# Explicit corrector in homogenization of monotone operators and its
application to nonlinear dielectric elastomer composites

Thuyen Dang111Department of Statistics/Committee on Computational and Applied
Mathematics, University of Chicago, 5747 S. Ellis Avenue, Chicago, Illinois
60637, USA (thuyend@uchicago.edu). Yuliya Gorb222National Science Foundation,
2415 Eisenhower Avenue, Alexandria, Virginia 22314, USA (ygorb@nsf.gov).
Silvia Jiménez Bolaños333Department of Mathematics, Colgate University, 13 Oak
Drive, Hamilton, New York 13346, USA (sjimenez@colgate.edu).

###### Abstract

This paper concerns the rigorous periodic homogenization for a weakly coupled
electroelastic system of a nonlinear electrostatic equation with an elastic
equation enriched with electrostriction. Such coupling is employed to describe
dielectric elastomers or deformable (elastic) dielectrics. It is shown that
the effective response of the system consists of a homogeneous dielectric
elastomer described by a nonlinear weakly coupled system of PDEs whose
coefficients depend on the coefficients of the original heterogeneous
material, the geometry of the composite, and the periodicity of the original
microstructure. The approach developed here for this nonlinear problem allows
us to obtain an explicit corrector result for the homogenization of monotone
operators with minimal regularity assumptions. Two $L^{p}-$gradient estimates
for elastic systems with discontinuous coefficients are also obtained.

##  1 Introduction

In recent years there has been a growing interest towards a class of materials
known as dielectric elastomers that can exhibit coupled electrical and
mechanical behavior, see cf. [14]. A unique property possessed by such
materials, known as electrostriction, which means they can respond to an
external electric field by changing their size and shape, makes them appealing
for various practical applications, e.g. soft robotics, artificial muscles,
active camouflage, haptic devices, energy harvesting, and others, see e.g.
[14] and [29]. Homogenization theory can be used to guide the design of
dielectric elastomers with enhanced electromechanical couplings, e.g. [35],
and this paper is devoted to the rigorous periodic homogenization of such a
coupling.

The governing equations describing the system under consideration consist of a
nonlinear (scalar) electrostatic equation in the presence of a bounded free
body charge weakly coupled with an elastic (vectorial) equation that involves
an electrostriction term. Here, weakly coupling means that the elastic
displacement does not enter the electrostatic equation. The PDE system is
posed on a heterogeneous bounded domain with periodic microstructure of size
$0<\varepsilon\ll 1$. For simplicity, we focus on Dirichlet boundary
conditions only. The goals of this paper are twofold. First, it aims at
determining the macroscopic or effective behavior of the considered periodic
composite under the assumption of scale separation. This amounts to developing
an asymptotic analysis of the limiting response of the given PDE system as
$\varepsilon$, the size of the microstructure, goes to zero. Second, since
many necessary facts (e.g. the regularity of the solution to the original
fine-scale problem) are not readily available, this paper presents a number of
stand-alone results that could be utilized in future contributions to the
topic of periodic homogenization for nonlinear electrostatic and/or elastic
composite materials under minimal regularity assumptions of the original
system.

The rigorous justification of a model for dielectric elastomers and the
derivation of its effective system, using the mathematical theory of
homogenization, were carried out in [51, 32] and references therein. To carry
out the homogenization of their system, the authors of [51] made a strict
integrability assumption, requiring that the solution of the electrostatic
equation belongs to a class of least $W^{1,4}$-functions. The results in [32]
showed that if the coefficients of the electrostatic equation are piecewise
Hölder continuous, then indeed its solution belongs to $W^{1,p}$, for any
$p\in[1,\infty)$. Later, in [25], this result was extended to the case
$p=\infty$. A similar model for the case of magnetic suspensions was
investigated in [24, 25].

In the contributions cited above [51, 32], the materials studied were linear,
i.e., the constitutive relationship between the electric field $\mathbf{E}$
and the electric displacement $\mathbf{D}$ was assumed linear. However, when
this relationship is nonlinear, e.g., as in the case of ferroelectric
materials, a new model, as well as a new approach, are required to obtain the
corresponding homogenized response. In this paper, we consider $\mathbf{E}$
and $\mathbf{D}$ satisfying a nonlinear constitutive relation that yields a
nonlinear divergence equation. Therefore, the improved gradient estimates,
obtained by the compactness method used in [32, 25], are no longer available.
Instead, in this paper, we derive a new approach that does not require such
estimates or the renormalization framework of [33, 44], which are the typical
techniques to deal with problems of low regularity source terms. To gain
additional regularity of the solution, we apply regularity theory to not only
the fine-scale or the effective systems but also to the two-scale homogenized
one. Two-scale convergence acts as an “intermediate” topology between weak and
strong convergence that allows canceling the dependence on the size of
microstructure, thus providing a system with rather nice coefficients, besides
the effective one. This idea enables us to relax several regularity
assumptions needed before in [32, 51, 25], and also to extend the cited works
to nonlinear cases.

To implement this idea for the current problem, several ingredients are
needed: a result from the nonlinear Caldéron-Zygmund theory [46] (see also
[39, 43]), an estimate for elliptic systems [40], the existence of the
(generalized) Green’s function [22], the SOLA technique (existence of the
solution by limit of approximations [8]), results from the theory of two-scale
convergence (for $L^{p}$ and $\mathrm{BV}$ functions) [1, 45, 2, 31], an
interpolation theorem [34], a corrector result in homogenization of monotone
operators [23], and a duality argument.

Along with the sought-after homogenized response, in this paper we obtain
three stand-alone results, namely: (i) Theorem 1, which provides an explicit
first-order corrector for the nonlinear electrostatic problem with minimal
regularity assumptions, while (ii) LABEL:sec:appendix-1 and (iii) Proposition
A.4 provide $L^{p}-$gradient estimates for the elasticity system via an
interpolation argument. Besides the fact that our approach is built for the
nonlinear problem with minimal regularity assumptions, combined with ideas
from [17, 27, 7] it could also be extended to the high-contrast case, for
which the minimal and maximal values of coefficients of the underlying PDEs
are vastly different. This case will be reported in future publications by the
authors elsewhere.

This paper is organized as follows. In Section 2, the main notations are
introduced and the formulation of the fine-scale problem is discussed. Our
main result is stated in Section 3, the conclusions are given in
LABEL:sec:conclusions, and auxiliary facts are discussed in LABEL:sec:an-
appendix.

##  2 Formulation

###  2.1 Notation and definitions

Throughout this paper, the scalar-valued functions, such as the pressure $p$,
are written in usual typefaces, while vector-valued or tensor-valued
functions, such as the displacement $\mathbf{u}$ and the stress tensor
$\boldsymbol{\sigma}$, are written in bold. Sequences are indexed by
superscripts ($\phi^{i}$), while elements of vectors or tensors are indexed by
numeric subscripts ($x_{i}$). Finally, the Einstein summation convention is
used whenever applicable; $\delta_{ij}$ is the Kronecker delta, and
$\epsilon_{ijk}$ is the permutation symbol. The letter $C$ represents a
constant, independent of $\varepsilon$, that can represent different values
from line to line.

Consider $\Omega\subset{\mathbb{R}}^{d}$, for $d\geq 2$, a simply connected
and bounded domain of class $C^{1,1}$, occupied by a deformable electroelastic
material, and let $Y\coloneqq\left[-\frac{1}{2},\frac{1}{2}\right]^{d}$ be the
unit cell in ${\mathbb{R}}^{d}$. The unit cell $Y$ is decomposed into:

$Y=Y_{s}\cup Y_{f}\cup\Gamma,$

where $Y_{s}$, representing the inclusion, and $Y_{f}$, representing the
matrix, are open sets in $\mathbb{R}^{d}$, and $\Gamma$ is the closed
$C^{1,1}$ interface that separates them.

Let $i=(i_{1},\ldots,i_{d})\in{\mathbb{Z}}^{d}$ be a vector of indices and
$\\{\mathrm{e}^{1},\ldots,\mathrm{e}^{d}\\}$ be the canonical basis of
${\mathbb{R}}^{d}$. For a fixed small $\varepsilon>0,$ we define the dilated
sets:

$\displaystyle Y^{\varepsilon}\coloneqq\varepsilon
Y,~{}~{}Y^{\varepsilon}_{i}\coloneqq\varepsilon(Y+i),~{}~{}Y^{\varepsilon}_{i,s}\coloneqq\varepsilon(Y_{s}+i),~{}~{}Y^{\varepsilon}_{i,f}\coloneqq\varepsilon(Y_{f}+i),~{}~{}\Gamma^{\varepsilon}_{i}\coloneqq\partial
Y^{\varepsilon}_{i,s}.$

Typically, in homogenization theory, the positive number $\varepsilon\ll 1$ is
referred to as the size of the microstructure. The effective or homogenized
response of the given suspension corresponds to the case $\varepsilon=0$,
whose derivation and justification is the main focus of this paper.

We denote by $\mathbf{n}_{i},~{}\mathbf{n}_{\Gamma}$ and
$\mathbf{n}_{\partial\Omega}$ the unit normal vectors to
$\Gamma^{\varepsilon}_{i}$ pointing outward $Y^{\varepsilon}_{i,s}$, on
$\Gamma$ pointing outward $Y_{s}$ and on $\partial\Omega$ pointing outward,
respectively; and also, we denote by $\operatorname{d\\!}\mathcal{H}^{d-1}$
the $(d-1)$-dimensional Hausdorff measure. In addition, we define the sets:

$\displaystyle\begin{split}I^{\varepsilon}&\coloneqq\\{i\in{\mathbb{Z}}^{d}\colon
Y^{\varepsilon}_{i}\subset\Omega\\},~{}~{}\Omega_{s}^{\varepsilon}\coloneqq\bigcup_{i\in
I^{\varepsilon}}Y_{i,s}^{\varepsilon},~{}~{}\Omega_{f}^{\varepsilon}\coloneqq\Omega\setminus\Omega_{s}^{\varepsilon},~{}~{}\Gamma^{\varepsilon}\coloneqq\bigcup_{i\in
I^{\varepsilon}}\Gamma^{\varepsilon}_{i},\\\
J^{\varepsilon}&\coloneqq\left\\{j\in{\mathbb{Z}}^{d}\colon
Y_{j}^{\varepsilon}\cap\left({\mathbb{R}}^{d}\setminus\Omega\right)\neq\varnothing\text{
and }Y_{j}^{\varepsilon}\cap\Omega\neq\varnothing\right\\},\\\
Z_{i}^{\varepsilon}&\coloneqq\begin{cases}Y_{i}^{\varepsilon}&\text{ if }i\in
I^{\varepsilon},\\\ Y_{i}^{\varepsilon}\cap\Omega&\text{ if }i\in
J^{\varepsilon},\end{cases}\end{split}$ (2.1)

see Figure 1.

$\mathbf{n}$
---
$Y_{s}$
---
$\varepsilon$
---
$\Omega^{\varepsilon}_{f}$
---
$\Omega^{\varepsilon}_{s}$
---
$\Omega$
---
$Y$
---
Figure 1: Reference cell $Y$ and domain $\Omega$.

The following spaces are used throughout this paper.

  * •

$C_{c}(\Omega)$ – the space of continuous functions with compact support in
$\Omega$;

  * •

$C_{\mathrm{per}}(Y)$ – the subspace of $C({\mathbb{R}}^{d})$ of $Y$-periodic
functions;

  * •

$C^{\infty}_{\mathrm{per}}(Y)$ – the subspace of
$C^{\infty}({\mathbb{R}}^{d})$ of $Y$-periodic functions;

  * •

$H^{1}_{\mathrm{per}}(Y)$ – the closure of $C^{\infty}_{\mathrm{per}}(Y)$ in
the $H^{1}$-norm;

  * •

$\mathcal{D}(\Omega,X)$ with $X$ being a Banach space – the space of
infinitely differentiable functions from $\Omega$ to $X$, whose support is a
compact set of $\mathbb{R}^{d}$ contained in $\Omega$.

  * •

$L^{p}(\Omega,X)$ with $X$ being a Banach space and $1\leq p\leq\infty$ – the
space of measurable functions $w\colon x\in\Omega\mapsto w(x)\in X$ such that
$\left\lVert
w\right\rVert_{L^{p}(\Omega,X)}\coloneqq\left(\int_{\Omega}\left\lVert
w(x)\right\rVert^{p}_{X}\operatorname{d\\!}x\right)^{\frac{1}{p}}<\infty.$

  * •

$L^{p}_{\mathrm{per}}\left(Y,C(\bar{\Omega})\right)$ – the space of measurable
functions $w\colon y\in Y\mapsto w(\cdot,y)\in C(\bar{\Omega})$, such that $w$
is $Y$-periodic with respect to $y$ and
$\int_{Y}\left(\sup_{x\in\bar{\Omega}}\left\lvert
w(x,y)\right\rvert\right)^{p}\operatorname{d\\!}y<\infty.$

  * •

$\mathcal{M}(\Omega)$ – the space of finite Radon measures on $\Omega$, i.e.,
the dual space of $C_{c}(\Omega)$. This space is equipped with the total
variation norm

$\displaystyle\left\lVert\mu\right\rVert_{\mathcal{M}}\coloneqq\left\lvert\mu\right\rvert(\Omega)=\int_{\Omega}\left\lvert\mu\right\rvert(\operatorname{d\\!}x)=\sup\left\\{\int_{\Omega}\phi(x)\mu(\operatorname{d\\!}x)\colon\phi\in
C_{c}(\Omega),\left\lVert\phi\right\rVert_{L^{\infty}(\Omega)}\leq
1\right\\}.$

We then define

$\displaystyle\mathcal{M}(\Omega,{\mathbb{R}}^{d})$
$\displaystyle\coloneqq\left\\{\mu=\mu_{i}\mathrm{e}^{i}\colon\mu_{i}\in\mathcal{M}(\Omega),\text{
for all }1\leq i\leq d\right\\},\text{ and }$
$\displaystyle\mathcal{M}\left(\Omega,X(Y,{\mathbb{R}}^{d})\right)$
$\displaystyle\coloneqq\left\\{\mu\in\mathcal{M}\left(\Omega\times
Y,{\mathbb{R}}^{d}\right)\colon\mu(x,\cdot)\in X(Y,{\mathbb{R}}^{d})\text{ for
}x\in\Omega\right\\},$

where $X(Y,{\mathbb{R}}^{d})$ is a given space of functions from
$Y\to{\mathbb{R}}^{d}$, cf. [31, 30, 2].

  * •

$\mathrm{BV}(\Omega,{\mathbb{R}}^{d})$ – the space of $d-$dimensional vector
valued $L^{1}$\- functions, whose Jacobians (in distributional sense) are
$d\times d-$matrices of finite Radon measures on $\Omega$, i.e,
$\mathbf{u}=u_{i}\mathrm{e}^{i}\in\mathrm{BV}(\Omega,{\mathbb{R}}^{d})$ iff
$u_{i}\in L^{1}(\Omega)$ and its distributional derivative $\frac{\partial
u_{i}}{\partial x_{j}}\in\mathcal{M}(\Omega)$ for all $1\leq i,j\leq d$. We
say a sequence of distributions $\left\\{T_{n}\right\\}$ in
$\mathcal{D}^{\prime}(\Omega)$ converges to $T$ in distributional sense if
$\left\langle
T_{n},\varphi\right\rangle\xrightarrow[\hphantom{~{}2~{}}]{}\left\langle
T_{n},\varphi\right\rangle$ for any $\varphi\in\mathcal{D}(\Omega)$.

Fix $p>1$ and $0\leq\alpha\leq\min\left\\{1,p-1\right\\}$. Suppose
$\mathbf{a}\colon Y\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ satisfies

  1. (A1)

Measurability and $Y-$periodicity: for any $\xi\in{\mathbb{R}}^{d}$, the
function $\xi\mapsto\mathbf{a}(\cdot,\xi)$ is measurable and

$\mathbf{a}(z+m\mathrm{e}^{k},\xi)=\mathbf{a}(z,\xi),$

for all $z\in{\mathbb{R}}^{d}$, $m\in{\mathbb{Z}}$, and
$k\in\\{1,\ldots,d\\}$.

  2. (A2)

Boundedness: for all $z\in Y$, there exists $\Lambda_{*}>0$, such that

$\left\lvert\mathbf{a}(z,0)\right\rvert\leq\Lambda_{*}.$

  3. (A3)

Continuity: for a.e. $z\in Y$, there exists $\Lambda_{o}>0$ such that the
function $\xi\mapsto\mathbf{a}\left(z,\xi\right)$ satisfies

$\displaystyle\left\lvert\mathbf{a}(z,\xi_{1})-\mathbf{a}(z,\xi_{2})\right\rvert\leq\Lambda_{o}\left(1+\left\lvert\xi_{1}\right\rvert^{2}+\left\lvert\xi_{2}\right\rvert^{2}\right)^{\frac{p-1-\alpha}{2}}\left\lvert\xi_{1}-\xi_{2}\right\rvert^{\alpha},$

for all $\xi_{1},\xi_{2}\in Y$, where $\alpha$ is described above.

  4. (A4)

Monotonicity: there exists $\lambda_{o}>0$ such that for a.e. $z\in Y$, we
have

$\displaystyle\left[\mathbf{a}(z,\xi_{1})-\mathbf{a}(z,\xi_{2})\right]\cdot(\xi_{1}-\xi_{2})\geq\lambda_{o}\left(1+\left\lvert\xi_{1}\right\rvert^{2}+\left\lvert\xi_{2}\right\rvert^{2}\right)^{\frac{p-2}{2}}\left\lvert\xi_{1}-\xi_{2}\right\rvert^{2},$

for all $\xi_{1},\xi_{2}\in{\mathbb{R}}^{d}$.

We also introduce a few more definitions that are necessary for the elasticity
equation. For $\mathbf{u}\in H^{1}(\Omega,{\mathbb{R}}^{d})$, we define the
symmetric gradient, also known as the linearized strain tensor,

$\displaystyle{\mathbb{D}}(\mathbf{u})\coloneqq\frac{\nabla\mathbf{u}+\nabla\mathbf{u}^{\top}}{2}.$
(2.2)

Let $0<\lambda_{e}<\Lambda_{e}<\infty$. We denote by
$\mathfrak{M}(\lambda_{e},\Lambda_{e})$ the set of all fourth-order tensors
$\mathbf{B}=\left(\mathbf{B}_{ijkh}\right)_{1\leq i,j,k,h\leq d}$ satisfying

  1. (B1)

Boundedness and measurability: there exists $\Lambda_{e}>0$ such that

$\left\lVert\mathbf{B}\right\rVert_{L^{\infty}}\leq\Lambda_{e}.$

  2. (B2)

Ellipticity: there exists $\lambda_{e}>0$ such that, for all $d\times
d-$matrices $\mathbf{c}$ and for all $x\in{\mathbb{R}}^{d}$, we have

$\mathbf{B}(x)\mathbf{c}:\mathbf{c}\geq\lambda_{e}\left\lvert\mathbf{c}\right\rvert^{2},$

where “$:$” represents the Frobenius inner product.

Here, we recall that for two matrices $\mathbf{c}$ and $\mathbf{d}$,
$\mathbf{B}\mathbf{c}\coloneqq\left((\mathbf{B}_{ijkh}\mathbf{c}_{kh})_{ij}\right)_{1\leq
i,j\leq d}$ and
$\mathbf{B}\mathbf{c}:\mathbf{d}\coloneqq\mathbf{B}_{ijkh}\mathbf{c}_{ij}\mathbf{d}_{kh}$,
with $1\leq i,j,k,h\leq d$. We say that a fourth-order tensor $\mathbf{B}$ is
symmetric if $\mathbf{B}_{ijkh}=\mathbf{B}_{jikh}=\mathbf{B}_{ijhk}$ for any
$1\leq i,j,k,h\leq d$. We denote by
$\mathfrak{M}_{\mathrm{per}}(\lambda_{e},\Lambda_{e})$ the subset of
$\mathfrak{M}(\lambda_{e},\Lambda_{e})$ consisting of $Y-$periodic tensors,
and by $\mathfrak{M}_{\mathrm{sym}}(\lambda_{e},\Lambda_{e})$ the subset of
$\mathfrak{M}(\lambda_{e},\Lambda_{e})$ consisting of symmetric tensors. We
define
$\mathfrak{M}_{\mathrm{mat}}\coloneqq\mathfrak{M}_{\mathrm{sym}}\cap\mathfrak{M}_{\mathrm{per}}$.

####  2.1.1 The two-scale convergence method

The proof of the main result in this paper will be based on the theory of two-
scale convergence that was first introduced by G. Nguetseng [45] and further
developed by G. Allaire [1]. In this section, we present important definitions
and results which are relevant to this paper, and whose proofs can be found in
[1, 45, 2, 31, 26, 41, 53, 20].

######  Definition 1 ($L^{p}$-admissible test function).

Let $1\leq p<+\infty$. A function $\psi\in L^{p}(\Omega\times Y)$,
$Y$-periodic in the second variable, is called an $L^{p}$-admissible test
function if, for all $\varepsilon>0$,
$\psi\left(\cdot,\frac{\cdot}{\varepsilon}\right)$ is measurable and

$\displaystyle\lim_{\varepsilon\to
0}\int_{\Omega}\left\lvert\psi\left(x,\frac{x}{\varepsilon}\right)\right\rvert^{p}\operatorname{d\\!}x=\frac{1}{\left\lvert
Y\right\rvert}\int_{\Omega}\int_{Y}\left\lvert\psi(x,y)\right\rvert^{p}\operatorname{d\\!}y\operatorname{d\\!}x.$
(2.3)

It is known that functions that belong to the spaces
$\mathcal{D}\left(\Omega,C_{\mathrm{per}}^{\infty}(Y)\right)$,
$C\left(\bar{\Omega},C_{\mathrm{per}}(Y)\right)$,
$L^{p}_{\mathrm{per}}\left(Y,C(\bar{\Omega})\right)$ or
$L^{p}\left(\Omega,C_{\mathrm{per}}(Y)\right)$ are admissible [1], but the
precise characterization of admissible test functions is still an open
question.

######  Definition 2 (Two-scale convergence).

A sequence $\\{v^{\varepsilon}\\}_{\varepsilon>0}$ in $L^{p}(\Omega)$
($\mathcal{M}(\Omega)$, respectively) is said to _(weakly) two-scale converge_
to $v=v(x,y)$ in $L^{p}(\Omega\times Y)$ ($\mathcal{M}(\Omega)$,
respectively), with $v\in L^{p}(\Omega\times Y)$ ($\mathcal{M}(\Omega)$,
respectively), and we write
$v^{\varepsilon}\xrightharpoonup[\hphantom{~{}2~{}}]{2}v$ in
$L^{p}(\Omega\times Y)$, if and only if:

$\displaystyle\lim_{\varepsilon\to
0}\int_{\Omega}v^{\varepsilon}(x)\psi\left(x,\frac{x}{\varepsilon}\right)\operatorname{d\\!}x=\frac{1}{\left\lvert
Y\right\rvert}\int_{\Omega}\int_{Y}v(x,y)\psi(x,y)\operatorname{d\\!}y\operatorname{d\\!}x,$
(2.4)

for any test function $\psi=\psi(x,y)$ with
$\psi\in\mathcal{D}\left(\Omega,C_{\mathrm{per}}^{\infty}(Y)\right)$.

In particular, if $v^{\varepsilon}\xrightharpoonup[\hphantom{~{}2~{}}]{2}v$
and $\left\lVert
v^{\varepsilon}\right\rVert_{L^{p}(\Omega)}\xrightarrow[\hphantom{~{}2~{}}]{}\left\lVert
v\right\rVert_{L^{p}(\Omega\times Y)}$, then we say that $v^{\varepsilon}$
_strongly two-scale converges_ to $v$ in $L^{p}(\Omega\times Y)$, and we write
$v^{\varepsilon}\xrightarrow[\hphantom{~{}2~{}}]{2}v$ in $L^{p}(\Omega\times
Y)$.

We note that any bounded sequence $v^{\varepsilon}\in L^{p}(\Omega)$, with
$1<p<\infty$, ($\mathcal{M}(\Omega)$, respectively) has a subsequence that
two-scale converges to a limit $v^{0}\in L^{p}(\Omega\times Y)$
($\mathcal{M}(\Omega\times Y)$, respectively), cf. [1, 45, 2, 31, 26, 30, 53,
20, 41].

The strong two-scale convergence plays an important role in establishing
corrector results in homogenization, cf. [1, Theorem 1.8] and [41, Theorem
11]:

######  Lemma 2.1.

Let $\left\\{v^{\varepsilon}\right\\}_{\varepsilon>0}$ be a sequence in
$L^{p}(\Omega)$ that strongly two-scale converges to $v\in L^{p}(\Omega\times
Y)$. Suppose further that $v$ is admissible, in the sense of Definition 1.
Then

$\displaystyle\lim_{\varepsilon\to 0}\left\lVert
v^{\varepsilon}(x)-v\left(x,\frac{x}{\varepsilon}\right)\right\rVert_{L^{p}(\Omega)}=0.$
(2.5)

For $v\in L^{p}(\Omega\times Y)$, the function
$v\left(x,\frac{x}{\varepsilon}\right)$ is not necessarily a measurable
function [1, 53]. This explains why it is required for $v$ to be admissible in
Lemma 2.1. We can circumvent this assumption by introducing the so-called
_coarse-scale averaging operator_ ${\mathbb{M}}^{\varepsilon}$, which will be
defined next. For each $z\in{\mathbb{R}}^{d}$, let $[z]_{Y}$ be the integer
part of $z$, more precisely, $[z]_{Y}\coloneqq k_{i}\mathrm{e}^{i}$ with
$k_{i}\in{\mathbb{Z}}$ such that $\left\\{z\right\\}_{Y}\coloneqq z-[z]_{Y}\in
Y$. For each function $v\in L^{p}(\Omega\times Y)$, let

$\displaystyle({\mathbb{M}}^{\varepsilon}v)(x,y)\coloneqq\int_{Y}v\left(\varepsilon\left[\frac{x}{\varepsilon}\right]_{Y}+\varepsilon\xi,y\right)\operatorname{d\\!}\xi,\quad\text{for
}(x,y)\in\Omega\times Y.$ (2.6)

It is known that ${\mathbb{M}}^{\varepsilon}$ is a bounded linear operator and
that $({\mathbb{M}}^{\varepsilon}v)(x,y)$ and
$({\mathbb{M}}^{\varepsilon}v)\left(x,\frac{x}{\varepsilon}\right)$ are both
measurable [53]. The following result is due to Visintin [53, Proposition
2.3]:

######  Lemma 2.2.

A sequence $\left\\{v^{\varepsilon}\right\\}_{\varepsilon>0}\subset
L^{p}(\Omega)$ strongly two-scale converges to $v\in L^{p}(\Omega\times Y)$ if
and only if

$\displaystyle\lim_{\varepsilon\to 0}\left\lVert
v^{\varepsilon}(x)-\left({\mathbb{M}}^{\varepsilon}v\right)\left(x,\frac{x}{\varepsilon}\right)\right\rVert_{L^{p}(\Omega)}=0.$

In particular, if $v$ is admissible, then the operator
${\mathbb{M}}^{\varepsilon}$ can be dropped.

###  2.2 Formulation of the problem: the fine-scale coupled system

In this section, we set up the fine-scale problem. Denote by
$\mathbf{C}\in\mathfrak{M}_{\mathrm{per}}(\lambda_{e},\Lambda_{e})$ the
electrostriction tensor and by
$\mathbf{B}\in\mathfrak{M}_{\mathrm{mat}}(\lambda_{e},\Lambda_{e})$ the
elasticity tensor. We assume further that $Y$ is a disjoint union of finite
subdomains with piecewise $C^{1,\alpha}-$boundaries and $\mathbf{B}$ is Hölder
continuous on the closure of each subdomain. Let $\mathbf{a}\colon
Y\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ satisfying (A1)–(A4), which
pertains to a (nonlinear) constitutive law between the electric displacement
and the electric field. Furthermore, let $\mathbf{g}\in
L^{r}(\Omega,{\mathbb{R}}^{d})$ for some $r>1$, and $f\in
L^{p^{\prime}}(\Omega)$, where $p^{\prime}$ is the Hölder conjugate of $p$,
i.e. $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. Then the fine-scale displacement
$\mathbf{u}^{\varepsilon}\in\mathrm{BV}(\Omega,{\mathbb{R}}^{d})$ and the
electrostatic potential $\varphi^{\varepsilon}\in W_{0}^{1,p}(\Omega)$ satisfy
the following coupled system [51, 32, 52]:

$\displaystyle-\operatorname{{div}}\left[\mathbf{a}\left(\frac{x}{\varepsilon},\nabla\varphi^{\varepsilon}\right)\right]$
$\displaystyle=f$ $\displaystyle\text{ in }\Omega,$ (2.7a)
$\displaystyle-\operatorname{{div}}\left[\mathbf{B}\left(\frac{x}{\varepsilon}\right)\nabla\mathbf{u}^{\varepsilon}+\mathbf{C}\left(\frac{x}{\varepsilon}\right)\left(\nabla\varphi^{\varepsilon}\otimes\nabla\varphi^{\varepsilon}\right)\right]$
$\displaystyle=\mathbf{g}$ $\displaystyle\text{ in }\Omega,$ (2.7b)

together with the balance equations:

$\displaystyle\int_{\Gamma_{i}^{\varepsilon}}\left\llbracket\mathbf{B}\left(\frac{x}{\varepsilon}\right){\mathbb{D}}(\mathbf{u}^{\varepsilon})+\mathbf{C}\left(\frac{x}{\varepsilon}\right)\left(\nabla\varphi^{\varepsilon}\otimes\nabla\varphi^{\varepsilon}\right)\right\rrbracket\mathbf{n}_{i}\operatorname{d\\!}\mathcal{H}^{d-1}$
$\displaystyle=0,$ (2.8a)
$\displaystyle\int_{\Gamma_{i}^{\varepsilon}}\left\\{\left\llbracket\mathbf{B}\left(\frac{x}{\varepsilon}\right){\mathbb{D}}(\mathbf{u}^{\varepsilon})+\mathbf{C}\left(\frac{x}{\varepsilon}\right)\left(\nabla\varphi^{\varepsilon}\otimes\nabla\varphi^{\varepsilon}\right)\right\rrbracket\mathbf{n}_{i}\right\\}\times\mathbf{n}_{i}\operatorname{d\\!}\mathcal{H}^{d-1}$
$\displaystyle=0,$ (2.8b)

and the boundary conditions:

$\displaystyle\varphi^{\varepsilon}$ $\displaystyle=0,\text{ on
}\partial\Omega,$ (2.9a) $\displaystyle\mathbf{u}^{\varepsilon}$
$\displaystyle=0,\text{ on }\partial\Omega.$ (2.9b)

Here, $\left\llbracket\,\cdot\,\right\rrbracket$ denotes the jump on the
interfaces $\Gamma^{\varepsilon}_{i}$.

Next, we introduce the variational formulation for (2.7)-(2.9). To simplify
the notation, we define

$\displaystyle\boldsymbol{\tau}^{\varepsilon}\coloneqq\nabla\varphi^{\varepsilon}\otimes\nabla\varphi^{\varepsilon},$

which will be called the Maxwell stress tensor (it is worth mentioning that
the most general version of the Maxwell stress tensor has three additional
terms [54]). Then, we seek for $\varphi^{\varepsilon}\in W_{0}^{1,p}(\Omega)$
and $\mathbf{u}^{\varepsilon}\in\mathrm{BV}(\Omega,{\mathbb{R}}^{d})$ such
that, for all $\eta\in W^{-1,p^{\prime}}(\Omega)$ and
$\mathbf{v}\in\mathcal{D}(\Omega,{\mathbb{R}}^{d})$, the following holds

$\displaystyle\begin{split}\int_{\Omega}\mathbf{a}\left(\frac{x}{\varepsilon},\nabla\varphi^{\varepsilon}\right){\color[rgb]{0,0,1}\cdot}\nabla\eta\operatorname{d\\!}x+\int_{\Omega}\mathbf{B}\left(\frac{x}{\varepsilon}\right){\mathbb{D}}(\mathbf{u}^{\varepsilon}):{\mathbb{D}}(\mathbf{v})\operatorname{d\\!}x+\int_{\Omega}\mathbf{C}\left(\frac{x}{\varepsilon}\right)\,\boldsymbol{\tau}^{\varepsilon}:{\mathbb{D}}(\mathbf{v})\operatorname{d\\!}x\\\
=\int_{\Omega}\mathbf{g}\cdot\mathbf{v}\operatorname{d\\!}x+\int_{\Omega}f\,\eta\operatorname{d\\!}x.\end{split}$
(2.10)

By setting $\mathbf{v}=0$, we seek for a unique solution
$\varphi^{\varepsilon}\in W_{0}^{1,p}(\Omega)$ of the electrostatic problem

$\displaystyle\int_{\Omega}\mathbf{a}\left(\frac{x}{\varepsilon},\nabla\varphi^{\varepsilon}\right)\cdot\nabla\eta\operatorname{d\\!}x=\int_{\Omega}f\,\eta\operatorname{d\\!}x,\quad\forall\eta\in
W^{-1,p^{\prime}}(\Omega).$ (2.11)

By setting $\eta=0$, we seek for
$\mathbf{u}^{\varepsilon}\in\mathrm{BV}(\Omega,{\mathbb{R}}^{d})$ such that

$\displaystyle\int_{\Omega}\mathbf{B}\left(\frac{x}{\varepsilon}\right)\,{\mathbb{D}}(\mathbf{u}^{\varepsilon}):{\mathbb{D}}(\mathbf{v})\operatorname{d\\!}x=\int_{\Omega}\mathbf{g}\cdot\mathbf{v}\operatorname{d\\!}x-\int_{\Omega}\mathbf{C}\left(\frac{x}{\varepsilon}\right)\,\boldsymbol{\tau}^{\varepsilon}:{\mathbb{D}}(\mathbf{v})\operatorname{d\\!}x,\quad\forall\,\mathbf{v}\in\mathcal{D}(\Omega,{\mathbb{R}}^{d}).$
(2.12)

##  3 Main results

This section is dedicated to presenting the main outcomes of this paper.
Before we state our main theorem, which provides the homogenization result for
the coupled system (2.7), we will introduce a _first-order corrector_ for the
solution $\varphi^{\varepsilon}\in W_{0}^{1,p}(\Omega)$ of the electrostatic
equation.

To begin, we consider the homogenization of the electrostatic equation using
its variational formulation (2.11). The proof can be found in Section 4.

######  Proposition 3.1.

Let $\varphi^{\varepsilon}\in W_{0}^{1,p}(\Omega)$ be the unique solution of
(2.11). There exist $\varphi^{0}\in W_{0}^{1,p}(\Omega)$ and $\varphi^{1}\in
L^{p}(\Omega,W_{\mathrm{per}}^{1,p}(Y)/{\mathbb{R}})$ such that

$\displaystyle\varphi^{\varepsilon}\xrightharpoonup[\hphantom{~{}2~{}}]{2}\varphi^{0},\text{
and
}\nabla\varphi^{\varepsilon}\xrightharpoonup[\hphantom{~{}2~{}}]{2}\nabla\varphi^{0}+\nabla_{y}\varphi^{1}$

where the two-scale limits $\varphi^{0}$ and $\varphi^{1}$ satisfy the
following system

$\displaystyle-\operatorname{{div}}\left[\frac{1}{\left\lvert
Y\right\rvert}\int_{Y}\mathbf{a}\left(y,\nabla\varphi^{0}+\nabla_{y}\varphi^{1}\right)\operatorname{d\\!}y\right]$
$\displaystyle=f,\text{ in }\Omega,$ (3.1a)
$\displaystyle-\operatorname{{div}}_{y}\mathbf{a}\left(y,\nabla\varphi^{0}+\nabla_{y}\varphi^{1}\right)$
$\displaystyle=0,\text{ in }\Omega\times Y.$ (3.1b)

We now state the first main result of this paper - an explicit _corrector
result_ for the electrostatic problem, whose proof can be found in
LABEL:sec:proof-crefs-results.

######  Theorem 1 (Explicit corrector result).

Suppose $p>1.$ Let $(\varphi^{0},\varphi^{1})\in W_{0}^{1,p}(\Omega)\times
L^{p}(\Omega,W^{1,p}_{\mathrm{per}}(Y)/{\mathbb{R}})$ be the solution of
(3.1). Then for $1\leq i\leq d$,

$\displaystyle\frac{\partial\varphi^{\varepsilon}}{\partial
x_{i}}\xrightarrow[\hphantom{~{}2~{}}]{2}\frac{\partial\varphi^{0}}{\partial
x_{i}}+\frac{\partial\varphi^{1}}{\partial y_{i}}\qquad\text{ in
}L^{p}(\Omega\times Y)$

As a consequence,

$\displaystyle\lim_{\varepsilon\to
0}\left\lVert\nabla\varphi^{\varepsilon}(x)-\nabla\varphi^{0}(x)-{\mathbb{M}}^{\varepsilon}(\nabla_{y}\varphi^{1})\left(x,\frac{x}{\varepsilon}\right)\right\rVert_{L^{p}(\Omega,{\mathbb{R}}^{d})}=0.$
(3.2)

In particular, if $\varphi^{1}$ is admissible, then

$\displaystyle\lim_{\varepsilon\to
0}\left\lVert\nabla\varphi^{\varepsilon}(x)-\nabla\varphi^{0}(x)-\nabla_{y}\varphi^{1}\left(x,\frac{x}{\varepsilon}\right)\right\rVert_{L^{p}(\Omega,{\mathbb{R}}^{d})}=0.$
(3.3)

######  Remark 3.2.

Theorem 1 implies the corrector result for the linear case, i.e., when
$\mathbf{a}(y,\xi)=\mathbf{b}(y)\xi$ for some bounded, elliptic, and
$Y$-periodic matrix $\mathbf{b}$. Indeed, fine-scale equation

$\displaystyle-\operatorname{{div}}\left(\mathbf{b}\left(\frac{x}{\varepsilon}\right)\nabla\varphi^{\varepsilon}\right)=f\text{
in }\Omega,\qquad\varphi^{\varepsilon}=0\text{ on }\partial\Omega,$ (3.4)

has a unique solution $\varphi^{\varepsilon}\in H^{1}_{0}(\Omega)$ such that

for some $\varphi^{0}\in H_{0}^{1}(\Omega)$ and $\varphi^{1}\in
L^{2}\left(\Omega,H_{\mathrm{per}}^{1}(Y)/{\mathbb{R}}\right)$ that satisfy

$\displaystyle-\operatorname{{div}}\left[\frac{1}{\left\lvert
Y\right\rvert}\mathbf{b}(y)\left(\nabla\varphi^{0}+\nabla_{y}\varphi^{1}\right)\right]$
$\displaystyle=f,\qquad\text{ in }\Omega,$
$\displaystyle-\operatorname{{div}}_{y}\left[\mathbf{b}(y)\left(\nabla\varphi^{0}+\nabla_{y}\varphi^{1}\right)\right]$
$\displaystyle=0,\qquad\text{ in }\Omega\times Y.$

By introducing the cell problems

$\displaystyle\omega^{i}\in
H_{\mathrm{per}}^{1}(Y)/{\mathbb{R}},\qquad-\operatorname{{div}}_{y}\left[\mathbf{b}(y)(\mathrm{e}^{i}+\nabla_{y}\omega^{i}(y))\right]=0\quad\text{
in }Y,\quad 1\leq i\leq d,$

we have $\varphi^{0}\in H^{1}_{0}(\Omega)$ and $\varphi^{1}\in
L^{2}\left(\Omega,H_{\mathrm{per}}^{1}(Y)/{\mathbb{R}}\right)$ satisfy

$\displaystyle\begin{split}-\operatorname{{div}}\left(\mathbf{b}^{\hom}\nabla\varphi^{0}\right)&=f\text{
in }\Omega,\\\ \varphi^{1}(x,y)&=\frac{\partial\varphi^{0}}{\partial
x_{i}}(x)\,\omega^{i}(y),\end{split}$ (3.5)

where

$\displaystyle\mathbf{b}^{\hom}_{jk}=\frac{1}{\left\lvert
Y\right\rvert}\int_{Y}\mathbf{b}(y)\left(\mathrm{e}^{k}+\nabla\omega^{k}(y)\right)\cdot\left(\mathrm{e}^{j}+\nabla\omega^{j}(y)\right)\operatorname{d\\!}y,\quad
1\leq j,k\leq d.$

Since $\mathbf{b}^{\hom}$ is a constant elliptic matrix, if $f$ is smooth
enough (e.g., $f\in L^{\infty}(\Omega)$), then $\varphi^{0}\in
C(\bar{\Omega})$, and therefore, $\varphi^{1}\in L^{2}(Y,C(\bar{\Omega}))$, or
$\varphi^{1}$ is admissible. Therefore, (3.3) holds, and we recover the
classical corrector result for the linear case.

In general, when $\mathbf{a}$ is nonlinear, $\varphi^{1}$ does not admit a
finite representation as in (3.5), and thus, the admisssibility assumption is
necessary.

######  Remark 3.3.

In contrast to our Theorem 1, in the existing literature, the corrector
results for monotone operators are not explicit, in the sense that they are
written in the following form:

$\displaystyle\lim_{\varepsilon\to
0}\left\lVert\nabla\varphi^{\varepsilon}-\mathbf{p}\left(\frac{\cdot}{\varepsilon},M^{\varepsilon}(\nabla\varphi^{0})\right)\right\rVert_{L^{p}(\Omega,{\mathbb{R}}^{d})}=0,$

where the functions $\mathbf{p}$ and $M^{\varepsilon}$ will be defined in
LABEL:sec:prel-estim, cf. e.g., [23] (see also [1, Remark 3.7]) and references
therein.

######  Remark 3.4.

In [1, Theorem 3.6], the author obtained (3.3) for $p=2$ under the assumption
that $\varphi^{1}(x,y)$ is admissible in the sense of Definition 1. In Theorem
1, we are able to prove that (3.3) holds for any $p>1$. The corrector result
obtained in Theorem 1 is fundamental to obtain the closed system in Theorem 2
below. It is also useful in numerical computation and large-scale regularity
of homogenization of monotone problems [28, 10, 11, 21].

Finally, the arguments used in Theorem 1 can be employed to improve several
classical results, which will be presented in LABEL:sec:some-inter-corr.

We now introduce some auxiliary problems and definitions that will be
necessary for the statement of the main theorem.

To write the homogenized electrostatic equation, in (3.1b), we replace
$\nabla\varphi^{0}$ by $\xi\in{\mathbb{R}}^{d}$ and let
$\varphi^{1}=\xi\eta_{\xi}$ for some $\eta_{\xi}\in
W^{1,p}_{\mathrm{per}}(Y)/{\mathbb{R}}$ to obtain the cell problem

$\displaystyle-\operatorname{{div}}_{y}\mathbf{a}\left(y,\xi+\nabla_{y}\eta_{\xi}\right)=0.$
(3.6)

For $1\leq i,j\leq d,$ denote by $\mathbf{U}^{ij}$ the vector defined by
$\mathbf{U}^{ij}_{k}\coloneqq y_{j}\delta_{ik}$ and consider
$\boldsymbol{\Upsilon}^{ij}\in
W^{1,p}_{\mathrm{per}}(Y,{\mathbb{R}}^{d})/{\mathbb{R}}$ solving

$\displaystyle\begin{split}\operatorname{{div}}_{y}\left[\mathbf{B}(y)\,{\mathbb{D}}_{y}\left(\mathbf{U}^{ij}-\boldsymbol{\Upsilon}^{ij}\right)\right]&=0\text{
in }Y,\\\
\int_{\Gamma}\left\llbracket\mathbf{B}(y)\,{\mathbb{D}}_{y}\left(\mathbf{U}^{ij}-\boldsymbol{\Upsilon}^{ij}\right)\right\rrbracket\mathbf{n}_{\Gamma}\operatorname{d\\!}\mathcal{H}^{d-1}&=0,\\\
\int_{\Gamma}\left\llbracket\mathbf{B}(y){\mathbb{D}}_{y}\left(\mathbf{U}^{ij}-\boldsymbol{\Upsilon}^{ij}\right)\right\rrbracket\mathbf{n}_{\Gamma}\times\mathbf{n}_{\Gamma}\operatorname{d\\!}\mathcal{H}^{d-1}&=0,\end{split}$
(3.7)

and, also, consider $\boldsymbol{\chi}^{ij}\in
W^{1,p}_{\mathrm{per}}(Y,{\mathbb{R}}^{d})/{\mathbb{R}}$ solving

$\displaystyle\begin{split}\operatorname{{div}}_{y}\left[\mathbf{C}(y)\,{\mathbb{D}}_{y}\left(\boldsymbol{\chi}^{ij}\right)+\boldsymbol{\zeta}^{ij}\right]&=0\text{
in }Y,\\\
\int_{\Gamma}\left\llbracket\mathbf{C}(y){\mathbb{D}}_{y}\left(\boldsymbol{\chi}^{ij}\right)+\boldsymbol{\zeta}^{ij}\right\rrbracket\mathbf{n}_{\Gamma}\operatorname{d\\!}\mathcal{H}^{d-1}&=0,\\\
\int_{\Gamma}\left\llbracket\mathbf{C}(y){\mathbb{D}}_{y}\left(\boldsymbol{\chi}^{ij}\right)+\boldsymbol{\zeta}^{ij}\right\rrbracket\mathbf{n}_{\Gamma}\times\mathbf{n}_{\Gamma}\operatorname{d\\!}\mathcal{H}^{d-1}&=0,\end{split}$
(3.8)

with the matrix $\boldsymbol{\zeta}^{ij}$ given by

$\boldsymbol{\zeta}^{ij}(y)\coloneqq(\mathrm{e}^{i}+\nabla_{y}\eta_{\mathrm{e}^{i}}(y))\otimes(\mathrm{e}^{j}+\nabla_{y}\eta_{\mathrm{e}^{j}}(y)),~{}y\in
Y,$ (3.9)

where $\eta_{\mathrm{e}^{j}}$ given by (3.6) denotes the microscopic electric
stress tensor on $Y$.

######  Remark 3.5.

The cell problem (3.7) describes the local behavior of the elastic
displacement without the electric effect. The electric effect is captured in
(3.8). Note that the above cell problems (or similar) were observed in [32,
24, 25].

We also define:

$\displaystyle\begin{split}\mathbf{a}^{\hom}(\xi)&\coloneqq\frac{1}{\left\lvert
Y\right\rvert}\int_{Y}\mathbf{a}\left(y,\xi+\nabla_{y}\eta_{\xi}(y)\right)\operatorname{d\\!}y,\quad\xi\in{\mathbb{R}}^{d},\\\
\mathbf{B}^{\hom}_{ijmn}&\coloneqq\frac{1}{\left\lvert
Y\right\rvert}\int_{Y}\mathbf{B}(y)\,{\mathbb{D}}_{y}(\mathbf{U}^{ij}-\boldsymbol{\Upsilon}^{ij}):{\mathbb{D}}_{y}(\mathbf{U}^{mn}-\boldsymbol{\Upsilon}^{mn})\operatorname{d\\!}y,\quad
1\leq i,j,m,n\leq d,\\\ \mathbf{C}^{\hom}_{ij}&\coloneqq\frac{1}{\left\lvert
Y\right\rvert}\int_{Y}\left(\mathbf{C}(y)\,{\mathbb{D}}_{y}(\boldsymbol{\chi}^{ij})+\boldsymbol{\zeta}^{ij}\right)\operatorname{d\\!}y,\quad
1\leq i,j\leq d,\end{split}$ (3.10)

where $\mathbf{a}^{\hom}$ is the _effective electric conductivity_ , which is
monotone. The tensor
$\mathbf{B}^{\hom}\coloneqq\left\\{\mathbf{B}^{\hom}_{ijmn}\right\\}_{1\leq
i,j,m,n\leq d}$ is the _effective elasticity_ , and it is a fourth-rank tensor
that is symmetric and elliptic. And lastly, $\mathbf{C}^{\hom}$ is the
_effective electrostriction tensor_.

We now state the main result of this paper, Theorem 2, which provides the
homogenization for the coupled system (2.7). The proof of Theorem 2 is carried
out in LABEL:sec:proof-crefthm:main.

######  Theorem 2.

Suppose further that $p\geq 2$. The solution
$(\varphi^{\varepsilon},\mathbf{u}^{\varepsilon})\in W_{0}^{1,p}(\Omega)\times
W_{0}^{1,1}(\Omega,{\mathbb{R}}^{d})$ of (2.7) satisfies

$\displaystyle\varphi^{\varepsilon}\xrightharpoonup[\hphantom{~{}2~{}}]{}\varphi^{0}\text{
in }W_{0}^{1,p}(\Omega),\qquad\text{ and
}\qquad\mathbf{u}^{\varepsilon}\xrightharpoonup[\hphantom{~{}2~{}}]{}\mathbf{u}^{0}\text{
in distribution}$

where $\mathbf{u}^{0}\in W_{0}^{1,1}(\Omega,{\mathbb{R}}^{d})$ and
$\varphi^{0}\in W_{0}^{1,p}(\Omega)$ are solutions of

$\displaystyle\begin{aligned}
-\operatorname{{div}}\left[\mathbf{a}^{\hom}(\nabla\varphi^{0})\right]&=f&&\text{
in }\Omega,\\\
-\operatorname{{div}}\left[\mathbf{B}^{\hom}\,{\mathbb{D}}\left(\mathbf{u}^{0}\right)+\mathbf{C}^{\hom}(\nabla\varphi^{0}\otimes\nabla\varphi^{0})\right]&=\mathbf{g}&&\text{
in }\Omega,\\\ \mathbf{u}^{0}=0,\quad\varphi^{0}&=0,&&\text{ on
}\partial\Omega,\end{aligned}$ (3.11)

with $\mathbf{a}^{\hom},\mathbf{B}^{\hom},$ and $\mathbf{C}^{\hom}$ defined in
(3.10).

Moreover,

  * •

$\mathbf{a}^{\hom}$ is continuous and monotone, i.e., for every
$\xi_{1},\xi_{2}\in{\mathbb{R}}^{d}$, the following holds

$\displaystyle\begin{split}\left\lvert\mathbf{a}^{\hom}(\xi_{1})-\mathbf{a}^{\hom}(\xi_{2})\right\rvert&\leq\Lambda_{\hom}\left(1+\left\lvert\xi_{1}\right\rvert^{2}+\left\lvert\xi_{2}\right\rvert^{2}\right)^{\frac{p-2-\theta}{2}}\left\lvert\xi_{1}-\xi_{2}\right\rvert^{\theta},\\\
\left[\mathbf{a}^{\hom}(\xi_{1})-\mathbf{a}^{\hom}(\xi_{2})\right]\cdot\left(\xi_{1}-\xi_{2}\right)&\geq\lambda_{\hom}\left(1+\left\lvert\xi_{1}\right\rvert^{2}+\left\lvert\xi_{2}\right\rvert^{2}\right)^{\frac{p-2}{2}}\left\lvert\xi_{1}-\xi_{2}\right\rvert^{2},\end{split}$
(3.12)

where $\theta=\frac{\alpha}{2-\alpha},$ and $\lambda_{\hom},\Lambda_{\hom}>0$
are constants depending on
$d,p,\alpha,\beta,\lambda_{o},\Lambda_{o},\Lambda_{*}$, which are defined in
Section 2.1.

  * •

$\mathbf{B}^{\hom}$ is symmetric and elliptic.

######  Remark 3.6.

The result in Theorem 2 holds also for non-homogeneous Dirichlet boundary
conditions. In fact, Theorem 2 holds if $\varphi^{\varepsilon}=\varphi_{b}$ on
$\partial\Omega$, for some $\varphi_{b}\in W^{1,p^{+}}(\Omega),p^{+}>p$, and
(A2) is replaced by

$\displaystyle\left\lvert\mathbf{a}(y,\nabla\varphi_{b}(x))\right\rvert\leq\Lambda_{*},\qquad\text{a.e.
}y\in Y,x\in\Omega;$ (3.13)

with all the estimates changed accordingly to adapt to this case. For example,
the boundedness of $\varphi^{\varepsilon}$ can be proven as follows. By the
monotonicity (A4) and the boundedness (A2) of $\mathbf{a}$, and using Hölder,
Young and Poincaré inequalities, we obtain

$\displaystyle\lambda_{o}\left\lVert\nabla\varphi^{\varepsilon}-\nabla\varphi_{b}\right\rVert^{p}_{L^{p}}$
$\displaystyle\leq\int_{\Omega}\left[\mathbf{a}\left(\frac{x}{\varepsilon},\nabla\varphi^{\varepsilon}\right)-\mathbf{a}\left(\frac{x}{\varepsilon},\nabla\varphi_{b}\right)\right]\cdot(\nabla\varphi^{\varepsilon}-\nabla\varphi_{b})\operatorname{d\\!}x$
$\displaystyle=\int_{\Omega}f\left(\varphi^{\varepsilon}-\varphi_{b}\right)\operatorname{d\\!}x-\int_{\Omega}\mathbf{a}\left(\frac{x}{\varepsilon},\nabla\varphi_{b}\right)\cdot\left(\nabla\varphi^{\varepsilon}-\nabla\varphi_{b}\right)\operatorname{d\\!}x$
$\displaystyle\leq\left\lVert
f\right\rVert_{L^{p^{\prime}}}\left\lVert\varphi^{\varepsilon}-\varphi_{b}\right\rVert_{L^{p}}+C\Lambda_{*}\left\lVert\nabla\varphi^{\varepsilon}-\nabla\varphi_{b}\right\rVert_{L^{p}}$
$\displaystyle\leq C(\left\lVert
f\right\rVert_{L^{p^{\prime}}}+\Lambda_{*})\left\lVert\nabla\varphi^{\varepsilon}-\nabla\varphi_{b}\right\rVert_{L^{p}}$
$\displaystyle\leq\frac{C}{p^{\prime}\mu}\left(\left\lVert
f\right\rVert_{L^{p^{\prime}}}+\Lambda_{*}\right)^{p^{\prime}}+C\,\frac{\mu}{p}\left\lVert\nabla\varphi^{\varepsilon}-\nabla\varphi_{b}\right\rVert_{L^{p}}^{p},$

for some $\mu>0$. Choosing $\mu$ small enough, we conclude

$\displaystyle\left\lVert\nabla\varphi^{\varepsilon}-\nabla\varphi_{b}\right\rVert^{p}_{L^{p}}\leq
C\left(\left\lVert
f\right\rVert_{L^{p^{\prime}}}+\Lambda_{*}\right)^{p^{\prime}}\leq
C\left(\left\lVert
f\right\rVert_{L^{p^{\prime}}}^{p^{\prime}}+\Lambda_{*}^{p^{\prime}}\right);$

and thus by Poincaré inequality,

$\displaystyle\left\lVert\varphi^{\varepsilon}\right\rVert_{W^{1,p}}^{p}$
$\displaystyle\leq
C\left\lVert\varphi^{\varepsilon}-\varphi_{b}\right\rVert_{W^{1,p}}^{p}+C\left\lVert\varphi_{b}\right\rVert_{W^{1,p}}^{p}$
$\displaystyle\leq
C\left\lVert\nabla\varphi^{\varepsilon}-\nabla\varphi_{b}\right\rVert^{p}_{L^{p}}+C\left\lVert\varphi_{b}\right\rVert_{W^{1,p}}^{p}$
$\displaystyle\leq C\left(\left\lVert
f\right\rVert^{p^{\prime}}_{L^{p^{\prime}}}+\left\lVert\varphi_{b}\right\rVert^{p}_{W^{1,p}}+\Lambda_{*}^{p^{\prime}}\right).$

######  Remark 3.7.

The homogenization results of the coupled systems in [32, 51] follow directly
as particular cases of Theorem 2, when $p=2$ and $\mathbf{a}$ is linear, i.e.,
$\mathbf{a}(y,\xi)=\mathbf{b}(y)\xi$ for some bounded, elliptic, and periodic
matrix $\mathbf{b}$.

######  Remark 3.8.

The enhancement effect, i.e., when the right hand side of (2.7a) has the form
$f(x)=\frac{1}{\varepsilon}f_{1}(x)f_{2}(\frac{x}{\varepsilon})$ for some
$f_{1}$ and $f_{2}$, as considered in [32], can be handled using the approach
developed in this paper.

######  Remark 3.9.

The proof in this paper can be straightforwardly extended to the case when
$\mathbf{B}$ is a function in $VMO(\Omega,{\mathbb{R}}^{d})$, see Remark 6.2.

##  4 Proof of Proposition 3.1

In (2.11), let $\eta=\varphi^{\varepsilon}\in W^{1,p}_{0}(\Omega)$ and use
(A2), (A4), Hölder inequality, and Poincaré inequality, to obtain

$\displaystyle\lambda_{o}\left\lVert\nabla\varphi^{\varepsilon}\right\rVert^{p}_{L^{p}(\Omega,{\mathbb{R}}^{d})}$
$\displaystyle\leq\int_{\Omega}\left[\mathbf{a}\left(\frac{x}{\varepsilon},\nabla\varphi^{\varepsilon}\right)-\mathbf{a}\left(\frac{x}{\varepsilon},0\right)\right]\cdot\nabla\varphi^{\varepsilon}\operatorname{d\\!}x$
$\displaystyle=\int_{\Omega}f\,\varphi^{\varepsilon}\operatorname{d\\!}x-\int_{\Omega}\mathbf{a}\left(\frac{x}{\varepsilon},0\right)\cdot\nabla\varphi^{\varepsilon}\operatorname{d\\!}x$
$\displaystyle\leq\left\lVert
f\right\rVert_{L^{p^{\prime}}(\Omega)}\left\lVert\varphi^{\varepsilon}\right\rVert_{L^{p}(\Omega)}+C\left\lVert\nabla\varphi^{\varepsilon}\right\rVert_{L^{p}(\Omega,{\mathbb{R}}^{d})}\leq
C\left\lVert
f\right\rVert_{L^{p^{\prime}}(\Omega)}\left\lVert\nabla\varphi^{\varepsilon}\right\rVert_{L^{p}(\Omega,{\mathbb{R}}^{d})}.$

The estimate above and Poincaré inequality deliver

$\displaystyle\left\lVert\varphi^{\varepsilon}\right\rVert^{p}_{W_{0}^{1,p}(\Omega)}\leq
C\left\lVert f\right\rVert^{p^{\prime}}_{L^{p^{\prime}}(\Omega)}.$ (4.1)

Therefore, there exist $\varphi^{0}\in W_{0}^{1,p}(\Omega)$ and
$\varphi^{1}\in L^{p}(\Omega,W_{\mathrm{per}}^{1,p}(Y)/{\mathbb{R}})$ such
that  For given functions $\mathbf{g}^{\prime}\in
L^{2}(\Omega,{\mathbb{R}}^{d})$ and $\mathbf{f}^{\prime}\in
L^{2}(\Omega,{\mathbb{R}}^{d\times d})$, let $\mathbf{w}^{\varepsilon}\in
W_{0}^{1,2}(\Omega,{\mathbb{R}}^{d})$ be the solution of

$\displaystyle\int_{\Omega}\mathbf{B}\left(\frac{x}{\varepsilon}\right)\,{\mathbb{D}}(\mathbf{w}^{\varepsilon}):{\mathbb{D}}(\mathbf{u})\operatorname{d\\!}x=\int_{\Omega}\mathbf{g}^{\prime}\cdot\mathbf{u}\operatorname{d\\!}x-\int_{\Omega}\mathbf{f}^{\prime}:{\mathbb{D}}(\mathbf{u})\operatorname{d\\!}x$
(6.1)

for all $\mathbf{u}\in\mathcal{D}(\Omega,{\mathbb{R}}^{d})$. This solution
$\mathbf{w}^{\varepsilon}\in W^{1,2}_{0}(\Omega,{\mathbb{R}}^{d})$ exists and
is unique by the Lax-Milgram theorem and

$\displaystyle\left\lVert{\mathbb{D}}\mathbf{w}^{\varepsilon}\right\rVert_{L^{2}(\Omega,{\mathbb{R}}^{d\times
d})}\leq
C_{2}\left(\left\lVert\mathbf{g}^{\prime}\right\rVert_{L^{2}(\Omega,{\mathbb{R}}^{d})}+\left\lVert\mathbf{f}^{\prime}\right\rVert_{L^{2}(\Omega,{\mathbb{R}}^{d\times
d})}\right),$ (6.2)

for some constant $C_{2}>0$ independent of $\varepsilon>0$. Then, Korn’s
inequality implies

$\displaystyle\left\lVert\mathbf{w}^{\varepsilon}\right\rVert_{W^{1,2}(\Omega,{\mathbb{R}}^{d})}\leq
C_{2}\left(\left\lVert\mathbf{g}^{\prime}\right\rVert_{L^{2}(\Omega,{\mathbb{R}}^{d})}+\left\lVert\mathbf{f}^{\prime}\right\rVert_{L^{2}(\Omega,{\mathbb{R}}^{d\times
d})}\right).$ (6.3)

We now consider the problem (6.1) with $\mathbf{g}^{\prime}\in
L^{\infty}(\Omega,{\mathbb{R}}^{d})$ and $\mathbf{f}^{\prime}\in
L^{\infty}(\Omega,{\mathbb{R}}^{d\times d})$, then by [40, Theorem 1.1 and
1.9] (see also [6, 5, 50, 25, 48]), ${\mathbb{D}}\mathbf{w}^{\varepsilon}$ is
piecewise continuous, and the following Lipschitz estimate holds:

$\displaystyle\begin{split}\left\lVert\mathbf{w}^{\varepsilon}\right\rVert_{W^{1,\infty}(\Omega^{\prime},{\mathbb{R}}^{d})}&\leq
C_{1}\left(\left\lVert\mathbf{w}^{\varepsilon}\right\rVert_{L^{2}(\Omega,{\mathbb{R}}^{d})}+\left\lVert\mathbf{g}^{\prime}\right\rVert_{L^{\infty}(\Omega,{\mathbb{R}}^{d})}+\left\lVert\mathbf{f}^{\prime}\right\rVert_{L^{\infty}(\Omega,{\mathbb{R}}^{d\times
d})}\right)\\\ &\leq
C_{1}\left(\left\lVert\mathbf{g}^{\prime}\right\rVert_{L^{\infty}(\Omega,{\mathbb{R}}^{d})}+\left\lVert\mathbf{f}^{\prime}\right\rVert_{L^{\infty}(\Omega,{\mathbb{R}}^{d\times
d})}\right),\end{split}$ (6.4)

for some
$C_{1}=C_{1}(d,\kappa,\lambda_{e},\Lambda_{e},\Omega^{\prime},\Omega)>0$
independent of $\varepsilon>0$. In the last estimate, we also use (6.3) and
the fact that $\Omega$ is bounded.

By (4.1) and an extension of Hölder inequality [9, Remark 2, p. 93], we obtain

$\displaystyle\left\lVert\boldsymbol{\tau}^{\varepsilon}\right\rVert_{L^{p/2}}=\left\lVert\nabla\varphi^{\varepsilon}\otimes\nabla\varphi^{\varepsilon}\right\rVert_{L^{p/2}}$
$\displaystyle\leq\left\lVert\nabla\varphi^{\varepsilon}\right\rVert_{L^{p}}\left\lVert\nabla\varphi^{\varepsilon}\right\rVert_{L^{p}}\leq
C\left\lVert f\right\rVert_{L^{p^{\prime}}}^{2}.$ (6.5)

Consider the problem (6.1) with
$\mathbf{g}^{\prime}\in\mathcal{D}(\Omega^{\prime},{\mathbb{R}}^{d}),\mathbf{f}^{\prime}\in\mathcal{D}(\Omega^{\prime},{\mathbb{R}}^{d\times
d})$, then $\mathbf{w}^{\varepsilon}=0$ on $\Omega\setminus\Omega^{\prime}$.
From (2.12), (6.1), (6.4), and (6.5), we have

$\displaystyle\begin{split}&\left\lvert\int_{\Omega^{\prime}}\mathbf{g}^{\prime}\cdot\mathbf{u}^{\varepsilon}\operatorname{d\\!}x-\int_{\Omega^{\prime}}\mathbf{f}^{\prime}:{\mathbb{D}}(\mathbf{u}^{\varepsilon})\operatorname{d\\!}x\right\rvert\\\
&=\left\lvert\int_{\Omega}\mathbf{g}^{\prime}\cdot\mathbf{u}^{\varepsilon}\operatorname{d\\!}x-\int_{\Omega}\mathbf{f}^{\prime}:{\mathbb{D}}(\mathbf{u}^{\varepsilon})\operatorname{d\\!}x\right\rvert\\\
&=\left\lvert\int_{\Omega}\mathbf{B}\left(\frac{x}{\varepsilon}\right){\mathbb{D}}(\mathbf{w}^{\varepsilon}):{\mathbb{D}}(\mathbf{u}^{\varepsilon})\operatorname{d\\!}x\right\rvert\\\
&=\left\lvert\int_{\Omega}\mathbf{g}\cdot\mathbf{w}^{\varepsilon}\operatorname{d\\!}x-\int_{\Omega}\mathbf{C}\left(\frac{x}{\varepsilon}\right)\boldsymbol{\tau}^{\varepsilon}:{\mathbb{D}}(\mathbf{w}^{\varepsilon})\operatorname{d\\!}x\right\rvert\\\
&=\left\lvert\int_{\Omega^{\prime}}\mathbf{g}\cdot\mathbf{w}^{\varepsilon}\operatorname{d\\!}x-\int_{\Omega^{\prime}}\mathbf{C}\left(\frac{x}{\varepsilon}\right)\boldsymbol{\tau}^{\varepsilon}:{\mathbb{D}}(\mathbf{w}^{\varepsilon})\operatorname{d\\!}x\right\rvert\\\
&\leq
C\left(\left\lVert\mathbf{g}^{\prime}\right\rVert_{L^{\infty}(\Omega^{\prime},{\mathbb{R}}^{d})}+\left\lVert\mathbf{f}^{\prime}\right\rVert_{L^{\infty}(\Omega^{\prime},{\mathbb{R}}^{d\times
d})}\right)\left(\left\lVert\mathbf{g}\right\rVert_{L^{1}(\Omega^{\prime},{\mathbb{R}}^{d})}+\left\lVert\boldsymbol{\tau}^{\varepsilon}\right\rVert_{L^{1}(\Omega^{\prime},{\mathbb{R}}^{d\times
d})}\right)\\\ &\leq
C\left(\left\lVert\mathbf{g}^{\prime}\right\rVert_{L^{\infty}(\Omega^{\prime},{\mathbb{R}}^{d})}+\left\lVert\mathbf{f}^{\prime}\right\rVert_{L^{\infty}(\Omega^{\prime},{\mathbb{R}}^{d\times
d})}\right)\left(\left\lVert\mathbf{g}\right\rVert_{L^{1}(\Omega,{\mathbb{R}}^{d})}+\left\lVert
f\right\rVert_{L^{p^{\prime}}(\Omega,{\mathbb{R}}^{d})}^{2}\right).\end{split}$
(6.6)

It follows that
$\mathbf{u}^{\varepsilon}\in\mathrm{BV}_{\mathrm{loc}}(\Omega,{\mathbb{R}}^{d})$,
see [4]. Moreover, by choosing $\mathbf{g}^{\prime}=0$ and $\mathbf{f}=0$
alternatively, then applying Riesz theorem, we conclude

$\displaystyle\left\lVert\mathbf{u}^{\varepsilon}\right\rVert_{\mathrm{BV}(\Omega^{\prime},{\mathbb{R}}^{d})}$
$\displaystyle\leq
C(d,\kappa,\lambda_{e},\Lambda_{e},\lambda_{o},\Lambda_{o},\Omega^{\prime},\Omega)\left(\left\lVert\mathbf{g}\right\rVert_{L^{r}(\Omega,{\mathbb{R}}^{d})}+\left\lVert
f\right\rVert_{L^{p^{\prime}}(\Omega,{\mathbb{R}}^{d\times d})}^{2}\right).$
(6.7)

###  6.2 Higher regularity and existence

The a priori estimate (6.7) shows that the solution $\mathbf{u}^{\varepsilon}$
of (2.7)-(2.9), if it exists, is merely a bounded variation function. However,
this result does not guarantee that the normal traces appearing in (2.8) are
well-defined, cf. e.g., [15, 47, 16]. Therefore, we will show next that
$\mathbf{u}^{\varepsilon}$ possesses higher regularity. Indeed, for each
$\varepsilon>0$, the (unique) solution $\mathbf{u}^{\varepsilon}$ of
(2.7)-(2.9) belongs to
$W_{\mathrm{loc}}^{1,q(\varepsilon)}(\Omega,{\mathbb{R}}^{d})$, for some
$1<q(\varepsilon)<2$ that will be given below.

To that end, we need to adapt a useful global gradient estimate, obtained by
N. C. Phuc [46, Theorem 1.1] to our current setting, as follows:

######  Proposition 6.1.

Suppose $2-\frac{1}{d}<p\leq d$ and $\mathbf{a}\colon
Y\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ satisfies (A1)–(A4). Let $\mu\in
L^{s}(\Omega)$ for some $d>s>\frac{1}{1-\frac{1}{p}+\frac{1}{d}}.$ Then there
exists $\delta=\delta(d,p,\lambda_{o},\Lambda_{o})>0$ such that for any weak
solution $\phi\in W_{0}^{1,p}(\Omega)$ of

$\displaystyle-\operatorname{{div}}\left(\mathbf{a}(x,\nabla\phi)\right)=\mu\text{
in }\Omega,\qquad\phi=0\text{ on }\partial\Omega,$ (6.8)

the following estimate holds:

$\displaystyle\int_{\Omega}\left\lvert\nabla\phi\right\rvert^{p+\delta}\operatorname{d\\!}x\leq
C\left(\int_{\Omega}\left\lvert\mu\right\rvert^{s}\operatorname{d\\!}x\right)^{\frac{p-1}{s(p+\delta)}}.$
(6.9)

where $C=C(d,p,s,\lambda_{o},\Lambda_{o},\operatorname{diam}(\Omega))>0.$

###### Proof of Proposition 6.1.

Since $s>\frac{1}{1-\frac{1}{p}+\frac{1}{d}}$, there exists
$\delta_{1}=\delta_{1}(p,d,s)>0$ such that
$s>\frac{1}{\frac{p-1}{p+\delta_{1}}+\frac{1}{d}}>\frac{1}{1-\frac{1}{p}+\frac{1}{d}}$,
then

$\displaystyle\frac{p+\delta_{1}}{p-1}<\frac{ds}{d-s}.$ (6.10)

It is clear that if $\mathbf{a}\colon
Y\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ satisfies (A1)–(A4) then it also
satisfies all the conditions in [46, Theorem 1.1] (note that we can extend
$\mathbf{a}$ to
$\tilde{\mathbf{a}}\colon{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$
by periodicity). Therefore, we have
$0<\delta=\delta(d,p,\lambda_{o},\Lambda_{o})<\delta_{1}$ small enough such
that

$\displaystyle\int_{\Omega}\left\lvert\nabla\phi\right\rvert^{p+\delta}\operatorname{d\\!}x\leq
C\int_{\Omega}\mathcal{N}_{1}\left(\left\lvert\mu\right\rvert\right)^{\frac{p+\delta}{p-1}}\operatorname{d\\!}x,$
(6.11)

for some $C=C(d,p,\delta,\operatorname{diam}(\Omega))>0,$ where
$\mathcal{N}_{1}$ is the fractional maximal function, defined as

$\displaystyle\mathcal{N}_{1}\left(\nu\right)(x)\coloneqq\sup_{r>0}\frac{r}{\left\lvert
B_{r}(x)\right\rvert}\nu\left(B_{r}(x)\right),$

for any nonnegative locally finite measure $\nu$ on ${\mathbb{R}}^{d}$.

Since $0<\delta<\delta_{1}$ and $\Omega$ is bounded, by (6.10) and Hölder’s
inequality, we have

$\displaystyle\int_{\Omega}\mathcal{N}_{1}\left(\left\lvert\mu\right\rvert\right)^{\frac{p+\delta}{p-1}}\operatorname{d\\!}x\leq
C\left(\int_{\Omega}\mathcal{N}_{1}\left(\left\lvert\mu\right\rvert\right)^{\frac{ds}{d-s}}\operatorname{d\\!}x\right)^{\frac{d-s}{ds}\cdot\frac{p-1}{p+\delta}}.$
(6.12)

From [38, Theorem 3.1], there exists $C=C(d,p)>0$ such that

$\displaystyle\int_{\Omega}\mathcal{N}_{1}\left(\left\lvert\mu\right\rvert\right)^{\frac{ds}{d-s}}\leq
C\left(\int_{\Omega}\left\lvert\mu\right\rvert^{s}\operatorname{d\\!}x\right)^{\frac{d}{d-s}}$
(6.13)

Combining (6.11), (6.12), and (6.13), we obtain (6.9).

∎

We are ready to show that the elastic displacement $\mathbf{u}^{\varepsilon}$
actually belongs to some Sobolev spaces.

Fix $\varepsilon>0$. Recall that $\mathbf{g}\in
L^{r}(\Omega,{\mathbb{R}}^{d})\subset W^{-1,r}(\Omega,{\mathbb{R}}^{d})$ for
some $r>1$. Applying Proposition 6.1 to the nonlinear divergence problem
(2.7a)–(2.9a), and using the fact that $\Omega$ is bounded, we obtain that
there exists $1<q(\varepsilon)<\min\left\\{r,2\right\\}$ small enough, such
that $\left\lvert\nabla\varphi^{\varepsilon}\right\rvert^{2}\in
L^{q(\varepsilon)}(\Omega)$ and $\mathbf{g}\in
W^{-1,q(\varepsilon)}(\Omega,{\mathbb{R}}^{d})$. It also follows that
$\boldsymbol{\tau}^{\varepsilon}\in
L^{q(\varepsilon)}(\Omega,{\mathbb{R}}^{d\times d})$.

Let $\mathbf{g}^{\prime}$ and $\mathbf{f}^{\prime}$ be some suitable Lebesgue
integrable functions that will be specified later. Let
$\mathbf{w}^{\varepsilon}\in
W_{0}^{1,q(\varepsilon)^{\prime}}(\Omega^{\prime},{\mathbb{R}}^{d})$, where
$q(\varepsilon)^{{}^{\prime}}$ is the Hölder conjugate of $q(\varepsilon)$, be
the (unique) solution of

$\displaystyle\int_{\Omega}\mathbf{B}\left(\frac{x}{\varepsilon}\right)\,{\mathbb{D}}(\mathbf{w}^{\varepsilon}):{\mathbb{D}}(\mathbf{u})\operatorname{d\\!}x=\int_{\Omega}\mathbf{g}^{\prime}\cdot\mathbf{u}\operatorname{d\\!}x-\int_{\Omega}\mathbf{f}^{\prime}:{\mathbb{D}}(\mathbf{u})\operatorname{d\\!}x$
(6.14)

for all $\mathbf{u}\in\mathcal{D}(\Omega,{\mathbb{R}}^{d})$.

For fixed $1\leq i,j\leq d$, define the interpolation map

$\displaystyle\mathrm{T}^{ij}\colon\left(L^{2}(\Omega,{\mathbb{R}}^{d})\times
L^{2}(\Omega,{\mathbb{R}}^{d\times
d})\right)+\left(L^{\infty}(\Omega,{\mathbb{R}}^{d})\times
L^{\infty}(\Omega,{\mathbb{R}}^{d\times d})\right)$ $\displaystyle\to
L^{2}(\Omega^{\prime})+L^{\infty}(\Omega^{\prime})$
$\displaystyle\left(\mathbf{g}_{2},\mathbf{f}_{2}\right)+\left(\mathbf{g}_{\infty},\mathbf{f}_{\infty}\right)$
$\displaystyle\mapsto\left[{\mathbb{D}}\mathbf{w}^{\varepsilon}_{2}+{\mathbb{D}}\mathbf{w}^{\varepsilon}_{\infty}\right]_{ij},$

where $\mathbf{w}^{\varepsilon}_{2}$ and $\mathbf{w}^{\varepsilon}_{\infty}$
are solutions of (6.14) with source terms
$(\mathbf{g}^{\prime},\mathbf{f}^{\prime})=(\mathbf{g}_{2},\mathbf{f}_{2})$
and
$(\mathbf{g}^{\prime},\mathbf{f}^{\prime})=(\mathbf{g}_{\infty},\mathbf{f}_{\infty})$,
respectively.

On the one hand, the estimates (6.2) and (6.4) imply the bounds of the
restriction maps

$\displaystyle\left\lVert\mathrm{T}^{ij}\bigg{|}_{L^{2}(\Omega,{\mathbb{R}}^{d})\times
L^{2}(\Omega,{\mathbb{R}}^{d\times d})}\right\rVert_{L^{2}\to L^{2}}$
$\displaystyle\qquad\coloneqq\sup\left\\{\left\lVert\mathrm{T}^{ij}(\mathbf{g}_{2},\mathbf{f}_{2})\right\rVert_{L^{2}(\Omega^{\prime})}\colon\left\lVert\mathbf{g}_{2}\right\rVert_{L^{2}(\Omega,{\mathbb{R}}^{d})}\leq
1,\left\lVert\mathbf{f}_{2}\right\rVert_{L^{2}(\Omega,{\mathbb{R}}^{d}{\mathbb{R}}^{d\times
d})}\leq 1\right\\}\leq 2C_{2},$
$\displaystyle\left\lVert\mathrm{T}^{ij}\bigg{|}_{L^{\infty}(\Omega,{\mathbb{R}}^{d})\times
L^{\infty}(\Omega,{\mathbb{R}}^{d\times d})}\right\rVert_{L^{\infty}\to
L^{\infty}}$ $\displaystyle\qquad\coloneqq\sup\left\\{\left\lVert
T^{ij}(\mathbf{g}_{\infty},\mathbf{f}_{\infty})\right\rVert_{L^{\infty}(\Omega^{\prime})}\colon\left\lVert\mathbf{g}_{\infty}\right\rVert_{L^{\infty}(\Omega,{\mathbb{R}}^{d})}\leq
1,\left\lVert\mathbf{f}_{\infty}\right\rVert_{L^{\infty}(\Omega,{\mathbb{R}}^{d}{\mathbb{R}}^{d\times
d})}\leq 1\right\\}\leq 2C_{1}.$

On the other hand, the identification

$\displaystyle\mathrm{Id}\colon L^{s}(\Omega,{\mathbb{R}}^{d})\times
L^{s}(\Omega,{\mathbb{R}}^{d\times d})$ $\displaystyle\to
L^{s}(\Omega,{\mathbb{R}}^{d^{3}})$ $\displaystyle(\mathbf{g},\mathbf{f})$
$\displaystyle\mapsto(\mathbf{g}_{1},\ldots,\mathbf{g}_{d},\mathbf{f}_{11},\ldots,\mathbf{f}_{1d},\ldots,\mathbf{f}_{d1},\ldots,\mathbf{f}_{\,dd})$

is an isomorphism with respect $L^{s}-$norm for any $2\leq s\leq\infty$.

Therefore, applying the Multilinear Riesz-Thorin Interpolation Theorem [34,
Corollary 7.2.11] to the map $\mathrm{T}^{ij}\circ\mathrm{Id}^{-1}$, there
exists $C_{3}(\varepsilon)>0$ such that the norm of the restriction of
$\mathrm{T}^{ij}$ on

$\displaystyle\left(L^{q(\varepsilon)}(\Omega,{\mathbb{R}}^{d})\times
L^{q(\varepsilon)}(\Omega,{\mathbb{R}}^{d\times
d})\right)\subset\left(L^{2}(\Omega,{\mathbb{R}}^{d})\times
L^{2}(\Omega,{\mathbb{R}}^{d\times
d})\right)+\left(L^{\infty}(\Omega,{\mathbb{R}}^{d})\times
L^{\infty}(\Omega,{\mathbb{R}}^{d\times d})\right)$

to $L^{q(\varepsilon)^{\prime}}(\Omega^{\prime})$ is bounded above by
$C_{3}(\varepsilon)$. In particular, the following
$L^{{q(\varepsilon)^{\prime}}}-$gradient estimate holds

$\displaystyle\left\lVert{\mathbb{D}}\mathbf{w}^{\varepsilon}\right\rVert_{L^{q(\varepsilon)^{\prime}}(\Omega^{\prime},{\mathbb{R}}^{d\times
d})}\leq
C_{3}(\varepsilon)d^{2}\left(\left\lVert\mathbf{g}^{\prime}\right\rVert_{L^{q(\varepsilon)^{\prime}}(\Omega,{\mathbb{R}}^{d})}+\left\lVert\mathbf{f}^{\prime}\right\rVert_{L^{q(\varepsilon)^{\prime}}(\Omega,{\mathbb{R}}^{d\times
d})}\right)$ (6.15)

whenever $\mathbf{w}^{\varepsilon}$ is the solution of (6.14) with source term
$(\mathbf{g}^{\prime},\mathbf{f}^{\prime})$ in $L^{q(\varepsilon)^{\prime}}$.
By (6.15), Korn’s inequality, and Hölder’s inequality, there exists
$C(\varepsilon)=C(\varepsilon,d,\kappa,\lambda_{e},\Lambda_{e},\lambda_{o},\Lambda_{o},\Omega^{\prime},\Omega)$
such that

$\displaystyle\left\lVert\mathbf{w}^{\varepsilon}\right\rVert_{W^{1,q(\varepsilon)^{\prime}}(\Omega^{\prime},{\mathbb{R}}^{d})}\leq
C(\varepsilon)\left(\left\lVert\mathbf{g}^{\prime}\right\rVert_{L^{q(\varepsilon)^{\prime}}(\Omega,{\mathbb{R}}^{d})}+\left\lVert\mathbf{f}^{\prime}\right\rVert_{L^{q(\varepsilon)^{\prime}}(\Omega,{\mathbb{R}}^{d\times
d})}\right).$ (6.16)

We now employ a duality argument to prove an a priori estimate for
$\mathbf{u}^{\varepsilon}\in
W_{\mathrm{loc}}^{1,q(\varepsilon)}(\Omega,{\mathbb{R}}^{d})\subset\mathrm{BV}_{\mathrm{loc}}(\Omega,{\mathbb{R}}^{d})$
satisfying (2.12).

Fix $\Omega^{\prime}\subset\subset\Omega$. In (6.14), let
$\mathbf{g}^{\prime}\in\mathcal{D}(\Omega^{\prime},{\mathbb{R}}^{d}),\mathbf{f}\in\mathcal{D}(\Omega^{\prime},{\mathbb{R}}^{d\times
d})$ and combine with (2.12), (6.16), we obtain

$\displaystyle\begin{split}&\left\lvert\int_{\Omega^{\prime}}\mathbf{g}^{\prime}\cdot\mathbf{u}^{\varepsilon}\operatorname{d\\!}x-\int_{\Omega^{\prime}}\mathbf{f}^{\prime}:{\mathbb{D}}(\mathbf{u}^{\varepsilon})\operatorname{d\\!}x\right\rvert\\\
&=\left\lvert\int_{\Omega^{\prime}}\mathbf{B}\left(\frac{x}{\varepsilon}\right){\mathbb{D}}(\mathbf{w}^{\varepsilon}):{\mathbb{D}}(\mathbf{u}^{\varepsilon})\operatorname{d\\!}x\right\rvert\\\
&=\left\lvert\int_{\Omega^{\prime}}\mathbf{g}\cdot\mathbf{w}^{\varepsilon}\operatorname{d\\!}x-\int_{\Omega^{\prime}}\mathbf{C}\left(\frac{x}{\varepsilon}\right)\boldsymbol{\tau}^{\varepsilon}:{\mathbb{D}}(\mathbf{w}^{\varepsilon})\operatorname{d\\!}x\right\rvert\\\
&\leq
C(\varepsilon)\left(\left\lVert\mathbf{g}^{\prime}\right\rVert_{L^{q(\varepsilon)^{\prime}}(\Omega^{\prime},{\mathbb{R}}^{d})}+\left\lVert\mathbf{f}^{\prime}\right\rVert_{L^{q(\varepsilon)^{\prime}}(\Omega^{\prime},{\mathbb{R}}^{d\times
d})}\right)\left(\left\lVert\mathbf{g}\right\rVert_{L^{q(\varepsilon)}(\Omega^{\prime},{\mathbb{R}}^{d})}+\left\lVert\boldsymbol{\tau}^{\varepsilon}\right\rVert_{L^{q(\varepsilon)}(\Omega^{\prime},{\mathbb{R}}^{d\times
d})}\right)\end{split}$ (6.17)

On the one hand, letting $\mathbf{f}^{\prime}=0$ in (6.17) implies

$\displaystyle\left\lvert\int_{\Omega^{\prime}}\mathbf{g}^{\prime}\cdot\mathbf{u}^{\varepsilon}\operatorname{d\\!}x\right\rvert\leq
C(\varepsilon)\left\lVert\mathbf{g}^{\prime}\right\rVert_{L^{q(\varepsilon)^{\prime}}(\Omega^{\prime},{\mathbb{R}}^{d})}\left(\left\lVert\mathbf{g}\right\rVert_{L^{q(\varepsilon)}(\Omega^{\prime},{\mathbb{R}}^{d})}+\left\lVert\boldsymbol{\tau}^{\varepsilon}\right\rVert_{L^{q(\varepsilon)}(\Omega^{\prime},{\mathbb{R}}^{d\times
d})}\right).$

Thus by Riesz Theorem, we obtain

$\displaystyle\left\lVert\mathbf{u}^{\varepsilon}\right\rVert_{L^{q(\varepsilon)}(\Omega^{\prime},{\mathbb{R}}^{d})}\leq
C(\varepsilon)\left(\left\lVert\mathbf{g}\right\rVert_{L^{q(\varepsilon)}(\Omega^{\prime},{\mathbb{R}}^{d})}+\left\lVert\boldsymbol{\tau}^{\varepsilon}\right\rVert_{L^{q(\varepsilon)}(\Omega^{\prime},{\mathbb{R}}^{d\times
d})}\right).$

On the other hand, letting $\mathbf{g}^{\prime}=0$ in (6.17) and arguing
similarly, we obtain

$\displaystyle\left\lVert{\mathbb{D}}(\mathbf{u}^{\varepsilon})\right\rVert_{L^{q(\varepsilon)}(\Omega^{\prime},{\mathbb{R}}^{d\times
d})}\leq
C(\varepsilon)\left(\left\lVert\mathbf{g}\right\rVert_{L^{q(\varepsilon)}(\Omega^{\prime},{\mathbb{R}}^{d})}+\left\lVert\boldsymbol{\tau}^{\varepsilon}\right\rVert_{L^{q(\varepsilon)}(\Omega^{\prime},{\mathbb{R}}^{d\times
d})}\right).$

Combining the two estimates above, we conclude that

$\displaystyle\left\lVert\mathbf{u}^{\varepsilon}\right\rVert_{W^{1,q(\varepsilon)}(\Omega^{\prime},{\mathbb{R}}^{d})}\leq
C(\varepsilon)\left(\left\lVert\mathbf{g}\right\rVert_{L^{q(\varepsilon)}(\Omega^{\prime},{\mathbb{R}}^{d})}+\left\lVert\boldsymbol{\tau}^{\varepsilon}\right\rVert_{L^{q(\varepsilon)}(\Omega^{\prime},{\mathbb{R}}^{d\times
d})}\right).$ (6.18)

We have shown that every distributional solution $\mathbf{u}^{\varepsilon}$ of
(2.12) belongs to
$W_{\mathrm{loc}}^{1,q(\varepsilon)}(\Omega,{\mathbb{R}}^{d})$, and thus the
normal traces appearing in (2.8) are well-defined. It remains to show that
(2.12) has a distributional solution $\mathbf{u}^{\varepsilon}\in
L^{1}_{\mathrm{loc}}(\Omega,{\mathbb{R}}^{d})$. This follows from the
existence and asymptotic decay of the Green’s function associated with the
operator
$\mathcal{L}^{\varepsilon}\coloneqq-\operatorname{{div}}\left(\mathbf{B}\left(\frac{x}{\varepsilon}\right){\mathbb{D}}[\cdot]\right)$
shown in [22, Theorem 1]. The uniqueness of the distributional solution
$\mathbf{u}^{\varepsilon}$ follows from a standard argument, by using density
and the fundamental lemma of calculus of variations.

######  Remark 6.2.

If we assume instead that $\mathbf{B}$ is in $VMO(\Omega,{\mathbb{R}}^{d})$,
then the $W^{1,p}-$estimate (6.18) would be obtained via a real variable
method by Caffarelli and Peral [13, 50], while the $W^{1,\infty}-$estimate
(6.2) was first obtained via the compactness method [6, 5]. In this case, the
existence of the solution $\mathbf{u}^{\varepsilon}$ can be shown by an
approximation argument (the SOLA method - Solutions Obtained by Limit of
Approximations, see [8]).

Indeed, from [13, 50], we have the global estimate

$\displaystyle\left\lVert\mathbf{u}^{\varepsilon}\right\rVert_{W^{1,q(\varepsilon)}(\Omega,{\mathbb{R}}^{d})}\leq
C\left(\left\lVert\mathbf{g}\right\rVert_{L^{q(\varepsilon)}(\Omega,{\mathbb{R}}^{d})}+\left\lVert\boldsymbol{\tau}\right\rVert_{L^{q(\varepsilon)}(\Omega,{\mathbb{R}}^{d\times
d})}\right).$ (6.19)

Let $\mathbf{g}_{n}\in\mathcal{D}(\Omega,{\mathbb{R}}^{d})$ and
$\mathbf{f}_{n}\in\mathcal{D}(\Omega,{\mathbb{R}}^{d\times d})$ that converge
to $\mathbf{g}$ and
$\mathbf{C}\left(\frac{x}{\varepsilon}\right)\boldsymbol{\tau}^{\varepsilon}$,
respectively, in the $L^{q(\varepsilon)}-$norm, as $n\to\infty$. Observe that
the variational problem

$\displaystyle\int_{\Omega}\mathbf{B}\left(\frac{x}{\varepsilon}\right){\mathbb{D}}(\mathbf{u}_{n}^{\varepsilon}):{\mathbb{D}}(\mathbf{v})\operatorname{d\\!}x=\int_{\Omega}\mathbf{g}_{n}\cdot\mathbf{v}\operatorname{d\\!}x-\int_{\Omega}\mathbf{f}_{n}:{\mathbb{D}}(\mathbf{v})\operatorname{d\\!}x,\quad\forall\mathbf{v}\in\mathcal{D}(\Omega,{\mathbb{R}}^{d})$
(6.20)

has a unique solution $\mathbf{u}^{\varepsilon}_{n}\in
W_{0}^{1,2}(\Omega,{\mathbb{R}}^{d})\subset
W_{0}^{1,q(\varepsilon)}(\Omega,{\mathbb{R}}^{d})$ by the Lax-Milgram Theorem.
Then, for $m,n\in{\mathbb{N}}$, we have
$\mathbf{u}^{\varepsilon}_{m}-\mathbf{u}^{\varepsilon}_{n}$ is the solution of

$\displaystyle\begin{split}\int_{\Omega}\mathbf{B}\left(\frac{x}{\varepsilon}\right){\mathbb{D}}(\mathbf{u}_{m}^{\varepsilon}-\mathbf{u}_{n}^{\varepsilon}):{\mathbb{D}}(\mathbf{v})\operatorname{d\\!}x=\int_{\Omega}\left(\mathbf{g}_{m}-\mathbf{g}_{n}\right)\cdot\mathbf{v}\operatorname{d\\!}x-\int_{\Omega}\left(\mathbf{f}_{m}-\mathbf{f}_{n}\right):{\mathbb{D}}(\mathbf{v})\operatorname{d\\!}x,\\\
\forall\mathbf{v}\in\mathcal{D}(\Omega,{\mathbb{R}}^{d}).\end{split}$ (6.21)

Because $\mathbf{g}_{m}-\mathbf{g}_{n}$ and $\mathbf{f}_{m}-\mathbf{f}_{n}$
are also in $L^{q(\varepsilon)}$, the estimate (6.18) applies to
$\mathbf{u}^{\varepsilon}_{m}-\mathbf{u}^{\varepsilon}_{n}$, so

$\displaystyle\left\lVert\mathbf{u}^{\varepsilon}_{m}-\mathbf{u}^{\varepsilon}_{n}\right\rVert_{W^{1,q(\varepsilon)}(\Omega,{\mathbb{R}}^{d})}\leq
C(\varepsilon)\left(\left\lVert\mathbf{g}_{m}-\mathbf{g}_{n}\right\rVert_{L^{q(\varepsilon)}(\Omega,{\mathbb{R}}^{d})}+\left\lVert\mathbf{f}_{m}-\mathbf{f}_{n}\right\rVert_{L^{q(\varepsilon)}(\Omega,{\mathbb{R}}^{d\times
d})}\right)\xrightarrow[\hphantom{~{}2~{}}]{}$

as $m,n\to\infty$. Therefore, $\mathbf{u}^{\varepsilon}_{n}$ is a Cauchy
sequence in $W^{1,q(\varepsilon)}(\Omega,{\mathbb{R}}^{d})$, and so there
exists $\mathbf{u}^{\varepsilon}\in
W^{1,q(\varepsilon)}(\Omega,{\mathbb{R}}^{d})$ such that
$\mathbf{u}^{\varepsilon}_{n}\xrightarrow[\hphantom{~{}2~{}}]{}\mathbf{u}^{\varepsilon}$,
as $n\to\infty$. By letting $n\to\infty$ in (6.20) and using density
arguments, we obtain $\mathbf{u}^{\varepsilon}$ is the solution of (2.12).

###  6.3 The two-scale homogenized system

Fix $K_{0}\subset\subset\Omega$, then there exists an open set $K$ such that
$K_{0}\subset\subset K\subset\subset\Omega$. By (6.7), there exists
$\mathbf{u}_{K_{0}}^{0}\in\mathrm{BV}(K,{\mathbb{R}}^{d})$ and
$\mathbf{u}_{K_{0}}^{1}\in\mathcal{M}\left(K,\mathrm{BV}_{\mathrm{per}}(Y,{\mathbb{R}}^{d})\right)$,
such that, up to a subsequence, see [2],

For $\Omega^{\prime}\subset\subset\Omega$, there exists
$C=C(d,\alpha,N_{\Omega},\lambda_{e},\Lambda_{e},r,\Omega^{\prime},\Omega)>0$
such that
$\displaystyle\left\lVert\mathbf{u}\right\rVert_{W^{1,r}(\Omega^{\prime},{\mathbb{R}}^{d})}\leq
C\left\lVert\mathbf{g}\right\rVert_{L^{r}(\Omega,{\mathbb{R}}^{d})}.$ (6.22i)

In the context of homogenization, we have the following result:

######  Proposition A.4.

Let $\mathbf{B}\in\mathfrak{M}_{\mathrm{mat}}(\lambda_{e},\Lambda_{e})$ and
$\Omega,\alpha$ be as above. Assume further that the unit cell $Y$ is a
disjoint union of $N_{Y}$ finite subdomains with piecewise
$C^{1,\alpha}-$boundaries and $\mathbf{B}$ is Hölder continuous on the closure
of each subdomain. Suppose $\mathbf{g}\in L^{r}(\Omega,{\mathbb{R}}^{d})$ for
$1\leq r\leq\infty$. Then the weak solution $\mathbf{u}^{\varepsilon}\in
W_{0}^{1,r}(\Omega,{\mathbb{R}}^{d})$ of

$\displaystyle-\operatorname{{div}}\left[\mathbf{B}\left(\frac{x}{\varepsilon}\right)\nabla\mathbf{u}^{\varepsilon}\right]$
$\displaystyle=\mathbf{g}\quad\text{ in }\Omega,$ (6.22j)

satisfies the following estimate:

For $\Omega^{\prime}\subset\subset\Omega$, there exists
$C=C(d,\alpha,N_{Y},\lambda_{e},\Lambda_{e},r,\Omega^{\prime},\Omega)>0$,
independent of $\varepsilon>0$, such that

$\displaystyle\left\lVert\mathbf{u}^{\varepsilon}\right\rVert_{W^{1,r}(\Omega^{\prime},{\mathbb{R}}^{d})}\leq
C\left\lVert\mathbf{g}\right\rVert_{L^{r}(\Omega,{\mathbb{R}}^{d})}.$ (6.22k)

This proposition holds because the constant $C_{1}$ in (6.4) can be improved,
so that it does not depend on $\varepsilon$, thanks to the celebrated
_compactness method_ , cf. e.g. [6, 5, 40, 25, 48, 50, 37, 3], which exploits
additional regularity information: (6.22j) is $H$-convergent to a system with
constant coefficients. Similar $W^{1,r}$-estimates were proved for the case
when $\mathbf{B}$ is in $VMO$ or $BMO$ spaces cf. e.g., [13, 50, 3] and
references therein.

##  Appendix B Acknowledgements

The work of the third author was supported by NSF grant DMS-2110036. This
material is based upon work supported by and while serving at the National
Science Foundation for the second author Yuliya Gorb. Any opinion, findings,
and conclusions or recommendations expressed in this material are those of the
authors and do not necessarily reflect views of the National Science
Foundation.

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