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International Journal of Differential Equations
Volume 2013 (2013), Article ID 693529, 16 pages
http://dx.doi.org/10.1155/2013/693529
Research Article

Homogenization in Sobolev Spaces with Nonstandard Growth: Brief Review of Methods and Applications

1Laboratoire de Mathėmatiques et de leurs Applications, CNRS-UMR 5142, UNIV Pau & Pays Adour, Avenue de l’Universitė, 64000 Pau, France
2Department of Mathematics, B. Verkin Institute for Low Temperature Physics and Engineering, 47, Avenue Lenin, Kharkov 61103, Ukraine

Received 12 July 2012; Accepted 22 January 2013

Academic Editor: Julio Rossi

Copyright © 2013 Brahim Amaziane and Leonid Pankratov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We review recent results on the homogenization in Sobolev spaces with variable exponents. In particular, we are dealing with the Γ-convergence of variational functionals with rapidly oscillating coefficients, the homogenization of the Dirichlet and Neumann variational problems in strongly perforated domains, as well as double porosity type problems. The growth functions also depend on the small parameter characterizing the scale of the microstructure. The homogenization results are obtained by the method of local energy characteristics. We also consider a parabolic double porosity type problem, which is studied by combining the variational homogenization approach and the two-scale convergence method. Results are illustrated with periodic examples, and the problem of stability in homogenization is discussed.

1. Introduction

In recent years, there has been an increasing interest in the study of the functionals with variable exponents or nonstandard -growth and the corresponding Sobolev spaces, see for instance [18] and the references therein. In particular, the conditions under which functions are dense in have been found. Also, Meyers estimates, which are used in the homogenization process, have been obtained in [6]. Let us mention that such partial differential equations arise in many engineering disciplines, such as electrorheological fluids, non-Newtonian fluids with thermoconvective effects, and nonlinear Darcy flow of compressible fluids in heterogeneous porous media, see for instance [1].

This paper discusses problems of homogenization in Sobolev spaces with variable exponents. Attention is focussed on the homogenization and minimization problems for variational functionals in the framework of Sobolev spaces with nonstandard growth. The material is essentially a review with some new results.

-convergence and minimization problems for functionals with periodic and locally periodic rapidly oscillating Lagrangians of -growth with a constant are well studied now, see for instance [9, 10] and the bibliography therein.

The works [1115] (see also [16]) focus on the variational functionals with nonstandard growth conditions. In particular, the homogenization and -convergence problems for Lagrangians with variable rapidly oscillating exponents were considered in [13, 14]. It was shown that the energy minimums and the homogenized Lagrangians in the spaces might depend on the value of (the so-called Lavrentiev phenomenon). For example, such a behavior can be observed for the Lagrangian with a periodic “chess board” exponent and a small parameter .

Another interesting example of Lagrangian with rapidly oscillating exponent was considered in [11]. Namely, for the functional with a smooth periodic such that , it was shown that the limit functional is bounded on Sobolev-Orlicz space of functions with gradient in an space, where is the fiber percolation level of .

Variational functionals with nonstandard growth conditions have also been considered in [9]. Chapter 21 of this book focuses on the -convergence of such functionals in spaces.

In this paper, we are dealing mainly with the -convergence of variational functionals with periodic rapidly oscillating coefficients, the homogenization of the variational problems in strongly perforated domains (Dirichlet and Neumann problems), and nonlinear double porosity type problems, that is, with the problems where the coefficient of the differential operator asymptotically degenerates on a some specially defined subset (e.g., the set of periodically distributed inclusions) of the domain under consideration. The paper is based on the results obtained in papers [1726].

The paper is organized 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. Then in auxiliary Section 3, we give some definitions which will be used in the paper. In Section 4, we study the question of the -convergence and homogenization of functionals with rapidly oscillating periodic coefficients. In Section 5, we are dealing with the homogenization of the Dirichlet variational problem in strongly perforated domains. The main result of the section (see Theorem 8) is then applied to the study of nonlocal effects in the homogenization (see Theorem 9). In Section 5.3, we give a periodic example, when all the conditions of Theorems 8 and 9 are satisfied and all the coefficients of the homogenized problem are calculated explicitly. Moreover, the question of stability in homogenization is also discussed here. Theorems 8 and 9 are proved by using the so-called method of local energy characteristics proposed earlier by Marchenko and Khruslov for linear homogenization problems (see [27] and the references herein). This method is close to the -convergence method. Briefly, it is based on the derivation of the lim-inf and lim-sup estimates for the variational functional under consideration along with the assumptions on the behavior of the local energy characteristics. In Section 6, the homogenization of the Neumann problem in strongly perforated domains is considered. In this section, the closeness of the method of local energy characteristics is shown directly. In Section 7, we are dealing with a variational problem with high contrast coefficients (nonlinear double porosity type model). The main result of the section (Theorem 23) is also obtained by the method of local energy characteristics. As an application of this result, we consider the periodic case, where we focus our attention on the question of stability in homogenization. Finally, in Section 8, we are dealing with the homogenization of a class of quasilinear parabolic equations with nonstandard growth. The main results of the section are obtained by combining the two-scale convergence method and the variational homogenization approach.

2. Sobolev Spaces with Variable Exponents

In this section, we introduce the function spaces used throughout the paper and describe their basic properties. We refer here to [1, 4, 5, 7, 8].

Let be a bounded Lipschitz domain in (). We introduce the function and assume that this function is bounded such that We also assume that the function satisfies the log-Hölder continuity property. Namely, for all , Notice that this property was introduced by Zhikov to avoid the so-called Lavrentiev phenomenon (see, e.g., [15]).(1)By we denote the space of measurable functions in such that The space equipped with the norm becomes a Banach space.(2)The following inequalities hold: (3)Let , with Then the Hölder’s inequality holds (4)According to (8), for every It is straightforward to check that for domains such that , (5)The space , , is defined by If condition (3) is satisfied, is the closure of the set with respect to the norm of . If the boundary of is Lipschitz continuous and satisfies (3), then is dense in . The norm in the space is defined by If the boundary of is Lipschitz and , then the norm is equivalent to the norm (6)If , then is separable and reflexive.(7)If , then the embedding is continuous and compact.(8)Friedrich’s inequality is valid in the following form: if satisfies conditions (2) and (3), then there exists a constant such that for every

3. Definitions

In this auxiliary section, we introduce the necessary definitions that will be used in the paper. We start by introducing the class of the variable exponents , where is a small positive parameter characterizing the microscopic length scale.

Definition 1 (class ). A sequence of functions is said to belong to the class if this sequence possesses the following properties:(i)for any , is bounded in the following sense: (ii)for any , satisfies the log-Hölder continuity property;(iii)the function converges uniformly in to a function , where the limit function is assumed to satisfy the log-Hölder continuity property.

We also recall the definition of the -convergence (see, e.g., [9, 10, 28] and the bibliography therein). In our case this definition takes the following form.

Definition 2 (-convergence). The functional is said to -converge to a functional if(a)(“”-inequality) for any and any sequence which converges to the function strongly in the space we have (b) (“”-inequality) for any , there exists a sequence such that converges to the function strongly in the space , and

We define the strong convergence in in the following way.

Definition 3 (strong convergence in ). The sequence is said to converge strongly in the space to a function if

Finally, we recall the definition of the two-scale convergence (see, e.g., [29]).

Definition 4. Let be a basic cell and . A function, , which is -periodic in and which satisfies is called an admissible test function.

Definition 5 (two-scale convergence). A sequence of functions two-scale converges to if for any admissible test function , This convergence is denoted by .

The method of the local energy characteristics, generally speaking, deals with nonperiodic structures. We often make use of the following definition.

Definition 6 (distribution in an asymptotically regular way). The set is said to be distributed in an asymptotically regular way in , if for any ball of radius centered at and small enough (), and .

4. -Convergence and Homogenization of Functionals with Rapidly Oscillating Coefficients in Sobolev Spaces with Variable Exponents

Let be a bounded domain in () with a sufficiently smooth boundary and denote that . We assume that a family of continuous functions belongs to the class, , and we also suppose that ) and are -periodic measurable functions such that ().

For the notational convenience, we set

In the space , we define the functional :

We study the asymptotic behavior of and its minimizers as . Our analysis relies on the -convergence approach in Sobolev spaces with variable exponents. The main result of the section is the following.

Theorem 7. Let assumptions () and () be fulfilled. Then (i)The functional , -converges to the functional given by where (ii)The minimizer of the functional converges to the minimizer of the functional strongly in the space .

The Scheme of the Proof of Theorem 7 (See [18] for More Details) Is as Follows. We start our analysis by proving the “”-inequality. The proof of this inequality is done in two main steps. First we introduce an auxiliary functional :

where and prove the “”-inequality for this functional. Then, at the second step, we show that the “”-inequality for the auxiliary functional implies the “”-inequality for . Then, using a special test function and the Meyers estimate (see [6]), we obtain the “”-inequality. Finally, we prove the convergence of the minimizers. This completes the proof of Theorem 7.

5. Homogenization of the Dirichlet Problem and Related Questions

5.1. Homogenization of the Dirichlet Problem

Let be a bounded domain in with sufficiently smooth boundary. Let be an open subset in . Here is a small parameter characterizing the scale of the microstructure. We assume that is distributed in an asymptotically regular way in and we set

Let be a continuous function defined in the domain . We assume that, for any , it satisfies the following conditions:() the function is bounded in the following sense: () the function satisfies the log-Hölder continuity property;() the function converges uniformly in to a function , where the limit function is assumed to satisfy the log-Hölder continuity property;() the function is such that in .

We consider the following variational problem: where and . It is known from [13, 6] that for each , there exists a unique solution (minimizer) of problem (30). Let us extend in by zero (keeping for it the same notation). Then we obtain the family . We study the asymptotic behavior of as .

Instead of the classical periodicity assumption on the microstructure of the perforated domain , we impose certain conditions on the local energy characteristic of the set . To this end we introduce an open cube centered at with length equal to (), and we set where , and the infimum is taken over that equal zero in . We assume that() there exists a continuous function such that for any , any , and a certain , () there exists a constant independent of such that for any ,

The first main result of Section 5 is the following.

Theorem 8. Let conditions (A.1.1)–(A.1.4) on the function and conditions (C.1.1) and (C.1.2) on the local characteristic be satisfied. Then the solution (minimizer) of the variational problem (30) (extended by zero in ) converges weakly in to the solution (minimizer) of where

The Scheme of the Proof of Theorem 8 (See [19] for More Details) Is as Follows. First, it follows from (6), (30), and the regularity properties of the functions that . We extend by zero to the set and consider as a sequence in . Then . Condition (A.1.4) immediately implies that Hence, one can extract a subsequence that converges weakly to a function . We will show that is the solution of the variational problem (35). The proof is done in two mains steps. On the first step, we prove the “”-inequality (the upper bound for the functional ). To this end, we cover the domain by the cubes of length centered at the points , where be a periodic grid in with a period (, ). We associate with this covering a partition of unity : ; for ; for ; for ; . Then we construct the function using the minimizers of (32) for . We show that, for any , and, therefore, On the second step, we prove the “”-inequality (the lower bound for the functional ) using the definition of the local energy characteristic and the convexity of our functional: Finally it follows from (39) and (40) that for any such that on . This means that any weak limit of the solution of problem (30) extended to the set by zero is the solution of the homogenized problem (35). This completes the proof of Theorem 8. The generalization of Theorem 8 is given in [17].

5.2. Nonlocal Effects in Homogenization of -Laplacian in Perforated Domains

Let be a domain in () defined in the previous section. Let be an open connected subset in like a net. We assume that the set satisfies the following conditions: (), where with independent of ;() is distributed in an asymptotically regular way in ;() the subdomain is a connected set.

We assume that for any , the function satisfies the conditions (A.1.1)–(A.1.4) from Section 5.1. In the space , we define the functional , where is defined in (31) and we consider the following variational problem: where is an unknown constant. It is known from [1, 2] that for each , there exists a unique solution of the variational problem (43). We extend by the equality in and keep for it the same notation. Thus, we obtain the family . We study the asymptotic behavior of the family as .

Let us introduce the functional , where is the characteristic function of the domain and is defined in (36).

The main result of the section is the following.

Theorem 9. Let be a solution of (43) extended by the equality in . Assume that the conditions (F.1.1)–(F.1.3), (A.1.1)–(A.1.4), and(C.1.1)-(C.1.2) on the set , the functions , and the local characteristic are satisfied. Then there is a subsequence that converges weakly in to a function such that the pair is a solution of

Remark 10. It is important to notice that the constant in (45) remains unknown. Suppose, in addition, that the function is differentiable with respect to the argument . Then it is easy to see that Euler’s equation for the homogenized problem (45) reads where denotes the partial derivative of the function with respect to . This means that the homogenized problem (45) is nonlocal.

Theorem 9 is proved by arguments similar to those from the proof of Theorem 8.

5.3. A Periodic Example

Theorems 8 and 9 provide sufficient conditions for the existence of the homogenized problem (45). The goal of this section is to prove that, for appropriate examples, all the conditions of Theorems 8 and 9 are satisfied and to compute the function and the constant in the homogenized problem (45) explicitly.

Let be a bounded Lipschitz domain in and let be the subdomain of defined in condition (F.1.1). Let be an -periodic coordinate lattice in formed by the intersecting circular cylinders of radius , We set and .

Let be a class of continuous functions defined in the domain . We assume that, for any , satisfies the -Hölder continuity condition and the following condition: () the functions are given by where denotes the cylindrical -neighborhood of the set and where and are smooth strictly positive functions in , moreover, as .

Consider the boundary value problem where is an unknown constant. It is known from [1, 2] that, for each , there exists a unique solution of problem (49). We extend by the equality in and keep for it the same notation. The asymptotic behavior of the solutions of problem (49) is given now by the following theorem.

Theorem 11. Let be a solution of (49) extended by the equality in . Assume that the condition (B.1.2) on the function is satisfied. Then the sequence converges weakly in to a function solution of where

Theorem 11 is proved in [23] and is based essentially on the calculation of the local energy characteristic for this periodic structure.

Remark 12. It is easy to see from the proof of Theorem 11 that if in (48) we replace by a function with , then as in the linear case.

Remark 13. It is shown in [27] Paragraph 3.3 that for the integrands of growth , the 3D lattice becomes extremely thin. Moreover, for the integrands of growth , where is a parameter independent of , there is no 3D lattice which admits nontrivial homogenization result, because the capacity of the lattice goes to infinity as . Theorem 11 shows the maximal possible polynomial growth of the integrand (in a small neighborhood of the lattice) which admits a nontrivial homogenization result.

6. Homogenization of the Neumann Problem

6.1. Statement of the Problem and Main Results

Let be a bounded Lipschitz domain in (). Let be a family of open subsets in ; in the sequel is a small positive parameter characterizing the microscopic length scale. We assume that the set consists of ( as ) small isolated components such that their diameters go to zero as and is distributed in an asymptotically regular way in . We set

We assume that a family of continuous functions belongs to the class . On the space , we define the functional : where is defined in (31) and .

We study the asymptotic behavior of and their minimizers as . 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 is a disperse medium; that is, the following assumptions hold: () the local concentration of the set has a positive continuous limit, that is, the indicator of converges weakly in to a continuous positive limit. This implies that there exists a continuous positive function such that for any open cube centered at with lengths equal to ;() for any , there exists a family of extension operators such that.global: for any , uniformly in , where and in ;local: for any there is such that for all , and any function , the estimates hold

Remark 14. Notice that in condition (C.2.2), we require the existence of extension operators only in usual Sobolev spaces with constant . In this case, the extension condition is well studied in the mathematical literature (see, e.g., [27, 3032]). For instance, it holds for a wide class of disperse media (see, for instance, [27]).

One more condition is imposed on the local characteristic of the set associated to the functional (53). In order to formulate this condition, we denote by an open cube centered at with edge length () and introduce the functional where , , and the infimum is taken over . Here stands for the scalar product in . We assume that() there is a continuous, with respect to , function and () such that for any , any and any ,

Remark 15. Condition (C.2.3) is always fulfilled for periodic and locally periodic structures.

Remark 16. It is crucial in condition (C.2.3) that the limit function does not depend on the particular choice of the sequence . 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.

Now we are in position to formulate the first convergence result of the section.

Theorem 17. Assume that , and let conditions (C.2.1)–(C.2.3) be satisfied. Then the functionals defined in (53), -converge to the functional given by

Now let us formulate the convergence result for the minimizers of the functionals . Consider the variational problem According to [13, 6], for each , problem (60) has a unique solution .

The following convergence result holds.

Theorem 18. Under the assumptions of Theorem 17, the solution of the variational problem (60) converges strongly in to a solution of the problem

The Scheme of the Proof of Theorems 17 and 18 (See [22] for More Details) Is as Follows. The “”-inequality is proved in two steps as in Theorem 7 by introducing an auxiliary functional. The “”-inequality is proved by the arguments similar to those from the proof of Theorem 8.

6.2. A Periodic Example

Theorems 17 and 18 of Section 6.1 provide sufficient conditions for the existence of the -limit functional (59) and for the convergence of minimizers of the variational problem (60) to the minimizer of the homogenized variational problem (61). 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 (59) in terms of solutions of auxiliary cell problems. In fact, conditions (C.2.1) and  (C.2.3) are always satisfied in the periodic case if the boundary of inclusions is regular enough, and that the extension condition (C.2.2) can also be replaced with the assumption on the regularity of the inclusions geometry.

Let be a bounded domain in () with sufficiently smooth boundary. We assume that in the periodic cell , there is an obstacle being an open set with a sufficiently smooth boundary . We assume that this geometry is repeated periodically in the whole . The geometric structure within the domain is then obtained by intersecting the -multiple of this geometry with . Let be an -periodic grid in : , . Then we define as the union of sets obtained from by translations with vectors , , that is, and . Notice that the geometry of the inclusions having a nontrivial intersection with the domain boundary might be rather complicated. In particular, the extension condition (C.2.2) might be violated for these inclusions. To avoid these technical difficulties, we assume that the domain is not perforated in a small neighborhood of its boundary . We set

Let a family of continuous functions belongs to the class . On the space , we define the functional : where .

We study the asymptotic behavior of the functional and its minimizer as . To formulate the main result of this section we will introduce some notation. We denote by a minimizer of the following variational problem: where , and is a vector in , and is a parameter.

If , then the solution coincides with a unique solution in of the following cell problem: here is the outward normal to .

The following result holds.

Theorem 19. The sequence of functionals defined in (64) -converges to the functional given by where Moreover, a minimizer of the functional (64) converges strongly in the space to the minimizer of the homogenized functional (67).

Notice that Theorem 19 (see [22]) can be proved in two different ways. One of them is to check that under the assumptions of Theorem 19 conditions (C.2.1)–(C.2.3) are satisfied and that the characteristics introduced in conditions (C.2.1) and (C.2.3) coincide with those defined in (68). On the other hand, in the periodic case, the direct -convergence techniques can be applied. This allows us to obtain formula (68) by means of -convergence approach used in periodic homogenization.

7. Homogenization of Quasilinear Elliptic Equations with Nonstandard Growth in High-Contrast Media

7.1. Statement of the Problem and the Main Result

Let be a bounded domain of () with Lipschitz boundary . Here is a family of open subsets in . We assume that the set is distributed in an asymptotically regular way in ; moreover, for the sake of simplicity, we suppose that .

Remark 20. In the framework of the method of local energy characteristics presented in the section, we do not specify the geometrical structure of the set . Generally speaking, it may consist of ( as ) small isolated components such that their diameters go to zero as or it may be defined as fibres becoming more and more dense as , such that the diameters of the fibers go to zero as .

We assume that the family of functions belongs to the class . Let be such that() there exist two real numbers and such that the function is bounded in the following sense: () the function satisfies the log-Hölder continuity property.

Let us now define the variational problem under consideration. To this end, we consider the functional , where Here the function is defined by , . We denote by the characteristic function of the set , . is a measurable function in such that() there exists a real number such that in ;() for any , there exists a real number such that and as .

We consider the following variational problem: It is known from [1] that for each , there exists a unique solution of the variational problem (72). We aim to study the asymptotic behavior of the family as , bearing in mind that the geometry of depends on . So, we have to specify this geometry. Most of the papers dealing with homogenization assume that is a periodic repetition of a standard cell. This classical periodicity assumption is here substituted by an abstract one covering a variety of concrete behaviors including periodicity and almost periodicity. We thus make the following assumptions: () the local concentration of the set has a positive continuous limit; that is, the indicator of converges weakly in to a continuous positive limit. This implies that there exists a continuous positive function such that for any open cube centered at with lengths equal to ;() for any , there is a constant such that if the function is defined by in , then: (i)the sequence belongs to , that is, ; (ii)there exists a family of extension operators : such that, for any , where is a strictly monotone continuous function in such that and as .

We also impose several conditions on the local characteristic of the set and associated to the functional (70). Let be an open cube centered at with lengths equal to (). We introduce the following functionals:(i)the functional associated to the energy in is defined in by for , , where and the infimum is taken over .(ii)The functional associated to the energy exchange between and is defined in by for , the infimum being taken over .

We assume that the local characteristics of are such that() for any and any , there is a continuous function and a real number () such that, for any , () for any and any , there is a continuous function and a real number () such that, for any ,

Remark 21. It is crucial in conditions (C.3.3) and (C.3.4) that the limit functions and do not depend on the particular choice of the sequence . It is proved in [26] that these assumptions are fulfilled for periodic and locally periodic media.

Remark 22. Contrary to the standard growth setting as considered in [20, 32], the local characteristic is not homogeneous with respect to the parameter . This induces the appearance of a nonlinear function in the homogenized functional (see Theorem 23 below).

The main result of the section is the following.

Theorem 23. Let be a solution of (72). Assume that and conditions (A.3.4)–(A.3.5), (K.1)-(K.2), and (C.3.1)–(C.3.4) are satisfied. Then , solution of the variational problem (72), converges strongly in to , solution of the following variational problem: the homogenized functional being defined by where Moreover, for any smooth function in , we have

The scheme of the proof of Theorem 23 (see [26] for more details) is similar to the scheme of the proof of Theorems 17 and 18.

7.2. Periodic Examples

Theorem 23 of Section 7.1 provides sufficient conditions for the existence of the homogenized functional (80) and for the convergence of minimizers of the variational problem (72) to the minimizer of the homogenized variational problem (79) under conditions (A.3.4)-(A.3.5), (K.1)-(K.2), and (C.3.1)–(C.3.4). It is important to show that the “intersection” of these conditions is not empty. The goal of this section is to prove that for periodic media all the conditions of the above-mentioned theorem are satisfied and to compute the coefficients of the homogenized functional (80) either in an explicit form or as usually by the solution of a corresponding cell problem.

Let be a bounded domain in () with Lipschitz boundary. We assume that in the standard periodic cell , there is an obstacle with Lipschitz boundary . We assume that this geometry is repeated periodically in the whole . The geometric structure within the domain is then obtained by intersecting the -multiple of this geometry with . Let be an -periodic grid in . Then we define as the union of sets () obtained from by translations of vectors , that is, where is the cube centered at the point and of length , , is the canonical basis of , and as .

Let be a log-Hölder continuous function such that Let be a sequence defined by where the function is such that as . The asymptotic behavior of will be specified below. On the space , we define the functional , where the function satisfies conditions (A.3.4) with , (A.3.5), the function is defined by , . Here and are strictly positive constants independent of .

Consider the following variational problem: We aim to study the asymptotic behavior of