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

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 𝑝(𝑥)-growth and the corresponding Sobolev spaces, see for instance [1–8] and the references therein. In particular, the conditions under which 𝐶^∞₀ functions are dense in 𝑊^1,𝑝(⋅) 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 𝑝-growth with a constant 𝑝 are well studied now, see for instance [9, 10] and the bibliography therein.

The works [11–15] (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 𝑝(𝑥) were considered in [13, 14]. It was shown that the energy minimums and the homogenized Lagrangians in the spaces 𝑊^1,𝑟 might depend on the value of 𝑟 (the so-called Lavrentiev phenomenon). For example, such a behavior can be observed for the Lagrangian |∇𝑢|^{𝑝(𝑥/𝜀)} with a periodic “chess board” exponent 𝑝(𝑦) and a small parameter 𝜀>0.

Another interesting example of Lagrangian with rapidly oscillating exponent was considered in [11]. Namely, for the functional 𝒥_𝜀[𝑢]=|∇𝑢|^{𝑝(𝑥/𝜀)}𝑑𝑥 (1) with a smooth periodic 𝑝(𝑦) such that 𝑝(𝑦)>1, it was shown that the limit functional is bounded on Sobolev-Orlicz space of functions with gradient in an 𝐿^𝛼log space, where 𝛼 is the fiber percolation level of 𝑝(𝑥).

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 𝐿^𝑝 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 [17–26].

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 Γ-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 Γ-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 Ω be a bounded Lipschitz domain in ℝ^𝑛 (𝑛≥2). We introduce the function 𝑝=𝑝(𝑥) and assume that this function is bounded such that 1<𝑝⁻=inf_Ω𝑝(𝑥)≤𝑝(𝑥)≤sup_Ω𝑝(𝑥)=𝑝⁺<+∞. (2) We also assume that the function 𝑝(𝑥) satisfies the log-Hölder continuity property. Namely, for all 𝑥,𝑦∈Ω, 𝑝(𝑥)−𝑝𝑦≤𝜔𝑥−𝑦withlim_𝜏→0𝜔(𝜏)1ln𝜏≤𝐶. (3) Notice that this property was introduced by Zhikov to avoid the so-called Lavrentiev phenomenon (see, e.g., [15]).(1)By 𝐿^𝑝(⋅)(Ω) we denote the space of measurable functions 𝑓 in Ω such that 𝐴_𝑝(⋅),Ω𝑓=_Ω𝑓(𝑥)^𝑝(𝑥)𝑑𝑥<+∞. (4) The space 𝐿^𝑝(⋅)(Ω) equipped with the norm 𝑓_{𝐿𝑝(⋅)(Ω)}=inf𝜆>0∶𝐴_𝑝(⋅),Ω𝑓𝜆≤1 (5) becomes a Banach space.(2)The following inequalities hold: min𝑓^𝑝−_{𝐿𝑝(⋅)(Ω)},𝑓^𝑝+_{𝐿𝑝(⋅)(Ω)}≤𝐴_𝑝(⋅),Ω𝑓≤max𝑓^𝑝−_{𝐿𝑝(⋅)(Ω)},𝑓^𝑝+_{𝐿𝑝(⋅)(Ω)},min𝐴^1/𝑝−_𝑝(⋅),Ω,𝐴^1/𝑝+_𝑝(⋅),Ω≤𝑓_{𝐿𝑝(⋅)(Ω)}≤max𝐴^1/𝑝−_𝑝(⋅),Ω,𝐴^1/𝑝+_𝑝(⋅),Ω. (6)(3)Let 𝑓∈𝐿^𝑝(⋅)(Ω), 𝑔∈𝐿^𝑞(⋅)(Ω) with 1𝑝(𝑥)+1𝑞(𝑥)=1,1<𝑝⁻≤𝑝(𝑥)≤𝑝⁺<∞,1<𝑞⁻≤𝑞(𝑥)≤𝑞⁺<+∞. (7) Then the Hölder’s inequality holds 𝑓𝑔𝑑𝑥≤2𝑓_{Ω𝐿𝑝(⋅)(Ω)}𝑔_{𝐿𝑞(⋅)(Ω)}. (8)(4)According to (8), for every 1≤𝑞=const<𝑝⁻≤𝑝(𝑥)<+∞𝑓_𝐿𝑞(Ω)≤𝐶𝑓_{𝐿𝑝(⋅)(Ω)}withtheconstant𝐶=2‖1‖_{𝑝(⋅)/(𝐿}^{𝑝(⋅)−𝑞}_(Ω)). (9) It is straightforward to check that for domains Ω such that measΩ<+∞, ‖1‖_𝐿^𝑝(⋅)_(Ω)≤2max[measΩ]^2/𝑝−,[measΩ]^1/2𝑝+. (10)(5)The space 𝑊^1,𝑝(⋅)(Ω), 𝑝(⋅)∈[𝑝⁻,𝑝⁺]⊂]1,+∞[, is defined by 𝑊^1,𝑝(⋅)(Ω)=𝑓∈𝐿^𝑝(⋅)(Ω)∶∇𝑓∈𝐿^𝑝(⋅)(Ω). (11) If condition (3) is satisfied, 𝑊₀^1,𝑝(⋅)(Ω) is the closure of the set 𝐶^∞₀(Ω) with respect to the norm of 𝑊^1,𝑝(⋅)(Ω). If the boundary of Ω is Lipschitz continuous and 𝑝(𝑥) satisfies (3), then 𝐶^∞₀(Ω) is dense in 𝑊₀^1,𝑝(⋅)(Ω). The norm in the space 𝑊₀^1,𝑝(⋅) is defined by ‖𝑢‖_𝑊0^1,𝑝(⋅)=_𝑖𝐷_𝑖𝑢_{𝐿𝑝(⋅)(Ω)}+‖𝑢‖_𝐿^𝑝(⋅)_(Ω). (12) If the boundary of Ω is Lipschitz and 𝑝∈𝐶⁰(Ω), then the norm ‖⋅‖_𝑊0^1,𝑝(⋅)_(Ω) is equivalent to the norm ‖𝑢^∼‖_𝑊0^1,𝑝(𝑥)_(Ω)=_𝑖𝐷_𝑖𝑢_{𝐿𝑝(⋅)(Ω)}. (13)(6)If 𝑝∈𝐶⁰(Ω), then 𝑊^1,𝑝(⋅)(Ω) is separable and reflexive.(7)If 𝑝,𝑞∈𝐶⁰(Ω), 𝑝_∗(𝑥)=𝑝(𝑥)𝑛(𝑥)(𝑥)(𝑥)𝑛−𝑝if𝑝<𝑛,and1<𝑞≤sup_Ω𝑞(𝑥)<inf_Ω𝑝_∗(𝑥),+∞if𝑝(𝑥)>𝑛, (14) then the embedding 𝑊₀^1,𝑝(⋅)(Ω)↪𝐿^𝑞(⋅)(Ω) is continuous and compact.(8)Friedrich’s inequality is valid in the following form: if 𝑝(𝑥) satisfies conditions (2) and (3), then there exists a constant 𝐶>0 such that for every 𝑓∈𝑊₀^1,𝑝(⋅)(Ω)𝑓_{𝐿𝑝(⋅)(Ω)}≤𝐶∇𝑓_{𝐿𝑝(⋅)(Ω)}. (15)

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 𝑝_𝜀, where 𝜀 is a small positive parameter characterizing the microscopic length scale.

Definition 1 (class 𝔏^𝜀_𝑝0(⋅)). 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:(i)for any 𝜀>0, 𝑝_𝜀 is bounded in the following sense: 1<𝑝⁻≤𝑝⁻_𝜀≡min_𝑥∈Ω𝑝_𝜀(𝑥)≤𝑝_𝜀(𝑥)≤max_𝑥∈Ω𝑝_𝜀(𝑥)≡𝑝⁺_𝜀≤𝑝⁺<+∞inΩ; (16)(ii)for any 𝜀>0, 𝑝_𝜀 satisfies the log-Hölder continuity property;(iii)the function 𝑝_𝜀 converges uniformly in Ω to a function 𝑝₀, where the limit function 𝑝₀ 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 2 (Γ_𝑝(⋅)-convergence). The functional 𝐼^𝜀∶𝐿^{𝑝𝜀(⋅)}(Ω)→ℝ∪{∞} is said to Γ_𝑝0(⋅)-converge to a functional 𝐼∶𝐿^𝑝0(⋅)(Ω)→ℝ∪{∞} if(a)(“liminf”-inequality) for any 𝑢∈𝐿^𝑝0(⋅)(Ω) and any sequence {𝑢^𝜀}_𝜀>0⊂𝐿^𝑝0(⋅)(Ω) which converges to the function 𝑢 strongly in the space 𝐿^𝑝0(⋅)(Ω) we have lim_𝜀→0𝐼^𝜀𝑢^𝜀≥𝐼[𝑢], (17)(b) (“limsup”-inequality) for any 𝑢∈𝐿^𝑝0(⋅)(Ω), there exists a sequence {𝑤^𝜀}_𝜀>0⊂𝐿^𝑝0(⋅)(Ω) such that {𝑤^𝜀}_𝜀>0 converges to the function 𝑢(⋅) strongly in the space 𝐿^𝑝0(⋅)(Ω), and lim_𝜀→0𝐼^𝜀𝑤^𝜀≤𝐼[𝑢]. (18)

We define the strong convergence in 𝐿^𝑝0(⋅)(Ω^𝜀) in the following way.

Definition 3 (strong convergence in 𝐿^𝑝0(⋅)(Ω^𝜀)). The sequence {𝜔^𝜀}_𝜀>0∈𝐿^𝑝0(⋅)(Ω^𝜀) is said to converge strongly in the space 𝐿^𝑝0(⋅)(Ω^𝜀) to a function 𝜔∈𝐿^𝑝0(⋅)(Ω) if lim_𝜀→0𝜔^𝜀−𝜔_{𝐿𝑝0(⋅)(Ω𝜀)}=0. (19)

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

Definition 4. Let 𝑌=]0,1[^𝑛 be a basic cell and Ω_𝑇=Ω×]0,𝑇[. A function, 𝜑∈𝐿²(Ω_𝑇;𝐶^∞_#(𝑌)), which is 𝑌-periodic in 𝑦 and which satisfies lim_{𝜀→0Ω𝑇}𝜑𝑥𝑥,𝜀,𝑡²𝑑𝑥𝑑𝑡=_Ω𝑇×𝑌𝜑𝑥,𝑦,𝑡²𝑑𝑦𝑑𝑥𝑑𝑡 (20) is called an admissible test function.

Definition 5 (two-scale convergence). A sequence of functions 𝑣^𝜀∈𝐿²(Ω_𝑇) two-scale converges to 𝑣∈𝐿²(Ω_𝑇×𝑌) if for any admissible test function 𝜑(𝑥,𝑦,𝑡), lim_{𝜀→0Ω𝑇}𝑣^𝜀()𝜑𝑥𝑥,𝑡𝑥,𝜀,𝑡𝑑𝑥𝑑𝑡=_Ω𝑇×𝑌𝑣𝑥,𝑦,𝑡𝜑𝑥,𝑦,𝑡𝑑𝑦𝑑𝑥𝑑𝑡. (21) This convergence is denoted by 𝑣^𝜀(𝑥,𝑡)⇀^2𝑠𝑣(𝑥,𝑦,𝑡).

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 ℱ^𝜀 is said to be distributed in an asymptotically regular way in Ω, if for any ball 𝑉(𝑦,𝑟) of radius 𝑟 centered at 𝑦∈Ω and 𝜀>0 small enough (𝜀≤𝜀₀(𝑟)), 𝑉(𝑦,𝑟)∩ℱ^𝜀≠∅ and 𝑉(𝑦,𝑟)∩(Ω⧵ℱ^𝜀)≠∅.

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

Let Ω be a bounded domain in ℝ^𝑛 (𝑛≥2) with a sufficiently smooth boundary and denote that 𝑌=]0,1[^𝑛. We assume that a family of continuous functions {𝑝_𝜀}_(𝜀>0) belongs to the class, 𝔏^𝜀_𝑝0(⋅), and we also suppose that (H.1)𝑎=𝑎(𝑦) and 𝑏=𝑏(𝑦) are 𝑌-periodic measurable functions such that 0<𝑎₀≤𝑎𝑦≤𝑎₁,0<𝑏₀≤𝑏𝑦≤𝑏₁. (22)(H.2)𝑓∈𝐶(Ω).

For the notational convenience, we set 𝑎^𝜀_𝑝𝜀(𝑥)=1𝑝_𝜀(𝑥)𝑎𝑥𝜀,𝑏^𝜀_𝑝𝜀(𝑥)=1𝑝_𝜀(𝑥)𝑏𝑥𝜀. (23)

In the space 𝐿^{𝑝𝜀(⋅)}(Ω), we define the functional 𝐽^𝜀∶𝐿^{𝑝𝜀(⋅)}(Ω)→ℝ: 𝐽^𝜀[𝑢]_def=∫_Ω𝑎^𝜀_𝑝𝜀(𝑥)|∇𝑢|^{𝑝𝜀(𝑥)}+𝑏^𝜀_𝑝𝜀(𝑥)|𝑢|^{𝑝𝜀(𝑥)}−𝑓(𝑥)𝑢𝑑𝑥,if𝑢∈𝑊^{1,𝑝𝜀(⋅)}(Ω),+∞,otherwise. (24)

We study the asymptotic behavior of 𝐽^𝜀 and its minimizers as 𝜀→0. Our analysis relies on the Γ-convergence approach in Sobolev spaces with variable exponents. The main result of the section is the following.

Theorem 7. Let assumptions (𝐻.1) and (𝐻.2) be fulfilled. Then (i)The functional 𝐽^𝜀, Γ_𝑝0(⋅)-converges to the functional 𝐽_ℎ𝑜𝑚∶𝐿^𝑝0(⋅)(Ω)→ℝ∪{+∞} given by 𝐽_ℎ𝑜𝑚[𝑢]=∫_Ω𝖳(𝑥,∇𝑢)+𝛽_𝑝(𝑥)|𝑢|^𝑝0(𝑥)−𝑓(𝑥)𝑢𝑑𝑥,𝑖𝑓𝑢∈𝑊^1,𝑝0(Ω),+∞,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒, (25) where 𝖳()1𝑥,𝐛=inf𝑝₀(𝑥)_𝑌𝑎𝑦∇𝑣𝑦+𝐛^𝑝0(𝑥)𝑑𝑦∶𝑣∈𝑊^1,𝑝0_#^(𝑥)(𝑌,𝛽_𝑝)(𝑥)=1𝑝₀(𝑥)_𝑌𝑏𝑦𝑑𝑦. (26)(ii)The minimizer 𝑢^𝜀 of the functional 𝐽^𝜀 converges to the minimizer 𝑢 of the functional 𝐽_ℎ𝑜𝑚 strongly in the space 𝐿^𝑝0(⋅)(Ω).

The Scheme of the Proof of Theorem 7 (See [18] for More Details) Is as Follows. We start our analysis by proving the “liminf”-inequality. The proof of this inequality is done in two main steps. First we introduce an auxiliary functional 𝐽^∼𝜀∶𝐿^{𝜋𝜀(⋅)}(Ω)→ℝ: 𝐽^∼𝜀[𝑢]=∫_Ω𝑎^𝜀_𝑝𝜀(𝑥)|∇𝑢|^{𝜋𝜀(𝑥)}+𝑏^𝜀_𝑝𝜀(𝑥)|𝑢|^{𝑝𝜀(𝑥)}−𝑓(𝑥)𝑢𝑑𝑥,𝑖𝑓𝑢∈𝑊^{1,𝜋𝜀(⋅)}(Ω);+∞,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒, (27)

where 𝜋_𝜀(𝑥)=min{𝑝_𝜀(𝑥),𝑝₀(𝑥)} and prove the “liminf”-inequality for this functional. Then, at the second step, we show that the “liminf”-inequality for the auxiliary functional 𝐽^∼𝜀 implies the “liminf”-inequality for 𝐽^𝜀. Then, using a special test function and the Meyers estimate (see [6]), we obtain the “limsup”-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 Ω be a bounded domain in ℝ^𝑛(𝑛≥2) with sufficiently smooth boundary. Let ℱ^𝜀 be an open subset in Ω. Here 𝜀 is a small parameter characterizing the scale of the microstructure. We assume that ℱ^𝜀 is distributed in an asymptotically regular way in Ω and we set Ω^𝜀=Ω⧵ℱ^𝜀. (28)

Let 𝑝_𝜀=𝑝_𝜀(𝑥) be a continuous function defined in the domain Ω. We assume that, for any 𝜀>0, it satisfies the following conditions:(A.1.1) the function 𝑝_𝜀(𝑥) is bounded in the following sense: 1<𝑝⁻≤𝑝⁻_𝜀≡min_𝑥∈Ω𝑝_𝜀(𝑥)≤𝑝_𝜀(𝑥)≤max_𝑥∈Ω𝑝_𝜀(𝑥)≡𝑝⁺_𝜀≤𝑝⁺≤𝑛inΩ; (29)(A.1.2) the function 𝑝_𝜀 satisfies the log-Hölder continuity property;(A.1.3) the function 𝑝_𝜀 converges uniformly in Ω to a function 𝑝₀, where the limit function 𝑝₀ is assumed to satisfy the log-Hölder continuity property;(A.1.4) the function 𝑝_𝜀 is such that 𝑝_𝜀(𝑥)≥𝑝₀(𝑥) in Ω.

We consider the following variational problem: 𝐽^𝜀[𝑢]=^def_Ω𝜀𝖥_𝜀(𝑥,𝑢,∇𝑢)𝑑𝑥⟶min,𝑢∈𝑊^1,𝑝𝜀₀^(⋅)Ω^𝜀, (30) where 𝖥_𝜀()=1𝑥,𝑢,∇𝑢𝑝_𝜀(𝑥)|∇𝑢|^{𝑝𝜀(𝑥)}+1𝑝_𝜀(𝑥)|𝑢|^{𝑝𝜀(𝑥)}−𝑓(𝑥)𝑢, (31) and 𝑓∈𝐶¹(Ω). It is known from [1–3, 6] that for each 𝜀>0, there exists a unique solution (minimizer) 𝑢^𝜀∈𝑊^{1,𝑝𝜀(⋅)}(Ω^𝜀) of problem (30). Let us extend 𝑢^𝜀 in ℱ^𝜀 by zero (keeping for it the same notation). Then we obtain the family {𝑢^𝜀}⊂𝑊^{1,𝑝𝜀(⋅)}(Ω). We study the asymptotic behavior of 𝑢^𝜀 as 𝜀→0.

Instead of the classical periodicity assumption on the microstructure of the perforated domain Ω^𝜀, we impose certain conditions on the local energy characteristic of the set ℱ^𝜀. To this end we introduce 𝐾^𝑧_ℎ an open cube centered at 𝑧∈Ω with length equal to ℎ (0<𝜀≪ℎ<1), and we set 𝑐^𝜀,ℎ(𝑧,𝑏)=inf_{𝑣𝜀𝐾𝑧ℎ}1𝑝_𝜀(𝑥)∇𝑣^{𝜀𝑝𝜀(𝑥)}+ℎ^{−𝑝+−𝛾}𝑣^𝜀−𝑏^{𝑝𝜀(𝑥)}𝑑𝑥, (32) where 𝛾>0, and the infimum is taken over 𝑣^𝜀∈𝑊^{1,𝑝𝜀(⋅)}(Ω) that equal zero in ℱ^𝜀. We assume that(C.1.1) there exists a continuous function 𝑐(𝑥,𝑏) such that for any 𝑥∈Ω, any 𝑏∈ℝ, and a certain 𝛾=𝛾₀>0, lim_ℎ→0lim_𝜀→0ℎ^−𝑛𝑐^𝜀,ℎ(𝑧,𝑏)=lim_ℎ→0lim_𝜀→0ℎ^−𝑛𝑐^𝜀,ℎ(𝑧,𝑏)=𝑐(𝑥,𝑏); (33)(C.1.2) there exists a constant 𝐴 independent of 𝜀 such that for any 𝑥∈Ω, lim_ℎ→0lim_𝜀→0ℎ^−𝑛𝑐^𝜀,ℎ(𝑧,𝑏)≤𝐴1+|𝑏|^𝑝0(𝑥). (34)

The first main result of Section 5 is the following.

Theorem 8. Let conditions (A.1.1)–(A.1.4) on the function 𝑝_𝜀 and conditions (C.1.1) and (C.1.2) on the local characteristic be satisfied. Then 𝑢^𝜀 the solution (minimizer) of the variational problem (30) (extended by zero in ℱ^𝜀) converges weakly in 𝑊^1,𝑝0(⋅)(Ω) to 𝑢 the solution (minimizer) of 𝐽_ℎ𝑜𝑚[𝑢]=^def_Ω𝖥₀(𝑥,𝑢,∇𝑢)+𝑐(𝑥,𝑢)−𝑓(𝑥)𝑢𝑑𝑥⟶min,𝑢∈𝑊^1,𝑝0₀^(⋅)(Ω), (35) where 𝖥₀(𝑥,𝑤,∇𝑤)=^def1𝑝₀(𝑥)|∇𝑤|^𝑝0(𝑥)+1𝑝₀(𝑥)|𝑤|^𝑝0(𝑥)−𝑔(𝑥)𝑤. (36)

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 𝑓,𝑝_𝜀 that ‖𝑢^𝜀‖_𝑊𝜀^1,𝑝(⋅)_(Ω^𝜀₎≤𝐶. We extend 𝑢^𝜀 by zero to the set ℱ^𝜀 and consider {𝑢^𝜀} as a sequence in 𝑊^{1,𝑝𝜀(⋅)}(Ω). Then ‖𝑢^𝜀‖_𝑊𝜀^1,𝑝(⋅)_(Ω)≤𝐶. Condition (A.1.4) immediately implies that 𝑢^𝜀_{𝑊01,𝑝(⋅)(Ω)}≤𝐶. (37) Hence, one can extract a subsequence {𝑢^𝜀,𝜀=𝜀_𝑘→0} that converges weakly to a function 𝑢∈𝑊^1,𝑝0(⋅)(Ω). We will show that 𝑢 is the solution of the variational problem (35). The proof is done in two mains steps. On the first step, we prove the “limsup”-inequality (the upper bound for the functional 𝐽^𝜀). To this end, we cover the domain Ω by the cubes 𝐾^𝛼_ℎ of length ℎ>0 centered at the points 𝑥^𝛼, where {𝑥^𝛼} be a periodic grid in Ω with a period ℎ^′=ℎ−ℎ^1+𝛾/𝑝+ (𝜀≪ℎ≪1, 0<𝛾<𝑝⁺). We associate with this covering a partition of unity {𝜑_𝛼}: 0≤𝜑_𝛼(𝑥)≤1; 𝜑_𝛼(𝑥)=0 for 𝑥∉𝐾^𝛼_ℎ; 𝜑_𝛼(𝑥)=1 for 𝑥∈𝐾^𝛼_ℎ⧵∪_𝛽≠𝛼𝐾^𝛽_ℎ; ∑_𝛼𝜑_𝛼(𝑥)=1 for 𝑥∈Ω; |∇𝜑_𝛼(𝑥)|≤𝐶ℎ^{−1−𝛾/𝑝+}. Then we construct the function 𝑤^𝜀_ℎ using the minimizers 𝑣^𝜀_𝛼=𝑣^𝜀_𝛼(𝑥) of (32) for 𝑧=𝑥^𝛼. We show that, for any 𝑤∈𝑊^1,𝑝0₀^(⋅)(Ω), lim_ℎ→0lim_𝜀→0𝐽^𝜀𝑤^𝜀_ℎ≤𝐽_hom[𝑤], (38) and, therefore, lim_𝜀→0𝐽^𝜀𝑢^𝜀≤𝐽_hom[𝑤]. (39) On the second step, we prove the “liminf”-inequality (the lower bound for the functional 𝐽^𝜀) using the definition of the local energy characteristic and the convexity of our functional: lim_{𝜀=𝜀𝑘→0}𝐽^𝜀𝑢^𝜀≥𝐽_hom[𝑢]. (40) Finally it follows from (39) and (40) that 𝐽_hom[𝑢]≤𝐽_hom[𝑤], (41) for any 𝑤∈𝑊^1,𝑝0(⋅)(Ω) such that 𝑤=0 on 𝜕Ω. This means that any weak limit of the solution of problem (30) extended to the set ℱ^𝜀 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 𝑝_𝜀(𝑥)-Laplacian in Perforated Domains

Let Ω be a domain in ℝ^𝑛 (𝑛≥3) defined in the previous section. Let ℱ^𝜀 be an open connected subset in Ω like a net. We assume that the set ℱ^𝜀 satisfies the following conditions: (F.1.1)ℱ^𝜀⋐Ω_𝑑, where Ω_𝑑={𝑥∈Ω∶dist(𝑥,𝜕Ω)≥𝑑} with 𝑑>0 independent of 𝜀;(F.1.2) ℱ^𝜀 is distributed in an asymptotically regular way in Ω_𝑑;(F.1.3) the subdomain Ω=^𝜀defΩ⧵ℱ^𝜀 is a connected set.

We assume that for any 𝜀>0, the function 𝑝_𝜀=𝑝_𝜀(𝑥) satisfies the conditions (A.1.1)–(A.1.4) from Section 5.1. In the space 𝑊^{1,𝑝𝜀(⋅)}(Ω^𝜀), we define the functional 𝐽^𝜀∶𝑊^{1,𝑝𝜀(⋅)}(Ω^𝜀)→ℝ, 𝐽^𝜀[𝑢]=∫_Ω𝜀𝖥_𝜀(𝑥,𝑢,∇𝑢)𝑑𝑥if𝑢∈𝑊^{1,𝑝𝜀(⋅)}Ω^𝜀,+∞otherwise, (42) where 𝖥_𝜀(𝑥,𝑢,∇𝑢) is defined in (31) and we consider the following variational problem: 𝐽^𝜀𝑢^𝜀⟶inf,𝑢^𝜀∈𝑊^{1,𝑝𝜀(⋅)}Ω^𝜀;𝑢^𝜀=𝐴^𝜀on𝜕ℱ^𝜀,𝑢^𝜀=0on𝜕Ω, (43) where 𝐴^𝜀 is an unknown constant. It is known from [1, 2] that for each 𝜀>0, there exists a unique solution 𝑢^𝜀∈𝑊^{1,𝑝𝜀(⋅)}(Ω^𝜀) of the variational problem (43). We extend 𝑢^𝜀 by the equality 𝑢^𝜀=𝐴^𝜀 in ℱ^𝜀 and keep for it the same notation. Thus, we obtain the family {𝑢^𝜀}_(𝜀>0)⊂𝑊^{1,𝑝𝜀(⋅)}(Ω). We study the asymptotic behavior of the family {𝑢^𝜀}_(𝜀>0) as 𝜀→0.

Let us introduce the functional 𝐽_hom∶𝑊^1,𝑝0(⋅)(Ω)→ℝ, 𝐽_hom[]=∫𝑢,𝐴_Ω𝖥₀(𝑥,𝑢,∇𝑢)+𝟏_𝑑(𝑥)𝑐(𝑥,𝑢−𝐴)𝑑𝑥if𝑢∈𝑊^1,𝑝0(⋅)(Ω),+∞otherwise, (44) where 𝟏_𝑑=𝟏_𝑑(𝑥) is the characteristic function of the domain Ω_𝑑 and 𝖥₀(𝑥,𝑢,∇𝑢) is defined in (36).

The main result of the section is the following.

Theorem 9. Let 𝑢^𝜀 be a solution of (43) extended by the equality 𝑢^𝜀(𝑥)=𝐴^𝜀 in ℱ^𝜀. 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 ℱ^𝜀, the functions 𝑝_𝜀, and the local characteristic 𝑐^𝜀,ℎ(𝑥,𝑏) are satisfied. Then there is a subsequence {𝑢^𝜀,𝜀=𝜀_𝑘→0} that converges weakly in 𝑊^1,𝑝0(⋅)(Ω) to a function 𝑢 such that the pair {𝑢,𝐴} is a solution of 𝐽_ℎ𝑜𝑚[𝑢,𝐴]⟶inf,𝑢∈𝑊^1,𝑝0₀^(⋅)(Ω)w𝑖𝑡ℎ𝐴=lim_𝜀→0𝐴^𝜀. (45)

Remark 10. It is important to notice that the constant 𝐴 in (45) remains unknown. Suppose, in addition, that the function 𝑐(𝑥,𝑏) is differentiable with respect to the argument 𝑏. Then it is easy to see that Euler’s equation for the homogenized problem (45) reads −div|∇𝑢|^{𝑝0(𝑥)−2}∇𝑢+|𝑢|^{𝑝0(𝑥)−2}𝑢+𝟏_𝑑(𝑥)𝑐^′_𝑢()(𝑥)𝑥,𝑢−𝐴=𝑓inΩ;𝑢=0on𝜕Ω,_Ω𝑑𝑐^′_𝑢(𝑥,𝑢−𝐴)𝑑𝑥=0, (46) where 𝑐^′_𝑢 denotes the partial derivative of the function 𝑐 with respect to 𝑢. 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 𝑐(𝑥,𝑢) and the constant 𝐴 in the homogenized problem (45) explicitly.

Let Ω be a bounded Lipschitz domain in ℝ³ and let Ω_𝑑 be the subdomain of Ω defined in condition (F.1.1). Let 𝔉^𝜀 be an 𝜀-periodic coordinate lattice in ℝ³ formed by the intersecting circular cylinders of radius 𝑟_𝜀, 𝑟_𝜀=𝑒^−1/𝜀2. (47) We set ℱ^𝜀=𝔉^𝜀∩Ω_𝑑 and Ω^𝜀=Ω⧵ℱ^𝜀.

Let {𝑝_𝜀}_(𝜀>0) be a class of continuous functions defined in the domain Ω. We assume that, for any 𝜀>0, 𝑝_𝜀 satisfies the log-Hölder continuity condition and the following condition: (B.1.2) the functions 𝑝_𝜀 are given by 𝑝_𝜀(𝑥)=2+ℓ(𝑥)𝜀²in𝒩ℱ^𝜀,𝜀²,2+ℓ_𝜀(𝑥)elsewhere, (48)where 𝒩(𝒪,𝛿) denotes the cylindrical 𝛿-neighborhood of the set 𝒪 and where ℓ and ℓ_𝜀 are smooth strictly positive functions in Ω, moreover, max_𝑥∈Ωℓ_𝜀(𝑥)=𝑜(1) as 𝜀→0.

Consider the boundary value problem −div∇𝑢^{𝜀𝑝𝜀(𝑥)−2}∇𝑢^𝜀+𝑢^{𝜀𝑝𝜀(𝑥)−2}𝑢^𝜀=𝑓(𝑥)inΩ^𝜀,𝑢^𝜀=𝐴^𝜀on𝜕ℱ^𝜀;𝑢^𝜀=0on𝜕Ω,_𝜕ℱ𝜀∇𝑢^{𝜀𝑝𝜀(𝑥)−2}𝜕𝑢^𝜀𝜕𝜈𝑑𝑠=0, (49) where 𝐴^𝜀 is an unknown constant. It is known from [1, 2] that, for each 𝜀>0, there exists a unique solution 𝑢^𝜀∈𝑊^{1,𝑝𝜀(⋅)}(Ω^𝜀) of problem (49). We extend 𝑢^𝜀 by the equality 𝑢^𝜀=𝐴^𝜀 in ℱ^𝜀 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 𝑢^𝜀 be a solution of (49) extended by the equality 𝑢^𝜀(𝑥)=𝐴^𝜀 in ℱ^𝜀. Assume that the condition (B.1.2) on the function 𝑝_𝜀 is satisfied. Then the sequence {𝑢^𝜀}_(𝜀>0) converges weakly in 𝐻¹(Ω) to a function 𝑢 solution of −Δ𝑢+𝑢+𝜇(𝑥)𝑢−𝖠_ℓ(𝑢)=𝑔(𝑥)𝑖𝑛Ω,𝑢=0𝑜𝑛𝜕Ω, (50) where 𝜇(𝑥)=6𝜋𝟏_𝑑(𝑥)𝑒^ℓ(𝑥)−1ℓ(𝑥),𝖠_ℓ(𝑢)=_Ω𝑑𝜇(𝑥)𝑑𝑥⁻¹_Ω𝑑𝜇(𝑥)𝑢(𝑥)𝑑𝑥. (51)

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(𝑥)𝜀² by a function ℓ^∼_𝜀 with maxℓ_𝑥∈Ω^∼_𝜀=𝑜(𝜀²), then 𝜇(𝑥)=6𝜋𝟏_𝑑(𝑥) as in the linear case.

Remark 13. It is shown in [27] Paragraph 3.3 that for the integrands of growth |∇𝑢^𝜀|²ln|∇𝑢^𝜀|, the 3D lattice becomes extremely thin. Moreover, for the integrands of growth |∇𝑢^𝜀|^2+𝛿, where 𝛿>0 is a parameter independent of 𝜀, there is no 3D lattice which admits nontrivial homogenization result, because the capacity of the lattice goes to infinity as 𝜀→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.

6. Homogenization of the Neumann Problem

6.1. 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 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 and ℱ^𝜀 is distributed in an asymptotically regular way in Ω. We set Ω^𝜀=Ω⧵ℱ^𝜀. (52)

We assume that a family of continuous functions {𝑝_𝜀}_(𝜀>0) belongs to the class 𝔏^𝜀_𝑝0(⋅). On the space 𝐿^{𝑝𝜀(⋅)}(Ω^𝜀), we define the functional 𝐽^𝜀∶𝐿^{𝑝𝜀(⋅)}(Ω^𝜀)→ℝ∪{+∞}: 𝐽^𝜀[𝑢]=∫_Ω𝜀𝖥_𝜀(𝑥,𝑢,∇𝑢)𝑑𝑥,if𝑢∈𝑊^{1,𝑝𝜀(⋅)}Ω^𝜀,+∞,otherwise, (53) where 𝖥_𝜀(𝑥,𝑢,∇𝑢) is defined in (31) and 𝑓∈𝐶(Ω).

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; that is, the following assumptions hold: (C.2.1) the local concentration of the set Ω^𝜀 has a positive continuous limit, that is, the indicator of Ω^𝜀 converges weakly in 𝐿²(Ω) to a continuous positive limit. This implies that there exists a continuous positive function 𝜌=𝜌(𝑥) such that lim_ℎ→0lim_𝜀→0ℎ^−𝑛meas𝐾^𝑥_ℎ∩Ω^𝜀=𝜌(𝑥), (54) for any open cube 𝐾^𝑥_ℎ centered at 𝑥∈Ω with lengths equal to ℎ>0;(C.2.2) 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), 𝗎^𝜀_𝐿𝑞(Ω)≤𝐶𝑢^𝜀_{𝐿𝑞(Ω𝜀)},∇𝗎^𝜀_𝐿𝑞(Ω)≤𝐶∇𝑢^𝜀_{𝐿𝑞(Ω𝜀)}, (55) uniformly in 𝜀>0, where 𝗎^𝜀=𝖯^𝜀𝑢^𝜀 and 𝗎^𝜀=𝑢^𝜀 in Ω^𝜀; local: for any ℎ>0 there is 𝜀₀(ℎ)>0 such that for all 𝜀<𝜀₀(ℎ), 𝑧∈Ω and any function 𝑢∈𝑊^1,𝑞((𝑧+[−2ℎ,2ℎ]^𝑛)∩Ω^𝜀), the estimates hold 𝗎^𝜀_{𝐿𝑞((𝑧+[−ℎ,ℎ]}^𝑛_)∩Ω)≤𝐶𝑢^𝜀_{𝐿𝑞((𝑧+[−2ℎ,2ℎ]}^𝑛_)∩Ω𝜀),∇𝗎^𝜀_{𝐿𝑞((𝑧+[−ℎ,ℎ]}^𝑛_)∩Ω)≤𝐶∇𝑢^𝜀_{𝐿𝑞((𝑧+[−2ℎ,2ℎ]}^𝑛_)∩Ω𝜀). (56)

Remark 14. Notice that in condition (C.2.2), 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 mathematical literature (see, e.g., [27, 30–32]). 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 ℱ^𝜀 associated to the functional (53). 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 𝑐_𝑝^𝜀,ℎ_𝜀(⋅)(𝑧,𝐛)=inf_{𝑣𝜀𝐾𝑧ℎ∩Ω𝜀}1𝑝_𝜀(𝑥)∇𝑣^{𝜀𝑝𝜀(𝑥)}+ℎ^{−𝑝𝜀(𝑥)−𝛾}𝑣^𝜀−(𝑥−𝑧,𝐛)^{𝑝𝜀(𝑥)}𝑑𝑥, (57) 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(C.2.3) there is a continuous, with respect to 𝑥∈Ω, function 𝑇(𝑥,𝐛) and 𝛾=𝛾₀ (0<𝛾₀<𝑝⁻) such that for any {𝑝_𝜀}_(𝜀>0)⊂𝔏^𝜀_𝑝0(⋅), any 𝑥∈Ω and any 𝐛∈ℝ^𝑛, lim_ℎ→0lim_𝜀→0ℎ^−𝑛𝑐_𝑝^𝜀,ℎ_𝜀(⋅)(𝑥,𝐛)=lim_ℎ→0lim_𝜀→0ℎ^−𝑛𝑐_𝑝^𝜀,ℎ_𝜀(⋅)(𝑥,𝐛)=𝑇(𝑥,𝐛). (58)

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 𝑇(𝑥,𝐛) does not depend on the particular choice of the sequence 𝑝_𝜀→𝑝₀. 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 {𝑝_𝜀}_(𝜀>0)⊂𝔏^𝜀_𝑝0(⋅), and let conditions (C.2.1)–(C.2.3) be satisfied. Then the functionals 𝐽^𝜀 defined in (53), Γ_𝑝0(⋅)-converge to the functional 𝐽_ℎ𝑜𝑚∶𝐿^𝑝0(⋅)(Ω)→ℝ∪{+∞} given by 𝐽_ℎ𝑜𝑚[𝑢]=∫_Ω𝑇()+𝜌(𝑥)𝑥,∇𝑢𝑝₀(𝑥)|𝑢|^𝑝0(𝑥)−𝜌(𝑥)𝑓(𝑥)𝑢𝑑𝑥,𝑖𝑓𝑢∈𝑊^1,𝑝0(⋅)(Ω),+∞,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (59)

Now let us formulate the convergence result for the minimizers of the functionals 𝐽^𝜀. Consider the variational problem 𝐽^𝜀𝑢^𝜀⟶min,𝑢^𝜀∈𝑊^{1,𝑝𝜀(⋅)}Ω^𝜀. (60) According to [1–3, 6], for each 𝜀>0, problem (60) has a unique solution 𝑢^𝜀∈𝑊^{1,𝑝𝜀(⋅)}(Ω^𝜀).

The following convergence result holds.

Theorem 18. Under the assumptions of Theorem 17, the solution 𝑢^𝜀 of the variational problem (60) converges strongly in 𝐿^𝑝0(⋅)(Ω^𝜀) to a solution of the problem 𝐽_ℎ𝑜𝑚[𝑢]⟶min,𝑢∈𝑊^1,𝑝0(⋅)(Ω). (61)

The Scheme of the Proof of Theorems 17 and 18 (See [22] for More Details) Is as Follows. The “liminf”-inequality is proved in two steps as in Theorem 7 by introducing an auxiliary functional. The “limsup”-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 Γ-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 Ω be a bounded domain in ℝ^𝑛 (𝑛≥2) with sufficiently smooth boundary. We assume that in the periodic cell 𝑌=]0,1[^𝑛, there is an obstacle 𝐹⋐𝑌 being an open set with a sufficiently smooth boundary 𝜕𝐹. 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. 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, that is, ℱ^𝜀_𝐤=𝜀𝐤+𝜀𝐹,ℱ^𝜀=_𝐤ℱ^𝜀_𝐤∩Ω, (62) 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 (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 Ω=^𝜀defΩ⧵ℱ^𝜀. (63)

Let a family of continuous functions {𝑝_𝜀}_(𝜀>0) belongs to the class 𝔏^𝜀_𝑝0(⋅). On the space 𝐿^{𝑝𝜀(⋅)}(Ω^𝜀), we define the functional 𝐽^𝜀∶𝐿^{𝑝𝜀(⋅)}(Ω^𝜀)→ℝ∪{+∞}: 𝐽^𝜀[𝑢]=∫_Ω𝜀1𝑝_𝜀(𝑥)|∇𝑢|^{𝑝𝜀(𝑥)}+|𝑢|^{𝑝𝜀(𝑥)}−𝑓(𝑥)𝑢𝑑𝑥,if𝑢∈𝑊^{1,𝑝𝜀(⋅)}Ω^𝜀,+∞,otherwise, (64) where 𝑓∈𝐶(Ω).

We study the asymptotic behavior of the functional 𝐽^𝜀 and its minimizer as 𝜀→0. To formulate the main result of this section we will introduce some notation. We denote by 𝑈^𝐛=𝑈^𝐛(𝑝,𝑦) a minimizer of the following variational problem: _𝑌⋆1𝑝∇_𝑦𝑈^𝐛−𝐛^𝑝𝑑𝑦⟶min,𝑢∈𝑊^1,𝑝_per𝑌^⋆, (65) where 𝑌^⋆=𝑌⧵𝐹, and 𝐛=(𝑏₁,𝑏₂,…,𝑏_𝑛) is a vector in ℝ^𝑛, and 𝑝>1 is a parameter.

If 𝑝≥2, then the solution 𝑈^𝐛 coincides with a unique solution in 𝑊^1,𝑝_per(𝑌^⋆) of the following cell problem: div_𝑦∇_𝑦𝑈^𝐛𝑝−2∇_𝑦𝑈^𝐛=0in𝑌^⋆,∇_𝑦𝑈^𝐛𝑝−2∇_𝑦𝑈^𝐛−𝐛,𝜈=0on𝜕𝐹,𝑦⟶𝑈^𝐛𝑦𝑌-periodic, (66) here 𝜈 is the outward normal to 𝜕𝐹.

The following result holds.

Theorem 19. The sequence of functionals {𝐽^𝜀}_(𝜀>0) defined in (64) Γ_𝑝0(⋅)-converges to the functional 𝐽_ℎ𝑜𝑚∶𝐿^𝑝0(⋅)(Ω)→ℝ∪{+∞} given by 𝐽_ℎ𝑜𝑚[𝑢]=∫_Ω𝑇()+𝜌𝑥,∇𝑢𝑝₀(𝑥)|𝑢|^𝑝0(𝑥)−𝜌𝑓(𝑥)𝑢𝑑𝑥,𝑖𝑓𝑢∈𝑊^1,𝑝0(⋅)(Ω),+∞,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒, (67) where 𝜌=meas𝑌^⋆,𝑇(𝑥,𝐛)=_𝑌⋆1𝑝₀(𝑥)∇_𝑦𝑈^𝐛𝑝₀(𝑥),𝑦−𝐛^𝑝0(𝑥)𝑑𝑦. (68) Moreover, a minimizer 𝑢^𝜀 of the functional (64) converges strongly in the space 𝐿^𝑝0(⋅)(Ω^𝜀) to 𝑢 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.

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

7.1. Statement of the Problem and the Main Result

Let Ω=Ω^𝜀_𝑓∪Ω^𝜀_𝑚 be a bounded domain of ℝ^𝑛 (𝑛≥2) with Lipschitz boundary 𝜕Ω. Here {Ω^𝜀_𝑚}_(𝜀>0) is a family of open subsets in Ω. We assume that the set Ω^𝜀_𝑚 is distributed in an asymptotically regular way in Ω; moreover, for the sake of simplicity, we suppose that Ω^𝜀_𝑚∩𝜕Ω=∅.

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^𝜀_𝑚. Generally speaking, it may consist of 𝑁_𝜀 (𝑁_𝜀→+∞ as 𝜀→0) small isolated components such that their diameters go to zero as 𝜀→0 or it may be defined as fibres becoming more and more dense as 𝜀→0, such that the diameters of the fibers go to zero as 𝜀→0.

We assume that the family of functions {𝑝_𝜀}_(𝜀>0) belongs to the class 𝔏^𝜀_𝑝0(⋅). Let 𝜎∈𝐶(Ω) be such that(A.3.4) there exist two real numbers 𝜎⁻ and 𝜎⁺ such that the function 𝜎 is bounded in the following sense: 0<𝜎⁻≡min_𝑥∈Ω𝜎(𝑥)≤𝜎(𝑥)≤max_𝑥∈Ω𝜎(𝑥)≡𝜎⁺<min_𝑥∈Ω𝑝₀(𝑥)𝑛𝑛−𝑝₀(𝑥)inΩ, (69)(A.3.5) the function 𝜎 satisfies the log-Hölder continuity property.

Let us now define the variational problem under consideration. To this end, we consider the functional 𝐽^𝜀∶𝑊^{1,𝑝𝜀(⋅)}(Ω)→ℝ∪{+∞}, 𝐽^𝜀[𝑢]=^def∫_Ω𝖦_𝜀(𝑥,𝑢,∇𝑢)𝑑𝑥if𝑢∈𝑊^{1,𝑝𝜀(⋅)}(Ω),+∞otherwise, (70) where 𝖦_𝜀(𝑥,𝑢,∇𝑢)=^def𝜘_𝜀(𝑥)|∇𝑢|^{𝑝𝜀(𝑥)}+1𝜎(𝑥)|𝑢|^𝜎(𝑥)−𝑔^𝜀(𝑥)𝑢with𝜘_𝜀(𝑥)=^def𝐾_𝜀(𝑥)𝑝_𝜀(𝑥). (71) Here the function 𝑔^𝜀 is defined by 𝑔^𝜀(𝑥)=^def𝟏^𝜀_𝑓(𝑥)𝑔(𝑥), 𝑔∈𝐶(Ω). We denote by 𝟏^𝜀_𝑘 the characteristic function of the set Ω^𝜀_𝑘, 𝑘=𝑓,𝑚. 𝐾_𝜀 is a measurable function in Ω such that(K.1) there exists a real number 𝑘₀ such that 0<𝑘₀≤𝐾_𝜀(𝑥)≤𝑘₀⁻¹ in Ω^𝜀_𝑓;(K.2) for any 𝜀>0, there exists a real number 𝚔_𝜀 such that sup_𝑥∈Ω^𝜀_𝑚𝐾_𝜀(𝑥)=𝚔_𝜀>0 and 𝚔_𝜀→0 as 𝜀→0.

We consider the following variational problem: 𝐽^𝜀𝑢^𝜀⟶min,𝑢^𝜀∈𝑊^1,𝑝𝜀₀^(⋅)(Ω). (72) It is known from [1] that for each 𝜀>0, there exists a unique solution 𝑢^𝜀∈𝑊^1,𝑝𝜀₀^(⋅)(Ω) of the variational problem (72). We aim to study the asymptotic behavior of the family {𝑢^𝜀} as 𝜀→0, bearing in mind that the geometry of Ω=Ω^𝜀_𝑓∪Ω^𝜀_𝑚 depends on 𝜀. 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^𝜀_𝑓 has a positive continuous limit; that is, the indicator of Ω^𝜀_𝑓 converges weakly in 𝐿²(Ω) to a continuous positive limit. This implies that there exists a continuous positive function 𝜌=𝜌(𝑥) such that lim_ℎ→0lim_𝜀→0ℎ^−𝑛meas𝐾^𝑥_ℎ∩Ω^𝜀_𝑓=𝜌(𝑥) (73) for any open cube 𝐾^𝑥_ℎ centered at 𝑥∈Ω with lengths equal to ℎ>0;(C.3.2) for any {𝑝_𝜀}_(𝜀>0)⊂𝔏^𝜀_𝑝0(⋅), there is a constant 𝐶_𝑝𝜀≥0 such that if the function 𝑝^⋆_𝜀 is defined by 𝑝^⋆_𝜀(𝑥)=𝑝_𝜀(𝑥)−𝐶_𝑝𝜀 in Ω, then: (i)the sequence {𝑝^⋆_𝜀}_(𝜀>0) belongs to 𝔏^𝜀_𝑝0(⋅), that is, lim_𝜀→0𝐶_𝑝𝜀=0; (ii)there exists a family of extension operators 𝐏^𝜀:𝑊^{1,𝑝⋆𝜀(⋅)}(Ω^𝜀_𝑓)→𝑊^{1,𝑝⋆𝜀(⋅)}(Ω) such that, for any 𝑣^𝜀∈𝑊^{1,𝑝𝜀(⋅)}(Ω^𝜀_𝑓), 𝐏^𝜀𝑣^𝜀=𝑣^𝜀inΩ^𝜀_𝑓,𝐏^𝜀𝑣^𝜀_𝑊^⋆_{𝜀1,𝑝(⋅)(Ω)}≤Φ𝑣^𝜀_{𝑊𝜀1,𝑝(⋅)(Ω𝜀𝑓)}, (74) where Φ=Φ(𝑡) is a strictly monotone continuous function in ℝ⁺ such that Φ(0)=0 and Φ(𝑡)→+∞ as 𝑡→+∞.

We also impose several conditions on the local characteristic of the set Ω^𝜀_𝑓 and Ω^𝜀_𝑚 associated to the functional (70). Let 𝐾^𝑧_ℎ be an open cube centered at 𝑧∈Ω with lengths equal to ℎ (0<𝜀≪ℎ≪1). We introduce the following functionals:(i)the functional 𝑐_𝑝^𝜀,ℎ_𝜀(⋅) associated to the energy in Ω^𝜀_𝑓 is defined in Ω×ℝ^𝑛 by 𝑐_𝑝^𝜀,ℎ_𝜀(⋅)(𝑧;𝐚_def)=inf_{𝑣𝜀𝐾𝑧ℎ∩Ω𝜀𝑓}𝜘_𝜀(𝑥)∇𝑣^𝜀(𝑥)^{𝑝𝜀(𝑥)}+ℎ^{−𝑝𝜀(𝑥)−𝛾}𝑣^𝜀(𝑥)−(𝑥−𝑧,𝐚)^{𝑝𝜀(𝑥)}𝑑𝑥, (75) for 𝑧∈Ω, 𝐚∈ℝ^𝑛, where 𝛾>0 and the infimum is taken over 𝑣^𝜀∈𝑊^{1,𝑝𝜀(⋅)}(𝐾^𝑧_ℎ∩Ω^𝜀_𝑓).(ii)The functional 𝑏_𝑝^𝜀,ℎ_𝜀(⋅) associated to the energy exchange between Ω^𝜀_𝑓 and Ω^𝜀_𝑚 is defined in Ω×ℝ^𝑛 by 𝑏_𝑝^𝜀,ℎ_𝜀(⋅)𝑧;𝛽_def=inf_{𝑤𝜀𝐾𝑧ℎ}𝜘_𝜀(𝑥)∇𝑤^{𝜀𝑝𝜀(𝑥)}+𝟏^𝜀_𝑚(𝑥)𝜎(𝑥)𝑤^{𝜀𝜎(𝑥)}+ℎ^{−𝑝𝜀(𝑥)−𝛾}𝟏^𝜀_𝑓(𝑥)𝑤^𝜀−𝛽^{𝑝𝜀(𝑥)}𝑑𝑥, (76)for 𝑧∈Ω,𝛽∈ℝ, the infimum being taken over 𝑤^𝜀∈𝑊^{1,𝑝𝜀(⋅)}(𝐾^𝑧_ℎ).

We assume that the local characteristics of Ω are such that(C.3.3) for any 𝑥∈Ω and any 𝐚∈ℝ^𝑛, there is a continuous function 𝑇(𝑥,𝐚) and a real number 𝛾=𝛾₀ (0<𝛾₀<𝑝⁻) such that, for any {𝑝_𝜀}_(𝜀>0)⊂𝔏^𝜀_𝑝0(⋅), lim_ℎ→0lim_𝜀→0ℎ^−𝑛𝑐_𝑝^𝜀,ℎ_𝜀(⋅)(𝑥,𝐚)=lim_ℎ→0lim_𝜀→0ℎ^−𝑛𝑐_𝑝^𝜀,ℎ_𝜀(⋅)(𝑥,𝐚)=𝑇(𝑥,𝐚). (77)(C.3.4) for any 𝑥∈Ω and any 𝛽∈ℝ, there is a continuous function 𝖻(𝑥,𝛽) and a real number 𝛾=𝛾₁ (0<𝛾₁<𝑝⁻) such that, for any {𝑝_𝜀}_(𝜀>0)⊂𝔏^𝜀_𝑝0(⋅), lim_ℎ→0lim_𝜀→0ℎ^−𝑛𝑏_𝑝^𝜀,ℎ_𝜀(⋅)𝑥,𝛽=lim_ℎ→0lim_𝜀→0ℎ^−𝑛𝑏_𝑝^𝜀,ℎ_𝜀(⋅)𝑥,𝛽=𝖻𝑥,𝛽. (78)

Remark 21. It is crucial in conditions (C.3.3) and (C.3.4) that the limit functions 𝑇(𝑥,𝐛) and 𝖻(𝑥,𝛽) do not depend on the particular choice of the sequence {𝑝_𝜀}_(𝜀>0)⊂𝔏^𝜀_𝑝0(⋅). 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 𝑏_𝑝^𝜀,ℎ_𝜀(⋅)(𝑧;𝛽) is not homogeneous with respect to the parameter 𝛽. This induces the appearance of a nonlinear function 𝖻(𝑥,𝑢) in the homogenized functional (see Theorem 23 below).

The main result of the section is the following.

Theorem 23. Let 𝑢^𝜀 be a solution of (72). Assume that 𝑝^𝜀∈𝔏^𝜀_𝑝0(⋅) and conditions (A.3.4)–(A.3.5), (K.1)-(K.2), and (C.3.1)–(C.3.4) are satisfied. Then 𝑢^𝜀, solution of the variational problem (72), converges strongly in 𝐿^𝑝0(⋅)(Ω^𝜀_𝑓) to 𝑢, solution of the following variational problem: 𝐽_ℎ𝑜𝑚[𝑢]⟶min,𝑢∈𝑊^1,𝑝0₀^(⋅)(Ω), (79) the homogenized functional 𝐽_ℎ𝑜𝑚∶𝑊^1,𝑝0₀^(⋅)(Ω)→ℝ∪{+∞} being defined by 𝐽_ℎ𝑜𝑚[𝑢]=^def∫_Ω𝖦₀(𝑥,𝑢,∇𝑢)𝑑𝑥𝑖𝑓𝑢∈𝑊^1,𝑝0₀^(⋅)(Ω),+∞𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒, (80) where 𝖦₀(𝑥,𝑢,∇𝑢)^def()+𝜌(𝑥)=𝑇𝑥,∇𝑢𝜎(𝑥)|𝑢|^𝜎(𝑥)+𝖻(𝑥,𝑢)−𝑔(𝑥)𝜌(𝑥)𝑢. (81) Moreover, for any smooth function 𝜁 in Ω, we have lim_{𝜀→0Ω𝜀𝑚}1𝑝_𝜀(𝑥)𝑢^{𝜀𝜎(𝑥)−2}𝑢(𝑥)𝑢^𝜀−𝑢^𝜀2+1𝜎(𝑥)𝑢^{𝜀𝜎(𝑥)}𝜁(𝑥)𝑑𝑥=_Ω𝖻(𝑥,𝑢)𝜁(𝑥)𝑑𝑥. (82)

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 Ω be a bounded domain in ℝ^𝑛 (𝑛≥2) with Lipschitz boundary. We assume that in the standard periodic cell 𝑌=^def]0,1[^𝑛, there is an obstacle 𝑀⊂𝑌 with Lipschitz boundary 𝜕𝑀. 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. Let {𝑥^𝑖,𝜀} be an 𝜀-periodic grid in Ω. Then we define Ω^𝜀_𝑚 as the union of sets 𝑀^𝜀_𝑖⊂𝐾^𝑖_𝜀 (𝑖=1,2,…,𝑁_𝜀) obtained from 𝜀𝑀 by translations of vectors 𝜀∑^𝑛_𝑗=1𝑘_𝑗𝐞_𝑗, that is, Ω^𝜀_𝑚=^𝑁𝜀_𝑖𝑀^𝜀_𝑖,Ω^𝜀_𝑓=Ω⧵Ω^𝜀_𝑚, (83) where 𝐾^𝑖_𝜀 is the cube centered at the point 𝑥^𝑖,𝜀 and of length 𝜀, 𝑘_𝑗∈ℤ, {𝐞_𝑗}^𝑛_𝑗=1 is the canonical basis of ℝ^𝑛, and 𝑁_𝜀→+∞ as 𝜀→0.

Let 𝑝₀=𝑝₀(𝑥) be a log-Hölder continuous function such that 2<𝑝⁻≡min_𝑥∈Ω𝑝₀(𝑥)≤𝑝₀(𝑥)≤max_𝑥∈Ω𝑝₀(𝑥)≡𝑝⁺<+∞inΩ. (84) Let {𝑝_𝜀}_(𝜀>0)⊂𝔏^𝜀_𝑝0(⋅) be a sequence defined by 𝑝_𝜀(𝑥)=^def𝑝₀(𝑥)+𝐝_𝜀(𝑥), (85) where the function 𝐝_𝜀 is such that 𝐝_𝜀=𝑜(1) as 𝜀→0. The asymptotic behavior of 𝐝_𝜀 will be specified below. On the space 𝑊^{1,𝑝𝜀(⋅)}(Ω), we define the functional 𝐽^𝜀∶𝑊^{1,𝑝𝜀(⋅)}(Ω)→ℝ∪{+∞}, 𝐽^𝜀[𝑢]_def=∫_Ω𝐾_𝜀(𝑥)𝑝_𝜀(𝑥)|∇𝑢|^{𝑝𝜀(𝑥)}+1𝜎(𝑥)|𝑢|^𝜎(𝑥)−𝑔^𝜀(𝑥)𝑢𝑑𝑥if𝑢∈𝑊^{1,𝑝𝜀(⋅)}(Ω),+∞otherwise, (86) where 𝐾^𝜀(𝑥)=𝑘_𝑓inΩ^𝜀_𝑓,𝑘_𝑚𝜀^𝑝0(𝑥)inΩ^𝜀_𝑚, (87) the function 𝜎 satisfies conditions (A.3.4) with 𝜎⁻>2, (A.3.5), the function 𝑔^𝜀 is defined by 𝑔^𝜀(𝑥)=^def𝟏^𝜀_𝑓(𝑥)𝑔(𝑥), 𝑔∈𝐶(Ω). Here 𝑘_𝑓 and 𝑘_𝑚 are strictly positive constants independent of 𝜀.

Consider the following variational problem: 𝐽^𝜀𝑢^𝜀⟶min,𝑢^𝜀∈𝑊^1,𝑝𝜀₀^(⋅)(Ω). (88) We aim to study the asymptotic behavior of 𝑢^𝜀 the solution of (88).

To formulate the main result of this section, we introduce some notations. We denote by 𝗎^𝑎=𝗎^𝑎(𝑥,𝑦) the unique solution in 𝑊^1,𝑝0_#^(⋅)(𝐹) of the following cell problem: div_𝑦𝑘_𝑓∇_𝑦𝗎^{𝑎𝑝0(𝑥)−2}∇_𝑦𝗎^𝑎𝑘=0in𝐹,_𝑓∇_𝑦𝗎^{𝑎𝑝0(𝑥)−2}∇_𝑦𝗎^𝑎−𝐚,𝜈=0on𝜕𝑀,𝑦⟶𝗎^𝑎𝑦𝑌-periodic, (89) where 𝐹=𝑌⧵𝑀, 𝜈 is the outward normal vector to 𝜕𝑀, and 𝐚∈ℝ^𝑛. We denote by 𝗐^𝛽=𝗐^𝛽(𝑥,𝑦) the unique solution in 𝑊^1,𝑝0_#^(⋅)(𝑀) of the following cell problem: −div_𝑦𝑘_𝑚𝐝(𝑥)∇_𝑦𝗐^{𝛽𝑝0(𝑥)−2}∇_𝑦𝗐^𝛽+𝗐^{𝛽𝜎(𝑥)−2}𝗐^𝛽𝗐=0in𝑀,_𝛽𝑦=𝛽on𝜕𝑀,𝑦⟶𝗐^𝛽𝑦𝑌-periodic. (90) Notice that in (89) and (90) 𝑥 is a parameter. Regularity results for 𝗎^𝑎 and 𝗐^𝛽 are thus easily deduced from [33] and [34]. We also introduce the homogenized functional 𝐽_hom∶𝑊^1,𝑝0(⋅)(Ω)→ℝ∪{+∞}: 𝐽_hom[𝑢]_def=∫_Ω𝑇()+𝜌𝑥,∇𝑢^⋆𝜎(𝑥)|𝑢|^𝜎(𝑥)+𝖻(𝑥,𝑢)−𝑔(𝑥)𝜌^⋆𝑢𝑑𝑥if𝑢∈𝑊^1,𝑝0₀^(⋅)(Ω),+∞otherwise, (91)

the following results hold.

Theorem 24. Let 𝑢^𝜀 be a solution of (88). Assume that lim_𝜀→0𝜀^{−𝐝𝜀(⋅)}=𝐝(⋅) (92) uniformly in Ω. Then 𝑢^𝜀 converges strongly in 𝐿^𝑝0(⋅)(Ω^𝜀_𝑓) to 𝑢 the solution of the variational problem: 𝐽_ℎ𝑜𝑚[𝑢]⟶min,𝑢∈𝑊^1,𝑝0₀^(⋅)(Ω), (93) where 𝜌^⋆def𝑇()=meas𝐹,𝑥,𝐚=^def1𝑝₀(𝑥)_𝐹∇_𝑦𝗎^𝑎𝑥,𝑦−𝐚^𝑝0(𝑥)𝖻𝑑𝑦,𝑥,𝛽=^def_ℳ1𝑝₀(𝑥)𝛽𝗐^𝛽𝗐^{𝛽𝜎(𝑥)−2}−𝗐^{𝛽𝜎(𝑥)}+1𝜎(𝑥)𝗐^{𝛽𝜎(𝑥)}𝑑𝑦. (94)(95)(96)

Theorem 25. Let 𝑢^𝜀 be a solution of (88). Assume that, for any 𝑥∈Ω, lim_𝜀→0𝜀^{−𝐝𝜀(𝑥)}=+∞. (97) Then 𝑢^𝜀 converges strongly in 𝐿^𝑝0(⋅)(Ω^𝜀_𝑓) to 𝑢 the solution of the variational problem (93), where 𝜌^⋆ and the function 𝑇(𝑥,𝐚) are given in (94), (95), and 𝖻𝑥,𝛽=^defmeas𝑀𝜎(𝑥)𝛽^𝜎(𝑥). (98)

Theorem 26. Let 𝑢^𝜀 be a solution of (88). Assume that, for any 𝑥∈Ω, lim_𝜀→0𝜀^{−𝐝𝜀(𝑥)}=0. (99) Then 𝑢^𝜀 converges strongly in 𝐿^𝑝0(⋅)(Ω^𝜀_𝑓) to 𝑢 the solution of the variational problem (93), where 𝜌^⋆ and the function 𝑇(𝑥,𝐚) are given in (94), (95), and 𝖻(𝑥,𝛽)=0.

Remark 27. Notice that if 𝐾_𝜀(𝑥)=𝟏^𝜀_𝑓𝑘_𝑓+𝟏^𝜀_𝑚𝑘_𝑚𝜀^{𝑝𝜀(𝑥)} then Theorem 24 holds true with 𝐝(𝑥)≡1.

8. Homogenization of a Class of Quasilinear Parabolic Equations with Nonstandard Growth

8.1. 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^𝑛 (𝑛≥2) to be a bounded connected domain with a periodic structure. More precisely, we will scale this periodic structure by a parameter 𝜀 which represents the ratio of the cell size to the size of the whole region Ω, and we will assume that 𝜀 is a parameter tending to zero. Let 𝑌=^def]0,1[^𝑛 represent the mesoscopic domain of the basic cell of a fractured porous medium. For the sake of simplicity and without loss of generality, we assume that 𝑌 is made up of two homogeneous porous media 𝑀⋐𝑌 and 𝐹 corresponding to parties of the mesoscopic domain occupied by the matrix block and the fracture, respectively. Thus 𝑌=𝑀∪Γ_𝑚,𝑓∪𝐹, where Γ_𝑚,𝑓 denotes the interface between the two media and the subscripts 𝑚 and 𝑓 refer to the matrix and fracture, respectively. Let Ω^𝜀_𝑖 with 𝑖=𝑚 or 𝑓 denotes the open set filled with the porous medium 𝑖. Then Ω=Ω^𝜀_𝑚∪Γ^𝜀_𝑚,𝑓∪Ω^𝜀_𝑓, where Γ^𝜀_𝑚,𝑓=𝜕Ω^𝜀_𝑚∩𝜕Ω^𝜀_𝑓. For the sake of simplicity, we will assume that 𝜕Ω∩Ω^𝜀_𝑚=∅.

Let us introduce the nonstandard growth function used in this section. We assume that a family of continuous functions 𝑝_𝜀=𝑝_𝜀(𝑥), 𝜀>0, is defined in Ω and satisfies the following conditions: (A.4.1) functions 𝑝_𝜀 are bounded from below such that: 𝑝_𝜀(𝑥)≥2i𝑛Ω;(A.4.2) the function 𝑝_𝜀(𝑥) satisfies the log-Hölder continuity property;(A.4.3) the function 𝛾_𝜀(𝑥)=^def𝑝_𝜀(𝑥)−2 converges uniformly to zero in Ω.

Now let us introduce the permeability coefficient and the porosity of the porous medium Ω. We set 𝑘^𝜀(𝑥)=𝑘_𝑚𝜀²𝟏^𝜀_𝑚(𝑥)+𝑘_𝑓𝟏^𝜀_𝑓(𝑥),𝜔^𝜀(𝑥)=𝜔_𝑚𝟏^𝜀_𝑚(𝑥)+𝜔_𝑓𝟏^𝜀_𝑓(𝑥), (100) where 𝑘_𝑓 is the permeability or the hydraulic conductivity of fissures, 𝑘_𝑚 is the permeability or the hydraulic conductivity of blocks, 𝜔_𝑓 is the porosity of fissures, and 𝜔_𝑚 is the porosity of blocks; 𝟏^𝜀_𝑓=𝟏^𝜀_𝑓(𝑥) and 𝟏^𝜀_𝑚=𝟏^𝜀_𝑚(𝑥) denote the characteristic functions of the sets Ω^𝜀_𝑓 and Ω^𝜀_𝑚, respectively. Here 0<𝑘_𝑓,𝑘_𝑚,𝜔_𝑓,𝜔_𝑚<+∞.

We consider the following initial boundary value problem for the function 𝑢^𝜀∶𝑄↦ℝ: 𝜔^𝜀(𝑥)𝜕𝑢^𝜀𝑘𝜕𝑡−div^𝜀(𝑥)∇𝑢^𝜀∇𝑢^{𝜀𝑝𝜀(𝑥)−2}()𝑢=𝑔𝑡,𝑥in𝑄,_𝜀()𝑢=0on0,𝑇×𝜕Ω,_𝜀(0,𝑥)=𝑢₀(𝑥)inΩ, (101) where 𝑄 denotes the cylinder (0,𝑇)×Ω, 𝑇>0 is given, and 𝑔,𝑢₀ 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, 𝐶 will denote a generic positive constant, independent of 𝜀, and may take different values for different occurrences.

8.2. Preliminary Results

The goal of this section is to obtain a priori estimates for the solution 𝑢^𝜀 of problem (101). We start by formulating the existence and uniqueness result for (101). It is given by the following theorem (see [35]).

Theorem 28. Let 𝑔∈𝐶(0,𝑇;𝐿²(Ω)) and 𝑢₀∈𝐻²(Ω). Then, for any 𝜀>0, there exists a unique solution 𝑢^𝜀=𝑢^𝜀(𝑡,𝑥) of the boundary value problem (101) in the space 𝐿^∞(0,𝑇;𝑊^{1,𝑝𝜀(⋅)}(Ω)). Furthermore, this solution satisfies the following a priori estimates, for a.e. 𝑡∈(0,𝑇): 𝑢^𝜀(𝑡)²_𝐿2(Ω)+^𝑡₀𝑑𝑡_Ω𝜕𝑢^𝜀𝜕𝑡²𝑑𝑥+_Ω𝑘^𝜀(𝑥)∇𝑢^{𝜀𝑝𝜀(𝑥)𝑑𝑥}𝑢≤𝐶,_𝜀(𝑡+𝛿𝑡)−𝑢^𝜀(𝑡)_𝐿2(Ω)≤𝐶(𝛿𝑡)^𝜅𝑤𝑖𝑡ℎ0<𝜅<1, (102)(103) where 𝛿𝑡>0 is a time step which tends to zero.

We study the asymptotic behavior of the solution 𝑢^𝜀 of problem (101) as 𝜀→0. For this it is convenient to introduce the following notation: 𝑢^𝜀=𝜌^𝜀inΩ^𝜀_𝑓,𝜎^𝜀inΩ^𝜀_𝑚, (104) and to rewrite (101) separately in the domains Ω^𝜀_𝑓,Ω^𝜀_𝑚 with appropriate interface conditions. Namely, in the domain Ω^𝜀_𝑓 (101) reads 𝜔_𝑓𝜕𝜌^𝜀𝑘𝜕𝑡−div_𝑓∇𝜌^𝜀∇𝜌^{𝜀𝑝𝜀(𝑥)−2}=𝑔(𝑡,𝑥)in(0,𝑇)×Ω^𝜀_𝑓,𝑘_𝑓∇𝜌^𝜀∇𝜌^{𝜀𝑝𝜀(𝑥)−2}⋅𝜈=𝑘_𝑚𝜀^{𝑝𝜀(𝑥)}∇𝜎^𝜀∇𝜎^{𝜀𝑝𝜀(𝑥)−2}⋅𝜈on(0,𝑇)×Γ^𝜀_𝑚,𝑓,𝜌^𝜀()𝜌=0on0,𝑇×𝜕Ω,_𝜀(0,𝑥)=𝑢₀(𝑥)inΩ^𝜀_𝑓, (105) where 𝜈 is the outward normal vector to Γ^𝜀_𝑚𝑓. In the domain Ω^𝜀_𝑚 (101) reads 𝜔_𝑚𝜕𝜎^𝜀𝑘𝜕𝑡−div_𝑚𝜀²∇𝜎^𝜀∇𝜎^{𝜀𝑝𝜀(𝑥)−2}=𝑔(𝑡,𝑥)in(0,𝑇)×Ω^𝜀_𝑚,𝜎^𝜀=𝜌^𝜀on(0,𝑇)×Γ^𝜀_𝑚𝑓,𝜎^𝜀(0,𝑥)=𝑢₀(𝑥)inΩ^𝜀_𝑚. (106)

To establish some preliminary compactness result, first we notice that the a priori estimate (102), conditions (A.4.1), (A.4.3) along with (110), and inequalities (106), and (107) imply the bound for a.e. 𝑡∈(0,𝑇): 𝑢^𝜀(𝑡)_𝐿2(Ω)+^𝑡₀𝑑𝑡_Ω𝜕𝑢^𝜀𝜕𝑡²+𝑑𝑥_Ω𝜀𝑓∇𝑢^𝜀2𝑑𝑥+𝜀²_Ω𝜀𝑚∇𝑢^𝜀2𝑑𝑥≤𝐶. (107) Therefore, from (105), (106) and (107), we have for a.e. 𝑡∈(0,𝑇): 𝜌^𝜀(𝑡)_{𝐿2(Ω𝜀𝑓)}+^𝑡₀𝑑𝑡_Ω𝜀𝑓𝜕𝜌^𝜀𝜕𝑡²𝑑𝑥+_Ω𝜀𝑓∇𝜌^𝜀2𝜎𝑑𝑥≤𝐶,_𝜀(𝑡)_{𝐿2(Ω𝜀𝑚)}+^𝑡₀𝑑𝑡_Ω𝜀𝑚𝜕𝜎^𝜀𝜕𝑡²𝑑𝑥+𝜀²_Ω𝜀𝑚∇𝜎^𝜀2𝑑𝑥≤𝐶. (108)

Now using the bounds (108), along with the extension result [30], it is easy to prove the following compactness result.

Lemma 29. Let 𝑢^𝜀=⟨𝜌^𝜀,𝜎^𝜀⟩ be the solution of problem (101). Then there exists a subsequence, still denoted by {𝑢^𝜀}_𝜀>0, and functions 𝑢_𝑓=𝑢_𝑓(𝑡,𝑥),𝑣_𝑓=𝑣_𝑓(𝑥,𝑦),𝑢_𝑚=𝑢_𝑚(𝑡,𝑥,𝑦) such that(a.1)𝑢_𝑓∈𝐻¹(0,𝑇;𝐿²(Ω))∩𝐿^∞(0,𝑇;𝐻¹(Ω)), 𝑣_𝑓∈𝐿²(Ω;𝐻¹_#(𝐹)⧵ℝ),(a.2)𝑢_𝑚∈𝐻¹(0,𝑇;𝐿²(Ω×𝑌))∩𝐿^∞(0,𝑇;𝐿²(Ω;𝐻¹(𝑀))), with 𝑢_𝑚(𝑡,𝑥,𝑦)=𝑢_𝑓(𝑡,𝑥) for 𝑦∈𝜕𝑀,(b) for any 𝑡∈(0,𝑇), 𝜌⇀^𝜀2𝑠𝑢_𝑓 and 𝜎⇀^𝜀2𝑠𝑢_𝑚,(c) for any 𝜑∈𝐿^∞(0,𝑇;𝐿²(Ω;𝐶_#(𝑌))), ^𝑇₀𝑑𝑡_Ω𝜀𝑓𝜕𝑢^𝜀𝜑𝑥𝜕𝑡𝑡,𝑥,𝜀⟶𝑑𝑥_𝑇0𝑑𝑡_Ω×𝐹𝜕𝑢_𝑓𝜑𝜕𝑡𝑡,𝑥,𝑦𝑑𝑥𝑑𝑦,_𝑇0𝑑𝑡_Ω𝜀𝑚𝜕𝑢^𝜀𝜑𝑥𝜕𝑡𝑡,𝑥,𝜀⟶𝑑𝑥_𝑇0𝑑𝑡_Ω×𝑀𝜕𝑢_𝑚𝜑𝜕𝑡𝑡,𝑥,𝑦𝑑𝑥𝑑𝑦, (109)(d)for a.e. 𝑡∈(0,𝑇), ∇𝜌⇀^𝜀2𝑠(∇_𝑥𝑢_𝑓+∇_𝑦𝑣_𝑓)𝟏_𝑓(𝑦) and ∇𝜎⇀^𝜀2𝑠∇_𝑦𝑢_𝑚𝟏_𝑚(𝑦).

8.3. Homogenization Results

In this section, we formulate the main results of the section. We give homogenization results for the problem (101). The convergence of the homogenization process is obtained by combining the technique of two-scale convergence and the variational homogenization method (see, e.g., [9, 16, 27] and the references therein).

The idea of the proof is the following. First we will reduce our parabolic problem to an elliptic one depending on the time variable as a parameter. Then we introduce a functional corresponding to this elliptic problem and study the minimization problem for it in the limit of small 𝜀. Then we obtain the limit functional corresponding to the homogenized problem. Regarding the variational technique, it is worth to mention one trick used in the paper. In order to obtain the lower bound for the original functional, we first replace the original exponent 𝑝_𝜀(𝑥) by a new one 𝑝₀=2 and consider the corresponding family of auxiliary functionals. Then the lower semicontinuity property of convex functionals with respect to the two-scale convergence implies the desired inequality. Finally, it is not difficult to show that the limit functional for the auxiliary family does not exceed the limiting functional for the original one.

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

Theorem 30. Let 𝑢^𝜀=𝑢^𝜀(𝑡,𝑥) be the solution of the boundary value problem (101) and let conditions (A.4.1)–(A.4.3) be satisfied. Moreover, we assume that there exists a function 𝛼∈𝐶(Ω) such that uniformly in 𝑥∈Ω, lim_𝜀→0𝜀^{−𝛾𝜀(𝑥)}=𝛼(𝑥). (110) Then, for a.e. 𝑡∈(0,𝑇), 𝑢^𝜀 two-scale converges to 𝑢^∗∈𝐿²((0,𝑇)×Ω×𝑌) such that 𝑢^∗=𝑢𝑡,𝑥,𝑦^𝑓()𝑢𝑡,𝑥𝑖𝑛𝑄×𝐹,_𝑚𝑡,𝑥,𝑦𝑖𝑛𝑄×𝑀, (111) where the couple (𝑢_𝑓,𝑢_𝑚)∈𝐿²(0,𝑇;𝐻¹(Ω))×𝐿^∞(0,𝑇;𝐿²(Ω;𝐻¹(𝑀))) is the unique solution of the homogenized problem 𝜔_𝑓|𝐹|𝜕𝑢_𝑓𝜕𝑡−div_𝑥𝐾^∗∇_𝑥𝑢_𝑓=𝖲𝑥,𝑢_𝑚𝑢𝑖𝑛𝑄,_𝑓=0on(0,𝑇)×𝜕Ω,𝑢_𝑓(0,𝑥)=𝑢₀(𝑥)𝜔𝑖𝑛Ω,_𝑚𝜕𝑢_𝑚−𝜕𝑡^∼𝐾^∗(𝑥)Δ_𝑦𝑢_𝑚()𝑢=𝑔𝑡,𝑥𝑖𝑛𝑄×𝑀,_𝑚𝑡,𝑥,𝑦=𝑢_𝑓()𝑢𝑡,𝑥𝑜𝑛𝑄×𝜕𝑀,_𝑚0,𝑥,𝑦=𝑢₀(𝑥)𝑖𝑛Ω×𝑀, (112) where |𝐹| is the measure of the set 𝐹 and 𝐾^∗={𝑘^∗_𝑖𝑗} is the homogenized permeability tensor defined by 𝑘^∗_𝑖𝑗=𝑘_𝑓𝐹𝑒_𝑖+∇_𝑦𝑤_𝑖⋅𝑒_𝑗+∇_𝑦𝑤_𝑗𝑑𝑦 (113) with {𝑒₁,𝑒₂,…,𝑒_𝑛} the canonical basis of ℝ^𝑛 and 𝑤_𝑖 being the unique solution in 𝐻¹_#(𝐹)⧵ℝ of −𝑘_𝑓Δ_𝑦𝑤_𝑖𝑒=0𝑖𝑛𝐹,_𝑖+∇_𝑦𝑤_𝑖⋅𝜈=0𝑜𝑛𝜕𝑀,𝑦⟶𝑤_𝑖𝑥,𝑦𝑌-𝑝𝑒𝑟𝑖𝑜𝑑𝑖𝑐, (114) where 𝜈=𝜈(𝑦) is the outer normal vector at 𝜕𝑀, and the coefficient 𝐾^∼∗ in the local problem is given by 𝐾^∼∗(𝑥)=𝛼(𝑥)𝑘_𝑚, (115) and the effective source term 𝖲(𝑥,𝑢_𝑚) is given by 𝖲𝑥,𝑢_𝑚=|𝐹|𝑔(𝑡,𝑥)−𝐾^∼∗(𝑥)_𝜕𝑀∇_𝑦𝑢_𝑚⋅𝜈𝑑𝑠_𝑦. (116)

Remark 31. The source term which appears in the right-hand side of the first equation in (112) is well defined, since 𝑢_𝑚∈𝐿^∞(0,𝑇;𝐿²(Ω;𝐻¹(𝑀))), and it follows from the third equation of (112) that −Δ_𝑦𝑢_𝑚∈𝐿²0,𝑇;𝐻^−1/2(𝜕𝑀), (117) which allows one to define (∇_𝑦𝑢_𝑚⋅𝜈) as an element of 𝐿²(0,𝑇;𝐻^−1/2(𝜕𝑀)).

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 𝑡 as a parameter. Namely, we consider the following boundary value problem, for a.e. 𝑡∈]0,𝑇[, −div𝑘^𝜀(𝑥)∇𝑢^𝜀∇𝑢^{𝜀𝑝𝜀(𝑥)−2}=𝐺^𝜀𝑢in𝑄,_𝜀=0on]0,𝑇[×𝜕Ω, (118) where the function 𝐺^𝜀, 𝐺^𝜀=𝐺^𝜀(𝑡,𝑥)=^def𝑔(𝑡,𝑥)−𝜔^𝜀(𝑥)𝜕𝑢^𝜀(),𝜕𝑡𝑡,𝑥 (119) is considered as a given function. Then, for any Δ_𝑡⊂[0,𝑇], 𝑢^𝜀 minimizes the functional 𝐽^𝜀[𝑢]=^def_Δ𝑡𝑑𝑡_Ω𝑘^𝜀(𝑥)𝑝_𝜀(𝑥)|∇𝑢|^{𝑝𝜀(𝑥)}−𝐺^𝜀𝑢𝑑𝑥 (120) over 𝑢∈𝐿^∞(0,𝑇;𝑊^{1,𝑝𝜀(⋅)}(Ω)). Using the two-scale convergence arguments, we obtain the “limsup”-inequality for the functional 𝐽^𝜀. The “liminf”-inequality is obtained in two steps. First, we introduce the auxiliary functional 𝐽^∼𝜀[𝑢]=^def_Δ𝑡𝑑𝑡_Ω^∼𝑘^𝜀(𝑥)𝑝_𝜀(𝑥)|∇𝑢|²−𝐺^𝜀𝑢𝑑𝑥with𝑘^∼𝜀(𝑥)=𝛼(𝑥)𝑘_𝑚𝜀²𝟏^𝜀_𝑚(𝑥)+𝑘_𝑓𝟏^𝜀_𝑓(𝑥) (121) and obtain the inequality lim^∼_𝜀→0𝐽^𝜀𝑢^𝜀≥𝐽_hom𝑢_𝑓,𝑢_𝑚, (122) where 𝐽_hom𝑢_𝑓,𝑢_𝑚=_Δ𝑡𝑑𝑡_Ω12𝐾^∗∇_𝑥𝑢_𝑓⋅∇_𝑥𝑢_𝑓−|𝐹|𝑔(𝑡,𝑥)−𝜔_𝑓𝜕𝑢_𝑓𝑢𝜕𝑡^𝑓+𝑑𝑥_Δ𝑡𝑑𝑡_Ω×𝑀𝛼(𝑥)𝑘_𝑚2∇_𝑦𝑢_𝑚²−𝑔(𝑡,𝑥)−𝜔_𝑚𝜕𝑢_𝑚𝑢𝜕𝑡^𝑚𝑑𝑥𝑑𝑦. (123)

Then in the second step, we obtain the desired “liminf”-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.

Theorem 32. Let 𝑢^𝜀=⟨𝜌^𝜀,𝜎^𝜀⟩ be the solution of the boundary value problem (101) and let conditions (A.4.1)–(A.4.3) be satisfied. Moreover, we assume that for any 𝑥∈Ω, lim_𝜀→0𝜀^{−𝛾𝜀(𝑥)}=+∞. (124) Then, for a.e. 𝑡∈(0,𝑇), 𝑢^𝜀 converges in 𝐿²(Ω), as 𝜀→0, to 𝑢⁰∈𝐿²(0,𝑇;𝐻¹(Ω)), the solution of 𝜔_𝑓|𝐹|+𝜔_𝑚|𝑀|𝜕𝑢⁰𝜕𝑡−div_𝑥𝐾^∗∇_𝑥𝑢⁰()𝑢=𝑔𝑡,𝑥𝑖𝑛𝑄,₀=0on(0,𝑇)×𝜕Ω,𝑢⁰(0,𝑥)=𝑢₀(𝑥)𝑖𝑛Ω, (125) where 𝐾^∗={𝑘^∗_𝑖𝑗} is the homogenized permeability tensor defined in (113)-(114).

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 (𝑝_𝜀(⋅)−2) to zero. The critical rate of convergence is 𝑝_𝜀(⋅)∼1−2||.ln𝜀 (126) More precisely, if lim_𝜀→0|ln𝜀|(𝑝_𝜀(𝑥)−2)<+∞, then the limit model is of a double porosity type. If lim_𝜀→0|ln𝜀|(𝑝_𝜀(𝑥)−2)=+∞, then in the limit we obtain a single porosity model.

The proofs of Theorems 30 and 32 are given in [21].