# Convergece rate and uniform Lipschitz estimate in periodic homogenization of
high-contrast elliptic systems

Xin Fu  Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084,
P.R. China<EMAIL_ADDRESS>and  Wenjia Jing  Yau Mathematical
Sciences Center, Tsinghua University, Beijing 100084 and Beijing Institute of
Mathematical Sciences and Applications, Beijing 101408, P.R. China
<EMAIL_ADDRESS>
###### Abstract.

We consider the Dirichlet problem for elliptic systems with periodically
distributed inclusions whose conduction parameter exhibits a significant
contrast compared to the background media. We develop a unified method to
quantify the convergence rates both as the periodicity of inclusions tends to
zero and as the parameter approaches either zero or infinity. Based on the
obtained convergence rates and a Campanato-type scheme, we also derive the
regularity estimates that are uniform both in the periodicity and the
contrast.

Key words: Periodic homogenization, layer potential, high-contrast media,
regularity estimates, perforated domains.

Mathematics subject classification (MSC 2020): 35B27, 35J70, 35P20

##  1\. Introduction

In this paper we study quantitative homogenization of elliptic systems with
rapidly oscillating and high-contrast periodic coefficients.

To state the problem in precisely, we need to impose two assumptions: the
coefficient condition ($\mathbf{H1}$) and the geometric condition
($\mathbf{{H2}}$). Throughout this paper, we will assume that $d\geq 2$ and
$m\geq 1$ are integers.

(H1) _Assumptions on the coefficients._ We assume the coefficient

$A(y)=\big{(}a_{ij}^{\alpha\beta}(y)\big{)}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{(dm)^{2}},\qquad
1\leq i,j\leq d,\ 1\leq\alpha,\beta\leq m$

satisfies

  * (a)

(Symmetry) For any $y\in\mathbb{R}^{d}$,

$a_{ij}^{\alpha\beta}(y)=a_{ji}^{\beta\alpha}(y),\qquad 1\leq i,j\leq d,\
1\leq\alpha,\beta\leq m.$ (1.1)

  * (b)

(Strong ellipticity) There exists $\mu>0$ such that

$\mu|\xi|^{2}\leq
a_{ij}^{\alpha\beta}(y)\xi_{i}^{\alpha}\xi_{j}^{\beta}\leq\mu^{-1}|\xi|^{2},\qquad\forall
y\in\mathbb{R}^{d},\ \forall\xi\in\mathbb{R}^{dm}.$ (1.2)

  * (c)

(Periodicity)

$A(y+\mathbf{n})=A(y),\qquad\forall y\in\mathbb{R}^{d},\
\forall\mathbf{n}\in\mathbb{Z}^{d}.$ (1.3)

  * (d)

(Hölder continuity) There exists $\lambda\in(0,1)$ and $\tau>0$ such that

$|A(y)-A(w)|\leq\tau|y-w|^{\lambda},\qquad\forall y,w\in\mathbb{R}^{d}.$ (1.4)

(H2) _Assumptions on the Geometric set-up._ Let $Y=(-1/2,1/2)^{d}$ denote the
unit cell, and let $\omega\subset Y$ be a simply connected domain with
Lipschitz boundary such that $\mathrm{dist}(\omega,\partial Y)>0$. Consider
$\Omega$ as a bounded Lipschitz domain in $\mathbb{R}^{d}$ (sometimes we may
impose more regularity on the boundary $\partial\Omega$ such that $\Omega$ is
a $C^{1,\alpha}$ domain). For each $\varepsilon\in(0,1)$ and
$\mathbf{n}\in\mathbb{Z}^{d}$, let
$\omega_{\varepsilon}^{\mathbf{n}}:=\varepsilon(\mathbf{n}+\omega)$ be a
rescaled and translated version of $\omega$, for a fixed $\kappa>0$, we then
define the indices set

$\mathcal{I}_{\varepsilon}=\\{\mathbf{n}\in\mathbb{Z}^{d}\,:\,\omega^{\mathbf{n}}_{\varepsilon}\subset\Omega\,\text{
and
}\,\mathrm{dist}(\omega^{\mathbf{n}}_{\varepsilon},\partial\Omega)>\kappa\varepsilon\\},$

and let

$D_{\varepsilon}:=\bigcup\limits_{\mathbf{n}\in\mathcal{I}_{\varepsilon}}\omega^{\mathbf{n}}_{\varepsilon}.$
(1.5)

The set $D_{\varepsilon}$ consists of periodic inclusions of size
$\varepsilon$ within $\Omega$ which are well-separated from $\partial\Omega$.
The complementary domain
$\Omega_{\varepsilon}:=\Omega\setminus\overline{D_{\varepsilon}}$ then
represents the background matrix surrounding these inclusions.

The pair $(\Omega,D_{\varepsilon})$ satisfying the geometric condition
($\mathbf{GC}$) is referred to as the type II perforated domain, as defined in
[20]. In comparison, the type I perforated domain refers to the following
construction:

$D_{\varepsilon}:=\Omega\cap\bigcup_{\mathbf{n}\in\mathbb{Z}^{d}}\omega^{\mathbf{n}}_{\varepsilon},\qquad\Omega_{\varepsilon}:=\Omega\setminus\overline{D_{\varepsilon}},$
(1.6)

see Figure 1.

(a) Type I domain

(b) Type II domain

Figure 1. Comparison between Type I and Type II domain

We are now in the position of describing our problem. For any
$\delta\in(0,\infty)$, let

$\Lambda_{\varepsilon,\delta}(x):=\delta\mathbbm{1}_{D_{\varepsilon}}(x)+\mathbbm{1}_{\Omega_{\varepsilon}}(x)$

be the contrast function, where $\delta$ denotes the contrast of physical
parameters between inclusions and background (in practice, $\delta$ could be
compressibility, permeability, conductivity and etc.). We study the
quantitative homogenization of the Dirichlet problem

$\left\\{\begin{aligned}
&\mathcal{L}_{\varepsilon,\delta}(u_{\varepsilon,\delta})=f&&\mathrm{in}\
\Omega,\\\ &u_{\varepsilon,\delta}=g&&\mathrm{on}\
\partial\Omega.\end{aligned}\right.$ (1.7)

where $\mathcal{L}_{\varepsilon,\delta}$ is given by

$\mathcal{L}_{\varepsilon,\delta}(u):=-\mathrm{div}\left[\Lambda_{\varepsilon,\delta}(x)A\left(\frac{x}{\varepsilon}\right)\nabla
u\right]=-\frac{\partial}{\partial
x_{i}}\left(\Lambda_{\varepsilon,\delta}(x)a^{\alpha\beta}_{ij}\left(x/\varepsilon\right)\frac{\partial}{\partial
x_{j}}u^{\beta}\right),$

which acts on functions defined on $\Omega\subset\mathbb{R}^{d}$ with values
in $\mathbb{R}^{m}$. Note that since we allow $\delta$ approaches extremal
value ($0$ or $\infty$), the coefficient
$\Lambda_{\varepsilon,\delta}(x)A(x/\varepsilon)$ exhibits high contrast
between inclusions and background, and thus $\mathcal{L}_{\varepsilon,\delta}$
models the media with high-contrast small inclusions.

For each fixed $\delta\in(0,\infty)$, it is well known that the periodic
homogenization is valid: as $\varepsilon\rightarrow 0$, the solution
$u_{\varepsilon,\delta}$ converges (e.g., in $L^{2}(\Omega)$) to
$\widehat{u}_{\delta}$, the solution of the homogenized problem

$\left\\{\begin{aligned}
&\widehat{\mathcal{L}}_{\delta}(\widehat{u}_{\delta})=-\mathrm{div}\,(\widehat{A}_{\delta}\nabla\widehat{u}_{\delta})=f&&\mathrm{in}\
\Omega,\\\ &\widehat{u}_{\delta}=g&&\mathrm{on}\
\partial\Omega,\end{aligned}\right.$ (1.8)

where
$\widehat{\mathcal{L}}_{\delta}=-\mathrm{div}\,(\widehat{A}_{\delta}\nabla)$
is the homogenized operator with coefficient
$\widehat{A}_{\delta}=\big{(}\widehat{a}_{\delta,ij}^{\alpha\beta}\big{)}$
defined by

$\widehat{a}_{\delta,ij}^{\alpha\beta}=\int_{Y}\Lambda_{\delta}(y)\Big{[}a_{ij}^{\alpha\beta}(y)+a_{ik}^{\alpha\gamma}(y)\frac{\partial}{\partial
y_{k}}\chi_{\delta,j}^{\gamma\beta}\Big{]}\,dy.$ (1.9)

Here
$\Lambda_{\delta}(y)=\delta\mathbbm{1}_{\omega}(y)+\mathbbm{1}_{Y\setminus\overline{\omega}}(y)$
is the contrast function in the unit cell, and for each $i$ and $\alpha$,
$\chi_{\delta,i}^{\alpha}=\big{(}\chi_{\delta,i}^{\alpha
1},\cdots,\chi_{\delta,i}^{\alpha m}\big{)}\in
H^{1}(\mathbb{T}^{d};\mathbb{R}^{m})$ is the solution of the cell problem

$\left\\{\begin{aligned}
&\mathcal{L}_{1,\delta}\big{(}\chi_{\delta,i}^{\alpha}+y_{i}e^{\alpha}\big{)}=0\qquad\mathrm{in}\
Y,\\\ &\chi_{\delta,i}^{\alpha}\ \mathrm{is\ }Y\mathrm{-periodic\ and\ mean\
zero},\end{aligned}\right.$ (1.10)

where $e^{\alpha}=(0,\cdots,1,\cdots,0)\in\mathbb{R}^{m}$ with $1$ in the
$\alpha^{\mathrm{th}}$ position. We denote the tensor
$\big{(}\chi_{\delta,i}^{\alpha\beta}\big{)}$ by $\chi_{\delta}$. Note that,
for $\delta=1$, $\chi_{1}$ is the solution to the classical cell problem in
standard periodic homogenization. The optimal $H^{1}$ convergence rate
$O(\varepsilon)$ with some proper correctors and the regularity estimates for
$u_{\varepsilon,\delta}$ are also obtained, see the monograph [25] and
references therein.

It is natural to ask whether the above quantitative results are uniform for
$\delta\in(0,\infty)$. Such problems are basic for understanding physical and
engineering models with rapidly oscillating and high-contrast coefficients.
For instance,

  * (a)

In biomedical imaging, there is a need to probe anomalous tissues, such as
tumors or thrombus. The elastic modulus of these anomalous tissues is often
significantly greater than that of the surrounding normal tissues, as observed
in [18].

  * (b)

Designing meta-materials exhibiting novel electromagnetic or elastic
properties. Meta-materials are composite materials typically composed of
arrays of small resonators structured on the microscale, the resonance occurs
when the electromagnetic or elastic modulus of resonators are high-contrast
compared to that of the background, see [23, 3].

  * (c)

The double porosity problem, studied by Arbogast, Douglas, and Hornung [4],
which models single phase flow in fractured porous media. In their context,
$u_{\varepsilon,\delta}$ is the fluid pressure, and $\delta=\varepsilon^{2}$
is the permeability that is much smaller in the porous rocks $D_{\varepsilon}$
than in the network of fractures $\Omega_{\varepsilon}$.

### Main results.

The first main result of this article concerns the uniform convergence rate of
$u_{\varepsilon,\delta}$ in $0<\delta<\infty$. As far as we know, this is the
first result in the full inhomogeneous boundary value problem (that is, with
nonzero source term and boundary data) in the high contrast setting.

######  Theorem 1.1.

Assume (1.1)-(1.3) and $\mathbf{(H1)}$, let $u_{\varepsilon,\delta}$ be the
solution of (1.7) and $\widehat{\mathcal{L}}_{\delta}^{-1}(f)$ be the solution
of (1.8). Let $v=\mathcal{L}_{D_{\varepsilon}}^{-1}(f)\in H^{1}_{0}(\Omega)$
be the unique solution of

$\left\\{\begin{aligned} &\mathcal{L}_{\varepsilon,1}(v)=f&\mathrm{in}\
D_{\varepsilon},\\\ &v=0&\mathrm{on}\ \Omega\setminus
D_{\varepsilon}.\end{aligned}\right.$ (1.11)

Then we have

$\displaystyle\Big{\|}u_{\varepsilon,\delta}-\widehat{u}_{\delta}-\varepsilon\chi_{\delta}\left(\frac{x}{\varepsilon}\right)S_{\varepsilon}\Big{(}\eta_{\varepsilon}\nabla\widehat{u}_{\delta}\Big{)}$
$\displaystyle-\delta^{-1}\mathcal{L}_{D_{\varepsilon}}^{-1}(f)\Big{\|}_{H^{1}(\Omega)}$
(1.12) $\displaystyle\leq
C\varepsilon^{1/2}\big{(}\|f\|_{L^{2}(\Omega)}+\|\nabla_{\mathrm{tan}}g\|_{L^{2}(\partial\Omega)}\big{)}.$

Here $C$ depends only on $d,m,\kappa,\mu,\Omega$ and $\omega$,
$S_{\varepsilon}$ is a smoothing operator defined in (2.6), and
$\eta_{\varepsilon}$ is a cut-off function supported away from
$\partial\Omega$ defined in (4.5).

######  Remark 1.

By a simple rescaling argument, it is clear that

$\varepsilon^{-2}\|\mathcal{L}^{-1}_{D_{\varepsilon}}(f)\|_{L^{2}(\Omega)}+\varepsilon^{-1}\|\nabla\mathcal{L}^{-1}_{D_{\varepsilon}}(f)\|_{L^{2}(\Omega)}\leq
C\|f\|_{L^{2}(\Omega)},$

so if $\delta\gg\varepsilon$, the term
$\delta^{-1}\mathcal{L}_{D_{\varepsilon}}^{-1}(f)$ is negligible as
$\varepsilon\rightarrow 0$. Similarly, if we modify $f$ in (1.7) by

$f_{\varepsilon,\delta}(x)=\begin{cases}f(x),\qquad&\text{for }\,\delta\geq
1,\\\ \Lambda_{\varepsilon,\delta}(x)f(x),\qquad&\text{for
}\,\delta\in(0,1),\end{cases}$ (1.13)

and replace $\widehat{u}_{\delta}$ by
$\widehat{\mathcal{L}}_{\delta}^{-1}(f_{\varepsilon,\delta})$ in (1.12), we
can drop the term $\delta^{-1}\mathcal{L}_{D_{\varepsilon}}^{-1}(f)$ there as
well.

The second main objective of this paper is to establish uniform regularity
estimates for $u_{\varepsilon,\delta}$ both in $0<\varepsilon<1$ and
$0<\delta<\infty$. From the aforementioned convergence rate result, it is
reasonable that we replace $f$ by $f_{\varepsilon,\delta}$ for
$\delta\in(0,1)$.

######  Theorem 1.2.

Assume $(\mathbf{H1})$ and $(\mathbf{H2})$, further assume $\Omega,\omega$ are
$C^{1,\alpha}$ domains for some $\alpha\in(0,1)$, $f\in L^{p}(\Omega)$ for
some $p\geq d$ and $g\in C^{1,\alpha}(\partial\Omega)$. For each
$\varepsilon\in(0,1)$ and $\delta\in(0,\infty)$, let $f_{\varepsilon,\delta}$
be defined by (1.13), in other words, for relatively soft inclusions, we
assume that the source term inside $D_{\varepsilon}$ is of order $O(\delta)$.
Then there is a constant $C>0$ that depends only on
$\kappa,\mu,\lambda,\tau,d,m,p,\alpha,\Omega$ and $\omega$ such that if
$u_{\varepsilon,\delta}$ solves (1.7) with $f$ replaced by
$f_{\varepsilon,\delta}$, then we have

$\|\nabla u_{\varepsilon,\delta}\|_{L^{\infty}(\Omega)}\leq
C\left(\|f\|_{L^{p}(\Omega)}+\|g\|_{C^{1,\alpha}(\partial\Omega)}\right).$
(1.14)

This is a global Lipschitz regularity estimate for $u_{\varepsilon,\delta}$
that is uniform both in $\varepsilon$ and $\delta$. The interior version was
essentially established in Shen’s work. We base our work on Shen’s and
establish the boundary regularity by carrying out the approximation scheme.

######  Remark 2.

Several remarks are in order.

(1) It is possible to include the two extreme settings, namely $\delta=0$ and
$\delta=\infty$ in our analysis. For $\delta=0$, we consider the heterogeneous
elliptic system posed on the perforated domain $\Omega_{\varepsilon}$ with
zero Neumann boundary condition on the interior boundary
$\Omega\cap\partial\Omega_{\varepsilon}$:

$\left\\{\begin{aligned}
&-\partial_{i}\left(a^{\alpha\beta}_{ij}(\tfrac{x}{\varepsilon})\partial_{j}u^{\beta}_{\varepsilon}\right)=f^{\alpha}&&\mathrm{in}\
\Omega_{\varepsilon},\\\
&\nu^{i}a^{\alpha\beta}_{ij}(\tfrac{x}{\varepsilon})\partial_{j}u^{\beta}_{\varepsilon}(x)=0&&\mathrm{on}\
\Omega\cap\partial\Omega_{\varepsilon},\\\
&u_{\varepsilon,\delta}=g&&\mathrm{on}\ \partial\Omega.\end{aligned}\right.$
(1.15)

For $\delta=\infty$, we consider the so-called stiff inclusion problem: find
$u_{\varepsilon}=(u_{\varepsilon}^{\beta})$ in $H^{1}_{0}(\Omega)$ so that
$u_{\varepsilon}$ is a constant vector in each of the inclusion
$\omega_{\varepsilon}^{\mathbf{n}}$ of $D_{\varepsilon}$, and
$u_{\varepsilon}$ satisfies

$\left\\{\begin{aligned}
&-\partial_{i}\left(a^{\alpha\beta}_{ij}(\tfrac{x}{\varepsilon})\partial_{j}u^{\beta}_{\varepsilon}\right)=f^{\alpha}&&\mathrm{in}\
\Omega_{\varepsilon},\\\
&\int_{\partial\omega_{\varepsilon}^{\mathbf{n}}}e^{\alpha}\cdot\left.\left(\frac{\partial
u_{\varepsilon}}{\partial\nu_{\varepsilon}}\right)\right\rvert_{+}\,d\sigma=\int_{\omega_{\varepsilon}^{\mathbf{n}}}e^{\alpha}\cdot
f&&\text{for each }\omega_{\varepsilon}^{\mathbf{n}}\subset
D_{\varepsilon},\\\ &u_{\varepsilon,\delta}=g&&\mathrm{on}\
\partial\Omega.\end{aligned}\right.$ (1.16)

The quantitative homogenization of (1.15) is similar to the analysis in [15,
22]; that for the stiff inclusion problem is similar to [10]. In particular,
the homogenized tensors are $\widehat{A}_{0}$ and $\widehat{A}_{\infty}$
defined by (3.12) and (3.13), respectively.

(2) The ellipticity condition (1.2) is often referred to as _strong_
ellipticity, to be distinguished from the _elasticity_ setting. It is possible
to combine the method of this paper and those of Shen to analyze the same
problem for the elasticity system. The main difference is, in the latter
setting $m=d$, and the ellipticity is checked only for symmetric $d\times d$
matrices $\xi$ (this is usually called weak ellipticity or Legendre–Hadamard
ellipticity). By using Korn’s inequalities, one may carry out most of the
analysis here in a similar way.

### History and strategy of the proof.

This is a problem in monograph by Jikov, Kozlov and Oleinik [13]. In the
scalar case with $0\leq\delta<1$, $A=I$ and $\omega$ being sufficiently
smooth, using the compactness method in [9], the $W^{1,p}$ and Lipschitz
estimates were obtained by Yeh [28, 29, 30, 31]. Also see earlier work by
Schweizer [24] and Masmoudi [19] for related uniform estimates in the case
$\delta=0$. Russell [21] established the large-scale interior Lipschitz
estimate for the system of elasticity with bounded measurable coefficients in
the case $\delta=0$, using an approximation method originated by Armstrong and
Smart [8]. The case $0<\delta<1$ was treated by Russell [22]. Shen [26]
obtained the boundary regularity estimates and nontangential-maximal-function
estimates for Hölder continuous $A$ with $0<\delta<1$. In the stochastic
setting with $\delta=0$, Armstrong and Dario [5] obtained quantitative
homogenization and large-scale regularity results for the random conductance
model on a supercritical percolation.

For the case $1<\delta<\infty$, Shen [27] established the interior Lipschitz
estimates by adapting a recent method proposed by Armstrong et al. [6], which
is based on a Caccioppoli type inequality and the idea that transfer the
higher-order regularity of $u$ in terms of the difference operator to higher-
order regularity of $u$ at a large scale through Caccioppoli and Poincaré’s
inequalities.

We briefly sketch our approach to the proof. In this paper, we adapt the
approximation method originated by Armstrong and Smart [8], which was used by
Russell [21, 22] for treating the case $0\leq\delta<1$. The main difficulty of
this method in the case $1<\delta<\infty$ is the establishment of uniform
$H^{1}$ convergence rate. To this end, we use the periodic layer potentials to
obtain the uniform controll for quantities associated to the cell problem,
i.e., the homogenized tensor $\widehat{A}$, the flux corrector and etc. This
combine with a standard elliptic estimate yield the $H^{1}$ convergence rate
in $D_{\varepsilon}$. Then a key observation is that the discrepancy function
could be decomposed to a regular one and a singular one, the regular part is
easy to be bounded, and $H^{1}(\Omega_{\varepsilon})$ norm of the singular one
is bounded by its $H^{1}(D_{\varepsilon})$ norm. These complete the proof of
$H^{1}(\Omega)$ convergence rate. Via the results of convergence rates, a
Caccippoli-type inequality by Shen [27] and its variants, and an approximation
method (also known as a Campanato-type scheme) by Armstrong and Smart [8], we
establish the regularity estimates.

Notations. We collect some notations used throughout this paper.

  * •

The boundary $t$-layer of $\Omega$ is denoted by
$\mathcal{O}_{t}=\\{x\in\Omega:\mathrm{dist}(x,\partial\Omega)<t\\}$.

  * •

We use the following notation for the product of $A$ with $\nabla u$:

$A\nabla u=(a^{\alpha\beta}_{ij}\partial_{j}u^{\beta})_{i}^{\alpha},\quad
A\nabla u\nabla
v=a^{\alpha\beta}_{ij}\partial_{j}u^{\beta}\partial_{i}v^{\alpha}.$

  * •

The open cube centered at $x_{0}$ with radius $R$ is denoted by
$Q_{R}(x_{0})=\\{x\in\mathbb{R}^{d}:\|x-x_{0}\|_{\infty}<R\\}$. Mention of the
center $x_{0}$ is omitted whenever this does not cause confusion.

  * •

We use $\mathcal{L}$ to denote the operator
$\mathcal{L}_{1,1}=-\mathrm{div}\,(A\nabla)$.

  * •

We use

$\|f\|_{\underline{L}^{p}(E)}=\Big{(}\fint_{E}|f|^{p}\,dx\Big{)}^{1/p}=\Big{(}|E|^{-1}\int_{E}|f|^{p}\,dx\Big{)}^{1/p}$
(1.17)

for $1<p<\infty$, to denote the $L^{p}$ mean of $f$ over a set $E$.

  * •

We use

$J^{E}(u,v)=\int_{E}a_{ij}^{\alpha\beta}(x)\frac{\partial u^{\beta}}{\partial
x_{j}}(x)\frac{\partial v^{\alpha}}{\partial x_{i}}(x)\,dx$ (1.18)

to denote the bilinear form of $u$ and $v$ over a set $E$, we write
$J^{E}(u)=J^{E}(u,u)$. By energy of $u$ in a set $E$ we usually mean the
$L^{2}(E)$ norm of $\nabla u$. Due to uniform ellipticity, it is roughly
$J^{E}(u)$.

  * •

We denote $\varepsilon(\omega+\mathbf{n})$ by
$\omega_{\varepsilon}^{\mathbf{n}}$ and $\varepsilon(Y+\mathbf{n})$ by
$Y_{\varepsilon}^{\mathbf{n}}$ for $\mathbf{n}\in\mathbb{Z}^{d}$.

  * •

$\widetilde{\omega}$ denotes an enlarging dilation of $\omega$ so that
$\overline{\omega}\subset\widetilde{\omega}\subset Q_{7/8}(0)$. Similarly,
$\widetilde{\omega}^{\mathbf{n}}_{\varepsilon}$ denotes
$\varepsilon(\mathbf{n}+\widetilde{\omega})$.

  * •

We use the flat-torus $\mathbb{T}^{d}$ to represent the unit cell $Y$ if the
periodic boundary condition is emphasised.

  * •

We use $C$ to denote a positive constant that may depends on
$d,m,\kappa,\mu,\lambda,\tau,\omega,\widetilde{\omega}$ and $\Omega$, but
independents on $\varepsilon$ and $\delta$. If $C$ depends also on other
parameters, it will be stated explicitly. The exact value of $C$ may change
from line to line.

Organization of the paper. The rest of the paper is organized as follows. In
section 2 we collect some results on periodic layer potentials and extension
operators that are useful to deal with functions on perforated domains,
together with basic elliptic estimates. In section 3 we obtain uniform
estimates for the cell problems in the high contrast setting, and establish
convergence rates in the high contrast limit. We then give the proof of
Theorem 1.1 in section 4. The Caccioppoli estimates in the high contrast
setting are established in section 5.2, and we prove in section 6 the uniform
regularity of the elliptic system in the high contrast setting.

##  2\. Layer potentials, Extension operators and basic elliptic estimates

###  2.1. Periodic layer potential

Let the coefficient $A$ satisfies (1.2)-(1.4), in view of the interior
Lipschitz regularity for $\mathcal{L}$, proved by Avellaneda and Lin [9], one
has the periodic fundamental solution
$\Gamma(x,y)=(\Gamma^{\alpha\beta}(x,y)):\mathbb{T}^{d}\times\mathbb{T}^{d}\rightarrow\mathbb{R}^{m\times
m}$, i.e.,

$\left\\{\begin{aligned}
&\mathcal{L}(x)\big{(}\Gamma^{\alpha}(x,y)\big{)}=(\delta_{y}(x)-1)e^{\alpha},&(x,y)\in
Y\times Y,\\\ &\int_{Y}\Gamma^{\alpha}(x,y)\,dx=0,&y\in
Y,\end{aligned}\right.$ (2.1)

where $\delta_{y}(x)$ is the Dirac measure massed at $y$, and the estimates

$\left\\{\begin{aligned} &|\Gamma(x,y)|\leq C|x-y|^{2-d},\\\
&|\nabla_{x}\Gamma(x,y)|+|\nabla_{y}\Gamma(x,y)|\leq
C|x-y|^{1-d}.\end{aligned}\right.$ (2.2)

The function $\Gamma(x,y)$ is called the periodic fundamental solution for the
operator $\mathcal{L}$ with pole at $y$. For construction of the periodic
fundamental solution and estimates (2.2), we refer to Hofmann and Kim [12],
where they treat the case of zero Dirichlet boundary condition, but their
methods apply also in the periodic setting.

For $f\in L^{2}(\partial\omega)$, based on the estimates (2.2), one can define
the single-layer potential $\mathcal{S}f$ by

$\big{(}\mathcal{S}f\big{)}^{\alpha}(x)=\int_{\partial\omega}\Gamma^{\alpha\beta}(x,y)f^{\beta}(y)\,d\sigma(y)$
(2.3)

while the Neumann-Poincaré (NP) operator $\mathcal{K}^{*}$ is defined by

$\big{(}\mathcal{K}^{*}f\big{)}^{\alpha}(x)=\mathrm{P.V.}\int_{\partial\omega}\left(\frac{\partial\Gamma^{\beta}}{\partial\nu_{x}}\right)^{\alpha}(x,y)f^{\beta}(y)\,d\sigma(y),$
(2.4)

where P.V. stands for the Cauchy principal value integral. For more regular
$\omega$, say $\partial\omega\in C^{1,\alpha}$ for some $\alpha\in(0,1)$, the
above can be replaced by the regular integral.

Denote by $L^{2}_{0}(\partial\omega)$ the space $\\{\phi\in
L^{2}(\partial\omega)\,:\,\int_{\partial\omega}\phi=0\\}$. The following
properties are standard:

  * (a)

For $\psi\in L^{2}(\partial\omega)$, $\mathcal{S}\psi\in
H^{1}(\mathbb{T}^{d})$.

  * (b)

For $\psi\in L^{2}(\partial\omega)$, jump relation holds, i.e.,

$\frac{\partial}{\partial\nu}\big{(}\mathcal{S}\psi\big{)}\Big{|}_{\omega\pm}=\Big{(}\pm\frac{1}{2}I+\mathcal{K}^{*}\Big{)}\psi.$
(2.5)

  * (c)

For $\psi\in L^{2}_{0}(\partial\omega)$, $\mathcal{L}(\mathcal{S}\psi)=0$ in
$\mathbb{T}^{d}\setminus\partial\omega$.

  * (d)

The spectrum of $\mathcal{K}^{*}$ on $L^{2}_{0}(\partial\omega)$ is contained
in $(-1/2,1/2)$.

We refer to [16, 14] and [2, Section 2.8] for more results on layer potential
theory with varying coefficients (i.e., for general elliptic operators). Note
that above and in the rest of the paper,

$\frac{\partial}{\partial\nu}w=\left((\partial_{\nu}w)^{\alpha}\right),\quad\text{with}\
\
(\partial_{\nu}w)^{\alpha}:=\sum_{i=1}^{d}\nu_{i}A^{\alpha\beta}_{ij}\partial_{j}w^{\beta}.$

###  2.2. Extension operators

The following extension operator is useful in the treatment of perforated
domains. The problem setting for high contrast inclusions is in a similar
situation. Recall that $\widetilde{\omega}$ denotes an enlarging dilation of
$\omega$ so that $\overline{\omega}\subset\widetilde{\omega}\subset
Q_{7/8}(0)$.

######  Lemma 2.1.

Assume $(\mathbf{H2})$, there exists a linear extension operator
$\mathcal{P}_{\varepsilon}:H^{1}(\Omega_{\varepsilon})\rightarrow
H^{1}(\Omega)$ such that for any $v\in H^{1}(\Omega_{\varepsilon})$,

$\mathcal{P}_{\varepsilon}v=v\quad\mathrm{in}\
\Omega_{\varepsilon}\qquad\mathrm{and}\qquad\|\nabla\mathcal{P}_{\varepsilon}v\|_{L^{2}(\Omega)}\leq
C\|\nabla v\|_{L^{2}(\Omega_{\varepsilon})}.$

###### Proof.

We refer to [10] or [13, Lemma 3.3]; see also [1, Appendix B]. The basic idea
is, at the unit scale around a typical obsctale $\omega$ and for a function
$v\in H^{1}(\widetilde{\omega}\setminus\overline{\omega})$ with mean value
$c=\fint_{\widetilde{\omega}\setminus\overline{\omega}}v$, we can apply the
usual extension operator to $v-c$ and obtain a $v_{1}\in
H^{1}(\widetilde{\omega})$ so that

$\|v_{1}\|_{H^{1}(\widetilde{\omega})}\leq\|v-c\|_{H^{1}(\widetilde{\omega}\setminus\overline{\omega})}\leq
C\|\nabla v\|_{L^{(}\widetilde{\omega}\setminus\overline{\omega})},$

where the last step results from Poincaré-Wirtinger inequality. Then
$\tilde{v}:=c+v_{1}$ is an extension of $v$ from
$\widetilde{\omega}\setminus\overline{\omega}$ to $\widetilde{\omega}$ and
satisfies

$\|\tilde{v}\|_{L^{2}(\widetilde{\omega})}\leq
C\|v\|_{L^{2}(\widetilde{\omega}\setminus\overline{\omega})},\quad\text{and}\quad\|\nabla\tilde{v}\|_{L^{2}(\widetilde{\omega})}\leq
C\|\nabla v\|_{L^{2}(\widetilde{\omega}\setminus\overline{\omega})}.$

Under $(\mathbf{H2})$, the desired extension $\mathcal{P}_{\varepsilon}$ in
the whole domain $\Omega_{\varepsilon}$ is by gluing the rescaled local
extensions together. ∎

As we will see, the extension above is useful for small $\delta$ in the high
contrast setting of this paper. For large $\delta$, we need the following
operator that extends functions defined in the inclusions with zero mean in
each inclusion to the outside.

######  Lemma 2.2.

Assume $(\mathbf{H2})$, there exists a linear map
$\mathcal{Q}_{\varepsilon}:H^{1}(D_{\varepsilon})\rightarrow
H^{1}_{0}(\Omega)$ such that $\mathcal{Q}_{\varepsilon}(v)$ is supported in
$\cup_{\mathbf{n}}\widetilde{\omega}^{\mathbf{n}}_{\varepsilon}$, and

$\mathcal{Q}_{\varepsilon}(v)=v-q^{\mathbf{n}}_{\varepsilon},\qquad
q^{\mathbf{n}}_{\varepsilon}=\int_{\omega^{\mathbf{n}}_{\varepsilon}}v,\qquad\mathrm{in\
each\ }\omega^{\mathbf{n}}_{\varepsilon}\subset D_{\varepsilon}.$

Moreover,

$\|\mathcal{Q}_{\varepsilon}(v)\|_{H^{1}(\Omega)}\leq C\|\nabla
v\|_{L^{2}(D_{\varepsilon})}.$

###### Proof.

As usual we first consider an extension operator in the unit scale. Suppose
$\omega$ is the model inclusion of unit length, and $\widetilde{\omega}$ is
the enlarged inclusion. Then there is a standard linear extension operator
$\mathcal{Q}$ from $H^{1}(\omega)$ to $H^{1}_{0}(\widetilde{\omega})$ so that

$\|\mathcal{Q}(v)\|_{H^{1}(\widetilde{\omega})}\leq C\|v\|_{H^{1}(\omega)}.$

Since the Poincaré-Wirtinger inequality holds for $v-q$, where
$q=\int_{\omega}v$, we get

$\|\mathcal{Q}(v-q)\|_{H^{1}(\widetilde{\omega})}\leq
C\|v-q\|_{H^{1}(\omega)}\leq C\|\nabla v\|_{L^{2}(\omega)}.$

In each small inclusion $\omega_{\varepsilon}^{\mathbf{n}}$ that is contained
in $\Omega$, we let

$\mathcal{Q}^{\mathbf{n}}_{\varepsilon}(v)(x):=\mathcal{Q}\Big{(}v(\varepsilon(\cdot+\mathbf{n}))-q^{\mathbf{n}}_{\varepsilon}\Big{)}\Big{(}\frac{x-\varepsilon\mathbf{n}}{\varepsilon}\Big{)},\qquad
x\in\widetilde{\omega}^{\mathbf{n}}_{\varepsilon}.$

Then we see that $\mathcal{Q}^{\mathbf{n}}_{\varepsilon}(v)$ belongs to
$H^{1}_{0}(\widetilde{\omega}_{\varepsilon}^{\mathbf{n}})$ and satisfies
$\mathcal{Q}^{\mathbf{n}}_{\varepsilon}(v)=v-q^{\mathbf{n}}_{\varepsilon}$,
and

$\|\nabla\mathcal{Q}^{\mathbf{n}}_{\varepsilon}(v)\|_{L^{2}(\widetilde{\omega}_{\varepsilon}^{\mathbf{n}})}\leq
C\|\nabla v\|_{L^{2}(\omega_{\varepsilon}^{\mathbf{n}})},$

where the constant $C$ does not change due to the scaling-invariant feature of
the estimate. For $v\in H^{1}(D_{\varepsilon})$, we use the map piecewise for
each component $\omega^{\mathbf{n}}_{\varepsilon}$ of $D_{\varepsilon}$ and
glue them together, we then get a linear map $\mathcal{Q}_{\varepsilon}$ with
the desired properties. ∎

###  2.3. Other useful tools and estimates

We will use the now standard smoothing operator: fix a function $\xi\in
C_{0}^{\infty}(B(0,\frac{1}{2}))$ such that $0\leq\xi(x)\leq 1$ and
$\int_{\mathbb{R}^{d}}\xi(x)\,dx=1$, for any $v\in L^{1}_{\rm
loc}(\mathbb{R}^{d})$, $S_{\varepsilon}(v)$ is the smooth function defined by

$S_{\varepsilon}(v)(x):=\varepsilon^{-d}\int_{\mathbb{R}^{d}}v(x-y)\xi(y/\varepsilon)\,dy.$
(2.6)

The operator is very useful in quantitative homogenization theory; it allows
one to use two-scale expansions because the regularization error can be well
controlled. In particular, we have the following.

######  Lemma 2.3.

Let $f\in L^{2}(\mathbb{T}^{d})$, then for any $g\in L^{2}(\mathbb{R}^{d})$,
we have

$\|f(x/\varepsilon)S_{\varepsilon}(g)\|_{L^{2}(\mathbb{R}^{d})}\leq
C\|f\|_{L^{2}(Y)}\|g\|_{L^{2}(\mathbb{R}^{d})}.$

We refer to Shen [25, Proposition 3.1.5] for the proof, together with more
useful properties of $S_{\varepsilon}$. We end the section by the following
useful but standard estimate for elliptic systems.

######  Lemma 2.4.

Let $A=\big{(}a_{ij}^{\alpha\beta}\big{)}$ be a constant tensor satisfying
(1.1)-(1.2), let $\Omega$ be a bounded Lipschitz domain with connected
boundary in $\mathbb{R}^{d}$, and let $u$ be the solution of

$\left\\{\begin{aligned} &-\mathrm{div}\,(A\nabla u)=f&\mathrm{in}\ \Omega,\\\
&u=g&\mathrm{on}\ \partial\Omega.\end{aligned}\right.$

then

$\|\nabla
u\|_{L^{2}(\mathcal{O}_{t})}+t\|\nabla^{2}u\|_{L^{2}(\Omega\setminus\mathcal{O}_{t})}\leq
Ct^{1/2}\big{(}\|f\|_{L^{2}(\Omega)}+\|\nabla_{\mathrm{tan}}g\|_{L^{2}(\partial\Omega)}\big{)},$

where $C$ depends only on $\mu$ and the Lipschitz character of $\Omega$.

###### Proof.

This follows from the interior estimates and the nontangential-maximal-
function estimate for second-order elliptic equations with constant
coefficients. See [17] for reference. ∎

######  Remark 3.

Note that $\Omega$ is a Lipschitz domain, it is impossible to control
$\|\nabla^{2}u\|_{L^{2}(\Omega)}$, since the solution may have a singular
behavior near the irregular points of $\partial\Omega$, even in the simplest
case of the Laplace equation, if $\Omega$ is a non-convex polygon in
$\mathbb{R}^{2}$, it is well known [11] that $u\notin H^{2}(\Omega)$ in
general, even if $f\in C^{\infty}(\overline{\Omega})$ and $g=0$.

##  3\. Limiting behavior associated to cell problem

Our overall strategy in this work is to treat $\varepsilon$ and $\delta$ as
two independent parameters. This approach was mentioned already in Chap 3 of
[13]. In general, we can consider the diagram

$\begin{CD}u_{\varepsilon,\delta}@>{\delta\to 0\text{ or
}\infty}>{}>u_{\varepsilon}\\\ @V{\varepsilon\to 0}V{}V@V{}V{\varepsilon\to
0}V\\\ \widehat{u}_{\delta}@>{\delta\to 0\text{ or
}\infty}>{}>\widehat{u}\end{CD}$ (3.1)

We show in this section that $\widehat{u}_{\delta}$, or the equation for it,
behaves quite well uniformly in $\delta$. In the scalar setting, such uniform
in $\delta$ results were established in Jikov using variational techniques.
Here, we obtain the results for the elliptic system using layer-potential
techniques.

Note that to use layer potentials, we require the coefficient $A$ satisfies
(1.2)-(1.4) (see Section 2.1), nevertheless, by using a ’weak’ form of layer
potentials, one can obtain all of results in this section, see Appendix A for
the construction of ’weak’ form of layer potentials. For simplicity, we assume
(1.2)-(1.4) for $A$ in this section, but as we have mentioned, the readers
should keep in mind that the results in this section also holds for $A$
satisfying (1.2)-(1.3).

###  3.1. Uniform bounds for the cell problems

Recall that

$\chi_{\delta}=(\chi_{\delta,j}^{\alpha})_{1\leq j\leq d,1\leq\alpha\leq m},$

denotes the solution to the cell problem that arises in the standard
homogenization of (1.7) for each fixed $\delta\in(0,1)$. What is most
important is to obtain estimates that are either uniform in $\delta$ or have
explicit dependence in $\delta$. We achieve this goal through the layer
potential representation of the cell problem.

######  Lemma 3.1.

For fixed $0<\delta<\infty$, let $\chi_{\delta}=(\chi_{\delta,i}^{\alpha})$,
$1\leq i\leq d$ and $1\leq\alpha\leq m$, be the solution of the cell problem
(1.10). Then for $\delta\neq 1$, $\chi_{\delta}$ has the folowing layer
potential representation:

$\chi_{\delta,i}^{\alpha}=\chi^{\alpha}_{1,i}+\mathcal{S}\left[\big{(}\lambda(\delta)I-\mathcal{K}^{*}\big{)}^{-1}\phi_{i}^{\alpha}\right]-q_{\delta,i}^{\alpha},\quad\lambda(\delta)=\frac{1}{2}+\frac{1}{\delta-1},$
(3.2)

where

$\phi_{i}^{\alpha}=\frac{\partial}{\partial\nu}\Big{|}_{+}\big{(}y_{i}e^{\alpha}+\chi_{1,i}^{\alpha}\big{)}=\frac{\partial}{\partial\nu}\Big{|}_{-}\big{(}y_{i}e^{\alpha}+\chi_{1,i}^{\alpha}\big{)}\in
H_{0}^{-1/2}(\partial\omega),$ (3.3)

and

$q_{\delta,i}^{\alpha}=\int_{Y}\mathcal{S}\left[\big{(}\lambda(\delta)I-\mathcal{K}^{*}\big{)}^{-1}\phi_{i}^{\alpha}\right](y)\,dy.$
(3.4)

Here, the periodic single-layer potential operator $\mathcal{S}$ and the
associated Neumann-Poincaré operator $\mathcal{K}^{*}$ are defined by (2.4).

###### Proof.

Let $u=\chi^{\alpha}_{\delta,i}-\chi^{\alpha}_{1,i}$, then $u$ is the solution
of the transmission problem

$\left\\{\begin{aligned} &\mathcal{L}u=0&\mathrm{in}\
\mathbb{T}^{d}\setminus\partial\omega,\\\ &\delta\frac{\partial
u}{\partial\nu}\Big{|}_{-}-\frac{\partial
u}{\partial\nu}\Big{|}_{+}=(1-\delta)\phi_{i}^{\alpha}&\mathrm{on}\
\partial\omega,\\\ &\int_{Y}u(y)\,dy=0.\end{aligned}\right.$

We aim to represent the solution by $u=\mathcal{S}\psi-q$, where
$q\in\mathbb{R}^{m}$. In view of the properties of layer potentials (See
Section 2.1), $\psi$ and $q$ must satisfy

$\big{(}\lambda(\delta)I-\mathcal{K}^{*}\big{)}\psi=\phi_{i}^{\alpha},\qquad
q=\int_{Y}\mathcal{S}\psi(y)\,dy.$

The integral system above is solvble for $\psi\in L_{0}^{2}(\partial\omega)$,
since $|\lambda(\delta)|>1/2$ when $0<\delta<\infty$ and the spectrum of
$\mathcal{K}^{*}$ on $L^{2}_{0}(\partial\omega)$ is contained in $(-1/2,1/2)$.
This completes the proof. ∎

Note that $\lambda(\delta)$ converges to $-1/2$ as $\delta\to 0+$, and
converges to $1/2$ as $\delta\to\infty$. We define the limit tensors
$\chi_{0}=(\chi^{\alpha}_{0,i})$ and
$\chi_{\infty}=(\chi^{\alpha}_{\infty,i})$, $1\leq i\leq d$, $1\leq\alpha\leq
m$, by

$\left\\{\begin{aligned}
&\chi^{\alpha}_{0,i}:=\chi^{\alpha}_{1,i}+\mathcal{S}\big{(}-I/2-\mathcal{K}^{*}\big{)}^{-1}\phi_{i}^{\alpha}-q^{\alpha}_{0,i},\\\
&\chi_{\infty,i}^{\alpha}:=\chi^{\alpha}_{1,i}+\mathcal{S}\big{(}I/2-\mathcal{K}^{*}\big{)}^{-1}\phi_{i}^{\alpha}-q^{\alpha}_{\infty,i},\end{aligned}\right.$
(3.5)

where $\phi_{i}^{\alpha}$ is defined in (3.3) and

$q_{0,i}^{\alpha}=\int_{Y}\mathcal{S}\left[\big{(}-I/2-\mathcal{K}^{*}\big{)}^{-1}\phi_{i}^{\alpha}\right](y)\,dy,\qquad
q_{\infty,i}^{\alpha}=\int_{Y}\mathcal{S}\left[\big{(}I/2-\mathcal{K}^{*}\big{)}^{-1}\phi_{i}^{\alpha}\right](y)\,dy.$
(3.6)

Indeed, as we show next, $\chi_{0}$ and $\chi_{\infty}$ are the limits of
$\chi_{\delta}$ as $\delta$ tends to $0$ and to $\infty$, respectively. In
fact, they correspond to the cell problem of the soft ($\delta=0$) and hard
($\delta=\infty$) inclusion problems, respectively.

######  Theorem 3.2.

For $1\leq i\leq d$ and $1\leq\alpha\leq m$, the function
$\chi_{0,i}^{\alpha}\in H^{1}(\mathbb{T}^{d})$ is the solution of the soft
problem

$\left\\{\begin{aligned}
&\mathcal{L}(\chi_{0,i}^{\alpha})=-\mathcal{L}(y_{i}e^{\alpha})&\mathrm{in}\
\mathbb{T}^{d}\setminus\partial\omega,\\\
&\frac{\partial\chi_{0,i}^{\alpha}}{\partial\nu}\Big{|}_{+}=-\frac{\partial(y_{i}e^{\alpha})}{\partial\nu}\Big{|}_{+}&\mathrm{on}\
\partial\omega,\\\ &\int_{Y}\chi_{0,i}^{\alpha}(y)\,dy=0.\end{aligned}\right.$
(3.7)

The function $\chi^{\alpha}_{\infty,i}\in H^{1}(\mathbb{T}^{d})$ is the
solution of the stiff problem

$\left\\{\begin{aligned}
&\mathcal{L}(\chi_{\infty,i}^{\alpha})=-\mathcal{L}(y_{i}e^{\alpha})&\mathrm{in}\
\mathbb{T}^{d}\setminus\partial\omega,\\\
&\frac{\partial\chi_{\infty,i}^{\alpha}}{\partial\nu}\Big{|}_{-}=-\frac{\partial(y_{i}e^{\alpha})}{\partial\nu}\Big{|}_{-}&\mathrm{on}\
\partial\omega,\\\
&\int_{Y}\chi_{\infty,i}^{\alpha}(y)\,dy=0.\end{aligned}\right.$ (3.8)

Moreover, there is a universal constant $C<\infty$ so that

$\|\chi_{\delta}-\chi_{0}\|_{H^{1}(Y)}\leq
C\delta\quad\mathrm{and}\quad\|\chi_{\delta}-\chi_{\infty}\|_{H^{1}(Y)}\leq
C\delta^{-1},$ (3.9)

and hence also $\|\chi_{\delta}\|_{H^{1}}\leq C$.

###### Proof.

In view of Section 2.1, especially the jump relation, it is clear that
$\chi^{\alpha}_{0,i}$ and $\chi^{\alpha}_{\infty,i}$ is the solution of
problem (3.7) and (3.8), respectively. We now prove (3.9).

Since
$\lambda(\delta)I-\mathcal{K}^{*}\rightarrow\mp\frac{1}{2}I-\mathcal{K}^{*}$
as $\delta$ tends to $0$ or $\infty$ and $\mp\frac{1}{2}I-\mathcal{K}^{*}$ are
bijections on $L^{2}_{0}(\partial\omega)$, by the basic theory of functional
analysis, we obtain that $\|(\lambda(\delta)I-\mathcal{K}^{*})^{-1}\|$ is
uniformly bounded in $0<\delta<\infty$. Therefore,

$\displaystyle\big{\|}\chi_{\delta,i}^{\alpha}-\chi_{0,i}^{\alpha}\big{\|}_{H^{1}(Y)}\leq$
$\displaystyle\big{\|}\mathcal{S}\big{[}(\lambda(\delta)I-\mathcal{K}^{*})^{-1}-(-I/2-\mathcal{K}^{*})^{-1}\big{]}\phi_{i}^{\alpha}\big{\|}_{H^{1}(Y)}+\big{|}q_{\delta,i}^{\alpha}-q_{0,i}^{\alpha}\big{|}$
$\displaystyle\leq$ $\displaystyle
2\big{\|}\mathcal{S}\big{[}(\lambda(\delta)I-\mathcal{K}^{*})^{-1}-(-I/2-\mathcal{K}^{*})^{-1}\big{]}\phi_{i}^{\alpha}\big{\|}_{H^{1}(Y)}$
$\displaystyle\leq$ $\displaystyle
C\big{(}\lambda(\delta)+1/2\big{)}\big{\|}\big{(}\lambda(\delta)I-\mathcal{K}^{*}\big{)}^{-1}\big{\|}\cdot\big{\|}(I/2+\mathcal{K}^{*})^{-1}\big{\|}$
$\displaystyle\leq$ $\displaystyle C\delta.$

This proves the convergence rate when $\delta\rightarrow 0$. The proof for
rate of $\delta\rightarrow\infty$ follows the same lines and is hence omitted.
The uniform estimates for $\|\chi_{\delta}\|_{H^{1}}$ follows also because
$\|\chi_{1}\|_{H^{1}}$ is easily bounded, and by (3.5) the bounds for
$\chi_{0},\chi_{\infty}$ also hold. ∎

######  Remark 4.

In Appendix A, we define a ’weak’ form of single layer potentials via Lax-
Milgram theorem, which maps from $H^{-1/2}$ to $H^{1/2}$, and the
corresponding NP operator without smoothness condition (1.4). Using the ’weak’
form of layer potentials, all above results can be recovered by the same
analysis.

###  3.2. Uniform bound for the homogenized tensor

Recall that, for each fixed $\delta>0$, the homogenized tensor for (1.7) is
$\hat{A}_{\delta}$ defined by (1.9). We show that $\widehat{A}_{\delta}$
enjoys uniform (in dependent of $\delta$) ellipticity.

######  Lemma 3.3.

For any $0<\delta<\infty$, $1\leq i,j\leq d$, $1\leq\alpha,\beta\leq m$, we
have

$\Big{\|}\Lambda_{\delta}\Big{(}a_{ij}^{\alpha\beta}+a_{ik}^{\alpha\gamma}\frac{\partial}{\partial
y_{k}}\chi_{\delta,j}^{\gamma\beta}\Big{)}\Big{\|}_{L^{2}(Y)}\leq C.$ (3.10)

###### Proof.

The aim is to bound the $L^{2}(Y)$ norm of

$\sum_{k=1}^{d}\sum_{\gamma=1}^{m}\Lambda_{\delta}(y)a_{ik}^{\alpha\gamma}(y)\frac{\partial}{\partial
y_{k}}\left(\chi^{\gamma\beta}_{\delta,j}(y)+y_{j}e^{\beta}\right),$

for each fixed $i,j,\alpha$ and $\beta$. By Theorem 3.2, we have

$\Big{\|}\frac{\partial}{\partial
y_{i}}(\chi_{\delta}-\chi_{0})\Big{\|}_{L^{2}(Y)}\leq
C\delta\quad\mathrm{and}\quad\Big{\|}\frac{\partial}{\partial
y_{i}}(\chi_{\delta}-\chi_{\infty})\Big{\|}_{L^{2}(Y)}\leq C\delta^{-1}.$
(3.11)

For $\delta\in(0,1]$, the desired estimates follow from the boundedness of
$a$, $\|\chi_{0}\|_{H^{1}}$ and $\delta\leq 1$. For $\delta\in(1,\infty)$,
after replacing $\chi_{\delta}$ by $\chi_{\infty}$ and applying the above
inequality, it remains to control

$\|\sum_{k=1}^{d}\sum_{\gamma=1}^{m}\Lambda_{\delta}(y)a_{ik}^{\alpha\gamma}(y)\frac{\partial}{\partial
y_{k}}\left(\chi^{\gamma\beta}_{\infty,j}(y)+y_{j}e^{\beta}\right)\|_{L^{2}(\omega)},$

because $\Lambda_{\delta}=1$ outside $\omega$ and
$\|\nabla\chi_{\infty}\|_{L^{2}}$ is bounded. Now in view of the stiff cell
problem (3.8) (for the part in $\omega$), we must have
$\chi_{\infty,j}^{\beta}+y_{j}e^{\beta}=c$ for some constant vector
$c\in\mathbb{R}^{d}$. Hence, the above quantity vanishes. This completes the
proof. ∎

######  Lemma 3.4.

The homogenized tensor $\widehat{A}_{\delta}$ is continuous as a function of
$\delta\in(0,\infty)$, we denote $\widehat{A}_{0}$ by $\lim_{\delta\rightarrow
0}\widehat{A}_{\delta}$ and $\widehat{A}_{\infty}$ by
$\lim_{\delta\rightarrow\infty}\widehat{A}_{\delta}$, then

$\widehat{a}_{0,ij}^{\alpha\beta}=\int_{Y\setminus\overline{\omega}}\Big{[}a_{ij}^{\alpha\beta}+a_{ik}^{\alpha\gamma}\frac{\partial}{\partial
y_{k}}\chi_{0,j}^{\gamma\beta}\Big{]}\,dy,$ (3.12)

and

$\widehat{a}_{\infty,ij}^{\alpha\beta}=\int_{\omega}a_{ik}^{\alpha\gamma}\frac{\partial
w_{j}^{\gamma\beta}}{\partial
y_{k}}\,dy+\int_{Y\setminus\overline{\omega}}\Big{[}a_{ij}^{\alpha\beta}+a_{ik}^{\alpha\gamma}\frac{\partial}{\partial
y_{k}}\chi_{\infty,j}^{\gamma\beta}\Big{]}\,dy,$ (3.13)

where for each fixed $j=1$ and $\alpha$, the function $w^{\alpha}_{j}\in
H^{1}(\omega;\mathbb{R}^{m})$ is the unique (up to an additive constant)
solution of

$\left\\{\begin{aligned} &\mathcal{L}(w_{j}^{\alpha})=0&\mathrm{in}\
\omega,\\\ &\frac{\partial
w_{j}^{\alpha}}{\partial\nu}\Big{|}_{-}=\frac{\partial}{\partial\nu}(\chi_{\infty,j}^{\alpha}+y_{j}e^{\alpha})\Big{|}_{+}&\mathrm{on}\
\partial\omega.\end{aligned}\right.$ (3.14)

It is worth to mention that, the tensor $\widehat{A}_{0}$ (respectively,
$\widehat{A}_{\infty}$) is the homogenized tensor for the soft inclusion
problem (respectively, stiff inclusion problem); see Remark 2.

###### Proof.

By the definition (1.9) of $\widehat{A}_{\delta}$ and the single-layer
potential representations of the solutions to cell problem, it is clear that
$\widehat{A}_{\delta}$ is continuous as a function of $\delta\in(0,\infty)$,
and (3.12) follows directly from the convergence
$\|\chi_{\delta}-\chi_{0}\|_{H^{1}(Y)}\leq C\delta$. It remains to verify
(3.13).

First we note that the problem for $w_{j}^{\alpha}$ is well posed. We need to
check that the Neumann data integrates to zero over $\partial\omega$. By the
definition (3.5) and the jump relation of layer potentials,

$\displaystyle\frac{\partial}{\partial\nu}(\chi_{\infty,j}^{\alpha}+y_{j}e^{\alpha})\Big{|}_{+}$
$\displaystyle=\frac{\partial}{\partial\nu}(\chi_{\infty,j}^{\alpha}+y_{j}e^{\alpha})\Big{|}_{-}+\big{(}I/2-\mathcal{K}^{*}\big{)}^{-1}\phi_{j}^{\alpha}$
$\displaystyle=\big{(}I/2-\mathcal{K}^{*}\big{)}^{-1}\phi_{j}^{\alpha}.$

The solvability of $w_{j}^{\delta}$ is verified. Moreover, we check that

$\displaystyle\mathcal{L}(\chi_{\delta,j}^{\alpha}-\chi_{\infty,j}^{\alpha}-\frac{1}{\delta-1}w_{j}^{\alpha})=0\;\text{
in }\omega,$
$\displaystyle\frac{\partial}{\partial\nu}(\chi_{\delta,j}^{\alpha}-\chi_{\infty,j}^{\alpha}-\frac{1}{\delta-1}w_{j}^{\alpha})\Big{\rvert}_{-}=\left[I+(-I/2+\mathcal{K}^{*})(\lambda(\delta)I-\mathcal{K}^{*})^{-1}-\frac{1}{\delta}(I/2-\mathcal{K}^{*})^{-1}\right]\phi_{j}^{\alpha}.$

The operators $(I/2-\mathcal{K}^{*})$ and its inverse act on
$L^{2}_{0}(\partial\omega)$. For $\delta>1$, the inverse operator can be
written as Neumann series, and the second line becomes

$\frac{\partial}{\partial\nu}(\chi_{\delta,j}^{\alpha}-\chi_{\infty,j}^{\alpha}-\frac{1}{\delta-1}w_{j}^{\alpha})\Big{\rvert}_{-}=\sum_{\ell=2}^{\infty}\frac{(-1)^{\ell}}{(\delta-1)^{\ell}}(I/2-\mathcal{K}^{*})^{-\ell}.$

By basic estimates of Neumann problem of elliptic system we get

$\|\nabla(\chi_{\delta,j}^{\alpha}-\chi^{\alpha}_{\infty,j}-\frac{1}{\delta-1}w^{\alpha}_{j})\|_{L^{2}(\omega)}\leq
C\delta^{-2},$ (3.15)

for all $\delta>1$ sufficiently large, and $C<\infty$ depends only on the norm
of $\|(I/2-\mathcal{K}^{*})^{-1}\|$. Using the fact that
$\chi_{\infty,j}^{\alpha}+y_{j}e^{\alpha}$ is a constant vector in $\omega$,
we rewrite

$\displaystyle\widehat{a}_{\delta,ij}^{\alpha\beta}-\widehat{a}_{\infty,ij}^{\alpha\beta}$
$\displaystyle=\delta\int_{\omega}a_{ik}^{\alpha\gamma}\frac{\partial}{\partial
y_{k}}\big{(}\chi_{\delta,j}^{\gamma\beta}-\chi^{\gamma\beta}_{\infty,j}-\frac{1}{\delta-1}w_{j}^{\beta}\big{)}\,dy+\int_{Y\setminus\overline{\omega}}a_{ik}^{\alpha\gamma}\frac{\partial}{\partial
y_{k}}\big{(}\chi_{\delta,j}^{\gamma\beta}-\chi_{\infty,j}^{\gamma\beta}\Big{)}\,dy$
$\displaystyle\qquad\qquad+\frac{1}{\delta-1}\int_{\omega}a_{ik}^{\alpha\gamma}\frac{\partial
u_{j}^{\gamma\beta}}{\partial y_{k}}\,dy.$

Send $\delta\to\infty$, due to the bounds (3.15) and (3.10), we obtain the
desired result. In fact, we get
$|\widehat{A}_{\delta}-\widehat{A}_{\infty}|\leq C\delta^{-1}$ for
sufficiently large $\delta$. ∎

As a corollary of Lemma 3.4, we show that $\widehat{A}_{\delta}$ is uniform
elliptic in $0<\delta<\infty$.

######  Theorem 3.5.

Assume $A$ satisfies (1.2)(1.3), then $\widehat{A}_{\delta}$ is uniformly
elliptic in $\delta$, i.e. there exists a universal positive number $\mu_{1}$,
so that for any $\delta\in(0,\infty)$, we have
$|\widehat{A}_{\delta}|\leq\mu_{1}^{-1}$ and

$\widehat{a}_{\delta,ij}^{\alpha\beta}\xi_{i}^{\alpha}\xi_{j}^{\beta}\geq\mu_{1}|\xi|^{2}\quad\mathrm{for}\
\mathrm{any}\ \xi\in\mathbb{R}^{dm}.$ (3.16)

###### Proof.

We have established a uniform in $\delta$ bound of $|\widehat{A}_{\delta}|$,
so we concentrate here on proving (3.16). Given
$\xi=(\xi_{i}^{\alpha})\in\mathbb{R}^{dm}$, let
$w_{\delta}=\xi^{\alpha}_{i}(y_{i}e^{\alpha}+\chi_{\delta,i}^{\alpha})$ be the
combination of the correctors $\chi_{\delta}$’s. We have

$\displaystyle\widehat{a}_{\delta,ij}^{\alpha\beta}\xi_{i}^{\alpha}\xi_{j}^{\beta}$
$\displaystyle=(J^{Y\setminus\overline{\omega}}+\delta
J^{\omega})(w_{\delta})$
$\displaystyle\geq\min\\{\delta,1\\}\mu\int_{Y}|\nabla w_{\delta}|^{2}\,dy$
$\displaystyle\geq\min\\{\delta,1\\}\mu\left|\int_{Y}\nabla
w_{\delta}\right|^{2}\,dy$ $\displaystyle=\min\\{\delta,1\\}\mu|\xi|^{2}.$

The last identity holds because the integral of
$\nabla\chi_{\delta,i}^{\alpha}$ on $\mathbb{T}^{d}$ vanishes. Therefore, for
$\delta\geq 1$, the lower ellipticity constant for $\widehat{A}_{\delta}$ is
uniformly bounded by $\mu$ from below. For $\delta\in(0,1)$, in view of the
continuity of $\widehat{A}_{\delta}$ with respect to $\delta$, it sufficies to
prove that $\widehat{A}_{0}$ is elliptic, but this follows by the same
reasoning as above. ∎

###  3.3. Uniform bound of the flux corrector

Given a tensor $F=(F^{\alpha\beta}_{ij})$ on $Y$, for $1\leq i,j\leq d$,
$1\leq\alpha,\beta\leq m$, such that

$\frac{\partial}{\partial
y_{i}}F^{\alpha\beta}_{ij}=0\quad\mathrm{and}\quad\int_{Y}F^{\alpha\beta}_{ij}(y)\,dy=0.$
(3.17)

It is proved, for example see [25], there exists a tensor
$\Psi=(\Psi^{\alpha\beta}_{kij})$ on $\mathbb{T}^{d}$, for $1\leq k,i,j\leq d$
and $1\leq\alpha,\beta\leq m$, such that

$\frac{\partial}{\partial
y_{k}}\Psi^{\alpha\beta}_{kij}=F_{ij}^{\alpha\beta}\quad\mathrm{and}\quad\Psi_{kij}^{\alpha\beta}=-\Psi_{ikj}^{\alpha\beta},$
(3.18)

and there exists a constant $C>0$ such that $\|\Psi\|_{L^{2}(Y)}\leq
C\|F\|_{L^{2}(Y)}$. $\Psi$ is called the flux corrector associated to $F$.

In this paper, we need to use the flux corrector
$\Psi_{\delta}=(\Psi_{\delta,kij}^{\alpha\beta})$ associated to
$F_{\delta}=(F_{\delta,ij}^{\alpha\beta})$, where

$F_{\delta,ij}^{\alpha\beta}(y)=\Lambda_{\delta}(y)\Big{(}a_{ij}^{\alpha\beta}(y)+a_{ik}^{\alpha\gamma}(y)\frac{\partial}{\partial
y_{k}}\chi_{\delta,j}^{\gamma\beta}(y)\Big{)}-\widehat{a}^{\alpha\beta}_{\delta,ij},\qquad
0<\delta<\infty.$ (3.19)

It is easy to check condition (3.17) for $F_{\delta}$, so $\Psi_{\delta}$
exists and $\|\Psi_{\delta}\|_{L^{2}(Y)}\leq C\|F_{\delta}\|_{L^{2}(Y)}$.

######  Theorem 3.6.

Let $F_{\delta}$ be defined by (3.19), and let $\Psi$ be the skew-symmetric
flux-tensor related to $F_{\delta}$ by (3.18). Then

$\|F_{\delta}\|_{L^{2}(Y)}+\|\Psi_{\delta}\|_{L^{2}(Y)}\leq C.$

This theorem follows directly from the uniform in $\delta$ bounds (3.10). The
uniform estimates for $F_{\delta}$ and $\Psi_{\delta}$ above will be the key
to obtain uniform in $\delta$ convergence rate.

##  4\. Proof of Theorem 1.1

In this section, we prove the uniform (in $\varepsilon$ and $\delta$)
convergence rate in $H^{1}(\Omega)$.

###  4.1. Uniform energy estimates

We first derive uniform $H^{1}$ energy estimates for the solution to the
heterogeneous equation in the high contrast setting. With the extension
operators of the previous section, the estimates become routine.

######  Theorem 4.1.

Assume (1.2)-(1.3) and $(\mathbf{H2})$, for $\delta\in(0,1]$ we have

$\|u_{\varepsilon}\|_{H^{1}(\Omega)}\leq
C\left(\|f\|_{L^{2}(\Omega)}+\varepsilon\delta^{-1}\|f\|_{L^{2}(D_{\varepsilon})}+\|g\|_{H^{\frac{1}{2}}(\partial\Omega)}\right).$
(4.1)

For $\delta>1$, the above still holds with the
$\varepsilon\delta^{-1}\|f\|_{L^{2}(D_{\varepsilon})}$ term removed from the
right hand side. In particular, for $\delta\in(0,1]$, if we replace $f$ on the
right hand side of (1.7) by $f_{\varepsilon,\delta}$ defined by (1.13) we may
ignore the second term on the right hand side above.

###### Proof.

Note that for fixed $\varepsilon,\delta$, the uniquely solvability of
$u_{\varepsilon}\in H^{1}(\Omega)$ is routine. Our goal is to establish the
above uniform energy estimates.

_Step 1: Control of the energy inside the inclusions_. Let
$\widetilde{u}_{\varepsilon}$ be the extension
$\mathcal{P}_{\varepsilon}(u_{\varepsilon}\rvert_{\Omega_{\varepsilon}})$
given by Lemma 2.1. Then $u_{\varepsilon}-\widetilde{u}_{\varepsilon}\in
H^{1}_{0}(\Omega)$, from the equation (1.7) and the fact that
$u_{\varepsilon}=\widetilde{u}_{\varepsilon}$ in $\Omega_{\varepsilon}$, we
get

$\delta\int_{D_{\varepsilon}}A(x/\varepsilon)\nabla
u_{\varepsilon}\cdot(\nabla
u_{\varepsilon}-\nabla\widetilde{u}_{\varepsilon})=\int_{D_{\varepsilon}}f\cdot(u_{\varepsilon}-\widetilde{u}_{\varepsilon}).$

The ellipticity of $A$ and Poincaré inequality
$\|u_{\varepsilon}-\widetilde{u}_{\varepsilon}\|_{L^{2}(D_{\varepsilon})}\leq
C\varepsilon\|\nabla
u_{\varepsilon}-\nabla\widetilde{u}_{\varepsilon}\|_{L^{2}(D_{\varepsilon})}$
yields

$\|\nabla u_{\varepsilon}\|^{2}_{L^{2}(D_{\varepsilon})}\leq C\|\nabla
u_{\varepsilon}\|_{L^{2}(D_{\varepsilon})}\|\nabla\widetilde{u}_{\varepsilon}\|_{L^{2}(D_{\varepsilon})}+C\varepsilon\delta^{-1}\|f\|_{L^{2}(D_{\varepsilon})}\|\nabla
u_{\varepsilon}-\nabla\widetilde{u}_{\varepsilon}\|_{L^{2}(D_{\varepsilon})},$

this together with Young’s inequality and
$\|\nabla\widetilde{u}_{\varepsilon}\|_{L^{2}(D_{\varepsilon})}\leq C\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$ leads to

$\|\nabla u_{\varepsilon}\|_{L^{2}(D_{\varepsilon})}\leq
C\Big{(}\varepsilon\delta^{-1}\|f\|_{L^{2}(D_{\varepsilon})}+\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\Big{)}$ (4.2)

with some bounding constant $C$ that is independent of $\varepsilon,\delta$
and $f$. This estimate will be useful for $\delta\in(0,1]$.

For $\delta>1$, we use Lemma 5.6 below (which is from Lemma 3.1 of [27]) to
get

$\delta\|\nabla u_{\varepsilon}\|_{L^{2}(D_{\varepsilon})}\leq
C(\|f\|_{L^{2}(D_{\varepsilon})}+\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}).$ (4.3)

_Step 2: Control of the total energy_. Let $G\in H^{1}(\Omega)$ be a function
such that $G=g$ on $\partial\Omega$ and $\|G\|_{H^{1}(\Omega)}\leq
C\|g\|_{H^{1/2}(\partial\Omega)}$. Test $u_{\varepsilon}-G\in
H^{1}_{0}(\partial\Omega)$ in (1.7), the energy estimate yields

$\|\nabla u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}^{2}\leq
C\|f\|_{L^{2}(\Omega)}\|\nabla u_{\varepsilon}-\nabla
G\|_{L^{2}(\Omega)}+C\big{(}\delta\|\nabla
u_{\varepsilon}\|_{L^{2}(D_{\varepsilon})}+\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\big{)}\|\nabla
G\|_{L^{2}(\Omega)}.$

First consider the case of $\delta\leq 1$. Using the fact that $\delta$ is not
large, we get

$\|\nabla u_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\varepsilon})}\leq
C\|f\|_{L^{2}(\Omega)}\left(\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega)}+\|\nabla
G\|_{L^{2}(\Omega)}\right)+C\|\nabla u_{\varepsilon}\|_{L^{2}(\Omega)}\|\nabla
G\|_{L^{2}(\Omega)}.$

Using (4.2) we then get

$\|\nabla u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\left(\|f\|_{L^{2}(\Omega)}+\varepsilon\delta^{-1}\|f\|_{L^{2}(D_{\varepsilon})}+\|\nabla
G\|_{L^{2}(\Omega)}\right).$

The above combined with (4.2) again yields the desired estimate (4.1).

For the case of $\delta>1$, we only need to use the bound (4.3) in the energy
estimate and the argument above, the desired inequality for $\delta>1$ then
follows. ∎

###  4.2. Convergence rates in homogenization

We consider the discrepancy function $w_{\varepsilon,\delta}$ defined by

$w_{\varepsilon,\delta}:=u_{\varepsilon,\delta}-\widehat{u}_{\delta}-\varepsilon\chi^{\alpha}_{\delta,i}(x/\varepsilon)S_{\varepsilon}\Big{(}\eta_{\varepsilon}\frac{\partial\widehat{u}_{\delta}^{\alpha}}{\partial
x_{i}}\Big{)},$ (4.4)

where $S_{\varepsilon}$ is the standard smoothing operator defined in section
2.3, and $\eta_{\varepsilon}$ is a smooth cut-off function supported away from
$\partial\Omega$. More precisely, $\eta_{\varepsilon}$ is chosen so that

$\left\\{\begin{aligned} &0\leq\eta_{\varepsilon}(x)\leq 1\quad\mathrm{for}\
x\in\Omega,\\\
&\mathrm{supp}\,\eta_{\varepsilon}\subset\Omega\setminus\mathcal{O}_{3\varepsilon},\\\
&\eta_{\varepsilon}=1\quad\mathrm{in}\
\Omega\setminus\mathcal{O}_{4\varepsilon},\\\
&\|\nabla\eta_{\varepsilon}\|_{\infty}\leq
C\varepsilon^{-1}.\end{aligned}\right.$ (4.5)

The following lemma is the start point of our proof.

######  Lemma 4.2.

Assume $A$ satisfies (1.2)-(1.3), then the discrepancy function
$w_{\varepsilon,\delta}$ defined by (4.4) satisfies

$\big{\\{}\mathcal{L}_{\varepsilon,\delta}(w_{\varepsilon,\delta})\big{\\}}^{\alpha}=-\frac{\partial}{\partial
x_{i}}\big{(}\Lambda_{\delta}(x/\varepsilon)P^{\alpha}_{\varepsilon,\delta,i}\big{)}-\frac{\partial}{\partial
x_{i}}Q_{\varepsilon,\delta,i}^{\alpha},$ (4.6)

where $P_{\varepsilon,\delta}=(P_{\varepsilon,\delta,i}^{\alpha})$ is defined
by

$P_{\varepsilon,\delta,i}^{\alpha}=a_{ij}^{\alpha\beta}\Big{(}\frac{x}{\varepsilon}\Big{)}\Big{[}S_{\varepsilon}\Big{(}\eta_{\varepsilon}\frac{\partial\widehat{u}_{\delta}^{\beta}}{\partial
x_{j}}\Big{)}-\frac{\partial\widehat{u}_{\delta}^{\beta}}{\partial
x_{j}}-\varepsilon\chi_{\delta,k}^{\gamma\beta}\Big{(}\frac{x}{\varepsilon}\Big{)}\frac{\partial}{\partial
x_{j}}S_{\varepsilon}\Big{(}\eta_{\varepsilon}\frac{\partial\widehat{u}_{\delta}^{\gamma}}{\partial
x_{k}}\Big{)}\Big{]}$ (4.7)

and $Q_{\varepsilon,\delta}=(Q_{\varepsilon,\delta,i}^{\alpha})$ is defined by

$Q_{\varepsilon,\delta,i}^{\alpha}=\varepsilon\Psi_{\delta,kij}^{\alpha\beta}\Big{(}\frac{x}{\varepsilon}\Big{)}\frac{\partial}{\partial
x_{k}}S_{\varepsilon}\Big{(}\eta_{\varepsilon}\frac{\partial\widehat{u}_{\delta}^{\beta}}{\partial
x_{j}}\Big{)}-\widehat{a}_{\delta,ij}^{\alpha\beta}\Big{[}S_{\varepsilon}\Big{(}\eta_{\varepsilon}\frac{\partial\widehat{u}_{\delta}^{\beta}}{\partial
x_{j}}\Big{)}-\frac{\partial\widehat{u}^{\beta}_{\delta}}{\partial
x_{j}}\Big{]}.$ (4.8)

Further assume $(\mathbf{H2})$, then they satisfy the following estimates

$\|\nabla w_{\varepsilon,\delta}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\big{(}\|P_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\|Q_{\varepsilon,\delta}\|_{L^{2}(\Omega)}\big{)},\qquad\mathrm{for}\
0<\delta<1,$ (4.9)

and

$\|\nabla w_{\varepsilon,\delta}\|_{L^{2}(D_{\varepsilon})}\leq
C\big{(}\|P_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\delta^{-\frac{1}{2}}\|Q_{\varepsilon,\delta}\|_{L^{2}(\Omega)}\big{)},\qquad\mathrm{for}\
1<\delta<\infty.$ (4.10)

###### Proof.

_Step 1: Verification of ( 4.6) in $H^{-1}(\Omega;\mathbb{R}^{m})$_. Let
$\psi\in H_{0}^{1}(\Omega;\mathbb{R}^{m})$, using

$\int_{\Omega}\Lambda_{\delta}\Big{(}\frac{x}{\varepsilon}\Big{)}a_{ij}^{\alpha\beta}\Big{(}\frac{x}{\varepsilon}\Big{)}\frac{\partial
u^{\beta}_{\varepsilon,\delta}}{\partial
x_{j}}\frac{\partial\psi^{\alpha}}{\partial
x_{i}}\,dx=\int_{\Omega}\widehat{a}_{\delta,ij}^{\alpha\beta}\frac{\partial\widehat{u}^{\beta}_{\delta}}{\partial
x_{j}}\frac{\partial\psi^{\alpha}}{\partial x_{i}}\,dx,$

and the definition of $w_{\varepsilon,\delta}$, by direct computation we
obtain

$\displaystyle\int_{\Omega}$
$\displaystyle\Lambda_{\delta}\Big{(}\frac{x}{\varepsilon}\Big{)}a_{ij}^{\alpha\beta}\Big{(}\frac{x}{\varepsilon}\Big{)}\frac{\partial
w^{\beta}_{\varepsilon,\delta}}{\partial
x_{j}}\frac{\partial\psi^{\alpha}}{\partial x_{i}}\,dx$ (4.11)
$\displaystyle=\int_{\Omega}\Big{(}\widehat{a}_{\delta,ij}^{\alpha\beta}-\Lambda_{\delta}\Big{(}\frac{x}{\varepsilon}\Big{)}a_{ij}^{\alpha\beta}\Big{(}\frac{x}{\varepsilon}\Big{)}\Big{)}\Big{[}\frac{\partial\widehat{u}^{\beta}_{\delta}}{\partial
x_{j}}-S_{\varepsilon}\Big{(}\eta_{\varepsilon}\frac{\partial\widehat{u}_{\delta}^{\beta}}{\partial
x_{j}}\Big{)}\Big{]}\frac{\partial\psi^{\alpha}}{\partial x_{i}}\,dx$
$\displaystyle\underbrace{-\int_{\Omega}\Big{[}\Lambda_{\delta}\Big{(}\frac{x}{\varepsilon}\Big{)}a_{ik}^{\alpha\gamma}\Big{(}\frac{x}{\varepsilon}\Big{)}+\Lambda_{\delta}\Big{(}\frac{x}{\varepsilon}\Big{)}a_{ij}^{\alpha\beta}\Big{(}\frac{x}{\varepsilon}\Big{)}\frac{\partial\chi_{\delta,k}^{\gamma\beta}}{\partial
x_{j}}\Big{(}\frac{x}{\varepsilon}\Big{)}-\widehat{a}_{\delta,ik}^{\alpha\gamma}\Big{]}S_{\varepsilon}\Big{(}\eta_{\varepsilon}\frac{\partial\widehat{u}_{\delta}^{\gamma}}{\partial
x_{k}}\Big{)}\frac{\partial\psi^{\alpha}}{\partial x_{i}}\,dx}_{I}$
$\displaystyle-\varepsilon\int_{\Omega}\Lambda_{\delta}\Big{(}\frac{x}{\varepsilon}\Big{)}a_{ij}^{\alpha\beta}\Big{(}\frac{x}{\varepsilon}\Big{)}\chi_{\delta,k}^{\gamma\beta}\Big{(}\frac{x}{\varepsilon}\Big{)}\frac{\partial}{\partial
x_{j}}S_{\varepsilon}\Big{(}\eta_{\varepsilon}\frac{\partial\widehat{u}_{\delta}^{\gamma}}{\partial
x_{k}}\Big{)}\frac{\partial\psi^{\alpha}}{\partial x_{i}}\,dx.$

For the terms grouped as $I$, we recall the definition of flux corrector $F$
in (3.19) and use (3.18) to compute

$\displaystyle I$
$\displaystyle=-\int_{\Omega}F^{\alpha\gamma}_{\delta,ik}\Big{(}\frac{x}{\varepsilon}\Big{)}S_{\varepsilon}\Big{(}\eta_{\varepsilon}\frac{\partial\widehat{u}_{\delta}^{\gamma}}{\partial
x_{k}}\Big{)}\frac{\partial\psi^{\alpha}}{\partial x_{i}}\,dx$ (4.12)
$\displaystyle=-\varepsilon\int_{\Omega}\frac{\partial}{\partial
x_{k}}\Big{(}\Psi_{\delta,kij}^{\alpha\beta}\Big{(}\frac{x}{\varepsilon}\Big{)}\Big{)}S_{\varepsilon}\Big{(}\eta_{\varepsilon}\frac{\partial\widehat{u}_{\delta}^{\beta}}{\partial
x_{j}}\Big{)}\frac{\partial\psi^{\alpha}}{\partial x_{i}}\,dx$
$\displaystyle=\varepsilon\int_{\Omega}\Psi_{\delta,kij}^{\alpha\beta}\Big{(}\frac{x}{\varepsilon}\Big{)}\frac{\partial}{\partial
x_{k}}S_{\varepsilon}\Big{(}\eta_{\varepsilon}\frac{\partial\widehat{u}_{\delta}^{\beta}}{\partial
x_{j}}\Big{)}\frac{\partial\psi^{\alpha}}{\partial x_{i}}\,dx.$

Then (4.6) immediately follows from the computations above and the definitions
of $P_{\varepsilon,\delta},Q_{\varepsilon,\delta}$ in (4.7) and (4.8).

_Step 2: Energy estimates for $w_{\varepsilon,\delta}$_. For any $\psi\in
H_{0}^{1}(\Omega)$, we have

$\int_{\Omega}\Lambda_{\delta}\Big{(}\frac{x}{\varepsilon}\Big{)}a_{ij}^{\alpha\beta}\Big{(}\frac{x}{\varepsilon}\Big{)}\frac{\partial
w^{\beta}_{\varepsilon,\delta}}{\partial
x_{j}}\frac{\partial\psi^{\alpha}}{\partial
x_{i}}\,dx=\int_{\Omega}\Lambda_{\delta}\Big{(}\frac{x}{\varepsilon}\Big{)}P^{\alpha}_{\varepsilon,\delta,i}\frac{\partial\psi^{\alpha}}{\partial
x_{i}}\,dx+\int_{\Omega}Q^{\alpha}_{\varepsilon,\delta,i}\frac{\partial\psi^{\alpha}}{\partial
x_{i}}\,dx.$ (4.13)

_Case 1: $0<\delta<1$_. Let $\psi=w_{\varepsilon,\delta}$ in the formulation
above, then by the ellipticity of $(a^{\alpha\beta}_{ij})$ we get

$\delta\int_{\Omega}|\nabla w_{\varepsilon,\delta}|^{2}\,dx\leq
C\big{(}\|P_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\|Q_{\varepsilon,\delta}\|_{L^{2}(\Omega)}\big{)}\|\nabla
w_{\varepsilon,\delta}\|_{L^{2}(\Omega)},$ (4.14)

and hence, for some universal constant $C<\infty$ independent of
$\varepsilon,\delta$,

$\|\nabla w_{\varepsilon,\delta}\|_{L^{2}(\Omega)}\leq
C\delta^{-1}\big{(}\|P_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\|Q_{\varepsilon,\delta}\|_{L^{2}(\Omega)}\big{)}.$

Let $\mathcal{P}_{\varepsilon}$ be the extension operator given in Lemma 2.1,
so that $\psi=\mathcal{P}_{\varepsilon}w_{\varepsilon,\delta}$ is an extension
of $w_{\varepsilon,\delta}\rvert_{\Omega_{\varepsilon}}$ to inside the
obstacles. Use this extended function as $\psi$ in (4.13) to get

$\displaystyle\mu\int_{\Omega_{\varepsilon}}$ $\displaystyle|\nabla
w_{\varepsilon,\delta}|^{2}\,dx\leq\int_{\Omega_{\varepsilon}}A(x/\varepsilon)\nabla
w_{\varepsilon,\delta}\cdot\nabla\mathcal{P}_{\varepsilon}(w_{\varepsilon,\delta})$
$\displaystyle=\int_{\Omega}\Lambda_{\delta}\left(\frac{x}{\varepsilon}\right)A\left(\frac{x}{\varepsilon}\right)\nabla
w_{\varepsilon,\delta}\cdot\nabla\mathcal{P}_{\varepsilon}w_{\varepsilon,\delta}\,dx-\delta\int_{D_{\varepsilon}}A\left(\frac{x}{\varepsilon}\right)\nabla
w_{\varepsilon,\delta}\cdot\nabla\mathcal{P}_{\varepsilon}w_{\varepsilon,\delta}\,dx$
$\displaystyle\leq\left|\int_{\Omega}\left(\Lambda_{\delta}(x/\varepsilon)P^{\alpha}_{\varepsilon,\delta}+Q^{\alpha}_{\varepsilon,\delta}\right)\cdot\nabla\mathcal{P}_{\varepsilon}(w_{\varepsilon,\delta}^{\alpha})\right|+\delta\left|\int_{D_{\varepsilon}}A\left(\frac{x}{\varepsilon}\right)\nabla
w_{\varepsilon,\delta}\cdot\nabla\mathcal{P}_{\varepsilon}w_{\varepsilon,\delta}\,dx\right|$
$\displaystyle\leq
C\big{(}\|P_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\|Q_{\varepsilon,\delta}\|_{L^{2}(\Omega)}\big{)}\|\nabla
w_{\varepsilon,\delta}\|_{L^{2}(\Omega_{\varepsilon})}+C\delta\|\nabla
w_{\varepsilon,\delta}\|_{L^{2}(\Omega)}\|\nabla
w_{\varepsilon,\delta}\|_{L^{2}(\Omega_{\varepsilon})}$ $\displaystyle\leq
C\big{(}\|P_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\|Q_{\varepsilon,\delta}\|_{L^{2}(\Omega)}\big{)}\|\nabla
w_{\varepsilon,\delta}\|_{L^{2}(\Omega_{\varepsilon})}.$

Note the second inequality follows from (4.13) and the boundedness of
$\mathcal{P}_{\varepsilon}$, and also $\delta\leq 1$. The last inequality
follows from (4.14). Estimate (4.9) follows, for $\delta\in(0,1]$.

_Case 2: $1<\delta<\infty$_. Let $\psi=w_{\varepsilon,\delta}$ in (4.13), by
Cauchy inequality we get

$\displaystyle\big{(}\|\nabla$ $\displaystyle
w_{\varepsilon,\delta}\|_{L^{2}(\Omega_{\varepsilon})}+\delta^{1/2}\|\nabla
w_{\varepsilon,\delta}\|_{L^{2}(D_{\varepsilon})}\big{)}^{2}$
$\displaystyle\leq
C\|P_{\varepsilon,\delta}\|_{L^{2}(\Omega_{\varepsilon})}\|\nabla
w_{\varepsilon,\delta}\|_{L^{2}(\Omega_{\varepsilon})}+C\delta\|P_{\varepsilon,\delta}\|_{L^{2}(D_{\varepsilon})}\|\nabla
w_{\varepsilon,\delta}\|_{L^{2}(D_{\varepsilon})}$ $\displaystyle\ \ \
+C\|Q_{\varepsilon,\delta}\|_{L^{2}(\Omega)}\|\nabla
w_{\varepsilon,\delta}\|_{L^{2}(\Omega)}$ $\displaystyle\leq
C\big{(}\delta^{1/2}\|P_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\|Q_{\varepsilon,\delta}\|_{L^{2}(\Omega)}\big{)}\big{(}\|\nabla
w_{\varepsilon,\delta}\|_{L^{2}(\Omega_{\varepsilon})}+\delta^{1/2}\|\nabla
w_{\varepsilon,\delta}\|_{L^{2}(D_{\varepsilon})}\big{)},$

so we get

$\|\nabla
w_{\varepsilon,\delta}\|_{L^{2}(\Omega_{\varepsilon})}+\delta^{1/2}\|\nabla
w_{\varepsilon,\delta}\|_{L^{2}(D_{\varepsilon})}\leq
C\big{(}\delta^{1/2}\|P_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\|Q_{\varepsilon,\delta}\|_{L^{2}(\Omega)}\big{)},$

which implies (4.10), for $\delta\in(1,\infty)$. ∎

###  4.3. A decomposition of the equations for the discrepancy function.

We rewrite the problem (4.6) for the discrepancy function
$w_{\varepsilon,\delta}$ as a transmission problem:

$\left\\{\begin{aligned}
&\big{\\{}\Lambda_{\delta}(x/\varepsilon)\mathcal{L}_{\varepsilon,1}(w_{\varepsilon,\delta})\big{\\}}^{\alpha}=-\Lambda_{\delta}(x/\varepsilon)\frac{\partial
P^{\alpha}_{\varepsilon,\delta,i}}{\partial x_{i}}-\frac{\partial
Q_{\varepsilon,\delta,i}^{\alpha}}{\partial x_{i}}&\mathrm{in}\
\Omega\setminus\partial D_{\varepsilon},\\\ &\Big{\\{}\delta\frac{\partial
w_{\varepsilon,\delta}}{\partial\nu}\Big{|}_{-}-\frac{\partial
w_{\varepsilon,\delta}}{\partial\nu}\Big{|}_{+}\Big{\\}}^{\alpha}=\delta
P^{\alpha}_{\varepsilon,\delta,i}n_{i}\big{|}_{-}-P^{\alpha}_{\varepsilon,\delta,i}n_{i}\big{|}_{+}&\mathrm{on}\
\partial D_{\varepsilon},\\\ &w_{\varepsilon,\delta}=0&\mathrm{on}\
\partial\Omega.\end{aligned}\right.$ (4.15)

Let $w_{\varepsilon,\delta}^{1}\in H^{1}_{0}(\Omega;\mathbb{R}^{m})$ be the
solution of

$\left\\{\begin{aligned}
&\big{\\{}\mathcal{L}_{\varepsilon,1}(w^{1}_{\varepsilon,\delta})\big{\\}}^{\alpha}=-\frac{\partial
P^{\alpha}_{\varepsilon,\delta,i}}{\partial x_{i}}-\frac{\partial
Q^{\alpha}_{\varepsilon,\delta,i}}{\partial x_{i}}&\mathrm{in}\ \Omega,\\\
&w^{1}_{\varepsilon,\delta}=0&\mathrm{on}\
\partial\Omega.\end{aligned}\right.$ (4.16)

Let $w_{\varepsilon,\delta}^{2}\in H^{1}(D_{\varepsilon};\mathbb{R}^{m})$ be
the solution of

$\left\\{\begin{aligned}
&\big{\\{}\mathcal{L}_{\varepsilon,1}(w^{2}_{\varepsilon,\delta})\big{\\}}^{\alpha}=-\frac{\partial
P^{\alpha}_{\varepsilon,\delta,i}}{\partial
x_{i}}-\frac{1}{\delta}\frac{\partial
Q^{\alpha}_{\varepsilon,\delta,i}}{\partial x_{i}}&\mathrm{in}\
D_{\varepsilon},\\\
&w^{2}_{\varepsilon,\delta}=w^{1}_{\varepsilon,\delta}&\mathrm{on}\ \partial
D_{\varepsilon}.\end{aligned}\right.$ (4.17)

Note that $w_{\varepsilon,\delta}^{2}$ is picewisely defined on each component
of $D_{\varepsilon}$.

######  Lemma 4.3.

Let $w^{1}_{\varepsilon,\delta}$ and $w^{2}_{\varepsilon,\delta}$ be defined
by (4.16) and (4.17), respectively. Let
$u=\mathcal{L}_{D_{\varepsilon}}^{-1}(f)$ be the unique solution (extended by
zero outside $D_{\varepsilon}$) of

$\left\\{\begin{aligned} &\mathcal{L}_{\varepsilon,1}(u)=f&\mathrm{in}\
D_{\varepsilon},\\\ &u=0&\mathrm{on}\ \Omega\setminus
D_{\varepsilon},\end{aligned}\right.$ (4.18)

Let $R_{\varepsilon,\delta}=(R_{\varepsilon,\delta}^{\alpha})$ be defined by

$R_{\varepsilon,\delta}^{\alpha}=\Big{[}a_{ij}^{\alpha\beta}\Big{(}\frac{x}{\varepsilon}\Big{)}+a_{ik}^{\alpha\gamma}\Big{(}\frac{x}{\varepsilon}\Big{)}\frac{\partial}{\partial
x_{k}}\chi_{\delta,j}^{\gamma\beta}\Big{(}\frac{x}{\varepsilon}\Big{)}\Big{]}\frac{\partial}{\partial
x_{i}}S_{\varepsilon}\Big{(}\eta_{\varepsilon}\frac{\partial\widehat{u}_{\delta}^{\beta}}{\partial
x_{j}}\Big{)}.$ (4.19)

Then there exits a universal constant $C$ that is independent of $\varepsilon$
and $\delta$ so that

$\displaystyle\|\nabla w^{1}_{\varepsilon,\delta}\|_{L^{2}(\Omega)}\leq
C\big{(}\|P_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\|Q_{\varepsilon,\delta}\|_{L^{2}(\Omega)}\big{)},$
(4.20)
$\displaystyle\big{\|}\nabla\big{(}w^{2}_{\varepsilon,\delta}-\delta^{-1}\mathcal{L}_{D_{\varepsilon}}^{-1}(f)\big{)}\big{\|}_{L^{2}(D_{\varepsilon})}\leq
C\big{(}\|P_{\varepsilon,\delta}\big{\|}_{L^{2}(\Omega)}+\|Q_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\varepsilon\|R_{\varepsilon,\delta}\|_{L^{2}(\Omega)}\big{)}.$
(4.21)

###### Proof.

Note that the differential operator $\mathcal{L}_{\varepsilon,1}$ has uniform
elliptic coefficients $(a^{\alpha\beta}_{ij}(\tfrac{\cdot}{\varepsilon}))$; as
a result (4.20) is due to the basic energy estimate for the problem (4.16) of
$w^{1}_{\varepsilon,\delta}$.

To obtain estimate for $w^{2}_{\varepsilon,\delta}$, using the definitions
(4.8) and (3.19), the relation (3.18) and the homogenized problem (1.8), we
first check by direct computation that

$-\frac{\partial Q_{\varepsilon,\delta,i}}{\partial x_{i}}=\delta
R^{\alpha}_{\varepsilon,\delta}+f\qquad\mathrm{in}\ D_{\varepsilon},$ (4.22)

where $R_{\varepsilon,\delta}$ is defined as in (4.19). Comparing the
equations (4.16)(4.23) and (4.18) and using the above formula, we get

$\left\\{\begin{aligned}
&\mathcal{L}_{\varepsilon,1}\big{(}w^{2}_{\varepsilon,\delta}-\delta^{-1}\mathcal{L}_{D_{\varepsilon}}^{-1}(F)-w^{1}_{\varepsilon,\delta}\big{)}=-\frac{\partial
Q_{\varepsilon,\delta,i}}{\partial x_{i}}+R_{\varepsilon,\delta}&\mathrm{in}\
D_{\varepsilon},\\\
&w^{2}_{\varepsilon,\delta}-\delta^{-1}\mathcal{L}_{D_{\varepsilon}}^{-1}(f)-w^{1}_{\varepsilon,\delta}=0&\mathrm{on}\
\partial D_{\varepsilon}.\end{aligned}\right.$ (4.23)

Multiply on both sides by
$v=w^{2}_{\varepsilon,\delta}-\delta^{-1}\mathcal{L}_{D_{\varepsilon}}^{-1}(F)-w^{1}_{\varepsilon,\delta}$,
which belongs to $H^{1}_{0}(D_{\varepsilon})$, and integrate by parts. We
obtain, using again the uniform ellipticity of the coefficients in
$\mathcal{L}_{\varepsilon,1}$,

$\displaystyle\mu\|\nabla
v\|_{L^{2}(D_{\varepsilon})}^{2}\leq\|Q_{\varepsilon,\delta}\|_{L^{2}(D_{\varepsilon})}\|\nabla
v\|_{L^{2}(D_{\varepsilon})}+\|R_{\varepsilon,\delta}\|_{L^{2}(D_{\varepsilon})}\|v\|_{L^{2}(D_{\varepsilon})}.$
(4.24)

Note that $D_{\varepsilon}$ consists of a collection of obstacles with
diameter of order $\varepsilon$. Using Poicaré inequality on each of the
obstacle we have

$\|v\|_{L^{2}(D_{\varepsilon})}\leq C\varepsilon\|\nabla
v\|_{L^{2}(D_{\varepsilon})}\qquad\mathrm{for\ any\ }u\in
H_{0}^{1}(D_{\varepsilon}).$

We hence get

$\|\nabla\left(w^{2}_{\varepsilon,\delta}-\delta^{-1}\mathcal{L}_{D_{\varepsilon}}^{-1}(f)-w^{1}_{\varepsilon,\delta}\right)\|_{L^{2}(D_{\varepsilon})}\leq
C\left(\|Q_{\varepsilon,\delta}\|_{L^{2}(D_{\varepsilon})}+\varepsilon\|R_{\varepsilon,\delta}\|_{L^{2}(D_{\varepsilon})}\right).$

The desired estimate (4.24) follows then from (4.20). ∎

######  Lemma 4.4.

Assume (1.2)-(1.3) and $(\mathbf{H2})$,

$\|P_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\|Q_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\varepsilon\|R_{\varepsilon,\delta}\|_{L^{2}(\Omega)}\leq
C\varepsilon^{1/2}\big{(}\|f\|_{L^{2}(\Omega)}+\|\nabla_{\mathrm{tan}}g\|_{L^{2}(\partial\Omega)}\big{)}.$
(4.25)

###### Proof.

By the definition of
$P_{\varepsilon,\delta},Q_{\varepsilon,\delta},R_{\varepsilon,\delta}$ and
Lemma 2.3 below, we have

$\displaystyle\|P_{\varepsilon,\delta}\|_{L^{2}(\Omega)}$
$\displaystyle+\|Q_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\varepsilon\|R_{\varepsilon,\delta}\|_{L^{2}(\Omega)}$
$\displaystyle\leq
C\|A\|_{L^{\infty}(Y)}\big{(}1+\|\widehat{A}_{\delta}|_{L^{\infty}(\Omega)}\big{)}\Big{(}\|S_{\varepsilon}(\eta_{\varepsilon}\nabla\widehat{u}_{\delta})-\nabla\widehat{u}_{\delta}\|_{L^{2}(\Omega)}$
$\displaystyle\ \ \
+\varepsilon\|\chi_{\delta}\|_{H^{1}(Y)}\|\nabla(\eta_{\varepsilon}\nabla\widehat{u}_{\delta})\|_{L^{2}(\Omega)}+\varepsilon\|\Psi_{\delta}\|_{L^{2}(\Omega)}\|\nabla(\eta_{\varepsilon}\nabla\widehat{u}_{\delta})\|_{L^{2}(\Omega)}\Big{)}.$

Since

$\|S_{\varepsilon}(\eta_{\varepsilon}\nabla\widehat{u}_{\delta})-\nabla\widehat{u}_{\delta}\|_{L^{2}(\Omega)}\leq
C\varepsilon\|\nabla(\eta_{\varepsilon}\nabla\widehat{u}_{\delta})\|_{L^{2}(\Omega)}+C\|\nabla\widehat{u}_{\delta}\|_{L^{2}(O_{4\varepsilon})},$

and

$\|\nabla(\eta_{\varepsilon}\nabla\widehat{u}_{\delta})\|_{L^{2}(\Omega)}\leq\varepsilon^{-1}\|\nabla\widehat{u}_{\delta}\|_{L^{2}(O_{4\varepsilon})}+\|\nabla^{2}\widehat{u}_{\delta}\|_{L^{2}(\Omega\setminus
O_{3\varepsilon})},$

(4.25) then follows from Lemma 2.4, the uniform ellipticity of
$\widehat{A}_{\delta}$, the uniform $L^{2}$ bound of $\Psi_{\delta}$ and the
uniform $H^{1}$ bound of $\chi_{\delta}$. ∎

We extend $w_{\varepsilon,\delta}^{2}$ to the outside $D_{\varepsilon}$ by
$w_{\varepsilon,\delta}^{1}$. In other words, define

$w^{\mathrm{R}}_{\varepsilon,\delta}:=w^{1}_{\varepsilon,\delta}\mathbbm{1}_{\Omega_{\varepsilon}}+w_{\varepsilon,\delta}^{2}\mathbbm{1}_{D_{\varepsilon}}.$
(4.26)

Then we check that $w^{\mathrm{R}}_{\varepsilon,\delta}\in
H^{1}_{0}(\Omega;\mathbb{R}^{m})$ and below it is referred to as the “regular”
part of $w_{\varepsilon,\delta}$. In view of the estimates (4.20) and (4.24),
we have

$\big{\|}\nabla\big{(}w^{\mathrm{R}}_{\varepsilon,\delta}-\delta^{-1}\mathcal{L}_{D_{\varepsilon}}^{-1}(F)\big{)}\big{\|}_{L^{2}(\Omega)}\leq
C\big{(}\|P_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\|Q_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\varepsilon\|R_{\varepsilon,\delta}\|_{L^{2}(\Omega)}\big{)}.$
(4.27)

Similarly, denote by the differece between
$w^{\mathrm{S}}_{\varepsilon,\delta}$ and $w_{\varepsilon,\delta}$ by

$w^{\mathrm{S}}_{\varepsilon,\delta}:=w_{\varepsilon,\delta}-w^{\mathrm{R}}_{\varepsilon,\delta}.$
(4.28)

Then $w^{\mathrm{S}}_{\varepsilon,\delta}\in H^{1}_{0}(\Omega;\mathbb{R}^{m})$
as well and below it is referred to as the “singular part”. Moreover, by
definition we have

$\mathcal{L}_{\varepsilon,1}(w^{\mathrm{S}}_{\varepsilon,\delta})=0\qquad\mathrm{in}\
\Omega\setminus\partial D_{\varepsilon}.$ (4.29)

The next lemma plays an important role in our proof of Theorem 1.1. It says
that the distributions of energy in $w^{\mathrm{S}}_{\varepsilon,\delta}$ (in
terms of the $L^{2}$ norm of its gradient) is balanced inside and outside of
the obstacles $D_{\varepsilon}$.

######  Lemma 4.5.

Assume (1.2)-(1.3) and $(\mathbf{H2})$, then

$C\int_{D_{\varepsilon}}|\nabla
w^{\mathrm{S}}_{\varepsilon,\delta}|^{2}\,dx\leq\int_{\Omega_{\varepsilon}}|\nabla
w^{\mathrm{S}}_{\varepsilon,\delta}|^{2}\,dx\leq
C^{-1}\int_{D_{\varepsilon}}|\nabla
w^{\mathrm{S}}_{\varepsilon,\delta}|^{2}\,dx.$ (4.30)

###### Proof.

For simplicity, we simply denote $w^{\mathrm{S}}_{\varepsilon,\delta}$ by $w$
in this proof. We divide the proof to several steps.

Step 1: We verify the first inequality by a standard extension argument.
Consider the extension operator
$\mathcal{P}_{\varepsilon}:H^{1}(\Omega_{\varepsilon})\to H^{1}(\Omega)$
defined in Lemma 2.1. Let
$\widetilde{w}=\mathcal{P}_{\varepsilon}(w\rvert_{\Omega_{\varepsilon}})$.
Then we have $\widetilde{w}-w$ vanishes at $\partial D_{\varepsilon}$.
Multiply on both sides of (4.29) and integrate in $D_{\varepsilon}$, we find

$\|\nabla w\|_{L^{2}(D_{\varepsilon})}\leq
C\|\widetilde{w}\|_{L^{2}(D_{\varepsilon})}\leq
C\|w\|_{L^{2}(\Omega_{\varepsilon})}.$

where $C$ only depends on the ellipticity bounds and the model domain
$\omega$.

Step 2. We verify that for each inclusion
$\omega^{\mathbf{n}}_{\varepsilon}\subset\Omega$,

$\int_{\partial\omega_{\varepsilon}^{\mathbf{n}}}\frac{\partial
w^{\mathrm{S}}_{\varepsilon,\delta}}{\partial\nu}\Big{|}_{+}\,d\sigma=0.$
(4.31)

The above is understood as the pairing of
$\partial_{\nu}w^{\mathrm{S}}_{\varepsilon,\delta}\rvert_{+}$ with constant
functions. By definition, along the boundary of each inclusion, we have

$\partial_{\nu}w^{\mathrm{S}}_{\varepsilon,\delta}\rvert_{+}=\partial_{\nu}w_{\varepsilon,\delta}\rvert_{+}-\partial_{\nu}w^{1}_{\varepsilon,\delta}\rvert_{+},$

where we used the fact that
$w^{\mathrm{R}}_{\varepsilon,\delta}=w^{1}_{\varepsilon,\delta}$ in
$\Omega_{\varepsilon}$, i.e., outside the inclusions. By an inspection of the
equations satisfied by $w_{\varepsilon,\delta}$ and
$w^{1}_{\varepsilon,\delta}$, we have

$\displaystyle\partial_{\nu}w_{\varepsilon,\delta}\rvert_{+}-N^{i}P_{\varepsilon,\delta,i}\rvert_{+}-N^{i}Q_{\varepsilon,\delta,i}\rvert_{+}=\delta\partial_{\nu}w_{\varepsilon,\delta}\rvert_{-}-\delta
N^{i}P_{\varepsilon,\delta,i}\rvert_{-}-N^{i}Q_{\varepsilon,\delta,i}\rvert_{-},$
$\displaystyle\partial_{\nu}w^{1}_{\varepsilon,\delta}\rvert_{+}-N^{i}P_{\varepsilon,\delta,i}\rvert_{+}-N^{i}Q_{\varepsilon,\delta,i}\rvert_{+}=\partial_{\nu}w^{1}_{\varepsilon,\delta}\rvert_{-}-N^{i}P_{\varepsilon,\delta,i}\rvert_{-}-N^{i}Q_{\varepsilon,\delta,i}\rvert_{-},$

which leads to

$\partial_{\nu}w^{\mathrm{S}}_{\varepsilon,\delta}\rvert_{+}=\delta\partial_{\nu}w_{\varepsilon,\delta}\rvert_{-}-\partial_{\nu}w^{1}_{\varepsilon,\delta}\rvert_{-}-(\delta-1)N^{i}P_{\varepsilon,\delta,i}\rvert_{-}.$

Integrate the equation of $w_{\varepsilon,\delta}$ and the equation of
$w^{1}_{\varepsilon,\delta}$ in $\omega^{\mathrm{n}}_{\varepsilon}$, we get

$\displaystyle\int_{\partial\omega^{\mathrm{n}}_{\varepsilon}}\delta\partial_{\nu}w_{\varepsilon,\delta}\rvert_{-}-\delta
N^{i}P_{\varepsilon,\delta,i}-N^{i}Q_{\varepsilon,\delta,i}\,d\sigma=0,$
$\displaystyle\int_{\partial\omega^{\mathrm{n}}_{\varepsilon}}\partial_{\nu}w^{1}_{\varepsilon,\delta}\rvert_{-}-N^{i}P_{\varepsilon,\delta,i}-N^{i}Q_{\varepsilon,\delta,i}\,d\sigma=0.$

Combining the results above we get (4.31).

Step 3. Proof of the second inequality. Consider the map
$\mathcal{Q}_{\varepsilon}$ defined in Lemma 2.2, and apply it to
$w\rvert_{D_{\varepsilon}}$ to define

$\widetilde{w}=\mathcal{Q}_{\varepsilon}(w\rvert_{D_{\varepsilon}}).$

Then $\widetilde{w}\in H^{1}_{0}(\Omega)$ (in fact, $\tilde{w}$ is supported
in the union of bubbles surrounding compoenents of $D_{\varepsilon}$).
Moreover,

$\widetilde{w}-w=q_{\varepsilon}^{\mathbf{n}},\qquad\text{in each
}\,\omega_{\varepsilon}^{\mathbf{n}}.$

Multiply by $\widetilde{w}-w$ on both sides of (4.29) and integrate by parts.
We obtain

$\int_{\Omega_{\varepsilon}}A^{\varepsilon}(x)\nabla
w\cdot\nabla(w-\widetilde{w})\,dx=\int_{\partial\Omega_{\varepsilon}}\frac{\partial
w}{\partial\nu}\cdot(w-\widetilde{w})=-\sum_{\mathbf{n}}\int_{\partial\omega_{\varepsilon}^{\mathbf{n}}}\frac{\partial
w}{\partial\nu}\Big{\rvert}_{+}\cdot(w-\widetilde{w})=0.$

From this we get by ellipticity that

$\|\nabla w\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\|\nabla\mathcal{Q}_{\varepsilon}(w)\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\|\nabla w\|_{L^{2}(D_{\varepsilon})}.$

The last inequality follows from Lemma 2.2. This completes the proof. ∎

We are in a position to complete the proof of Theorem 1.1.

###### Proof of Theorem 1.1.

By the decomposition above (see (4.26) ad (4.28)), we have

$w_{\varepsilon,\delta}=w_{\varepsilon,\delta}^{\mathrm{R}}+w_{\varepsilon,\delta}^{\mathrm{S}}.$

In view of (4.27) we only need to estimate $\|\nabla
w^{\mathrm{S}}_{\varepsilon,\delta}\|_{L^{2}(\Omega)}$.

_Case 1: $0<\delta<1$_. By Lemma 4.5, we have

$\displaystyle\|\nabla w^{\mathrm{S}}_{\varepsilon,\delta}\|_{L^{2}(\Omega)}$
$\displaystyle\leq C\|\nabla
w^{\mathrm{S}}_{\varepsilon,\delta}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\|\nabla
w^{\mathrm{R}}_{\varepsilon,\delta}\|_{L^{2}(\Omega_{\varepsilon})}+C\|\nabla
w_{\varepsilon,\delta}\|_{L^{2}(\Omega_{\varepsilon})}$ (4.32)
$\displaystyle\leq
C\big{(}\|P_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\|Q_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\varepsilon\|R_{\varepsilon,\delta}\|_{L^{2}(\Omega)}\big{)},$

where the last equality follows from (4.9) and (4.27) (noting that
$\mathcal{L}_{D_{\varepsilon}}^{-1}(f)$ is supported outside
$\Omega_{\varepsilon}$).

_Case 2: $1\leq\delta<\infty$_. By Lemma 4.5, we have

$\displaystyle\|\nabla w^{\mathrm{S}}_{\varepsilon,\delta}\|_{L^{2}(\Omega)}$
$\displaystyle\leq C\|\nabla
w^{\mathrm{S}}_{\varepsilon,\delta}\|_{L^{2}(D_{\varepsilon})}$
$\displaystyle\leq
C\|\nabla\big{(}w^{\mathrm{R}}_{\varepsilon,\delta}-\delta^{-1}\mathcal{L}_{D_{\varepsilon}}^{-1}(f)\big{)}\|_{L^{2}(D_{\varepsilon})}+C\delta^{-1}\|\nabla\mathcal{L}_{D_{\varepsilon}}^{-1}(f)\|_{L^{2}(D_{\varepsilon})}+C\|\nabla
w_{\varepsilon,\delta}\|_{L^{2}(D_{\varepsilon})}$ $\displaystyle\leq
C\big{(}\|P_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\|Q_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\varepsilon\|R_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\varepsilon\|f\|_{L^{2}(\Omega)}\big{)},$

where the last equality follows from (4.27), (4.9), a standard energy estimate
for (4.18), and the fact that $\delta\geq 1$.

Combine the above estimates with (4.27). We conclude that, for all
$\delta\in(0,\infty)$, we have found a universal constant $C<\infty$ such that

$\displaystyle\|\nabla\big{(}w_{\varepsilon,\delta}$
$\displaystyle-\delta^{-1}\mathcal{L}_{D_{\varepsilon}}^{-1}(F)\big{)}\|_{L^{2}(\Omega)}\leq
C\big{(}\|P_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\|Q_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\varepsilon\|R_{\varepsilon,\delta}\|_{L^{2}(\Omega)}+\varepsilon\|F\|_{L^{2}(\Omega)}\big{)}.$

The desired estimate (1.12) then follows from (4.25). ∎

######  Remark 5.

Note that using a boundary cut-off, for Type I domain, the same method of
analysis still yields a similar $H^{1}$ convergence rate result away the
boundary:

$\displaystyle\Big{\|}u_{\varepsilon,\delta}-\widehat{u}_{\delta}-\varepsilon\chi_{\delta}\left(\frac{x}{\varepsilon}\right)S_{\varepsilon}\Big{(}\eta_{\varepsilon}\nabla\widehat{u}_{\delta}\Big{)}$
$\displaystyle-\delta^{-1}\mathcal{L}_{D_{\varepsilon}}^{-1}(f)\Big{\|}_{H^{1}(\Omega\setminus\mathcal{O}_{c})}$
(4.33) $\displaystyle\leq
C\varepsilon^{1/2}\big{(}\|f\|_{L^{2}(\Omega)}+\|\nabla_{\mathrm{tan}}g\|_{L^{2}(\partial\Omega)}\big{)}$

where $c>0$ is a fixed number independent of $\varepsilon$.

##  5\. Caccioppoli-type inequalities

We will prove the uniform Lipschitz estimate for

$\left\\{\begin{aligned}
&\mathcal{L}_{\varepsilon,\delta}(u_{\varepsilon})=f_{\varepsilon,\delta},\qquad&&x\in\Omega,\\\
&u_{\varepsilon,\delta}=g,\qquad&&x\in\partial\Omega.\end{aligned}\right.$
(5.1)

Here, $f_{\varepsilon,\delta}=f$ for $\delta\in(1,\infty)$, and
$f_{\varepsilon,\delta}=\Lambda_{\varepsilon,\delta}f$ if $\delta\in(0,1]$. We
follow the now standard approach by Shen [25]. A key step in the scheme is to
establish (interior and boundary) Caccioppoli inequalities. In the high
contrast setting, this turns out to be a non-trivial task especially for large
$\delta$. The following rescaling property of
$\mathcal{L}_{\varepsilon,\delta}$ will be used throughout the rest of the
paper: if we define $\tilde{u}(\cdot)=u(r\cdot)$ and
$\tilde{f}(\cdot)=r^{2}f(r\cdot)$,

$\mathcal{L}_{\varepsilon,\delta}(u)=f\quad\Leftrightarrow\quad\mathcal{L}_{\varepsilon/r,\delta}(\tilde{u})=\tilde{f}.$

In particular when $r=\varepsilon$, we relate the operator
$\mathcal{L}_{\varepsilon,\delta}$ to $\mathcal{L}_{1,\delta}$. The latter
operator was studied by Shen in [27], where Caccioppoli inequalities and large
scale interior Lipschitz regularity were proved. We modify Shen’s proof to
obtain Caccioppoli inequalities for large $\delta$ allowing the presence of
source term $f$ and boundary data $g$. Those combined with the convergence
rate result of the previous section allows us to obtain boundary Lipschitz
regularity estimate in the high contrast setting.

From here, we always assume assumptions $(\mathbf{H1})$ and $(\mathbf{H2})$
hold. We start from two basic energy estimates with cut-off functions.

######  Lemma 5.1.

If $u\in H^{1}(\Omega;\mathbb{R}^{m})$ is a solution to
$\mathcal{L}_{\varepsilon,\delta}(u)=f$ in $\Omega$. Then for any $\varphi\in
C^{1}_{0}(\Omega)$,

$\int_{\Omega}\Lambda_{\varepsilon,\delta}(x)\varphi^{2}|\nabla u|^{2}\leq
C\int_{\Omega}\Lambda_{\varepsilon,\delta}(x)|u|^{2}|\nabla\varphi|^{2}+C\int_{\Omega}|f|\varphi^{2}|u|.$
(5.2)

###### Proof.

Let $v=u\varphi^{2}\in H^{1}_{0}(\Omega)$. Then we have

$\int_{\Omega}\Lambda_{\varepsilon,\delta}(x)A(\tfrac{x}{\varepsilon})\nabla
u\nabla v=\int_{\Omega}f\cdot v,$

the conclusion follows by integration by parts and the ellipticity of $A$, we
refer [25, Theorem 2.1.3] to the readers for details. ∎

For a boundary version of Lemma 5.1, we use the following standard notation:
let $x_{0}\in\partial\Omega$ and $r>0$, let
$\mathbf{D}_{r}=\mathbf{D}_{r}(x_{0},\psi)$ be the boundary cylinder

$\mathbf{D}_{r}=\Phi\\{(x^{\prime},z)\,:\,|x^{\prime}-x_{0}|<r,0<z-\psi(x^{\prime})<M_{0}r\\},$

where $(x^{\prime},z)\in\mathbb{R}^{d-1}\times\mathbb{R}$ denotes local
coordinate close to $x_{0}$, $\Phi=\Phi_{x_{0}}$ is an affine transform that
maps $(0,0)$ in the new coordinate system to $x_{0}$, and allign the subspace
$\\{(x^{\prime},0)\,:\,x^{\prime}\in\mathbb{R}^{d-1}\\}$ to the tangent space
of $T_{x_{0}}\partial\Omega$, and $\psi:\mathbb{R}^{d-1}\to\mathbb{R}$ is a
$C^{1,\alpha}$ (or Lipschitz) function in the new coordinate whose graph
(after mapped by $\Phi$) corresponds to $\partial\Omega$, and $M_{0}$ is a
regularity character of $\psi$. Let

$\Delta_{r}=\partial\mathbf{D}_{r}\cap\partial\Omega.$

Note that $\Delta_{r}$ is part of $\partial\Omega$ and is a proper subset of
$\partial\mathbf{D}_{r}$. It is clear that the boundary cylinders
$\\{\mathbf{D}_{r}(x_{0})\\}$ are comparable with
$\\{B_{r}(x_{0})\cap\Omega\\}$, in the sense that one class can cover the
other.

######  Lemma 5.2.

If $u\in H^{1}(\Omega;\mathbb{R}^{m})$ be a weak solution of
$\mathcal{L}_{\varepsilon,\delta}(u)=f$ in $\Omega$ with $u=0$ on
$\partial\Omega$, then for any $\varphi\in C^{1}(\mathbf{D}_{R})$ with support
of $\varphi$ away from $\partial\mathbf{D}_{R}\cap\Omega$, we have

$\int_{\mathbf{D}_{R}}\Lambda_{\varepsilon,\delta}(x)\varphi^{2}|\nabla
u|^{2}\leq
C\int_{\mathbf{D}_{R}}\Lambda_{\varepsilon,\delta}(x)|u|^{2}|\nabla\varphi|^{2}+C\int_{\mathbf{D}_{R}}|f|\varphi^{2}|u|.$
(5.3)

The proof is exactly as in the interior case, because, thanks to the zero
Dirichlet data $u=0$ on $\Delta_{R}$, the function $\varphi^{2}u$ belongs to
$H^{1}_{0}(\mathbf{D}_{R})$. We hence omit the details.

###  5.1. Caccioppoli inequalities for small $\delta$

For relatively soft inclusions, the Caccioppoli inequality can be obtained by
the standard method provided that the source term $f$ is of order $\delta$ in
the inclusions. Recall that for $\delta\in(0,1]$, $f_{\varepsilon,\delta}$ is
defined by (1.13).

Recall that $\widetilde{\omega}$ denotes an enlarging dilation of $\omega$ so
that $\overline{\omega}\subset\widetilde{\omega}\subset Q_{7/8}(0)$, and
$\widetilde{\omega}^{\mathbf{n}}_{\varepsilon}=\varepsilon(\mathbf{n}+\widetilde{\omega})$
(simplified by $\widetilde{\omega}_{\varepsilon}$) is the dilation of
$\omega^{\mathbf{n}}_{\varepsilon}$ (simplified by $\omega_{\varepsilon}$).
The next lemma provides a way to control the energy _inside_
$\omega_{\varepsilon}$ of a solution to the elliptic system by its energy
_outside_ , namely the energy in
$\widetilde{\omega}\setminus\overline{\omega}_{\varepsilon}$.

######  Lemma 5.3.

If $u\in H^{1}(\widetilde{\omega}_{\varepsilon};\mathbb{R}^{m})$ solves
$\mathcal{L}_{\varepsilon,1}(u)=f$ in $\omega_{\varepsilon}$, then

$\|\nabla u\|_{L^{2}(\omega_{\varepsilon})}\leq
C\left(\varepsilon\|f\|_{L^{2}(\omega_{\varepsilon})}+\|\nabla
u\|_{L^{2}(\widetilde{\omega}_{\varepsilon}\setminus\overline{\omega}_{\varepsilon})}\right).$

###### Proof.

This result is essentially Lemma 2.3 of [27] and it is proved by a simple
extension argument. By rescaling we may assume $\varepsilon=1$. Let $v$ be the
extension of $u-\fint_{\widetilde{\omega}\setminus\overline{\omega}}u$ from
$\widetilde{\omega}\setminus\overline{\omega}$ to $\widetilde{\omega}$, then
$\|v\|_{L^{2}(\widetilde{\omega})}\leq C\|\nabla
u\|_{L^{2}(\widetilde{\omega}\setminus\overline{\omega})}$. The desired result
then follows by testing $u-v\in H^{1}_{0}(\omega)$ against the equation
$\mathcal{L}_{1,1}(u)=f$. ∎

######  Theorem 5.4 (Caccioppoli inequalities for small $\delta$).

Suppose $\delta\in(0,1]$ and let $f_{\varepsilon,\delta}$ be defined by
(1.13).

  * (i)

_(Interior estimate)_ Let $R\geq 4\varepsilon$ and $Q_{2R}$ be a cube in
$\Omega$. If $\mathcal{L}_{\varepsilon,\delta}(u)=f_{\varepsilon,\delta}$ in
$Q_{2R}$, then

$\fint_{Q_{R}}|\nabla u|^{2}\leq
C\left(\frac{1}{R^{2}}\fint_{Q_{2R}}|u|^{2}+R^{2}\fint_{Q_{2R}}|f|^{2}\right).$
(5.4)

  * (ii)

_(Boundary estimate)_ Let $R\geq 4\varepsilon$ and $\mathbf{D}_{R}$ be a
boundary cylinder. If
$\mathcal{L}_{\varepsilon,\delta}(u)=f_{\varepsilon,\delta}$ in
$\mathbf{D}_{2R}$ and $u=g$ in $\Delta_{2R}$, then

$\fint_{\mathbf{D}_{3R/2}}|\nabla u|^{2}\leq
C\left\\{\frac{1}{R^{2}}\fint_{\mathbf{D}_{2R}}|u|^{2}+R^{2}\fint_{\mathbf{D}_{2R}}|f|^{2}+\frac{1}{R^{2}}\|g\|^{2}_{L^{\infty}(\Delta_{2R})}+\|\nabla_{\rm
tan}g\|^{2}_{L^{\infty}(\Delta_{2R})}\right\\}.$ (5.5)

###### Proof of (i) – the interior case.

Let $\varphi\in C_{0}^{1}(Q_{2R};[0,1])$ such that $\varphi\equiv 1$ in
$Q_{3R/2}$ and $\|\nabla\varphi\|_{L^{\infty}}\leq C/R$, plug it in (5.2) and
replacing $f$ there by $f_{\varepsilon,\delta}$. Using Young’s inequality, we
obtain

$\displaystyle\int_{Q_{3R/2}}\Lambda_{\varepsilon,\delta}(x)|\nabla
u|^{2}\leq$
$\displaystyle\frac{C}{R^{2}}\int_{Q_{2R}}\Lambda_{\varepsilon,\delta}(x)|u|^{2}+CR^{2}\int_{Q_{2R}}\Lambda_{\varepsilon,\delta}(x)|f|^{2}$
$\displaystyle\leq$
$\displaystyle\frac{C}{R^{2}}\int_{Q_{2R}}|u|^{2}+CR^{2}\int_{Q_{2R}}|f|^{2}.$

Now for each $\omega_{\varepsilon}\subset D_{\varepsilon}$ that has non-empty
intersection with $Q_{R}$, its enlarged domain
$\widetilde{\omega}_{\varepsilon}$ is contained in $Q_{R+2\varepsilon}\subset
Q_{3R/2}$. By Lemma 5.3, we have

$\int_{Q_{R}\cap D_{\varepsilon}}|\nabla u|^{2}\leq
C\left(\int_{Q_{3R/2}\cap\Omega_{\varepsilon}}|\nabla
u|^{2}+\varepsilon^{2}\int_{Q_{R}\cap
D_{\varepsilon}}\delta^{-2}|f_{\varepsilon,\delta}|^{2}\right).$

We combine the two estimates above and get

$\displaystyle\int_{Q_{R}}|\nabla u|^{2}\,dx\leq$ $\displaystyle
C\left(\frac{1}{R^{2}}\int_{Q_{2R}}|u|^{2}+R^{2}\int_{Q_{2R}}|f|^{2}+\varepsilon^{2}\int_{Q_{R}\cap
D_{\varepsilon}}\delta^{-2}|f_{\varepsilon,\delta}|^{2}\right)$
$\displaystyle\leq$ $\displaystyle
C\left(\frac{1}{R^{2}}\int_{Q_{2R}}|u|^{2}+R^{2}\int_{Q_{2R}}|f|^{2}\right),$

here we used the fact that $\varepsilon<R$ to deduce the second line. This
completes the proof. ∎

###### Proof of (ii) – the boundary case.

By rescaling we assume $R=1$. We first solve the problem

$\mathcal{L}_{\varepsilon,\delta}(v)=h\ \ \text{in
}\,\mathbf{D}_{2+4\varepsilon},\qquad v=\widetilde{g}\ \ \text{on
}\,\partial\mathbf{D}_{2}.$

Here, we let the inclusions in $\mathbf{D}_{2+4\varepsilon}$ be exactly those
of $D_{\varepsilon}$ that had non-empty intersections with $\mathbf{D}_{2}$.
Let $F$ denote the union of these inclusions, then the pair
$(\mathbf{D}_{2+4\varepsilon},F)$ satisfying Assumption $\mathbf{G}$ is a
type-II perforated domain. The source term $h$ is still
$\Lambda_{\varepsilon,\delta}(x)f(x)$ except that the weight
$\Lambda_{\varepsilon,\delta}$ are determined by the locations of the
inclusions just described. The boundary data $\widetilde{g}$ is an extension
of $g\rvert_{\Delta_{1}}$ that is compactly supported in $\Delta_{3/2}$ and is
zero on $\partial\mathbf{D}_{2}\cap\Omega$. We can apply Theorem 4.1 to get

$\|\nabla v\|_{L^{2}(\mathbf{D}_{2})}\leq
C\left(\|f\|_{L^{2}(\mathbf{D}_{2})}+\|g\|_{H^{1/2}(\Delta_{2})}\right).$

Note that $\mathcal{L}_{\varepsilon,\delta}(u-v)=0$ in $\mathbf{D}_{3/2}$ and
$u-v=0$ in $\Delta_{3/2}$, so the general setting is reduced to the special
setting of $g=0$ (in fact also $f=0$). Thanks to Lemma 5.2, the special
setting is exactly the same as in the interior setting. ∎

###  5.2. Caccioppoli inequalities for large $\delta$

For relatively hard obstacles, the Caccioppoli inequalities are harder to get.
Shen [27] nevertheless established the interior version with zero source term
$f$. We use his method to prove the boundary version and in a general setting.

######  Theorem 5.5 (Caccioppoli inequality for large $\delta$ without source
term).

Suppose $\delta\in(1,\infty)$. Let $R\geq 16\varepsilon$, if $u\in
H^{1}(Q_{2R};\mathbb{R}^{m})$ that solves
$\mathcal{L}_{\varepsilon,\delta}(u)=0$ in $Q_{2R}$, and for any positive
integer $\ell\in\mathbb{N}^{*}$, we have

$\int_{Q_{R}}|\nabla u|^{2}\leq
C_{\ell}\left(\frac{1}{R^{2}}\int_{Q_{2R}}|u|^{2}+\left(\frac{\varepsilon}{R}\right)^{2\ell}\int_{Q_{2R}}|\nabla
u|^{2}\right).$ (5.6)

This result is precisely Theorem 3.3 of Shen [27] after the rescaling
$u\to\tilde{u}(\cdot)=u(\varepsilon\cdot)$. Note that the source term is zero
in the equation. Using the method in the proof of item (ii) of Theorem 5.4, we
can get the general setting with a nonzero source $f$.

On the other hand, we may repeat the argument of Shen while allowing the
presence of the source term. The next three lemmas provide the detailed
process.

######  Lemma 5.6.

Let $\delta\in(1,\infty)$ and let $\omega_{\varepsilon}\subset
D_{\varepsilon}$ be a typical obstacle. If $u\in
H^{1}(\widetilde{\omega}_{\varepsilon};\mathbb{R}^{m})$ solves
$\mathcal{L}_{\varepsilon,\delta}(u)=f$ in $\widetilde{\omega}_{\varepsilon}$,
we have

$\delta\|\nabla u\|_{L^{2}(\omega_{\varepsilon})}\leq
C\left\\{\|f\|_{L^{p}(\widetilde{\omega}_{\varepsilon})}+\|\nabla
u\|_{L^{2}(\widetilde{\omega}_{\varepsilon}\setminus\overline{\omega}_{\varepsilon})}\right\\},$

for $p=\frac{2d}{d+2}$ if $d\geq 3$ and $p>1$ for $d=2$.

###### Proof.

The choice of $p$ guarantees $\varepsilon^{2-d/p}\leq\varepsilon^{1-d/2}$. By
rescaling, we only need to consider the case $\varepsilon=1$ and then the
result is Lemma 3.1 of Shen [27]. We refer to the proof there. The key idea is
to extend the function $u$ from the inside of $\omega_{\varepsilon}$ to a
function $v\in H^{1}_{0}(\widetilde{\omega}_{\varepsilon};\mathbb{R}^{m})$ and
use it as the test function of the equation. ∎

######  Lemma 5.7.

Let $\delta\in(1,\infty)$, $\varepsilon,h\in(0,1)$, $R\geq 16\varepsilon$ and
$Q_{R}$ be a cube inside $\Omega$. If $u\in H^{1}(Q_{R};\mathbb{R}^{m})$
solves $\mathcal{L}_{\varepsilon,\delta}(u)=f$ in $Q_{R}$, then for any
$r\in[R/2,R-8\varepsilon]$, we have

$\int_{Q_{r}}|\nabla
u|^{2}\,dx\leq\frac{C}{h(R-r)^{2}}\int_{Q_{R}}|u|^{2}\,dx+\left(h+\frac{C}{\delta}\right)\int_{Q_{R}}|\nabla
u|^{2}\,dx+CR^{2}\int_{Q_{R}}|f|^{2}.$

###### Proof.

First note that the result only concerns interior of $\Omega$. By rescaling we
may assume $\varepsilon=1$. The result is then essentially Lemma 3.2 of Shen
[27] except that a nonzero source term $f$ is allowed. We repeat the proof for
the sake of completeness.

Fix a cut-off function $\varphi\in C^{1}_{0}(Q_{R-2};[0,1])$ such that
$\varphi\equiv 1$ in $Q_{r+3}$. For each $k\in\mathbb{Z}^{d}$ so that
$\widetilde{\omega}^{k}=\widetilde{\omega}+k$ is still in $Q_{R}$, let $w_{k}$
be an extension of $(u\varphi^{2}-c_{k})\rvert_{\omega^{k}}$ to
$H^{1}_{0}(\widetilde{\omega}^{k})$, where $c_{k}$ is the mean of
$u\varphi^{2}$ in $\omega^{k}$. Then we have

$\displaystyle\|w_{k}\|_{H^{1}(\widetilde{\omega}^{k})}$ $\displaystyle\leq
C\|\nabla(u\varphi^{2})\|_{L^{2}(\omega^{k})}\leq C\|\nabla
u\|_{L^{2}(\omega^{k})}+C\|u\nabla\varphi\|_{L^{2}(\omega^{k})}$ (5.7)
$\displaystyle\leq C\delta^{-1}\|\nabla
u\|_{L^{2}(\widetilde{\omega}^{k}\setminus\overline{\omega}^{k})}+C\delta^{-1}\|f\|_{L^{2}(\widetilde{\omega}^{k})}+C\|u\nabla\varphi\|_{L^{2}(\omega^{k})}.$

Here the first inequality results from the extension estimate and the
Poincaré-Wirtinger inequality. The second line is due to Lemma 5.6 (by noting
$p<2$ for $d\geq 3$ and by choosing $p=2$ for $d=2$).

Now let $\phi=u\varphi^{2}-\sum_{k}w_{k}\in H^{1}_{0}(Q_{R})$, where $k$ is
summed over the range described above. Let $F$ denote the closure of the union
of such $\omega^{k}$’s. Then because $\phi=c_{k}$ are constants in
$\omega^{k}$, we get

$\int_{Q_{R}\setminus F}A\nabla u\cdot\nabla\phi\,dx=\int_{Q_{R}}f\cdot\phi.$

In view of the definition of $\phi$, we obtain

$\left|\int_{Q_{R}\setminus F}A\nabla
u\cdot\nabla(u\varphi^{2})\,dx\right|\leq\int_{Q_{R}}|f||\phi|+C\sum_{k}\|\nabla
u\|_{L^{2}(\widetilde{\omega}^{k}\setminus\overline{\omega}^{k})}\|\nabla
w_{k}\|_{L^{2}(\widetilde{\omega}^{k}\setminus\overline{\omega}^{k})}.$ (5.8)

For the term involving $w_{k}$, we use (5.7) and Young’s inequality and bound
the sum by

$\displaystyle\sum_{k}\left([h+C\delta^{-1}]\|\nabla
u\|_{L^{2}(\widetilde{\omega}^{k}\setminus\overline{\omega}^{k})}^{2}+C\delta^{-1}\|f\|^{2}_{L^{2}(\omega_{k})}+Ch^{-1}\int_{\omega^{k}}|u|^{2}|\nabla\varphi|^{2}\right)$
$\displaystyle\leq$
$\displaystyle\left(\frac{C}{\delta}+h\right)\int_{Q_{R}\setminus F}|\nabla
u|^{2}+CR^{2}\int_{Q_{R}}|f|^{2}+\frac{C}{h(R-r)^{2}}\int_{Q_{R}\setminus\overline{Q}_{r}}|u|^{2},$

where we use the relation $R^{2}>16^{2}>\delta^{-1}$, and the fact that
$|\nabla\varphi|$ is supported in $Q_{R}\setminus Q_{r+3}$ and bounded by
$C|R-r|^{-1}$.

For the integral $\int_{Q_{R}}|f||\phi|$, we have

$\int_{Q_{R}}|f||\phi|\leq\|f\|_{L^{2}(Q_{R})}\|u\|_{L^{2}(Q_{R})}+\sum_{k}\|f\|_{L^{2}(\widetilde{\omega}^{k})}\|w_{k}\|_{L^{2}(\widetilde{\omega}_{k})},$

using Poincaré inequality for $w_{k}$ and (5.7), we conclude that

$\int_{Q_{R}}|f||\phi|\leq
CR^{2}\int_{Q_{R}}|f|^{2}+\frac{C}{\delta}\int_{Q_{R}}|\nabla
u|^{2}+\frac{C}{(R-r)^{2}}\int_{Q_{R}}|u|^{2}.$

By plugging $\nabla(u\varphi^{2})=\varphi^{2}\nabla u+2\varphi
u\otimes\nabla\varphi$ in the integral on the left of (5.8), using the fact
$\varphi=1$ in $Q_{r+3}$ and the estimate

$\left|\int_{Q_{R}\setminus F}2\varphi A\nabla
u:(u\otimes\nabla\varphi)\right|\leq h\int_{Q_{R}}|\nabla
u|^{2}+\frac{C}{h(R-r)^{2}}\int_{Q_{R}\setminus Q_{r}}|u|^{2},$

which is easily checked as before, we obtain

$\int_{Q_{r+3}\setminus F}|\nabla
u|^{2}\leq\frac{C}{h(R-r)^{2}}\int_{Q_{R}}|u|^{2}\,dx+\left(h+\frac{C}{\delta}\right)\int_{Q_{R}}|\nabla
u|^{2}\,dx+CR^{2}\int_{Q_{R}}|f|^{2}.$

Finally we conclude the proof by using the estimate

$\int_{Q_{r}}|\nabla u|^{2}\leq
CR^{2}\int_{Q_{R}}|f|^{2}+C\int_{Q_{r+3}\setminus F}|\nabla u|^{2},$

which is easily implied by Lemma 5.6. ∎

######  Theorem 5.8 (Caccioppoli inequality for large $\delta$: interior
estimate).

Let $\delta\in(1,\infty)$. Let $u\in H^{1}(Q_{R};\mathbb{R}^{m})$ be a weak
solution of $\mathcal{L}_{\varepsilon,\delta}(u)=f$ in $Q_{R}$ for some $R\geq
32\varepsilon$. Then for any $\ell\geq 1$,

$\int_{Q_{R/2}}|\nabla u|^{2}\leq
C_{\ell}\left\\{\frac{1}{R^{2}}\int_{Q_{R}}|u|^{2}+\left(\frac{\varepsilon}{R}\right)^{2\ell}\int_{Q_{R}}|\nabla
u|^{2}+R^{2}\int_{Q_{R}}|f|^{2}\right\\}.$

###### Proof.

We modify the proof of Shen [27, Theorem 3.3] and add $f$ to the right hand
side. By rescaling we assume $\varepsilon=1$. Given $R\geq 32$, and an integer
$\ell\geq 1$, let $k$ be the largest integer so that $R2^{-k-1}\geq 8$. For
$1\leq i\leq k$, let $r_{i}=R(1-2^{-i})$. Then we verify that, for all $1\leq
i\leq k$,

$r_{i+1}\leq 16,\quad\tfrac{1}{2}r_{i+1}\leq r_{i}\leq r_{i+1}-8.$

So, for any $h\in(0,1)$, we can apply the estimate in the previous Lemma 5.7
and obtain

$\int_{Q_{r_{i}}}|\nabla
u|^{2}\leq\frac{C}{h(r_{i+1}-r_{i})^{2}}\int_{Q_{r_{i+1}}}|u|^{2}+(h+C\delta^{-1})\int_{Q_{r_{i+1}}}|\nabla
u|^{2}+Cr_{i+1}^{2}\int_{Q_{r_{i+1}}}|f|^{2}.$

By iteration, we finally get

$\displaystyle\int_{Q_{R/2}}|\nabla u|^{2}\leq$
$\displaystyle\frac{C}{h}\sum_{i=1}^{k}\frac{(h+C\delta^{-1})^{i-1}}{(r_{i+1}-r_{i})^{2}}\int_{Q_{R}}|u|^{2}+C\sum_{i=1}^{k}(h+C\delta^{-1})^{i-1}r_{i+1}^{2}\int_{Q_{R}}|f|^{2}$
$\displaystyle+(h+C\delta^{-1})^{k}\int_{Q_{R}}|\nabla u|^{2}.$

Plug in the value $r_{i+1}-r_{i}=2^{-i-1}R$, we have

$\displaystyle\int_{Q_{R/2}}|\nabla u|^{2}\leq$
$\displaystyle\frac{4C}{hR^{2}(h+C\delta^{-1})}\sum_{i=1}^{\infty}(4h+4C\delta^{-1})^{i}\int_{Q_{R}}|u|^{2}$
$\displaystyle+CR^{2}\sum_{i=1}^{\infty}(h+C\delta^{-1})^{i-1}\int_{Q_{R}}|f|^{2}+(h+C\delta^{-1})^{k}\int_{Q_{R}}|\nabla
u|^{2}.$

We now fix $\ell\geq 1$, note that if $\delta\in[1,2^{2\ell+2}C)$, the
conclusion directly follows from the classical setting. Hereafter we assume
$\delta\geq 2^{2\ell+2}C$. We choose $h=2^{-2\ell-2}$ so that
$4h+4C\delta^{-1}\leq 2^{-2\ell+1}$. Hence,
$\sum_{i=1}^{\infty}(4h+4C\delta^{-1})^{i}\leq 1$, and the estimate above
yields

$\int_{Q_{R/2}}|\nabla
u|^{2}\leq\frac{C_{\ell}}{R^{2}}\int_{Q_{R}}|u|^{2}+C_{\ell}R^{2}\int_{Q_{R}}|f|^{2}+(2^{-2\ell+1})^{k}\int_{Q_{R}}|\nabla
u|^{2}.$

Note that we fixed $R\geq 32$ first, and then choose $k$ so that $2^{k+4}\leq
R<2^{k+5}$, so $(2^{-2\ell+1})^{k}\approx R^{-2\ell}$. This completes the
proof. ∎

By modifying the estimate a little bit, more precisely, by carrying the
boundary terms in the interation steps (like the source term $f$ above), we
can prove a boundary version of the result above.

######  Theorem 5.9 (Caccioppoli inequality for large $\delta$: boundary
estimate).

Let $\delta\in(1,\infty)$. Let $u\in H^{1}(\mathbf{D}_{R};\mathbb{R}^{m})$ be
a weak solution of $\mathcal{L}_{\varepsilon,\delta}(u)=f$ in $\mathbf{D}_{R}$
and $u=g$ on $\Delta_{R}$ for some $R\geq 32\varepsilon$. Then for any
$\ell\geq 1$,

$\displaystyle\int_{\mathbf{D}_{3R/2}}|\nabla u|^{2}\leq$
$\displaystyle\frac{C_{\ell}}{R^{2}}\int_{\mathbf{D}_{2R}}|u|^{2}+C_{\ell}\left(\frac{\varepsilon}{R}\right)^{2\ell}\int_{\mathbf{D}_{2R}}|\nabla
u|^{2}$ (5.9)
$\displaystyle+C_{\ell}\left(R^{2}\int_{\mathbf{D}_{2R}}|f|^{2}+\frac{1}{R^{2}}\|g\|_{L^{\infty}(\Delta_{2R})}^{2}+\|\nabla_{\rm
tan}g\|_{L^{\infty}(\Delta_{2R})}^{2}\right).$

We only outline the proof as it is very similar to the proof of item (ii) of
Theorem 5.4. First, in the special setting of $g=0$, the estimate (5.9) can be
proved just like in the interior setting, following the steps of Lemma 5.6,
Lemma 5.7 and Theorem 5.8. Indeed, given a cut-off function $\varphi$ that
vanishes near $\partial\mathbf{D}_{R}\cap\Omega$, the function $\varphi^{2}u$
belongs to $H^{1}_{0}(\mathbf{D}_{R})$ as $u=0$ on $\Delta_{R}$, and hence it
can be used to get a boundary version of Lemma 5.7 and then an “iteration”
argument like the one used in Theorem 5.8 yield the result. The more general
setting can be reduced to the special setting using the idea in the proof of
item (ii) of Theorem 5.4; we omit the details.

##  6\. Uniform Lipschitz regularity

In this section we prove Theorem 1.2 and hence establish the uniform (in
$\varepsilon$ and $\delta$) Lipschitz regularity of the solution of
$\mathcal{L}_{\varepsilon,\delta}(u)=f$ provided that for small $\delta$ the
source term $f$ is small inside the obstacles. It is well known that via the
routine covering and rescaling arguments, the global uniform Lipschitz
regularity follows from a large-scale (down to $\varepsilon$-scale) Lipschitz
estimate and a local Lipschitz estimate (in the blown-up domain).

The main difference compared with Shen’s work is that, Shen did not use the
fact that for each fixed $\delta$, there is a homogenized problem and,
moreover, the homogenized problem behaves quite well in the sense that it
enjoys estimates that is uniform in $\delta$, for $\delta\in(1,\infty)$. For
$\delta\ll 1$, although the homogenized problem is no longer a good
approximation, a simple additional corrector is sufficient.

######  Theorem 6.1 (Large-scale boundary Lipschitz estimate).

Assume that $A$ satisfies (1.2)(1.3), $f\in L^{p}(\Omega;\mathbb{R}^{m})$ with
$p\geq d$. For $\varepsilon\in(0,1)$ and $\delta\in(0,\infty)$, let
$f_{\varepsilon,\delta}$ is defined by (1.13). Let $u_{\varepsilon,\delta}$ be
a weak solution of
$\mathcal{L}_{\varepsilon,\delta}(u_{\varepsilon,\delta})=f_{\varepsilon,\delta}$
in $\mathbf{D}_{2}$ and $u=g$ in $\Delta_{2}$. Then there is $c>1$ so that for
any $r\in[c\varepsilon,1]$ we have

$\|\nabla u_{\varepsilon,\delta}\|_{\underline{L}^{2}(\mathbf{D}_{r})}\leq
C\left(\|\nabla
u_{\varepsilon,\delta}\|_{\underline{L}^{2}(\mathbf{D}_{2})}+\|f\|_{\underline{L}^{p}(\mathbf{D}_{2})}+\|g\|_{C^{1,\sigma}(\Delta_{2})}\right).$
(6.1)

Clearly, an interior version of the above large-scale Lipschitz estimate can
be similarly formulated and proved. We focus only on the boundary setting
because the interior version is relatively easier and because it is already
available in Shen [27, Theorem 7.3]. We point out that Shen’s method is
different and is not based on the approximation scheme.

The large-scale Lipschitz estimate above combined with a blow-up argument and
a local Lipschitz estimate yield the following full-scale Lipschitz
regularity. Again we only present and prove the boundary version.

######  Theorem 6.2 (Boundary Lipschitz estimate).

Assume the conditions of Theorem 6.1 and futher assume that $A$ satisfies
(1.4) and that $\Omega$ and $\Omega_{\varepsilon}$ satisfies the geometric
condition. Let $\mathbf{D}_{5/2}$ be a boundary cylinder of $\Omega$, and let
$u_{\varepsilon,\delta}\in H^{1}(\mathbf{D}_{5/2};\mathbb{R}^{m})$ bs a weak
solution of
$\mathcal{L}_{\varepsilon,\delta}(u_{\varepsilon,\delta})=f_{\varepsilon,\delta}$
in $\mathbf{D}_{5/2}$ with boundary condition $u_{\varepsilon,\delta}=g$ on
$\Delta_{5/2}$. Then

$\|\nabla u_{\varepsilon,\delta}\|_{L^{\infty}(\mathbf{D}_{1/2})}\leq
C\Big{\\{}\|\nabla
u_{\varepsilon,\delta}\|_{L^{2}(\mathbf{D}_{5/2})}+\|f\|_{L^{p}(\mathbf{D}_{5/2})}+\|\nabla_{\rm
tan}g\|_{C^{0,\alpha}(\Delta_{5/2})}\Big{\\}}.$ (6.2)

We only detail the proof of boundary Lipschitz regularity. Shen has
essentially proved the interior Lipschitz regularity in [27, Theorem 7.3], but
as far as we know, the uniform boundary Lipschitz regularity of (1.7) is new.
We prove it using the approximation scheme which is now standard, and this
method is easier to carry out for the interior Lipschitz estimate. We mention
that, even for interior estimate, our method differs from [27] because there
Shen used a method that relies on the special structure of homogeneous
equations, combined with Green’s function estimates. It seems that the
approximation scheme has a large scope for applications, and what was missing
for this scheme, before our present work, was the lack of the convergece rate
result in (1.12) for $\delta\gg 1$.

We give the proof of the full-scale boundary Lipschitz estimate, assuming the
large-scale Lipschitz estimate in Theorem 6.1.

###### Proof of Theorem 6.2.

For simplicity we write $u$ for $u_{\varepsilon,\delta}$. Let $c\geq 8$ in
Theorem 6.1, we first assume that $\varepsilon<1/c$. Then for any
$x_{0}\in\Delta_{1/2}$, we have

$\mathbf{D}_{c\varepsilon}(x_{0})\subset\mathbf{D}_{2}(x_{0})\subset\mathbf{D}_{5/2}.$
(6.3)

On the other hand, we assume that $\mathbf{D}_{2\varepsilon}(x_{0})$ is
contained in $\Omega_{\varepsilon}$ (that is, there is no obstacles inside
$\mathbf{D}_{2\varepsilon}(x_{0})$). In view of the geometric condition (GC),
this assumption is always valid if we further shrink the domain
$\mathbf{D}_{2\varepsilon}(x_{0})$ here.

Next we use the standard method by blow-uping the domain $D_{2\varepsilon}$
via the mapping $x\mapsto y$ with $y(x)=\varepsilon^{-1}(x-x_{0})$, we denote
the blown-up domain by $\widetilde{\mathbf{D}}_{2}$. Without loss of
generality assume $x_{0}=0$ and consider the rescaled function
$w(y)=\varepsilon^{-1}u(\varepsilon y)$, then a rescaling computation shows

$\mathcal{L}_{1,1}(w)=\widetilde{f}\quad\text{in}\quad\widetilde{\mathbf{D}}_{2},\qquad\text{and}\quad
w=\widetilde{g}\quad\text{in}\quad\widetilde{\Delta}_{2},$

with $\widetilde{f}(y)=\varepsilon f(\varepsilon y)$,
$\widetilde{g}(y)=\varepsilon^{-1}g(\varepsilon y)$. Note that $\delta=1$ in
the domain involved because there is no inclusion inside
$\mathbf{D}_{\varepsilon}(x_{0})$. Moreover, $\widetilde{\Delta}_{2}$ is now
the graph of the function $\varepsilon^{-1}\psi(\varepsilon y)$ where $\psi$
is the $C^{1,\alpha}$ function that defines $\Delta_{2\varepsilon}(x_{0})$.
Note that $\widetilde{\Delta}_{2}$ satisfies the same regularity bounds of
$\partial\Omega$. Then we can apply the standard local $C^{1,\alpha}$
regularity estimate for the operator $\mathcal{L}_{1,1}$ with Hölder
continuous coefficients in $C^{1,\alpha}$ domains, and get

$\|\nabla w\|_{L^{\infty}(\widetilde{\mathbf{D}}_{1})}\leq C\left\\{\|\nabla
w\|_{L^{2}(\widetilde{\mathbf{D}}_{2})}+\|\widetilde{f}\|_{L^{p}(\widetilde{\mathbf{D}}_{2})}+\|\nabla_{\rm
tan}\widetilde{g}\|_{C^{0,\alpha}(\widetilde{\Delta}_{2})}\right\\}.$ (6.4)

By the change of variable $y=\varepsilon x$, we get

$\displaystyle\|\nabla u\|_{L^{\infty}(\mathbf{D}_{\varepsilon}(x_{0}))}$
$\displaystyle\leq C\Big{\\{}\|\nabla
u\|_{\underline{L}^{2}(\mathbf{D}_{2\varepsilon}(x_{0}))}+\varepsilon\|f\|_{\underline{L}^{p}(\mathbf{D}_{2\varepsilon}(x_{0}))}$
(6.5) $\displaystyle\qquad+\|\nabla_{\rm
tan}g\|_{L^{\infty}(\Delta_{2\varepsilon}(x_{0}))}+\varepsilon^{\alpha}\lvert\nabla_{\rm
tan}g\rvert_{C^{0,\alpha}(\Delta_{2\varepsilon}(x_{0}))}\Big{\\}}$
$\displaystyle\leq C\Big{\\{}\|\nabla
u\|_{\underline{L}^{2}(\mathbf{D}_{c\varepsilon}(x_{0}))}+\|f\|_{{L}^{p}(\mathbf{D}_{2}(x_{0}))}+\|\nabla_{\rm
tan}g\|_{C^{0,\alpha}(\Delta_{2}(x_{0}))}\Big{\\}}$ $\displaystyle\leq
C\Big{\\{}\|\nabla
u\|_{L^{2}(\mathbf{D}_{2}(x_{0}))}+\|f\|_{{L}^{p}(\mathbf{D}_{2}(x_{0}))}+\|\nabla_{\rm
tan}g\|_{C^{0,\alpha}(\Delta_{2}(x_{0}))}\Big{\\}}.$

Here to get the second inequality we used the fact that $p>d$ and
$\varepsilon<1$. The third inequality follows from Theorem 6.1. In view of
(6.3), we have bound $|\nabla u(x)|$ by the right hand side of (6.2) for all
$x\in\mathbf{D}_{1/2}$ in the ’boundary layer’, i.e., those can be covered by
$\mathbf{D}_{\varepsilon}(x_{0})$ for some $x_{0}\in\Delta_{1/2}$.

For any other point $x$ in $\mathbf{D}_{1/2}$, we can find some
$Y^{\mathbf{n}}_{\varepsilon}$ such that $x\in Y^{\mathbf{n}}_{\varepsilon}$,
in the blown-up domain, the operator is $\mathcal{L}_{1,\delta}$, so we need a
local regularity estimate that is uniform for $\delta\in(0,\infty)$: for $w\in
H^{1}(Q_{1};\mathbb{R}^{m})$ solving

$\mathcal{L}_{1,\delta}(w)=\widetilde{f}_{\delta}\quad\text{in}\quad Q_{1},$

where $\widetilde{f}_{\delta}$ is modifed to be $\delta\widetilde{f}$ in the
inclusions for $\delta\in(0,1)$. Then for any $\delta\in(0,\infty)$, one has

$\|\nabla w\|_{L^{\infty}(Q_{1/2})}\leq C\left\\{\|\nabla
w\|_{L^{2}(Q_{1})}+\|\widetilde{f}_{\delta}\|_{L^{p}(Q_{1})}\right\\}.$ (6.6)

It plays the role of (6.4). Fortunately, this is proved by Shen; see [27,
Lemma 5.1] or equations (5.9) and (5.10) there. Hence, the proof of Theorem
6.2 is done by (6.6) and the large-scale interior Lipschitz estimate for
$\varepsilon<1/c$.

Finally, for $\varepsilon\geq 1/c$, the tensor $A(x/\varepsilon)$ is uniformly
(in $\varepsilon$) Hölder continous, and the result boils down to the local
regularity estimates mentioned above. ∎

The interior version of the full-scale Lipschitz estimate can similarly be
proved. The proof of Theorem 2.4 is then complete by the usual covering
argument. Hence it remains only to prove the large-scale Lipschitz regularity.

###  6.1. Proof of the large-scale Lipschitz estimate

We focus on the boundary version, i.e., the proof of Theorem 6.1. Our approach
is the now more or less standard approximation scheme. Although the approach
originates from the work of Armstrong and Smart, we refer to the monograph
[25, Chapter 6, section 5] for the improved and standardized steps which we
follow.

The first step is the following approximation result.

######  Lemma 6.3.

Suppose that $u_{\varepsilon,\delta}$ is a weak solution of
$\mathcal{L}_{\varepsilon,\delta}(u_{\varepsilon,\delta})=f_{\varepsilon,\delta}$
in $\mathbf{D}_{2r}$ with $u_{\varepsilon,\delta}=g$ on $\Delta_{2r}$ for some
$r\geq 4\varepsilon$. There exists $v\in H^{1}(\mathbf{D}_{r})$ such that
$\widehat{\mathcal{L}}_{\delta}(v)=f_{\varepsilon,\delta}$ in $\mathbf{D}_{r}$
with $v=g$ on $\Delta_{r}$, and

$\displaystyle\|u_{\varepsilon,\delta}-v\|_{\underline{L}^{2}(\mathbf{D}_{r})}\leq$
$\displaystyle
C\left(\frac{\varepsilon}{r}\right)^{1/2}\left\\{\|u_{\varepsilon,\delta}\|_{\underline{L}^{2}(\mathbf{D}_{2r})}+r^{2}\|f\|_{\underline{L}^{2}(\mathbf{D}_{2r})}+\|g\|_{L^{\infty}(\Delta_{2r})}+r\|\nabla_{\mathrm{tan}}g\|_{L^{\infty}(\mathbf{D}_{2r})}\right\\}$
(6.7) $\displaystyle+C\left(\frac{\varepsilon}{r}\right)^{3/2}r\|\nabla
u_{\varepsilon,\delta}\|_{\underline{L}^{2}(\mathbf{D}_{2r})}.$

###### Proof.

By rescaling we may assume $r=1$. For notational simplicity we write $u$ for
$u_{\varepsilon,\delta}$. For $\delta>1$, Let $R=1$ and $\ell=1$ in Theorem
5.9, we get

$\displaystyle\int_{\mathbf{D}_{3/2}}|\nabla u|^{2}\,dx\leq$ $\displaystyle
C\left(\int_{\mathbf{D}_{2}}|u|^{2}\,dx+\int_{\mathbf{D}_{2}}|f|^{2}\,dx+\|g\|_{L^{\infty}(\Delta_{2})}^{2}+\|\nabla_{\rm
tan}g\|_{L^{2}(\Delta_{2})}^{2}\right.$
$\displaystyle+\left.\varepsilon^{2}\int_{\mathbf{D}_{2}}|\nabla
u|^{2}\,dx\right).$

For $\delta\in(0,1]$, we use (5.5) instead, and the above still holds (even
without the last $O(\varepsilon^{2})$ term). By the co-area formula and
arguing by contradiction, we can find some $t\in[5/4,3/2]$ for which

$\|\nabla u\|_{L^{2}(\partial\mathbf{D}_{t}\setminus\Delta_{2})}\leq
C\left(\|u\|_{L^{2}(\mathbf{D}_{2})}+\|f\|_{L^{2}(\mathbf{D}_{t})}+\|g\|_{L^{\infty}(\Delta_{2})}+\|\nabla_{\rm
tan}g\|_{L^{\infty}(\Delta_{2})}+\varepsilon\|\nabla
u\|_{L^{2}(\mathbf{D}_{2})}\right).$ (6.8)

Let $h=u\rvert_{\partial\mathbf{D}_{t}\setminus\Delta_{2}}$ and $h=g$ in
$\Delta_{2}$. Let $v\in H^{1}(\mathbf{D}_{t})$ solve

$\widehat{\mathcal{L}}_{\delta}(v)=f_{\varepsilon,\delta}\quad\text{in
}\;\mathbf{D}_{t},\qquad v=h\quad\text{on }\,\partial\mathbf{D}_{t}.$

Then by Theorem 1.1 and Remark 5 (note that $\mathbf{D}_{t}$ may be a Type-I
domain) we have

$\displaystyle\Big{\|}u-v-\delta^{-1}\mathcal{L}^{-1}_{D_{\varepsilon}}(f_{\varepsilon,\delta})-\varepsilon\chi_{\delta}\left(\frac{x}{\varepsilon}\right)S_{\varepsilon}\big{(}\eta_{\varepsilon}\nabla
v\big{)}\Big{\|}_{H^{1}(\mathbf{D}_{t-1/4})}$ $\displaystyle\leq
C\varepsilon^{1/2}\left(\|f\|_{L^{2}(\mathbf{D}_{t})}+\|\nabla_{\mathrm{tan}}h\|_{L^{2}(\partial\mathbf{D}_{t})}\right)$
$\displaystyle\leq C\varepsilon^{1/2}E(u),$

where $E(u)$ denote the quantity inside the parathesis in the right hand side
of (6.8). It is easy to check the following estimates (see Remark 1 and Lemma
2.3)

$\displaystyle\delta^{-1}\|\mathcal{L}_{D_{\varepsilon}}^{-1}(f_{\varepsilon,\delta})\|_{L^{2}(\mathbf{D}_{t})}$
$\displaystyle\leq C\varepsilon^{2}\|f\|_{L^{2}(\mathbf{D}_{2})},$
$\displaystyle\Big{\|}\chi_{\delta}\left(\frac{x}{\varepsilon}\right)S_{\varepsilon}\big{(}\eta_{\varepsilon}\nabla
v\big{)}\Big{\|}_{L^{2}(\mathbf{D}_{t})}$ $\displaystyle\leq
C\|\chi_{\delta}\|_{L^{2}(Y)}\|\nabla v\|_{L^{2}(\mathbf{D}_{t})}$
$\displaystyle\leq
C\|\chi_{\delta}\|_{L^{2}(Y)}\|\left(\|f\|_{L^{2}(\mathbf{D}_{t})}+\|\nabla_{\mathrm{tan}}h\|_{L^{2}(\partial\mathbf{D}_{t})}\right).$

Using the estimate in (6.8) we hence obtain

$\displaystyle\|u-v\|_{L^{2}(\mathbf{D}_{1})}$ $\displaystyle\leq
C(\varepsilon^{1/2}+\varepsilon\|\chi_{\delta}\|_{L^{2}(Y)})E(u)$ (6.9)
$\displaystyle\leq
C\varepsilon^{1/2}\left(\|u\|_{L^{2}(\mathbf{D}_{2})}+\|f\|_{L^{2}(\mathbf{D}_{2})}+\|\nabla_{\rm\tan}g\|_{L^{\infty}(\Delta_{2})})\right)+C\varepsilon^{3/2}\|\nabla
u\|_{L^{2}(\mathbf{D}_{2})}.$

The proof is hence completed. ∎

Note that for $\delta\in(0,1)$ (with the modified
$f_{\varepsilon,\delta}=\delta f$ inside $D_{\varepsilon}$), we do not need
the last term on the right hand side of (6.7) and the proof of Theorem 6.1
reduces to the standard procedure; see e.g. [25, Section 6.4]. The proof below
is a modification of the standard method to handel the extra gradient term.

Let $f$ and $g$ be as in Theorem 6.1. For $t\leq 2$, $p>d$, and for functions
$w\in H^{1}(\mathbf{D}_{2})$, we define the following notations:

$\displaystyle H[w](t)=$
$\displaystyle\frac{1}{t}\inf_{\begin{subarray}{c}E\in\mathbb{R}^{m\times
d}\\\
q\in\mathbb{R}^{m}\end{subarray}}\Bigg{\\{}\left(\fint_{\mathbf{D}_{t}}|w-Ex-q|^{2}\right)^{\frac{1}{2}}+t^{2}\left(\fint_{\mathbf{D}_{t}}|f|^{p}\right)^{\frac{1}{p}}+\|g-Ex-q\|_{L^{\infty}(\Delta_{t})}$
(6.10) $\displaystyle+t\|\nabla_{\rm
tan}(g-Ex)\|_{L^{\infty}(\Delta_{t})}+t^{1+\rho}\lvert\nabla_{\rm
tan}(g-Ex)\rvert_{C^{0,\rho}(\Delta_{t})}\Bigg{\\}},$

where $\rho$ is a number in $(0,\alpha)$ and $\lvert\cdot\rvert_{C^{0,\rho}}$
denotes the usual Hölder semi-norm. We also defined $h[w](t)$ to be the norm
$|E_{t}|$ of a matrix $E_{t}$ for which

$\displaystyle H[w](t)=$
$\displaystyle\frac{1}{t}\inf_{q\in\mathbb{R}^{m}}\Bigg{\\{}\left(\fint_{\mathbf{D}_{t}}|w-E_{t}x-q|^{2}\right)^{\frac{1}{2}}+t^{2}\left(\fint_{\mathbf{D}_{t}}|f|^{p}\right)^{\frac{1}{p}}+\|g-E_{t}x-q\|_{L^{\infty}(\Delta_{t})}$
(6.11) $\displaystyle\qquad+t\|\nabla_{\rm
tan}(g-E_{t}x)\|_{L^{\infty}(\Delta_{t})}+t^{1+\rho}|\nabla_{\rm
tan}(g-E_{t}x)\rvert_{C^{\rho}(\Delta_{t})}\Bigg{\\}}.$

######  Lemma 6.4.

Suppose that $\widehat{\mathcal{L}}_{\delta}(w)=f$ in $\mathbf{D}_{r}$ with
$w=g$ on $\Delta_{r}$ for some $r>0$. Then there exists a universal constant
$\theta\in(0,1/4)$ (independent of $r,f$ or $g$) such that

$H[w](\theta r)\leq\frac{1}{2}H[w](r).$ (6.12)

###### Proof.

We have shown in Theorem 3.5 that the homogenized operator
$\widehat{\mathcal{L}}_{\delta}$ has constant elliptic tensor with uniform in
$\delta$ ellipticity bounds. The result of this lemma then follows from
$C^{1,\alpha}$ regularity estimate of constant coefficients. We refer to [7,
Lemma 7.1] for the proof with $f=0$. The setting with $f\in L^{p}$ and $p>d$
is the same. ∎

######  Lemma 6.5.

Let $\theta$ be determined in Lemma 6.4 and $c>0$ be a number so that
$c\theta\geq 4$. Suppose $u_{\varepsilon,\delta}$ is a weak solution of
$\mathcal{L}_{\varepsilon,\delta}(u_{\varepsilon,\delta})=f_{\varepsilon,\delta}$
in $\mathbf{D}_{2r}$ with $u_{\varepsilon,\delta}=g$ on $\Delta_{2r}$ for some
$r\geq c\varepsilon$, then

$\displaystyle H[u_{\varepsilon,\delta}](\theta r)$
$\displaystyle\leq\frac{1}{2}H[u_{\varepsilon,\delta}](r)$ (6.13)
$\displaystyle\ \ \
+C\left(\frac{\varepsilon}{r}\right)^{1/2}\Big{(}H[u_{\varepsilon,\delta}](2r)+h[u_{\varepsilon,\delta}](2r)\Big{)}+C\left(\frac{\varepsilon}{r}\right)^{3/2}\|\nabla
u_{\varepsilon,\delta}\|_{\underline{L}^{2}(\mathbf{D}_{2r})}.$

###### Proof.

Let $v$ be the function chosen in Lemma 6.3. By Lemma 6.4

$H[v](\theta r)\leq\frac{1}{2}H[v](r).$ (6.14)

Let $c$ be a positive number so that $\theta c\geq 4$, then by (6.14), we have

$\displaystyle H[u_{\varepsilon,\delta}](\theta r)$ $\displaystyle\leq
H[v](\theta
r)+\frac{\|u_{\varepsilon,\delta}-v\|_{\underline{L}^{2}(\mathbf{D}_{\theta
r})}}{\theta r}$
$\displaystyle\leq\frac{1}{2}H[u_{\varepsilon,\delta}](r)+\frac{1}{2r}\|u_{\varepsilon,\delta}-v\|_{\underline{L}^{2}(\mathbf{D}_{r})}+\frac{1}{\theta
r}\|u_{\varepsilon,\delta}-v\|_{\underline{L}^{2}(\mathbf{D}_{\theta r})}.$

By Lemma 6.3, we obtain

$\displaystyle H[u_{\varepsilon,\delta}](\theta r)\leq$
$\displaystyle\frac{1}{2}H[u_{\varepsilon,\delta}](r)+C\left(\frac{\varepsilon}{r}\right)^{3/2}\|\nabla
u_{\varepsilon,\delta}\|_{\underline{L}^{2}(\mathbf{D}_{2r})}$
$\displaystyle+C\left(\frac{\varepsilon}{r}\right)^{1/2}\frac{1}{r}\left\\{\|u_{\varepsilon,\delta}\|_{\underline{L}^{2}(\mathbf{D}_{2r})}+r^{2}\|f\|_{\underline{L}^{2}(\mathbf{D}_{2r})}+\|g\|_{L^{\infty}(\Delta_{2r})}+r\|\nabla_{\mathrm{tan}}g\|_{L^{\infty}(\mathbf{D}_{2r})}\right\\}.$

Since adding a constant to $u_{\varepsilon,\delta}$ and $g$ does not change
the setting, we in fact get

$\displaystyle H[u_{\varepsilon,\delta}](\theta r)$
$\displaystyle\leq\frac{1}{2}H[u_{\varepsilon,\delta}](r)+C\left(\frac{\varepsilon}{r}\right)^{3/2}\|\nabla
u_{\varepsilon,\delta}\|_{\underline{L}^{2}(\mathbf{D}_{2r})}$ $\displaystyle\
\ \
\,+C\left(\frac{\varepsilon}{r}\right)^{1/2}\inf_{q\in\mathbb{R}^{m}}\frac{1}{r}\Big{\\{}\|u_{\varepsilon,\delta}-q\|_{\underline{L}^{2}(\mathbf{D}_{2r})}+r^{2}\|f\|_{\underline{L}^{p}(\mathbf{D}_{2r})}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\,+\|g-q\|_{L^{\infty}(\Delta_{2r})}+r\|\nabla_{\mathrm{tan}}(g-q)\|_{L^{\infty}(\mathbf{D}_{2r})}\Big{\\}}.$

Then the desired estimate follows because the infimum in the last line is
clearly dominated by

$H[u_{\varepsilon,\delta}](2r)+h[u_{\varepsilon,\delta}](2r).$

The proof is complete. ∎

###### Proof of Theorem 6.1.

For notational simplicity we will write $u$ for $u_{\varepsilon,\delta}$,
$H(r)=H[u](r)$, and $h(r)=h[u](r)$. Let $c$ be a number so that $c\geq 8$ and
$c\theta\geq 4$. We consider below $r\geq c\varepsilon$. We divide the proof
to several steps.

Step 1. Let $\ell=1$ in Theorem 5.9, we have

$\|\nabla u\|_{\underline{L}^{2}(\mathbf{D}_{r})}\leq
C\left\\{\frac{\varepsilon}{r}\|\nabla
u\|_{\underline{L}^{2}(\mathbf{D}_{2r})}+\frac{1}{r}\|u\|_{\underline{L}^{2}(\mathbf{D}_{2r})}+r\|f\|_{\underline{L}^{2}(\mathbf{D}_{2r})}+\frac{1}{r}\|g\|_{L^{\infty}(\Delta_{2r})}+\|\nabla_{\rm
tan}g\|_{L^{\infty}(\Delta_{2r})}\right\\}.$

Because we can insert any constant $q$ simultaneously to $u$ and $g$ above,
and note that
$\|f\|_{\underline{L}^{2}(\mathbf{D}_{2r})}\leq\|f\|_{\underline{L}^{p}(\mathbf{D}_{2r})}$
for $p>2$, we actually get

$\|\nabla
u\|_{\underline{L}^{2}(\mathbf{D}_{r})}\leq\frac{C\varepsilon}{r}\|\nabla
u\|_{\underline{L}^{2}(\mathbf{D}_{2r})}+C\Phi(2r),$

where

$\Phi(2r)=\inf_{q\in\mathbb{R}^{m}}\frac{1}{r}\left\\{\|u-q\|_{\underline{L}^{2}(\mathbf{D}_{2r})}+r^{2}\|f\|_{\underline{L}^{p}(\mathbf{D}_{2r})}+\|g-q\|_{L^{\infty}(\Delta_{2r})}+r\|\nabla_{\rm
tan}g\|_{L^{\infty}(\Delta_{2r})}\right\\}.$

From the definitions of $H$ and $h$ we verify $\Phi(2r)\leq C(H(2r)+h(2r))$,
and hence we conclude that

$\|\nabla u\|_{\underline{L}^{2}(\mathbf{D}_{r})}\leq
C\left(H(2r)+h(2r)\right)+\frac{C\varepsilon}{r}\|\nabla
u\|_{\underline{L}^{2}(\mathbf{D}_{2r})}.$

Iterating the inequality above, for the largest integer $M\in\mathbb{N}^{*}$
such that $2^{M}r\leq 2$, we get

$\displaystyle\|\nabla u\|_{\underline{L}^{2}(\mathbf{D}_{r})}$
$\displaystyle\leq\sum_{n=1}^{M}\frac{C^{n}\varepsilon^{n-1}}{2^{(n-1)(n-2)/2}r^{n-1}}\Big{(}H(2^{n}r)+h(2^{n}r)\Big{)}$
$\displaystyle\ \ \ +\frac{C^{M}\varepsilon^{M}}{2^{M(M-1)/2}r^{M}}\|\nabla
u\|_{\underline{L}^{2}(\mathbf{D}_{2^{M}r})}.$

Since for any $\varepsilon>0$ and $r>\varepsilon$,

$\sup_{n\in\mathbb{N}^{*}}2^{n}\cdot\frac{C^{n}\varepsilon^{n-1}}{2^{(n-1)(n-2)/2}r^{n-1}}+\sup_{M\in\mathbb{N}^{*}}\frac{C^{M}\varepsilon^{M}}{2^{M(M-1)/2}r^{M}}<\infty,$

we obtain

$\|\nabla u\|_{\underline{L}^{2}(\mathbf{D}_{r})}\leq
C\sum_{n=1}^{M}\frac{H(2^{n}r)+h(2^{n}r)}{2^{n}}+C\|\nabla
u\|_{\underline{L}^{2}(\mathbf{D}_{2})}.$ (6.15)

Step 2. As is standard, from the definitions (6.10) and (6.11) it is not hard
to verify that

$\displaystyle H(t)\leq CH(2r)\qquad$ $\displaystyle\forall t\in[r,2r],$
(6.16) $\displaystyle|h(t)-h(s)|\leq CH(2r)\qquad$ $\displaystyle\forall
t,s\in[r,2r].$

Moreover, those inequalities lead to

$h(a)+H(a)\leq
C\left\\{H(2)+h(2)+\int_{a}^{2}\frac{H(s)}{s}\,ds\right\\},\quad\forall
a\in[\varepsilon,2].$ (6.17)

We refer to [25, page 157] for the verification of the last estimate. We then
get

$\displaystyle\sum_{n=1}^{M}\frac{H(2^{n}r)+h(2^{n}r)}{2^{n}}$
$\displaystyle\leq
C\sum_{n=1}^{M}\frac{1}{2^{n}}\left[H(2)+h(2)+\int_{2^{n}r}^{2}\frac{H(t)}{t}\,dt\right]$
$\displaystyle\leq C\left(H(2)+h(2)+\int_{r}^{2}\frac{H(t)}{t}\,dt\right).$

By taking $E=0$ and $q=0$ in (6.10) we have

$H(2)\leq\|u\|_{\underline{L}^{2}(\mathbf{D}_{2})}+\|f\|_{\underline{L}^{p}(\mathbf{D}_{2})}+\|g\|_{C^{1,\rho}(\Delta_{2})},$
(6.18)

and since $E_{2}x+q$ is harmonic for any $q\in\mathbb{R}^{m}$, we also have

$h(2)\leq
C\inf_{q\in\mathbb{R}^{m}}\|E_{1}x+q\|_{\underline{L}^{2}(\mathbf{D}_{2})}\leq
C\\{H(2)+\|u\|_{\underline{L}^{2}(\mathbf{D}_{2})}\\}.$

We hence get

$H(2)+h(2)\leq C\left(\|\nabla
u\|_{\underline{L}^{2}(\mathbf{D}_{2})}+\|f\|_{\underline{L}^{p}(\mathbf{D}_{2})}+\|\nabla_{\rm
tan}g\|_{C^{0,\rho}(\Delta_{2})}\right).$

In the last line we used the freedom to insert any constant $q$ to $u$ and
$g$, and applied Poincaré’s inequality.

_Step 3_ : Combine the estimates above. We get, for $c\varepsilon\leq r\leq
1$,

$\|\nabla u\|_{\underline{L}^{2}(\mathbf{D}_{r})}\leq C\left(\|\nabla
u\|_{\underline{L}^{2}(\mathbf{D}_{2})}+\|f\|_{\underline{L}^{p}(\mathbf{D}_{2})}+\|\nabla_{\rm
tan}g\|_{C^{0,\rho}(\Delta_{2})}\right)+C\int_{r}^{2}\frac{H(t)}{t}\,dt.$
(6.19)

Step 4. We estimate the integral $\int_{r}^{2}\frac{H(t)}{t}\,dt$. By Lemma
6.5, (6.17) and (6.19), we get

$H(\theta
r)\leq\frac{1}{2}H(r)+C\Big{(}\frac{\varepsilon}{r}\Big{)}^{1/2}\Phi_{2}+C\Big{(}\frac{\varepsilon}{r}\Big{)}^{1/2}\int_{r}^{2}\frac{H(t)}{t}\,dt,$
(6.20)

for any $c\varepsilon\leq r\leq 1/2$, where $\Phi_{2}=\|\nabla
u\|_{\underline{L}^{2}(\mathbf{D}_{2})}+\|f\|_{\underline{L}^{p}(\mathbf{D}_{2})}+\|\nabla_{\rm
tan}g\|_{C^{0,\rho}(\Delta_{2})}$. Let $c^{\prime}>c$ be a large number so
that

$\int_{0}^{1/c^{\prime}}\sqrt{t}\leq\frac{1}{4C}$ (6.21)

where $C$ is the constant in front of the integral $\int_{r}^{2}H(t)/t\,dt$ in
(6.20). We compute

$\displaystyle\int_{c^{\prime}\theta\varepsilon}^{2\theta}\frac{H(r)}{r}\,dr=\int_{c^{\prime}\varepsilon}^{2}\frac{H(\theta
r)}{r}\,dr$
$\displaystyle\leq\frac{1}{2}\int_{c^{\prime}\varepsilon}^{1/2}\frac{H(r)}{r}\,dr+\int_{1/2}^{2}\frac{H(\theta
r)}{r}\,dr+C\Phi_{2}\int_{c^{\prime}\varepsilon}^{1/2}\left(\frac{\varepsilon}{r}\right)^{\frac{1}{2}}\frac{1}{r}\,dr$
(6.22) $\displaystyle\ \ \
+C\int_{c^{\prime}\varepsilon}^{1/2}\left(\frac{\varepsilon}{r}\right)^{1/2}\int_{r}^{2}\frac{H(t)}{t}\,dt\,\frac{dr}{r}.$

The second integral on the right hand side above can be bounded by
$C_{\theta}H(2)$ where $C_{\theta}$ is a constant only depending on $\theta$.
For the third integral, we can bound using the fact that

$\int_{c^{\prime}\varepsilon}^{1}\left(\frac{\varepsilon}{r}\right)^{1/2}\frac{dr}{r}=\int_{\varepsilon}^{1/c^{\prime}}t^{-1/2}\,dt\leq\frac{1}{4C}.$

For the last integral on the right hand side of (6.22), by Fubini’s Theorem,
we have

$\displaystyle\int_{c^{\prime}\varepsilon}^{2}\left(\frac{\varepsilon}{r}\right)^{1/2}\int_{r}^{2}\frac{H(t)}{t}\,dt\,\frac{dr}{r}$
$\displaystyle=\int_{c^{\prime}\varepsilon}^{2}\int_{c^{\prime}\varepsilon}^{t}\left(\frac{\varepsilon}{r}\right)^{1/2}\frac{dr}{r}\frac{H(t)}{t}\,dt$
$\displaystyle=\int_{c^{\prime}\varepsilon}^{2}H(t)\int_{\varepsilon/t}^{1/c^{\prime}}s^{-1/2}\,ds\,\frac{dt}{t}$
$\displaystyle\leq\int_{0}^{1/c^{\prime}}s^{-1/2}\,ds\int_{c^{\prime}\varepsilon}^{2}\frac{H(t)}{t}\,dt$
$\displaystyle\leq\frac{1}{4C}\int_{c^{\prime}\varepsilon}^{2}\frac{H(t)}{t}\,dt.$

Apply those estimates above in (6.22), we get

$\int_{c^{\prime}\theta\varepsilon}^{2\theta}\frac{H(r)}{r}\,dr\leq\frac{3}{4}\int_{c^{\prime}\varepsilon}^{2}\frac{H(r)}{r}\,dr+C\Phi_{2}\leq\frac{3}{4}\int_{c^{\prime}\varepsilon}^{2\theta}\frac{H(r)}{r}\,dr+C\Phi_{2},$

where the second inequality follows from (6.16) and (6.18). It follows that

$\int_{c^{\prime}\theta\varepsilon}^{2}\frac{H(t)}{t}\,dt\leq\int_{c^{\prime}\theta\varepsilon}^{2\theta}\frac{H(t)}{t}\,dt+\int_{2\theta}^{2}\frac{H(t)}{t}\,dt\leq
C\Phi_{2}.$

By another iterative usage of (6.16) (up to finitely many steps independent of
$\varepsilon$), we get

$\int_{r}^{2}\frac{H(t)}{t}\,dt\leq C\Phi_{2}$

for all $r\in[c\varepsilon,1]$. The desired estimate (6.1) follows from above
and (6.19). ∎

##  Appendix A A weak form of layer potentials

In this appendix, we construct the ‘weak’ form of single layer potential and
the NP operator, and explain how does the results in Section 3 are still valid
provided the Hölder continuity condition (1.4) is removed.

Let $\psi\in H_{0}^{-1/2}(\partial\omega)$, we consider the following
variational problem:

$\int_{Y}A\nabla u\cdot\nabla
v\,dy+\langle\psi,v\rangle_{-1/2,1/2}=0,\qquad\mathrm{for\ any}\ v\in
H^{1}(\mathbb{T}^{d}),$ (A.1)

where $\langle\cdot,\cdot\rangle_{-1/2,1/2}$ denotes $H^{-1/2}-H^{1/2}$ pair.
By Lax-Milgram theorem and Poincaré inequality, there exists a unique solution
$u\in H^{1}(\mathbb{T}^{d})$ such that $\int_{Y}u(y)\,dy=0$, we denote such
$u$ by $\mathcal{S}\psi$. It is clear that $\mathcal{L}\mathcal{S}\psi=0$ in
$\mathbb{T}^{d}\setminus\partial\omega$ and

$I=\left.\frac{\partial\mathcal{S}}{\partial\nu_{A}}\right|_{+}-\left.\frac{\partial\mathcal{S}}{\partial\nu_{A}}\right|_{-}.$
(A.2)

We then define the boundary operator $K$ by

$K:H\rightarrow
H,\quad\psi\mapsto\left.\frac{\partial\mathcal{S}\psi}{\partial\nu_{A}}\right|_{+}-\frac{1}{2}\psi,$
(A.3)

where $H$ is the Hilbert space $H_{0}^{-1/2}(\partial\omega)$ with a new inner
product: for any $\phi,\psi\in H_{0}^{-1/2}(\partial\omega)$,

$\langle\phi,\psi\rangle_{H}:=\int_{Y}A\nabla\mathcal{S}\phi\cdot\nabla\mathcal{S}\psi\,dy.$
(A.4)

By definition of $K$ and (A.2), we have