# Darcy’s Law for Porous Media
with Multiple Microstructures

Zhongwei Shen Supported in part by NSF grant DMS-2153585.

###### Abstract

In this paper we study the homogenization of the Dirichlet problem for the
Stokes equations in a perforated domain with multiple microstructures. First,
under the assumption that the interface between subdomains is a union of
Lipschitz surfaces, we show that the effective velocity and pressure are
governed by a Darcy law, where the permeability matrix is piecewise constant.
The key step is to prove that the effective pressure is continuous across the
interface, using Tartar’s method of test functions. Secondly, we establish the
sharp error estimates for the convergence of the velocity and pressure,
assuming the interface satisfies certain smoothness and geometric conditions.
This is achieved by constructing two correctors. One of them is used to
correct the discontinuity of the two-scale approximation on the interface,
while the other is used to correct the discrepancy between boundary values of
the solution and its approximation.

Keywords: Homogenization; Stokes Equations; Perforated Domain; Convergence
Rate.

MR (2020) Subject Classification: 35Q35; 35B27; 76D07.

##  1 Introduction

In this paper we study the homogenization of the Dirichlet problem for the
Stokes equations in a perforated domain $\Omega_{\varepsilon}$,

$\left\\{\begin{aligned} -\varepsilon^{2}\mu\Delta u_{\varepsilon}+\nabla
p_{\varepsilon}&=f&\quad&\text{ in }\Omega_{\varepsilon},\\\ \text{\rm
div}(u_{\varepsilon})&=0&\quad&\text{ in }\Omega_{\varepsilon},\\\
u_{\varepsilon}&=0&\quad&\text{ on
}\partial\Omega_{\varepsilon},\end{aligned}\right.$ (1.1)

where $0<\varepsilon<1$ and $\mu>0$ is a constant. Let $\Omega$ be a bounded
Lipschitz domain in $\mathbb{R}^{d}$, $d\geq 2$. Let
$\\{\Omega^{\ell}:1\leq\ell\leq L\\}$ be a finite number of disjoint
subdomains of $\Omega$, each with a Lipschitz boundary, such that

$\overline{\Omega}=\bigcup_{\ell=1}^{L}\overline{\Omega^{\ell}}.$ (1.2)

To describe the porous domain $\Omega_{\varepsilon}$, let $Y=[0,1]^{d}$ be a
closed unit cube and $\\{Y_{s}^{\ell}:1\leq\ell\leq L\\}$ open subsets (solid
parts) of $Y$ with Lipschitz boundaries. Assume that for $1\leq\ell\leq L$,
dist$(\partial Y,\partial Y_{s}^{\ell})>0$ and
$Y^{\ell}_{f}=Y\setminus\overline{Y_{s}^{\ell}}$ (the fluid part) is
connected. For $0<\varepsilon<1$ and $1\leq\ell\leq L$, define

$\Omega_{\varepsilon}^{\ell}=\Omega^{\ell}\setminus\bigcup_{z}\varepsilon(\overline{Y_{s}^{\ell}}+z),$
(1.3)

where $z\in\mathbb{Z}^{d}$ and the union is taken over those $z$’s for which
$\varepsilon(Y+z)\subset\Omega^{\ell}$. Thus the subdomain $\Omega^{\ell}$ is
perforated periodically, using the solid obstacle $Y_{s}^{\ell}$. Let

$\Omega_{\varepsilon}=\Sigma\cup\bigcup_{\ell=1}^{L}\Omega_{\varepsilon}^{\ell}=\Omega\setminus\bigcup_{\ell=1}^{L}\bigcup_{z}\varepsilon(\overline{Y_{s}^{\ell}}+z),$
(1.4)

where $\Sigma$ is the interface between subdomains, given by

$\Sigma=\Omega\setminus\bigcup_{\ell=1}^{L}\Omega^{\ell}=\bigcup_{\ell=1}^{L}\partial\Omega^{\ell}\setminus\partial\Omega.$
(1.5)

For $f\in L^{2}(\Omega;\mathbb{R}^{d})$, let
$(u_{\varepsilon},p_{\varepsilon})\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})\times
L^{2}(\Omega_{\varepsilon})$ be the weak solution of (1.1) with
$\int_{\Omega_{\varepsilon}}p_{\varepsilon}\,dx=0$. We extend
$u_{\varepsilon}$ to the whole domain $\Omega$ by zero. Let $P_{\varepsilon}$
denote the extension of $p_{\varepsilon}$ to $\Omega$, defined by (2.21). In
the case $L=1$, where $\Omega$ is perforated periodically with small holes of
same shape, it is well known that as $\varepsilon\to 0$, $u_{\varepsilon}\to
u_{0}$ weakly in $L^{2}(\Omega;\mathbb{R}^{d})$ and $P_{\varepsilon}\to P_{0}$
strongly in $L^{2}(\Omega)$, where the effective velocity and pressure
$(u_{0},P_{0})$ are governed by the Darcy law,

$\left\\{\begin{aligned} u_{0}&=\mu^{-1}K(f-\nabla P_{0})&\quad&\text{ in
}\Omega,\\\ \text{div}(u_{0})&=0&\quad&\text{ in }\Omega,\\\ u_{0}\cdot
n&=0&\quad&\text{ on }\partial\Omega,\end{aligned}\right.$ (1.6)

with $\int_{\Omega}P_{0}\,dx=0$. Note that in (1.1) we have normalized the
velocity vector by a factor $\varepsilon^{2}$, where $\varepsilon$ is the
period. For references on the Darcy law we refer to the reader to [13, 10, 1,
3, 4].

In (1.6) the permeability matrix $K$ is a $d\times d$ positive definite,
constant and symmetric matrix and $n$ denotes the outward unit normal to
$\partial\Omega$. It was observed in [3] by G. Allaire that as $\varepsilon\to
0$,

$u_{\varepsilon}-\mu^{-1}W(x/\varepsilon)(f-\nabla P_{0})\to
0\quad\text{strongly in }L^{2}(\Omega;\mathbb{R}^{d}),$ (1.7)

where $W(y)$ is an 1-periodic $d\times d$ matrix defined by a cell problem and
$\fint_{Y}W(y)\,dy=K$. Recently, it was proved in [15] by the present author
that

$\|u_{\varepsilon}-\mu^{-1}W(x/\varepsilon)(f-\nabla
P_{0})\|_{L^{2}(\Omega)}+\|P_{\varepsilon}-P_{0}\|_{L^{2}(\Omega)}\leq
C\sqrt{\varepsilon}\|f\|_{C^{1,1/2}(\Omega)},$ (1.8)

and that

$\|\varepsilon\nabla u_{\varepsilon}-\mu^{-1}\nabla W(x/\varepsilon)(f-\nabla
P_{0})\|_{L^{2}(\Omega)}\leq C\sqrt{\varepsilon}\|f\|_{C^{1,1/2}(\Omega)}.$
(1.9)

We point out that due to the discrepancy between boundary values of
$\mu^{-1}W(x/\varepsilon)(f-\nabla P_{0})$ and $u_{\varepsilon}$ on
$\partial\Omega$, the $O(\varepsilon^{1/2})$ convergence rates in (1.8)-(1.9)
are sharp. See [11] for an earlier partial result on solutions with periodic
boundary conditions.

The primary purpose of this paper is to study the Darcy law for the case
$L\geq 2$, where the domain $\Omega$ is divided into several subdomains and
different subdomains are perforated with small holes of different shapes.

######  Theorem 1.1.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{d}$, $d\geq 2$, and
$\Omega_{\varepsilon}$ be given by (1.4). Let
$(u_{\varepsilon},p_{\varepsilon})\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})\times
L^{2}(\Omega_{\varepsilon})$ be a weak solution of (1.1), where $f\in
L^{2}(\Omega;\mathbb{R}^{d})$ and
$\int_{\Omega_{\varepsilon}}p_{\varepsilon}\,dx=0$. Let $P_{\varepsilon}$ be
the extension of $p_{\varepsilon}$, defined by (2.21). Then
$u_{\varepsilon}\to u_{0}$ weakly in $L^{2}(\Omega;\mathbb{R}^{d})$ and
$P_{\varepsilon}-\fint_{\Omega}P_{\varepsilon}\to P_{0}$ strongly in
$L^{2}(\Omega)$, as $\varepsilon\to 0$, where $P_{0}\in H^{1}(\Omega)$ and
$(u_{0},P_{0})$ is governed by the Darcy law (1.6) with the matrix

$K=\sum_{\ell=1}^{L}K^{\ell}\chi_{\Omega^{\ell}}\qquad\text{ in }\Omega.$
(1.10)

The matrix $K^{\ell}$ in (1.10) is the (constant) permeability matrix
associated with the solid obstacle $Y_{s}^{\ell}$. Thus, the matrix $K$ is
piecewise constant in $\Omega$, taking value $K^{\ell}$ in the subdomain
$\Omega^{\ell}$, and

$u_{0}=K^{\ell}(f-\nabla P_{0})\quad\text{ in }\Omega^{\ell}.$ (1.11)

Since $\text{\rm div}(u_{0})=0$ in $\Omega$ and $P_{0}\in H^{1}(\Omega)$, both
the normal component $u_{0}\cdot n$ and $P_{0}$ are continuous across the
interface $\Sigma$ (in the sense of trace) between subdomains. However, the
tangential components of $u_{0}$ may not be continuous across $\Sigma$.

The Dirichlet problem for the Stokes equations (1.1) is used to model fluid
flows in porous media with different microstructures in different subdomains.
The continuity of the effective pressure $P_{0}$ and the normal component
$u_{0}\cdot n$ of the effective velocity across the interface is generally
accepted in engineering [6, 9]. Theorem 1.1 is probably known to experts.
However, to the best of the author’s knowledge, the existing literatures on
rigorous proofs only treat the case of flat interfaces. In particular, the
result was proved in [9] under the assumption that $d=2$, the interface
$\Gamma=\mathbb{R}\times\\{0\\}$ and the solutions are 1-periodic in the
direction $x_{1}$. Also see related work in [5, 12]. We provide a proof here
for the general case, where the interface is a union of Lipschitz surfaces,
using Tartar’s method of test functions. We point out that the proof for
(1.11) and $P_{0}\in H^{1}(\Omega^{\ell})$ for each $\ell$ is the same as in
the classical case $L=1$. The challenge is to show that the effective pressure
$P_{0}$ is continuous across the interface and thus $P_{0}\in H^{1}(\Omega)$,
which is essential for proving the uniqueness of the limits of subsequences of
$\\{u_{\varepsilon}\\}$.

Our main contribution in this paper is on the sharp convergence rates and
error estimates for $u_{\varepsilon}$ and $P_{\varepsilon}$. We are able to
extend the results in [14] for the case $L=1$ to the case $L\geq 2$ under some
smoothness and geometric conditions on subdomains. More specifically, we
assume that each subdomain is a bounded $C^{2,1/2}$ domain, and that there
exists $r_{0}>0$ such that if
$x_{0}\in\partial\Omega^{k}\cap\partial\Omega^{m}$ for some $1\leq k,m\leq L$
and $k\neq m$, there exists a coordinate system, obtained from the standard
one by translation and rotation, such that

$\displaystyle B(x_{0},r_{0})\cap\Omega^{k}$
$\displaystyle=B(x_{0},r_{0})\cap\big{\\{}(x^{\prime},x_{d})\in\mathbb{R}^{d}:x_{d}>\psi(x^{\prime})\big{\\}},$
(1.12) $\displaystyle B(x_{0},r_{0})\cap\Omega^{m}$
$\displaystyle=B(x_{0},r_{0})\cap\big{\\{}(x^{\prime},x_{d})\in\mathbb{R}^{d}:x_{d}<\psi(x^{\prime})\big{\\}},$

where $\psi:\mathbb{R}^{d-1}\to\mathbb{R}$ is a $C^{2,1/2}$ function. Roughly
speaking, this means that inside a small ball centered on the interface
$\Sigma$, the domain $\Omega$ is divided by $\Sigma$ into exactly two
subdomains. In particular, the condition excludes the cases where the
interface intersects with each other or with the boundary of $\Omega$.

The following is the main result of the paper. The matrix $W^{\ell}(y)$ in
(1.13)-(1.14) is the 1-periodic matrix associated with the solid obstacle
$Y_{s}^{\ell}$.

######  Theorem 1.2.

Let $\Omega$ be a bounded $C^{2,1/2}$ domain and $\Omega_{\varepsilon}$ be
given by (1.4). Assume that the subdomains $\\{\Omega^{\ell}\\}$ are bounded
$C^{2,1/2}$ domains satisfying the condition (1.12). Let
$(u_{\varepsilon},P_{\varepsilon})$ and $(u_{0},P_{0})$ be the same as in
Theorem 1.1. Then, for $f\in C^{1,1/2}(\Omega;\mathbb{R}^{d})$,

$\sum_{\ell=1}^{L}\|u_{\varepsilon}-\mu^{-1}W^{\ell}(x/\varepsilon)(f-\nabla
P_{0})\|_{L^{2}(\Omega^{\ell})}+\|P_{\varepsilon}-\fint_{\Omega}P_{\varepsilon}-P_{0}\|_{L^{2}(\Omega)}\leq
C\sqrt{\varepsilon}\|f\|_{C^{1,1/2}(\Omega)},$ (1.13)

and

$\sum_{\ell=1}^{L}\|\varepsilon\nabla u_{\varepsilon}-\mu^{-1}\nabla
W^{\ell}(x/\varepsilon)(f-\nabla P_{0})\|_{L^{2}(\Omega^{\ell})}\leq
C\sqrt{\varepsilon}\|f\|_{C^{1,1/2}(\Omega)},$ (1.14)

where $C$ depends on $d$, $\mu$, $\Omega$, $\\{\Omega^{\ell}\\}$ and
$\\{Y_{s}^{\ell}\\}$.

As we mentioned earlier, the sharp convergence rates in (1.13) and (1.14) were
proved in [15] for the case $L=1$. In the case of two porous media with a flat
interface, partial results were obtained in [9] for solutions with periodic
boundary conditions. Theorem 1.2 is the first result that treats the general
case of smooth interfaces.

As in [9], the basic idea in our approach to Theorem 1.2 is to use

$V_{\varepsilon}(x)=\sum_{\ell=1}^{L}W^{\ell}(x/\varepsilon)(f-\nabla
P_{0})\chi_{\Omega_{\varepsilon}^{\ell}}$ (1.15)

to approximate the solution $u_{\varepsilon}$ and obtain the error estimates
by the energy method. Observe that $V_{\varepsilon}=0$ on
$\Gamma_{\varepsilon}=\partial\Omega_{\varepsilon}\setminus\partial\Omega$.
There are three main issues with this approach: (1) the divergence of
$V_{\varepsilon}$ is not small in $L^{2}$; (2) $V_{\varepsilon}$ does not
agree with $u_{\varepsilon}$ on $\partial\Omega$; and (3) $V_{\varepsilon}$ is
not in $H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$, as it is not continuous
across the interface. To overcome these difficulties, we introduce three
corresponding correctors: $\Phi_{\varepsilon}^{(1)}$,
$\Phi_{\varepsilon}^{(2)}$, and $\Phi_{\varepsilon}^{(3)}$. To correct the
divergence of $V_{\varepsilon}$, we construct $\Phi_{\varepsilon}^{(1)}\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ with the property that

$\varepsilon\|\nabla\Phi_{\varepsilon}^{(1)}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}+\|\text{\rm
div}(\Phi_{\varepsilon}^{(1)}+V_{\varepsilon})\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\leq
C\sqrt{\varepsilon}\|f\|_{C^{1,1/2}(\Omega)}$ (1.16)

for $1\leq\ell\leq L$. The construction of $\Phi_{\varepsilon}^{(1)}$ is
similar to that in [11, 9, 15]. Next, we correct the boundary data of
$V_{\varepsilon}$ on $\partial\Omega$ by constructing
$\Phi_{\varepsilon}^{(2)}\in H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ such
that $\Phi_{\varepsilon}^{(2)}+V_{\varepsilon}=0$ on $\partial\Omega$,
$\Phi_{\varepsilon}^{(2)}=0$ on $\Gamma_{\varepsilon}$, and that

$\varepsilon\|\nabla\Phi_{\varepsilon}^{(2)}\|_{L^{2}(\Omega_{\varepsilon})}+\|\text{\rm
div}(\Phi_{\varepsilon}^{(2)})\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\sqrt{\varepsilon}\|f\|_{C^{1,1/2}(\Omega)}.$ (1.17)

The construction of $\Phi_{\varepsilon}^{(2)}$ is similar to that in [15] for
the case $L=1$. The key observation is that the normal component of
$V_{\varepsilon}$ on $\partial\Omega$ can be written in the form

$\varepsilon\nabla_{\tan}\left(\phi(x/\varepsilon)\right)\cdot g,$ (1.18)

where $\nabla_{\tan}$ denotes the tangential gradient on $\partial\Omega$. We
remark that a similar observation is also used in the proof of Theorem 1.1.
Finally, to correct the discontinuity of $V_{\varepsilon}$ across the
interface, we introduce

$\Phi_{\varepsilon}^{(3)}=\sum_{\ell=1}^{L}I_{\varepsilon}^{\ell}(x)(f-\nabla
P_{0})\chi_{\Omega_{\varepsilon}^{\ell}},$ (1.19)

with the properties that $V+\Phi_{\varepsilon}^{(3)}\in
H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$, $\Phi_{\varepsilon}^{(3)}=0$ on
$\partial\Omega_{\varepsilon}$, and that

$\varepsilon\|\nabla\Phi_{\varepsilon}^{(3)}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}+\|\text{\rm
div}(\Phi_{\varepsilon}^{(3)})\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\leq
C\sqrt{\varepsilon}\|f\|_{C^{1,1/2}(\Omega)}.$ (1.20)

More specifically, for each $1\leq\ell\leq L$, the matrix-valued function
$I_{\varepsilon}^{\ell}$ is a solution of the Stokes equations in
$\Omega_{\varepsilon}^{\ell}$ with $I_{\varepsilon}^{\ell}=0$ on
$\partial\Omega_{\varepsilon}^{\ell}\setminus\partial\Omega^{\ell}$. On each
connected component $\Sigma^{k}$ of the interface $\Sigma$, the boundary value
of $I_{\varepsilon}^{\ell}$ is either $0$ or given by

$W_{j}^{-}(x/\varepsilon)-W_{j}^{+}(x/\varepsilon)-W_{i}^{-}(x/\varepsilon)(K_{mj}^{-}-K_{mj}^{+})\frac{n_{i}n_{m}}{\langle
nK^{-},n\rangle},$ (1.21)

where the repeated indices $i$ and $m$ are summed from $1$ to $d$. Here the
subdomains $\Omega^{\pm}$ are separated by $\Sigma^{k}$, and
$(W^{\pm},K^{\pm})$ denote the corresponding 1-periodic matrices for
$\Omega^{\pm}$ and their averages over $Y$, respectively. To show
$V+\Phi_{\varepsilon}^{(3)}$ is continuous across $\Sigma$, we use the fact
that $(\nabla_{\tan}P_{0})^{+}=(\nabla_{\tan}P_{0})^{-}$ and

$n\cdot K^{+}(f-\nabla P_{0})^{+}=n\cdot K^{-}(f-\nabla P_{0})^{-},$ (1.22)

where $(v)^{\pm}$ denote the trace of $v$ taken from $\Omega^{\pm}$,
respectively. The proof of the estimate (1.20) again relies on the observation
that the normal component of (1.21) is of form (1.18).

Theorem 1.2 is proved under the assumption that $\\{Y_{s}^{\ell}:1\leq\ell\leq
L\\}$ are subdomains of $Y$ with Lipschitz boundaries. The $C^{2,1/2}$
condition and the geometric condition (1.12) for $\Omega$ and subdomains
$\\{\Omega^{\ell}\\}$ are dictated by the smoothness requirement in its proof
for $P_{0}$ in each subdomain. Note that $P_{0}$ is a solution of an elliptic
equation with piecewise constant coefficients in $\Omega$. Not much is known
about the boundary regularity of $P_{0}$ if the interface intersects with the
boundary $\partial\Omega$ or with each other.

The paper is organized as follows. In Section 2 we collect several useful
estimates that are more or less known. In Section 3 we establish the energy
estimates for the Dirichlet problem (1.1). Theorem 1.1 is proved in Section 4.
In Section 5 we give the proof of Theorem 1.2, assuming the existence of
suitable correctors. Finally, we construct correctors
$\Phi_{\varepsilon}^{(1)}$, $\Phi_{\varepsilon}^{(2)}$, and
$\Phi_{\varepsilon}^{(3)}$, described above, in the last three sections of the
paper. Throughout the paper we will use $C$ to denote constants that may
depend on $d$, $\mu$, $\Omega$, $\\{\Omega^{\ell}\\}$, and
$\\{Y_{s}^{\ell}\\}$. In fact, since the viscosity constant $\mu$ is
irrelevant in our study, we will assume $\mu=1$ in the rest of the paper.

Acknowledgement. The author is indebt to Professor Xiaoming Wang for raising
the question that is addressed in this paper and for several stimulating
discussions.

##  2 Preliminaries

Let $Y=[0,1]^{d}$ and $\\{Y_{s}^{\ell}:1\leq\ell\leq L\\}$ be a finite number
of open subsets of $Y$ with Lipschitz boundaries. We assume that
dist$(\partial Y,\partial Y^{\ell}_{s})>0$ and that
$Y_{f}^{\ell}=Y\setminus\overline{Y_{s}^{\ell}}$ is connected. Let

$\omega^{\ell}=\bigcup_{z\in\mathbb{Z}^{d}}(Y_{f}^{\ell}+z)$

be the periodic repetition of $Y_{f}^{\ell}$. For $1\leq j\leq d$ and
$1\leq\ell\leq L$, let

$\left(W_{j}^{\ell}(y),\pi_{j}^{\ell}(y)\right)=\left(W_{1j}^{\ell}(y),\dots,W_{dj}^{\ell}(y),\pi_{j}^{\ell}(y)\right)\in
H^{1}_{\text{loc}}(\omega^{\ell};\mathbb{R}^{d})\times
L^{2}_{\text{loc}}(\omega^{\ell})$

be the 1-periodic solution of

$\left\\{\begin{aligned} -\Delta
W_{j}^{\ell}+\nabla\pi_{j}^{\ell}&=e_{j}&\quad&\text{ in }\omega^{\ell},\\\
\text{\rm div}(W_{j}^{\ell})&=0&\quad&\text{ in }\omega^{\ell},\\\
W_{j}^{\ell}&=0&\quad&\text{ on }\partial\omega^{\ell},\end{aligned}\right.$
(2.1)

with $\int_{Y^{\ell}_{f}}\pi_{j}^{\ell}\,dy=0$, where
$e_{j}=(0,\dots,1,\dots,0)$ with $1$ in the $j^{th}$ place. We extend the
matrix $W^{\ell}=(W_{j}^{\ell})$ to $\mathbb{R}^{d}$ by zero and define

$K^{\ell}_{ij}=\int_{Y}W_{ij}^{\ell}(y)\,dy.$ (2.2)

Since

$K_{ij}^{\ell}=\int_{Y}\nabla W_{ik}^{\ell}\cdot\nabla W^{\ell}_{jk}\,dy$

(the repeated index $k$ is summed from $1$ to $d$), it follows that the
$d\times d$ matrix $K^{\ell}=(K^{\ell}_{ij})$ is symmetric and positive
definite.

The existence and uniqueness of solutions to (2.1) can be proved by applying
the Lax-Milgram Theorem on the closure of the set,

$\left\\{u\in C^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d}):u\text{ is 1-periodic,
}u=0\text{ in }Y_{s}^{\ell},\text{ and }\text{\rm div}(u)=0\text{ in
}\mathbb{R}^{d}\right\\},$

in $H^{1}(Y;\mathbb{R}^{d})$. By energy estimates,

$\int_{Y}\left(|\nabla
W^{\ell}|^{2}+|W^{\ell}|^{2}+|\pi^{\ell}|^{2}\right)dy\leq C,$ (2.3)

where we have also extended $\pi^{\ell}$ to $\mathbb{R}^{d}$ by zero. By
periodicity this implies that

$\int_{D}\left(|\nabla
W^{\ell}(x/\varepsilon)|^{2}+|W^{\ell}(x/\varepsilon)|^{2}+|\pi^{\ell}(x/\varepsilon)|^{2}\right)dx\leq
C,$ (2.4)

where $D$ is a bounded domain and $C$ depends on diam($D$).

######  Lemma 2.1.

Let $D$ is a bounded Lipschitz domain in $\mathbb{R}^{d}$. Then

$\int_{\partial D}\left(|\nabla
W^{\ell}(x/\varepsilon)|^{2}+|W^{\ell}(x/\varepsilon)|^{2}+|\pi^{\ell}(x/\varepsilon)|^{2}\right)d\sigma\leq
C,$ (2.5)

where $C$ depends on $D$.

###### Proof.

If $Y_{s}^{\ell}$ is of $C^{1,\alpha}$, the inequality above follows directly
from the fact that $\nabla W^{\ell}$ and $\pi^{\ell}$ are bounded in $Y$. To
treat the case where $\partial Y_{s}^{\ell}$ is merely Lipschitz, by
periodicity, we may assume that $\varepsilon=1$ and $D$ is a subdomain of $Y$.
Note that the bound for the integral of $|W^{\ell}|^{2}$ on $\partial D$
follows from (2.3). Indeed, if $D$ is a subdomain of $Y$ with Lipschitz
boundary,

$\int_{\partial D}|W^{\ell}|^{2}\,d\sigma\leq C\int_{D}\left(|\nabla
W^{\ell}|^{2}+|W^{\ell}|^{2}\right)dy.$

The estimates for $\nabla W^{\ell}$ and $\pi^{\ell}$ are a bit more involved.
By using the fundamental solutions for the Stokes equations in
$\mathbb{R}^{d}$, we may reduce the problem to the estimate

$\|\nabla u\|_{L^{2}(\partial D)}+\|p\|_{L^{2}(\partial D)}\leq
C\left\\{\|\nabla u\|_{L^{2}(\widetilde{Y}\setminus
Y_{s}^{\ell})}+\|p\|_{L^{2}(\widetilde{Y}\setminus
Y_{s}^{\ell})}+\|h\|_{H^{1}(\partial Y_{s}^{\ell})}\right\\},$

for solutions of the Stokes equations,

$\left\\{\begin{aligned} -\Delta u+\nabla p&=0&\quad&\text{ in
}\widetilde{Y}\setminus\overline{Y_{s}^{\ell}},\\\ \text{\rm
div}(u)&=0&\quad&\text{ in }\widetilde{Y}\setminus\overline{Y_{s}^{\ell}},\\\
u&=h&\quad&\text{ on }\partial Y_{s}^{\ell},\end{aligned}\right.$

where $h\in H^{1}(\partial Y_{s}^{\ell};\mathbb{R}^{d})$ and
$\widetilde{Y}=(1+c)Y$. The desired estimates follow from the interior
estimates as well as the nontangential-maximal-function estimate,

$\|(\nabla u)^{*}\|_{L^{2}(\partial Y_{s}^{\ell})}+\|(p)^{*}\|_{L^{2}(\partial
Y_{s}^{\ell})}\leq C\left\\{\|h\|_{H^{1}(\partial
Y_{s}^{\ell})}+\|u\|_{L^{2}(\widetilde{Y}\setminus
Y_{s}^{\ell})}+\|p\|_{L^{2}(\widetilde{Y}\setminus Y_{s}^{\ell})}\right\\},$
(2.6)

where the nontangential maximal function $(v)^{*}$ is defined by

$(v)^{*}(x)=\sup\left\\{|v(y)|:\ y\in Y\setminus Y_{s}^{\ell}\text{ and
}|y-x|<C_{0}\,\text{\rm dist}(y,\partial Y_{s}^{\ell})\right\\}$

for $x\in\partial Y_{s}^{\ell}$. The estimate (2.6) is a consequence of the
nontangential-maximal-function estimates, established in [7], for solutions of
the Dirichlet problem for the Stokes equations in a bounded Lipschitz domain.
∎

######  Lemma 2.2.

Fix $1\leq j\leq d$ and $1\leq\ell\leq L$. There exist 1-periodic functions
$\phi_{kij}^{\ell}(y)$, $i,m=1,2,\dots,d$, such that $\phi_{kij}^{\ell}\in
H^{1}(Y)$, $\int_{Y}\phi_{kij}^{\ell}\,dy=0$,

$\frac{\partial}{\partial
y_{k}}\left(\phi_{kij}^{\ell}\right)=W^{\ell}_{ij}-K^{\ell}_{ij}\quad\text{
and }\quad\phi_{kij}^{\ell}=-\phi_{ikj}^{\ell},$ (2.7)

where the repeated index $k$ is summed from $1$ to $d$. Moreover,

$\int_{\partial D}|\phi^{\ell}_{kij}(x/\varepsilon)|^{2}\,d\sigma\leq C,$
(2.8)

where $D$ is a bounded Lipschitz domain in $\mathbb{R}^{d}$ and $C$ depends on
$D$.

###### Proof.

See [15, Lemma 5.3] for the proof of (2.7). The estimate (2.8) follows from
the observation,

$\displaystyle\|\nabla\phi_{kij}^{\ell}\|_{L^{2}(Y)}+\|\phi_{kij}^{\ell}\|_{L^{2}(Y)}$
$\displaystyle\leq
C\|\nabla^{2}h_{ij}^{\ell}\|_{L^{2}(Y)}+C\|\nabla^{2}h_{kj}^{\ell}\|_{L^{2}(Y)}$
$\displaystyle\leq C\|W_{j}^{\ell}\|_{L^{2}(Y)}\leq C.$

∎

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{d}$ and
$\\{\Omega^{\ell}:1\leq\ell\leq L\\}$ be disjoint subdomains of $\Omega$, each
with Lipschitz boundary, and satisfying the condition,

$\overline{\Omega}=\cup_{\ell=1}^{L}\overline{\Omega^{\ell}}.$ (2.9)

Define

$K=\sum_{\ell=1}^{L}K^{\ell}\chi_{\Omega^{\ell}},$ (2.10)

where $K^{\ell}$ is given by (2.2) and $\chi_{\Omega^{\ell}}$ denotes the
characteristic function of $\Omega^{\ell}$.

######  Lemma 2.3.

Let $f\in L^{2}(\Omega;\mathbb{R}^{d})$. Then there exists $P_{0}\in
H^{1}(\Omega)$, unique up to constants, such that

$\left\\{\begin{aligned} \text{\rm div}\left(K(f-\nabla
P_{0})\right)&=0&\quad&\text{ in }\Omega,\\\ n\cdot K(f-\nabla
P_{0})&=0&\quad&\text{ on }\partial\Omega,\end{aligned}\right.$ (2.11)

in the sense that

$\int_{\Omega}K(f-\nabla P_{0})\cdot\nabla\varphi\,dx=0$ (2.12)

for any $\varphi\in H^{1}(\Omega)$.

###### Proof.

This is standard since the coefficient matrix $K$ is positive definite in each
subdomain $\Omega^{\ell}$ and thus in $\Omega$. ∎

For each $1\leq\ell\leq L$ and $0<\varepsilon<1$, let
$\Omega^{\ell}_{\varepsilon}$ be the perforated domain defined by (1.3), using
$Y_{s}^{\ell}$. Let $\Omega_{\varepsilon}$ be given by (1.4). Note that

$\partial\Omega_{\varepsilon}=\partial\Omega\cup\Gamma_{\varepsilon},$ (2.13)

where $\Gamma_{\varepsilon}=\cup_{\ell=1}^{L}{\Gamma_{\varepsilon}^{\ell}}$
and $\Gamma_{\varepsilon}^{\ell}$ consists of the boundaries of holes
$\varepsilon(Y_{s}^{\ell}+z)$ that are removed from $\Omega^{\ell}$.

######  Lemma 2.4.

Let $u\in H^{1}(\Omega_{\varepsilon})$ with $u=0$ on $\Gamma_{\varepsilon}$.
Assume $\Gamma^{\ell}_{\varepsilon}\neq\emptyset$ for all $1\leq\ell\leq L$.
Then

$\|u\|_{L^{2}(\Omega_{\varepsilon})}\leq C\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}.$ (2.14)

###### Proof.

It follows from Lemma 2.2 in [15] that for $1\leq\ell\leq L$,

$\|u\|^{2}_{L^{2}(\Omega_{\varepsilon}^{\ell})}\leq C\varepsilon^{2}\|\nabla
u\|^{2}_{L^{2}(\Omega_{\varepsilon}^{\ell})},$

which yields (2.14) by summation. Note that we do not assume $u=0$ on
$\partial\Omega^{\ell}$. ∎

From now on we will assume that $\varepsilon>0$ is sufficiently small so that
$\Gamma_{\varepsilon}^{\ell}\neq\emptyset$ for all $1\leq\ell\leq L$. The main
results in this paper are only relevant for small $\varepsilon$.

######  Lemma 2.5.

Let $\Omega$ be a bounded Lipschitz domain and $\Omega_{\varepsilon}$ be given
by (1.4). There exists a bounded linear operator,

$R_{\varepsilon}:H^{1}(\Omega;\mathbb{R}^{d})\to
H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d}),$ (2.15)

such that

$\left\\{\begin{aligned} &R_{\varepsilon}(u)=0\quad\text{ on
}\Gamma_{\varepsilon}\quad\text{ and }\quad R_{\varepsilon}(u)=u\quad\text{ on
}\partial\Omega,\\\ &R_{\varepsilon}(u)\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})\quad\text{ if }\ u\in
H_{0}^{1}(\Omega;\mathbb{R}^{d}),\\\ &R_{\varepsilon}(u)=u\quad\text{ in
}\Omega\quad\text{ if }\ u=0\quad\text{ on }\Gamma_{\varepsilon},\\\
&\text{\rm div}(R_{\varepsilon}(u))=\text{\rm div}(u)\quad\text{ in
}\Omega_{\varepsilon}\ \text{ if \ \ }\text{\rm div}(u)=0\quad\text{ in
}\Omega\setminus\Omega_{\varepsilon},\end{aligned}\right.$ (2.16)

and

$\varepsilon\|\nabla
R_{\varepsilon}(u)\|_{L^{2}(\Omega_{\varepsilon})}+\|R_{\varepsilon}(u)\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\left\\{\varepsilon\|\nabla
u\|_{L^{2}(\Omega)}+\|u\|_{L^{2}(\Omega)}\right\\}.$ (2.17)

Moreover,

$\|\text{\rm div}(R_{\varepsilon}(u))\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\|\text{\rm div}(u)\|_{L^{2}(\Omega)}.$ (2.18)

###### Proof.

A proof for the case $L=1$, which is similar to that of a lemma due to Tartar
(in an appendix of [13]), may be found in [15, Lemma 2.3]. Also see [10, 1].
The same proof works equally well for the case $L\geq 2$. Indeed, let $u\in
H^{1}(\Omega;\mathbb{R}^{d})$. For each $\varepsilon(Y+z)\subset\Omega^{\ell}$
with $1\leq\ell\leq L$ and $z\in\mathbb{Z}^{d}$, we define
$R_{\varepsilon}(u)$ on $\varepsilon(Y_{f}^{\ell}+z)$ by the Dirichlet
problem,

$\left\\{\begin{aligned} -\varepsilon^{2}\Delta R_{\varepsilon}(u)+\nabla
q&=-\varepsilon^{2}\Delta u&\quad&\text{ in }\varepsilon(Y_{f}^{\ell}+z),\\\
\text{\rm div}(R_{\varepsilon}(u))&=\text{\rm
div}(u)+\frac{1}{|\varepsilon(Y_{f}^{\ell}+z)|}\int_{\varepsilon(Y_{s}^{\ell}+z)}\text{\rm
div}(u)\,dx&\quad&\text{ in }\varepsilon(Y_{f}^{\ell}+z),\\\
R_{\varepsilon}(u)&=0&\quad&\text{ on
}\partial(\varepsilon(Y_{s}^{\ell}+z)),\\\ R_{\varepsilon}(u)&=u&\quad&\text{
on }\partial(\varepsilon(Y+z)).\end{aligned}\right.$ (2.19)

If $x\in\Omega_{\varepsilon}$ and $x\notin\varepsilon(Y_{f}+z)$ for any
$\varepsilon(Y+z)\subset\Omega^{\ell}$, we let $R_{\varepsilon}(u)=u$. ∎

######  Lemma 2.6.

Let $f\in L^{2}(\Omega_{\varepsilon})$ with
$\int_{\Omega_{\varepsilon}}f\,dx=0$. Then there exists $u_{\varepsilon}\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ such that $\text{\rm
div}(u_{\varepsilon})=f$ in $\Omega_{\varepsilon}$ and

$\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\|f\|_{L^{2}(\Omega_{\varepsilon})}.$ (2.20)

###### Proof.

Let $F$ be the zero extension of $f$ to $\Omega$. Since $F\in L^{2}(\Omega)$
and $\int_{\Omega}F\,dx=0$, there exists $u\in
H_{0}^{1}(\Omega;\mathbb{R}^{d})$ such that div$(u)=F$ in $\Omega$ and
$\|u\|_{L^{2}(\Omega)}+\|\nabla u\|_{L^{2}(\Omega)}\leq
C\|F\|_{L^{2}(\Omega)}$. Let $u_{\varepsilon}=R_{\varepsilon}(u)$. Then
$u_{\varepsilon}\in H_{0}^{1}(\Omega_{\varepsilon},\mathbb{R}^{d})$, and by
(2.17),

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

Since div$(u)=F=0$ in $\Omega\setminus\Omega_{\varepsilon}$, by the last line
in (2.16), we obtain div$(u_{\varepsilon})=\text{\rm div}(u)=f$ in
$\Omega_{\varepsilon}$. ∎

For $p\in L^{2}(\Omega_{\varepsilon})$, as in [10], we define an extension $P$
of $p$ to $L^{2}(\Omega)$ by

$P(x)=\left\\{\begin{aligned} &p(x)&\quad&\text{ if
}x\in\Omega_{\varepsilon},\\\
&\fint_{\varepsilon(Y_{f}^{\ell}+z)}p&\quad&\text{ if
}x\in\varepsilon(Y_{s}^{\ell}+z)\subset\varepsilon(Y+z)\subset\Omega^{\ell}\text{
for some }1\leq\ell\leq L\text{ and }z\in\mathbb{Z}^{d}.\end{aligned}\right.$
(2.21)

######  Lemma 2.7.

Let $p\in L^{2}(\Omega_{\varepsilon})$ and $P$ be its extension given by
(2.21). Then

$\langle\nabla p,R_{\varepsilon}(u)\rangle_{H^{-1}(\Omega_{\varepsilon})\times
H_{0}^{1}(\Omega_{\varepsilon})}=\langle\nabla
P,u\rangle_{H^{-1}(\Omega)\times H_{0}^{1}(\Omega)},$ (2.22)

where $u\in H_{0}^{1}(\Omega;\mathbb{R}^{d})$ and $R_{\varepsilon}(u)$ is
given by Lemma 2.5.

###### Proof.

We use an argument found in [10, 1, 2]. Note that if $u\in
H_{0}^{1}(\Omega;\mathbb{R}^{d})$, we have $R_{\varepsilon}(u)\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ and

$\displaystyle|\langle\nabla
p,R_{\varepsilon}(u)\rangle_{H^{-1}(\Omega_{\varepsilon})\times
H^{1}_{0}(\Omega_{\varepsilon})}|$ $\displaystyle=|\langle p,\text{\rm
div}(R_{\varepsilon}(u))\rangle_{L^{2}(\Omega_{\varepsilon})\times
L^{2}(\Omega_{\varepsilon})}|$
$\displaystyle\leq\|p\|_{L^{2}(\Omega_{\varepsilon})}\|\text{\rm
div}(R_{\varepsilon}(u))\|_{L^{2}(\Omega_{\varepsilon})}$ $\displaystyle\leq
C\|p\|_{L^{2}(\Omega_{\varepsilon})}\|\text{\rm div}(u)\|_{L^{2}(\Omega)},$

where we have used the estimate (2.18) for the last inequality. Thus there
exists $\Lambda\in H^{-1}(\Omega;\mathbb{R}^{d})$ such that

$\langle\nabla p,R_{\varepsilon}(u)\rangle_{H^{-1}(\Omega_{\varepsilon})\times
H_{0}^{1}(\Omega_{\varepsilon})}=\langle\Lambda,u\rangle_{H^{-1}(\Omega)\times
H_{0}^{1}(\Omega)}$

for any $u\in H_{0}^{1}(\Omega;\mathbb{R}^{d})$. Since
$\langle\Lambda,u\rangle=0$ if div$(u)=0$ in $\Omega$, it follows that
$\Lambda=\nabla Q$ for some $Q\in L^{2}(\Omega)$.

Next, using the fact that $R_{\varepsilon}(u)=u$ for $u\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$, we obtain

$\langle\nabla p-\nabla Q,u\rangle_{H^{-1}(\Omega_{\varepsilon})\times
H_{0}^{1}(\Omega_{\varepsilon})}=0$

for any $u\in H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$. This implies
that $p-Q$ is constant in $\Omega_{\varepsilon}$. Since $Q$ is only determined
up to a constant, we may assume that $Q=p$ in $\Omega_{\varepsilon}$.
Moreover, we note that if $\varepsilon(Y+z)\subset\Omega^{\ell}$ for some
$1\leq\ell\leq L$ and $z\in\mathbb{Z}^{d}$, and $u\in
C_{0}^{1}(\varepsilon(Y_{s}^{\ell}+z),\mathbb{R}^{d})$, then
$R_{\varepsilon}(u)=0$ in $\Omega_{\varepsilon}$. It follows that $\nabla Q=0$
in $\varepsilon(Y_{s}^{\ell}+z)$. Thus $Q$ is constant in each
$\varepsilon(Y_{s}^{\ell}+z)$.

Finally, for any $u\in C_{0}^{1}(\varepsilon(Y+z);\mathbb{R}^{d})$ with
$\varepsilon(Y+z)\subset\Omega^{\ell}$, we have

$R_{\varepsilon}(u)\in H_{0}^{1}(\varepsilon(Y_{f}^{\ell}+z);\mathbb{R}^{d}),$

and by (2.19),

$\text{\rm div}(R_{\varepsilon}(u))=\text{\rm
div}(u)+\frac{1}{|\varepsilon(Y_{f}^{\ell}+z)|}\int_{\varepsilon(Y_{s}^{\ell}+z)}\text{\rm
div}(u)\,dx$

in $\varepsilon(Y_{f}^{\ell}+z)$. This, together with

$\int_{\varepsilon(Y_{f}^{\ell}+z)}p\cdot\text{\rm
div}(R_{\varepsilon}(u))\,dx=\int_{\varepsilon(Y+z)}Q\cdot\text{\rm
div}(u)\,dx$

and the fact that $Q=p$ in $\Omega_{\varepsilon}$, yields

$\int_{\varepsilon(Y_{s}^{\ell}+z)}\Big{(}Q-\fint_{\varepsilon(Y_{f}^{\ell}+z)}p\Big{)}\text{\rm
div}(u)\,dx=0.$

Consequently,

$Q=\fint_{\varepsilon(Y_{f}^{\ell}+z)}p\qquad\text{ in
}\varepsilon(Y_{s}^{\ell}+z).$

As a result, we have proved that $Q=P$, an extension of $p$ given by (2.21). ∎

##  3 Energy estimates

Let $\Omega_{\varepsilon}$ be given by (1.4). Recall that
$\partial\Omega_{\varepsilon}=\partial\Omega\cup\Gamma_{\varepsilon}$, where
$\Gamma_{\varepsilon}$ consists of the boundaries of the holes of size
$\varepsilon$ that are removed from $\Omega$. In this section we establish the
energy estimates for the Dirichlet problem,

$\left\\{\begin{aligned} -\varepsilon^{2}\Delta u_{\varepsilon}+\nabla
p_{\varepsilon}&=f+\varepsilon\,\text{\rm div}(F)&\quad&\text{ in
}\Omega_{\varepsilon},\\\ \text{\rm div}(u_{\varepsilon})&=g&\quad&\text{ in
}\Omega_{\varepsilon},\\\ u_{\varepsilon}&=0&\quad&\text{ on
}\Gamma_{\varepsilon},\\\ u_{\varepsilon}&=h&\quad&\text{ on
}\partial\Omega,\end{aligned}\right.$ (3.1)

where $(g,h)$ satisfies the compatibility condition,

$\int_{\Omega_{\varepsilon}}g\,dx=\int_{\partial\Omega}h\cdot n\,d\sigma.$
(3.2)

Throughout this section we assume that $\Omega$, $\Omega^{\ell}$ and
$Y_{s}^{\ell}$ for $1\leq\ell\leq L$ are domains with Lipschitz boundaries. We
use $L^{2}_{0}(\Omega_{\varepsilon})$ to denote the subspace of functions in
$L^{2}(\Omega_{\varepsilon})$ with mean value zero.

######  Theorem 3.1.

Let $f\in L^{2}(\Omega_{\varepsilon};\mathbb{R}^{d})$ and $F\in
L^{2}(\Omega_{\varepsilon};\mathbb{R}^{d\times d})$. Let $g\in
L^{2}(\Omega_{\varepsilon})$ and $h\in H^{1/2}(\partial\Omega;\mathbb{R}^{d})$
satisfy the condition (3.2). Let $(u_{\varepsilon},p_{\varepsilon})\in
H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})\times
L^{2}_{0}(\Omega_{\varepsilon})$ be a weak solution of (3.1). Then

$\displaystyle\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$
(3.3) $\displaystyle\leq
C\left\\{\|f\|_{L^{2}(\Omega_{\varepsilon})}+\|F\|_{L^{2}(\Omega_{\varepsilon})}+\|g\|_{L^{2}(\Omega_{\varepsilon})}+\|H\|_{L^{2}(\Omega)}+\|\text{\rm
div}(H)\|_{L^{2}(\Omega)}+\varepsilon\|\nabla H\|_{L^{2}(\Omega)}\right\\},$

where $H$ is any function in $H^{1}(\Omega;\mathbb{R}^{d})$ with the property
$H=h$ on $\partial\Omega$.

###### Proof.

This theorem was proved in [15, Section 3] for the case $L=1$. The proof for
the case $L\geq 2$ is similar. We provide a proof here for the reader’s
convenience.

Step 1. We show that

$\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\left\\{\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|f\|_{L^{2}(\Omega_{\varepsilon})}+\|F\|_{L^{2}(\Omega_{\varepsilon})}\right\\}.$
(3.4)

To this end we use Lemma 2.6 to find $v_{\varepsilon}\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ such that $\text{\rm
div}(v_{\varepsilon})=p_{\varepsilon}$ in $\Omega_{\varepsilon}$ and

$\varepsilon\|\nabla
v_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|v_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}.$ (3.5)

By using $v_{\varepsilon}$ as a test function we obtain

$\displaystyle\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}^{2}$
$\displaystyle\leq\varepsilon^{2}\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\|\nabla
v_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|f\|_{L^{2}(\Omega_{\varepsilon})}\|v_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\varepsilon\|F\|_{L^{2}(\Omega_{\varepsilon})}\|\nabla
v_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$ $\displaystyle\leq
C\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\left\\{\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|f\|_{L^{2}(\Omega_{\varepsilon})}+\|F\|_{L^{2}(\Omega_{\varepsilon})}\right\\},$

where we have used (3.5) for the last inequality. This yields (3.4).

Step 2. We prove (3.3) in the case $h=0$. In this case we may use
$u_{\varepsilon}\in H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ as a test
function to obtain

$\displaystyle\varepsilon^{2}\|\nabla
u_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\varepsilon})}\leq\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\|g\|_{L^{2}(\Omega_{\varepsilon})}+\|f\|_{L^{2}(\Omega_{\varepsilon})}\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\varepsilon\|F\|_{L^{2}(\Omega_{\varepsilon})}\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}.$

By using the Cauchy inequality as well as the estimate
$\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq C\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$, we deduce that

$\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\left\\{\|p_{\varepsilon}\|^{1/2}_{L^{2}(\Omega_{\varepsilon})}\|g\|^{1/2}_{L^{2}(\Omega_{\varepsilon})}+\|f\|_{L^{2}(\Omega_{\varepsilon})}+\|F\|_{L^{2}(\Omega_{\varepsilon})}\right\\}.$

This, together with (3.4), gives (3.3) for the case $h=0$.

Step 3. We consider the general case $h\in
H^{1/2}(\partial\Omega;\mathbb{R}^{d})$. Let $H$ be a function in
$H^{1}(\Omega;\mathbb{R}^{d})$ such that $H=h$ on $\partial\Omega$. Let
$w_{\varepsilon}=u_{\varepsilon}-R_{\varepsilon}(H)$, where
$R_{\varepsilon}(H)$ is given by Lemma 2.5. Then $w_{\varepsilon}\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ and

$\left\\{\begin{aligned} -\varepsilon^{2}\Delta w_{\varepsilon}+\nabla
p_{\varepsilon}&=f+\varepsilon\,\text{\rm div}(F)+\varepsilon^{2}\Delta
R_{\varepsilon}(H),\\\ \text{\rm div}(w_{\varepsilon})&=g-\text{\rm
div}(R_{\varepsilon}(H)),\end{aligned}\right.$

in $\Omega_{\varepsilon}$. By Step 2 we obtain

$\displaystyle\varepsilon\|\nabla
w_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|w_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$
$\displaystyle\leq
C\left\\{\|f\|_{L^{2}(\Omega_{\varepsilon})}+\|F\|_{L^{2}(\Omega_{\varepsilon})}+\varepsilon\|\nabla
R_{\varepsilon}(H)\|_{L^{2}(\Omega_{\varepsilon})}+\|g\|_{L^{2}(\Omega_{\varepsilon})}+\|\text{\rm
div}(R_{\varepsilon}(H))\|_{L^{2}(\Omega_{\varepsilon})}\right\\}.$

It follows that

$\displaystyle\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$
$\displaystyle\leq
C\Big{\\{}\|f\|_{L^{2}(\Omega_{\varepsilon})}+\|F\|_{L^{2}(\Omega_{\varepsilon})}+\|g\|_{L^{2}(\Omega_{\varepsilon})}$
$\displaystyle\qquad\qquad+\varepsilon\|\nabla
R_{\varepsilon}(H)\|_{L^{2}(\Omega_{\varepsilon})}+\|R_{\varepsilon}(H)\|_{L^{2}(\Omega_{\varepsilon})}+\|\text{\rm
div}(R_{\varepsilon}(H))\|_{L^{2}(\Omega_{\varepsilon})}\Big{\\}}$
$\displaystyle\leq
C\Big{\\{}\|f\|_{L^{2}(\Omega_{\varepsilon})}+\|F\|_{L^{2}(\Omega_{\varepsilon})}+\|g\|_{L^{2}(\Omega_{\varepsilon})}+\varepsilon\|\nabla
H\|_{L^{2}(\Omega)}+\|H\|_{L^{2}(\Omega)}+\|\text{\rm
div}(H)\|_{L^{2}(\Omega)}\Big{\\}},$

where we have used estimates (2.17) and (2.18) for the last inequality. ∎

######  Corollary 3.2.

Let $(u_{\varepsilon},p_{\varepsilon})$ be the same as in Theorem 3.1. Then

$\displaystyle\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$
(3.6) $\displaystyle\leq
C\Big{\\{}\|f\|_{L^{2}(\Omega_{\varepsilon})}+\|F\|_{L^{2}(\Omega_{\varepsilon})}+\|g\|_{L^{2}(\Omega_{\varepsilon})}+\|h\|_{L^{2}(\partial\Omega)}+\varepsilon\|h\|_{H^{1/2}(\partial\Omega)}\Big{\\}}.$

###### Proof.

For $h\in H^{1/2}(\partial\Omega;\mathbb{R}^{d})$, let $H$ be the weak
solution in $H^{1}(\Omega;\mathbb{R}^{d})$ of the Dirichlet problem,

$\left\\{\begin{aligned} -\Delta H+\nabla q&=0&\quad&\text{ in }\Omega,\\\
\text{\rm div}(H)&=\gamma&\quad&\text{ in }\Omega,\\\ u&=h&\quad&\text{ on
}\partial\Omega,\end{aligned}\right.$

where the constant

$\gamma=\frac{1}{|\Omega|}\int_{\partial\Omega}h\cdot n\,d\sigma$

is chosen so that the compatibility condition (3.2) is satisfied. Note that

$\|\text{\rm div}(H)\|_{L^{2}(\Omega)}=C|\gamma|\leq
C\|h\|_{L^{2}(\partial\Omega)},$

and by the standard energy estimates, $\|\nabla H\|_{L^{2}(\Omega)}\leq
C\|h\|_{H^{1/2}(\partial\Omega)}.$ In view of (3.3) we only need to show that

$\|H\|_{L^{2}(\Omega)}\leq C\|h\|_{L^{2}(\partial\Omega)}.$ (3.7)

To this end, let

$H_{1}=H-\gamma(x-x_{0})/d,$

where $x_{0}\in\Omega$. Since $-\Delta H_{1}+\nabla q=0$ and div$(H_{1})=0$ in
$\Omega$, it follows from [7] that

$\displaystyle\|H_{1}\|_{L^{2}(\Omega)}$ $\displaystyle\leq
C\|(H_{1})^{*}\|_{L^{2}(\partial\Omega)}$ $\displaystyle\leq
C\|H_{1}\|_{L^{2}(\partial\Omega)}\leq C\|h\|_{L^{2}(\partial\Omega)},$

where $(H_{1})^{*}$ denotes the nontangential maximal function of $H_{1}$. As
a result, we obtain

$\displaystyle\|H\|_{L^{2}(\Omega)}$
$\displaystyle\leq\|H_{1}\|_{L^{2}(\Omega)}+C|\gamma|$ $\displaystyle\leq
C\|h\|_{L^{2}(\partial\Omega)},$

which completes the proof. ∎

######  Corollary 3.3.

Let $(u_{\varepsilon},p_{\varepsilon})$ be the same as in Theorem 3.1. Let
$P_{\varepsilon}$ be the extension of $p_{\varepsilon}$, defined by (2.21).
Then

$\|P_{\varepsilon}\|_{L^{2}(\Omega)}\leq
C\left\\{\|f\|_{L^{2}(\Omega_{\varepsilon})}+\|F\|_{L^{2}(\Omega_{\varepsilon})}+\|g\|_{L^{2}(\Omega_{\varepsilon})}+\|h\|_{L^{2}(\partial\Omega)}+\varepsilon\|h\|_{H^{1/2}(\partial\Omega)}\right\\}.$
(3.8)

###### Proof.

By the definition of $P_{\varepsilon}$, we have

$\displaystyle\int_{\Omega}|P_{\varepsilon}|^{2}\,dx$
$\displaystyle=\int_{\Omega_{\varepsilon}}|p_{\varepsilon}|^{2}\,dx+\sum_{\ell=1}^{L}\sum_{z}|\varepsilon(Y_{s}^{\ell}+z)|\Big{(}\fint_{\varepsilon(Y^{\ell}_{f}+z)}p_{\varepsilon}\Big{)}^{2}$
$\displaystyle\leq\sum_{\ell=1}^{L}\frac{1}{|Y_{f}^{\ell}|}\int_{\Omega_{\varepsilon}^{\ell}}|p_{\varepsilon}|^{2}\,dx,$

which, together with (3.6), gives (3.8). ∎

##  4 Homogenization and proof of Theorem 1.1

Let $f\in L^{2}(\Omega;\mathbb{R}^{d})$ and $h\in
H^{1/2}(\partial\Omega;\mathbb{R}^{d})$ with $\int_{\partial\Omega}h\cdot
n\,d\sigma=0$, where $n$ denotes the outward unit normal to $\partial\Omega$.
Consider the Dirichlet problem,

$\left\\{\begin{aligned} -\varepsilon^{2}\Delta u_{\varepsilon}+\nabla
p_{\varepsilon}&=f&\quad&\text{ in }\Omega_{\varepsilon},\\\ \text{\rm
div}(u_{\varepsilon})&=0&\quad&\text{ in }\Omega_{\varepsilon},\\\
u_{\varepsilon}&=0&\quad&\text{ on }\Gamma_{\varepsilon},\\\
u_{\varepsilon}&=h&\quad&\text{ on }\partial\Omega,\end{aligned}\right.$ (4.1)

where $\Omega_{\varepsilon}$ is given by (1.4) and
$\partial\Omega_{\varepsilon}=\partial\Omega\cup\Gamma_{\varepsilon}$.
Throughout the section we assume that $\Omega$, $\Omega^{\ell}$ and
$Y_{s}^{\ell}$ for $1\leq\ell\leq L$, are domains with Lipschitz boundaries.
As before, we extend $u_{\varepsilon}$ to $\Omega$ by zero and still denote
the extension by $u_{\varepsilon}$. We use $P_{\varepsilon}$ to denote the
extension of $p_{\varepsilon}$ to $\Omega$, given by (2.21). The goal of this
section is to prove the following theorem, which contains Theorem 1.1 as a
special case $h=0$.

######  Theorem 4.1.

Let $f\in L^{2}(\Omega;\mathbb{R}^{d})$ and $h\in
H^{1/2}(\partial\Omega;\mathbb{R}^{d})$ with $\int_{\partial\Omega}h\cdot
n\,d\sigma=0$. Let $(u_{\varepsilon},p_{\varepsilon})\in
H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})\times
L_{0}^{2}(\Omega_{\varepsilon})$ be the weak solution of (4.1). Let
$(u_{\varepsilon},P_{\varepsilon})$ be the extension of
$(u_{\varepsilon},p_{\varepsilon})$. Then $u_{\varepsilon}\to u_{0}$ weakly in
$L^{2}(\Omega;\mathbb{R}^{d})$ and
$P_{\varepsilon}-\fint_{\Omega}P_{\varepsilon}\to P_{0}$ strongly in
$L^{2}(\Omega)$, as $\varepsilon\to 0$, where $P_{0}\in H^{1}(\Omega)$,
$\int_{\Omega}P_{0}\,dx=0$, $(u_{0},P_{0})$ is governed by a Darcy law,

$\left\\{\begin{aligned} u_{0}&=K(f-\nabla P_{0})&\quad&\text{ in }\Omega,\\\
\text{\rm div}(u_{0})&=0&\quad&\text{ in }\Omega,\\\ u_{0}\cdot n&=h\cdot
n&\quad&\text{ on }\partial\Omega,\end{aligned}\right.$ (4.2)

with the permeability matrix $K$ given by (1.10).

We begin with the strong convergence of $P_{\varepsilon}$.

######  Lemma 4.2.

Let $(u_{\varepsilon_{k}},p_{\varepsilon_{k}})$ be a weak solution of (4.1)
with $\varepsilon=\varepsilon_{k}$. Suppose that as $\varepsilon_{k}\to 0$,
$P_{\varepsilon_{k}}\to P$ weakly in $L^{2}(\Omega)$ for some $P\in
L^{2}(\Omega)$. Then $P_{\varepsilon_{k}}\to P$ strongly in $L^{2}(\Omega)$.

###### Proof.

The proof is similar to that for the classical case $L=1$ (see e.g. [4]). One
argues by contradiction. Suppose that $P_{\varepsilon_{k}}$ does not converge
strongly to $P$ in $L^{2}(\Omega)$. Since

$\|\nabla P_{\varepsilon_{k}}-\nabla
P\|_{H^{-1}(\Omega)}\sim\|P_{\varepsilon_{k}}-P-\fint_{\Omega}(P_{\varepsilon_{k}}-P)\|_{L^{2}(\Omega)}$

and $\int_{\Omega}P_{\varepsilon_{k}}\,dx\to\int_{\Omega}P\,dx$, it follows
that $\nabla P_{\varepsilon_{k}}$ does not converge to $\nabla P$ strongly in
$H^{-1}(\Omega;\mathbb{R}^{d})$. By passing to a subsequence, this implies
that there exists a sequence $\\{\psi_{k}\\}\subset
H_{0}^{1}(\Omega;\mathbb{R}^{d})$ such that
$\|\psi_{k}\|_{H^{1}_{0}(\Omega)}=1$ and

$|\langle\nabla P_{\varepsilon_{k}}-\nabla
P,\psi_{k}\rangle_{H^{-1}(\Omega)\times H^{1}_{0}(\Omega)}|\geq c_{0}>0.$

By passing to another subsequence, we may assume that $\psi_{k}\to\psi_{0}$
weakly in $H_{0}^{1}(\Omega;\mathbb{R}^{d})$. Let
$\varphi_{k}=\psi_{k}-\psi_{0}$. Using $P_{\varepsilon_{k}}\to P$ weakly in
$L^{2}(\Omega)$, we obtain

$|\langle\nabla P_{\varepsilon_{k}}-\nabla
P,\varphi_{k}\rangle_{H^{-1}(\Omega)\times H^{1}_{0}(\Omega)}|\geq c_{0}/2,$
(4.3)

if $k$ is sufficiently large. Since $\varphi_{k}\to 0$ weakly in
$H^{1}_{0}(\Omega;\mathbb{R}^{d})$, we may conclude further that

$|\langle\nabla P_{\varepsilon_{k}},\varphi_{k}\rangle_{H^{-1}(\Omega)\times
H^{1}_{0}(\Omega)}|\geq c_{0}/4,$ (4.4)

if $k$ is sufficiently large. On the other hand, by (2.7), we have

$\displaystyle|\langle\nabla
P_{\varepsilon_{k}},\varphi_{k}\rangle_{H^{-1}(\Omega)\times
H_{0}^{1}(\Omega)}|=|\langle\nabla
p_{\varepsilon_{k}},R_{\varepsilon_{k}}(\varphi_{k})\rangle_{H^{-1}(\Omega_{\varepsilon_{k}})\times
H_{0}^{1}(\Omega_{\varepsilon_{k}})}|$ (4.5)
$\displaystyle=|\langle\varepsilon_{k}^{2}\Delta
u_{\varepsilon_{k}}+f,R_{\varepsilon_{k}}(\varphi_{k})\rangle_{H^{-1}(\Omega_{\varepsilon_{k}})\times
H_{0}^{1}(\Omega_{\varepsilon_{k}})}|$
$\displaystyle\leq\varepsilon_{k}^{2}\|\nabla
u_{\varepsilon_{k}}\|_{L^{2}(\Omega_{\varepsilon_{k}})}\|\nabla
R_{\varepsilon_{k}}(\varphi_{k})\|_{L^{2}(\Omega_{\varepsilon_{k}})}+\|f\|_{L^{2}(\Omega)}\|R_{\varepsilon_{k}}(\varphi_{k})\|_{L^{2}(\Omega_{\varepsilon_{k}})}$
$\displaystyle\leq
C\left(\|f\|_{L^{2}(\Omega)}+\|h\|_{H^{1/2}(\partial\Omega)}\right)\left(\varepsilon_{k}\|\nabla
R_{\varepsilon_{k}}(\varphi_{k})\|_{L^{2}(\Omega_{\varepsilon_{k}})}+\|R_{\varepsilon_{k}}(\varphi_{k})\|_{L^{2}(\Omega_{\varepsilon_{k}})}\right)$
$\displaystyle\leq
C\left(\|f\|_{L^{2}(\Omega)}+\|h\|_{H^{1/2}(\partial\Omega)}\right)\left(\varepsilon_{k}\|\nabla\varphi_{k}\|_{L^{2}(\Omega)}+\|\varphi_{k}\|_{L^{2}(\Omega)}\right),$

where we have used the estimate (3.6) for the second inequality and (2.17) for
the last. This contradicts with (4.4) as the right-hand side of (4.5) goes to
zero. ∎

By Corollaries 3.2 and 3.3, the sets $\\{u_{\varepsilon}:0<\varepsilon<1\\}$
and $\\{P_{\varepsilon}:0<\varepsilon<1\\}$ are bounded in
$L^{2}(\Omega;\mathbb{R}^{d})$ and $L^{2}(\Omega)$, respectively. It follows
that for any sequence $\varepsilon_{k}\to 0$, there exists a subsequence,
still denoted by $\\{\varepsilon_{k}\\}$, such that $u_{\varepsilon_{k}}\to u$
and $P_{\varepsilon_{k}}\to P$ weakly in $L^{2}(\Omega;\mathbb{R}^{d})$ and
$L^{2}(\Omega)$, respectively. By Lemma 4.2, $P_{\varepsilon_{k}}\to P$
strongly in $L^{2}(\Omega)$. Thus, as in the classical case $L=1$, to prove
Theorem 4.1, it suffices to show that if $\varepsilon_{k}\to 0$,
$u_{\varepsilon_{k}}\to u$ weakly in $L^{2}(\Omega;\mathbb{R}^{d})$, and
$P_{\varepsilon_{k}}\to P$ strongly in $L^{2}(\Omega)$, then $P\in
H^{1}(\Omega)$ and $(u,P)$ is a weak solution of (4.2). Since the solution of
(4.2) is unique under the conditions that $P_{0}\in H^{1}(\Omega)$ and
$\int_{\Omega}P_{0}\,dx=0$, one concludes that as $\varepsilon\to 0$,
$u_{\varepsilon}\to u_{0}$ weakly in $L^{2}(\Omega;\mathbb{R}^{d})$ and
$P_{\varepsilon}-\fint_{\Omega}P_{\varepsilon}\to P_{0}$ strongly in
$L^{2}(\Omega)$, where $(u_{0},P_{0})$ is the unique solution of (4.2) with
the property $P_{0}\in H^{1}(\Omega)$ and $\int_{\Omega}P_{0}\,dx=0$.

######  Lemma 4.3.

Let $\\{\varepsilon_{k}\\}$ be a sequence such that $\varepsilon_{k}\to 0$.
Suppose that $u_{\varepsilon_{k}}\to u$ weakly in
$L^{2}(\Omega;\mathbb{R}^{d})$ and $P_{\varepsilon_{k}}\to P$ strongly in
$L^{2}(\Omega)$. Then $P\in H^{1}(\Omega^{\ell})$ for $1\leq\ell\leq L$ and
$(u,P)$ is a solution of $\eqref{Darcy-4}$.

###### Proof.

Since

$\int_{\Omega}u_{\varepsilon_{k}}\cdot\nabla\varphi\,dx=\int_{\partial\Omega}(h\cdot
n)\varphi\,d\sigma$

for any $\varphi\in C^{\infty}(\mathbb{R}^{d})$, by letting $k\to\infty$, we
see that

$\int_{\Omega}u\cdot\nabla\varphi\,dx=\int_{\partial\Omega}(h\cdot
n)\varphi\,d\sigma$

for any $\varphi\in C^{\infty}(\mathbb{R}^{d})$. It follows that div$(u)=0$ in
$\Omega$ and $u\cdot n=h\cdot n$ on $\partial\Omega$.

Next, we show that $P\in H^{1}(\Omega^{\ell})$ for each subdomain
$\Omega^{\ell}$ and that

$u=K^{\ell}(f-\nabla P)\quad\text{ in }\Omega^{\ell},$ (4.6)

where $K^{\ell}=(K^{\ell}_{ij})$ is defined by (2.2). The argument is the same
as that of Tartar for the case $L=1$ (see [13]). Fix $1\leq\ell\leq L$, $1\leq
j\leq d$, and $\varphi\in C_{0}^{\infty}(\Omega^{\ell})$. We assume $k>1$ is
sufficiently large that
supp$(\varphi)\subset\\{x\in\Omega^{\ell}:\text{dist}(x,\partial\Omega^{\ell})\geq
C_{d}\varepsilon_{k}\\}$. Let $(W_{j}^{\ell}(y),\pi_{j}^{\ell}(y))$ be the
1-periodic functions given by (2.1). By using
$W_{j}^{\ell}(x/\varepsilon_{k})\varphi$ as a test function, we obtain

$\displaystyle\varepsilon_{k}\int_{\Omega^{\ell}}\nabla
u_{\varepsilon_{k}}\cdot\nabla
W_{j}^{\ell}(x/\varepsilon_{k})\varphi\,dx+\varepsilon^{2}_{k}\int_{\Omega^{\ell}}\nabla
u_{\varepsilon_{k}}\cdot
W_{j}^{\ell}(x/\varepsilon_{k})\nabla\varphi\,dx-\int_{\Omega^{\ell}}P_{\varepsilon_{k}}W_{j}^{\ell}(x/\varepsilon_{k})\cdot\nabla\varphi\,dx$
(4.7) $\displaystyle=\int_{\Omega^{\ell}}f\cdot
W_{j}^{\ell}(x/\varepsilon_{k})\varphi\,dx,$

where we have used the facts that $\text{\rm
div}(W^{\ell}_{j}(x/\varepsilon))=0$ in $\mathbb{R}^{d}$ and
$W_{j}^{\ell}(x/\varepsilon)=0$ on $\Gamma_{\varepsilon}$. Since
$W_{ij}^{\ell}(x/\varepsilon_{k})\to K^{\ell}_{ij}$ weakly in
$L^{2}(\Omega^{\ell})$ and $P_{\varepsilon_{k}}\to P$ strongly in
$L^{2}(\Omega^{\ell})$, we deduce from (4.7) that

$\lim_{k\to\infty}\varepsilon_{k}\int_{\Omega^{\ell}}\nabla
u_{\varepsilon_{k}}\cdot\nabla
W_{j}^{\ell}(x/\varepsilon_{k})\varphi\,dx=\int_{\Omega^{\ell}}PK^{\ell}_{ij}\frac{\partial\varphi}{\partial
x_{i}}\,dx+\int_{\Omega^{\ell}}f_{i}K_{ij}^{\ell}\varphi\,dx,$ (4.8)

where the repeated index $i$ is summed from $1$ to $d$.

Note that

$\displaystyle-\varepsilon^{2}\Delta\left(W_{j}^{\ell}(x/\varepsilon)\right)+\nabla\left(\varepsilon\pi_{j}^{\ell}(x/\varepsilon)\right)$
$\displaystyle=e_{j}$

in the set $\\{x\in\Omega^{\ell}_{\varepsilon}:\text{\rm
dist}(x,\partial\Omega^{\ell})\geq c_{d}\varepsilon\\}$. By using
$u_{\varepsilon_{k}}\varphi$ as a test function, we see that

$\displaystyle\varepsilon_{k}\int_{\Omega^{\ell}}\nabla
W_{j}^{\ell}(x/\varepsilon_{k})\cdot(\nabla
u_{\varepsilon_{k}})\varphi\,dx+\varepsilon_{k}\int_{\Omega^{\ell}}\nabla
W_{j}^{\ell}(x/\varepsilon_{k})\cdot u_{\varepsilon_{k}}(\nabla\varphi)\,dx$
(4.9)
$\displaystyle\qquad\qquad-\varepsilon_{k}\int_{\Omega^{\ell}}\pi_{j}^{\ell}(x/\varepsilon_{k})u_{\varepsilon_{k}}(\nabla\varphi)\,dx=\int_{\Omega^{\ell}}e_{j}\cdot
u_{\varepsilon_{k}}\varphi\,dx,$

which leads to

$\lim_{k\to\infty}\varepsilon_{k}\int_{\Omega^{\ell}}\nabla
W_{j}^{\ell}(x/\varepsilon_{k})\cdot(\nabla
u_{\varepsilon_{k}})\varphi\,dx.=\int_{\Omega^{\ell}}e_{j}\cdot u\varphi\,dx.$
(4.10)

In view of (4.8) and (4.10) we obtain

$\int_{\Omega^{\ell}}e_{j}\cdot
u\varphi\,dx=\int_{\Omega^{\ell}}PK^{\ell}_{ij}\frac{\partial\varphi}{\partial
x_{i}}\,dx+\int_{\Omega^{\ell}}f_{i}K_{ij}^{\ell}\varphi\,dx.$

Since $\varphi\in C_{0}^{\infty}(\Omega^{\ell})$ is arbitrary and the constant
matrix $K^{\ell}=(K_{ij}^{\ell})$ is invertible, we conclude that $P\in
H^{1}(\Omega^{\ell})$ and

$u_{j}=K_{ij}^{\ell}\Big{(}f_{i}-\frac{\partial P}{\partial x_{i}}\Big{)}$

in $\Omega^{\ell}$. Since $K^{\ell}$ is also symmetric, this gives (4.6). ∎

To prove the effective pressure in Lemma 4.3 $P\in H^{1}(\Omega)$, it remains
to show that $P$ is continuous across the interface
$\Sigma=\Omega\setminus\cup_{\ell=1}^{L}\Omega^{\ell}$ between subdomains.

######  Lemma 4.4.

Let $f\in C^{m}(B(x_{0},2c\varepsilon);\mathbb{R}^{d})$ for some
$x_{0}\in\mathbb{R}^{d}$, $m\geq 0$ and $c>0$. Suppose that

$\left\\{\begin{aligned} -\varepsilon^{2}\Delta u_{\varepsilon}+\nabla
p_{\varepsilon}&=f&\quad&\text{ in }B(x_{0},2c\varepsilon),\\\ \text{\rm
div}(u_{\varepsilon})&=0&\quad&\text{ in
}B(x_{0},2c\varepsilon).\end{aligned}\right.$ (4.11)

Then

$\displaystyle\varepsilon^{m+2}\left(\fint_{B(x_{0},c\varepsilon)}|\nabla^{m+2}u_{\varepsilon}|^{2}\right)^{1/2}\leq
C\left(\fint_{B(x_{0},2c\varepsilon)}|u_{\varepsilon}|^{2}\right)^{1/2}+C\sum_{k=0}^{m}\varepsilon^{k}\|\nabla^{k}f\|_{\infty},$
(4.12)

where $C$ depends only on $d$, $m$ and $c$.

###### Proof.

The case $\varepsilon=1$ is given by interior estimates for the Stokes
equations. The general case follows by a simple rescaling argument. ∎

Define

$\gamma_{\varepsilon}=\big{\\{}x\in\Sigma:\ \text{\rm
dist}(x,\partial\Omega)\geq\varepsilon\big{\\}},$ (4.13)

where $\Sigma$ is the interface given by (1.5)

######  Lemma 4.5.

Let $(u_{\varepsilon},p_{\varepsilon})$ be a solution of (4.1) with $f\in
C^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})$ and $h\in
H^{1/2}(\partial\Omega;\mathbb{R}^{d})$. Then, for $m\geq 0$,

$\displaystyle\|\nabla^{m}u_{\varepsilon}\|_{L^{2}(\gamma_{\varepsilon})}$
$\displaystyle\leq C(f,h)\varepsilon^{-m-\frac{1}{2}},$ (4.14)
$\displaystyle\|p_{\varepsilon}\|_{L^{2}(\gamma_{\varepsilon})}$
$\displaystyle\leq C(f,h)\varepsilon^{-\frac{1}{2}},$ $\displaystyle\|\nabla
p_{\varepsilon}\|_{L^{2}(\gamma_{\varepsilon})}$ $\displaystyle\leq
C(f,h)\varepsilon^{-\frac{1}{2}},$

where $C(f,h)$ depends on $m$, $f$ and $h$, but not on $\varepsilon$.

###### Proof.

Recall that

$\Sigma=\cup_{\ell=1}^{L}\partial\Omega^{\ell}\setminus\partial\Omega.$

It follows that
$\gamma_{\varepsilon}=\cup_{\ell=1}^{L}\gamma_{\varepsilon}^{\ell}$, where

$\gamma_{\varepsilon}^{\ell}=\big{\\{}x\in\partial\Omega^{\ell}:\
\text{dist}(x,\partial\Omega)\geq\varepsilon\big{\\}}.$

Thus, it suffices to prove (4.14) with $\gamma_{\varepsilon}^{\ell}$ in the
place of $\gamma_{\varepsilon}$. Let

$D^{\ell}_{\varepsilon}=\left\\{x\in\Omega^{\ell}:\text{
dist}(x,\gamma_{\varepsilon}^{\ell})<c\,\varepsilon\right\\}.$

Using the assumption that $\Omega^{\ell}$ is a bounded Lipschitz domain, one
may show that

$\displaystyle\int_{\gamma_{\varepsilon}^{\ell}}|\nabla^{m}u_{\varepsilon}|^{2}\,d\sigma$
$\displaystyle\leq\frac{C}{\varepsilon}\int_{D_{\varepsilon}^{\ell}}|\nabla^{m}u_{\varepsilon}|^{2}\,dx+C\varepsilon\int_{D_{\varepsilon}^{\ell}}|\nabla^{m+1}u_{\varepsilon}|^{2}\,dx$
(4.15)
$\displaystyle\leq\frac{C}{\varepsilon^{1+2m}}\left\\{\int_{\Omega_{\varepsilon}}|u_{\varepsilon}|^{2}\,dx+C(f)\right\\},$

where $C(f)$ depends on $f$. We point out that the second inequality in (4.15)
follows by covering $D_{\varepsilon}^{\ell}$ with balls of radius
$c\varepsilon$ and using (4.12). This, together with the energy estimate
(3.6), yields

$\|\nabla^{m}u_{\varepsilon}\|_{L^{2}(\gamma_{\varepsilon}^{\ell})}\leq
C(f,h)\varepsilon^{-m-\frac{1}{2}},$

where $C(f,h)$ depends on $f$ and $h$. Next, using the equation
$-\varepsilon^{2}\Delta u_{\varepsilon}+\nabla p_{\varepsilon}=f$, we obtain

$\displaystyle\|\nabla p_{\varepsilon}\|_{L^{2}(\gamma_{\varepsilon}^{\ell})}$
$\displaystyle\leq\varepsilon^{2}\|\Delta
u_{\varepsilon}\|_{L^{2}(\gamma_{\varepsilon}^{\ell})}+\|f\|_{L^{2}(\gamma_{\varepsilon}^{\ell})}$
$\displaystyle\leq C(f,h)\varepsilon^{-1/2}.$

Finally, observe that

$\displaystyle\int_{\gamma_{\varepsilon}^{\ell}}|p_{\varepsilon}|^{2}\,d\sigma$
$\displaystyle\leq\frac{C}{\varepsilon}\int_{D_{\varepsilon}^{\ell}}|p_{\varepsilon}|^{2}\,dx+C\varepsilon\int_{D_{\varepsilon}^{\ell}}|\nabla
p_{\varepsilon}|^{2}\,dx$
$\displaystyle\leq\frac{C}{\varepsilon}\int_{\Omega_{\varepsilon}}|p_{\varepsilon}|^{2}\,dx+C\varepsilon^{5}\int_{D_{\varepsilon}^{\ell}}|\Delta
u_{\varepsilon}|^{2}\,dx+C(f),$
$\displaystyle\leq\frac{C}{\varepsilon}\int_{\Omega_{\varepsilon}}|p_{\varepsilon}|^{2}\,dx+C\varepsilon\int_{\Omega_{\varepsilon}}|u_{\varepsilon}|^{2}\,dx+C(f).$

This, together with the energy estimate (3.6), yields the second inequality in
(4.14). ∎

The following is the main technical lemma in the proof of Theorem 4.1.

######  Lemma 4.6.

Let $(u_{\varepsilon_{k}},p_{\varepsilon_{k}})$, $P_{\varepsilon_{k}}$, and
$(u,P)$ be the same as in Lemma 4.3. Also assume that $f\in
C^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})$. Let $P^{\ell}$ denote the trace of
$P$, as a function in $H^{1}(\Omega^{\ell})$, on $\partial\Omega^{\ell}$.
Then, for any $\varphi\in C_{0}^{\infty}(\Omega)$,

$\int_{\partial\Omega^{\ell}}n_{j}P^{\ell}\varphi\,dx=\lim_{k\to\infty}\int_{\partial\Omega^{\ell}}n_{j}p_{\varepsilon_{k}}\varphi\,d\sigma,$
(4.16)

where $1\leq\ell\leq L$, $1\leq j\leq d$, and $n=(n_{1},n_{2},\dots,n_{d})$
denotes the outward unit normal to $\partial\Omega^{\ell}$.

###### Proof.

For notational simplicity we use $\varepsilon$ to denote $\varepsilon_{k}$.
Fix $1\leq j\leq d$ and $1\leq\ell\leq L$. Let $\varphi\in
C_{0}^{\infty}(\Omega)$. Then

$\displaystyle\varepsilon^{2}\int_{\Omega_{\varepsilon}^{\ell}}\nabla
u_{\varepsilon}\cdot\nabla\left(W_{j}^{\ell}(x/\varepsilon)\varphi\right)\,dx$
$\displaystyle\qquad=\varepsilon\int_{\Omega_{\varepsilon}^{\ell}}\nabla
u_{\varepsilon}\cdot\nabla
W_{j}^{\ell}(x/\varepsilon)\varphi\,dx+\varepsilon^{2}\int_{\Omega_{\varepsilon}^{\ell}}\nabla
u_{\varepsilon}\cdot W_{j}^{\ell}(x/\varepsilon)(\nabla\varphi)\,dx,$

and by integration by parts,

$\displaystyle\varepsilon^{2}\int_{\Omega_{\varepsilon}^{\ell}}\nabla
u_{\varepsilon}\cdot\nabla\left(W_{j}^{\ell}(x/\varepsilon)\varphi\right)\,dx$
$\displaystyle=\int_{\Omega^{\ell}}f\cdot
W_{j}^{\ell}(x/\varepsilon)\varphi\,dx+\int_{\Omega^{\ell}}P_{\varepsilon}W_{j}^{\ell}(x/\varepsilon)\cdot\nabla\varphi\,dx+\int_{\partial\Omega^{\ell}}\frac{\partial
u_{\varepsilon}}{\partial\nu}\cdot
W_{j}^{\ell}(x/\varepsilon)\varphi\,d\sigma,$

where

$\frac{\partial u_{\varepsilon}}{\partial\nu}=\varepsilon^{2}\frac{\partial
u_{\varepsilon}}{\partial n}-p_{\varepsilon}n.$

By letting $\varepsilon\to 0$ we obtain

$\displaystyle\lim_{\varepsilon\to
0}\varepsilon\int_{\Omega_{\varepsilon}^{\ell}}\nabla
u_{\varepsilon}\cdot\nabla W_{j}^{\ell}(x/\varepsilon)\varphi\,dx$ (4.17)
$\displaystyle=\int_{\Omega^{\ell}}f\cdot
K_{j}^{\ell}\varphi\,dx+\int_{\Omega^{\ell}}PK_{j}^{\ell}\cdot\nabla\varphi\,dx+\lim_{\varepsilon\to
0}\int_{\partial\Omega^{\ell}}\frac{\partial
u_{\varepsilon}}{\partial\nu}\cdot
W_{j}^{\ell}(x/\varepsilon)\varphi\,d\sigma.$

It follows by Lemma 2.1 that
$\|W_{j}^{\ell}(x/\varepsilon)\|_{L^{2}(\partial\Omega^{\ell})}\leq C$. This,
together with the first inequality in (4.14) with $m=1$, show that

$\Big{|}\varepsilon^{2}\int_{\partial\Omega^{\ell}}\frac{\partial
u_{\varepsilon}}{\partial n}\cdot
W_{j}^{\ell}(x/\varepsilon)\varphi\,d\sigma\Big{|}\leq
C\varepsilon^{2}\|(\nabla
u_{\varepsilon})\varphi\|_{L^{2}(\partial\Omega^{\ell})}=O(\varepsilon^{1/2}).$

Hence, by (4.17),

$\displaystyle\lim_{\varepsilon\to
0}\varepsilon\int_{\Omega_{\varepsilon}^{\ell}}\nabla
u_{\varepsilon}\cdot\nabla W_{j}^{\ell}(x/\varepsilon)\varphi\,dx$ (4.18)
$\displaystyle=\int_{\Omega^{\ell}}f\cdot
K_{j}^{\ell}\varphi\,dx+\int_{\Omega^{\ell}}PK_{j}^{\ell}\cdot\nabla\varphi\,dx-\lim_{\varepsilon\to
0}\int_{\partial\Omega^{\ell}}p_{\varepsilon}n\cdot
W_{j}^{\ell}(x/\varepsilon)\varphi\,d\sigma.$

Next, note that

$\displaystyle\varepsilon^{2}\int_{\Omega_{\varepsilon}^{\ell}}\nabla\left(W_{j}^{\ell}(x/\varepsilon)\right)\cdot\nabla(u_{\varepsilon}\varphi)\,dx$
(4.19) $\displaystyle=\varepsilon\int_{\Omega_{\varepsilon}^{\ell}}\nabla
W_{j}^{\ell}(x/\varepsilon)\cdot(\nabla
u_{\varepsilon})\varphi\,dx+\varepsilon\int_{\Omega_{\varepsilon}^{\ell}}\nabla
W_{j}^{\ell}(x/\varepsilon)\cdot u_{\varepsilon}(\nabla\varphi)\,dx.$

Choose a cut-off function $\eta_{\varepsilon}$ such that
supp$(\eta_{\varepsilon})\subset\\{x\in\mathbb{R}^{d}:\text{dist}(x,\partial\Omega^{\ell})\leq
2C\varepsilon\\}$, $\eta_{\varepsilon}(x)=1$ if
dist$(x,\partial\Omega^{\ell})\leq C\varepsilon$, and
$|\nabla\eta_{\varepsilon}|\leq C\varepsilon^{-1}$. Then

$\displaystyle\varepsilon^{2}\int_{\Omega_{\varepsilon}^{\ell}}\nabla\left(W_{j}^{\ell}(x/\varepsilon)\right)\cdot\nabla(u_{\varepsilon}\varphi)\,dx$
(4.20)
$\displaystyle=\varepsilon^{2}\int_{\Omega_{\varepsilon}^{\ell}}\nabla\left(W_{j}^{\ell}(x/\varepsilon)\right)\cdot\nabla(u_{\varepsilon}(1-\eta_{\varepsilon})\varphi)\,dx+\varepsilon^{2}\int_{\Omega_{\varepsilon}^{\ell}}\nabla\left(W_{j}^{\ell}(x/\varepsilon)\right)\cdot\nabla(u_{\varepsilon}\eta_{\varepsilon}\varphi)\,dx$
$\displaystyle=J_{1}+J_{2}.$

Using (4.19), (4.20), and

$\displaystyle|J_{2}|$ $\displaystyle\leq
C\varepsilon\left(\int_{\\{x\in\mathbb{R}^{d}:\,\text{dist}(x,\partial\Omega^{\ell})\leq
C\varepsilon\\}}|\nabla
W_{j}^{\ell}(x/\varepsilon)|^{2}\,dx\right)^{1/2}\left\\{\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega)}+\varepsilon^{-1}\|u_{\varepsilon}\|_{L^{2}(\Omega)}\right\\}$
$\displaystyle\leq C\varepsilon^{3/2}\left\\{\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega)}+\varepsilon^{-1}\|u_{\varepsilon}\|_{L^{2}(\Omega)}\right\\}$
$\displaystyle\leq\varepsilon^{1/2}C(f,h),$

we obtain

$\lim_{\varepsilon\to 0}\varepsilon\int_{\Omega_{\varepsilon}^{\ell}}\nabla
W_{j}^{\ell}(x/\varepsilon)\cdot(\nabla
u_{\varepsilon})\varphi\,dx=\lim_{\varepsilon\to 0}J_{1}$ (4.21)

To handle the term $J_{1}$, we use integration by parts as well as the fact
that

$-\varepsilon^{2}\Delta\left(W_{j}^{\ell}(x/\varepsilon)\right)+\nabla\left(\varepsilon\pi_{j}^{\ell}(x/\varepsilon)\right)=e_{j}$

in the set $\\{x\in\Omega_{\varepsilon}^{\ell}:\text{\rm
dist}(x,\partial\Omega^{\ell})\geq C\varepsilon\\}$, to obtain

$\displaystyle J_{1}$
$\displaystyle=\int_{\Omega_{\varepsilon}^{\ell}}\varepsilon\pi_{j}^{\ell}(x/\varepsilon)u_{\varepsilon}\cdot\nabla((1-\eta_{\varepsilon})\varphi)\,dx+\int_{\Omega^{\ell}}e_{j}\cdot
u_{\varepsilon}\varphi(1-\eta_{\varepsilon})\,dx$
$\displaystyle=J_{11}+J_{12},$

where we have used the fact div$(u_{\varepsilon})=0$ in
$\Omega_{\varepsilon}$. Since

$\displaystyle|J_{11}|$ $\displaystyle\leq
C\left(\int_{\\{x\in\mathbb{R}^{d}:\,\text{dist}(x,\partial\Omega^{\ell})\leq
C\varepsilon\\}}|\pi_{j}^{\ell}(x/\varepsilon)|^{2}\,dx\right)^{1/2}\|u_{\varepsilon}\|_{L^{2}(\Omega^{\ell})}+C\varepsilon\|u_{\varepsilon}\|_{L^{2}(\Omega^{\ell})}$
$\displaystyle\leq C\varepsilon^{1/2}C(f,h),$

we see that

$\lim_{\varepsilon\to 0}J_{1}=\lim_{\varepsilon\to
0}J_{12}=\int_{\Omega^{\ell}}e_{j}\cdot u\varphi\,dx.$ (4.22)

In view of (4.18), (4.21) and (4.22), we have proved that

$\lim_{\varepsilon\to 0}\int_{\partial\Omega^{\ell}}p_{\varepsilon}n\cdot
W_{j}^{\ell}(x/\varepsilon)\varphi\,d\sigma=\int_{\Omega^{\ell}}f\cdot
K_{j}^{\ell}\varphi\,dx+\int_{\Omega^{\ell}}PK_{j}^{\ell}\cdot\nabla\varphi\,dx-\int_{\Omega^{\ell}}e_{j}\cdot
u\varphi\,dx.$ (4.23)

Recall that $K^{\ell}=(K^{\ell}_{ij})$ is symmetric and by Lemma 4.3,

$u=K^{\ell}(f-\nabla P)\quad\text{ in }\Omega^{\ell}.$

Thus, by (4.23),

$\lim_{\varepsilon\to 0}\int_{\partial\Omega^{\ell}}p_{\varepsilon}n\cdot
W_{j}^{\ell}(x/\varepsilon)\varphi\,d\sigma=\int_{\partial\Omega^{\ell}}P^{\ell}(n\cdot
K_{j}^{\ell})\varphi\,d\sigma,$ (4.24)

where $P^{\ell}$ denotes the trace of $P$ on $\partial\Omega^{\ell}$.

Finally, we use Lemma 2.2 to obtain

$n\cdot\left(W^{\ell}_{j}(x/\varepsilon)-K^{\ell}_{j}\right)=\frac{\varepsilon}{2}\left(n_{\beta}\frac{\partial}{\partial
x_{\alpha}}-n_{\alpha}\frac{\partial}{\partial
x_{\beta}}\right)\left(\phi^{\ell}_{\alpha\beta j}(x/\varepsilon)\right),$
(4.25)

where the repeated indices $\alpha$ and $\beta$ are summed from $1$ to $d$.
Since $n_{\beta}\frac{\partial}{\partial
x_{\alpha}}-n_{\alpha}\frac{\partial}{\partial x_{\beta}}$ is a tangential
derivative on $\partial\Omega^{\ell}$, we obtain

$\displaystyle\Big{|}\int_{\partial\Omega^{\ell}}p_{\varepsilon}n\cdot\left(W_{j}^{\ell}(x/\varepsilon)-K^{\ell}_{j}\right)\varphi\,d\sigma\Big{|}$
$\displaystyle=\frac{\varepsilon}{2}\Big{|}\int_{\partial\Omega^{\ell}}\phi^{\ell}_{\alpha\beta
j}(x/\varepsilon)\left(n_{\beta}\frac{\partial}{\partial
x_{\alpha}}-n_{\alpha}\frac{\partial}{\partial
x_{\beta}}\right)(p_{\varepsilon}\varphi)\,d\sigma\Big{|}$ $\displaystyle\leq
C\varepsilon\|\nabla(p_{\varepsilon}\varphi)\|_{L^{2}(\partial\Omega^{\ell})}$
$\displaystyle\leq C(f,h)\varepsilon^{1/2},$

where we have used (2.8) for the first inequality and (4.14) for the last.
This, together with (4.24), yields

$\lim_{\varepsilon\to 0}\int_{\partial\Omega^{\ell}}p_{\varepsilon}(n\cdot
K_{j}^{\ell})\varphi\,d\sigma=\int_{\partial\Omega^{\ell}}P^{\ell}(n\cdot
K_{j}^{\ell})\varphi\,d\sigma.$ (4.26)

Since the constant matrix $K^{\ell}=(K_{ij}^{\ell})$ is invertible, the
desired equation (4.16) follows readily from (4.26). ∎

We are now in a position to give the proof of Theorem 4.1.

###### Proof of Theorem 4.1.

We first prove Theorem 4.1 under the additional assumption $f\in
C^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})$. Let $\\{\varepsilon_{k}\\}$ be a
sequence such that $\varepsilon_{k}\to 0$, $u_{\varepsilon_{k}}\to u$ weakly
in $L^{2}(\Omega;\mathbb{R}^{d})$ and $P_{\varepsilon_{k}}\to P$ strongly in
$L^{2}(\Omega)$. By Lemma 4.3, $P\in H^{1}(\Omega^{\ell})$ and
$u=K^{\ell}(f-\nabla P)$ in $\Omega^{\ell}$ for $1\leq\ell\leq L$. It suffices
to show that $P\in H^{1}(\Omega)$. This would imply that $P$ is a weak
solution of the Neumann problem,

$\left\\{\begin{aligned} \text{\rm div}(K(f-\nabla P))&=0&\quad&\text{ in
}\Omega,\\\ n\cdot K(f-\nabla P)&=n\cdot h&\quad&\text{ on
}\partial\Omega.\end{aligned}\right.$ (4.27)

As a result, we may deduce that as $\varepsilon\to 0$, $u_{\varepsilon}\to
u_{0}$ weakly in $L^{2}(\Omega;\mathbb{R}^{d})$ and
$P_{\varepsilon}-\fint_{\Omega}P_{\varepsilon}\to P_{0}$ strongly in
$L^{2}(\Omega)$, where $u_{0}=K(f-\nabla P_{0})$ in $\Omega$ and $P_{0}$ is
the unique weak solution of (4.27) with $\int_{\Omega}P_{0}\,dx=0$.

To prove $u\in H^{1}(\Omega)$, we use the assumption $f\in
C^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})$ and Lemma 4.6 to obtain

$\sum_{\ell=1}^{L}\int_{\partial\Omega^{\ell}}n_{j}P^{\ell}\varphi\,d\sigma=\lim_{k\to\infty}\sum_{\ell=1}^{L}\int_{\partial\Omega^{\ell}}n_{j}p_{\varepsilon_{k}}\varphi\,d\sigma,$

for any $\varphi\in C_{0}^{\infty}(\Omega)$ and $1\leq j\leq d$, where
$P^{\ell}$ denotes the trace of $P$, as a function in $H^{1}(\Omega^{\ell})$,
on $\partial\Omega^{\ell}$. Since $p_{\varepsilon}$ is continuous in
$\Omega_{\varepsilon}$, we have

$\sum_{\ell=1}^{L}\int_{\partial\Omega^{\ell}}n_{j}p_{\varepsilon}\varphi\,d\sigma=0.$

It follows that

$\sum_{\ell=1}^{L}\int_{\partial\Omega^{\ell}}n_{j}P^{\ell}\varphi\,d\sigma=0$

for $1\leq j\leq d$ and for any $\varphi\in C_{0}^{\infty}(\Omega)$. This,
together with the fact that $P\in H^{1}(\Omega^{\ell})$ for $1\leq\ell\leq L$,
gives

$\displaystyle\int_{\Omega}P\frac{\partial\varphi}{\partial x_{j}}\,dx$
$\displaystyle=\sum_{\ell=1}^{L}\int_{\Omega^{\ell}}P\frac{\partial\varphi}{\partial
x_{j}}\,dx$
$\displaystyle=-\sum_{\ell=1}^{L}\int_{\Omega^{\ell}}\frac{\partial
P}{\partial
x_{j}}\varphi\,dx+\sum_{\ell=1}^{L}\int_{\partial\Omega^{\ell}}n_{j}P^{\ell}\varphi\,d\sigma$
$\displaystyle=-\sum_{\ell=1}^{L}\int_{\Omega^{\ell}}\frac{\partial
P}{\partial x_{j}}\varphi\,dx.$

As a result, we obtain $P\in H^{1}(\Omega)$.

In the general case $f\in L^{2}(\Omega;\mathbb{R}^{d})$, we choose a sequence
of functions $\\{f_{m}\\}$ in $C^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})$ such
that $\|f_{m}-f\|_{L^{2}(\Omega)}\to 0$ as $m\to\infty$. Let
$(u_{\varepsilon,m},p_{\varepsilon,m})$ denote the weak solution of (4.1) with
$f_{m}$ in the place of $f$ and with
$\int_{\Omega_{\varepsilon}}p_{\varepsilon,m}\,dx=0$. By the energy estimates
(3.6) and (3.8) we obtain

$\|u_{\varepsilon}-u_{\varepsilon,m}\|_{L^{2}(\Omega)}+\|P_{\varepsilon}-P_{\varepsilon,m}\|_{L^{2}(\Omega)}\leq
C\|f-f_{m}\|_{L^{2}(\Omega)},$ (4.28)

where $P_{\varepsilon,m}$ denotes the extension of $p_{\varepsilon,m}$,
defined by (2.21). Let $u_{0,m}=K(f_{m}-\nabla P_{0,m})$, where $P_{0,m}$ is
the unique solution of (4.27) with $f_{m}$ in the place of $f$ and with
$\int_{\Omega}P_{0,m}\,dx=0$. Note that

$\displaystyle\|P_{\varepsilon}-\fint_{\Omega}P_{\varepsilon}-P_{0}\|_{L^{2}(\Omega)}\leq$
$\displaystyle\|P_{\varepsilon}-P_{\varepsilon,m}-\fint_{\Omega}(P_{\varepsilon}-P_{\varepsilon,m})\|_{L^{2}(\Omega)}$
$\displaystyle\qquad+\|P_{\varepsilon,m}-\fint_{\Omega}P_{\varepsilon,m}-P_{0,m}\|_{L^{2}(\Omega)}+\|P_{0,m}-P_{0}\|_{L^{2}(\Omega)}$
$\displaystyle\leq
C\|f-f_{m}\|_{L^{2}(\Omega)}+\|P_{\varepsilon,m}-\fint_{\Omega}P_{\varepsilon,m}-P_{0,m}\|_{L^{2}(\Omega)}.$

Since $P_{\varepsilon,m}-\fint_{\Omega}P_{\varepsilon,m}\to P_{0,m}$ in
$L^{2}(\Omega)$, as $\varepsilon\to 0$, we see that

$\limsup_{\varepsilon\to
0}\|P_{\varepsilon}-\fint_{\Omega}P_{\varepsilon}-P_{0}\|_{L^{2}(\Omega)}\leq
C\|f-f_{m}\|_{L^{2}(\Omega)}.$

By letting $m\to\infty$, we obtain
$P_{\varepsilon}-\fint_{\Omega}P_{\varepsilon}\to P_{0}$ in $L^{2}(\Omega)$,
as $\varepsilon\to 0$.

Finally, let $v\in L^{2}(\Omega;\mathbb{R}^{d})$. Note that

$\displaystyle\Big{|}\int_{\Omega}(u_{\varepsilon}-u_{0})v\,dx\Big{|}$
$\displaystyle\leq\Big{|}\int_{\Omega}(u_{\varepsilon}-u_{\varepsilon,m})v\,dx\Big{|}+\Big{|}\int_{\Omega}(u_{\varepsilon,m}-u_{0,m})v\,dx\Big{|}+\Big{|}\int_{\Omega}(u_{0,m}-u_{0})v\,dx\Big{|}$
$\displaystyle\leq\|u_{\varepsilon}-u_{\varepsilon,m}\|_{L^{2}(\Omega)}\|v\|_{L^{2}(\Omega)}+\Big{|}\int_{\Omega}(u_{\varepsilon,m}-u_{0,m})v\,dx\Big{|}+\|u_{0,m}-u_{0}\|_{L^{2}(\Omega)}\|v\|_{L^{2}(\Omega)}$
$\displaystyle\leq
C\|f-f_{m}\|_{L^{2}(\Omega)}\|v\|_{L^{2}(\Omega)}+\Big{|}\int_{\Omega}(u_{\varepsilon,m}-u_{0,m})v\,dx\Big{|}.$

By letting $\varepsilon\to 0$ and then $m\to\infty$, we see that
$u_{\varepsilon}\to u_{0}$ weakly in $L^{2}(\Omega;\mathbb{R}^{d})$. ∎

##  5 Convergence rates and proof of Theorem 1.2

Throughout the rest of this paper, unless indicated otherwise, we will assume
that $\Omega^{\ell},1\leq\ell\leq L$, are $C^{2,1/2}$ domains satisfying the
interface condition (1.12). Given $f\in L^{2}(\Omega;\mathbb{R}^{d})$, let
$P_{0}\in H^{1}(\Omega)$ be the weak solution of

$\left\\{\begin{aligned} -\text{\rm div}\left(K(f-\nabla
P_{0})\right)&=0&\quad&\text{ in }\Omega,\\\ n\cdot K(f-\nabla
P_{0})&=0&\quad&\text{ on }\partial\Omega,\end{aligned}\right.$ (5.1)

with $\int_{\Omega}P_{0}\,dx=0$, where the coefficient matrix $K$ is given by
(1.10). Since the interface $\Sigma$ and $\partial\Omega$ are of $C^{2,1/2}$,
it follows from [16, Theorem 1.1] that

$\displaystyle\|\nabla P_{0}\|_{C^{\alpha}(\Omega)}$ $\displaystyle\leq
C\|f\|_{C^{\alpha}(\Omega)},$ (5.2) $\displaystyle\|\nabla
P_{0}\|_{C^{1,\beta}(\Omega)}$ $\displaystyle\leq
C\|f\|_{C^{1,\beta}(\Omega)},$

for $0<\alpha<1$ and $0<\beta\leq 1/2$.

Let

$V_{\varepsilon}(x)=\sum_{\ell=1}^{L}W^{\ell}(x/\varepsilon)(f-\nabla
P_{0})\chi_{\Omega^{\ell}}\quad\text{ in }\Omega,$ (5.3)

where the 1-periodic matrix $W^{\ell}(y)$ is defined by (2.1) . Note that
$V_{\varepsilon}=0$ in $\Gamma_{\varepsilon}$. For each $\ell$, using

$-\varepsilon^{2}\Delta\left\\{W_{j}^{\ell}(x/\varepsilon)\right\\}+\nabla\left\\{\varepsilon\pi_{j}^{\ell}(x/\varepsilon)\right\\}=e_{j}\quad\text{
in }\bigcup_{z\in\mathbb{Z}^{d}}\varepsilon(z+{Y^{\ell}_{f}}),$ (5.4)

one may show that for any $\psi\in
H^{1}(\Omega^{\ell}_{\varepsilon};\mathbb{R}^{d})$ with $\psi=0$ on
$\Gamma^{\ell}_{\varepsilon}$,

$\displaystyle\Big{|}\varepsilon\int_{\Omega^{\ell}_{\varepsilon}}\nabla
W_{j}^{\ell}(x/\varepsilon)\cdot\nabla\psi\,dx-\varepsilon\int_{\Omega_{\varepsilon}^{\ell}}\pi_{j}^{\ell}(x/\varepsilon)\,\text{div}(\psi)\,dx-\int_{\Omega_{\varepsilon}^{\ell}}\psi_{j}\,dx\Big{|}$
(5.5) $\displaystyle\leq
C\varepsilon^{3/2}\|\nabla\psi\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}.$

To see (5.5), let

$\mathcal{O}_{\varepsilon}^{\ell}=\bigcup_{z}\varepsilon(z+Y_{f}^{\ell}),$

where $z\in\mathbb{Z}^{d}$ and the union is taken over those $z$’s for which
$\varepsilon(z+Y)\subset\Omega^{\ell}$. Using
$|\Omega_{\varepsilon}^{\ell}\setminus\mathcal{O}_{\varepsilon}^{\ell}|\leq
C\varepsilon^{1/2}$ and $\|\psi\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}\leq
C\varepsilon\|\nabla\psi\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}$, one may show
that each integral in the left-hand side of (5.5), with
$\Omega_{\varepsilon}^{\ell}\setminus\mathcal{O}^{\ell}_{\varepsilon}$ in the
place of $\Omega_{\varepsilon}^{\ell}$, is bounded by the right-hand side of
(5.5). By using integration by parts and (5.4), it follows that the left-hand
side of (5.5) with $\mathcal{O}_{\varepsilon}^{\ell}$ in the place of
$\Omega_{\varepsilon}^{\ell}$ is bounded by

$\displaystyle
C\varepsilon\left(\int_{\partial\mathcal{O}_{\varepsilon}^{\ell}}\left(|\nabla
W^{\ell}(x/\varepsilon)|+|\pi^{\ell}(x/\varepsilon)|\right)^{2}\,d\sigma\right)^{1/2}\left(\int_{\partial\mathcal{O}_{\varepsilon}^{\ell}}|\psi|^{2}\,d\sigma\right)^{1/2}$
$\displaystyle\leq
C\varepsilon^{3/2}\|\nabla\psi\|_{L^{2}(\Omega^{\ell}_{\varepsilon})},$

where we have used (2.5) and the observation,

$\displaystyle\|\psi\|_{L^{2}(\partial\mathcal{O}_{\varepsilon}^{\ell})}$
$\displaystyle\leq
C\varepsilon^{-1/2}\|\psi\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}+C\varepsilon^{1/2}\|\nabla\psi\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}$
$\displaystyle\leq
C\varepsilon^{1/2}\|\nabla\psi\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}.$

From (5.5) we deduce further that

$\displaystyle\Big{|}\varepsilon\int_{\Omega^{\ell}_{\varepsilon}}\nabla
W_{j}^{\ell}(x/\varepsilon)\cdot\nabla\psi\,dx-\int_{\Omega_{\varepsilon}^{\ell}}\psi_{j}\,dx\Big{|}$
(5.6) $\displaystyle\qquad\leq
C\varepsilon^{1/2}\left\\{\varepsilon\|\nabla\psi\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}+\varepsilon^{1/2}\|\text{\rm
div}(\psi)\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}\right\\}$

for any $\psi\in H^{1}(\Omega^{\ell}_{\varepsilon};\mathbb{R}^{d})$ with
$\psi=0$ on $\Gamma^{\ell}_{\varepsilon}$.

######  Theorem 5.1.

Let $(u_{\varepsilon},p_{\varepsilon})\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})\times
L^{2}_{0}(\Omega_{\varepsilon})$ be a weak solution of (1.1). Let
$V_{\varepsilon}$ be given by (5.3). Then

$\displaystyle\Big{|}\varepsilon^{2}\sum_{\ell=1}^{L}\int_{\Omega^{\ell}_{\varepsilon}}(\nabla
u_{\varepsilon}-\nabla
V_{\varepsilon})\cdot\nabla\psi\,dx-\int_{\Omega_{\varepsilon}}(p_{\varepsilon}-P_{0})\,\text{\rm
div}(\psi)\,dx\Big{|}$ (5.7) $\displaystyle\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}\left\\{\varepsilon\|\nabla\psi\|_{L^{2}(\Omega_{\varepsilon})}+\varepsilon^{1/2}\|\text{\rm
div}(\psi)\|_{L^{2}(\Omega_{\varepsilon})}\right\\},$

for any $\psi\in H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$.

###### Proof.

We apply (5.6) with $\psi(f_{j}-\frac{\partial P_{0}}{\partial x_{j}})$ in the
place of $\psi$. Using

$\displaystyle|\varepsilon^{2}\nabla
V_{\varepsilon}\cdot\nabla\psi-\varepsilon\nabla
W^{\ell}(x/\varepsilon)\cdot\nabla\left(\psi(f-\nabla P_{0})\right)|$
$\displaystyle\leq
C\left\\{\varepsilon^{2}|W^{\ell}(x/\varepsilon)||\nabla\psi|+C\varepsilon|\nabla
W^{\ell}(x/\varepsilon)||\psi|\right\\}|\nabla(f-\nabla P_{0})|$

in $\Omega^{\ell}_{\varepsilon}$, we obtain

$\displaystyle\Big{|}\varepsilon^{2}\int_{\Omega_{\varepsilon}^{\ell}}\nabla
V_{\varepsilon}\cdot\nabla\psi\,dx-\int_{\Omega_{\varepsilon}^{\ell}}(f-\nabla
P_{0})\cdot\psi\,dx\Big{|}$ $\displaystyle\leq
C\varepsilon^{3/2}\left(\|f\|_{\infty}+\|\nabla f\|_{\infty}+\|\nabla
P_{0}\|_{\infty}+\|\nabla^{2}P_{0}\|_{\infty}\right)\|\nabla\psi\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}$
$\displaystyle\qquad+C\varepsilon(\|f\|_{\infty}+\|\nabla
P_{0}\|_{\infty})\|\text{\rm
div}(\psi)\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}.$

This, together with

$\int_{\Omega_{\varepsilon}}(f-\nabla
P_{0})\cdot\psi\,dx=\varepsilon^{2}\int_{\Omega_{\varepsilon}}\nabla
u_{\varepsilon}\cdot\nabla\psi\,dx-\int_{\Omega_{\varepsilon}}(p_{\varepsilon}-P_{0})\,\text{\rm
div}(\psi)\,dx,$

gives (5.7). ∎

Let

$U_{\varepsilon}=V_{\varepsilon}+\Phi_{\varepsilon},$ (5.8)

where $\Phi_{\varepsilon}$ is a corrector to be constructed so that
$U_{\varepsilon}\in H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$,

$\|\text{\rm div}(U_{\varepsilon})\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)},$ (5.9)

and that

$\varepsilon\|\nabla\Phi_{\varepsilon}\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}$ (5.10)

for $1\leq\ell\leq L$.

Assuming that such corrector $\Phi_{\varepsilon}$ exists, we give the proof of
Theorem 1.2.

###### Proof of Theorem 1.2.

By letting
$\psi=u_{\varepsilon}-U_{\varepsilon}=u_{\varepsilon}-V_{\varepsilon}-\Phi_{\varepsilon}\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ in (5.7), we obtain

$\displaystyle\varepsilon^{2}\|\nabla u_{\varepsilon}-\nabla
V_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\varepsilon})}$
$\displaystyle\leq\varepsilon^{2}\|\nabla u_{\varepsilon}-\nabla
V_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\|\nabla\Phi_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}-P_{0}-\beta\|_{L^{2}(\Omega_{\varepsilon})}\|\text{\rm
div}(U_{\varepsilon})\|_{L^{2}(\Omega_{\varepsilon})}$
$\displaystyle\qquad+C\varepsilon^{1/2}\
\|f\|_{C^{1,1/2}(\Omega)}\left\\{\varepsilon\|\nabla u_{\varepsilon}-\nabla
V_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\varepsilon\|\nabla\Phi_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\varepsilon^{1/2}\|\text{\rm
div}(U_{\varepsilon})\|_{L^{2}(\Omega_{\varepsilon})}\right\\}$
$\displaystyle\leq C\varepsilon^{3/2}\|f\|_{C^{1,1/2}(\Omega)}\|\nabla
u_{\varepsilon}-\nabla V_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$
$\displaystyle\qquad+C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}\|p_{\varepsilon}-P_{0}-\beta\|_{L^{2}(\Omega_{\varepsilon})}+C\varepsilon\|f\|_{C^{1,1/2}(\Omega)}^{2},$

for any $\beta\in\mathbb{R}$, where we have used (5.9) and (5.10) for the last
inequality. By the Cauchy inequality, this implies that

$\varepsilon^{2}\|\nabla u_{\varepsilon}-\nabla
V_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\varepsilon})}\leq
C\varepsilon\|f\|^{2}_{C^{1,1/2}(\Omega)}+C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}\|p_{\varepsilon}-P_{0}-\beta\|_{L^{2}(\Omega_{\varepsilon})}.$
(5.11)

We should point out that both $V_{\varepsilon}$ and $\Phi_{\varepsilon}$ are
not in $H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$. In the estimates above
(and thereafter) we have used the convention that

$\|\
\nabla\psi\|_{L^{2}(\Omega_{\varepsilon})}=\left(\sum_{\ell=1}^{L}\|\nabla\psi\|^{2}_{L^{2}(\Omega_{\varepsilon}^{\ell})}\right)^{1/2},$

where $\psi\in H^{1}(\Omega_{\varepsilon}^{\ell})$ for $1\leq\ell\leq L$.

Next, we choose $\beta=\fint_{\Omega_{\varepsilon}}(p_{\varepsilon}-P_{0})$.
By Lemma 2.6, there exists $v_{\varepsilon}\in
H^{1}_{0}(\Omega_{\varepsilon};\mathbb{R}^{d})$ such that

$\displaystyle\text{\rm div}(v_{\varepsilon})$
$\displaystyle=p_{\varepsilon}-P_{0}-\beta\quad\text{ in
}\Omega_{\varepsilon},$ $\displaystyle\varepsilon\|\nabla
v_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$ $\displaystyle\leq
C\|p_{\varepsilon}-P_{0}-\beta\|_{L^{2}(\Omega_{\varepsilon})}.$

By letting $\psi_{\varepsilon}=v_{\varepsilon}$ in (5.7), we obtain

$\|p_{\varepsilon}-P_{0}-\beta\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\varepsilon\|\nabla u_{\varepsilon}-\nabla
V_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}.$
(5.12)

By combining (5.11) with (5.12), it is not hard to see that

$\varepsilon\|\nabla u_{\varepsilon}-\nabla
V_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}-P_{0}-\beta\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}.$ (5.13)

This, together with
$\|u_{\varepsilon}-V_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\leq
C\varepsilon\|\nabla u_{\varepsilon}-\nabla
V_{\varepsilon}\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}$, gives the bound for
the first term in (1.13). Also, note that

$\|\varepsilon\nabla V_{\varepsilon}-\nabla W^{\ell}(x/\varepsilon)(f-\nabla
P_{0})\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\leq
C\varepsilon\|\nabla(f-\nabla P_{0})\|_{\infty}.$

Thus,

$\|\varepsilon\nabla u_{\varepsilon}-\nabla W^{\ell}(x/\varepsilon)(f-\nabla
P_{0})\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}.$

Finally, to estimate the pressure, we let $Q_{\varepsilon}$ be the extension
of $(P_{0}+\beta)|_{\Omega_{\varepsilon}}$ to $\Omega$, using the formula in
(2.21). Note that

$\|Q_{\varepsilon}-(P_{0}+\beta)\|^{2}_{L^{2}(\Omega)}=\sum_{\ell,z}\int_{\varepsilon(Y_{s}^{\ell}+z)}\Big{|}P_{0}-\fint_{\varepsilon(Y_{f}^{\ell}+z)}P_{0}\Big{|}^{2}\,dx,$

where the sum is taken over those $(\ell,z)$’s for which $z\in\mathbb{Z}^{d}$
and $\varepsilon(Y+z)\subset\Omega^{\ell}$. It follows that

$\displaystyle\|Q_{\varepsilon}-(P_{0}+\beta)\|_{L^{2}(\Omega)}$
$\displaystyle\leq C\varepsilon\|\nabla P_{0}\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq C\varepsilon\|f\|_{C^{1,1/2}(\Omega)}.$

As a result, by (5.13), we obtain

$\displaystyle\|P_{\varepsilon}-P_{0}-\beta\|_{L^{2}(\Omega)}$
$\displaystyle\leq\|P_{\varepsilon}-Q_{\varepsilon}\|_{L^{2}(\Omega)}+\|Q_{\varepsilon}-(P_{0}+\beta)\|_{L^{2}(\Omega)}$
$\displaystyle\leq
C\|p_{\varepsilon}-P_{0}-\beta\|_{L^{2}(\Omega_{\varepsilon})}+C\varepsilon\|f\|_{C^{1,1/2}(\Omega)}$
$\displaystyle\leq C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)},$

where $\beta=-\fint_{\Omega_{\varepsilon}}P_{0}$. Clearly, we may replace
$\beta$ by
$\fint_{\Omega}(P_{\varepsilon}-P_{0})=\fint_{\Omega}P_{\varepsilon}$. This
gives the bound for the second term in (1.13). ∎

To complete the proof of Theorem 1.2, it remains to construct a corrector
$\Phi_{\varepsilon}$ such that $V_{\varepsilon}+\Phi_{\varepsilon}\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ and (5.9)-(5.10) hold. This
will be done in the next three sections. More precisely, we let

$\Phi_{\varepsilon}=\Phi_{\varepsilon}^{(1)}+\Phi_{\varepsilon}^{(2)}+\Phi_{\varepsilon}^{(3)},$
(5.14)

where $\Phi_{\varepsilon}^{(1)}$ is a corrector for the divergence operator
with the properties that

$\left\\{\begin{aligned} &\Phi_{\varepsilon}^{(1)}\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d}),\\\
&\varepsilon\|\nabla\Phi_{\varepsilon}^{(1)}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)},\\\ &\|\text{\rm
div}(\Phi_{\varepsilon}^{(1)}+V_{\varepsilon})\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)},\end{aligned}\right.$ (5.15)

$\Phi_{\varepsilon}^{(2)}$ is a corrector for the boundary data of
$V_{\varepsilon}$ on $\partial\Omega$ with the properties that

$\left\\{\begin{aligned} &\Phi_{\varepsilon}^{(2)}\in
H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})\quad\text{ and
}\quad\Phi^{(2)}_{\varepsilon}=0\quad\text{ on }\Gamma_{\varepsilon},\\\
&\Phi_{\varepsilon}^{(2)}+V_{\varepsilon}=0\quad\text{ on }\partial\Omega,\\\
&\varepsilon\|\nabla\Phi_{\varepsilon}^{(2)}\|_{L^{2}(\Omega_{\varepsilon})}+\|\text{\rm
div}(\Phi_{\varepsilon}^{(2)})\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)},\end{aligned}\right.$ (5.16)

and $\Phi_{\varepsilon}^{(3)}$ is a corrector for the interface $\Sigma$ with
the properties that

$\left\\{\begin{aligned} &\Phi_{\varepsilon}^{(3)}\in
H^{1}(\Omega^{\ell}_{\varepsilon};\mathbb{R}^{d})\quad\text{ and
}\quad\Phi_{\varepsilon}^{(3)}=0\quad\text{ on
}\partial\Omega_{\varepsilon},\\\ &V_{\varepsilon}+\Phi_{\varepsilon}^{(3)}\in
H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d}),\\\
&\varepsilon\|\nabla\Phi_{\varepsilon}^{(3)}\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}+\|\text{\rm
div}(\Phi_{\varepsilon}^{(3)})\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)},\end{aligned}\right.$ (5.17)

for $1\leq\ell\leq L$. It is not hard to verify that the desired property
$V_{\varepsilon}+\Phi_{\varepsilon}\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ as well as the estimates
(5.9)-(5.10) follows from (5.15)-(5.17).

##  6 Correctors for the divergence operator

Let $V_{\varepsilon}$ be given by (5.3). Note that since
div$(W_{j}^{\ell}(x/\varepsilon))=0$ in $\mathbb{R}^{d}$,

$\text{\rm div}(V_{\varepsilon})=W^{\ell}(x/\varepsilon)\nabla(f-\nabla
P_{0})\quad\text{ in }\Omega^{\ell}_{\varepsilon}.$ (6.1)

In this section we construct a corrector $\Phi_{\varepsilon}^{(1)}$ that
satisfies (5.15). The approach is similar to that used in [11, 15].

For $1\leq\ell\leq L$ and $1\leq\,i,j\leq d$, let
$\Theta_{ij}^{\ell}=(\Theta^{\ell}_{1ij},\dots,\Theta^{\ell}_{dij})$ be a
1-periodic function in $H^{1}_{loc}(\mathbb{R}^{d};\mathbb{R}^{d})$ such that

$\left\\{\begin{aligned} \text{\rm
div}(\Theta^{\ell}_{ij})&=-W^{\ell}_{ij}+|Y_{f}|^{-1}K^{\ell}_{ij}&\quad&\text{
in }Y_{f},\\\ \Theta_{ij}^{\ell}&=0&\quad&\text{ in
}Y_{s}.\end{aligned}\right.$ (6.2)

Fix $\varphi\in C_{0}^{\infty}(B(0,1/8))$ such that $\varphi\geq 0$ and
$\int_{\mathbb{R}^{d}}\varphi\,dx=1$. Define

$S_{\varepsilon}(\psi)(x)=\psi*\varphi_{\varepsilon}(x)=\int_{\mathbb{R}^{d}}\psi(y)\varphi_{\varepsilon}(x-y)\,dy,$
(6.3)

where $\varphi_{\varepsilon}(x)=\varepsilon^{-d}\varphi(x)$. Let
$\Phi_{\varepsilon}^{(1)}=(\Phi_{\varepsilon,1}^{(1)},\dots,\Phi_{\varepsilon,d}^{(1)})$,
where, for $x\in\Omega_{\varepsilon}^{\ell}$,

$\Phi_{\varepsilon,k}^{(1)}(x)=\varepsilon\eta^{\ell}_{\varepsilon}(x)\Theta^{\ell}_{kij}(x/\varepsilon)\frac{\partial}{\partial
x_{i}}S_{\varepsilon}\left(f_{j}-\frac{\partial P_{0}}{\partial
x_{j}}\right),$ (6.4)

and $P_{0}$ is the solution of (5.1). The function $\eta_{\varepsilon}^{\ell}$
in (6.4) is a cut-off function in $C_{0}^{\infty}(\Omega^{\ell})$ with the
properties that $|\nabla\eta_{\varepsilon}^{\ell}|\leq C\varepsilon^{-1}$ and

$\left\\{\begin{aligned} &\eta_{\varepsilon}^{\ell}(x)=0&\quad&\text{ if
dist}(x,\partial\Omega^{\ell})\leq 2d\varepsilon,\\\
&\eta^{\ell}_{\varepsilon}(x)=1&\quad&\text{ if }x\in\Omega^{\ell}\text{ and
dist}(x,\partial\Omega^{\ell})\geq 3d\varepsilon.\end{aligned}\right.$

As a result, $\Phi_{\varepsilon}^{(1)}$ vanishes near $\partial\Omega^{\ell}$.

######  Theorem 6.1.

Let $\Phi_{\varepsilon}^{(1)}$ be defined by (6.4). Then (5.15) holds.

###### Proof.

Clearly, $\Phi_{\varepsilon}^{(1)}\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$. Note that

$\displaystyle\|\nabla\Phi_{\varepsilon}^{(1)}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}$
$\displaystyle\leq C\varepsilon^{1/2}\|\nabla S_{\varepsilon}(f-\nabla
P_{0})\|_{L^{\infty}(N_{3d\varepsilon}\setminus N_{2d\varepsilon})}+C\|\nabla
S_{\varepsilon}(f-\nabla P_{0})\|_{L^{\infty}(\Omega^{\ell}\setminus
N_{2d\varepsilon})}$
$\displaystyle\qquad\qquad+C\varepsilon\|\nabla^{2}S_{\varepsilon}(f-\nabla
P_{0})\|_{L^{\infty}(\Omega^{\ell}\setminus N_{2d\varepsilon})},$

where $N_{r}=\\{x\in\Omega^{\ell}:\text{\rm
dist}(x,\partial\Omega^{\ell})<r\\}$. This, together with the observation that
$\nabla S_{\varepsilon}(\psi)=S_{\varepsilon}(\nabla\psi)$ and

$|S_{\varepsilon}(\psi)(x)|+\varepsilon|\nabla S_{\varepsilon}(\psi)(x)|\leq
C\fint_{B(x,\varepsilon/8)}|\psi|,$

yields

$\displaystyle\varepsilon\|\nabla\Phi_{\varepsilon}^{(1)}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}$
$\displaystyle\leq C\varepsilon\|\nabla(f-\nabla
P_{0})\|_{L^{\infty}(\Omega^{\ell})}$ $\displaystyle\leq
C\varepsilon\|f\|_{C^{1,1/2}(\Omega)}.$

Next, note that in $\Omega^{\ell}_{\varepsilon}$,

$\displaystyle\text{\rm div}(\Phi_{\varepsilon}^{(1)})$
$\displaystyle=\varepsilon(\nabla\eta_{\varepsilon}^{\ell})\Theta^{\ell}(x/\varepsilon)\nabla
S_{\varepsilon}(f-\nabla
P_{0})-\eta_{\varepsilon}^{\ell}W^{\ell}(x/\varepsilon)\nabla
S_{\varepsilon}(f-\nabla P_{0})$
$\displaystyle\qquad\qquad+\varepsilon\eta_{\varepsilon}^{\ell}\Theta^{\ell}(x/\varepsilon)\nabla^{2}S_{\varepsilon}(f-\nabla
P_{0}),$

where we have used the fact that $\text{\rm div}(K^{\ell}(f-\nabla P_{0}))=0$
in $\Omega_{\varepsilon}^{\ell}$. It follows that

$\displaystyle\|\text{\rm
div}(\Phi_{\varepsilon}^{(1)})+W^{\ell}(x/\varepsilon)\nabla(f-\nabla
P_{0})\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}$ $\displaystyle\leq
C\varepsilon^{1/2}\|\nabla(f-\nabla
P_{0})\|_{L^{\infty}(\Omega^{\ell})}+\|W^{\ell}(x/\varepsilon)\left\\{\nabla(f-\nabla
P_{0})-\eta_{\varepsilon}^{\ell}\nabla S_{\varepsilon}(f-\nabla
P_{0})\right\\}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}$
$\displaystyle\qquad\qquad+C\varepsilon\|\nabla^{2}S_{\varepsilon}(f-\nabla
P_{0})\|_{L^{\infty}(\Omega_{\varepsilon}^{\ell}\setminus N_{2d\varepsilon})}$
$\displaystyle\leq C\varepsilon^{1/2}\|\nabla(f-\nabla
P_{0})\|_{L^{\infty}(\Omega^{\ell})}+\|\nabla(f-\nabla P_{0})-\nabla
S_{\varepsilon}(f-\nabla P_{0})\|_{L^{\infty}(\Omega^{\ell}\setminus
N_{2d\varepsilon})}$
$\displaystyle\qquad\qquad+C\varepsilon\|\nabla^{2}S_{\varepsilon}(f-\nabla
P_{0})\|_{L^{\infty}(\Omega^{\ell}\setminus N_{2d\varepsilon})}$
$\displaystyle\leq C\varepsilon^{1/2}\|\nabla(f-\nabla
P_{0})\|_{C^{1/2}(\Omega^{\ell})}$ $\displaystyle\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)},$

where we have used (5.2) for the last inequality. In the third inequality
above we also used the observation that

$\nabla
S_{\varepsilon}(\psi)(x)=-\int_{\mathbb{R}^{d}}\left(\psi(x-y)-\psi(x)\right)\nabla_{y}(\varphi_{\varepsilon}(y))\,dy,$

which gives

$|\nabla S_{\varepsilon}(\psi)(x)|\leq
C\varepsilon^{\alpha-1}\|\psi\|_{C^{0,\alpha}(B(x,\varepsilon))}.$

This completes the proof of (5.15). ∎

##  7 Boundary correctors

To construct the boundary corrector $\Phi_{\varepsilon}^{(2)}$, we consider
the Dirichlet problem,

$\left\\{\begin{aligned} -\varepsilon^{2}\Delta u_{\varepsilon}+\nabla
p_{\varepsilon}&=0&\quad&\text{ in }\Omega_{\varepsilon}\\\ \text{\rm
div}(u_{\varepsilon})&=\gamma&\quad&\text{ in }\Omega_{\varepsilon},\\\
u_{\varepsilon}&=0&\quad&\text{ on }\Gamma_{\varepsilon},\\\
u_{\varepsilon}&=h&\quad&\text{ on }\partial\Omega,\end{aligned}\right.$ (7.1)

where $\Omega_{\varepsilon}$ is given by (1.4) and

$\gamma=\frac{1}{|\Omega_{\varepsilon}|}\int_{\partial\Omega}h\cdot
n\,d\sigma.$ (7.2)

Let $\Phi_{\varepsilon}^{(2)}\in H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$
be the solution of (7.1) with boundary value,

$h=-V_{\varepsilon}\quad\text{ on }\partial\Omega,$ (7.3)

where $V_{\varepsilon}$ is given by (5.3). Thus, if
$\partial\Omega\cap\partial\Omega^{\ell}\neq\emptyset$ for some $1\leq\ell\leq
L$,

$\Phi_{\varepsilon}^{(2)}=-W^{\ell}(x/\varepsilon)(f-\nabla P_{0})\quad\text{
on }\partial\Omega\cap\partial\Omega^{\ell}.$ (7.4)

######  Theorem 7.1.

Let $\Phi_{\varepsilon}^{(2)}$ be defined as above. Then
$\Phi_{\varepsilon}^{(2)}$ satisfies (5.16).

To show Theorem 7.1, we first prove some general results, which will be used
also in the construction of correctors for the interface.

######  Theorem 7.2.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{d}$, $d\geq 2$.
Assume that $\Omega^{\ell}$ and $Y_{s}^{\ell}$ with $1\leq\ell\leq L$ are
subdomains of $\Omega$ and $Y$, respectively, with Lipschitz boundaries. Let
$(u_{\varepsilon},p_{\varepsilon})$ be a weak solution in
$H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})\times
L_{0}^{2}(\Omega_{\varepsilon})$ of (7.1), where $h\in
H^{1}(\partial\Omega;\mathbb{R}^{d})$ and

$u\cdot n=0\quad\text{ on }\partial\Omega.$ (7.5)

Then

$\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\sqrt{\varepsilon}\left\\{\|h\|_{L^{2}(\partial\Omega)}+\varepsilon\|\nabla_{\tan}h\|_{L^{2}(\partial\Omega)}\right\\},$
(7.6)

where $\nabla_{\tan}h$ denotes the tangential gradient of $h$ on
$\partial\Omega$.

###### Proof.

This theorem was proved in [15, Theorem 4.1] for the case $L=1$. The proof
only uses the energy estimate (3.6) and the fact that

$-\varepsilon^{2}\Delta u_{\varepsilon}+\nabla p_{\varepsilon}=0\quad\text{
and }\quad\text{\rm div}(u_{\varepsilon})=0$

in the set $\\{x\in\Omega:\ \text{\rm
dist}(x,\partial\Omega)<c\,\varepsilon\\}$. As a result, the same proof works
equally well for the case $L\geq 2$. We mention that the argument relies on
the Rellich estimates in [7] for the Stokes equations in Lipschitz domains.
The condition (7.5) allows us to drop the pressure $p_{\varepsilon}$ term in
the conormal derivative $\partial u_{\varepsilon}/{\partial\nu}$ for
$u_{\varepsilon}$ on $\partial\Omega$. We omit the details. ∎

In the next theorem we consider the case where

$h\cdot n=\varepsilon\ (\nabla_{\tan}\phi_{\varepsilon})\cdot g)\quad\text{ on
}\partial\Omega.$ (7.7)

By using integration by parts on $\partial\Omega$, we see that

$\displaystyle|\gamma|$ $\displaystyle\leq C\Big{|}\int_{\partial\Omega}h\cdot
n\,d\sigma\Big{|}$ (7.8) $\displaystyle\leq
C\varepsilon\|\phi_{\varepsilon}\nabla_{\tan}g\|_{L^{2}(\partial\Omega)}.$

######  Theorem 7.3.

Let $\Omega$ be a bounded $C^{2,\alpha}$ domain in $\mathbb{R}^{d}$, $d\geq
2$. Let $(u_{\varepsilon},p_{\varepsilon})$ be a weak solution in
$H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})\times L_{0}^{2}(\Omega)$ of (7.1),
where $h\in H^{1}(\partial\Omega)$ and $h\cdot n$ is given (7.7). Then

$\displaystyle\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$
(7.9) $\displaystyle\qquad\leq
C\sqrt{\varepsilon}\left\\{\|h\|_{L^{2}(\partial\Omega)}+\varepsilon\|\nabla_{\tan}h\|_{L^{2}(\partial\Omega)}+\|\phi_{\varepsilon}g\|_{L^{2}(\partial\Omega)}+\varepsilon^{1/2}\|\phi_{\varepsilon}\nabla_{\tan}g\|_{L^{2}(\partial\Omega)}\right\\}.$

###### Proof.

A version of this theorem was proved in [15, Theorem 5.1] for the case $L=1$.
We give the proof for the general case, using a somewhat different argument.

We first note that by writing

$h=(h-(h\cdot n)n)+(h\cdot n)n$

and applying Theorem 7.2 to the solution of (7.1) with boundary data
$h-(h\cdot n)n$, we may reduce the problem to case where $h=(h\cdot n)n$ on
$\partial\Omega$.

Next, by the energy estimate (3.3) and (7.8),

$\displaystyle\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$
(7.10) $\displaystyle\qquad\leq C\left\\{\|H\|_{L^{2}(\Omega)}+\|\text{\rm
div}(H)\|_{L^{2}(\Omega)}+\varepsilon\|\nabla
H\|_{L^{2}(\Omega)}+\varepsilon\|\phi_{\varepsilon}\nabla_{\tan}g\|_{L^{2}(\partial\Omega)}\right\\},$

where $H$ is any function in $H^{1}(\Omega;\mathbb{R}^{d})$ with $H=h$ on
$\partial\Omega$. We choose $H=H_{1}+\gamma(x-x_{0})/d$, where
$x_{0}\in\Omega$ and $H_{1}$ is the weak solution of

$-\Delta H_{1}+\nabla q=0\quad\text{ and }\quad\text{\rm
div}(H_{1})=0\quad\text{ in }\Omega,$

with the boundary value $H_{1}=h-\gamma(x-x_{0})/d$ on $\partial\Omega$. It
follows that

$\displaystyle\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega)}+\|u_{\varepsilon}\|_{L^{2}(\partial\Omega)}+\|p_{\varepsilon}\|_{L^{2}(\Omega)}$
(7.11) $\displaystyle\qquad\leq
C\left\\{\|H_{1}\|_{L^{2}(\Omega)}+\varepsilon\|\nabla
H_{1}\|_{L^{2}(\Omega)}+\varepsilon\|\phi_{\varepsilon}\nabla_{\tan}g\|_{L^{2}(\partial\Omega)}\right\\},$

where we have used (7.8). By the energy estimates for the Stokes equations in
$\Omega$,

$\displaystyle\|\nabla H_{1}\|_{L^{2}(\Omega)}$ $\displaystyle\leq
C\left\\{\|h\|_{H^{1/2}(\partial\Omega)}+|\gamma|\right\\}$ $\displaystyle\leq
C\left\\{\|h\|_{L^{2}(\partial\Omega)}^{1/2}\|h\|^{1/2}_{H^{1}(\partial\Omega)}+|\gamma|\right\\}$
$\displaystyle\leq
C\left\\{\varepsilon^{-1/2}\|h\|_{L^{2}(\partial\Omega)}+\varepsilon^{1/2}\|\nabla_{\tan}h\|_{L^{2}(\partial\Omega)}+|\gamma|\right\\}.$

It follows that

$\varepsilon\|\nabla H_{1}\|_{L^{2}(\Omega)}\leq
C\sqrt{\varepsilon}\left\\{\|h\|_{L^{2}(\partial\Omega)}+\varepsilon\|\nabla_{\tan}h\|_{L^{2}(\partial\Omega)}+\varepsilon\|\phi_{\varepsilon}\nabla_{\tan}g\|_{L^{2}(\partial\Omega)}\right\\}.$
(7.12)

To bound $\|H_{1}\|_{L^{2}(\Omega)}$, we use the following nontangential-
maximal-function estimate,

$\|(H_{1})^{*}\|_{L^{2}(\partial\Omega)}\leq
C\|H_{1}\|_{L^{2}(\partial\Omega)},$ (7.13)

where the nontangential maximal function $(H_{1})^{*}$ on $\partial\Omega$ is
defined by

$(H_{1})^{*}(x)=\sup\big{\\{}|H_{1}(y)|:\ y\in\Omega\text{ and
}|y-x|<C_{0}\,\text{dist}(y,\partial\Omega)\big{\\}}$

for $x\in\partial\Omega$. The estimate (7.13) was proved in [7] for a bounded
Lipschitz domain $\Omega$. Let

$N_{r}=\big{\\{}x\in\Omega:\ \text{dist}(x,\partial\Omega)<r\big{\\}}.$

It follows from (7.13) that

$\displaystyle\|H_{1}\|_{L^{2}(N_{\varepsilon})}$ $\displaystyle\leq
C\sqrt{\varepsilon}\|(H_{1})^{*}\|_{L^{2}(\partial\Omega)}$ (7.14)
$\displaystyle\leq
C\sqrt{\varepsilon}\left\\{\|h\|_{L^{2}(\partial\Omega)}+\varepsilon\|\phi_{\varepsilon}\nabla_{\tan}g\|_{L^{2}(\partial\Omega)}\right\\}.$

It remains to bound $\|H_{1}\|_{L^{2}(\Omega\setminus N_{\varepsilon})}$. To
this end, we consider the Dirichlet problem,

$\left\\{\begin{aligned} -\Delta G+\nabla\pi&=F&\quad&\text{ in }\Omega,\\\
\text{\rm div}(G)&=0&\quad&\text{ in }\Omega,\\\ G&=0&\quad&\text{ on
}\partial\Omega,\end{aligned}\right.$

where $F\in C_{0}^{\infty}(\Omega\setminus N_{\varepsilon})$ and
$\int_{\Omega}\pi\,dx=0$. Under the assumption that $\partial\Omega$ is of
$C^{2,\alpha}$, we have the $W^{2,2}$ estimates,

$\|G\|_{H^{2}(\Omega)}+\|\pi\|_{H^{1}(\Omega)}\leq C\|F\|_{L^{2}(\Omega)}.$
(7.15)

This implies that

$\|\nabla G\|_{L^{2}(\partial\Omega)}+\|\pi\|_{L^{2}(\partial\Omega)}\leq
C\|F\|_{L^{2}(\Omega)}.$ (7.16)

Moreover, since $F=0$ in $N_{\varepsilon}$, by covering $\partial\Omega$ with
balls of radius $c\varepsilon$, one may show that

$\displaystyle\int_{\partial\Omega}\left(|\nabla^{2}G|^{2}+|\nabla\pi|^{2}\right)\,d\sigma$
$\displaystyle\leq C\varepsilon^{-1}\|F\|_{L^{2}(\Omega)}^{2}.$ (7.17)

To see this, we use the Green function representation for $G$ to obtain

$|\nabla^{2}G(x)|\leq C\int_{\Omega\setminus
N_{\varepsilon}}\frac{|F(y)|}{|x-y|^{d}}\,dy$ (7.18)

for $x\in\partial\Omega$. See e.g. [8] for estimates of Green functions for
the Stokes equations. Choose $\alpha,\beta\in(0,1)$ such that
$\alpha+\beta=1$, $\alpha>(1/2)$ and $\beta>(1/2)-(1/2d)$. It follows by the
Cauchy inequality that for $x\in\partial\Omega$,

$\displaystyle|\nabla^{2}G(x)|^{2}$ $\displaystyle\leq
C\left(\int_{\Omega\setminus
N_{\varepsilon}}\frac{dy}{|x-y|^{2d\alpha}}\right)\left(\int_{\Omega\setminus
N_{\varepsilon}}\frac{|F(y)|^{2}}{|x-y|^{2d\beta}}\,dy\right)$
$\displaystyle\leq C\varepsilon^{d-2d\alpha}\int_{\Omega\setminus
N_{\varepsilon}}\frac{|F(y)|^{2}}{|x-y|^{2d\beta}}\,dy,$

where we have used the conditions $\alpha+\beta=1$ and $\alpha>(1/2)$. Hence,

$\displaystyle\int_{\partial\Omega}|\nabla^{2}G|^{2}\,d\sigma$
$\displaystyle\leq C\varepsilon^{d-2d\alpha}\int_{\Omega\setminus
N_{\varepsilon}}|F(y)|^{2}\,dy\sup_{y\in\Omega\setminus
N_{\varepsilon}}\int_{\partial\Omega}\frac{d\sigma(x)}{|x-y|^{2d\beta}}$
$\displaystyle\leq C\varepsilon^{-1}\int_{\Omega}|F(y)|^{2}\,dy,$

where we have used the condition $\beta>(1/2)-(1/2d)$. This gives the estimate
for $|\nabla^{2}G|$ in (7.17). The estimate for $\nabla\pi$ follows from the
equation $-\Delta G+\nabla\pi=0$ near $\partial\Omega$.

Finally, using integration by parts, we see that

$\displaystyle\int_{\Omega}H_{1}\cdot F\,dx$
$\displaystyle=\int_{\Omega}H_{1}\cdot(-\Delta G+\nabla\pi)\,dx$
$\displaystyle=-\int_{\partial\Omega}H_{1}\cdot\Big{(}\frac{\partial
G}{\partial n}-n\pi\Big{)}\,d\sigma$
$\displaystyle=-\int_{\partial\Omega}\Big{(}\varepsilon((\nabla_{\tan}\phi_{\varepsilon})\cdot
g)n-\gamma(x-x_{0})/d\Big{)}\cdot\Big{(}\frac{\partial G}{\partial
n}-n\pi\Big{)}\,d\sigma.$

It follows by using integration by parts on $\partial\Omega$ that

$\displaystyle\Big{|}\int_{\Omega}H_{1}\cdot F\,dx\Big{|}$ $\displaystyle\leq
C\varepsilon\int_{\partial\Omega}|\phi_{\varepsilon}|\left(|\nabla g||\nabla
G|+|g||\nabla^{2}G|+|g||\nabla G|+|\nabla
g||\pi|+|g||\nabla\pi|+|g||\pi|\right)\,d\sigma$
$\displaystyle\qquad\qquad+|\gamma|\int_{\partial\Omega}\left(|\nabla
G|+|\pi|\right)\,d\sigma$ $\displaystyle\leq
C\varepsilon\|\phi_{\varepsilon}g\|_{L^{2}(\partial\Omega)}\left\\{\|\nabla^{2}G\|_{L^{2}(\partial\Omega)}+\|\nabla
G\|_{L^{2}(\partial\Omega)}+\|\nabla\pi\|_{L^{2}(\partial\Omega)}+\|\pi\|_{L^{2}(\partial\Omega)}\right\\}$
$\displaystyle\qquad\qquad+C\varepsilon\|\phi_{\varepsilon}\nabla_{\tan}g\|_{L^{2}(\partial\Omega)}\left\\{\|\nabla
G\|_{L^{2}(\partial\Omega)}+\|\pi\|_{L^{2}(\partial\Omega)}\right\\},$

where we have used the Cauchy inequality and (7.8). This, together with (7.16)
and (7.17), gives

$\Big{|}\int_{\Omega}H_{1}\cdot F\,dx\Big{|}\leq
C\varepsilon^{1/2}\|F\|_{L^{2}(\Omega)}\left\\{\|\phi_{\varepsilon}g\|_{L^{2}(\partial\Omega)}+\varepsilon^{1/2}\|\phi_{\varepsilon}\nabla_{\tan}g\|_{L^{2}(\partial\Omega)}\right\\}.$

By duality we obtain

$\|H_{1}\|_{L^{2}(\Omega\setminus N_{\varepsilon})}\leq
C\varepsilon^{1/2}\left\\{\|\phi_{\varepsilon}g\|_{L^{2}(\partial\Omega)}+\varepsilon^{1/2}\|\phi_{\varepsilon}\nabla_{\tan}g\|_{L^{2}(\partial\Omega)}\right\\}.$
(7.19)

The desired estimate (7.9) follows from (7.10), (7.12), (7.14) and (7.19). ∎

###### Proof of Theorem 7.1.

Clearly, by its definition, $\Phi_{\varepsilon}^{(2)}\in
H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$, $\Phi_{\varepsilon}^{(2)}=0$ on
$\Gamma_{\varepsilon}$, and $\Phi_{\varepsilon}^{(2)}+V_{\varepsilon}=0$ on
$\partial\Omega$. Using the fact that $n\cdot K^{\ell}(f-\nabla P_{0})=0$ on
$\partial\Omega\cap\partial\Omega^{\ell}$, we obtain

$\displaystyle n\cdot h$ $\displaystyle=-n\cdot
W^{\ell}(x/\varepsilon)(f-\nabla P_{0})$ (7.20)
$\displaystyle=-n\cdot(W^{\ell}(x/\varepsilon)-K^{\ell})(f-\nabla P_{0})$
$\displaystyle=-\frac{\varepsilon}{2}\left(n_{i}\frac{\partial}{\partial
x_{k}}-n_{k}\frac{\partial}{\partial
x_{i}}\right)\left(\phi^{\ell}_{kij}(x/\varepsilon)\right)\left(f_{j}-\frac{\partial
P_{0}}{\partial x_{j}}\right)$

on $\partial\Omega\cap\partial\Omega^{\ell}$. It follows that

$\Big{|}\int_{\partial\Omega}n\cdot h\,d\sigma\Big{|}\leq
C\varepsilon\|\nabla(f-\nabla P_{0})\|_{L^{\infty}(\partial\Omega)}.$

Hence,

$\displaystyle\|\text{\rm
div}(\Phi_{\varepsilon}^{(2))}\|_{L^{2}(\Omega_{\varepsilon})}$
$\displaystyle\leq C|\gamma|\leq C\varepsilon\|\nabla(f-\nabla
P_{0})\|_{L^{\infty}(\partial\Omega)}$ $\displaystyle\leq
C\varepsilon\|f\|_{C^{1,1/2}(\Omega)}.$

Finally, in view of (7.20), we apply Theorem 7.3 to obtain

$\displaystyle\varepsilon\|\nabla\Phi_{\varepsilon}^{(2)}\|_{L^{2}(\Omega)}$
$\displaystyle\leq C\varepsilon^{1/2}\left\\{\|f-\nabla
P_{0}\|_{L^{\infty}(\partial\Omega)}+\varepsilon^{1/2}\|\nabla(f-\nabla
P_{0})\|_{L^{\infty}(\partial\Omega)}\right\\}$ $\displaystyle\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}.$

∎

##  8 Interface correctors

In this section we construct a corrector $\Phi_{\varepsilon}^{(3)}$ for the
interface $\Sigma$ and thus completes the proof of Theorem 1.2. Let
$D=\Omega^{\ell}$ and $D_{\varepsilon}=\Omega^{\ell}_{\varepsilon}$ for some
$1\leq\ell\leq L$. Assume that $\partial D$ has no intersection with the
boundary of the unbounded connected component of
$\mathbb{R}^{d}\setminus\overline{\Omega}$. Consider the Dirichlet problem,

$\left\\{\begin{aligned} -\Delta u_{\varepsilon}+\nabla
p_{\varepsilon}&=0&\quad&\text{ in }D_{\varepsilon},\\\ \text{\rm
div}(u_{\varepsilon})&=\gamma&\quad&\text{ in }D_{\varepsilon},\\\
u_{\varepsilon}&=0&\quad&\text{ on }\Gamma_{\varepsilon}^{\ell},\\\
u_{\varepsilon}&=h&\quad&\text{ on }\partial D,\end{aligned}\right.$ (8.1)

where $\Gamma_{\varepsilon}^{\ell}=\Gamma_{\varepsilon}\cap D$ and

$\gamma=\frac{1}{|D_{\varepsilon}|}\int_{\partial D}h\cdot n\,d\sigma.$

Let $W^{+}(y)=W^{\ell}(y)$. Fix $1\leq j\leq d$. The boundary data $h$ on
$\partial D$ in (8.1) is given as follows. Let $\partial
D=\cup_{k=1}^{k_{0}}\Sigma^{k}$, where $\Sigma^{k}$ are the connected
component of $\partial D$. On each $\Sigma^{k}$, either

$h=0\quad$ (8.2)

or

$h=W^{-}_{j}(x/\varepsilon)-W^{+}_{j}(x/\varepsilon)-W^{-}_{i}(x/\varepsilon)(K^{-}_{mj}-K^{+}_{mj})\frac{n_{i}n_{m}}{\langle
nK^{-},n\rangle},$ (8.3)

where $W^{-}(y)$ denotes the 1-periodic matrix defined by (2.1) for the
subdomain on the other side of $\Sigma^{k}$, and

$K^{+}=\int_{Y}W^{+}(y)\,dy,\qquad K^{-}=\int_{Y}W^{-}(y)\,dy.$

In particular, if $\Sigma^{k}\subset\partial\Omega$, we let $h=0$ on
$\Sigma^{k}$. Note that the repeated indices $i,m$ in (8.3) are summed from
$1$ to $d$.

######  Lemma 8.1.

Let $D$ be a bounded $C^{2,\alpha}$ domain in $\mathbb{R}^{d}$, $d\geq 2$. Let
$(u_{\varepsilon},p_{\varepsilon})$ be a weak solution of (8.1) with
$\int_{D_{\varepsilon}}p_{\varepsilon}\,dx=0$, where $h$ is given by
(8.2)-(8.3). Then

$\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(D_{\varepsilon})}+\|u_{\varepsilon}\|_{L^{2}(D_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(D_{\varepsilon})}\leq
C\sqrt{\varepsilon},$ (8.4)

and

$\|\text{\rm div}(u_{\varepsilon})\|_{L^{2}(D_{\varepsilon})}\leq
C\varepsilon.$ (8.5)

###### Proof.

We apply Theorem 7.3 with $\Omega=D$ to establish (8.4). First, observe that
by (2.4),

$\|h\|_{L^{2}(\partial D)}+\varepsilon\|\nabla_{\tan}h\|_{L^{2}(\partial
D)}\leq C.$ (8.6)

Next, we compute $u\cdot n$ on $\Sigma^{k}$, assuming $h$ is given by (8.3).
Note that

$\displaystyle h\cdot n$
$\displaystyle=n_{t}W_{tj}^{-}(x/\varepsilon)-n_{t}W^{+}_{tj}(x/\varepsilon)-n_{t}W^{-}_{ti}(x/\varepsilon)(K_{mj}^{-}-K_{mj}^{+})\frac{n_{i}n_{m}}{\langle
nK^{-},n\rangle}$ (8.7)
$\displaystyle=n_{t}\left(W_{tj}^{-}(x/\varepsilon)-K_{tj}^{-}\right)-n_{t}\left(W^{+}_{tj}(x/\varepsilon)-K_{tj}^{+}\right)$
$\displaystyle\qquad\qquad-
n_{t}\left(W^{-}_{ti}(x/\varepsilon)-K_{ti}^{-}\right)(K_{mj}^{-}-K_{mj}^{+})\frac{n_{i}n_{m}}{\langle
nK^{-},n\rangle},$

where the repeated indices $t,i,m$ are summed from $1$ to $d$. We use Lemma
2.2 to write

$n_{t}\left(W_{ti}^{\pm}(x/\varepsilon)-K_{ti}^{\pm}\right)=\frac{\varepsilon}{2}\left(n_{t}\frac{\partial}{\partial
x_{s}}-n_{s}\frac{\partial}{\partial
x_{t}}\right)\left(\phi^{\pm}_{sti}(x/\varepsilon)\right).$ (8.8)

As a result, the function in the right-hand side of (8.7) may be written in
the form $\varepsilon(\nabla_{\tan}\phi_{\varepsilon})\cdot g$ with
$(\phi_{\varepsilon},g)$ satisfying

$\|\phi_{\varepsilon}\|_{L^{2}(\partial
D)}+\|g\|_{\infty}+\|\nabla_{\tan}g\|_{\infty}\leq C.$

Consequently, the estimate (8.4) follows from (7.9) in Theorem 7.3. Finally,
note that (8.7) and (8.8) yield

$\displaystyle\|\text{\rm div}(u_{\varepsilon})\|_{L^{2}(D_{\varepsilon})}$
$\displaystyle\leq C\Big{|}\int_{\partial D}h\cdot n\,d\sigma\Big{|}$
$\displaystyle\leq C\varepsilon.$

∎

Define

$\Phi^{(3)}_{\varepsilon}=\sum_{\ell=1}^{L}I_{\varepsilon}^{\ell}(x)(f-\nabla
P_{0})\chi_{\Omega_{\varepsilon}^{\ell}}\quad\text{ in }\Omega_{\varepsilon},$
(8.9)

where
$I^{\ell}_{\varepsilon}=(I_{\varepsilon,1}^{\ell},\dots,I_{\varepsilon,d}^{\ell})$
is a solution of (8.1) in $D_{\varepsilon}=\Omega^{\ell}_{\varepsilon}$ with
$h$ given by (8.2)-(8.3). To fix the boundary value $h$ for each subdomain, we
assume that the unbounded connected component of
$\mathbb{R}^{d}\setminus\overline{\Omega}$ shares boundary with $\Omega^{1}$,
and let $h=0$ on $\partial\Omega^{1}$. Thus, $I^{1}_{\varepsilon}(x)=0$ and
$\Phi_{\varepsilon}^{(3)}=0$ in $\Omega^{1}$. Next, for each subdomain
$\Omega^{\ell}$ that shares boundaries with $\partial\Omega^{1}$, we use the
boundary data (8.3) for the common boundary with $\partial\Omega^{1}$ and let
$h=0$ on other components of $\partial\Omega^{\ell}$. We continue this
process. More precisely, at each step, we use (8.3) on the connected component
$\Sigma^{k}$ of $\partial\Omega^{\ell}$ if $\Sigma^{k}$ is also the connected
component of the boundary of a subdomain considered in the previous step, and
let $h=0$ on the remaining components. We point out that at each interface
$\Sigma^{k}$, the nonzero data (8.3) is used only once. Also, $h=0$ on
$\partial\Omega$.

######  Lemma 8.2.

Let $\Phi^{(3)}_{\varepsilon}$ be given by (8.9) with $f\in
C^{1,1/2}({\Omega;\mathbb{R}^{d}})$. Then
$V_{\varepsilon}+\Phi_{\varepsilon}^{(3)}\in
H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$.

###### Proof.

Let $\Psi_{\varepsilon}=V_{\varepsilon}+\Phi_{\varepsilon}^{(3)}$. Since $f\in
C^{1,1/2}(\Omega)$ implies that $\nabla^{2}P_{0}$ is bounded in each
subdomain, it follows that $\Psi_{\varepsilon}\in
H^{1}(\Omega_{\varepsilon}^{\ell};\mathbb{R}^{d})$ for $1\leq\ell\leq L$.
Thus, to show $\Psi_{\varepsilon}\in
H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$, it suffices to show that the
trace of $\Psi_{\varepsilon}$ is continuous across each interface
$\Sigma^{k}$.

Suppose that $\Sigma^{k}$ is the common boundary of subdomains $\Omega^{+}$
and $\Omega^{-}$. Let $\Psi_{\varepsilon}^{\pm}$ denote the trace of
$\Psi_{\varepsilon}$ on $\Sigma^{k}$, taken from $\Omega^{\pm}$ respectively.
Recall that in the definition of $\\{I_{\varepsilon}^{\ell}\\}$, the non-zero
data (8.3) is used once on each interface. Assume that the non-zero data on
$\Sigma^{k}$ is used for $\Omega^{+}$. Then

$\Psi_{\varepsilon}^{+}-\Psi_{\varepsilon}^{-}=\left(W^{+}(x/\varepsilon)+I^{+}_{\varepsilon}(x)\right)(f-\nabla
P_{0})^{+}-W^{-}(x/\varepsilon)(f-\nabla P_{0})^{-},$

where $I^{+}_{\varepsilon}$ is given by (8.3). It follows that

$\displaystyle\Psi_{\varepsilon}^{+}-\Psi_{\varepsilon}^{-}$
$\displaystyle=\left(W_{j}^{-}(x/\varepsilon)-W_{i}^{-}(x/\varepsilon)(K_{mj}^{-}-K_{mj}^{+})\frac{n_{i}n_{m}}{\langle
nK^{-},n\rangle}\right)\left(f_{j}-\frac{\partial P_{0}}{\partial
x_{j}}\right)^{+}$
$\displaystyle\qquad\qquad-W^{-}_{j}(x/\varepsilon)\left(f_{j}-\frac{\partial
P_{0}}{\partial x_{j}}\right)^{-}$
$\displaystyle=W_{j}^{-}(x/\varepsilon)\left\\{\left(\frac{\partial
P_{0}}{\partial x_{j}}\right)^{-}-\left(\frac{\partial P_{0}}{\partial
x_{j}}\right)^{+}-\frac{n_{j}n_{m}}{\langle
nK^{-},n\rangle}K_{mi}^{-}\left(f_{i}-\frac{\partial P_{0}}{\partial
x_{i}}\right)^{+}\right\\}$
$\displaystyle\qquad\qquad+W_{j}^{-}(x/\varepsilon)\frac{n_{j}n_{m}}{\langle
nK^{-},n\rangle}K_{mi}^{-}\left(f_{i}-\frac{\partial P_{0}}{\partial
x_{i}}\right)^{-},$

where we have used the observation that

$n_{m}K_{mi}^{+}\left(f_{i}-\frac{\partial P_{0}}{\partial
x_{i}}\right)^{+}=n_{m}K_{mi}^{-}\left(f_{i}-\frac{\partial P_{0}}{\partial
x_{i}}\right)^{-}$ (8.10)

on the interface. Thus,

$\displaystyle\Psi_{\varepsilon}^{+}-\Psi_{\varepsilon}^{-}$
$\displaystyle=W_{j}^{-}(x/\varepsilon)\left\\{\left(\frac{\partial
P_{0}}{\partial x_{j}}\right)^{-}-\left(\frac{\partial P_{0}}{\partial
x_{j}}\right)^{+}-\frac{n_{j}n_{m}}{\langle
nK^{-},n\rangle}K_{mi}^{-}\left(\left(\frac{\partial P_{0}}{\partial
x_{i}}\right)^{-}-\left(\frac{\partial P_{0}}{\partial
x_{i}}\right)^{+}\right)\right\\}$
$\displaystyle=W_{j}^{-}(x/\varepsilon)\left\\{\delta_{ij}-\frac{n_{j}n_{m}}{\langle
nK^{-},n\rangle}K_{mi}^{-}\right\\}\left(\left(\frac{\partial P_{0}}{\partial
x_{i}}\right)^{-}-\left(\frac{\partial P_{0}}{\partial
x_{i}}\right)^{+}\right).$

Since

$n_{i}\left\\{\delta_{ij}-\frac{n_{j}n_{m}}{\langle
nK^{-},n\rangle}K_{mi}^{-}\right\\}=0$

and $(\nabla_{\tan}P_{0})^{+}=(\nabla_{\tan}P_{0})^{-}$ on $\Sigma^{k}$, we
obtain $\Psi_{\varepsilon}^{+}=\Psi_{\varepsilon}^{-}$ on $\Sigma^{k}$. ∎

######  Theorem 8.3.

Let $\Phi_{\varepsilon}^{(3)}$ be defined by (8.9) with $f\in
C^{1,1/2}(\Omega;\mathbb{R}^{d})$. Then
$V_{\varepsilon}+\Phi_{\varepsilon}^{(3)}\in H^{1}(\Omega;\mathbb{R}^{d})$ and

$\varepsilon\|\nabla\Phi_{\varepsilon}^{(3)}\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}+\|\text{\rm
div}(\Phi_{\varepsilon}^{(3)})\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}$ (8.11)

for $1\leq\ell\leq L$.

###### Proof.

By Lemma 8.2, we have $V_{\varepsilon}+\Phi_{\varepsilon}^{(3)}\in
H^{1}(\Omega;\mathbb{R}^{d})$. Note that by Lemma 8.1,

$\varepsilon\|\nabla
I_{\varepsilon}^{\ell}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}+\|I_{\varepsilon}^{\ell}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}+\|\text{\rm
div}(I_{\varepsilon}^{\ell})\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\leq
C\varepsilon^{1/2}$

for $1\leq\ell\leq L$. It follows that

$\displaystyle\varepsilon\|\nabla\Phi_{\varepsilon}^{(3)}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}$
$\displaystyle\leq\varepsilon\|\nabla
I_{\varepsilon}^{\ell}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\|f-\nabla
P_{0}\|_{L^{\infty}(\Omega_{\varepsilon}^{\ell})}+\varepsilon\|I_{\varepsilon}^{\ell}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\|\nabla(f-\nabla
P_{0})\|_{L^{\infty}(\Omega_{\varepsilon}^{\ell})}$ $\displaystyle\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)},$

and

$\displaystyle\|\text{\rm
div}(\Phi_{\varepsilon}^{(3)})\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}$
$\displaystyle\leq\|\text{\rm
div}(I_{\varepsilon}^{\ell})\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}\|f-\nabla
P_{0}\|_{L^{\infty}(\Omega^{\ell})}+\|I^{\ell}_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\|\nabla(f-\nabla
P_{0})\|_{L^{\infty}(\Omega_{\varepsilon}^{\ell})}$ $\displaystyle\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}.$

∎

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Zhongwei Shen, Department of Mathematics, University of Kentucky, Lexington,
Kentucky 40506, USA.

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