k (u i − g) → T k (u − g) a.e. in Ω for all k > 0. Then we deduce for all k > 0. Again using (3.5), we can also assume that (Ω) as i → ∞, and so in particular u ∈ T 1,p(·) g (Ω). Now we choose any continuous function r(·) on Ω with (3.4) and write for fixed i, j ∈ N, Taking a continuous functionr(·) such that r(·) <r(·) < min n(p(·)−1) n−1 , p(·) , we discover (3.14) for any ε > 0. Here the two integrals on the right-hand side above are bounded independently from i, j ∈ N by (3.6). Also, (3.11) implies that {u i } is a Cauchy sequence in L 1 (Ω). Therefore This and (3.14) yield Next, for the estimate II where Ω + ij := {x ∈ Ω : |u i − u j | ≤ k and p(x) ≥ 2} and Ω − ij := {x ∈ Ω : |u i − u j | ≤ k and p(x) < 2} . Now we use a comparison function ϕ = u i + T k (u j − u i ) in the variational inequality (3.3) for u i , while we choose ϕ = u j + T k (u i − u j ) in the version for u j . Then we infer that for any k > 0 It follows from (1.7) and (3.2) that Keeping in mind r(·) < p(·), the first integral on the right-hand side in (3.16) yields from Young's inequality and (3.17) that for any ε > 0 For the second integral on the right-hand side in (3.16), we employ (3.18) and Young's inequality to havê for any ε > 0. In the last inequality we used that r(x)(2−p(x))

2−r(x)
< r(x) for x ∈ Ω − ij . Then it follows from (3.6) that which implies that {Du i } i∈N is a Cauchy sequence in L r(·) (Ω). In view of (3.12), we infer Du i → Du in L r(·) (Ω), which completes the proof.
In this section, we establish comparison estimates in L 1 -sense between the weak solution u of (4.1) and the weak solutions of some reference problems. Throughout this section we assume that p − ≤ n and (p(·), a, Ω) is (δ, R)-vanishing. Moreover, we give the two obstacles ψ 1 and ψ 2 with (1.2). We only focus on the comparisons near boundary regions of Ω, since the interior case can be derived with the same spirit. Let 0 < r ≤ R0 8 , where R 0 ∈ (0, 1) is determined later. We assume the following geometric setting: In addition, we denote that for a measurable set D ⊂ R n , We now introduce the reference problems. Defining the admissible set (Ω 8r ) : ϕ ≥ ψ 1 a.e. in Ω 8r , we consider the weak solution z ∈ A ψ1 (Ω 8r ) to the variational inequality for all ϕ ∈ A ψ1 (Ω 8r ), where u is the weak solution of the variational inequality (4.1). We next take into account the following equations, sequentially, where z ∈ A ψ1 (Ω 8r ) is the weak solution of the variational inequality (4.5), and where h ∈ W 1,p(·) (Ω 8r ) is the weak solution of the equation (4.6). We first derive the comparison estimates between (4.1) and (4.5) as follows: Let u ∈ A ψ1,ψ2 and z ∈ A ψ1 (Ω 8r ) be the weak solutions to the variational inequalities (4.1) and (4.5), respectively. Then there exists a constant c = c(data) > 0 such that Proof.
Like similar arguments in the proof of Lemma 4.1, we can obtain the following comparison estimates alongside (4.5), (4.6), and (4.7).
From Lemma 4.1, 4.4 and 4.5, we directly compute Adopting this M 1 and applying Lemma 4.6, we obtain the higher integrability result for the problem (4.7) as follows: with M as in (4.4) and M 1 as in (4.16). Let w be the weak solution of (4.7). Then w belongs to W 1,p2 (Ω 3r ) with the estimate for some c = c(data) > 0.
Next, defining a new vector field B = B(ξ, x) : R n × Ω 8r → R n by we directly check the following growth and ellipticity conditions: for all η ∈ R n , ξ ∈ R n \ {0} and x ∈ Ω 8r , and for Λ 1 and Λ 2 as in (1.6), whenever We now consider the homogeneous frozen problem where w is the weak solution of (4.7). Indeed, w ∈ W 1,p2 (Ω 3r ) from Lemma 4.7, and from the standard energy estimate we directly obtain v ∈ W 1,p2 (Ω 3r ). We need to study the comparison estimate between (4.7) and (4.18). for some c = c(data) > 0.
Then this problem has the following Lipschitz regularity, and we also find a proper comparison estimate between (4.18) and (4.19).
with M as in (4.4), M 1 as in (4.16) and τ 0 as in Lemma 4.6. Then for any 0 < ε < 1, there exists a small constant 0 < δ < 1, depending only on data and ε, such that if (p(·), a, Ω) is (δ, R)-vanishing and Ω 8r forces (4.2), and if u ∈ A ψ1,ψ2 is the weak solution of (4.1) with  In this section, we prove our main theorem (Theorem 1.5). We first construct sequences {u i } and {µ i } satisfying (1.10)-(1.12). For any µ ∈ M b (Ω) we may regard that µ is defined on R n by the zero extension to R n \ Ω. Taking φ ∈ C ∞ 0 (B 1 ) as the standard mollifier, we define φ i (x) := i n φ(ix) for i ∈ N and x ∈ R n . We now consider Then µ i ∈ C ∞ c (R n ), in particular, µ i ∈ W −1,p ′ (·) (Ω) ∩ L 1 (Ω) satisfying (1.10) and (5.1) Having such a function µ i , we construct the corresponding weak solution u i ∈ A ψ1,ψ2 to the variational inequality (4.1) with µ replaced by µ i such that for all continuous functions r(·) on Ω with (3.4), as in Lemma 3.3.
Throughout this section, we suppose that (p(·), a, Ω) is (δ, R)-vanishing. Moreover, we assume that R 0 > 0 satisfies where M , M 1 and τ 0 are given in (4.4), (4.16) and Lemma 4.6, respectively. According to (5.1) and (5.2), we see that R 0 satisfies (4.22) and (4.23) with u and µ replaced by u i and µ i , respectively, for sufficiently large i. Our strategy for proving Theorem 1.5 is to apply a Vitali type covering lemma (Lemma 2.2), under the L 1 -comparison estimates (Lemma 4.11) in Section 4 and Lemma 5.1 below, to obtain the power decay estimate for upper level sets of Du (see (5.13) below). Combining this decay estimate, Lemma 2.1 and the standard energy L 1 -estimate (Section 5.2), we finally derive the desired regularity estimate (1.16) in Theorem 1.5.