An extension of a Theorem of V. \v{S}ver\'ak to variable exponent spaces

In 1993, V. \v{S}ver\'ak proved that if a sequence of uniformly bounded domains $\Omega_n\subset {\mathbb R}^2$ such that $\Omega_n\to \Omega$ in the sense of the Hausdorff complementary topology, verify that the number of connected components of its complements are bounded, then the solutions of the Dirichlet problem for the Laplacian with source $f\in L^2({\mathbb R}^2)$ converges to the solution of the limit domain with same source. In this paper, we extend \v{S}ver\'ak result to variable exponent spaces.


Introduction
One important problem in partial differential equations is the stability of solutions with respect to perturbations on the domain. This problem has fundamental applications in numerical computations of the solutions and is also fundamental in optimal shape design problems. See [1,12,20] and references therein.
The famous example of Cioranescu and Murat [6] shows that this problem presents severe difficulties when treated in full generality. In fact, in [6] the authors take D = [0, 1] × [0, 1] ⊂ R 2 and define the domains Ω n = D \ ∪ n−1 i,j=1 B rn (x n i,j ) where the centers of the balls x n i,j = (i/n, j/n), 1 ≤ i, j ≤ n − 1 and the radius r n = n −2 . Then these domains Ω n converge to the empty set in the Hausdorff complementary topology, but if u n ∈ H 1 0 (Ω n ) is the solution to −∆u n = f in Ω n , u n = 0 on ∂Ω n , then u n ⇀ u * weakly in H 1 0 (D) to the solution of −∆u * + 2 π u * = f in D, u * = 0 on ∂D.
This example can be generalized to other space dimensions, to different bounded sets D and also to different types of holes. See the original work [6] and also [24].
There are some simple cases where the continuity can be granted. For instance, if Ω is convex and {Ω n } n∈N is an increasing sequence of convex polygons such that Ω = ∪ n∈N Ω n , then the solutions of the approximating domains Ω n converges to the one of Ω. This fact can be traced back to the late 50's and the beginning of the 60's, see [2,13,14,15]. Then, this result can be generalized in terms of the capacity of the symmetric differences of Ω and Ω n . See the book of Henrot, [12].
In practical applications, when one does not have control on the sequence of approximating domains, this hypothesis is uncheckable, so a different condition is needed.Šverák in [23] gave such a condition. In fact, given a bounded domain D ⊂ R 2 and a sequence of domains Ω n ⊂ D such that Ω n → Ω in the sense of the Hausdorff complementary topology the condition that guaranty the convergence of the solutions in Ω n to the one in Ω is that the number of connected components of D \ Ω n be bounded. c.f. with the example of Cioranescu-Murat.
The reason whyŠverák's result holds in dimension 2 is because the capacity of curves in dimension 2 is positive, while in higher dimension curves have zero capacity.
Sverák's result was later generalized to nonlinear elliptic equations of p−Laplace type. In fact, in [4], the authors prove the continuity of the solutions of −∆ p u n = f in Ω n ⊂ R N , u n = 0 on ∂Ω n , when the domains Ω n converges to Ω in the Hausdorff complementary topology under the assumption that the number of connected components of its complements remains bounded. The idea of the proof is similar to the original one ofŠverák and so they end up with the restriction p > N − 1 that is needed for the curves to have positive p−capacity.
Recall that ∆ p u = div(|∇u| p−2 ∇u) is the so-called p−laplace operator. In recent years a lot of attention have been put in nonlinear elliptic equations with nonstandard growth. One of the most representative of such equations is the so-called p(x)−laplacian, that is defined as ∆ p(x) u = div(|∇u| p(x)−2 ∇u). This operator became very popular due to many new interesting applications, for instance in the mathematical modeling of electrorheological fluids (see [21]) and also in image processing (see [5]). Here, the exponent p(x) is assumed to be measurable and bounded away from 1 and infinity.
So, the purpose of this paper is the extension of the result ofŠverák (and also the results of [4]) to the variable exponent setting.
Organization of the paper. The rest of the paper is organized as follows. In section 2 we collect some preliminaries on variable exponent spaces that are needed in this paper. The standard reference for this is the book [7]. Some results are slight variations of the ones found in [7] and in these cases we present full proofs of those facts (c.f. Theorem 2.24).
In section 3, we study the Dirichlet problem for the p(x)−laplacian, the main result being the continuity of the solution with respect to the source. Although some of the results are well known, we decided to present the proofs of all of the results since we were unable to find a reference for these.
In section 4 we analyze the dependence of the solution of the Dirichlet problem for the p(x)−laplacian with respect to variations on the domain. Our two main theorems here are Theorem 4.7 where a capacity condition on the sequence of approximating domains is given in order for the continuity of solutions to hold, and Theorem 4.8 where it is shown that the continuity only depends on the approximating domains and not on the source term.
In section 5 after giving some capacity estimates that are needed in the remaining of the paper, collect all of our results and prove the main result of the paper, namely the extension of Sverák's result to the variable exponent setting, i.e. Theorem 5.9. The variable exponent Lebesgue space L p(x) (Ω) is defined by This space is endowed with the luxemburg norm The infimum and the supremum of the exponent p play an important role in the estimates as the next elementary proposition shows. For further references, the following notation will be imposed The proof of the following proposition can be found in [9, Theorem 1.3, p.p. 427].
We will use the following form of Hölder's inequality for variable exponents. The proof, which is an easy consequence of Young's inequality, can be found in [7,Lemma 3.2.20].
The variable exponent Sobolev space W 1,p(x) is defined by This space posses a natural modular given by The corresponding luxenburg norm associated to this modular is Observe that this norm turns out to be equivalent to u := u p(x) + ∇u p(x) .
One important subspace of W 1,p(x) (Ω) is the functions with zero boundary values. This is the content of the next definition.
(Ω) as the closure in W 1,p(x) (Ω) of functions with compact support.
In most applications is very helpful to have test functions to be dense in W 1,p(x) 0 (Ω). It is well known, see [7], that this property fails in general, even for continuous exponents p(x). In order to have this desired property one need to impose some regularity conditions on the exponent p(x).
The proof of the following theorem can be found in [ (Ω) the following norm will be used, This norm, is equivalent to the usual norm in W 1,p(x) (Ω) for functions u ∈ W 1,p(x) 0 (Ω).
The norm in this space will be denoted by (Ω), ∇u p(x) ≤ 1}.
We now present a result which we will find most useful later.
Proof. By Hölder's inequality we have that W (Ω) and p ∈ P log (Ω) we have the embeddings are dense. Therefore, Analogous to the constant exponent case, we have the following characterization of W −1,p ′ (x) (Ω).
We will then say that Proof. The characterization of W −1,p ′ (x) (Ω) follows exactly as in the constant exponent case. It remains to see the equivalence of the norms · −1,p ′ (x) and · * .
By Hölder's inequality (Proposition 2.3) and Poincaré's inequality (Theorem 2.7), we have Therefore, (D), the inclusion being canonical, extending by zero. This inclusion (Ω) there is no ambiguity in the notation f, u . Definition 2.13. Given E ⊂ R N , we consider the set Definition 2.14. Let p ∈ P log (Ω) and K ⊂ Ω compact, we define the p(x)−relative capacity as Finally, if E ⊂ Ω is arbitrary, we define the p(x)− relative capacity of E with respect to Ω as The main advantage of the relativa capacity is the fact that is possible to obtain a capacitary potential, i.e. a function whose modular gives the capacity of a set.
To this end, let A ⊂ D and consider the class (D) is closed and convex (the closure of a convex set is convex), it follows that is weakly convex. This fact will be used in the next proposition. Now we show that the relative capacity of a set is realized by a function in Γ A .

in an open set containing A and
By Theorem 2.7 and Proposition 2.1, we have Since the reverse inequality is obvious, the first part of the Proposition is proved. The uniqueness is an immediate consequence of the strict convexity of the modular, since p − > 1. We leave the details to the reader.
We can now give the definition of capacitary potential.
Definition 2.17. We define the capacitary potential of A such as the only u A that verifies It is well known that when dealing with pointwise properties of Sobolev functions, the concept of almost everywhere needs to be changed to quasi everywhere. This is the content of the next definition.
The proof of the next theorem can be found in [7, Corollary 11.1.5].
Remark 2.22. It is easy to see that two p(x)−quasi continuous representatives of a given function u ∈ W 1,p(x) (Ω) can only differ in a set of zero p(x)−capacity. Therefore, the unique p(x)−quasi continuous representative (defined p(x)−q.e.) of u ∈ W 1,p(x) (Ω) will be denoted byũ.
The proof of the next proposition can be found in [7,Section 11.1.11]. (Ω) as the restriction of quasi continuous functions that vanishes quasi everywhere on R N \ Ω. This theorem is esentialy contained in [7, Corollary 11.2.5, Theorem 11.2.5]. We include here the proof since a minor modification of the above mentioned result is needed and for the reader's convenience.
To see the converse, let us assume that D = R N (or else, we extend by zero). Since u = u + − u − , we can assume that u ≥ 0. Moreover, since min{u, n} ∈ W 1,p(x) (R N ) converges to u in W 1,p(x) (R N ), we can assume that u is bounded. Finally, let us consider ξ ∈ C ∞ c (B(0, 2)) such that 0 ≤ ξ ≤ 1 and ξ ≡ 1 in B(0, 1). Setting ξ n (x) = ξ( x n ), we have that ξ n u converges to u in W 1,p(x) (R N ). Therefore we can assume that u(x) = 0 for every x ∈ (B(0, R)) c with R large enough.
Therefore, we need to prove the converse for bounded, compactly supported and nonnegative functions u ∈ W 1,p(x) 0 (R N ) such thatũ = 0 p(x)−q.e. in Ω c . Sinceũ is p(x)−quasi continuous, there is a decreasing sequence of open sets {W n } n∈N such that cap p(x) (W n , D) → 0 andũ| R N \Wn is continuous.
We can assume that W n contains the set of null capacity of R N \ Ω whereũ = 0. Therefore, u = 0 in (Ω ∪ W n ) c = Ω c ∩ W c n . Given δ > 0, set V n = {x :ũ(x) < δ} ∪ W n . Sinceũ is continuous in R N \ W n , V n is an open set. Therefore, V c n is a closed set. It is also bounded since V c n ⊂ B(0, R). Then, V c n is compact. Let u Wn be the capacitary potential of W n , then (u − δ) + (1 − u Wn ) = 0 a.e. in Ω \ V c n . Consider now a regularizing sequence {φ j } j∈N . Therefore, for j sufficiently large we have that By Proposition 2.1, we can conclude that ∇u Wn p(x) → 0 and, by Poincaré's inequality, u Wn 1,p(x) → 0. Therefore, 1 − u Wn → 1 in W 1,p(x) (D) when n → ∞.
We end this subsection with a lemma that will be much helpful in the sequel.
Proof. It is enough to see that v + ∈ W 1,p(x) 0 (D) (for v − we prodece similarly and haven shown this result for v + and v − , we can state that is valid for Since w ≥ 0, by density we can consider {w n } n∈N ⊂ C ∞ c (D) + such that {w n } n∈N converges to w in W 1,p(x) (D).
Therefore, inf{w n , v + }, which has compact support in D (for each w n has so) converges to inf{w, v + } which coincides with v + since |v| ≤ w a.e. in D.
Then, taking an adequate regularizing sequence, we obtain a sequence of C ∞ c (D) convergent to v + , which completes the proof.
Observe that when p(x) = 2 this operator agrees with the classical Laplace operator, and when p(x) = p is constant is the well-known p−laplacian.
By standard methods, we obtain the following result (Ω) and a unique weak solution of (3.1) u ∈ W 1,p(x) 0 (Ω).
Proof. The proof is standard and uses the direct method of the calculus of variations. We omit the details.
Remark 3.2. The unique weak solution of (3.1) will be denoted by u f Ω .
Proof. Let us assume that ∇u f Ω p(x) > 1 (otherwise, the result is clear). By Proposition 2.1, Therefore , which completes the proof.
In what follows, the monontonicity of the p(x)−laplacian is crucial. This fact is a consequence of the following well-known lemma that is proved in [22, p.p. 210].
The following proposition gives the monotonicity property of the solution with respect to the domain. The proof follows the ideas of [12,Theorem 3.2.5.] where the linear case p(x) = 2 is treated. Nevertheless, since the p(x)−laplacian is nonlinear, the monotonicity property of this operator comes into play replacing linearity in the argument.
We now end this section with an stability result for solutions of the Dirichlet problem where the constant β > 0 depends only on p − and p + .
Theorem 3.10 immediately implies the following Corollary. (Ω), we have that In particular, considering ϕ = u 1 − u 2 ∈ W 1,p(x) 0 (Ω) and subtracting, we obtain where we have used Proposition 3.3 in the last inequality.
for some constants α and β depending only on p − and p + . Let us observe that for the first inequality we took into account Hölder's inequality and for the second one, Observation 2.2.
Let us now find a bound for the first factor. In fact, by Proposition 3.3.
Observe that, by Lemma 3.4, we are able to find a bound for the second factor. Then, So we can conclude that This finishes the proof. 4. Continuity of the Dirichlet problem with respect to perturbations on the domain.
In this section we investigate the dependence of the solutions of the Dirichlet problem u f Ω with respect to perturbations on the domain. We will analyze a rather general problem considering a sequence of uniformly bounded domains Ω n converging to a limiting domain Ω in the Haussdorf complementary topology. Then we study whether u f Ωn converges to u f Ω or not. We begin this section by defining a notion of convergence of domains that will be essential for our next results. For an study and properties of this topology of open sets, we refer to the book [12]. We now present the one property that will be essential for our purposes. Proof. The proof is immediate from the definition. See [12]. Now we state a couple of corollaries of Proposition 3.3 that will be most useful.
We now extend to variable exponent spaces Proposition 3.7 in [4]. Let Ω ⊂ D be such that for every compact subset K ⊂ Ω, there is an integer n 0 such that K ⊂ Ω n for every n ≥ n 0 . Then, there holds that Remark 4.6. Observe that in order to conclude that u * = u f Ω it remains to see that u * ∈ W 1,p(x) 0 (Ω).
We will, from now on, work with n ≥ max{n 0 , n 1 }.

Standard computations now give us
(Ω) and so f, φ n → 0.
The next result shows that the continuity of the solutions of the Dirichlet problem for the p(x)−laplacian with respect to the domain is independent of the second member f .
For constant exponents, this result was obtained in [4,Lemma 4.1]. The proof that we present here, in the non-constant exponent case, follows closely the one in [12,Theorem 3.2.5] where the linear case p(x) ≡ 2 is studied.
Proof. Let us assume first that f ∈ L ∞ (D). Therefore, there is a constant M > 0 such that −M ≤ f ≤ M a.e. We can also assume that M > 1.
We will name u f n = u f Ωn and u f = u f Ω . Given k > 1, since u 1 n is the solution of the equation with f ≡ 1, Considering k = M 1 p − −1 , we have that f ≤ M = k p − −1 ≤ −∆ p(x) (ku 1 n ), therefore, u is a supersolution. Since we also have that 0 = u f n | ∂D ≤ ku 1 n | ∂D = 0, by Proposition 3.9, we can conclude that u f n ≤ ku 1 n . On the other hand, since −∆ p(x) (−ku 1 n ) = ∆ p(x) (ku 1 n ) ≤ −k p − −1 = −M ≤ f , we obtain that −ku 1 n ≤ u f n . Therefore (D). Then, by Alaoglu's Theorem, there is a subsequence, which will remaine denoted {u f n } n∈N such that u f n ⇀ u * en W 1,p(x) 0 (D).
Since, by Rellich-Kondrachov's Theorem, we know that W 1,p(x) 0 (D) is compactedly embbeded in L p(x) (D), we have that u f n → u * in L p(x) (D). Then, taking into account the convergence in L p(x) (D) in (4.4), we have that −ku 1 ≤ u * ≤ ku 1 .
Let us assume now that f ∈ W −1,p ′ (x) (D). By density, there is a sequence {f j } j∈N ⊂ L ∞ (D) such that f j → f in W −1,p ′ (x) (D).
Given ϕ ∈ W −1,p ′ (x) (D), ϕ, u f n − u f = ϕ, u f n − u f j n + ϕ, u f j n − u f j + ϕ, u f j − u f . Now, by Theorem 3.10, given ε > 0, there exists j 0 ∈ N such that ∇u f n − ∇u f j n p(x) ≤ ε and ∇u f − ∇u f j p(x) ≤ ε, uniformly in n ∈ N for every j ≥ j 0 . By the first part of the proof, ϕ, u f j n − u f j 0 → 0 as n → ∞. This completes the proof.

Extension of a result ofŠverák.
In this section, we apply our results to prove the extension of the theorems ofŠverák discussed in the introduction. Our main result being Theorem 5.9.
We begin by establishing some capacity estimate from below for compact connected sets. This was obtained for p(x) ≡ 2 byŠverák in [23]. See the book [12] for a proof. For general constant exponents, this estimate was obtained in [4]. Our extension to variable exponents will rely on Bucur and Trebeschi's result [4]. In fact, we use the following proposition.