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

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 variationalproblemsinstronglyperforateddomains,aswellasdoubleporositytypeproblems.Thegrowthfunctionsalsodependonthesmallparametercharacterizingthescaleofthemicrostructure.Thehomogenizationresultsareobtainedbythemethodoflocalenergycharacteristics.Wealsoconsideraparabolicdoubleporositytypeproblem,whichisstudiedbycombiningthevariationalhomogenizationapproachandthetwo-scaleconvergencemethod.Resultsareillustratedwithperiodicexamples,andtheproblemofstabilityinhomogenizationisdiscussed.


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 [1][2][3][4][5][6][7][8] and the references therein. In particular, the conditions under which ∞ 0 functions are dense in 1, (⋅) 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.
Γ-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 [11][12][13][14][15] (see also [16]) focus on the variational functionals with nonstandard growth conditions. In particular, the homogenization and Γ-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 1, 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 > 0.
Another interesting example of Lagrangian with rapidly oscillating exponent was considered in [11]. Namely, for the functional 2 International Journal of Differential Equations with a smooth periodic ( ) such that ( ) > 1, it was shown that the limit functional is bounded on Sobolev-Orlicz space of functions with gradient in an log 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 Γ-convergence of such functionals in spaces.
In this paper, we are dealing mainly with the Γ-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 [17][18][19][20][21][22][23][24][25][26].
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 Γconvergence 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 Γconvergence 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.

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 Ω be a bounded Lipschitz domain in R ( ≥ 2). 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 , ∈ Ω, Notice that this property was introduced by Zhikov to avoid the so-called Lavrentiev phenomenon (see, e.g., [15]).
(1) By (⋅) (Ω) we denote the space of measurable functions in Ω such that The space (⋅) (Ω) equipped with the norm becomes a Banach space.

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 L 0 (⋅) ). A sequence of functions { } ( >0) is said to belong to the class L 0 (⋅) if this sequence possesses the following properties: (i) for any > 0, is bounded in the following sense: (ii) for any > 0, satisfies the log-Hölder continuity property; (iii) the function converges uniformly in Ω to a function 0 , where the limit function 0 is assumed to satisfy the log-Hölder continuity property.
We also recall the definition of the Γ-convergence (see, e.g., [9,10,28] and the bibliography therein). In our case this definition takes the following form.
Definition 5 (two-scale convergence). A sequence of functions V ∈ 2 (Ω ) two-scale converges to V ∈ 2 (Ω × ) if for any admissible test function ( , , ), This convergence is denoted by V ( , ) The method of the local energy characteristics, generally speaking, deals with nonperiodic structures. We often make use of the following definition.

Γ-Convergence and Homogenization of Functionals with Rapidly Oscillating Coefficients in Sobolev Spaces with Variable Exponents
Let Ω be a bounded domain in R ( ≥ 2) with a sufficiently smooth boundary and denote that = ]0, 1[ . We assume that a family of continuous functions { } ( >0) belongs to the class, L 0 (⋅) , and we also suppose that (H.1) = ( ) and = ( ) are -periodic measurable functions such that (H.2) ∈ (Ω).
For the notational convenience, we set In the space (⋅) (Ω), we define the functional : We study the asymptotic behavior of and its minimizers as → 0. Our analysis relies on the Γ-convergence approach in Sobolev spaces with variable exponents. The main result of the section is the following. where (ii) The minimizer of the functional converges to the minimizer of the functional ℎ 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 "lim inf"-inequality. The proof of this inequality is done in two main steps. First we introduce an auxiliary functional International Journal of Differential Equations 5 where ( ) = min{ ( ), 0 ( )} and prove the "lim inf"inequality for this functional. Then, at the second step, we show that the "lim inf"-inequality for the auxiliary functional implies the "lim inf"-inequality for . Then, using a special test function and the Meyers estimate (see [6]), we obtain the "lim sup"-inequality. Finally, we prove the convergence of the minimizers. This completes the proof of Theorem 7.

Homogenization of the Dirichlet Problem.
Let Ω be a bounded domain in R ( ≥ 2) with sufficiently smooth boundary. Let F be an open subset in Ω.
Here is a small parameter characterizing the scale of the microstructure. We assume that F is distributed in an asymptotically regular way in Ω and we set Let = ( ) be a continuous function defined in the domain Ω. We assume that, for any > 0, it satisfies the following conditions: (A.1.1) the function ( ) is bounded in the following sense: (A.1.2) the function satisfies the log-Hölder continuity property; (A.1.3) the function converges uniformly in Ω to a function 0 , where the limit function 0 is assumed to satisfy the log-Hölder continuity property; We consider the following variational problem: where and ∈ 1 (Ω). It is known from [1][2][3]6] that for each > 0, there exists a unique solution (minimizer) ∈ 1, (⋅) (Ω ) of problem (30). Let us extend in F by zero (keeping for it the same notation). Then we obtain the family { } ⊂ 1, (⋅) (Ω). We study the asymptotic behavior of as → 0.
Instead of the classical periodicity assumption on the microstructure of the perforated domain Ω , we impose certain conditions on the local energy characteristic of the set F . To this end we introduce ℎ an open cube centered at ∈ Ω with length equal to ℎ (0 < ≪ ℎ < 1), and we set where > 0, and the infimum is taken over V ∈ 1, (⋅) (Ω) that equal zero in F . We assume that (C.1.1) there exists a continuous function ( , ) such that for any ∈ Ω, any ∈ R, and a certain = 0 > 0, (C.1.2) there exists a constant independent of such that for any ∈ Ω, 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 F ) converges weakly in
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 ‖ ‖ 1, (⋅) (Ω ) ≤ . We extend by zero to the set F and consider { } as a sequence Hence, one can extract a subsequence { , = → 0} that converges weakly to a function ∈ 1, 0 (⋅) (Ω). 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 "lim sup"-inequality (the upper bound for the functional ). To this end, we cover the domain Ω by the cubes ℎ of length ℎ > 0 centered at the points , where { } be a 6 International Journal of Differential Equations periodic grid in Ω with a period ℎ = ℎ − ℎ 1+ / + ( ≪ ℎ ≪ 1, 0 < < + ). We associate with this covering a partition of unity { }: 0 ≤ ( ) ≤ 1; ( ) = 0 for ∉ ℎ ; |∇ ( )| ≤ ℎ −1− / + . Then we construct the function ℎ using the minimizers V = V ( ) of (32) for = . We show that, for any ∈ and, therefore, On the second step, we prove the "lim inf"-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 ∈ 1, 0 (⋅) (Ω) such that = 0 on Ω. This means that any weak limit of the solution of problem (30) extended to the set F 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].

Nonlocal Effects in Homogenization of ( )-Laplacian in Perforated Domains.
Let Ω be a domain in R ( ≥ 3) defined in the previous section. Let F be an open connected subset in Ω like a net. We assume that the set F satisfies the following conditions: We assume that for any > 0, the function = ( ) satisfies the conditions (A.1.1)-(A.1.4) from Section 5.1. In the space 1, (⋅) (Ω ), we define the functional where F ( , , ∇ ) 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 > 0, there exists a unique solution ∈ 1, (⋅) (Ω ) of the variational problem (43). We extend by the equality = in F and keep for it the same notation. Thus, we obtain the family { } ( >0) ⊂ 1, (⋅) (Ω). We study the asymptotic behavior of the family { } ( >0) as → 0.
Let us introduce the functional hom : The main result of the section is the following.
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. Let Ω be a bounded Lipschitz domain in R 3 and let Ω be the subdomain of Ω defined in condition (F.1.1). Let F be an International Journal of Differential Equations 7 -periodic coordinate lattice in R 3 formed by the intersecting circular cylinders of radius ,

A Periodic
We set F = F ∩ Ω and Ω = Ω \ F . Let { } ( >0) be a class of continuous functions defined in the domain Ω. We assume that, for any > 0, satisfies the log-Hölder continuity condition and the following condition: (B.1.2) the functions are given by where N(O, ) denotes the cylindrical -neighborhood of the set O and where ℓ and ℓ are smooth strictly positive functions in Ω, moreover, max ∈Ω ℓ ( ) = (1) as → 0.
Consider the boundary value problem where is an unknown constant. It is known from [1,2] that, for each > 0, there exists a unique solution ∈ 1, (⋅) (Ω ) of problem (49). We extend by the equality = in F and keep for it the same notation. The asymptotic behavior of the solutions of problem (49) is given now by the following theorem. 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 ℓ( ) 2 by a functionl with max ∈Ωl = ( 2 ), then ( ) = 6 1 ( ) as in the linear case.

Remark 13.
It is shown in [27] Paragraph 3.3 that for the integrands of growth |∇ | 2 ln |∇ |, the 3D lattice becomes extremely thin. Moreover, for the integrands of growth |∇ | 2+ , where > 0 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 → 0. Theorem 11 shows the maximal possible polynomial growth of the integrand (in a small neighborhood of the lattice) which admits a nontrivial homogenization result. We assume that the set F consists of ( → +∞ as → 0) small isolated components such that their diameters go to zero as → 0 and F is distributed in an asymptotically regular way in Ω. We set
We study the asymptotic behavior of and their minimizers as → 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 ⊂ Ω is a disperse medium; that is, the following assumptions hold: (C.2.1) the local concentration of the set Ω has a positive continuous limit, that is, the indicator of Ω converges weakly in 2 (Ω) to a continuous positive limit. This implies that there exists a continuous positive function = ( ) such that for any open cube ℎ centered at ∈ Ω with lengths equal to ℎ > 0; (C.2.2) for any ∈ [ − , + ], there exists a family of extension operators P : 1, (Ω ) → 1, (Ω) such that.
One more condition is imposed on the local characteristic of the set F associated to the functional (53). In order to formulate this condition, we denote by ℎ an open cube centered at ∈ Ω with edge length ℎ (0 < ≪ ℎ ≪ 1) and introduce the functional where > 0, b ∈ R , and the infimum is taken over V ∈ 1, (⋅) ( ℎ ∩ Ω ). Here (⋅, ⋅) stands for the scalar product in R . We assume that (C.2.3) there is a continuous, with respect to ∈ Ω, function ( , b) and = 0 (0 < 0 < − ) such that for any { } ( >0) ⊂ L 0 (⋅) , any ∈ Ω and any b ∈ R ,

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 ( , b) does not depend on the particular choice of the sequence → 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.
The following convergence result holds.

Theorem 18. Under the assumptions of Theorem 17, the solution of the variational problem (60) converges strongly in 0 (⋅) (Ω ) 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 "lim inf"-inequality is proved in two steps as in Theorem 7 by introducing an auxiliary functional. The "lim sup"-inequality is proved by the arguments similar to those from the proof of Theorem 8.

A Periodic Example.
Theorems 17 and 18 of Section 6.1 provide sufficient conditions for the existence of the Γlimit 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 Ω be a bounded domain in R ( ≥ 2) with sufficiently smooth boundary. We assume that in the periodic cell = ]0, 1[ , there is an obstacle ⋐ being an open set with a sufficiently smooth boundary . We assume that this geometry is repeated periodically in the whole R . The geometric structure within the domain Ω is then obtained by intersecting the -multiple of this geometry with Ω. Let { k, } be an -periodic grid in R : k, = k, k ∈ Z . Then we International Journal of Differential Equations 9 define F as the union of sets F k ⊂ k obtained from by translations with vectors k, k ∈ Z , that is, and k = k + . 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 Ω is not perforated in a small neighborhood of its boundary Ω. We set Let a family of continuous functions { } ( >0) belongs to the class L 0 (⋅) . On the space (⋅) (Ω ), we define the func- where ∈ (Ω).
We study the asymptotic behavior of the functional and its minimizer as → 0. To formulate the main result of this section we will introduce some notation. We denote by b = b ( , ) a minimizer of the following variational problem: where ⋆ = \ , and b = ( 1 , 2 , . . . , ) is a vector in R , and > 1 is a parameter. If ≥ 2, then the solution b coincides with a unique solution in 1, per ( ⋆ ) of the following cell problem: here ⃗ ] is the outward normal to .
The following result holds.

Theorem 19. The sequence of functionals
where = meas ⋆ , Moreover, a minimizer of the functional (64) converges strongly in the space 0 (⋅) (Ω ) 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 Γ-convergence techniques can be applied. This allows us to obtain formula (68) by means of Γ-convergence approach used in periodic homogenization. We assume that the set Ω is distributed in an asymptotically regular way in Ω; moreover, for the sake of simplicity, we suppose that Ω ∩ Ω = 0.

Homogenization of Quasilinear Elliptic Equations with Nonstandard Growth in High-Contrast Media
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 Ω . Generally speaking, it may consist of ( → +∞ as → 0) small isolated components such that their diameters go to zero as → 0 or it may be defined as fibres becoming more and more dense as → 0, such that the diameters of the fibers go to zero as → 0.
We assume that the family of functions { } ( >0) belongs to the class L 0 (⋅) . Let ∈ (Ω) be such that 10 International Journal of Differential Equations (A.3.4) there exist two real numbers − and + such that the function is bounded in the following sense: Let us now define the variational problem under consideration. To this end, we consider the functional where Here the function is defined by ( ) def = 1 ( ) ( ), ∈ (Ω). We denote by 1 the characteristic function of the set Ω , = , .
We consider the following variational problem: It is known from [1] that for each > 0, there exists a unique solution ∈ 1, (⋅) 0 (Ω) of the variational problem (72). We aim to study the asymptotic behavior of the family { } as → 0, bearing in mind that the geometry of Ω = Ω ∪ Ω depends on . So, we have to specify this geometry. Most of the papers dealing with homogenization assume that Ω 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: (C.3.1) the local concentration of the set Ω has a positive continuous limit; that is, the indicator of Ω converges weakly in 2 (Ω) to a continuous positive limit. This implies that there exists a continuous positive function = ( ) such that for any open cube ℎ centered at ∈ Ω with lengths equal to ℎ > 0; (C.3.2) for any { } ( >0) ⊂ L 0 (⋅) , there is a constant ≥ 0 such that if the function ⋆ is defined by ⋆ ( ) = ( ) − in Ω, then: (ii) there exists a family of extension operators P : where Φ = Φ( ) is a strictly monotone continuous function in R + such that Φ(0) = 0 and Φ( ) → +∞ as → +∞.
We also impose several conditions on the local characteristic of the set Ω and Ω associated to the functional (70). Let ℎ be an open cube centered at ∈ Ω with lengths equal to ℎ (0 < ≪ ℎ ≪ 1). We introduce the following functionals: for ∈ Ω, a ∈ R , where > 0 and the infimum is taken over V ∈ 1, (⋅) ( ℎ ∩ Ω ).
(ii) The functional ,ℎ (⋅) associated to the energy exchange between Ω and Ω is defined in Ω × R by ,ℎ for ∈ Ω, ∈ R, the infimum being taken over ∈ 1, (⋅) ( ℎ ). We assume that the local characteristics of Ω are such that International Journal of Differential Equations 11 (C.3.3) for any ∈ Ω and any a ∈ R , there is a continuous function ( , a) and a real number = 0 (0 < 0 < − ) such that, for any { } ( >0) ⊂ L 0 (⋅) , (C.3.4) for any ∈ Ω and any ∈ R, there is a continuous function b( , ) and a real number . 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 ,ℎ (⋅) ( ; ) is not homogeneous with respect to the parameter . This induces the appearance of a nonlinear function b( , ) in the homogenized functional (see Theorem 23 below).
The main result of the section is the following.
the homogenized functional ℎ : where Moreover, for any smooth function in Ω, 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. ⊂ with Lipschitz boundary . We assume that this geometry is repeated periodically in the whole R . The geometric structure within the domain Ω is then obtained by intersecting the -multiple of this geometry with Ω. Let { , } be an -periodic grid in Ω. Then we define Ω as the union of sets ⊂ ( = 1, 2, . . . , ) obtained from by translations of vectors ∑ =1 e , that is,

Periodic
where is the cube centered at the point , and of length , ∈ Z, {e } =1 is the canonical basis of R , and → +∞ as → 0.
Let 0 = 0 ( ) be a log-Hölder continuous function such that where the function d is such that d = (1) as → 0. The asymptotic behavior of d will be specified below. On the space 1, (⋅) (Ω), we define the functional Consider the following variational problem: We aim to study the asymptotic behavior of the solution of (88).
To formulate the main result of this section, we introduce some notations. We denote by u = u ( , ) the unique solution in where = \ , ⃗ ] is the outward normal vector to , and a ∈ R . We denote by w = w ( , ) the unique solution in 1, 0 (⋅) # ( ) of the following cell problem: Notice that in (89) and (90) is a parameter. Regularity results for u and w are thus easily deduced from [33] and [34]. We also introduce the homogenized functional hom : the following results hold.

Theorem 24.
Let be a solution of (88). Assume that uniformly in Ω. Then converges strongly in 0 (⋅) (Ω ) to the solution of the variational problem:

Statement of the Problem and Assumptions.
In this section, we describe a mesoscopic double porosity model in a periodic fractured medium. We consider a reservoir Ω ⊂ R ( ≥ 2) to be a bounded connected domain with a periodic structure. More precisely, we will scale this periodic structure by a parameter which represents the ratio of the cell size to the size of the whole region Ω, and we will assume that is a parameter tending to zero. Let to parties of the mesoscopic domain occupied by the matrix block and the fracture, respectively. Thus = ∪ Γ , ∪ , where Γ , denotes the interface between the two media and the subscripts and refer to the matrix and fracture, respectively. Let Ω with = or denotes the open set filled with the porous medium . Then Ω = Ω ∪ Γ , ∪ Ω , where Γ , = Ω ∩ Ω . For the sake of simplicity, we will assume that Ω ∩ Ω = 0.
Let us introduce the nonstandard growth function used in this section. We assume that a family of continuous functions = ( ), > 0, is defined in Ω and satisfies the following conditions: Now let us introduce the permeability coefficient and the porosity of the porous medium Ω. We set where is the permeability or the hydraulic conductivity of fissures, is the permeability or the hydraulic conductivity of blocks, is the porosity of fissures, and is the porosity of blocks; 1 = 1 ( ) and 1 = 1 ( ) denote the characteristic functions of the sets Ω and Ω , respectively. Here 0 < , , , < +∞. We consider the following initial boundary value problem for the function : → R: where denotes the cylinder (0, ) × Ω, > 0 is given, and , 0 are given functions. For simplicity and without loss of generality, we restrict the presentation to a homogeneous Dirichlet boundary condition on Ω, but it is easy to see that all results also hold for other boundary conditions. Throughout the section, will denote a generic positive constant, independent of , and may take different values for different occurrences.

Preliminary
Results. The goal of this section is to obtain a priori estimates for the solution of problem (101). We start by formulating the existence and uniqueness result for (101). It is given by the following theorem (see [35]).
We study the asymptotic behavior of the solution of problem (101) as → 0. For this it is convenient to introduce the following notation: and to rewrite (101) separately in the domains Ω , Ω with appropriate interface conditions. Namely, in the domain Ω where ⃗ ] is the outward normal vector to Γ . In the domain Ω (101) reads Remark 31. The source term which appears in the righthand side of the first equation in (112) is well defined, since ∈ ∞ (0, ; 2 (Ω; 1 ( ))), and it follows from the third equation of (112) that −Δ ∈ 2 (0, ; −1/2 ( )) , which allows one to define (∇ ⋅ ⃗ ]) as an element of 2 (0, ; −1/2 ( )).
The Scheme of the Proof of Theorem 30 Is as Follows. We consider our parabolic boundary value problem (101) as an elliptic one depending on the time variable as a parameter. Namely, we consider the following boundary value problem, for a.e. ∈]0, [, where the function , is considered as a given function. Then, for any Δ ⊂ [0, ], minimizes the functional over ∈ ∞ (0, ; 1, (⋅) (Ω)). Using the two-scale convergence arguments, we obtain the "lim sup"-inequality for the functional . The "lim inf"-inequality is obtained in two steps. First, we introduce the auxiliary functional Then in the second step, we obtain the desired "lim inf"inequality for the initial functional. This completes the proof of Theorem 30.
The macroscopic model corresponding to the second situation is given by the following convergence result.
The proof of Theorem 32 is similar to that of Theorem 30.
Remark 33. Notice that the structure of the limit problem depends crucially on the rate of convergence of ( (⋅) − 2) to zero. The critical rate of convergence is More precisely, if lim → 0 | ln |( ( ) − 2) < +∞, then the limit model is of a double porosity type. If lim → 0 | ln | ( ( ) − 2) = +∞, then in the limit we obtain a single porosity model.
The proofs of Theorems 30 and 32 are given in [21].