H-convergence of a class of quasilinear equations in perforated domains beyond periodic setting

In this paper, we aim to study the asymptotic behavior (when ε→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \;\rightarrow \; 0$$\end{document}) of the solution of a quasilinear problem of the form -div(Aε(·,uε)∇uε)=f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\mathrm{{div}}\;(A^{\varepsilon }(\cdot ,u^{\varepsilon }) \nabla u^{\varepsilon })=f$$\end{document} given in a perforated domain Ω\Tε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \backslash T_{\varepsilon }$$\end{document} with a Neumann boundary condition on the holes Tε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon }$$\end{document} and a Dirichlet boundary condition on ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document}. We show that, if the holes are admissible in certain sense (without any periodicity condition) and if the family of matrices (x,d)↦Aε(x,d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,d)\mapsto A^{\varepsilon }(x,d)$$\end{document} is uniformly coercive, uniformly bounded and uniformly equicontinuous in the real variable d, the homogenization of the problem considered can be done in two steps. First, we fix the variable d and we homogenize the linear problem associated to Aε(·,d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^{\varepsilon }(\cdot ,d)$$\end{document} in the perforated domain. Once the H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{0}$$\end{document}-limit A0(·,d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^{0}(\cdot ,d)$$\end{document} of the pair (Aε,Tε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(A^{\varepsilon },T^{\varepsilon })$$\end{document} is determined, in the second step, we deduce that the solution uε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{\varepsilon }$$\end{document} converges in some sense to the unique solution u0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{0}$$\end{document} in H01(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{1}_{0}(\Omega )$$\end{document} of the quasilinear equation -div(A0(·,u0)∇u)=χ0f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\mathrm{{div}}\;(A^{0}(\cdot ,u^{0})\nabla u )=\chi ^{0}f$$\end{document} (where χ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \chi ^{0}$$\end{document} is L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\infty }$$\end{document} weak ⋆\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\star }$$\end{document} limit of the characteristic function of the perforated domain). We complete our study by giving two applications, one to the classical periodic case and the second one to a non-periodic one.

We complete our study by giving two applications of the established compactness results. The first application is for the classical periodic case, where the obtained result coincides (in our framework) with a result given in [7]. While, the second one which concerns a non-periodic case introduced in [5] is an original result.
Our work generalizes that of Murat-Bocardo given in [4] which treated in the general framework of Hconvergence the same type of quasilinear equations in fixed domains without holes. The periodic case with Lipschitz continuous coefficients was subsequently processed by Artola-Duvaut in [1]. On the other hand, for periodically perforated domains, the same type of quasilinear equations was firstly studied in Bendib [2] and Bendib-Tcheugoué Teboué [3], with Lipschitz continuous coefficients and linear Robin conditions. After this Cabarrubias-Donato have studied in [7] this equation with a nonlinear Robin condition boundary of the holes and the module of equicontinuity satisfies a suitable assumption introduced by Chipot in [9], but not assumed to be Lipschitz continuous. For the homogenization of other type of Neumann quasilinear equations in perforated domains with data satisfying a general assumptions of abstract homogenization, see for example [8,13] among others.
This article is organized as follows: Sect. 2 is devoted to some preliminary results on the H 0 -convergence as introduced by [5]. This notion generalizes that of H -convergence in fixed domains due to Murat-Tartar (see [12,14]). We give at the end of this section, a new result about a pointwise estimate of the dierence of two H 0 -limits. In Sect. 3, we present our main compactness results for a class of quasilinear equations in perforated domains in the general framework of H 0 -convergence. Section 4 is devoted to the proofs of our results. Finally, in Sect. 5, we give two applications of the obtained compactness results, namely the classical periodic case and a certain non-periodic case.

Notations
• {ε} denotes a strictly decreasing sequence converging to zero, • if ζ = (ζ i ) 1≤i≤n and ξ = (ξ i ) 1≤i≤n are two vectors, we set • for two real numbers α and β such that 0 < α < β, M (α, β; ) is the set of the matrix fields A = A i j 1≤i, j≤n defined on such that almost everywhere in , we have

Preliminary results on the H-convergence for perforated domains
Since we work in the framework of the H 0 -convergence, we recall in this subsection some preliminary results about this notion and we give at the end a useful new result on the pointwise estimate of the dierence of two H 0 -limits. We introduce the perforated domain by ε = \ T ε , where {T ε } is a sequence of compact subsets of and set We denote by · the extension by 0 from ε to and set χ ε = χ ε . In the following ν denotes the outward normal unit vector to the boundary of ε .
We denote by P ε the adjoint operator of P ε , which is defined from and T ε be admissible in . We say that the pair (A ε , T ε ) H 0converges to the matrix A 0 ∈ M α , β ; and we write (A ε , T ε ) A 0 in if and only if for every function g of L 2 ( ), and every subsequence of ε (still denoted by ε) such that χ ε χ 0 weakly in L ∞ ( ) (χ 0 depending upon the subsequence), the solution v ε of satisfies the weak convergence where v 0 is the unique solution of the problem (2) In the case of T ε = ∅, this definition reduces to the definition of H -convergence.
The main properties of the H 0 -convergence are given by the results below.
Finally, we complete the preliminary results by giving a pointwise estimate of the dierence of two H 0 -limits. This result needs the following lemma (which is a directly consequence of [5, Proposition 1.14]): where v ε and v 0 are solutions of (2.1) and (2.3) respectively.
We are now able to give a pointwise estimate of the dierence of two H 0 -limits.
Assume that Then, Proof The proof is obtained by using Lemma 2.6 and Proposition 2.5, and by following the same techniques used to prove a similar result given for the elasticity case in [11,Theorem 28].

Remark 2.8 Assumptions
(i)-(iii) of Theorem 2.7 are reasonable. Indeed, -(i) is obviously satisfied when T 1 ε = T 2 ε for every ε, -(ii) is satisfied when there exists a bounded domain O in R n in which is relatively compact and for which T ε is admissible (see the proof of [5, Proposition 1.15]), -(iii) is satisfied for the classical periodic case and also for the non-periodic case considered in [5].

Statement of compactness results
In this section, we give our compactness results for the H 0 -convergence of a class of elliptic and uniformly equicontinuous operators in perforated domains. Firstly, we introduce the set M Equi (α, β, ω; ) in the following definition : Definition 3.1 For two real numbers α, β such that 0 < α < β and ω a function defined from R + to R + nondecreasing and continuous at 0, M Equi (α, β, ω; ) denotes the set of all Caratheodory functions satisfying the following assumptions: (ii) for almost every x in and for every d, d ∈ R, one has Our first main result is the following:

Then, there exists a subsequence of {ε} (still denoted by {ε}), and an element
Moreover, if we suppose that there exists a bounded domain O in R n in which is relatively compact and for which T ε is also admissible, we have

Remark 3.3 (i)
A similar property to (3.2) is given in [14] in the case of fixed domain when the mapping d → A ε (·, d) is of class C k (or real analytic) from an open set D of R p into L ∞ ( ; L(R n ; R n )) for every p ∈ N * .
(ii) Theorem 3.2 still holds if d ∈ R p and v ∈ L 1 ( ) p for every p ∈ N * .
As a consequence of Theorem 3.2, we obtained a general homogenization result for some quasilinear equations in perforated domain beyond periodic setting.
with A 0 the family of matrices given by Theorem 3.2.

Proofs of compactness results
We give in this section the proofs of our main results. The proofs are an adaptation of the similar ones given in [4] for fixed domains.
Proof of Theorem 3. 2 We give the proof in two steps.
Step 1. Let us prove that there exists A 0 ∈ M Equi ( α C 2 , β, β α ω; ) which satisfies convergence (3.2) up to subsequence. Using Theorem 2.4 and the diagonal subsequence procedure, we extract a subsequence of {ε} (still denoted by {ε}) such that, for every d ∈ Q, we will have Hence, by the fact that A ε ∈ M Equi (α, β, ω; ), Assumption (3.1) and Theorem 2.7, we obtain Thus, the mapping is uniformly continuous. Hence, it is extensible to a mapping (denoted again by A 0 ) defined and uniformly continuous on all R (since Q is dense in R), namely On the other hand, let d ∈ R and {d m } be a sequence in Q which converges to d as m → ∞. Thanks to Theorem 2.4, there exists a subsequence of {ε} (still denoted by {ε}) such that Since, for every ε > 0, we have then from this, (4.1), (4.3), Assumption (3.1) and Theorem 2.7, it comes This, with (4.2) and by the triangle inequality, we deduce that for almost every x in Using the continuity of ω at 0, passing to the limit in this inequality as m → ∞, we find A 0 (x, d), a.e. x ∈ .
Step 2. We now show property (3.3). Let v ∈ L 1 ( ). Then, A ε (v(·)) . = A ε (·, v(·)) belongs to M(α, β; ). Hence, taking into account Theorem 2.4, there exists B 0 ∈ M( α C 2 , β; ) such that to up a subsequence, we have where {Y i } 1≤i≤k is a family of disjoint rectangles of R n included in and l m i real constants. Set (4.6) and (3.2) gives Hence, using (4.4), (4.6), (4.7), Assumption (3.1), point (ii) of Remark 2.8 and by Theorem 2.7, we obtain which implies that for almost every x in Moreover, thanks to (4.2), we have Hence, from this two latter inequalities, it follows from triangle inequality that Since ω is continuous at 0, passing to the limit in this inequality when m → ∞, one obtains which, with (4.4), gives ( 3.3).
Proof of Theorem 3.4 First, note that problem (3.5) (respect. (3.7) has a unique solution in H 1 0 ( ε ) (respect. H 1 0 ( )) thanks to [6]. Second, taking u ε as a test function in the variational formulation of (3.5), we obtain Hence, we can extract a subsequence of {ε} (still denoted by {ε}), such that This implies, for every m, that where {v m } is a sequence of functions introduced in (4.5) such that v m → u 0 strongly in L 1 (R). So, thanks to continuity of ω, we get On the other hand, since A ε (P ε u ε ) . = A ε (·, P ε u ε (·)) ∈ M(α, β; ), there exists a subsequence of {ε} (still denoted by {ε}) and C 0 ∈ M( α C 2 , β; ), such that hence by this last inequality, (4.8), (4.9), Theorem 3.2, Assumption (3.1), point (ii) of Remark 2.8 and Theorem 2.7, it comes This gives Since lim d→0 ω(d) = 0, passing to the limit in this inequality when m → ∞, we obtain Then, from this and (4.9), we find which implies by the definition of the H 0 -convergence and the uniqueness of the solutions of (3.5) and (3.7) that P ε u ε u 0 weakly in H 1 0 ( ), where u 0 is the solution of (3.7), which completes proof of (i) and (iii) of (3.6). Finally, we deduce (3.6)ii) from the fact that χ ε χ 0 weakly in L ∞ ( ) and P ε u ε → u 0 strongly in L 2 ( ).

Applications
As an application of our results, we consider the classical periodic case and a non-periodic case.
Let θ a diffeomorphism of class C 2 from R n onto R n and introduce the holes T ε defined by In what follows, the spherical geometry of the holes can be generalized to the case where a regular boundary hole with a finite number of connected components replace a ball.

Classical periodic case
Take here θ = I d R n and δ = 1 3 . Then, the pair (A ε (·, ·), T ε ) satisfies all assumptions of Theorem 3.4 and it is well-known that in this case (see [10]) with A 0 (d) is independent of x and given by and for all λ ∈ R n , y −→ v λ (y, d) be the solution of In this framework, we have the following result about the convergence of problem (3.5): and where A 0 defined by (5.1) belongs to M Equi ( α C 2 , β, β α ω; ).

Remark 5.2
In the geometric framework of this example, Proposition 5.1 coincides with a result given in [7] by using the periodic unfolding, when the nonlinear Robin boundary condition on the holes reduces to the homogeneous Neumann condition.

Non-periodic case
Consider here the non-periodic perforated domain introduced in [5, Section 3] when studying the corresponding linear case. We suppose that θ −1 has a Lipschitz constant κ −1 with κ > 2 and take δ = 1. In this case, from [5, Sections 3-4], we deduce easily that for every d ∈ R, the pair (A ε , T ε ) satisfies all assumptions of Theorem 3.4 and Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.