HOMOGENIZATION OF VARIATIONAL FUNCTIONALS WITH NONSTANDARD GROWTH IN PERFORATED DOMAINS

The aim of the paper is to study the asymptotic behavior of solutions to a Neumann boundary value problem for a nonlinear elliptic equation with nonstandard growth condition of the form −div ( |∇u|ε ∇u ) + |u|ε u = f(x) in a perforated domain Ω, ε being a small parameter that characterizes the microscopic length scale of the microstructure. Under the assumption that the functions pε(x) converge uniformly to a limit function p0(x) and that p0 satisfy certain logarithmic uniform continuity condition, it is shown that u converges, as ε → 0, to a solution of homogenized equation whose coefficients are calculated in terms of local energy characteristics of the domain Ω. This result is then illustrated with periodic and locally periodic examples.


1.
Introduction. In recent years, increasing attention has been paid to the study of the so called differential equations and variational problems with nonstandard p(x)-growth motivated by their applications to the mathematical modeling in continuum mechanics. Such equations arise, for example, from the modeling of non-Newtonian fluids with thermo-convective effects (see, e.g., [9,10]), the modeling of electro-rheological fluids (see, e.g., [28,29]), the thermistor problem (see, e.g., [37]), the problem of image recovery (see, e.g., [16]), and the motion of a compressible fluid in a heterogeneous anisotropic porous medium obeying to the nonlinear Darcy law (see, e.g., [11,14]). There is an extensive literature on this subject. We will not attempt a review of the literature here, but merely mention a few references, Following the approach developed in [26], instead of a classical periodicity assumption on the structure of the perforated domain Ω ε , we impose certain conditions on the so-called local energy characteristics associated with the equation (1). It will be shown that the asymptotic behavior, as ε → 0, of the solution u ε is described by the Neumann problem for the following nonlinear equation: where the functions a i (i = 1, 2, .., n), a 0 , and ρ are defined in terms of the above mentioned local characteristics. The proof of the main result is based on the application of the notion of Γconvergence and the variational homogenization technique which is nowadays widely used in the homogenization theory (see, e.g., [15,26,38] and references therein).
The outline of the rest of the paper is as follows. In Section 2, for the sake of completeness, we recall the definition and main results on the Lebesgue and Sobolev spaces with variable exponents which will be used in the sequel. In Section 3 all necessary mathematical notation is defined, the microscopic problem is formulated, the general assumptions are stated, and the main result is formulated. The proof of the convergence result is carried out in Section 5; it relies on auxiliary results from Section 4. Two examples of periodic and locally periodic structures are considered in Section 6.

Notational convention.
In what follows C, C 1 , C 2 , etc. are generic constants independent of ε.
2. Sobolev spaces with variable exponents. For the reader's convenience, we recall some basic facts concerning Sobolev spaces with variable exponents, see for instance [13] or [20] and the bibliography herein.
Let Ω be a bounded Lipschitz domain in R n (n 2) and the exponent function p(x) satisfies the following conditions: For all x, y ∈ Ω, where C is a constant. 1. By L p(·) (Ω) we denote the space of measurable functions f in Ω such that The space L p(·) (Ω) equipped with the norm f L p(·) (Ω) = inf λ > 0 : A p(·),Ω f λ 1 (5) becomes a Banach space.
8. Friedrichs' inequality is valid in the following form: if p(x) satisfies conditions (3)-(4), then there exists a constant C > 0 such that for every f ∈ W 1,p(·) 0 (Ω) 3. Statement of the problem and main results. Let Ω be a bounded Lipschitz domain in R n (n 2). Let {F ε } (ε>0) be a family of open subsets in Ω; in the sequel ε is a small positive parameter characterizing the microscopic length scale. We assume that: (F1) the set F ε consists of N ε (N ε → +∞ as ε → 0) small isolated components such that their diameters go to zero as ε → 0; (F2) the set F ε is distributed in an asymptotically regular way in Ω, i.e., for any ball B(y, r) of radius r centered at y ∈ Ω and sufficiently small ε > 0 (ε ε 0 (r)), we have that B(y, r) ∩ F ε = ∅ and B(y, r) ∩ (Ω \ F ε ) = ∅. We set Ω ε = Ω \ F ε . (12) A sequence of functions {p ε } (ε>0) is said to belong to the class L ε p0(·) if this sequence possesses the following properties: (A1) for any ε > 0 , p ε is bounded in the following sense: for any x, y ∈ Ω and any ε > 0, the function p ε satisfies the local log-Hölder continuity property, i.e., (A3) the function p ε converges uniformly in Ω to a function p 0 , i.e., where the limit function p 0 is assumed to satisfy (4); Notice that the sequence p ε = p 0 for any ε > 0 belongs to the family L ε p0(·) . On the space L pε(·) (Ω ε ) we define the functional J ε : L pε(·) (Ω ε ) −→ R ∪ {+∞} : We study the asymptotic behavior of J ε and their minimizers as ε → 0. The classical periodicity assumption is here substituted by an abstract one covering a variety of concrete behaviors such as the periodicity, the almost periodicity, and many more besides. For this, we assume that Ω ε ⊂ Ω is a disperse medium, i.e., the following assumptions hold: (C1) the local concentration of the set Ω ε has a positive continuous limit, that is the indicator of Ω ε converges weakly in L 2 (Ω) to a continuous positive limit. This implies that there exists a continuous positive function ρ = ρ(x) such that lim for any open cube K x h centered at x ∈ Ω with lengths equal to h > 0; (C2) for any q ∈ [p − , p + ] there exists a family of extension operators P ε q : W 1,q (Ω ε ) → W 1,q (Ω) such that global: for any u ε ∈ W 1,q (Ω ε ), uniformly in ε > 0, where u ε = P ε q u ε and u ε = u ε in Ω ε . local: for any h > 0 there is ε 0 (h) > 0 such that for all ε < ε 0 (h), z ∈ Ω and any function u ∈ W 1,q ((z + [−2h, 2h] n ) ∩ Ω ε ) the estimates hold Remark 1. Notice that in condition (C2) we require the existence of extension operators only in usual Sobolev spaces W 1,q with constant q. In this case the extension condition is well studied in the existing mathematical literature (see, e.g., [1,6,17,26,27]). For instance, it holds for a wide class of disperse media (see, for instance, [26]).
One more condition is imposed on the so called local characteristic of the set F ε associated to the functional (16). In order to formulate this condition we denote by K z h an open cube centered at z ∈ Ω with edge length h (0 < ε ≪ h ≪ 1), and introduce the functional: where γ > 0, b ∈ R n , and the infimum is taken over v ε ∈ W 1,pε(·) (K z h ∩ Ω ε ). Here ( · , · ) stands for the scalar product in R n . We assume that: (C3) there is a continuous, with respect to x ∈ Ω, function T (x, b) and γ = γ 0 (0 < γ 0 < p − ) such that, for any {p ε } (ε>0) ⊂ L ε p0(·) , any x ∈ Ω and any b ∈ R n , Remark 2. Condition (C3) is always fulfilled for periodic and locally periodic structures.
Remark 3. It is crucial in condition (C3) that the limit function T (x, b) does not depend on the particular choice of the sequence p ε → p 0 . Notice that this is always the case for periodic and locally periodic perforated media. These media will be considered in detail in the last section of the paper.
We define the strong convergence in L p0(·) (Ω ε ) in the following way.
We also recall the definition of the Γ-convergence (see, e.g., [15,18]). In our case it reads.
The following convergence result holds.
Theorem 3.4. Under the assumptions of Theorem 3.3, the solution u ε of the variational problem (23) converges strongly in L p0(·) (Ω ε ) to a solution of the problem: 4. Properties of the homogenized problem. In this Section we deal with the properties of the homogenized problem (24). First, we will describe the properties of the function T (x, b) defined in (21). Then using this result we will show the continuity of the homogenized functional J hom in the space W 1,p0(·) (Ω). Finally, we will prove that the variational problem (24) has a unique solution u ∈ W 1,p0(·) (Ω). Properties of the function T (x, b) are given by the following lemma. (i) it is convex with respect to the variable b, i.e., (ii) it satisfies the bound: (iii) it is locally Lipschitz in the following sense: Proof of Lemma 4.1. We begin by proving the statement (i) of Lemma 4.1.
Let v ε 1 , v ε 2 , and v ε 1,2 be minimizers of the functional in (20) where Then from (28) we get: Now the statement (i) of Lemma 4.1 immediately follows from (29) and the condition (C3).
We turn to the statement (ii) of Lemma 4.1. Let v ε be the minimizer of the functional in (20). (20) we get: This inequality and (13) immediately imply that Then we have: Now using the assumption (A3) we obtain This inequality and the assumption (A2) imply that, for ε sufficiently small, Now the statement (ii) of Lemma 4.1 immediately follows from (32) and the condition (C3). It remains to prove the statement (iii) of the lemma. Let τ be defined by Consider the functional c ε,h pε(·) (z, b 1 ). It can be represented as follows: Then it follows from (29) that We apply the inequality (32) to estimate the second term of the right hand side of (34). For ε sufficiently small and h → 0, from (33), we have: (33)-(35), for ε sufficiently small and h → 0, we obtain: as h → 0. In the same way, for ε sufficiently small and h → 0, Our next aim is to show that the homogenized functional J hom is continuous in the space W 1,p0(·) (Ω).
Proof of Lemma 4.2. From the definition of the homogenized functional J hom and regularity properties of functions p 0 , ρ, f , we get: Let us estimate the right hand side of (39). For the first term, by (27), we have: To estimate the integral on the right hand side of (40), we apply Hölder's inequality (7) and inequalities (6). Then we obtain: where ,Ω (1 + |∇u| + |∇v|) and 1 In a similar way one can estimate the second and the third terms on the right hand side of (39). Finally, this yields the desired inequality (38). Lemma 4.2 is proved.

5.
Proof of main results. The proof of main results is based on the Γ-convergence and variational homogenization techniques (see for instance [26]). The proof of Theorem 3.3 is given below in Sections 5.1 and 5.2. First, we show that the Γ-limit functional takes on finite values only for u ∈ W 1,p0(·) (Ω). Then we obtain the "lim inf"-inequality and the "lim sup"-inequality.
The assertion of Theorem 3.4 is then a consequence of Theorem 3.3. It is shown in Section 5.3.
The following statement characterizes the domain of the Γ-limit functional.
Then there is a function u 0 ∈ W 1,p0(·) (Ω) such that along a subsequence Proof of Lemma 5.1. Considering the coercive properties of the functional J ε it is easy to see that, for some subsequence ε k , the uniform bound holds Exploiting the extension condition (C2), the continuity of p 0 (x) and the fact that p ε converges to p 0 uniformly in Ω, we conclude that for any δ > 0 there exist h = h(δ) > 0 and piece-wise constant functionp δ such that for all sufficiently small ε > 0: By the Sobolev embedding theorem, the sequence {u ε k } is compact in L p0(·) (Ω). Thus, there is u 0 ∈ L p0(·) (Ω) such that (43) holds, after probably taking another subsequence.
We assert that u 0 ∈ W 1,p0(·) (Ω). Indeed, it follows from the uniform bound (iii) that ∇u 0 C with a constant C being independent of δ. Passing to the limit as δ → 0 and applying the Fatou theorem, we arrive at the desired statement.
5.1. Proof of the "lim inf"-inequality. The proof of the "lim inf"-inequality is done in two main steps. At the first step we introduce an auxiliary functional J ε and prove the "lim inf"-inequality for this functional. On the second step, using the condition (C3), we obtain this inequality for the functional J ε .
(44) Notice that the functional J ε is continuous in W 1,πε(·) (Ω ε ). More precisely, the following inequality holds: where the exponent q 0 = q 0 (x) and the value q − 0 are defined in (42). Notice also, that the statement of Lemma 5.1 remains valid for the functional J ε . Now let u be an arbitrary C ∞ (Ω) function and {u ε } be a sequence which converges to the function u strongly in L p0(·) (Ω ε ) and such that J ε [u ε ] C. We will show that lim Let {x α } be a set of points in the domain Ω that form an h-periodic space lattice. Let us cover the domain Ω by the cubes K α h with nonintersecting interiors and introduce the notation: Consider now J ε [u ε ]. It is clear that where Consider, first, the second term on the right hand side of (48). It follows from the strong convergence of the sequence {u ε } to u ∈ C ∞ (Ω) in the space L p0(·) (Ω ε ) and (47) that Consider now the first term on the right hand side of (48). We have:

BRAHIM AMAZIANE, LEONID PANKRATOV AND ANDREY PIATNITSKI
For any α such that and consider the first term on the right hand side of (51). Bearing in mind (20), as ε → 0, we have: Let us estimate the second integral on the right hand side of (53). It follows from the regularity of the function u and assumptions (A1), (A3) that, for any b ∈ R n and any ε > 0, we have: It is easy to see that, for h → 0, Now it follows from (54) that We set b = b α = ∇u(x α ). Then it follows from the strong convergence of the sequence {u ε } to u in the space L p0(·) (Ω ε ) and (55) that as h → 0. Finally, from the definition (20) and relations (53), (56) we get: Therefore, by (57), the first term on the right hand side of (48) can be estimated as follows: Finally, from (50), (58) we have: We pass to the limit in the inequality (59) first as ε → 0 and then as h → 0. Taking into account the strong convergence of the sequence {u ε } to u in the space L p0(·) (Ω ε ), the regularity of the function f , the properties of the function p ε , and conditions (C1), (C3) we obtain the desired inequality (46). By the definition of π ε (x) we have π ε (x) p 0 (x) in Ω. Therefore, the family { J ε } is uniformly in ε continuous in W 1,p0(·) (Ω ε ) topology. In addition, by Lemma 4.2 the functional J hom is continuous in W 1,p0(·) (Ω) topology. Then the fact that inequality (46) holds for any u ∈ C ∞ (Ω) implies that (46) holds for all u ∈ W 1,p0(·) (Ω). This completes the proof of the "lim inf"-inequality for the functional J ε .
Step 2. "Lim inf"-inequality for the initial functional. Let u be an arbitrary function from L p0(·) (Ω) and {u ε } be a sequence which converges to the function u strongly in L p0(·) (Ω ε ) and such that J ε [u ε ] C. First we remark that one can prove the inequality in the same way as the inequality (46). Notice that in contrast with J ε , the functional J ε is not continuous in W 1,p0(·) topology. Therefore, the fact that (60) holds for any C ∞ -function does not imply this inequality for any u ∈ W 1,p0(·) (Ω). To prove (60) for any u ∈ W 1,p0(·) (Ω) we use another technique based on the assumption (C3). Namely, let u ∈ W 1,p0(·) (Ω)\ C ∞ (Ω). Consider the value It is easy to see that for all

BRAHIM AMAZIANE, LEONID PANKRATOV AND ANDREY PIATNITSKI
with a constant C that does not depend on x and ε. Then Now it follows from (62), (63) that Inequalities (60), (64) mean that if u is an arbitrary function from W 1,p0(·) (Ω) and {u ε } is a sequence converging to the function u strongly in L p0(·) (Ω ε ) then and the "lim inf"-inequality is proved.
Step 1. Upper bound. Let {x α } be a periodic grid in Ω with a period h ′ = h−h 1+γ/p + (ε ≪ h ≪ 1, 0 < γ < p − ). Let us cover the domain Ω by cubes K α h of length h > 0 centered at points x α . We associate with this covering a partition of unity be a function minimizing functional (20) with b = b α and z = x α , where b α is a constant vector which will be specified later on. It follows from conditions (A1) and (C3) that, as h → 0, Denote by K α h ′ the cube of length h ′ centered at the point x α , and by Π α h the set K α h \ K α h ′ . By (66) and condition (A1) we have: as h → 0. Then from the definition of c ε,h pε(·) (z, b) (see (20)) it follows that, for sufficiently small ε, the bound holds Considering now condition (C3), we obtain that, as h → 0, Let u be a smooth function in Ω. In the domain Ω ε we define It is clear that w ε h ∈ W 1,pε(·) (Ω ε ) and that where the function F ε is defined by First, we consider the second sum on the right hand side of (70). Considering the properties of the partition of unity {ϕ α }, it is not difficult to check that for any α and β the number of terms which are nontrivial in K α h ∩ K β h is finite and does not depend on ε. Then in order to estimate the second term on the right hand side of (70) it is sufficient to estimate the following integral: For the first term on the right hand side of (72) we have: By the condition (A1), taking into account the relation meas (K α h ∩ K β h ) = o(h n ) as h → 0, and the fact that u is a smooth function in Ω and b α is a constant vector, we obtain It also follows from (68) that Due to the properties of {ϕ α }, for the last term on the right hand side of (73), we have: It is clear that Therefore, from the second estimate in (68) we deduce that Finally, (74)-(76) yield: lim In a similar way we can show that lim h→0 lim ε→0 j ε 2 [w ε h ] = 0, and lim h→0 lim ε→0 j ε 3 [w ε h ] = 0. This implies that the contribution of the second term on the right hand side of (70) is asymptotically negligible, that is Consider now the first term on the right hand side of (70). We set b α = ∇u(x α ). It follows from the definition of the function w ε h that, for any α, and (80) Then we have that In order to estimate the second term on the right-hand side of this relation we apply the inequality: where ξ, η 0 and A = A(p (−) , p (+) ) is a constant. Then from condition (A3), (66) with b α = ∇u(x α ), (79), (81), and the regularity of u, f, p ε , p 0 , for sufficiently small ε and h → 0, we get: Now it follows from (80), (82), the regularity properties of functions u, f , and the definition (20) that, for sufficiently small ε and h → 0, Now we take the union in (83) over all cubes and pass to the limit first as ε → 0 and then as h → 0. Taking into account (78) and the condition (C3), we obtain that for any smooth function u This inequality also holds true for any u ∈ W 1,p0(·) (Ω). This fact immediately follows from density arguments and the continuity of the homogenized functional in W 1,p0(·) (Ω) (cf. Lemma 4.2).
Step 2. Construction of the recovery sequence. Consider the sequence {w ε h } defined by (69). Letε(h) be a decreasing function such that lim h→0ε (h) = 0. We set .
It is clear that the sequence w ε converges strongly in L p0(·) (Ω ε ) to the function u = u(x) and satisfies the inequality:

BRAHIM AMAZIANE, LEONID PANKRATOV AND ANDREY PIATNITSKI
This completes the proof of the "lim sup"-inequality and of Theorem 3.3.

5.3.
Proof of Theorem 3.4. Let u ε be a solution of the variational problem (23). Then from (6), (7), (23), and the properties of the functions f, p ε we have: It follows from the properties (F1) and (F2) of the set F ε (see the beginning of Section 3) and the continuity of the function p 0 that we can cover the domain Ω by the finite number of subdomains Θ α (α = 1, 2, .., M ) with nonintersecting interiors such that, for any α, ∂Θ α ∩ F ε = ∅ and Since the number of the subdomains Θ α is finite, then inequalities (86) imply that where p α min = min α p − α . Now it follows from condition (C3) that u ε and inequalities (87), (89) imply that the family {u ε } is a compact set in the space L p0(·) (Ω). Hence, one can extract a subsequence {u ε , ε = ε k → 0} that converges strongly in L p0(·) (Ω) to a function u ∈ W 1,p0(·) (Ω). Let us show that u = u(x) is a solution of the variational problem (24). First, it is clear that since u ε is the solution of the variational problem (23), then , where the function w ε h is given by (69). Now the "lim sup"-inequality (84) immediately implies that, for any w ∈ W 1,p0(·) (Ω), On the other hand, from the "lim inf"-inequality, we have: Now inequalities (90) and (91) imply that if a subsequence of solutions of problem (23) converges strongly in L p0(·) (Ω ε ) to a function u = u(x), then, for any w ∈ W 1,p0(·) (Ω), and u is the solution of (24). Since this problem has a unique solution, then the whole sequence of solutions of problem (23) converges strongly in L p0(·) (Ω ε ) to the function u.
This completes the proof of Theorem 3.4.
6. Periodic and locally periodic examples. Theorems 3.3 and 3.4 of Section 3 provide sufficient conditions for the existence of the Γ-limit functional (22) and for the convergence of minimizers of the variational problem (23) to the minimizer of the homogenized variational problem (24). It is important to show that the class of functions which satisfy the conditions of these theorems is not empty. The goal of this Section is to prove that for periodic and locally periodic media all conditions of the above mentioned theorems are satisfied and to compute the coefficients of the homogenized functional (22) in terms of solutions of auxiliary cell problems.
In fact, we will prove that conditions (C1), (C3) are always satisfied in the periodic case if the boundary of inclusions is regular enough (see Proposition 1), and that the extension condition (C2) can also be replaced with the assumption on the regularity of the inclusions geometry (see the beginning of Appendix).
6.1. A periodic example. Let Ω be a bounded domain in R n (n 2) with sufficiently smooth boundary. We assume that, in the standard periodic cell Y = (−1/2, 1/2) n , there is an obstacle F ⊂ Y being an open set with a sufficiently smooth boundary ∂F such thatF ⊂ Y . We assume that this geometry is repeated periodically in the whole R n . The geometric structure within the domain Ω is then obtained by intersecting the ε-multiple of this geometry with Ω, ε being a small positive parameter. Let {x k,ε } be an ε-periodic grid in R n : x k,ε = εk, k ∈ Z n . Then we define F ε as the union of sets F ε k ⊂ K k ε obtained from εF by translations with vectors εk, k ∈ Z n , i.e., and K k ε = εk + εY . Notice that the geometry of the inclusions having a nontrivial intersection with the domain boundary, might be rather complicated. In particular, the extension condition (C2) might be violated for these inclusions. To avoid these technical difficulties we will often assume below that the domain Ω is not perforated in a small neighbourhood of its boundary ∂Ω.
Denote by K ε the union of k ∈ Z n such that K k ε ⊂ Ω, and set Let a family of continuous functions {p ε } (ε>0) and a function p 0 satisfy conditions (A1)-(A3) from Section 3.

BRAHIM AMAZIANE, LEONID PANKRATOV AND ANDREY PIATNITSKI
If p 2 then the solution U b coincides with a unique solution in W 1,p per (Y ⋆ ) of the following cell problem: here ν is the outward normal to ∂F . The following result holds.
Remark 4. In general, the existence of a minimizer of the functional (96) is a complicated problem because the geometry of Ω ε B might be rather complex. 6.1.1. Proof of Theorem 6.1. Theorem 6.1 can be proved in two different ways. One of them is to check that under the assumptions of Theorem 6.1 conditions (C1) -(C3) are satisfied and that the characteristics introduced in conditions (C1) and (C3) coincide with those defined in (100). In order to make the results of Theorems 3.3 and 6.1 compatible, we will prove in this section that the mentioned characteristics do coincide.
On the other hand, in the periodic case the direct Γ-convergence techniques apply. This allows us to simplify the proof and to obtain formula (100) by means of Γ-convergence approach used in periodic homogenization. In this connection, we will provide below the proof of "liminf" inequality for the stated in Theorem 6.1 Γ-convergence. Since the proof of "limsup" inequality and of the convergence of minimizers is standard, it will be omitted.
Let us show that conditions (C1)-(C3) are satisfied in the periodic case under consideration. The following result holds.