#  Dirichlet Problems in Perforated Domains

Robert Righi Zhongwei Shen Supported in part by NSF grants DMS-1856235 and
DMS-2153585.Supported in part by NSF grants DMS-1856235 and DMS-2153585.

###### Abstract

In this paper we establish $W^{1,p}$ estimates for solutions $u_{\varepsilon}$
to Laplace’s equation with the Dirichlet condition in a bounded and
perforated, not necessarily periodically, $C^{1}$ domain
$\Omega_{\varepsilon,\eta}$ in $\mathbb{R}^{d}$. The bounding constants depend
explicitly on two small parameters $\varepsilon$ and $\eta$, where
$\varepsilon$ represents the scale of the minimal distance between holes, and
$\eta$ denotes the ratio between the size of the holes and $\varepsilon$. The
proof relies on a large-scale $L^{p}$ estimate for $\nabla u_{\varepsilon}$,
whose proof is divided into two parts. In the first part, we show that as
$\varepsilon,\eta$ approach zero, harmonic functions in
$\Omega_{\varepsilon,\eta}$ may be approximated by solutions of an
intermediate problem for a Schrödinger operator in $\Omega$. In the second
part, a real-variable method is employed to establish the large-scale $L^{p}$
estimate for $\nabla u_{\varepsilon}$ by using the approximation at scales
above $\varepsilon$. The results are sharp except in the case $d\geq 3$ and
$p=d$ or $d^{\prime}$.

Keywords: Uniform Estimates; Dirichlet Problem; Perforated Domain;
Homogenization.

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

##  1 Introduction

In this paper we continue the study of the Dirichlet problem for Laplace’s
equation,

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

where $\Omega_{\varepsilon,\eta}$ is a domain perforated, not necessarily
periodically, with a large number of tiny holes. Given $F\in
L^{p}(\Omega_{\varepsilon,\eta})$ and $f\in
L^{p}(\Omega_{\varepsilon,\eta};\mathbb{R}^{d})$, it is well known that the
Dirichlet problem (1.1) possesses a unique solution
$u_{\varepsilon}=u_{\varepsilon,\eta}$ in
$W_{0}^{1,p}(\Omega_{\varepsilon,\eta})$, if $1<p<\infty$ and
$\Omega_{\varepsilon,\eta}$ is a bounded $C^{1}$ domain in $\mathbb{R}^{d}$,
$d\geq 2$. Let $A_{p}(\Omega_{\varepsilon,\eta})$ and
$B_{p}(\Omega_{\varepsilon,\eta})$ denote the smallest constants for which the
$W^{1,p}$ estimate,

$\|\nabla u_{\varepsilon}\|_{L^{p}(\Omega_{\varepsilon,\eta})}\leq
A_{p}(\Omega_{\varepsilon,\eta})\|f\|_{L^{p}(\Omega_{\varepsilon,\eta})}+B_{p}(\Omega_{\varepsilon,\eta})\|F\|_{L^{p}(\Omega_{\varepsilon,\eta})},$
(1.2)

holds. We are interested in bounds of $A_{p}(\Omega_{\varepsilon,\eta})$ and
$B_{p}(\Omega_{\varepsilon,\eta})$ that exhibit explicit and sharp dependence
on the sizes of the holes as well as on the distances between them.

Let $Q(x,r)$ denote the open cube centered at $x$ with side length $r$. To
describe the perforated domain $\Omega_{\varepsilon,\eta}$, let
$\\{Y_{z}^{s}:z\in\mathbb{Z}^{d}\\}$ be a family of bounded domains with
connected and uniform $C^{1}$ boundaries such that

$B(0,c_{0})\subset Y_{z}^{s}\subset B(0,1/8)$ (1.3)

for some $c_{0}>0$. Let $\\{x_{z}:z\in\mathbb{Z}^{d}\\}$ be a sequence of
points in $Q(0,1/2)$ and

$T_{z,\eta}=z+x_{z}+\eta\overline{Y_{z}^{s}},$ (1.4)

where $\eta\in(0,1/2)$. For a domain $\Omega$ in $\mathbb{R}^{d}$ and
$0<\varepsilon\leq 1$, define

$\Omega_{\varepsilon,\eta}=\Omega\setminus\bigcup_{z}\varepsilon{T_{z,\eta}},$
(1.5)

where the union is taken over those $z$’s in $\mathbb{Z}^{d}$ for which
$\varepsilon(z+Y)\subset\Omega$, where $Y=Q(0,1)$. Thus, the perforated domain
$\Omega_{\varepsilon,\eta}$ is obtained from $\Omega$ by removing a hole
$\varepsilon{T_{z,\eta}}$, centered at $\varepsilon(z+x_{z})$ and of size
$\varepsilon\eta$, from each cell $\varepsilon(z+Y)$ of size $\varepsilon$ and
contained in $\Omega$. Roughly speaking, the parameter $\varepsilon$
represents the scale of the minimal distances between holes, while the
parameter $\eta$ represents the scale of the ratios between the sizes of the
holes and $\varepsilon$. We point out that the holes need not to be identical,
nor they are placed periodically, unless the sequences $\\{x_{z}\\}$, and
$\\{Y_{z}^{s}\\}$ are independent of $z$.

To state the main results for $A_{p}(\Omega_{\varepsilon,\eta})$, we note that
$A_{2}(\Omega_{\varepsilon,\eta})=1$, and that
$A_{p}(\Omega_{\varepsilon,\eta})=A_{p^{\prime}}(\Omega_{\varepsilon,\eta})$,
where $p^{\prime}=\frac{p}{p-1}$, by duality. As a result, it suffices to
consider the case $2<p<\infty$. Let

$\sigma_{\varepsilon}=\sigma_{\varepsilon,\eta}=\left\\{\begin{aligned}
&\varepsilon\eta^{1-\frac{d}{2}}&\quad&\text{ if }d\geq 3,\\\
&\varepsilon|\ln\eta|^{\frac{1}{2}}&\quad&\text{ if }d=2.\end{aligned}\right.$
(1.6)

The asymptotic behaviors of $A_{p}(\Omega_{\varepsilon,\eta})$ and
$B_{p}(\Omega_{\varepsilon,\eta})$, as $\varepsilon,\eta\to 0$, depend on
$\sigma_{\varepsilon}$. Our first theorem treats the case of relatively large
holes, where $\sigma_{\varepsilon}\leq 1$, while our second theorem handles
the case of relatively small holes, where $\sigma_{\varepsilon}\geq 1$.

######  Theorem 1.1.

Suppose that $0<\sigma_{\varepsilon}\leq 1$ and $2<p<\infty$. Let $\Omega$ be
a bounded $C^{1}$ domain in $\mathbb{R}^{d}$ and $\Omega_{\varepsilon,\eta}$
be given by (1.5). Then

$A_{p}(\Omega_{\varepsilon,\eta})\leq\left\\{\begin{aligned}
&C\eta^{-d|\frac{1}{2}-\frac{1}{p}|}&\quad&\text{ if }d\geq 3,\\\
&C\eta^{-2|\frac{1}{2}-\frac{1}{p}|}|\ln\eta|^{-\frac{1}{2}}&\quad&\text{ if
}d=2,\end{aligned}\right.$ (1.7)

where $C$ depends only on $d$, $p$, $\Omega$, and $\\{Y_{z}^{s}\\}$.

######  Theorem 1.2.

Suppose that $\sigma_{\varepsilon}\geq 1$ and $2<p<\infty$. Let $\Omega$ be a
bounded $C^{1}$ domain in $\mathbb{R}^{d}$ and $\Omega_{\varepsilon,\eta}$ be
given by (1.5). Then

$A_{p}(\Omega_{\varepsilon,\eta})\leq\left\\{\begin{aligned}
&C(1+\varepsilon^{-1}\eta^{\frac{d}{p}-1})&\quad&\text{ if }d\geq 3\text{ and
}2<p<d,\\\ &C(\varepsilon^{-1}+|\ln\eta|^{1-\frac{1}{d}})&\quad&\text{ if
}d\geq 3\text{ and }p=d,\\\
&C\varepsilon^{-1}\eta^{\frac{d}{p}-1}&\quad&\text{ if }d\geq 3\text{ and
}d<p<\infty,\\\
&C\varepsilon^{-1}\eta^{\frac{2}{p}-1}|\ln\eta|^{-1}&\quad&\text{ if
}d=2\text{ and }2<p<\infty,\end{aligned}\right.$ (1.8)

where $C$ depends only on $d$, $p$, $\Omega$, and $\\{Y_{z}^{s}\\}$.

We point out that the upper bounds for $A_{p}(\Omega_{\varepsilon,\eta})$ in
(1.7) are sharp. The upper bounds of $A_{p}(\Omega_{\varepsilon,\eta})$ in
(1.8) are also sharp for $d=2$ as well as for $d\geq 3$ and $p\neq d$. Whether
the upper bound is sharp for the remaining case, where $d\geq 3$ and $p=d$ (or
$d^{\prime}$), is not known. Indeed, if $\Omega_{\varepsilon,\eta}$ is a
periodically perforated domain, given by (1.5) with the sequences
$\\{x_{z}\\}$ and $\\{Y_{z}^{s}\\}$ independent of $z$, we will show that

$A_{p}(\Omega_{\varepsilon,\eta})\geq\left\\{\begin{aligned}
&c\,\eta^{-d|\frac{1}{2}-\frac{1}{p}|}&\quad&\text{ if }d\geq 3,\\\
&c\,\eta^{-2|\frac{1}{2}-\frac{1}{p}|}|\ln\eta|^{-\frac{1}{2}}&\quad&\text{ if
}d=2,\end{aligned}\right.$ (1.9)

in the case $\sigma_{\varepsilon}\leq 1$, and that

$A_{p}(\Omega_{\varepsilon,\eta})\geq\left\\{\begin{aligned}
&c(1+\varepsilon^{-1}\eta^{\frac{d}{p}-1})&\quad&\text{ if }d\geq 3\text{ and
}2<p<d,\\\ &c\varepsilon^{-1}\eta^{\frac{d}{p}-1}&\quad&\text{ if }d\geq
3\text{ and }d\leq p<\infty,\\\
&c\varepsilon^{-1}\eta^{\frac{2}{p}-1}|\ln\eta|^{-1}&\quad&\text{ if
}d=2\text{ and }2<p<\infty,\end{aligned}\right.$ (1.10)

in the case $\sigma_{\varepsilon}\geq 1$. The constants $c>0$ in (1.9)-(1.10)
are independent of $\varepsilon$ and $\eta$. See Theorem 9.6. Note that in the
case $d\geq 3$ and $p=d$, our upper and lower bounds differ by a term
$C|\ln\eta|^{1-\frac{1}{d}}$.

The main results for $B_{p}(\Omega_{\varepsilon,\eta})$ are given in Theorems
3.1, 3.2 and 3.3. The question of their sharpness is addressed in Section 9.

Motivated by the homogenization theory of boundary value problems for elliptic
equations in perforated domains [8, 5, 1, 2, 7, 4], the study of the uniform
$W^{1,p}$ estimate (1.2) in $\Omega_{\varepsilon,\eta}$ was initiated by N.
Masmoudi [9] in the case $\eta=1$. The general case $0<\eta<1$ was first
studied by the second author of the present paper in [13]. Both the bounded
domain $\Omega_{\varepsilon,\eta}$ and the unbounded perforated domain,

$\omega_{\varepsilon,\eta}=\mathbb{R}^{d}\setminus\bigcup_{z\in\mathbb{Z}^{d}}\varepsilon(z+\eta\overline{Y_{z}^{s}}),$
(1.11)

were considered in [13], where $\Omega_{\varepsilon,\eta}$ and
$\omega_{\varepsilon,\eta}$ need not to be periodically perforated. The upper
bounds obtained in [13] for $A_{p}$ are not sharp, although they are only off
by an arbitrary small power of $\eta$. In [14] J. Wallace and the second
author studied the case of the unbounded domain $\omega_{\varepsilon,\eta}$.
Under the additional assumption that the holes $\\{Y^{s}_{z}\\}$ are identical
and thus $\omega_{\varepsilon,\eta}$ is periodic, they were able to obtain the
estimate (1.2) with sharp constants $A_{p}$ and $B_{p}$. The argument in [14]
uses a large-scale Lipschitz estimate for harmonic functions $u_{\varepsilon}$
in perforated domains, whose proof relies on the observation that the
difference $u_{\varepsilon}(x+\varepsilon e_{j})-u_{\varepsilon}(x)$ is also
harmonic. It is not clear how to extend this proof to the non-periodic case
and to the case of bounded domains.

In this paper we introduce a new approach to the study of $W^{1,p}$ estimates
in perforated domains, which allows us to handle the case of non-periodic
holes as well as boundary estimates for bounded perforated domains. To show
Theorems 1.1 and 1.2, we start with a localization argument as in [14] and
reduce the $L^{p}$ estimates of $\nabla u_{\varepsilon}$ to an $L^{p}$
estimate of the operator $S_{\varepsilon}$ defined by

$S_{\varepsilon}(F,f)=\left(\fint_{Q(x,2\varepsilon)}|\nabla
u_{\varepsilon}|^{2}\right)^{1/2},$ (1.12)

where $u_{\varepsilon}$ is the solution of (1.1), extended to $\mathbb{R}^{d}$
by zero. It is not hard to see that

$\|S_{\varepsilon}(F,f)\|_{L^{2}(\mathbb{R}^{d})}=\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon,\eta})}.$

Thus, by energy estimates,

$\|S_{\varepsilon}(F,f)\|_{L^{2}(\mathbb{R}^{d})}\leq\|f\|_{L^{2}(\Omega_{\varepsilon,\eta})}+C\min(\sigma_{\varepsilon},1)\|F\|_{L^{2}(\Omega_{\varepsilon,\eta})}.$
(1.13)

######  Theorem 1.3.

Let $\Omega$ be a bounded $C^{1}$ domain in $\mathbb{R}^{d}$ and
$\Omega_{\varepsilon,\eta}$ be given by (1.5). Then

$\|S_{\varepsilon}(F,f)\|_{L^{p}(\mathbb{R}^{d})}\leq
C\left\\{\|f\|_{L^{p}(\Omega_{\varepsilon,\eta})}+\min(\sigma_{\varepsilon},1)\|F\|_{L^{p}(\Omega_{\varepsilon,\eta})}\right\\}$
(1.14)

for $2<p<\infty$, where $C$ depends only on $d$, $p$, $\\{Y_{z}^{s}\\}$ and
$\Omega$.

By a localization argument as well as $L^{p}$ estimates for $u_{\varepsilon}$
in [13], Theorems 1.1 and 1.2 follow from Theorem 1.3. Since $S_{\varepsilon}$
is defined by averaging $\nabla u_{\varepsilon}$ over a cell of size
$2\varepsilon$, the estimate (1.14) should be regarded as a large-scale
$W^{1,p}$ estimate for $u_{\varepsilon}$. Theorem 1.3 shows that such average
of $\nabla u_{\varepsilon}$ behaves much better than $\nabla u_{\varepsilon}$
in $L^{p}$ spaces for $p\neq 2$. Indeed, as $\varepsilon,\eta\to 0$,
$A_{p}(\Omega_{\varepsilon,\eta})\to\infty$ for $p\neq 2$ and
$\sigma_{\varepsilon}\leq 1$, while the operator norm
$\|S_{\varepsilon}(0,\cdot)\|_{L^{p}\to L^{p}}$ remains bounded.

To prove Theorem 1.3, we use a real-variable method from [10], which reduces
the problem to a reverse Hölder inequality for harmonic functions
$u_{\varepsilon}$ in perforated domains. The proof of the reverse Hölder
inequality again relies on the same real-variable argument. As such, for each
subdomain $D$ of size greater than $\varepsilon$, one needs to approximate
$\nabla u_{\varepsilon}$ by a function that behaves well in the $L^{p}$ norm.
To this end, for each hole in the domain, we introduce an associated
nonnegative potential, supported in a neighborhood of the hole. Let
$V_{\varepsilon}$ denote the sum of these potentials. A key observation of
this paper is that a harmonic function $u_{\varepsilon}$ in a perforated
domain $D_{\varepsilon,\eta}$ is well approximated by
$\chi_{\varepsilon,\eta}v_{\varepsilon}$, where $\chi_{\varepsilon,\eta}$ is a
corrector for $D_{\varepsilon,\eta}$ with $\chi_{\varepsilon,\eta}=1$ on
$\partial D$, and $v_{\varepsilon}$ is the solution of an intermediate
equation,

$(-\Delta+\sigma_{\varepsilon}^{-2}V_{\varepsilon})v_{\varepsilon}=0$ (1.15)

in the non-perforated domain $D$ with boundary data
$v_{\varepsilon}=u_{\varepsilon}$ on $\partial D$. We note that even though
the holes may not be periodically placed, the potential $V_{\varepsilon}$
behaves like a constant. As a result, $\nabla v_{\varepsilon}$ satisfies the
reverse Hölder inequality for any $p>2$.

The rest of the paper is organized as follows. In Section 2 we give the proof
of Theorems 1.1 and 1.2, using Theorem 1.3. The estimates of bounding
constants $B_{p}(\Omega_{\varepsilon,\eta})$ in (1.2) are given in Section 3.
Sections 4 \- 8 are devoted to the proof of Theorem 1.3. Specifically, the
construction and estimates of the correctors $\chi_{\varepsilon,\eta}$ are
given in Sections 4 and 5. The estimates for solutions of the intermediate
problem (1.15) are given in Section 7. A convergence rate for
$\nabla(u_{\varepsilon}-\chi_{\varepsilon,\eta}v_{\varepsilon})$ in $L^{2}$ is
established in Section 6, and the proof of Theorem 1.3 is given in Section 8.
Finally, we address the question of sharpness of Theorems 1.1 and 1.2 in
Section 9.

##  2 Reduction to the large-scale estimates

Recall $Q(x,r)$ denotes the open cube centered at $x$ with size length $r$.
Let $\\{Y_{z}^{s}:z\in\mathbb{Z}^{d}\\}$ be a family of bounded domains with
connected and uniform $C^{1}$ boundaries such that $B(0,c_{0})\subset
Y_{z}^{s}\subset B(0,1/8)$. Let $\\{x_{z}:z\in\mathbb{Z}^{d}\\}$ be a sequence
of points in $Q(0,1/2)$. For $z\in\mathbb{Z}^{d}$, we define $Q_{z}=z+Q(0,1)$
and $T_{z,\eta}=z+x_{z}+\eta\overline{Y_{z}^{s}}$, where $\eta\in(0,1/2)$.
Note that $B(z+x_{z},c_{0}\eta)\subset T_{z,\eta}\subset Q_{z}$ and
dist$(T_{z,\eta},\partial Q_{z})\geq c>0$.

The following lemma will be useful to us. Its proof for $p=2$ is well known
[1]. The proof for $p\neq 2$ is similar and can be found in [13, Lemma 2.1].

######  Lemma 2.1.

Let $d\geq 2$ and $1\leq p<\infty$. Suppose that $u\in
W^{1,p}(Q(0,\varepsilon))$ and $u=0$ on $B(x_{0},\varepsilon\eta)$ for some
$x_{0}\in Q(0,\varepsilon/2)$ and $0<\eta<1/4$. Then

$\int_{Q(0,\varepsilon)}|u|^{p}\,dx\leq C\int_{Q(0,\varepsilon)}|\nabla
u|^{p}\,dx\cdot\begin{cases}\begin{aligned}
&\varepsilon^{p}\eta^{p-d}\qquad&&\text{if}\ 1\leq p<d,\\\
&\varepsilon^{p}|\ln\eta|^{d-1}\qquad&&\text{if}\ p=d,\\\
&\varepsilon^{p}\qquad&&\text{if}\ d<p<\infty,\end{aligned}\end{cases}$ (2.1)

where $C$ depends on $d$ and $p$.

######  Lemma 2.2.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{d}$ and
$\Omega_{\varepsilon,\eta}$ be given by (1.5). Then

$\|u\|_{L^{2}(\Omega_{\varepsilon,\eta})}\leq
C\min(\sigma_{\varepsilon},1)\|\nabla u\|_{L^{2}(\Omega_{\varepsilon,\eta})}$
(2.2)

for any $u\in H_{0}^{1}(\Omega_{\varepsilon,\eta})$, where
$\sigma_{\varepsilon}$ is given by (1.6) and $C$ is independent of
$\varepsilon$ and $\eta$.

###### Proof.

It follows from Lemma 2.1 that if $u\in H^{1}(\Omega_{\varepsilon,\eta})$ and
$u=0$ on $\partial\Omega_{\varepsilon,\eta}\setminus\partial\Omega$, then

$\|u\|_{L^{2}(\Omega_{\varepsilon,\eta})}\leq C\sigma_{\varepsilon}\|\nabla
u\|_{L^{2}(\Omega_{\varepsilon,\eta})}.$ (2.3)

This, together with the Poincaré inequality $\|u\|_{L^{2}(\Omega)}\leq
C\|\nabla u\|_{L^{2}(\Omega)}$ for $u\in H_{0}^{1}(\Omega)$, yields (2.2) for
$u\in H_{0}^{1}(\Omega_{\varepsilon,\eta})$. ∎

For $F\in L^{2}(\Omega)$ and $f\in L^{2}(\Omega;\mathbb{R}^{d})$, let
$u_{\varepsilon}\in H_{0}^{1}(\Omega_{\varepsilon,\eta})$ be the solution of
the Dirichlet problem (1.1). Using (2.2), it is not hard to show that

$\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega)}\leq\|f\|_{L^{2}(\Omega)}+C\min(\sigma_{\varepsilon},1)\|F\|_{L^{2}(\Omega)},$
(2.4)

where we have extended $u_{\varepsilon}$ to $\Omega$ by zero. The following
theorem gives the $L^{p}$ estimate of $u_{\varepsilon}$ for $p>2$.

######  Theorem 2.3.

Let $F\in L^{p}(\Omega)$ and $f\in L^{p}(\Omega;\mathbb{R}^{d})$ for some
$p\geq 2$. Let $u_{\varepsilon}$ be the solution of (1.1), where $\Omega$ is a
bounded Lipschitz domain. Then

$\|u_{\varepsilon}\|_{L^{p}(\Omega)}\leq
C\min(\sigma_{\varepsilon},1)\|f\|_{L^{p}(\Omega)}+C\min(\sigma_{\varepsilon}^{2},1)\|F\|_{L^{p}(\Omega)},$
(2.5)

where $C$ depends only on $d$, $p$, $c_{0}$ and $\Omega$.

###### Proof.

The case $p=2$ follows readily from the energy estimate (2.4) and (2.2). The
case $p>2$ was proved in [13]. ∎

To establish the $L^{p}$ estimates for $\nabla u_{\varepsilon}$, we consider
the local $W^{1,p}$ estimates for solutions of

$\left\\{\begin{aligned} -\Delta u_{\varepsilon}&=F+\text{\rm
div}(f)&\quad&\text{ in }\varepsilon(\widetilde{Q}_{z}\setminus
T_{z,\eta}),\\\ u_{\varepsilon}&=0&\quad&\text{ in }\varepsilon
T_{z,\eta},\end{aligned}\right.$ (2.6)

where $z\in\mathbb{Z}^{d}$ and $\widetilde{Q}_{z}=Q(z,17/16)$. Let

$\Phi_{p}(R)=\begin{cases}1&\text{if}\ d\geq 3\ \text{and}\ 2<p<d,\\\ (\ln
R)^{1-\frac{1}{d}}&\text{if}\ d\geq 3\ \text{and}\ p=d,\\\
R^{1-\frac{d}{p}}&\text{if}\ d\geq 3\ \text{and}\ d<p<\infty,\\\
R^{1-\frac{2}{p}}(\ln R)^{-1}&\text{if}\ d=2\ \text{and}\
2<p<\infty,\end{cases}$ (2.7)

for $R>2$.

######  Lemma 2.4.

Let $2<p<\infty$. Let $u_{\varepsilon}$ be a solution of (2.6) with $F\in
L^{p}(\varepsilon\widetilde{Q}_{z})$ and $f\in
L^{p}(\varepsilon\widetilde{Q}_{z};\mathbb{R}^{d})$. Then for $d\geq 3$,

$\begin{split}\left(\fint_{\varepsilon Q_{z}}|\nabla
u_{\varepsilon}|^{p}\right)^{1/p}\leq
C|\alpha|\varepsilon^{-1}\eta^{\frac{d}{p}-1}+C\Phi_{p}(\eta^{-1})\left(\fint_{\varepsilon\widetilde{Q}_{z}}(|\varepsilon
F|+|f|)^{p}\right)^{1/p}\\\
+C\varepsilon^{-1}\Phi_{p}(\eta^{-1})\left(\fint_{\varepsilon\widetilde{Q}_{z}\backslash
Q(y_{z},\varepsilon/3)}|u_{\varepsilon}-\alpha|^{2}dx\right)^{1/2},\end{split}$
(2.8)

and for $d=2$

$\begin{split}\left(\fint_{\varepsilon Q_{z}}|\nabla
u_{\varepsilon}|^{p}\right)^{1/p}\leq
C|\alpha|\varepsilon^{-1}\Phi_{p}(\eta^{-1})+C\Phi_{p}(\eta^{-1})\left(\fint_{\varepsilon\widetilde{Q}_{z}}(|\varepsilon
F|+|f|)^{p}\right)^{1/p}\\\
+C\varepsilon^{-1}\Phi_{p}(\eta^{-1})\left(\fint_{\varepsilon\widetilde{Q}_{z}\backslash
Q(y_{z},\varepsilon/3)}|u_{\varepsilon}-\alpha|^{2}dx\right)^{1/2},\end{split}$
(2.9)

where $\alpha\in\mathbb{R}$, $y_{z}=z+x_{z}$ and $C$ depends only on $d$, $p$
and $\\{Y_{z}^{s}\\}$.

###### Proof.

The case where $\varepsilon=1$ and $x_{z}=0$ was proved in [14, Theorem 6.1],
using a result in [3]. The same argument, together with the correctors
constructed in this paper, also gives (2.8)-(2.9) for $\varepsilon=1$ and
$x_{z}\in Q(0,1/2)$ with constants independent of $x_{z}$. We point out that
there is a minor flaw in the proof of Theorem 6.1 in [14]. In the case $d=2$,
the corrector defined in [14, (6.6)] does not satisfy the condition in [14,
(6.11)]. The mistake can be fixed easily by using the corrector in Section 5.

The general case $0<\varepsilon<1$ follows readily from the case
$\varepsilon=1$ by dilation. Note that the parameter $\eta$ remains invariant
under dilation. ∎

######  Remark 2.5.

Let $u_{\varepsilon}$ be the solution of (1.1) with $F\in L^{p}(\Omega)$ and
$f\in L^{p}(\Omega;\mathbb{R}^{d})$ for some $p>2$. Let

$\Omega^{\prime}_{\varepsilon,\eta}=\bigcup_{z}\varepsilon(Q_{z}\setminus
T_{z,\eta}),$ (2.10)

where the union is taken over those $z$’s in $\mathbb{Z}^{d}$ for which
$\varepsilon Q_{z}\subset\Omega$. It follows from (2.8)-(2.9) with $\alpha=0$
and a covering argument that

$\int_{\Omega^{\prime\prime}_{\varepsilon,\eta}}|\nabla
u_{\varepsilon}|^{p}\leq
C|\Phi_{p}(\eta^{-1})|^{p}\int_{\Omega}(\varepsilon|F|+|f|)^{p}+C\varepsilon^{-p}|\Phi_{p}(\eta^{-1})|^{p}\int_{\Omega}|u_{\varepsilon}|^{p},$
(2.11)

where
$\Omega_{\varepsilon,\eta}^{\prime\prime}=\\{x\in\Omega^{\prime}_{\varepsilon,\eta}:\text{\rm
dist}(x,\partial\Omega)\geq c\varepsilon\\}$. For the region near
$\partial\Omega$ and away from the holes, we use the boundary $W^{1,p}$
estimates for Laplace’s equation in $C^{1}$ domains [6] to obtain

$\int_{B(x,c\varepsilon)}|\nabla u_{\varepsilon}|^{p}\leq
C\varepsilon^{-p}\int_{B(x,2c\varepsilon)}|u_{\varepsilon}|^{p}+C\int_{B(x,2c\varepsilon)}(\varepsilon|F|+|f|)^{p},$
(2.12)

where $x\in\Omega$ and
$B(x,2c\varepsilon)\cap\Omega=B(x,2c\varepsilon)\cap\Omega_{\varepsilon,\eta}$.
By a covering argument we obtain

$\int_{\Omega_{\varepsilon,\eta}\setminus\Omega_{\varepsilon,\eta}^{\prime\prime}}|\nabla
u_{\varepsilon}|^{p}\leq
C\int_{\Omega}(\varepsilon|F|+|f|)^{p}+C\varepsilon^{-p}\int_{\Omega_{\varepsilon,\eta}}|u_{\varepsilon}|^{p}.$
(2.13)

This, together with (2.11), gives

$\displaystyle\int_{\Omega}|\nabla u_{\varepsilon}|^{p}$ $\displaystyle\leq
C|\Phi_{p}(\eta^{-1})|^{p}\int_{\Omega}(\varepsilon|F|+|f|)^{p}+C\varepsilon^{-p}|\Phi_{p}(\eta^{-1})|^{p}\int_{\Omega}|u_{\varepsilon}|^{p}$
$\displaystyle\leq
C|\Phi_{p}(\eta^{-1})|^{p}\left(1+\varepsilon^{-1}\min(\sigma_{\varepsilon},1)\right)^{p}\int_{\Omega}|f|^{p}$
$\displaystyle\qquad\qquad+C|\Phi_{p}(\eta^{-1})|^{p}\left(1+\varepsilon^{-1}\min(\sigma_{\varepsilon},1)\right)^{2p}\int_{\Omega}|\varepsilon
F|^{p},$

where we have used (2.5) for the last inequality. As a result, we have proved
that

$\displaystyle\|\nabla u_{\varepsilon}\|_{L^{p}(\Omega)}$ $\displaystyle\leq
C\Phi_{p}(\eta^{-1})(1+\varepsilon^{-1}\min(\sigma_{\varepsilon},1))\|f\|_{L^{p}(\Omega)}$
(2.14)
$\displaystyle\quad\qquad+C\Phi_{p}(\eta^{-1})\left(1+\varepsilon^{-1}\min(\sigma_{\varepsilon},1)\right)^{2}\|\varepsilon
F\|_{L^{p}(\Omega)}$

for $p>2$ and $d\geq 2$. Suppose $F=0$. In the case
$0<\sigma_{\varepsilon}\leq 1$, $d\geq 3$ and $p>d$, it follows from (2.14)
that

$\|\nabla u_{\varepsilon}\|_{L^{p}(\Omega)}\leq
C\eta^{\frac{d}{p}-\frac{d}{2}}\|f\|_{L^{p}(\Omega)}.$ (2.15)

Since (2.15) also holds for $p=2$, by Riesz Thorin Interpolation Theorem,
(2.15) holds for $2<p<\infty$. This gives the proof of Theorem 3.1 for the
case $d\geq 3$.

######  Remark 2.6.

Suppose $\sigma_{\varepsilon}\geq 1$. It follows from (2.14) that if $F=0$,

$\|\nabla u_{\varepsilon}\|_{L^{p}(\Omega)}\leq
C\varepsilon^{-1}\Phi_{p}(\eta^{-1})\|f\|_{L^{p}(\Omega)}.$ (2.16)

It follows that

$\|\nabla u_{\varepsilon}\|_{L^{p}(\Omega)}\leq\left\\{\begin{aligned}
&C\varepsilon^{-1}\eta^{\frac{d}{p}-1}\|f\|_{L^{p}(\Omega)}&\quad&\text{ if
}d\geq 3\text{ and }p>d,\\\
&C\varepsilon^{-1}\eta^{\frac{2}{p}-1}|\ln\eta|^{-1}\|f\|_{L^{p}(\Omega)}&\quad&\text{
if }d=2\text{ and }2<p<\infty.\end{aligned}\right.$ (2.17)

This gives the estimates in Theorem 1.2 for the case $d\geq 2$ and
$d<p<\infty$.

To deal with the remaining cases in Theorem 1.1 and 1.2, we let

$\alpha=\fint_{\varepsilon\widetilde{Q}_{z}\setminus
Q(y_{z},\varepsilon/3)}u_{\varepsilon}$

in (2.8)-(2.9). By Poincaré’s inequality we obtain

$\displaystyle\left(\fint_{\varepsilon Q_{z}}|\nabla
u_{\varepsilon}|^{p}\right)^{1/p}\leq
C\varepsilon^{-1}\eta^{\frac{d}{p}-1}\fint_{\varepsilon\widetilde{Q}_{z}}|u_{\varepsilon}|$
$\displaystyle+C\Phi_{p}(\eta^{-1})\left(\fint_{\varepsilon\widetilde{Q}_{z}}(|\varepsilon
F|+|f|)^{p}\right)^{1/p}$ (2.18)
$\displaystyle+C\Phi_{p}(\eta^{-1})\left(\fint_{\varepsilon\widetilde{Q}_{z}}|\nabla
u_{\varepsilon}|^{2}\right)^{1/2}$

for $d\geq 3$, and

$\displaystyle\left(\fint_{\varepsilon Q_{z}}|\nabla
u_{\varepsilon}|^{p}\right)^{1/p}\leq
C\varepsilon^{-1}\Phi_{p}(\eta^{-1})\fint_{\varepsilon\widetilde{Q}_{z}}|u_{\varepsilon}|$
$\displaystyle+C\Phi_{p}(\eta^{-1})\left(\fint_{\varepsilon\widetilde{Q}_{z}}(|\varepsilon
F|+|f|)^{p}\right)^{1/p}$ (2.19)
$\displaystyle+C\Phi_{p}(\eta^{-1})\left(\fint_{\varepsilon\widetilde{Q}_{z}}|\nabla
u_{\varepsilon}|^{2}\right)^{1/2}$

for $d=2$. Define

$S_{\varepsilon}(F,f)(x)=\left(\fint_{Q(x,2\varepsilon)}|\nabla
u_{\varepsilon}|^{2}\right)^{1/2}.$

It follows from (2.18)-(2.19) by a covering argument that

$\displaystyle\int_{\Omega_{\varepsilon,\eta}^{\prime\prime}}|\nabla
u_{\varepsilon}|^{p}\leq
C\varepsilon^{-p}\eta^{d-p}\int_{\Omega}|u_{\varepsilon}|^{p}$
$\displaystyle+C|\Phi_{p}(\eta^{-1})|^{p}\int_{\Omega}(|\varepsilon
F|+|f|)^{p}$ (2.20)
$\displaystyle+C|\Phi_{p}(\eta^{-1})|^{p}\int_{\Omega}|S_{\varepsilon}(F,f)|^{p}$

for $d\geq 3$, and that

$\displaystyle\int_{\Omega_{\varepsilon,\eta}^{\prime\prime}}|\nabla
u_{\varepsilon}|^{p}\leq
C\varepsilon^{-p}|\Phi_{p}(\eta^{-1})|^{p}\int_{\Omega}|u_{\varepsilon}|^{p}$
$\displaystyle+C|\Phi_{p}(\eta^{-1})|^{p}\int_{\Omega}(|\varepsilon
F|+|f|)^{p}$ (2.21)
$\displaystyle+C\Phi_{p}(\eta^{-1})|^{p}\int_{\Omega}|S_{\varepsilon}(F,f)|^{p}$

for $d=2$, where $\Omega^{\prime\prime}_{\varepsilon,\eta}$ is the same as in
(2.11). For the region near $\partial\Omega$ and away from the holes, we use
the local $W^{1,p}$ estimate in $C^{1}$ domains,

$\left(\fint_{Q(x,c\varepsilon)}|\nabla u_{\varepsilon}|^{p}\right)^{1/p}\leq
C\left(\fint_{Q(x,2\varepsilon)}|\nabla
u_{\varepsilon}|^{2}\right)^{1/2}+C\left(\fint_{Q(x,2\varepsilon)}(|\varepsilon
F|+|f|)^{p}\right)^{1p},$ (2.22)

where $x\in\Omega$, dist$(x,\partial\Omega)\leq C\varepsilon$ and
$B(x,2c\varepsilon)\cap\Omega=B(x,2c\varepsilon)\cap\Omega_{\varepsilon,\eta}$.
By a covering argument, this leads to

$\int_{\Omega_{\varepsilon,\eta}\setminus\Omega_{\varepsilon,\eta}^{\prime\prime}}|\nabla
u_{\varepsilon}|^{p}\leq
C\int_{\Omega}|S_{\varepsilon}(F,f)|^{p}+C\int_{\Omega}(|\varepsilon
F|+|f|)^{p}.$ (2.23)

######  Lemma 2.7.

Let $\Omega$ be a bounded $C^{1}$ domain in $\mathbb{R}^{d}$. Let
$u_{\varepsilon}\in H_{0}^{1}(\Omega_{\varepsilon,\eta})$ be the solution of
(1.1) with $F\in L^{p}(\Omega)$ and $f\in L^{p}(\Omega;\mathbb{R}^{d})$ for
some $p>2$. Then

$\displaystyle\|\nabla u_{\varepsilon}\|_{L^{p}(\Omega)}$ $\displaystyle\leq
C\varepsilon^{-1}\eta^{\frac{d}{p}-1}\|u_{\varepsilon}\|_{L^{p}(\Omega)}$
(2.24)
$\displaystyle\qquad+C\Phi_{p}(\eta^{-1})\left\\{\|\varepsilon|F|+|f|\|_{L^{p}(\Omega)}+\|S_{\varepsilon}(F,f)\|_{L^{p}(\Omega)}\right\\}$

for $d\geq 3$, and

$\displaystyle\|\nabla u_{\varepsilon}\|_{L^{p}(\Omega)}$ $\displaystyle\leq
C\varepsilon^{-1}\Phi_{p}(\eta^{-1})\|u_{\varepsilon}\|_{L^{p}(\Omega)}$
(2.25)
$\displaystyle\qquad+C\Phi_{p}(\eta^{-1})\left\\{\|\varepsilon|F|+|f|\|_{L^{p}(\Omega)}+\|S_{\varepsilon}(F,f)\|_{L^{p}(\Omega)}\right\\}$

for $d=2$, where $C$ depends on $d$, $p$, $\\{Y_{z}^{s}\\}$ and $\Omega$.

###### Proof.

This follows from (2.20)- (2.21) and (2.23). ∎

Using Theorem 1.3, we now give the proof of Theorems 1.1 and 1.2.

###### Proof of Theorem 1.1.

Suppose $0<\sigma_{\varepsilon}\leq 1$ and $2<p<\infty$. The case $d\geq 3$ is
treated already in Remark 2.5. Assume $d=2$. Then
$\sigma_{\varepsilon}=\varepsilon|\ln\eta|^{1/2}$ and
$\Phi_{p}(\eta^{-1})=\eta^{\frac{2}{p}-1}|\ln\eta|^{-1}$. Let
$u_{\varepsilon}$ be the solution of (1.1) with $F=0$. By (2.25) and Theorem
1.3,

$\displaystyle\|\nabla u_{\varepsilon}\|_{L^{p}(\Omega)}$ $\displaystyle\leq
C\varepsilon^{-1}\Phi_{p}(\eta^{-1})\|u_{\varepsilon}\|_{L^{p}(\Omega)}+C\Phi_{p}(\eta^{-1})\|f\|_{L^{p}(\Omega)}$
(2.26) $\displaystyle\leq
C\Phi_{p}(\eta^{-1})\varepsilon^{-1}\sigma_{\varepsilon}\|f\|_{L^{p}(\Omega)}$
$\displaystyle=C\eta^{\frac{2}{p}-1}|\ln\eta|^{-\frac{1}{2}}\|f\|_{L^{p}(\Omega)},$

where we have used Theorem 2.3 for the second inequality. ∎

###### Proof of Theorem 1.2.

Suppose $\sigma_{\varepsilon}\geq 1$ and $2<p<\infty$. The case for $d\geq 2$
and $d<p<\infty$ is treated in Remark 2.6. Assume $d\geq 3$ and $2<p\leq d$.
Let $u_{\varepsilon}$ be the solution of (1.1) with $F=0$. It follows from
(2.24) and Theorem 1.3 that

$\displaystyle\|\nabla u_{\varepsilon}\|_{L^{p}(\Omega)}$ $\displaystyle\leq
C\varepsilon^{-1}\eta^{\frac{d}{p}-1}\|u_{\varepsilon}\|_{L^{p}(\Omega)}+C\Phi_{p}(\eta^{-1})\|f\|_{L^{p}(\Omega)}$
(2.27) $\displaystyle\leq
C\left(\varepsilon^{-1}\eta^{\frac{d}{p}-1}+\Phi_{p}(\eta^{-1})\right)\|f\|_{L^{p}(\Omega)},$

where we have used Theorem 2.3 for the last inequality. In view of (2.7) we
obtain

$\|\nabla
u_{\varepsilon}\|_{L^{p}(\Omega)}\leq\left(\varepsilon^{-1}\eta^{\frac{d}{p}-1}+1\right)\|f\|_{L^{p}(\Omega)},$

for $2<p<d$, and

$\|\nabla
u_{\varepsilon}\|_{L^{p}(\Omega)}\leq\left(\varepsilon^{-1}+|\ln\eta|^{1-\frac{1}{d}}\right)\|f\|_{L^{p}(\Omega)}$

for $p=d$. This gives the estimates in Theorem 1.2 for the case $2<p\leq d$. ∎

##  3 Estimates of $B_{p}(\Omega_{\varepsilon,\eta})$

In this section we establish upper bounds for
$B_{p}(\Omega_{\varepsilon,\eta})$ in (1.2). We begin with the case $1<p\leq
2$. The estimate (3.1) below is sharp. See Theorem 9.4.

######  Theorem 3.1.

Suppose $1<p\leq 2$. Let $\Omega$ be a bounded Lipschitz domain in
$\mathbb{R}^{d}$, $d\geq 2$, and $\Omega_{\varepsilon,\eta}$ be given by
(1.5). Then

$\displaystyle B_{p}(\Omega_{\varepsilon,\eta})\leq
C\min(\sigma_{\varepsilon},1),$ (3.1)

where $C$ depends only on $d$, $p$, $\Omega$, and $\\{Y_{z}^{s}\\}$.

###### Proof.

This was proved in [13], using (2.5) and a duality argument. ∎

Next, we consider the case $2<p<\infty$ and $0<\sigma_{\varepsilon}\leq 1$.
The estimate (3.2) below is sharp for $d\geq 3$ and $2<p<\infty$. See Theorem
9.5.

######  Theorem 3.2.

Suppose that $0<\sigma_{\varepsilon}\leq 1$ and $2<p<\infty$. Let $\Omega$ be
a bounded $C^{1}$ domain in $\mathbb{R}^{d}$ and $\Omega_{\varepsilon}$ be
given by (1.5). Then

$B_{p}(\Omega_{\varepsilon,\eta})\leq C\varepsilon\eta^{1-d+\frac{d}{p}},$
(3.2)

where $C$ depends only on $d$, $p$, $\Omega$, and $\\{Y_{z}^{s}\\}$.

###### Proof.

It follows from (2.14) that

$B_{p}(\Omega_{\varepsilon,\eta})\leq
C\Phi_{p}(\eta^{-1})\left\\{1+\varepsilon^{-1}\min(\sigma_{\varepsilon},1)\right\\}^{2}\varepsilon.$
(3.3)

Using $0<\sigma_{\varepsilon}\leq 1$, we obtain

$B_{p}(\Omega_{\varepsilon,\eta})\leq
C\Phi_{p}(\eta^{-1})\sigma_{\varepsilon}^{2}\varepsilon^{-1}.$ (3.4)

In the case $d=2$, since
$\Phi_{p}(\eta^{-1})=\eta^{\frac{2}{p}-1}|\ln\eta|^{-1}$ and
$\sigma_{\varepsilon}=\varepsilon|\ln\eta|^{1/2}$, this gives (3.2).

In the case $d\geq 3$ and $p>d$, we have
$\sigma_{\varepsilon}=\varepsilon\eta^{1-\frac{d}{2}}$ and
$\Phi_{p}(\eta^{-1})=\eta^{\frac{d}{p}-1}$. It follows from (3.4) that (3.2)
holds. Since (3.2) also holds for $p=2$, by Riesz-Thorin Interpolation
Theorem, the estimate (3.2) holds for all $2<p<\infty$. ∎

Finally, we treat the case $2<p<\infty$ and $\sigma_{\varepsilon}\geq 1$. The
estimate (3.5) below is sharp in the case $d\geq 3$ and $2<p<d$. See Theorem
9.5.

######  Theorem 3.3.

Suppose that $\sigma_{\varepsilon}\geq 1$ and $2<p<\infty$. Let $\Omega$ be a
bounded $C^{1}$ domain in $\mathbb{R}^{d}$ and $\Omega_{\varepsilon,\eta}$ be
given by (1.5). Then

$B_{p}(\Omega_{\varepsilon,\eta})\leq\left\\{\begin{aligned}
&C(1+\varepsilon^{-1}\eta^{\frac{d}{p}-1})&\quad&\text{ if }d\geq 3\text{ and
}2<p<d,\\\ &C(\varepsilon^{-1}+|\ln\eta|^{1-\frac{1}{d}})&\quad&\text{ if
}d\geq 3\text{ and }p=d,\\\
&C\varepsilon^{-1}\eta^{\frac{d}{p}-1}&\quad&\text{ if }d\geq 3\text{ and
}d<p<\infty,\\\
&C\varepsilon^{-1}\eta^{\frac{2}{p}-1}|\ln\eta|^{-1}&\quad&\text{ if
}d=2\text{ and }2<p<\infty,\end{aligned}\right.$ (3.5)

where $C$ depends only on $d$, $p$, $\Omega$, and $\\{Y_{z}^{s}\\}$.

###### Proof.

Using $\sigma_{\varepsilon}\geq 1$, we obtain from (3.3) that

$B_{p}(\Omega_{\varepsilon,\eta})\leq C\Phi_{p}(\eta^{-1})\varepsilon^{-1}.$
(3.6)

In view of (2.7), this gives (3.5) if $d\geq 2$ and $p>d$. For the remaining
case $d\geq 3$ and $2<p\leq d$, we let $u_{\varepsilon}\in
W^{1,p}_{0}(\Omega_{\varepsilon,\eta})$ be a solution of $-\Delta
u_{\varepsilon}=F$ in $\Omega_{\varepsilon,\eta}$, where $F\in L^{p}(\Omega)$.
We extend $u_{\varepsilon}$ to $\mathbb{R}^{d}$ by zero. It follows from
(2.24) and Theorem 1.3 that

$\displaystyle\|\nabla u_{\varepsilon}\|_{L^{p}(\Omega)}$ $\displaystyle\leq
C\varepsilon^{-1}\eta^{\frac{d}{p}-1}\|u_{\varepsilon}\|_{L^{p}(\Omega)}+C\Phi_{p}(\eta^{-1})\|F\|_{L^{p}(\Omega)}$
(3.7) $\displaystyle\leq
C\left(\varepsilon^{-1}\eta^{\frac{d}{p}-1}+\Phi_{p}(\eta^{-1})\right)\|F\|_{L^{p}(\Omega)},$

where we have used Theorem 2.3 for the last inequality. In view of (2.7) we
obtain (3.5) for $d\geq 3$ and $2<p\leq d$. ∎

##  4 Correctors: the case $d\geq 3$

This section is dedicated to constructing and establishing estimates for the
corrector $\chi_{\varepsilon,\eta}$ in the case $d\geq 3$. The two-dimensional
case will be handled separately in the next section.

Let $\Omega_{\varepsilon,\eta}$ be given by (1.5), where $T_{z,\eta}$ be given
by (1.4). The corrector $\chi_{\varepsilon,\eta}$ for
$\Omega_{\varepsilon,\eta}$ is defined in a piecewise manner, following an
idea in [1]. For $z\in\mathbb{Z}^{d}$, let $y_{z}=z+x_{z}$ and
$Q_{z}=z+Q(0,1)$. If $\varepsilon Q_{z}\subset\Omega$, we define

$\chi_{\varepsilon,\eta}(x)=\begin{cases}1&\text{in}\ \ \varepsilon
Q_{z}\setminus\varepsilon B(y_{z},1/5),\\\ \phi^{z}_{*}(\frac{x-\varepsilon
y_{z}}{\varepsilon\eta})&\text{in}\ \ \varepsilon
B(y_{z},1/6)\setminus\varepsilon T_{z,\eta},\\\ 0&\text{in}\ \ \varepsilon
T_{z,\eta},\end{cases}$ (4.1)

where $\phi^{z}_{*}$ is a solution to the exterior problem,

$\begin{cases}\begin{aligned} -\Delta\phi^{z}_{*}&=0\qquad&&\text{in}\ \
\mathbb{R}^{d}\setminus\overline{Y^{s}_{z}},\\\
\phi^{z}_{*}&=0\qquad&&\text{on}\ \ \partial Y^{s}_{z},\\\
\phi^{z}_{*}&\rightarrow 1\qquad&&\text{as}\ \
|x|\rightarrow\infty.\end{aligned}\end{cases}$ (4.2)

In the region $\varepsilon B(y_{z},1/5)\setminus\varepsilon B(y_{z},1/6)$ we
let $\chi_{\varepsilon,\eta}$ solve the following Dirichlet problem,

$\begin{cases}\begin{aligned}
-\Delta\chi_{\varepsilon,\eta}&=0\qquad&&\text{in}\ \ \varepsilon
B(y_{z},1/5)\setminus\varepsilon\overline{B(y_{z},1/6)},\\\
\chi_{\varepsilon,\eta}&=\phi_{*}^{z}\Big{(}\frac{x-\varepsilon
y_{z}}{\varepsilon\eta}\Big{)}\qquad&&\text{on}\ \ \partial B(\varepsilon
y_{z},\varepsilon/6),\\\ \chi_{\varepsilon,\eta}&=1\qquad&&\text{on}\ \
\partial B(\varepsilon y_{z},\varepsilon/5).\end{aligned}\end{cases}$ (4.3)

Finally, if $\varepsilon Q_{z}$ does not lie entirely in $\Omega$, we define
$\chi_{\varepsilon,\eta}=1$ in $\varepsilon Q_{z}$. As a result, the corrector
$\chi_{\varepsilon,\eta}$ is defined in $\mathbb{R}^{d}$, taking value $1$ in
$\Omega^{c}$ and $0$ in the holes $\varepsilon T_{z,\eta}$ inside $\Omega$.
The definition in (4.3) bridges the gap between the boundary data of the
exterior problem and 1. Thus, we see that $\chi_{\varepsilon,\eta}\in
H^{1}(\Omega)$, $\chi_{\varepsilon,\eta}=1$ on $\partial\Omega$ and
$\chi_{\varepsilon,\eta}=0$ on
$\partial\Omega_{\varepsilon,\eta}\setminus\partial\Omega$.

It is known that for each $\phi_{*}^{z}$ that satisfies (4.2),

$\begin{cases}\begin{aligned}
\phi^{z}_{*}({x})&=1-c^{z}_{*}|x|^{2-d}+O(|x|^{1-d}),\\\
\nabla\phi^{z}_{*}({x})&=-c^{z}_{*}\nabla(|x|^{2-d})+O(|x|^{-d}),\\\
\nabla^{2}\phi^{z}_{*}({x})&=-c^{z}_{*}\nabla^{2}(|x|^{2-d})+O(|x|^{-d-1}),\end{aligned}\end{cases}$
(4.4)

as $|x|\to\infty$, where $c^{z}_{*}=C_{d}\mu^{z}_{*}$ is the Newtonian
capacity of $Y_{z}^{s}$ with

$\mu^{z}_{*}=\int_{\partial Y^{s}_{z}}n\cdot\nabla\phi^{z}_{*}\,d\sigma$ (4.5)

and $C_{d}=\frac{1}{(d-2)|\partial B(0,1)|}$. See [3, 15]. The condition (1.3)
on $Y_{z}^{s}$ ensures that there exist $\mu_{0},\mu_{1}>0$ such that
$\mu_{0}\leq\mu_{*}^{z}\leq\mu_{1}$ for any $z\in\mathbb{Z}^{d}$.

######  Lemma 4.1.

Let $\chi_{\varepsilon,\eta}$ be defined as above. Then

$\|\chi_{\varepsilon,\eta}-1\|_{L^{p}(\Omega_{\varepsilon,\eta})}\leq\begin{cases}C\eta^{d-2}&\text{for
}\ \ 1\leq p<\frac{d}{d-2},\\\ C\eta^{d-2}|\ln\eta|^{d-2}&\text{for}\ \
p=\frac{d}{d-2},\\\ C\eta^{\frac{d}{p}}&\text{for}\ \
p>\frac{d}{d-2},\end{cases}$ (4.6)

where $C$ does not depend on $\varepsilon$ or $\eta$.

###### Proof.

Suppose $\varepsilon Q_{z}\subset\Omega$. Note that
$\chi_{\varepsilon,\eta}-1=0$ in $\varepsilon Q_{z}\setminus B(\varepsilon
y_{z},\varepsilon/5)$ and is harmonic in $B(\varepsilon
y_{z},\varepsilon/{5})\setminus\overline{B(\varepsilon
y_{z},\varepsilon/{6})}$. Hence, by the maximum principle,

$\displaystyle\|\chi_{\varepsilon,\eta}-1\|_{L^{\infty}(B(\varepsilon
y_{z},\varepsilon/{5})\backslash B(\varepsilon y_{z},\varepsilon/{6}))}$
$\displaystyle\leq\|\chi_{\varepsilon,\eta}-1\|_{L^{\infty}(\partial
B(\varepsilon y_{z},\varepsilon/{6}))}.$

Since $\chi_{\varepsilon,\eta}(x)=\phi_{*}^{z}((x-\varepsilon
y_{z})/(\varepsilon\eta))$ on $\partial B(\varepsilon y_{z},\varepsilon/6)$,
it follows from (4.4) that

$\|\chi_{\varepsilon,\eta}-1\|_{L^{\infty}(B(\varepsilon
y_{z},\varepsilon/{5})\backslash B(\varepsilon y_{z},\varepsilon/{6}))}\leq
C\eta^{d-2}.$

Next, we note that

$\displaystyle\int_{B(\varepsilon
y_{z},\frac{\varepsilon}{6})\backslash\varepsilon
T_{z,\eta}}|\chi_{\varepsilon,\eta}-1|^{p}dx$
$\displaystyle=\int_{B(\varepsilon
y_{z},\frac{\varepsilon}{6})\backslash\varepsilon
T_{z,\eta}}|\phi^{z}_{*}\Big{(}\frac{x-\varepsilon
y_{z}}{\varepsilon\eta}\Big{)}-1|^{p}\,dx$
$\displaystyle=\varepsilon^{d}\eta^{d}\int_{B(0,\frac{1}{6\eta})\backslash
Y^{s}_{z}}|\phi^{z}_{*}(y)-1|^{p}\,dy.$

In view of (4.4) we obtain

$\displaystyle\int_{B(\varepsilon
y_{z},\frac{\varepsilon}{6})\backslash\varepsilon
T_{z,\eta}}|\chi_{\varepsilon,\eta}-1|^{p}\,dx$ $\displaystyle\leq
C\varepsilon^{d}\eta^{d}\int_{B(0,\frac{1}{6\eta})\backslash
Y^{s}_{z}}\frac{dy}{|y|^{(d-2)p}}$ $\displaystyle\leq
C\varepsilon^{d}\eta^{d}\int_{c}^{\frac{1}{6\eta}}r^{d-1-dp+2p}\,dr.$

Thus,

$\|\chi_{\varepsilon,\eta}-1\|^{p}_{L^{p}(\varepsilon
Q_{z}\setminus\varepsilon
T_{z,\eta})}\leq\begin{cases}C\varepsilon^{d}\eta^{p(d-2)}&\text{for }\ \
1\leq p<\frac{d}{d-2},\\\ C\varepsilon^{d}\eta^{d}|\ln\eta|&\text{for}\ \
p=\frac{d}{d-2},\\\ C\varepsilon^{d}\eta^{d}&\text{for}\ \
p>\frac{d}{d-2}.\end{cases}$ (4.7)

Summing over all cubes $\varepsilon Q_{z}$ in $\Omega$ gives the desired
result, as the number of such cubes is bounded by $C\varepsilon^{-d}$. ∎

######  Remark 4.2.

Suppose $\varepsilon Q_{z}\subset\Omega$. Consider
$\xi(x)=\chi_{\varepsilon,\eta}(\varepsilon x+\varepsilon y_{z})-1$. The
function $\xi$ is harmonic in $B(0,1/5)\setminus\overline{B(0,1/6)}$ with
$\xi=0$ on $\partial B(0,1/5)$ and $\xi=\phi^{z}_{*}(\eta^{-1}x)-1$ on
$\partial B(0,1/6)$. It follows that in $B(0,1/5)\setminus B(0,1/6)$,

$|\nabla\xi|\leq C\|\phi_{*}^{z}(\eta^{-1}x)-1\|_{C^{1,1}(\partial
B(0,1/6))}\leq C\eta^{d-2},$

where we have used (4.4) for the last inequality. This implies that

$|\nabla\chi_{\varepsilon,\eta}|\leq C\varepsilon^{-1}\eta^{d-2}$

in $B(\varepsilon y_{z},\varepsilon/3)\backslash B(\varepsilon
y_{z},\varepsilon/4)$.

######  Lemma 4.3.

Suppose $\varepsilon Q_{z}\subset\Omega$. Then

$\left(\fint_{\varepsilon
Q_{z}}|\nabla\chi_{\varepsilon,\eta}|^{p}\right)^{1/p}\leq\begin{cases}C\varepsilon^{-1}\eta^{\frac{d-p}{p}}&\text{if
}p>d^{\prime},\\\ C\varepsilon^{-1}\eta^{d-2}|\ln\eta|^{\frac{1}{p}}&\text{if
}p=d^{\prime},\\\ C\varepsilon^{-1}\eta^{d-2}&\text{if
}p<d^{\prime},\end{cases}$ (4.8)

where $d^{\prime}=\frac{d}{d-1}$ and the constant $C$ does not depend on
$\varepsilon$ or $\eta$.

###### Proof.

We start by decomposing the integral along the sub-regions within the cube to
obtain

$\int_{\varepsilon
Q_{z}}|\nabla\chi_{\varepsilon,\eta}|^{p}=\int_{B(\varepsilon
y_{z},\frac{\varepsilon}{5})\setminus B(\varepsilon
y_{z},\frac{\varepsilon}{6})}|\nabla\chi_{\varepsilon,\eta}|^{p}+\int_{B(\varepsilon
y_{z},\frac{\varepsilon}{6})\setminus\varepsilon
T_{z,\eta}}|\nabla\chi_{\varepsilon,\eta}|^{p}.$ (4.9)

By Remark 4.2, the first term in the right-hand side of (4.9) is bounded by
$C\varepsilon^{d}(\varepsilon^{-1}\eta^{d-2})^{p}$. For the second term, we
have

$\displaystyle\int_{B(\varepsilon
y_{z},\frac{\varepsilon}{6})\backslash\varepsilon
T_{z,\eta}}|\nabla\chi_{\varepsilon,\eta}|^{p}$
$\displaystyle=\int_{B(\varepsilon
y_{z},\frac{\varepsilon}{6})\backslash\varepsilon
T_{z,\eta}}|\nabla(\phi_{*}^{z}\Big{(}\frac{x-\varepsilon
y_{z}}{\varepsilon\eta}\Big{)})|^{p}\,dx$ (4.10)
$\displaystyle=(\varepsilon\eta)^{-p}\int_{B(\varepsilon
y_{z},\frac{\varepsilon}{6})\backslash\varepsilon
T_{z,\eta}}|\nabla\phi_{*}^{z}(\frac{x-\varepsilon
y_{z}}{\epsilon\eta})|^{p}\,dx$
$\displaystyle=(\varepsilon\eta)^{d-p}\int_{B(0,\frac{1}{6\eta})\backslash
Y_{z}^{s}}|\nabla\phi_{*}^{z}(y)|^{p}\,dy$ $\displaystyle\leq
C(\varepsilon\eta)^{d-p}\int_{c}^{\frac{1}{6\eta}}r^{(1-d)p+d-1}dr,$

where we have used (4.4) as well as the fact that $\nabla\phi_{*}^{z}$ is
$L^{p}$ integrable near $\partial Y_{z}^{s}$ under the condition that
$Y_{z}^{s}$ is uniformly $C^{1}$. It follows that

$\left(\fint_{\varepsilon
B(y_{z},1/6)}|\nabla\chi_{\varepsilon,\eta}|^{p}\right)^{1/p}\leq\begin{cases}C\varepsilon^{-1}\eta^{\frac{d-p}{p}}&\text{if
}p>d^{\prime},\\\
C\varepsilon^{-1}\eta^{\frac{d-p}{p}}|\ln\eta|^{\frac{1}{p}}&\text{if
}p=d^{\prime},\\\ C\varepsilon^{-1}\eta^{d-2}&\text{if
}p<d^{\prime}.\end{cases}$ (4.11)

Combining (4.11) with the estimate for $B(\varepsilon
y_{z},\varepsilon/5)\setminus B(\varepsilon y_{z},\varepsilon/6)$ gives (4.8).
∎

Recall that $\sigma_{\varepsilon}=\varepsilon\eta^{-\frac{d-2}{2}}$ for $d\geq
3$ and $\mu_{*}^{z}$ is given by (4.5).

######  Lemma 4.4.

Suppose $\varepsilon Q_{z}\subset\Omega$. Then

$\left|\int_{\varepsilon
Q_{z}}\nabla\chi_{\varepsilon,\eta}\cdot\nabla\phi-\int_{\varepsilon
Q_{z}}\sigma_{\varepsilon}^{-2}\mu_{*}^{z}\phi\right|\leq
C\varepsilon^{-1}\eta^{d-2}\int_{\varepsilon Q_{z}}|\nabla\phi|,$ (4.12)

where $\phi\in H^{1}(\varepsilon Q_{z})$ and $\phi=0$ in $\varepsilon
T_{z,\eta}$.

###### Proof.

Note that

$\displaystyle\int_{\varepsilon
Q_{z}}\nabla\chi_{\varepsilon,\eta}\cdot\nabla\phi-\int_{\varepsilon
Q_{z}}\sigma_{\varepsilon}^{-2}\mu_{*}^{z}\phi$ (4.13)
$\displaystyle=\int_{\varepsilon Q_{z}\setminus B(\varepsilon
y_{z},\varepsilon/6)}\nabla\chi_{\varepsilon,\eta}\cdot\nabla\phi+\int_{B(\varepsilon
y_{z},\varepsilon/6)\setminus\varepsilon
T_{z,\eta}}\nabla\chi_{\varepsilon,\eta}\cdot\nabla\phi-\int_{\varepsilon
Q_{z}}\sigma_{\varepsilon}^{-2}\mu_{*}^{z}\phi.$

By Remark 4.2,

$\left|\int_{\varepsilon Q_{z}\setminus B(\varepsilon
y_{z},\varepsilon/6)}\nabla\chi_{\varepsilon,\eta}\cdot\nabla\phi\right|\leq
C\varepsilon^{-1}\eta^{d-2}\int_{\varepsilon Q_{z}}|\nabla\phi|.$ (4.14)

For the remaining two terms in the right-hand side of (4.13), we use
integration by parts to get

$\displaystyle\int_{\partial B(\varepsilon
y_{z},\varepsilon/6)}\left(\frac{\partial\chi_{\varepsilon,\eta}}{\partial
n}-\fint_{\partial B(\varepsilon
y_{z},\varepsilon/6)}\frac{\partial\chi_{\varepsilon,\eta}}{\partial
n}\right)\cdot\left(\phi-\alpha\right)$ $\displaystyle+\fint_{\partial
B(\varepsilon
y_{z},\varepsilon/6)}\frac{\partial\chi_{\varepsilon,\eta}}{\partial
n}\int_{\partial B(\varepsilon y_{z},\varepsilon/6)}\phi$ (4.15)
$\displaystyle-\int_{\varepsilon
Q_{z}}\sigma_{\varepsilon}^{-2}\mu_{*}^{z}\phi,$

where $n$ denotes the outward unit normal and $\alpha$ is a constant to be
determined. Here we also use the fact that $\chi_{\varepsilon,\eta}$ is
harmonic in $B(\varepsilon y_{z},\varepsilon/6)\setminus\varepsilon
T_{z,\eta}$. Using (4.4), we obtain

$\displaystyle\left|\int_{\partial B(\varepsilon
y_{z},\varepsilon/6)}\left(\frac{\partial\chi_{\varepsilon,\eta}}{\partial
n}-\fint_{\partial B(\varepsilon
y_{z},\varepsilon/6)}\frac{\partial\chi_{\varepsilon,\eta}}{\partial
n}\right)\cdot\left(\phi-\alpha\right)\right|$ (4.16) $\displaystyle\leq
C\varepsilon^{-1}\eta^{d-2}\int_{\partial B(\varepsilon
y_{z},\varepsilon/6)}|\phi-\alpha|$ $\displaystyle\leq
C\varepsilon^{-1}\eta^{d-2}\left(\frac{1}{\varepsilon}\int_{B(\varepsilon
y_{z},\varepsilon/6)}|\phi-\alpha|+\int_{B(\varepsilon
y_{z},\varepsilon/6)}|\nabla\phi|\right),$

where we have used a trace inequality for the last step (see Remark 4.5).
Picking $\alpha=\fint_{B(\varepsilon y_{z},\epsilon/6)}\phi$ and applying the
Poincaré inequality gives the desired bound.

To bound

$\left|\fint_{\partial B(\varepsilon
y_{z},\varepsilon/6)}\frac{\partial\chi_{\varepsilon,\eta}}{\partial
n}\int_{\partial B(\varepsilon y_{z},\varepsilon/6)}\phi-\int_{\varepsilon
Q_{z}}\sigma_{\varepsilon}^{-2}\mu_{*}^{z}\phi\right|,$ (4.17)

we move the average onto the integral of $\phi$ and note that

$\displaystyle\int_{\partial B(\varepsilon
y_{z},\varepsilon/6)}\frac{\partial\chi_{\varepsilon,\eta}}{\partial n}$
$\displaystyle=(\varepsilon\eta)^{-1}\int_{\partial B(\varepsilon
y_{z},\varepsilon/6)}\frac{\partial\phi^{z}_{*}}{\partial
n}\Big{(}\frac{x-\varepsilon y_{z}}{\varepsilon\eta}\Big{)}d\sigma(x)$
$\displaystyle=(\varepsilon\eta)^{d-2}\int_{\partial
B(0,\frac{1}{6\eta})}\frac{\partial\phi^{z}_{*}}{\partial n}d\sigma(y)$
$\displaystyle=(\varepsilon\eta)^{d-2}\int_{\partial
Y^{s}_{z}}\frac{\partial\phi^{z}_{*}}{\partial n}d\sigma(y)$
$\displaystyle=\mu_{*}^{z}(\varepsilon\eta)^{d-2}.$

As a result, (4.17) becomes

$\mu_{*}^{z}(\varepsilon\eta)^{d-2}\left|\fint_{\partial B(\varepsilon
y_{z},\varepsilon/4)}\phi-\fint_{\varepsilon Q_{z}}\phi\right|,$

which, in view of (4.21), is bounded by

$C\varepsilon^{-1}\eta^{d-2}\int_{\varepsilon Q_{z}}|\nabla\phi|.$

This completes the proof. ∎

######  Remark 4.5.

In (4.16) we have used the trace inequality,

$\int_{\partial
B(x_{0},r)}|\phi|\leq\frac{d}{r}\int_{B(x_{0},r)}|\phi|+\int_{B(x_{0},r)}|\nabla\phi|,$
(4.18)

for $\phi\in H^{1}(B(x_{0},r))$. This follows by writing

$\int_{\partial B(x_{0},r)}|\phi|=\frac{1}{r}\int_{\partial
B(x_{0},r)}|\phi|((x-x_{0})\cdot n)\,d\sigma(x)$

and applying the divergence theorem. Replacing $\phi$ in (4.18) by
$\phi-\alpha$, where $\alpha=\fint_{\varepsilon Q_{z}}\phi$, we obtain

$\int_{\partial B(\varepsilon y_{z},c\varepsilon)}|\phi-\fint_{\varepsilon
Q_{z}}\phi|\leq C\int_{\varepsilon Q_{z}}|\nabla\phi|,$ (4.19)

where we have used a Poincaré inequality,

$\int_{\varepsilon Q_{z}}|\phi-\fint_{\varepsilon Q_{z}}\phi|\leq
C\varepsilon\int_{\varepsilon Q_{z}}|\nabla\phi|.$ (4.20)

It follows from (4.19) that

$\left|\fint_{\partial B(\varepsilon
y_{z},c\varepsilon)}\phi-\fint_{\varepsilon Q_{z}}\phi\right|\leq
C\varepsilon\fint_{\varepsilon Q_{z}}|\nabla\phi|.$ (4.21)

##  5 Correctors: the two-dimensional case

In this section we introduce a corrector $\chi_{\varepsilon,\eta}$ for the
case $d=2$ and establish its estimates analogous to those in Lemmas 4.1, 4.3
and 4.4.

For $z\in\mathbb{Z}^{2}$, let $\phi^{z}_{*}$ be a solution to the exterior
problem,

$\begin{cases}\begin{aligned} -\Delta\phi^{z}_{*}&=0\qquad&&\text{in}\ \
\mathbb{R}^{2}\setminus\overline{Y^{s}_{z}},\\\
\phi^{z}_{*}&=0\qquad&&\text{on}\ \ \partial Y^{s}_{z},\\\
\phi^{z}_{*}(x)-\ln|x|&=O(1)\qquad&&\text{as}\ \
|x|\rightarrow\infty.\end{aligned}\end{cases}$ (5.1)

It is known that

$\left\\{\begin{aligned}
\nabla\phi_{*}^{z}(x)&=\frac{x}{|x|^{2}}+O(|x|^{-2}),\\\
\nabla^{2}\phi_{*}^{z}(x)&=O(|x|^{-2}),\end{aligned}\right.$ (5.2)

as $|x|\to\infty$. Recall that $y_{z}=z+x_{z}$ and $Q_{z}=z+Q(0,1)$. If
$\varepsilon Q_{z}\subset\Omega$, we let

$\chi_{\varepsilon,\eta}(x)=\begin{cases}1&\text{in}\ \ \varepsilon
Q_{z}\setminus\varepsilon B(y_{z},1/5),\\\ \phi^{z}_{*}(\frac{x-\varepsilon
y_{z}}{\varepsilon\eta})/|\ln\eta|&\text{in}\ \ \varepsilon
B(y_{z},1/6)\setminus\varepsilon T_{z,\eta},\\\ 0&\text{in}\ \ \varepsilon
T_{z,\eta}.\end{cases}$ (5.3)

In the region $\varepsilon B(y_{z},1/5)\setminus\varepsilon B(y_{z},1/6)$, we
let $\chi_{\varepsilon,\eta}$ solve the following Dirichlet problem,

$\begin{cases}\begin{aligned}
-\Delta\chi_{\varepsilon,\eta}&=0\qquad&&\text{in}\ \ \varepsilon
B(y_{z},1/5)\setminus\varepsilon\overline{B(y_{z},1/6)},\\\
\chi_{\varepsilon,\eta}&=\phi_{*}^{z}\Big{(}\frac{x-\varepsilon
y_{z}}{\varepsilon\eta}\Big{)}/|\ln\eta|\qquad&&\text{on}\ \ \partial
B(\varepsilon y_{z},\varepsilon/6),\\\
\chi_{\varepsilon,\eta}&=1\qquad&&\text{on}\ \ \partial B(\varepsilon
y_{z},\varepsilon/5).\end{aligned}\end{cases}$ (5.4)

Finally, if $\varepsilon Q_{z}$ is not contained in $\Omega$, let
$\chi_{\varepsilon,\eta}=1$ in $\varepsilon Q_{z}$, as in the case of $d\geq
3$. By construction, we have $\chi_{\varepsilon,\eta}\in H^{1}(\Omega)$.
Moreover, $\chi_{\varepsilon,\eta}=1$ in $\Omega^{c}$ and
$\chi_{\varepsilon,\eta}=0$ in $\Omega\setminus\Omega_{\varepsilon,\eta}$.
Furthermore, in the region $\varepsilon B(y_{z},1/5)\setminus B(y_{z},1/6)$,
we have $L^{\infty}$ estimates,

$|\chi_{\varepsilon,\eta}-1|\leq C|\ln\eta|^{-1}\quad\text{ and
}\quad|\nabla\chi_{\varepsilon,\eta}|\leq C\varepsilon^{-1}|\ln\eta|^{-1}.$
(5.5)

As in the case $d\geq 3$, the estimates in (5.5) follow from the maximum
principle and the Lipschitz estimates for harmonic functions.

######  Lemma 5.1.

Let $\chi_{\varepsilon,\eta}$ be defined as above. Then for $1\leq p<\infty$,

$\|\chi_{\varepsilon,\eta}-1\|_{L^{p}(\Omega_{\varepsilon,\eta})}\leq
C|\ln\eta|^{-1},$ (5.6)

where $C$ does not depend on $\varepsilon$ or $\eta$.

###### Proof.

Note that if $\varepsilon Q_{z}\subset\Omega$,

$\displaystyle\int_{\varepsilon Q_{z}\setminus\varepsilon
T_{z,\eta}}|\chi_{\varepsilon,\eta}-1|^{p}$ $\displaystyle\leq
C\varepsilon^{2}|\ln\eta|^{-p}+\int_{B(\varepsilon
y_{z},\varepsilon/6)\setminus\varepsilon
T_{z,\eta}}\left|\frac{\ln|(x-\varepsilon
y_{z})/\varepsilon|}{\ln\eta}\right|^{p}\,dx$
$\displaystyle\leq\frac{C\varepsilon^{2}}{|\ln\eta|^{p}}\left\\{\int_{B(0,1)}|\ln|x||^{p}\,dx+1\right\\}$
$\displaystyle\leq\frac{C\varepsilon^{2}}{|\ln\eta|^{p}},$

where we have used (5.5) as well as the fact that
$\phi_{*}^{z}(x)=\ln|x|+O(1)$ as $|x|\to\infty$. Summing over all such cubes
yields

$\|\chi_{\varepsilon,\eta}-1\|^{p}_{L^{p}(\Omega_{\varepsilon,\eta})}\leq
C|\ln\eta|^{-p},$

which yields (5.6). ∎

Recall that $\sigma_{\varepsilon}=\varepsilon|\ln\eta|^{1/2}$ for $d=2$.

######  Lemma 5.2.

Suppose $\varepsilon Q_{z}\subset\Omega$. Then

$\left(\fint_{\varepsilon
Q_{z}}|\nabla\chi_{\varepsilon,\eta}|^{p}\right)^{1/p}\leq\begin{cases}C\sigma_{\varepsilon}^{-1}|\ln\eta|^{-1/2}&\text{if
}1<p<2,\\\ C\sigma_{\varepsilon}^{-1}&\text{if }p=2,\\\
C\sigma_{\varepsilon}^{-1}\eta^{\frac{2-p}{p}}|\ln\eta|^{-1/2}&\text{if
}2<p<\infty,\end{cases}$ (5.7)

where the constant $C$ does not depend on $\varepsilon$ or $\eta$.

###### Proof.

Note that

$\displaystyle\int_{\varepsilon Q_{z}}|\nabla\chi_{\varepsilon,\eta}|^{p}$
$\displaystyle\leq C\varepsilon^{2-p}|\ln\eta|^{-p}+\int_{B(\varepsilon
y_{z},\varepsilon/6)\setminus\varepsilon
T_{z,\eta}}|\nabla\chi_{\varepsilon,\eta}|^{p}dx$
$\displaystyle=C\varepsilon^{2-p}|\ln\eta|^{-p}+\int_{B(\varepsilon
y_{z},\varepsilon/6)\setminus\varepsilon
T_{z,\eta}}\left|\left(\frac{C}{|\ln\eta||x-\varepsilon
y_{z}|}\right)\right|^{p}dx$
$\displaystyle=C\varepsilon^{2-p}|\ln\eta|^{-p}\left\\{1+\int_{c\eta}^{1}r^{1-p}dr\right\\},$

where we have used (5.5) and (5.2). It follows that

$\displaystyle\left(\fint_{\varepsilon
Q_{z}}|\nabla\chi_{\varepsilon,\eta}|^{p}\right)^{1/p}\leq\begin{cases}C\varepsilon^{-1}|\ln\eta|^{-1}&\text{if
}1<p<2,\\\ C\varepsilon^{-1}|\ln\eta|^{-\frac{1}{2}}&\text{if }p=2,\\\
C\varepsilon^{-1}\eta^{\frac{2-p}{p}}|\ln\eta|^{-1}&\text{if
}2<p<\infty,\end{cases}$

which gives (5.7) ∎

######  Lemma 5.3.

Suppose $\varepsilon Q_{z}\subset\Omega$. Then

$\left|\int_{\varepsilon
Q_{z}}\nabla\chi_{\varepsilon,\eta}\cdot\nabla\phi-2\pi\sigma_{\varepsilon}^{-2}\int_{\varepsilon
Q_{z}}\phi\right|\leq C\varepsilon^{-1}|\ln\eta|^{-1}\int_{\varepsilon
Q_{z}}|\nabla\phi|,$ (5.8)

where $\phi\in H^{1}(\varepsilon Q_{z})$ and $\phi=0$ in $\varepsilon
T_{z,\eta}$.

###### Proof.

The proof is similar to that of Lemma 4.4. Note that

$\displaystyle\int_{\varepsilon
Q_{z}}\nabla\chi_{\varepsilon,\eta}\cdot\nabla\phi-2\pi\sigma_{\varepsilon}^{-2}\int_{\varepsilon
Q_{z}}\phi$ (5.9) $\displaystyle=\int_{B(\varepsilon
y_{z},\varepsilon/5)\setminus B(\varepsilon
y_{z},\varepsilon/6)}\nabla\chi_{\varepsilon,\eta}\cdot\nabla\phi+\int_{B(\varepsilon
y_{z},\varepsilon/6)\setminus\varepsilon
T_{z,\eta}}\nabla\chi_{\varepsilon,\eta}\cdot\nabla\phi-\frac{2\pi}{|\ln\eta|}\fint_{\varepsilon
Q_{z}}\phi$ $\displaystyle=\int_{B(\varepsilon y_{z},\varepsilon/5)\setminus
B(\varepsilon
y_{z},\varepsilon/6)}\nabla\chi_{\varepsilon,\eta}\cdot\nabla\phi+\left\\{\int_{\partial
B(\varepsilon
y_{z},\varepsilon/6)}\frac{\partial\chi_{\varepsilon,\eta}}{\partial
n}\phi-\frac{2\pi}{|\ln\eta|}\fint_{\varepsilon Q_{z}}\phi\right\\}$
$\displaystyle=I_{1}+I_{2},$

where we have used integration by parts and the fact that
$\chi_{\varepsilon,\eta}$ is harmonic in $B(\varepsilon
y_{z},\varepsilon/6)\setminus\varepsilon T_{z,\eta}$. By (5.5), it is easy to
see that $|I_{1}|$ is bounded by the right-hand side of (5.8).

To bound $I_{2}$, we write it as

$\displaystyle I_{2}=$ $\displaystyle\int_{\partial B(\varepsilon
y_{z},\varepsilon/6)}\left(\frac{\partial\chi_{\varepsilon,\eta}}{\partial
n}-\fint_{\partial B(\varepsilon
y_{z},\varepsilon/6)}\frac{\partial\chi_{\varepsilon,\eta}}{\partial
n}\right)\cdot\left(\phi-\alpha\right)$ (5.10)
$\displaystyle\qquad+\left\\{\int_{\partial B(\varepsilon
y_{z},\varepsilon/6)}\frac{\partial\chi_{\varepsilon,\eta}}{\partial
n}\fint_{\partial B(\varepsilon
y_{z},\varepsilon/6)}\phi-\frac{2\pi}{|\ln\eta|}\fint_{\varepsilon
Q_{z}}\phi\right\\}$ $\displaystyle=I_{21}+I_{22},$

where $\alpha$ is a constant to be determined. Using (5.2), one may show that

$\displaystyle|I_{21}|$
$\displaystyle\leq\frac{C}{\varepsilon|\ln\eta|}\int_{\partial B(\varepsilon
y_{z},\varepsilon/6)}|\phi-\alpha|$ (5.11)
$\displaystyle\leq\frac{C}{\varepsilon|\ln\eta|}\int_{\varepsilon
Q_{z}}|\nabla\phi|,$

where we have choose $\alpha$ to be the average of $\phi$ over $\varepsilon
Q_{z}$, as in the proof of Lemma 4.4.

Finally, to handle $I_{22}$, we note that by (5.2),

$\int_{\partial B(\varepsilon
y_{z},\varepsilon/6)}\frac{\partial\chi_{\varepsilon,\eta}}{\partial
n}=\frac{1}{|\ln\eta|}\left\\{2\pi+O(\eta)\right\\}.$

This, together with (4.21), shows that

$\displaystyle|I_{22}|$
$\displaystyle\leq\frac{C}{\varepsilon|\ln\eta|}\int_{\varepsilon
Q_{z}}|\nabla\phi|+\frac{C\eta}{\varepsilon|\ln\eta|}\int_{\varepsilon
Q_{z}}|\phi|$ $\displaystyle\leq
C\frac{C}{\varepsilon|\ln\eta|}\int_{\varepsilon Q_{z}}|\nabla\phi|,$

where we have used Lemma 2.1 with $p=1$ for the last inequality. This
completes the proof. ∎

##  6 Convergence Rates

In this section we study the Dirichlet problem with nonhomogeneous boundary
conditions,

$\begin{cases}\begin{aligned} -\Delta u_{\varepsilon}&=0\qquad&&\text{in}\ \
\Omega_{\varepsilon,\eta},\\\ u_{\varepsilon}&=h\qquad&&\text{on}\ \
\partial\Omega,\\\ u_{\varepsilon}&=0\qquad&&\text{on}\ \
\partial\Omega_{\varepsilon,\eta}\setminus\partial\Omega,\end{aligned}\end{cases}$
(6.1)

where $\Omega_{\varepsilon,\eta}$ is given in (1.5) and $h\in
H^{1/2}(\partial\Omega)$. Let $V_{\varepsilon}(x)=V(x/\varepsilon)$, where

$V(y)=\sum_{z\in\mathbb{Z}^{d}}\mu_{*}^{z}\chi_{Q(z,1)}(y),$ (6.2)

and $\mu_{*}^{z}$ is given by (4.5) for $d\geq 3$ and $\mu_{*}^{z}=2\pi$ for
$d=2$. In (6.2) we have used $\chi_{Q(z,1)}$ to denote the characteristic
function of the cell $Q(z,1)$. Note that $c_{0}\leq V(y)\leq c_{1}$ for some
$c_{0},c_{1}>0$. Let $\chi_{\varepsilon,\eta}$ be the corrector constructed in
the last two sections for the domain $\Omega_{\varepsilon,\eta}$. We will show
that the solution $u_{\varepsilon}$ of (6.1) is well approximated by
$\chi_{\varepsilon,\eta}u_{0,\varepsilon}$, where $u_{0,\varepsilon}$ is the
solution of a boundary value problem in $\Omega$,

$\left\\{\begin{aligned} -\Delta
u_{0,\varepsilon}+\sigma_{\varepsilon}^{-2}V_{\varepsilon}(x)u_{0,\varepsilon}&=0&\quad&\text{in}\
\ \Omega,\\\ u_{0,\varepsilon}&=h&\quad&\text{on}\ \
\partial\Omega,\end{aligned}\right.$ (6.3)

which we will call the intermediate problem, for a Schrödinger operator
$-\Delta+\sigma_{\varepsilon}^{-2}V_{\varepsilon}$. The parameter
$\sigma_{\varepsilon}$, which depends on $\varepsilon$ and $\eta$, is given by
(1.6). The rest of this section will be divided into two parts. The first part
deals with the case $d\geq 3$, while the second treats the case $d=2$.

###  6.1 Dimension $d\geq 3$

The goal of this subsection is to prove the following.

######  Theorem 6.1.

Let $u_{\varepsilon}$ be a solution of (6.1) and $u_{0,\varepsilon}$ a
solution of (6.3). Let

$r_{\varepsilon}=u_{\varepsilon}-\chi_{\varepsilon,\eta}u_{0,\varepsilon},$
(6.4)

where $\chi_{\varepsilon,\eta}$ is the corrector constructed in Section 4.
Assume that $u_{0,\varepsilon}\in W^{1,p}(\Omega)$ for some $2<p<\infty$. Then
for $d\geq 3$,

$\|\nabla
r_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon,\eta})}\leq\begin{cases}C\eta^{\frac{d-2}{2}}\left(\sigma_{\varepsilon}^{-1}\|u_{0,\varepsilon}\|_{L^{p}(\Omega)}+\|\nabla
u_{0,\varepsilon}\|_{L^{p}(\Omega)}\right)&\text{if}\ \ p\geq d,\\\
C\eta^{d/q}\left(\sigma_{\varepsilon}^{-1}\|u_{0,\varepsilon}\|_{L^{p}(\Omega)}+\|\nabla
u_{0,\varepsilon}\|_{L^{p}(\Omega)}\right)&\text{if}\ 2<p<d,\end{cases}$ (6.5)

where $\frac{1}{2}=\frac{1}{p}+\frac{1}{q}$ and $C$ does not depend on
$\varepsilon$ or $\eta$.

Note that $\chi_{\varepsilon,\eta}\in W^{1,p}(\Omega)$ for any $p>2$.
Moreover, $\chi_{\varepsilon,\eta}=1$ on $\partial\Omega$ and
$\chi_{\varepsilon,\eta}=0$ on
$\partial\Omega_{\varepsilon,\eta}\setminus\partial\Omega$. It follows that
$r_{\varepsilon}\in H_{0}^{1}(\Omega_{\varepsilon,\eta})$. The proof of
Theorem 6.1 will rely heavily on the estimates for $\chi_{\varepsilon,\eta}$
in Section 4.

######  Lemma 6.2.

Let $\phi\in H_{0}^{1}(\Omega_{\varepsilon,\eta})$. Then

$\left|\int_{\Omega_{\varepsilon,\eta}}\nabla\chi_{\varepsilon,\eta}\cdot\nabla\phi-\int_{\Omega_{\varepsilon,\eta}}\sigma_{\varepsilon}^{-2}V_{\varepsilon}\phi\right|\leq
C\varepsilon^{-1}\eta^{d-2}\int_{\Omega_{\varepsilon,\eta}}|\nabla\phi|,$
(6.6)

where $V_{\varepsilon}=V(x/\varepsilon)$ and $V(y)$ is defined in (6.2).

###### Proof.

We extend $\phi$ to $\mathbb{R}^{d}$ by zero. Note that

$\displaystyle\left|\int_{\Omega_{\varepsilon,\eta}}\nabla\chi_{\varepsilon,\eta}\cdot\nabla\phi-\int_{\Omega_{\varepsilon,\eta}}\sigma_{\varepsilon}^{-2}V_{\varepsilon}\phi\right|$
(6.7)
$\displaystyle\leq\sum_{z\in\Omega^{\prime}_{\varepsilon}}\left|\int_{\varepsilon
Q_{z}}\nabla\chi_{\varepsilon,\eta}\cdot\nabla\phi-\int_{\varepsilon
Q_{z}}\sigma_{\varepsilon}^{-2}V_{\varepsilon}\phi\right|+\left|\int_{\mathcal{F}}\sigma_{\varepsilon}^{-2}V_{\varepsilon}\phi\right|,$

where $\Omega^{\prime}_{\varepsilon}$ is the union of cubes $\varepsilon
Q_{z}$ lying entirely inside of $\Omega$ and $\mathcal{F}$ is the union of
cubes with non-empty intersection with the boundary of $\Omega$. By Lemma 4.4,
the first term in the right-hand side of (6.7) is bounded by the right-hand
side of (6.6). For the second term we have

$\displaystyle\left|\int_{\mathcal{F}}\sigma_{\varepsilon}^{-2}V_{\varepsilon}\phi\right|$
$\displaystyle\leq C\sigma_{\varepsilon}^{-2}\int_{\text{\rm
dist(}x,\partial\Omega)\leq C\varepsilon}|\phi|$ $\displaystyle\leq
C\varepsilon\sigma_{\varepsilon}^{-2}\int_{\text{\rm
dist}(x,\partial\Omega)\leq C\varepsilon}|\nabla\phi|\leq
C\varepsilon^{-1}\eta^{d-2}\int_{\Omega_{\varepsilon,\eta}}|\nabla\phi|,$

where we have used a Poincaré-Sobolev inequality for the second inequality. ∎

###### Proof of Theorem 6.1.

Let $u_{\varepsilon}$ be a solution of (6.1) and $u_{0,\varepsilon}$ a
solution of (6.3). Assume further that $u_{0,\varepsilon}\in W^{1,p}(\Omega)$
for some $p>2$. Let
$r_{\varepsilon}=u_{\varepsilon}-\chi_{\varepsilon,\eta}u_{0,\varepsilon}$.
Note that for any $\phi\in H_{0}^{1}(\Omega_{\varepsilon,\eta})$,

$\displaystyle\int_{\Omega_{\varepsilon,\eta}}\nabla
r_{\varepsilon}\cdot\nabla\phi$
$\displaystyle=-\int_{\Omega_{\varepsilon,\eta}}(\nabla\chi_{\varepsilon,\eta}\cdot\nabla\phi)u_{0,\varepsilon}-\int_{\Omega_{\varepsilon,\eta}}\chi_{\varepsilon,\eta}\left(\nabla
u_{0,\varepsilon}\cdot\nabla\phi\right)$ (6.8)
$\displaystyle=-\int_{\Omega_{\varepsilon,\eta}}\nabla\chi_{\varepsilon,\eta}\cdot\nabla(u_{0,\varepsilon}\phi)+\int_{\Omega_{\varepsilon,\eta}}[\nabla(\chi_{\varepsilon,\eta}-1)\cdot\nabla
u_{0,\varepsilon}]\phi$
$\displaystyle\qquad\qquad\qquad-\int_{\Omega_{\varepsilon,\eta}}(\chi_{\varepsilon,\eta}-1)\left(\nabla
u_{0,\varepsilon}\cdot\nabla\phi\right)-\int_{\Omega_{\varepsilon,\eta}}\left(\nabla
u_{0,\varepsilon}\cdot\nabla\phi\right).$

By using integration by parts, the right-hand side of (6.8) becomes

$\displaystyle-\int_{\Omega_{\varepsilon,\eta}}\nabla\chi_{\varepsilon,\eta}\cdot\nabla(u_{0,\varepsilon}\phi)$
$\displaystyle-2\int_{\Omega_{\varepsilon,\eta}}(\chi_{\varepsilon,\eta}-1)\left(\nabla
u_{0,\varepsilon}\cdot\nabla\phi\right)$
$\displaystyle-\int_{\Omega_{\varepsilon,\eta}}(\chi_{\varepsilon,\eta}-1)\Delta
u_{0,\varepsilon}\cdot\phi+\int_{\Omega_{\varepsilon,\eta}}\left(\Delta
u_{0,\varepsilon}\cdot\phi\right),$

which, in view of (6.3), equals to

$\displaystyle-\int_{\Omega_{\varepsilon,\eta}}\nabla\chi_{\varepsilon,\eta}\cdot\nabla(u_{0,\varepsilon}\phi)+\sigma_{\varepsilon}^{-2}\int_{\Omega_{\varepsilon,\eta}}V_{\varepsilon}u_{0,\varepsilon}\phi$
$\displaystyle-2\int_{\Omega_{\varepsilon,\eta}}(\chi_{\varepsilon,\eta}-1)\left(\nabla
u_{0,\varepsilon}\cdot\nabla\phi\right)$
$\displaystyle-\sigma_{\varepsilon}^{-2}\int_{\Omega_{\varepsilon,\eta}}(\chi_{\varepsilon,\eta}-1)V_{\varepsilon}u_{0,\varepsilon}\cdot\phi.$

Note that $V_{\varepsilon}(x)\leq c_{2}$. We now apply Lemma 6.2 to obtain

$\displaystyle\left|\int_{\Omega_{\varepsilon,\eta}}\nabla
r_{\varepsilon}\cdot\nabla\phi\right|$ $\displaystyle\leq
C\varepsilon^{-1}\eta^{d-2}\int_{\Omega_{\varepsilon,\eta}}|\nabla(u_{0,\varepsilon}\phi)|+2\int_{\Omega_{\varepsilon,\eta}}|\chi_{\varepsilon,\eta}-1||\nabla
u_{0,\varepsilon}||\nabla\phi|$
$\displaystyle\qquad+C\sigma_{\varepsilon}^{-2}\int_{\Omega_{\varepsilon,\eta}}|\chi_{\varepsilon,\eta}-1||u_{0,\varepsilon}||\phi|$
$\displaystyle\leq
C\varepsilon^{-1}\eta^{d-2}\int_{\Omega_{\varepsilon,\eta}}|\nabla
u_{0,\varepsilon}||\phi|+C\varepsilon^{-1}\eta^{d-2}\int_{\Omega_{\varepsilon,\eta}}|\nabla\phi||u_{0,\varepsilon}|$
$\displaystyle\qquad+2\int_{\Omega_{\varepsilon,\eta}}|\chi_{\varepsilon,\eta}-1||\nabla
u_{0,\varepsilon}||\nabla\phi|+C\sigma_{\varepsilon}^{-2}\int_{\Omega_{\varepsilon,\eta}}|\chi_{\varepsilon,\eta}-1||u_{0,\varepsilon}||\phi|.$

Applying Hölder’s inequality yields

$\displaystyle\left|\int_{\Omega_{\varepsilon,\eta}}\nabla
r_{\varepsilon}\cdot\nabla\phi\right|\leq$ $\displaystyle
C\sigma_{\varepsilon}^{-1}\eta^{\frac{d-2}{2}}\|\nabla\phi\|_{L^{2}(\Omega_{\varepsilon,\eta})}||u_{0,\varepsilon}\|_{L^{2}(\Omega_{\varepsilon,\eta})}$
(6.9)
$\displaystyle\qquad+C\sigma_{\varepsilon}^{-1}\eta^{\frac{d-2}{2}}\|\phi\|_{L^{2}(\Omega_{\varepsilon,\eta})}\|\nabla
u_{0,\epsilon}\|_{L^{2}(\Omega_{\varepsilon,\eta})}$
$\displaystyle\qquad\qquad+C\|\chi_{\varepsilon,\eta}-1\|_{L^{q}(\Omega_{\varepsilon,\eta})}\|\nabla\phi\|_{L^{2}(\Omega_{\varepsilon,\eta})}\|\nabla
u_{0,\varepsilon}\|_{L^{p}(\Omega_{\varepsilon,\eta})}$
$\displaystyle\qquad\qquad\qquad+C\sigma_{\varepsilon}^{-2}\|\chi_{\varepsilon,\eta}-1\|_{L^{q}(\Omega_{\varepsilon,\eta})}\|\phi\|_{L^{2}(\Omega_{\varepsilon,\eta})}\|u_{0,\varepsilon}\|_{L^{p}(\Omega_{\varepsilon,\eta})},$

where $\frac{1}{2}=\frac{1}{q}+\frac{1}{p}$. By applying the Poincaré
inequality $\|\phi\|_{L^{2}(\Omega_{\varepsilon,\eta})}\leq
C\sigma_{\varepsilon}\|\nabla\phi\|_{L^{2}(\Omega_{\varepsilon,\eta})}$ in
(6.9) we have

$\displaystyle\left|\int_{\Omega_{\varepsilon,\eta}}\nabla
r_{\varepsilon}\cdot\nabla\phi\right|$ $\displaystyle\leq
C\eta^{\frac{d-2}{2}}\|\nabla\phi\|_{L^{2}(\Omega_{\varepsilon,\eta})}\left(\sigma_{\varepsilon}^{-1}\|u_{0,\varepsilon}\|_{L^{2}(\Omega)}+\|\nabla
u_{0,\varepsilon}\|_{L^{2}(\Omega)}\right)$
$\displaystyle+C\|\chi_{\varepsilon,\eta}-1\|_{L^{q}(\Omega_{\varepsilon,\eta})}||\nabla\phi\|_{L^{2}(\Omega_{\varepsilon,\eta})}\left(\|\nabla
u_{0,\varepsilon}\|_{L^{p}(\Omega)}+\sigma_{\varepsilon}^{-1}\|u_{0,\varepsilon}\|_{L^{p}(\Omega)}\right).$

Choosing $\phi=r_{\varepsilon}$, we obtain

$\|\nabla r_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon,\eta})}\leq
C\left(\eta^{\frac{d-2}{2}}+\|\chi_{\varepsilon,\eta}-1\|_{L^{q}(\Omega_{\varepsilon,\eta})}\right)\left(\|\nabla
u_{0,\varepsilon}||_{L^{p}(\Omega)}+\sigma_{\varepsilon}^{-1}\|u_{0,\varepsilon}\|_{L^{p}(\Omega)}\right).$

Applying Lemma 4.1 gives (6.5). ∎

###  6.2 Dimension $d=2$

We now consider the case $d=2$.

######  Theorem 6.3.

Let $u_{\varepsilon}$ be a solution of (6.1) amd $u_{0,\varepsilon}$ a
solution of (6.3) Assume that $u_{0,\varepsilon}\in W^{1,p}(\Omega)$ for some
$2<p<\infty$. Let
$r_{\varepsilon}=u_{\varepsilon}-\chi_{\varepsilon,\eta}u_{0,\varepsilon}$.
Then

$\|\nabla r_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon,\eta})}\leq
C|\ln\eta|^{-1/2}\left(\sigma_{\varepsilon}^{-1}\|u_{0,\varepsilon}\|_{L^{p}(\Omega)}+\|\nabla
u_{0,\varepsilon}\|_{L^{p}(\Omega)}\right),$ (6.10)

where $C$ does not depend on $\varepsilon$ or $\eta$.

The following lemma plays the role of Lemma 6.2.

######  Lemma 6.4.

Let $\chi_{\varepsilon,\eta}$ be the corrector constructed in Section 5. Then,
for any $\phi\in H^{1}_{0}(\Omega_{\varepsilon,\eta})$,

$\left|\int_{\Omega_{\varepsilon,\eta}}\nabla\chi_{\varepsilon,\eta}\cdot\nabla\phi-2\pi\sigma_{\varepsilon}^{-2}\int_{\Omega_{\varepsilon,\eta}}\phi\right|\leq\frac{C}{\varepsilon|\ln\eta|}\int_{\Omega_{\varepsilon,\eta}}|\nabla\phi|.$
(6.11)

###### Proof.

The proof is similar to that of Lemma 6.2. Let $\phi\in
H_{0}^{1}(\Omega_{\varepsilon,\eta})$ and extend it to $\mathbb{R}^{2}$ by
zero. The left-hand side of (6.11) is bounded by

$\sum_{z}\left|\int_{\varepsilon
Q_{z}}\nabla\chi_{\varepsilon,\eta}\cdot\nabla\phi-2\pi\sigma_{\varepsilon}^{-2}\int_{\varepsilon
Q_{z}}\phi\right|+\left|\int_{\mathcal{F}}2\pi\sigma_{\varepsilon}^{-2}\phi\right|,$
(6.12)

where the sum is taken over those $z$’s for which $\varepsilon
Q_{z}\subset\Omega$ and $\mathcal{F}$ is the union of cubes $\varepsilon
Q_{z}$ with non-empty intersections with $\partial\Omega$. As in the proof of
Lemma 6.2, the second term in (6.12) can be handled readily by a Poincaré
inequality, since $\phi=0$ on $\partial\Omega$. For the first term we use
Lemma 5.3 to bound it by the right-hand side of (6.11). ∎

###### Proof of Theorem 6.3.

Let $u_{\varepsilon}$ be a solution of (6.1) and $u_{\varepsilon,0}$ a
solution of (6.3). Assume further that $u_{0,\varepsilon}\in W^{1,p}(\Omega)$.
Let
$r_{\varepsilon}=u_{\varepsilon}-\chi_{\varepsilon,\eta}u_{0,\varepsilon}$. As
in the proof of Theorem 6.1, for $\phi\in
H^{1}_{0}(\Omega_{\varepsilon,\eta})$, we have

$\displaystyle\int_{\Omega_{\varepsilon,\eta}}\nabla
r_{\varepsilon}\cdot\nabla\phi$
$\displaystyle=-\int_{\Omega_{\varepsilon,\eta}}\nabla\chi_{\varepsilon,\eta}\cdot\nabla(u_{0,\varepsilon}\phi)+2\pi\sigma_{\varepsilon}^{-2}\int_{\Omega_{\varepsilon,\eta}}u_{0,\varepsilon}\phi$
$\displaystyle\qquad-2\int_{\Omega_{\varepsilon,\eta}}(\chi_{\varepsilon,\eta}-1)\left(\nabla
u_{0,\varepsilon}\cdot\nabla\phi\right)-2\pi\sigma_{\varepsilon}^{-2}\int_{\Omega_{\varepsilon,\eta}}(\chi_{\varepsilon,\eta}-1)u_{0,\varepsilon}\phi.$

By applying Lemma 6.4 we obtain

$\displaystyle\left|\int_{\Omega_{\varepsilon,\eta}}\nabla
r_{\varepsilon}\cdot\nabla\phi\right|$
$\displaystyle\leq\frac{C}{\varepsilon|\ln\eta|}\int_{\Omega_{\varepsilon,\eta}}|\nabla(u_{0,\varepsilon}\phi)|+2\int_{\Omega_{\varepsilon,\eta}}|\chi_{\varepsilon,\eta}-1||\nabla
u_{0,\varepsilon}||\nabla\phi|$
$\displaystyle\qquad+2\pi\sigma_{\varepsilon}^{-2}\int_{\Omega_{\varepsilon,\eta}}|\chi_{\varepsilon,\eta}-1||u_{0,\varepsilon}||\phi|$
$\displaystyle=I.$

Next, by Hölder’s inequality,

$\displaystyle I$ $\displaystyle\leq
C\sigma_{\varepsilon}^{-1}|\ln\eta|^{-\frac{1}{2}}\left(\|\nabla\phi\|_{L^{2}(\Omega_{\varepsilon,\eta})}\|u_{0,\varepsilon}\|_{L^{2}(\Omega_{\varepsilon,\eta})}+\|\phi\|_{L^{2}(\Omega_{\varepsilon,\eta})}\|\nabla
u_{0,\varepsilon}\|_{L^{2}(\Omega_{\varepsilon,\eta})}\right)$
$\displaystyle\quad+C\|\chi_{\varepsilon,\eta}-1\|_{L^{q}(\Omega_{\varepsilon,\eta})}\left(\|\nabla\phi\|_{L^{2}(\Omega_{\varepsilon,\eta})}\|\nabla
u_{0,\varepsilon}\|_{L^{p}(\Omega_{\varepsilon,\eta})}+\sigma_{\varepsilon}^{-2}\|\phi\|_{L^{2}(\Omega_{\varepsilon,\eta})}\|u_{0,\varepsilon}\|_{L^{p}(\Omega_{\varepsilon,\eta})}\right),$

where $\frac{1}{2}=\frac{1}{p}+\frac{1}{q}$. By applying the Poincaré
inequality $\|\phi\|_{L^{2}(\Omega_{\varepsilon,\eta})}\leq
C\sigma_{\varepsilon}\|\nabla\phi\|_{L^{2}(\Omega_{\varepsilon,\eta})}$ and
Lemma 5.1, we obtain

$\displaystyle I\leq
C|\ln\eta|^{-\frac{1}{2}}\|\nabla\phi\|_{L^{2}(\Omega_{\varepsilon,\eta})}\left(\|\nabla
u_{0,\varepsilon}\|_{L^{p}(\Omega)}+\sigma_{\varepsilon}^{-1}\|u_{0,\varepsilon}\|_{L^{p}(\Omega)}\right).$

By choosing $\phi=r_{\varepsilon}$, we obtain (6.10). ∎

##  7 An Intermediate Problem

In this section we consider the boundary value problem for a Schrödinger
operator,

$\left\\{\begin{aligned} -\Delta u+\lambda^{2}V(x)u&=F&\quad&\text{ in
}\Omega,\\\ u&=g&\quad&\text{ on }\partial\Omega,\end{aligned}\right.$ (7.1)

where $\lambda>0$ and $V=V(x)$ is a potential satisfying the condition
$0<\mu_{0}\leq V\leq\mu_{1}$.

######  Lemma 7.1.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{d}$ and
$\lambda>0$. Suppose that $u\in H^{1}(B(x_{0},2r)\cap\Omega)$ and

$\left\\{\begin{aligned} -\Delta u+\lambda^{2}V(x)u&=0&\quad&\text{ in
}B(x_{0},2r)\cap\Omega,\\\ u&=0&\quad&\text{ on
}B(x_{0},2r)\cap\partial\Omega,\end{aligned}\right.$ (7.2)

where $x_{0}\in\partial\Omega$ and $0<r<r_{0}$. Then

$\sup_{B(x_{0},r)\cap\Omega}|u|\leq
C\left(\fint_{B(x_{0},2r)\cap\Omega}|u|^{2}\right)^{1/2},$ (7.3)

where $C$ depends on $d$, $\Omega$ and $(\mu_{0},\mu_{1})$. Moreover,

$\left(\fint_{B(x_{0},r)\cap\Omega}|\nabla u|^{p}\right)^{1/p}\leq
C\left(\fint_{B(x_{0},2r)\cap\Omega}|\nabla u|^{2}\right)^{1/2},$ (7.4)

where $2<p<3+\delta$ for $d\geq 3$, $2<p<4+\delta$ for $d=2$, and $\delta>0$
depends on $d$ and $\Omega$. The constant $C$ in (7.4) depends on $d$, $p$,
$\Omega$ and $(\mu_{0},\mu_{1})$. If $\Omega$ is a bounded $C^{1}$ domain, the
estimate (7.4) holds for any $2<p<\infty$.

###### Proof.

Let $\phi\in C_{0}^{\infty}(B(x_{0},2r))$. Then

$\int_{B(x_{0},2r)\cap\Omega}\nabla
u\cdot\nabla(u\phi^{2})+\int_{B(x_{0},2r)\cap\Omega}\lambda^{2}Vu\cdot
u\phi^{2}=0.$

It follows by Cauchy’s inequality that

$\int_{B(x_{0},2r)\cap\Omega}|\nabla
u|^{2}\phi^{2}+\int_{B(x_{0},2r)\cap\Omega}\lambda^{2}Vu^{2}\phi^{2}\leq
C\int_{B(x_{0},2r)\cap\Omega}|u|^{2}|\nabla\phi|^{2}.$

By choosing a suitable cut-off function $\phi$, we see that

$\lambda^{2}\int_{B(x_{0},tr)\cap\Omega}u^{2}\leq\frac{C}{r^{2}(s-t)^{2}}\int_{B(x_{0},sr)\cap\Omega}u^{2}$

for $1\leq t<s\leq 2$. This implies that

$\int_{B(x_{0},tr)\cap\Omega}u^{2}\leq\frac{C}{1+(s-t)^{2}\lambda^{2}r^{2}}\int_{B(x_{0},sr)\cap\Omega}u^{2}.$
(7.5)

By iterating (7.5) $k$ times we obtain

$\left(\fint_{B(x_{0},tr)\cap\Omega}u^{2}\right)^{1/2}\leq\frac{C_{k,t,s}}{(1+\lambda
r)^{2k}}\left(\int_{B(x_{0},sr)\cap\Omega}u^{2}\right)^{1/2}$ (7.6)

for any $k\geq 1$, where $1\leq t<s\leq 2$.

Now, since

$\Delta u=\lambda^{2}Vu\ \ \ \text{in}\ B(x_{0},2r)\cap\Omega\quad\text{ and
}\quad u=0\quad\text{ on }B(x_{0},2r)\cap\partial\Omega,$

the boundary $L^{\infty}$ estimates for Laplace’s equation in Lipschitz
domains give

$\displaystyle\left(\fint_{B(x_{0},tr)\cap\Omega}|u|^{p}\right)^{1/p}$
$\displaystyle\leq
C_{t,s}\left(\fint_{B(x_{0},sr)\cap\Omega}|u|^{2}\right)^{1/2}+C_{t,s}r^{2}\lambda^{2}\left(\fint_{B(x_{0},sr)\cap\Omega}|u|^{q}\right)^{1/q},$

where $0<\frac{1}{q}-\frac{1}{p}<\frac{2}{d}$, and

$\displaystyle\sup_{B(x_{0},tr)\cap\Omega}|u|$ $\displaystyle\leq
C_{t,s}\left(\fint_{B(x_{0},sr)\cap\Omega}|u|^{2}\right)^{1/2}+C_{t,s}r^{2}\lambda^{2}\left(\fint_{B(x_{0},sr)\cap\Omega}|u|^{q}\right)^{1/q},$

where $q>(d/2)$. These two estimates, together with (7.6), yield (7.3) by a
bootstrapping argument.

Note that the $W^{1,p}$ estimate for Laplace’s equation in $C^{1}$ domains [6]
gives

$\left(\fint_{B(x_{0},ts)\cap\Omega}|\nabla u|^{p}\right)^{1/p}\leq
C_{t,s}\left(\fint_{B(x_{0},sr)\cap\Omega}|\nabla
u|^{2}\right)^{1/2}+C_{t,s}r\lambda^{2}\left(\fint_{B(x_{0},sr)\cap\Omega}|u|^{q}\right)^{1/q},$

where $1<t<s<2$ and $0<\frac{1}{q}-\frac{1}{p}<\frac{1}{d}$. If $\Omega$ is a
Lipschitz domain, we need to impose the additional conditions that
$2<p<3+\delta$ for $d\geq 3$, and $2<p<4+\delta$ for $d=2$, where $\delta>0$
depends on $d$ and $\Omega$. Thus, by (7.6) and (7.3),

$\displaystyle\left(\fint_{B(x_{0},r)\cap\Omega}|\nabla u|^{p}\right)^{1/p}$
$\displaystyle\leq C\left(\fint_{B(x_{0},2r)\cap\Omega}|\nabla
u|^{2}\right)^{1/2}+Cr^{-1}\left(\fint_{B(x_{0},2r)\cap\Omega}|u|^{2}\right)^{1/2}$
(7.7) $\displaystyle\leq C\left(\fint_{B(x_{0},2r)\cap\Omega}|\nabla
u|^{2}\right)^{1/2},$

where we have used a Poincaré inequality and the fact that $u=0$ on
$B(x_{0},2r)\cap\partial\Omega$ for the last step. ∎

######  Remark 7.2.

Suppose that $u\in H^{1}(B(x_{0},2r))$ and $-\Delta u+\lambda^{2}Vu=0$ in
$B(x_{0},2r)$. Then

$\sup_{B(x_{0},r)}|u|\leq C\left(\fint_{B(x_{0},2r)}|u|^{2}\right)^{1/2},$
(7.8) $\sup_{B(x_{0},r)}|\nabla u|\leq C\left(\fint_{B(x_{0},2r)}(|\nabla
u|^{2}+\lambda^{2}|u|^{2})\right)^{1/2},$ (7.9)

where $C$ depends on $d$ and $(\mu_{0},\mu_{1})$. The proof is similar to that
of Lemma 7.1.

An operator $T$ is called sublinear if there exists $K\geq 1$ such that

$|T(f+g)|\leq K\left(|T(f)|+|T(g)|\right).$ (7.10)

The following theorem was proved in [10].

######  Theorem 7.3.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{d}$. Let $T$ be a
bounded sublinear operator on $L^{2}(\Omega)$ with

$\|T\|_{L^{2}\rightarrow L^{2}}\leq C_{0}.$ (7.11)

Let $q>2$. Suppose that

$\left(\fint_{\Omega\cap B(x_{0},r)}|T(f)|^{q}\right)^{1/q}\leq
N\left(\fint_{\Omega\cap B(x_{0},2r)}|T(f)|^{2}\right)^{1/2}$ (7.12)

for any ball $B(x_{0},r)$ with the property that $0<r<r_{0}$ and either
$B(x_{0},4r)\subset\Omega$ or $x_{0}\in\partial\Omega$ and for any $f\in
C^{\infty}_{0}(\Omega)$ with supp$(f)\subset\Omega\backslash B(x_{0},4r)$.
Then for any $F\in L^{p}(\Omega)$,

$\|T(F)\|_{L^{p}(\Omega)}\leq C\|F\|_{L^{p}(\Omega)},$

where $2<p<q$ and $C$ depends at most on $d,p,q,C_{0},N,r_{0},\Omega$ and $K$.

######  Theorem 7.4.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{d}$. Let $u\in
W^{1,2}_{0}(\Omega)$ be a solution of

$\left\\{\begin{aligned} -\Delta u+\lambda^{2}V(x)u&=F&\quad&\text{ in
}\Omega,\\\ u&=0&\quad&\text{ on }\partial\Omega,\end{aligned}\right.$ (7.13)

where $F\in L^{2}(\Omega)$. Suppose $F\in L^{p}(\Omega)$, where $2<p<3+\delta$
for $d\geq 3$, and $2<p<4+\delta$ for $d=2$. Then

$\lambda\|\nabla u\|_{L^{p}(\Omega)}+\lambda^{2}\|u\|_{L^{p}(\Omega)}\leq
C\|F\|_{L^{p}(\Omega)},$ (7.14)

where $C$ depends on $d$, $p$, $\Omega$ and $(\mu_{0},\mu_{1})$. If $\Omega$
is a bounded $C^{1}$ domain, the estimate (7.14) holds for any $2<p<\infty$.

###### Proof.

To prove (7.14), we consider the sublinear operator,

$T_{\lambda}(F)=\lambda|\nabla u|+\lambda^{2}|u|,$

where $u$ is a solution of (7.13), and apply Theorem 7.3. Note that by the
energy estimate, $\|T_{\lambda}\|_{L^{2}(\Omega)\to L^{2}(\Omega)}\leq C$,
where $C$ depends on $d$, $\Omega$ and $(\mu_{0},\mu_{1})$. To verify the
condition (7.12), let $f\in C_{0}^{\infty}(\Omega)$ such that $f=0$ in
$B(x_{0},2r)$, where $0<r<r_{0}$ and either $x_{0}\in\partial\Omega$ or
$B(x_{0},4r)\subset\Omega$. Let $w$ be the solution of (7.13) with $f$ in the
place of $F$. Then $-\Delta w+\lambda^{2}Vw=0$ in $B(x_{0},4r)\cap\Omega$ and
$w=0$ on $B(x_{0},4r)\cap\partial\Omega$ (if $x_{0}\in\partial\Omega$). If
$\Omega$ is a bounded $C^{1}$ domain, it follows from Lemma 7.1 and Remark 7.2
that

$\left(\fint_{B(x_{0},r)\cap\Omega}(\lambda|\nabla
w|+\lambda^{2}|w|)^{q}\right)^{1/q}\leq
C\left(\fint_{B(x_{0},2r)\cap\Omega}(\lambda|\nabla
w|+\lambda^{2}|w|)^{2}\right)^{1/2}$ (7.15)

for any $2<q<\infty$, where $C$ depends on $d$, $q$, $\Omega$, and
$(\mu_{0},\mu_{1})$. As a result, by Theorem 7.3, we obtain
$\|T_{\lambda}(F)\|_{L^{p}(\Omega)}\leq C\|F\|_{L^{p}(\Omega)}$ for any
$2<p<\infty$. If $\Omega$ is a bounded Lipschitz domain, the estimate (7.15)
holds for $2<q<3+\delta$ if $d\geq 3$, and for $2<q<4+\delta$ if $d=2$. As a
result, we obtain (7.14) for $2<p<3+\delta$ if $d\geq 3$, and for
$2<p<4+\delta$ if $d=2$, where $\delta>0$ depends on $d$ and $\Omega$. ∎

######  Theorem 7.5.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{d}$ with connected
boundary. Let $u\in W^{1,2}(\Omega)$ be a solution of

$\left\\{\begin{aligned} -\Delta u+\lambda^{2}V(x)u&=0&\quad&\text{ in
}\Omega,\\\ u&=g&\quad&\text{ on }\partial\Omega,\end{aligned}\right.$

where $g\in W^{1,2}(\partial\Omega)$. Then

$\left\\{\begin{aligned} \|u\|_{L^{p}(\Omega)}&\leq
C\|g\|_{L^{2}(\partial\Omega)},\\\ \|\nabla u\|_{L^{p}(\Omega)}&\leq
C\left\\{\|\nabla_{t}g\|_{L^{2}(\partial\Omega)}+\lambda\|g\|_{L^{2}(\partial\Omega)}\right\\},\end{aligned}\right.$
(7.16)

where $p=\frac{2d}{d-1}$, $\nabla_{t}g$ denotes the tangential gradient of $g$
on $\partial\Omega$, and $C$ depends on $d$, $\Omega$ and $(\mu_{0},\mu_{1})$.

###### Proof.

We start by solving the Dirichlet problem,

$\begin{cases}\begin{aligned} -\Delta G&=0\qquad&&\text{in}\ \Omega,\\\
G&=g\qquad&&\text{on}\ \partial\Omega.\end{aligned}\end{cases}$

By the well known nontangential-maximal-function estimates for harmonic
functions in Lipschitz domains (see e.g. [15]),

$\|(\nabla G)^{*}||_{L^{2}(\partial\Omega)}\leq
C\|\nabla_{t}g\|_{L^{2}(\partial\Omega)}\quad\text{ and
}\quad\|(G)^{*}\|_{L^{2}(\Omega)}\leq\|g\|_{L^{2}(\partial\Omega)},$

where $\nabla_{t}g$ denotes the tangential gradient of $g$ on
$\partial\Omega$. It follows that

$\displaystyle\|\nabla G\|_{L^{p}(\Omega)}\leq C\|(\nabla
G)^{*}\|_{L^{2}(\partial\Omega)}\leq
C\|\nabla_{t}g\|_{L^{2}(\partial\Omega)},$
$\displaystyle\|G\|_{L^{p}(\Omega)}\leq
C\|(G)^{*}\|_{L^{2}(\partial\Omega)}\leq C\|g\|_{L^{2}(\partial\Omega)},$

where $p=\frac{2d}{d-1}$ and we have used the inequality
$\|w\|_{L^{p}(\Omega)}\leq C\|(w)^{*}\|_{L^{2}(\partial\Omega)}$ for functions
$w$ in $\Omega$.

Next, let $v=u-G$. Then $v$ satisfies the following equation,

$\left\\{\begin{aligned} -\Delta v+\lambda^{2}V(x)v&=F&\quad&\text{ in
}\Omega,\\\ v&=0&\quad&\text{ on }\partial\Omega,\end{aligned}\right.$

where $F=-\lambda^{2}V(x)G$. Let $p=\frac{2d}{d-1}$. Note that $p=4$ for
$d=2$, $p=3$ for $d=3$, and $p<3$ for $d\geq 4$. In view of Theorem 7.4 we
have

$\lambda^{2}\|u-G\|_{L^{p}(\Omega)}\leq C\|F\|_{L^{p}(\Omega)}\leq
C\lambda^{2}\|G\|_{L^{p}(\Omega)}.$

This implies

$\|u\|_{L^{p}(\Omega)}\leq C\|G\|_{L^{p}(\Omega)}\leq
C\|g\|_{L^{2}(\partial\Omega)}$ (7.17)

for $p=\frac{2d}{d-1}$. Moreover, by Theorem 7.4,

$\lambda\|\nabla(u-G)\|_{L^{p}(\Omega)}\leq
C\|\lambda^{2}VG\|_{L^{p}(\Omega)}.$

It follows that

$\displaystyle\|\nabla u\|_{L^{p}(\Omega)}$ $\displaystyle\leq\|\nabla
G\|_{L^{p}(\Omega)}+C\lambda\|G\|_{L^{p}(\Omega)}$ (7.18) $\displaystyle\leq
C\|\nabla_{t}g\|_{L^{2}(\partial\Omega)}+C\lambda\|g\|_{L^{2}(\partial\Omega)}$

for $p=\frac{2d}{d-1}$. ∎

##  8 Large-scale $W^{1,p}$ estimates

Let $u_{\varepsilon}$ be a solution of

$\left\\{\begin{aligned} -\Delta u_{\varepsilon}&=F+\text{\rm
div}(f)&\quad&\text{ in }\Omega_{\varepsilon,\eta},\\\
u_{\varepsilon}&=0&\quad&\text{ on
}\partial\Omega_{\varepsilon,\eta},\end{aligned}\right.$ (8.1)

where $\Omega_{\varepsilon,\eta}$ is given by (1.5). Let

$S_{\varepsilon}(F,f)=\left(\fint_{Q(x,2\varepsilon)}|\nabla
u_{\varepsilon}|^{2}\right)^{1/2},$

where we have extended $u_{\varepsilon}$ to $\mathbb{R}^{d}$ by zero. Note
that

$\|S_{\varepsilon}(F,f)\|_{L^{2}(\mathbb{R}^{d})}=\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon,\eta})}.$

It follows that

$\|S_{\varepsilon}(F,f)\|_{L^{2}(\mathbb{R}^{d})}\leq\|f\|_{L^{2}(\Omega_{\varepsilon,\eta})}+C\min(\sigma_{\varepsilon},1)\|F\|_{L^{2}(\Omega_{\varepsilon,\eta})}.$
(8.2)

This section is dedicated to the $L^{p}$ estimates for $S_{\varepsilon}(F,f)$
in Theorem 1.3.

######  Theorem 8.1.

Let $\Omega$ be a bounded $C^{1}$ domain in $\mathbb{R}^{d}$ and
$\Omega_{\varepsilon,\eta}$ be given by (1.5). Then,

$\|S_{\varepsilon}(0,f)\|_{L^{p}(\Omega)}\leq C\|f\|_{L^{p}(\Omega)}$ (8.3)

for $2<p<\infty$, where $C$ depends only on $d$, $p$, $\Omega$, and
$\\{Y_{z}^{s}\\}$.

In the following we will assume $0<\eta<\eta_{0}$, where $\eta_{0}>0$ is
sufficiently small. The case $\eta\geq\eta_{0}$ with any fixed $\eta_{0}>0$ is
the same as the case $\eta=1$, which is contained in [9]. To prove Theorem
8.1, we use Theorem 7.3. The $L^{2}$ boundedness condition (7.11) is given by
(8.2). To verify the reverse Hölder condition (7.12), it suffices to show that
if $\Delta u_{\varepsilon}=0$ in $Q(x_{0},4R)\cap\Omega_{\varepsilon,\eta}$
and $u_{\varepsilon}=0$ on $Q(x_{0},4R)\cap\partial\Omega_{\varepsilon,\eta}$,
where $0<R<c_{0}$ and either $Q(x_{0},4R)\subset\Omega$ or
$x_{0}\in\partial\Omega$, then

$\left(\fint_{Q(x_{0},R)\cap\Omega}\left(\fint_{Q(x,2\varepsilon)}|\nabla
u_{\varepsilon}|^{2}\right)^{p/2}dx\right)^{1/p}\leq
C\left(\fint_{Q(x_{0},2R)\cap\Omega}\left(\fint_{Q(x,2\varepsilon)}|\nabla
u_{\varepsilon}|^{2}\right)dx\right)^{1/2}$ (8.4)

for $p>2$. It is not hard to see that (8.4) holds for $0<R<C\varepsilon$. The
proof of (8.4) for the large-scale case $R\geq C\varepsilon$ uses a real-
variable argument and relies on the estimates from the previous two sections.

We begin with the interior case $Q(x_{0},4R)\subset\Omega$. The following
theorem was proved in [11, Theorem 3.2].

######  Theorem 8.2.

Let $F\in L^{2}(Q(x_{0},2R))$ and $2<p<q$. Suppose that for each cube
$Q=Q(y,r)$ with $y\in Q(x_{0},R)$ and $0<r<c_{0}R$, there exists two functions
$R_{Q}$ and $F_{Q}$ such that

$|F|\leq|R_{Q}|+|F_{Q}|\quad\text{ in }2Q,$ (8.5)
$\left(\fint_{2Q}|R_{Q}|^{q}\right)^{1/q}\leq
N\left(\fint_{8Q}|F|^{2}\right)^{1/2},$ (8.6)
$\left(\fint_{2Q}|F_{Q}|^{2}\right)^{1/2}\leq\delta\left(\fint_{8Q}|F|^{2}\right)^{1/2},$
(8.7)

where $N>1$ and $0<c_{0}<1$. Then there exists $\delta_{0}>0$, depending only
on $d,N,c_{0},p,q$, with the property that if $0\leq\delta<\delta_{0}$, then
$F\in L^{p}(Q(x_{0},R))$ and

$\left(\fint_{Q(x_{0},R)}|F|^{p}\right)^{1/p}\leq
C\left(\fint_{Q(x_{0},2R)}|F|^{2}\right)^{1/2},$ (8.8)

where $C$ depends at most on $d,c_{0},p,q,N$.

We use $Q_{\varepsilon,\eta}(x_{0},r)$ to denote $\Omega_{\varepsilon,\eta}$,
where $\Omega=Q(x_{0},r)$. We point out that in general,
$Q_{\varepsilon,\eta}(x_{0},r)\neq Q(x_{0},r)\cap\Omega_{\varepsilon,\eta}$,
as there is no hole near the boundary of $Q(x_{0},r)$ in
$Q_{\varepsilon,\eta}(x_{0},r)$. However, under the assumption $r\geq
C\varepsilon$, it is possible to find $Q(y_{0},t)$ such that
$Q(x_{0},r)\subset Q(y_{0},t)\subset Q(x_{0},2r)$ and
$Q_{\varepsilon,\eta}(y_{0},t)=\Omega_{\varepsilon,\eta}\cap Q(y_{0},t)$. This
observation is used in the proof of the following lemma.

######  Lemma 8.3.

Let $u\in H^{1}(Q(x_{0},4r))$, where $r\geq 8\varepsilon$ and
$Q(x_{0},8r)\subset\Omega$. Suppose that $\Delta u=0$ in
$Q(x_{0},4r)\cap\Omega_{\varepsilon,\eta}$ and $u=0$ on
$Q(x_{0},4r)\setminus\Omega_{\varepsilon,\eta}$. Then there exists $v\in
H^{1}(Q(x_{0},2r))$ such that

$\left(\fint_{Q(x_{0},r)}|\nabla(u-v)|^{2}\right)^{1/2}\leq
C\psi(\eta)\left(\fint_{Q(x_{0},3r)}|\nabla u|^{2}\right)^{1/2},$ (8.9)
$\max_{x\in Q(x_{0},r)}\left(\fint_{Q(x,2\varepsilon)}|\nabla
v|^{2}\right)^{1/2}\leq C\left(\fint_{Q(x_{0},3r)}|\nabla
u|^{2}\right)^{1/2},$ (8.10)

where $\psi(\eta)=\eta^{1/2}$ for $d\geq 3$ and $\psi(\eta)=|\ln\eta|^{-1/2}$
for $d=2$. The constant $C$ depends only on $d$ and $\\{Y_{z}^{s}\\}$.

###### Proof.

Since $r\geq 8\varepsilon$, without loss of generality, we may assume that
$x_{0}=0$ and $r=2^{j}\varepsilon$ for some $j\geq 2$. By dilation we may also
assume $r=1$. Note that the parameter $\eta$ remains invariant under dilation.
Since $u=0$ on $Q(0,3)\setminus\Omega_{\varepsilon,\eta}$, it follows by Lemma
2.1 that

$\sigma_{\varepsilon}^{-2}\int_{Q(0,3)}|u|^{2}\leq C\int_{Q(0,3)}|\nabla
u|^{2}.$ (8.11)

We claim that there exists $t\in[2,3]$ such that
$Q_{\varepsilon,\eta}(0,t)=Q(0,t)\cap\Omega_{\varepsilon,\eta}$ and

$\sigma_{\varepsilon}^{-2}\int_{\partial Q(0,t)}|u|^{2}+\int_{\partial
Q(0,t)}|\nabla u|^{2}\leq C\int_{Q(0,3)}|\nabla u|^{2}.$ (8.12)

To see (8.12), we consider the set $E=\\{t\in[2,3]:|t-k\varepsilon|\leq
c_{0}\varepsilon\text{ for some }1\leq k\leq 2^{j+2}\\}$. Note that $|E|\geq
c>0$ and $Q_{\varepsilon,\eta}(0,t)=Q(0,t)\cap\Omega_{\varepsilon,\eta}$ for
$t\in E$. If $\eqref{L-1-3}$ fails for all $t\in E$, we may integrate the
left-hand side of (8.12) with respect to $t$ over $E$ and use (8.11) to obtain
a contradiction.

Next, we let $w$ be the solution of

$\left\\{\begin{aligned} -\Delta
w+\sigma_{\varepsilon}^{-2}V(x/\varepsilon)w&=0&\quad&\text{ in }Q(0,t),\\\
w&=u&\quad&\text{ on }\partial Q(0,t),\end{aligned}\right.$ (8.13)

where $V(x)$ is given by (6.2). It follows by Theorems 6.1 and 6.3 that

$\|\nabla(u-\chi_{\varepsilon,\eta}w)\|_{L^{2}(Q(0,t))}\leq
C\psi(\eta)\left\\{\sigma_{\varepsilon}^{-1}\|w\|_{L^{p}(Q(0,t))}+\|\nabla
w\|_{L^{p}(Q(0,t))}\right\\},$ (8.14)

where $p=\frac{2d}{d-1}$ and $\chi_{\varepsilon,\eta}$ is the corrector for
the domain $Q_{\varepsilon,\eta}(0,t)$. This, together with Theorem 7.5,
yields

$\displaystyle\|\nabla(u-\chi_{\varepsilon,\eta}w)\|_{L^{2}(Q(0,t))}$
$\displaystyle\leq
C\psi(\eta)\left\\{\sigma_{\varepsilon}^{-1}\|u\|_{L^{2}(\partial
Q(0,t))}+\|\nabla u\|_{L^{2}(\partial Q(0,t))}\right\\}$ (8.15)
$\displaystyle\leq C\psi(\eta)\|\nabla u\|_{L^{2}(Q(0,3))},$

where we have used (8.12) for the last step.

Finally, we let $v=\chi_{\varepsilon,\eta}w$. Note that the estimate (8.9) is
given by (8.15). To see (8.10), we note that

$\displaystyle\left(\fint_{Q(x,2\varepsilon)}|\nabla v|^{2}\right)^{1/2}$
$\displaystyle\leq\max_{Q(x,2\varepsilon)}|w|\left(\fint_{Q(x,2\varepsilon)}|\nabla\chi_{\varepsilon,\eta}|^{2}\right)^{1/2}+\max_{Q(x,2\varepsilon)}|\chi_{\varepsilon,\eta}||\nabla
w|$ $\displaystyle\leq
C\max_{Q(x,2\varepsilon)}(\sigma_{\varepsilon}^{-1}|w|+|\nabla w|),$

where we have used Lemmas 4.3 and 5.2. This, together with Remark 7.2, gives

$\displaystyle\left(\fint_{Q(x,2\varepsilon)}|\nabla v|^{2}\right)^{1/2}$
$\displaystyle\leq C\max_{Q(0,3/2)}(\sigma_{\varepsilon}^{-1}|w|+|\nabla w|)$
$\displaystyle\leq
C\left\\{\sigma_{\varepsilon}^{-1}\|w\|_{L^{2}(Q(0,2))}+\|\nabla
w\|_{L^{2}(Q(0,2))}\right\\}$ $\displaystyle\leq C\|\nabla
u\|_{L^{2}(Q(0,3))}$

for any $x\in Q(0,1)$. ∎

######  Lemma 8.4.

Let $u\in H^{1}(B(x_{0},4R))$, where $R\geq 8\varepsilon$ and
$B(x_{0},8R)\subset\Omega$. Suppose that $\Delta u=0$ in
$Q(x_{0},4R)\cap\Omega_{\varepsilon,\eta}$ and $u=0$ in
$Q(x_{0},4R)\setminus\Omega_{\varepsilon,\eta}$. Let

$v(x)=\left(\fint_{Q(x,2\varepsilon)}|\nabla u|^{2}\right)^{1/2}.$ (8.16)

Then for $2<p<\infty$,

$\left(\fint_{Q(x_{0},R)}|v|^{p}\right)^{1/p}\leq
C\left(\fint_{Q(x_{0},2R)}|v|^{2}\right)^{1/2},$ (8.17)

where $C$ depends only on $d$, $p$ and $\\{Y_{z}^{s}\\}$.

###### Proof.

Without loss of generality we may assume $x_{0}=0$. By dilation we may assume
$R=1$. To prove (8.17), we apply Theorem 8.2 with $F=v$. Let $Q=Q(y_{0},r)$,
where $y_{0}\in Q(0,1)$ and $0<r<(1/8)$. If $0<r<8\varepsilon$, we let
$R_{Q}=v$ and $F_{Q}=0$. Note that

$\max_{2Q}R_{Q}\leq C\left(\fint_{Q(y_{0},2r+2\varepsilon)}|\nabla
u|^{2}\right)^{1/2}\leq C\left(\fint_{4Q}|v|^{2}\right)^{1/2}.$

If $r\geq 8\varepsilon$, we let $R_{Q}=|\nabla v|$ and $F_{Q}=|\nabla(u-v)|$,
where $v$ is given by Lemma 8.3. In view of (8.9) and (8.10), we obtain (8.6)
and (8.7) with $\delta=C\psi(\eta)$. It follows by Theorem 8.2 that the
estimate (8.17) holds if $\eta<\eta_{0}$, where $\eta_{0}>0$ depends only on
$d$, $p$ and $\\{Y_{z}^{s}\\}$. ∎

Next, we treat the boundary case where $x_{0}\in\partial\Omega$. Since
$\Omega$ is Lipschitz, there exists $r_{0}>0$ such that
$B(x_{0},r_{0})\cap\Omega=B(x_{0},r_{0})\cap D$ and
$B(x_{0},r_{0})\cap\partial\Omega=B(x_{0},r_{0})\cap\partial D$, where, after
a rotation of the coordinate system, $D$ is given by

$D=\left\\{(x^{\prime},x_{d})\in\mathbb{R}^{d}:\
x^{\prime}\in\mathbb{R}^{d-1}\text{ and }x_{d}>\varphi(x^{\prime})\right\\},$
(8.18)

for some Lipschitz function $\varphi:\mathbb{R}^{d-1}\to\mathbb{R}$.

######  Theorem 8.5.

Let $2<p<q$ and $D$ be given by (8.18). Let $x_{0}\in\partial D$ and $F\in
L^{2}(Q(x_{0},2R)\cap D)$. Suppose that for each cube $Q=Q(y,r)$ with the
property that $0<r<c_{0}R$ and either $y\in Q(x_{0},R)\cap\partial D$ or
$4Q\subset Q(x_{0},2R)\cap D$, there exist two measurable functions $F_{Q}$
and $R_{Q}$ in $2Q\cap D$ such that

$|F|\leq|R_{Q}|+|F_{Q}|\quad\text{ in }2Q\cap D,$ (8.19) $\left(\fint_{2Q\cap
D}|R_{Q}|^{q}\right)^{1/q}\leq N\left(\fint_{4Q\cap D}|F|^{2}\right)^{1/2},$
(8.20) $\left(\fint_{2Q\cap
D}|F_{Q}|^{2}\right)^{1/2}\leq\eta\left(\fint_{4Q\cap D}|F|^{2}\right)^{1/2},$
(8.21)

where $N>1$ and $0<c_{0}<1$. Then there exists $\eta_{0}>0$, depending only on
$d,N,c_{0},,p,q$ and $\|\nabla\varphi\|_{\infty}$, with the property that if
$0\leq\eta<\eta_{0}$, then $F\in L^{p}(Q(x_{0},R)\cap D)$ and

$\left(\fint_{Q(x_{0},R)\cap D}|F|^{p}\right)^{1/p}\leq
C\left(\fint_{Q(x_{0},2R)\cap D}|F|^{2}\right)^{1/2},$ (8.22)

where $C$ depends at most on d, $N$, $c_{0}$, $p$, $q$ and
$\|\nabla\varphi\|_{\infty}$.

The proof of the above theorem can be found in [11] or [12, pp. 79-82].

######  Lemma 8.6.

Let $\Omega$ be a bounded $C^{1}$ domain. Let $u\in
H^{1}(Q(x_{0},4r)\cap\Omega)$, where $r\geq 8\varepsilon$ and
$x_{0}\in\partial\Omega$. Suppose that $\Delta u=0$ in
$Q(x_{0},4r)\cap\Omega_{\varepsilon,\eta}$ and $u=0$ on
$Q(x_{0},4r)\cap\partial\Omega_{\varepsilon,\eta}$. Then there exists $v\in
H^{1}(Q(x_{0},2r)\cap\Omega)$ such that

$\left(\fint_{Q(x_{0},r)\cap\Omega}|\nabla(u-v)|^{2}\right)^{1/2}\leq
C\psi(\eta)\left(\fint_{Q(x_{0},3r)\cap\Omega}|\nabla u|^{2}\right)^{1/2},$
(8.23)
$\left(\fint_{Q(x_{0},r)\cap\Omega}\left(\fint_{Q(x,2\varepsilon)\cap\Omega}|\nabla
v|^{2}\right)^{p/2}\right)^{1/p}\leq
C\left(\fint_{Q(x_{0},3r)\cap\Omega}|\nabla u|^{2}\right)^{1/2},$ (8.24)

for any $p>2$, where $\psi(\eta)=\eta^{1/2}$ for $d\geq 3$ and
$\psi(\eta)=|\ln\eta|^{-1/2}$ for $d=2$. The constant $C$ depends only on $d$,
$p$, $\\{Y_{z}^{s}\\}$ and $\Omega$.

###### Proof.

The proof is similar to that of Lemma 8.3. Without loss of generality we may
assume $x_{0}=0$. By dilation we may also assume $r=1$. Choose $t\in[2,3]$
such that
$(Q(0,t)\cap\Omega)_{\varepsilon,\eta}=Q(0,t)\cap\Omega_{\varepsilon,\eta}$
and

$\sigma_{\varepsilon}^{-2}\int_{\Omega\cap\partial
Q(0,t)}|u|^{2}+\int_{\Omega\cap\partial Q(0,t)}|\nabla u|^{2}\leq
C\int_{\Omega\cap Q(0,3)}|\nabla u|^{2}.$ (8.25)

Let $w$ be the solution of

$\left\\{\begin{aligned} -\Delta
w+\sigma_{\varepsilon}^{-2}V(x/\varepsilon)w&=0&\quad&\text{ in
}Q(0,t)\cap\Omega,\\\ w&=u&\quad&\text{ on
}\partial(Q(0,t)\cap\Omega).\end{aligned}\right.$ (8.26)

As in the proof of Theorem 8.3, we have

$\|\nabla(u-\chi_{\varepsilon,\eta}w)\|_{L^{2}(Q(0,t)\cap\Omega)}\leq
C\psi(\eta)\|\nabla u\|_{L^{2}(Q(0,3)\cap\Omega)},$ (8.27)

where $\chi_{\varepsilon,\eta}$ denotes the corrector for the domain
$Q_{\varepsilon,\eta}(0,t)=Q(0,t)\cap\Omega_{\varepsilon,\eta}$. Let
$v=\chi_{\varepsilon,\eta}w$. Then

$\left(\fint_{Q(x,2\varepsilon)\cap\Omega}|\nabla v|^{2}\right)^{1/2}\leq
C\sigma_{\varepsilon}^{-1}\max_{Q(x,2\varepsilon)\cap\Omega}|w|+C\left(\fint_{Q(x,2\varepsilon)\cap\Omega}|\nabla
w|^{2}\right)^{1/2}.$ (8.28)

Since $w=u=0$ on $Q(0,t)\cap\partial\Omega$, it follows from Lemma 7.1 that

$\displaystyle\sigma_{\varepsilon}^{-1}\max_{Q(0,3/2)\cap\Omega}|w|$
$\displaystyle\leq C\left(\fint_{Q(0,2)\cap\Omega}|\nabla
w|^{2}\right)^{1/2},$ (8.29)
$\displaystyle\left(\fint_{Q(0,3/2)\cap\Omega}|\nabla w|^{p}\right)^{1/p}$
$\displaystyle\leq C\left(\fint_{Q(0,2)\cap\Omega}|\nabla w|^{2}\right)^{1/2}$

for $2<p<\infty$. As a result, we obtain

$\displaystyle\int_{Q(0,1)\cap\Omega}\left(\fint_{Q(x,2\varepsilon)\cap\Omega}|\nabla
v|^{2}\right)^{p/2}$ $\displaystyle\leq C\left(\fint_{Q(0,2)\cap\Omega}|\nabla
w|^{2}\right)^{p/2}$ $\displaystyle\leq C\big{\\{}\|\nabla
u\|_{L^{2}(\partial(Q(0,t)\cap\Omega))}+\sigma_{\varepsilon}^{-1}\|u\|_{L^{2}(\partial
Q(0,t)\cap\Omega)}\big{\\}}^{p},$

This, together with (8.25), gives (8.24). ∎

###### Proof of Theorem 8.1..

As we discussed earlier, it suffices to prove the reverse Hölder estimate
(8.4), where $0<R<c_{0}$ and either $Q(x_{0},4R)\subset\Omega$ or
$x_{0}\in\partial\Omega$. Note that the interior case
$Q(x_{0},4R)\subset\Omega$ is given by Lemma 8.4. For the boundary case where
$x_{0}\in\partial\Omega$, we apply Theorem 8.5. To this end, for each cube
$Q=Q(y,r)$ with the property that $0<r<c_{0}R$ and either $4Q\subset
Q(x_{0},2R)\cap\Omega$ or $y\in Q(x_{0},2R)\cap\partial\Omega$, we need to
construct two functions $F_{Q}$ and $R_{Q}$ such that (8.19)-(8.21) hold.
Again, the case $4Q\subset Q(x_{0},2R)\cap\Omega$ is given by Lemma 8.4. For
the case $y\in\partial\Omega$ and $0<r<8\varepsilon$, we let $F_{Q}=F$ and
$R_{Q}=0$. Finally, for $r\geq 8\varepsilon$, we use Lemma 8.6. ∎

######  Theorem 8.7.

Let $\Omega$ be a bounded $C^{1}$ domain and $\Omega_{\varepsilon,\eta}$ be
given by (1.5). Then,

$\|S_{\varepsilon}(F,0)\|_{L^{p}(\Omega)}\leq
C\min(\sigma_{\varepsilon},1)\|F\|_{L^{p}(\Omega)}$ (8.30)

for any $2<p<\infty$, where $C$ depends only on $d$, $p$, $\Omega$, and
$\\{Y_{z}^{s}\\}$.

###### Proof.

To prove (8.30), we consider the operator

$T_{\varepsilon}(F)=\left(\min(\sigma_{\varepsilon},1)\right)^{-1}S_{\varepsilon}(F,0)$

and apply Theorem 7.3. Note that $\|T_{\varepsilon}\|_{L^{2}\to L^{2}}\leq C$.
Moreover, the reverse Hölder condition (7.12) is reduced to the estimate
(8.4), as in the case of $S_{\varepsilon}(0,f)$. As a result, by Theorem 7.3,
we obtain $\|T_{\varepsilon}\|_{L^{p}\to L^{p}}\leq C$ for any $p>2$. This
gives the estimate (8.30) for $p>2$. ∎

###### Proof of Theorem 1.3.

Since

$S_{\varepsilon}(F,f)\leq S_{\varepsilon}(F,0)+S_{\varepsilon}(0,f),$

Theorem 1.3 follows readily from Theorems 8.1 and 8.7. ∎

##  9 Lower bounds for $A_{p}(\Omega_{\varepsilon,\eta})$ and
$B_{p}(\Omega_{\varepsilon,\eta})$

In this section we discuss the sharpness of the estimates in Theorems 1.1 and
1.2 for $A_{p}(\Omega_{\varepsilon,\eta})$ as well as those in Section 3 for
$B_{p}(\Omega_{\varepsilon,\eta})$. Our approach is similar to that used in
[13] for an unbounded and periodically perforated domain,

$\omega_{\epsilon,\eta}=\mathbb{R}^{d}\backslash\bigcup_{z\in\mathbb{Z}^{d}}\varepsilon(z+\eta\overline{Y^{s}}),$
(9.1)

where $Y^{s}$ is a bounded domain with connected $C^{1}$ boundary such that
$B(0,c_{0})\subset Y^{s}\subset B(0,1/4)$. Following an idea of [7], we
consider an $\varepsilon$-periodic corrector defined by

$-\Delta\psi_{\varepsilon,\eta}=\varepsilon^{-2}\eta^{d-2}\qquad\text{in }\
\omega_{\varepsilon,\eta}\qquad\text{and}\qquad\psi_{\varepsilon,\eta}=0\qquad\text{on}\
\ \mathbb{R}^{d}\setminus\omega_{\varepsilon,\eta}.$ (9.2)

######  Lemma 9.1.

Let $\psi_{\varepsilon,\eta}$ be the $\varepsilon$-periodic corrector defined
in (9.2). Then if $d\geq 3$,

$\fint_{Q(0,\varepsilon)}\psi_{\varepsilon,\eta}\approx
1\qquad\text{and}\qquad\left(\fint_{Q(0,\varepsilon)}|\nabla\psi_{\varepsilon,\eta}|^{2}\right)^{1/2}\approx\varepsilon^{-1}\eta^{\frac{d-2}{2}}.$
(9.3)

If $d=2$, we have

$\fint_{Q(0,\varepsilon)}\psi_{\varepsilon,\eta}\approx|\ln\eta|\qquad\text{and}\qquad\left(\fint_{Q(0,\varepsilon)}|\nabla\psi_{\varepsilon,\eta}|^{2}\right)^{1/2}\approx\varepsilon^{-1}|\ln\eta|^{1/2}.$
(9.4)

###### Proof.

The case $\varepsilon=1$ was proved in [13, Lemma 4.4]. The case
$0<\varepsilon<1$ follows by rescaling. ∎

######  Lemma 9.2.

Let $\psi_{\varepsilon,\eta}$ be the same as in Lemma 9.1. Suppose
$2<p<\infty$. Then if $d\geq 3$,

$\left(\fint_{Q(0,\varepsilon)}|\psi_{\varepsilon,\eta}|^{p}\right)^{1/p}\leq
C\qquad\text{and}\qquad\left(\fint_{Q(0,\varepsilon)}|\nabla\psi_{\varepsilon,\eta}|^{p}\right)^{1/p}\geq
C\varepsilon^{-1}\eta^{\frac{d}{p}-1}.$ (9.5)

If $d=2$, we have

$\left(\fint_{Q(0,\varepsilon)}|\psi_{\varepsilon,\eta}|^{p}\right)^{1/p}\leq
C|\ln\eta|\qquad\text{and}\qquad\left(\fint_{Q(0,\varepsilon)}|\nabla\psi_{\varepsilon,\eta}|^{p}\right)^{1/p}\geq
C\varepsilon^{-1}\eta^{\frac{2}{p}-1}.$ (9.6)

###### Proof.

The case $\varepsilon=1$ was proved in [13, Lemma 5.3], while the general case
follows by rescaling. ∎

For a bounded Lipschitz domain $\Omega$, define

$\Omega_{\varepsilon,\eta}=\Omega\setminus\bigcup_{z}\varepsilon(z+\eta\overline{Y^{s}}),$
(9.7)

where the union is taken over those $z$’s in $\mathbb{Z}^{d}$ for which
$\varepsilon Q(z,1)\subset\Omega$; i.e., $\Omega_{\varepsilon,\eta}$ is given
by (1.5) with $x_{z}=0$ and $Y_{z}^{s}=Y^{s}$ for all $z\in\mathbb{Z}^{d}$.
Recall that $A_{p}(\Omega_{\varepsilon,\eta})$ and
$B_{p}(\Omega_{\varepsilon,\eta})$ denote the smallest bounding constants for
which (1.2) holds, where $u_{\varepsilon}$ is the solution of the Dirichlet
problem (1.1) in $\Omega_{\varepsilon,\eta}$. Similarly, we use
$C_{p}(\Omega_{\varepsilon,\eta})$ and $D_{p}(\Omega_{\varepsilon,\eta})$ to
denote the smallest bounding constants for which the $L^{p}$ estimate,

$\|u_{\varepsilon}\|_{L^{p}(\Omega_{\varepsilon,\eta})}\leq
C_{p}(\Omega_{\varepsilon,\eta})\|f\|_{L^{p}(\Omega_{\varepsilon,\eta})}+D_{p}(\Omega_{\varepsilon,\eta})\|F\|_{L^{p}(\Omega_{\varepsilon,\eta})},$
(9.8)

holds. By a duality argument we have

$A_{p}(\Omega_{\varepsilon,\eta})=A_{p^{\prime}}(\Omega_{\varepsilon,\eta}),\quad
D_{p}(\Omega_{\varepsilon,\eta})=D_{p^{\prime}}(\Omega_{\varepsilon,\eta})\quad\text{
and }\quad
B_{p}(\Omega_{\varepsilon,\eta})=C_{p^{\prime}}(\Omega_{\varepsilon,\eta}),$
(9.9)

where $1<p<\infty$ and $p^{\prime}=\frac{p}{p-1}$.

We begin with the sharpness of $D_{p}(\Omega_{\varepsilon,\eta})$.

######  Theorem 9.3.

Let $\Omega$ be a bounded Lipschitz domain and $\Omega_{\varepsilon,\eta}$ be
given by (9.7). Then, for $d\geq 2$ and $1<p<\infty$,

$D_{p}(\Omega_{\varepsilon,\eta})\approx\min(\sigma_{\varepsilon}^{2},1).$
(9.10)

###### Proof.

The upper bound $D_{p}(\Omega_{\varepsilon,\eta})\leq
C\min(\sigma_{\varepsilon}^{2},1)$ is contained in Theorem 2.3. To show the
lower bound $D_{p}(\Omega_{\varepsilon,\eta})\geq
c\min(\sigma_{\varepsilon}^{2},1)$, by (9.9), we may assume $1<p\leq 2$.
Without loss of generality we may also assume $Q(0,4r_{0})\subset\Omega$.

Suppose $\varepsilon$ is sufficiently small and $R\varepsilon\leq r_{0}$ for
some $R\in\mathbb{N}$. Let $\phi\in C_{0}^{\infty}(Q(0,2R\varepsilon))$ be a
cut-off function such that $0\leq\phi\leq 1$, $\phi=1$ in $Q(0,R\varepsilon)$,
$|\nabla\phi|\leq C/(R\varepsilon)$ and $|\nabla^{2}\phi|\leq
C/(R\varepsilon)^{2}$. Note that

$-\Delta(\psi_{\varepsilon,\eta}\phi)=\varepsilon^{-2}\eta^{d-2}\phi-2\nabla\psi_{\varepsilon,\eta}\cdot\nabla\phi-\psi_{\varepsilon,\eta}\Delta\phi$
(9.11)

in $\Omega_{\varepsilon,\eta}$ and $\psi_{\varepsilon,\eta}\phi=0$ on
$\partial\Omega_{\varepsilon,\eta}$. It follows that

$\displaystyle
cR^{d/p}\|\psi_{\varepsilon,\eta}\|_{L^{p}(Q(0,\varepsilon))}\leq\|\psi_{\varepsilon,\eta}\phi\|_{L^{p}(\Omega_{\varepsilon,\eta})}$
$\displaystyle\leq
CD_{p}(\Omega_{\varepsilon,\eta})\left\\{\varepsilon^{-2+\frac{d}{p}}\eta^{d-2}R^{\frac{d}{p}}+\varepsilon^{-1}R^{\frac{d}{p}-1}\|\nabla\psi_{\varepsilon,\eta}\|_{L^{p}(Q(0,\varepsilon))}+\varepsilon^{-2}R^{\frac{d}{p}-2}\|\psi_{\varepsilon,\eta}\|_{L^{p}(Q(0,\varepsilon))}\right\\},$

where we have used the periodicity of $\psi_{\epsilon,\eta}$. This gives

$\displaystyle\varepsilon^{\frac{d}{p}}\fint_{Q(0,\varepsilon)}|\psi_{\varepsilon,\eta}|$
(9.12) $\displaystyle\leq
CD_{p}{(\Omega_{\varepsilon,\eta})}\left\\{\varepsilon^{\frac{d}{p}-2}\eta^{d-2}+R^{-1}\varepsilon^{\frac{d}{p}-1}\left(\fint_{Q(0,\varepsilon)}|\nabla\psi_{\varepsilon,\eta}|^{2}\right)^{1/2}+(\varepsilon
R)^{-2}\|\psi_{\varepsilon,\eta}\|_{L^{p}(Q(0,\varepsilon))}\right\\},$

where we have used the fact $1<p\leq 2$.

Case 1: $d\geq 3$ and $0<\sigma_{\varepsilon}\leq 1.$

By applying Lemmas 9.1 and 9.2 to (9.12), we obtain

$c\leq
D_{p}{(\Omega_{\varepsilon,\eta})}\left\\{\varepsilon^{-2}\eta^{d-2}+R^{-1}\varepsilon^{-2}\eta^{\frac{d-2}{2}}+(\varepsilon
R)^{-2}\right\\}.$ (9.13)

Since $\sigma_{\varepsilon}=\varepsilon\eta^{1-\frac{d}{2}}\leq 1$, we may
choose $R\in\mathbb{N}$ such that $R\approx\eta^{1-\frac{d}{2}}$ in (9.13) to
get

$c\leq D_{p}{(\Omega_{\varepsilon,\eta})}\cdot\varepsilon^{-2}\eta^{d-2}.$

Hence, $D_{p}(\Omega_{\varepsilon,\eta})\geq
c\sigma_{\varepsilon}^{2}=c\min(\sigma_{\varepsilon}^{2},1).$

Case 2: $d\geq 3$ and $\sigma_{\varepsilon}\geq 1$.

In this case we choose $R\in\mathbb{N}$ such that $R\approx\varepsilon^{-1}$
in (9.13). It follows that

$c\leq
D_{p}(\Omega_{\varepsilon,\eta})\left\\{\varepsilon^{-2}\eta^{d-2}+\varepsilon^{-1}\eta^{\frac{d-2}{2}}+1\right\\}.$

Since $\sigma_{\varepsilon}\geq 1$, we obtain
$D_{p}(\Omega_{\varepsilon,\eta})\geq c=c\min(\sigma_{\varepsilon}^{2},1).$

Case 3: $d=2$ and $\sigma_{\varepsilon}\leq 1$.

Applying Lemmas 9.1 and 9.2 to (9.12) now yields

$c|\ln\eta|\leq
D_{p}(\Omega_{\varepsilon,\eta})\left\\{\varepsilon^{-2}+R^{-1}\varepsilon^{-2}|\ln\eta|^{\frac{1}{2}}+(\varepsilon
R)^{-2}|\ln\eta|\right\\}.$ (9.14)

Picking $R\in\mathbb{N}$ such that $R\approx|\ln\eta|^{\frac{1}{2}}$ in (9.14)
gives

$D_{p}(\Omega_{\varepsilon,\eta})\geq
c\,\varepsilon^{2}|\ln\eta|=c\min(\sigma_{\varepsilon}^{2},1).$

Case 4: $d=2$ and $\sigma_{\varepsilon}\geq 1$.

Choosing $R\in\mathbb{N}$ such that $R\approx\varepsilon^{-1}$ in (9.14)
yields

$\displaystyle c|\ln\eta|$ $\displaystyle\leq
D_{p}(\Omega_{\varepsilon,\eta})\left\\{\varepsilon^{-2}+\varepsilon^{-1}|\ln\eta|^{\frac{1}{2}}+|\ln\eta|\right\\}$
$\displaystyle\leq CD_{p}(\Omega_{\varepsilon,\eta})|\ln\eta|.$

It follows that $D_{p}(\Omega_{\epsilon})\geq
c=c\min(\sigma_{\varepsilon}^{2},1)$. ∎

Next, we consider the case $B_{p}(\Omega_{\varepsilon,\eta})$ for $1<p\leq 2$.

######  Theorem 9.4.

Let $\Omega$ be a bounded Lipschitz domain and $\Omega_{\varepsilon,\eta}$ be
given by (9.7). Then

$B_{p}(\Omega_{\varepsilon,\eta})\approx\min(\sigma_{\varepsilon},1)$ (9.15)

for $d\geq 2$ and $1<p\leq 2$,

###### Proof.

It follows from Theorem 2.3 that

$C_{p}(\Omega_{\varepsilon,\eta})\leq C\min(\sigma_{\varepsilon},1)$

for $2\leq p<\infty$. Since
$B_{p}(\Omega_{\varepsilon,\eta})=C_{p^{\prime}}(\Omega_{\varepsilon,\eta})$,
we obtain the upper bound

$B_{p}(\Omega_{\varepsilon,\eta})\leq C\min(\sigma_{\varepsilon},1)$

for $1<p\leq 2$. To establish the lower bounds, we first note that
$D_{2}(\Omega_{\varepsilon,\eta})\leq
C\min(\sigma_{\varepsilon},1)B_{2}(\Omega_{\varepsilon,\eta})$, which follows
(2.2). As a result, by Theorem 9.3, $B_{2}(\Omega_{\varepsilon,\eta})\geq
c\min(\sigma_{\varepsilon},1)$.

For the case $1<p<2$, we proceed as in the proof of Theorem 9.3. It follows
from (9.11) that

$\displaystyle\|\psi_{\varepsilon,\eta}\phi\|_{L^{q}(\Omega_{\varepsilon,\eta})}\leq
C\|\nabla(\psi_{\varepsilon,\eta}\phi)\|_{L^{p}(\Omega_{\varepsilon,\eta})}$
$\displaystyle\leq
CB_{p}(\Omega_{\varepsilon,\eta})\left\\{\varepsilon^{\frac{d}{p}-2}\eta^{d-2}R^{\frac{d}{p}}+\varepsilon^{-1}R^{\frac{d}{p}-1}\|\nabla\psi_{\varepsilon,\eta}\|_{L^{p}(Q(0,\varepsilon))}+\varepsilon^{-2}R^{\frac{d}{p}-2}\|\psi_{\varepsilon,\eta}\|_{L^{p}(Q(0,\varepsilon))}\right\\},$

where $\frac{1}{q}=\frac{1}{p}-\frac{1}{d}$. This implies that

$\fint_{Q(0,\varepsilon)}|\psi_{\varepsilon,\eta}|\leq
CB_{p}(\Omega_{\varepsilon,\eta})\left\\{\varepsilon^{-1}\eta^{d-2}R+\left(\fint_{Q(0,\varepsilon)}|\nabla\psi_{\varepsilon,\eta}|^{2}\right)^{{1}/{2}}+(\varepsilon
R)^{-1}\left(\fint_{Q(0,\varepsilon)}|\psi_{\varepsilon,\eta}|^{p}\right)^{1/p}\right\\}.$
(9.16)

Case 1: $d\geq 3$ and $\sigma_{\varepsilon}\leq 1.$

We apply Lemmas 9.1 and 9.2 to (9.16) to obtain

$c\leq
B_{p}(\Omega_{\varepsilon,\eta})\left\\{\varepsilon^{-1}\eta^{d-2}R+\varepsilon^{-1}\eta^{\frac{d-2}{2}}+\varepsilon^{-1}R^{-1}\right\\}.$
(9.17)

Letting $R\approx\eta^{-\frac{d-2}{2}}$ in (9.17) yields

$c\leq B_{p}(\Omega_{\epsilon})\cdot\varepsilon^{-1}\eta^{\frac{d-2}{2}}.$

Thus, $B_{p}(\Omega_{\varepsilon,\eta})\geq
c\sigma_{\varepsilon}=c\min(\sigma_{\varepsilon},1)$.

Case 2: $d\geq 3$ and $\sigma_{\varepsilon}\geq 1.$

Let $R\approx\varepsilon^{-1}$ in (9.17). Then

$c\leq
B_{p}(\Omega_{\varepsilon,\eta})\left\\{\varepsilon^{-2}+\varepsilon^{-1}\eta^{\frac{d-2}{2}}+C\right\\}.$

Thus, $B_{p}(\Omega_{\varepsilon,\eta})\geq c=c\min(\sigma_{\varepsilon},1)$.

Case 3: $d=2$ and $\sigma_{\varepsilon}\leq 1$.

Applying Lemmas 9.1 and 9.2 to (9.16) now yields

$c|\ln\eta|\leq
B_{p}(\Omega_{\varepsilon,\eta})\left\\{\varepsilon^{-1}R+\varepsilon^{-1}|\ln\eta|^{\frac{1}{2}}+(\varepsilon
R)^{-1}|\ln\eta|\right\\}.$ (9.18)

Picking $R\approx|\ln\eta|^{\frac{1}{2}}$ in (9.18) gives

$c|\ln\eta|\leq
B_{p}(\Omega_{\varepsilon,\eta})\cdot\varepsilon^{-1}|\ln\eta|^{\frac{1}{2}}.$

Thus, $B_{p}(\Omega_{\varepsilon,\eta})\geq
c\varepsilon|\ln\eta|^{1/2}=c\min(\sigma_{\varepsilon},1)$.

Case 4: $d=2$ and $\sigma_{\varepsilon}\geq 1$.

Taking $R\approx\varepsilon^{-1}$ in (9.18) leads to

$\displaystyle c|\ln\eta|$ $\displaystyle\leq
B_{p}(\Omega_{\varepsilon,\eta})\left\\{\varepsilon^{-2}+\varepsilon^{-1}|\ln\eta|^{\frac{1}{2}}+|\ln\eta|\right\\}$
$\displaystyle\leq CB_{p}(\Omega_{\varepsilon,\eta})|\ln\eta|.$

This implies that $B_{p}(\Omega_{\varepsilon,\eta})\geq
c=c\min(\sigma_{\varepsilon},1)$. ∎

We now consider $B_{p}(\Omega_{\varepsilon,\eta})$ for $2<p<\infty$.

######  Theorem 9.5.

Let $\Omega_{\varepsilon,\eta}$ be given by (9.7). Assume $d\geq 3$. Then if
$0<\sigma_{\varepsilon}\leq 1$ and $2<p<\infty$,

$B_{p}(\Omega_{\varepsilon,\eta})\approx\varepsilon\eta^{1-d+\frac{d}{p}}.$
(9.19)

If $\sigma_{\varepsilon}\geq 1$ and $2<p<d$, we have

$B_{p}(\Omega_{\varepsilon,\eta})\approx
1+\varepsilon^{-1}\eta^{\frac{d}{p}-1}.$ (9.20)

###### Proof.

Suppose $0<\sigma_{\varepsilon}\leq 1$. The upper bound
$B_{p}(\Omega_{\varepsilon,\eta})\leq C\varepsilon\eta^{1-d+\frac{d}{p}}$ is
contained in Theorem 3.2. To prove the lower bound, we use the Poincaré
inequality,

$\|u\|_{L^{p}(\Omega_{\varepsilon,\eta})}\leq
C\min(\varepsilon\eta^{1-\frac{d}{p}},1)\|\nabla
u\|_{L^{p}(\Omega_{\varepsilon,\eta})}$ (9.21)

for $u\in W^{1,p}_{0}(\Omega_{\varepsilon,\eta})$, where $2<p<d$. The
inequality (9.21) follows readily from Lemma 2.1. As a result, we see that

$D_{p}(\Omega_{\varepsilon,\eta})\leq
CB_{p}(\Omega_{\varepsilon,\eta})\min(\varepsilon\eta^{1-\frac{d}{p}},1).$
(9.22)

By Theorem 9.3, we have $D_{p}(\Omega_{\varepsilon,\eta})\geq
c\sigma_{\varepsilon}^{2}=c\varepsilon^{2}\eta^{2-d}$. This, together with
(9.22), gives (9.19) for the case $2<p<d$.

Next, we use a convexity argument to handle the case
$0<\sigma_{\varepsilon}\leq 1$ and $d\leq p<\infty$. Choose $2<q<d$ and
$t\in(0,1)$ so that

$\frac{1}{q}=\frac{1-t}{2}+\frac{t}{p}.$

By Riesz-Thorin Theorem we have

$B_{q}(\Omega_{\varepsilon,\eta})\leq\left[B_{2}(\Omega_{\varepsilon,\eta})\right]^{1-t}\left[B_{p}(\Omega_{\varepsilon,\eta})\right]^{t}.$

It follows that

$B_{p}(\Omega_{\varepsilon,\eta})\geq\left[B_{q}(\Omega_{\varepsilon,\eta})\right]^{\frac{1}{t}}\left[B_{2}(\Omega_{\varepsilon,\eta})\right]^{1-\frac{1}{t}}.$

The desired estimate follows by using $B_{2}(\Omega_{\varepsilon,\eta})\leq
C\sigma_{\varepsilon}$ and $B_{q}(\Omega_{\varepsilon,\eta})\geq
c\varepsilon\eta^{1-d+\frac{d}{q}}$.

Finally, we consider the case $\sigma_{\varepsilon}\geq 1$ and $2<p<d$. The
upper bound is contained in Theorem 3.3. Since
$D_{p}(\Omega_{\varepsilon,\eta})\geq c$, by (9.22), we obtain

$B_{p}(\Omega_{\varepsilon,\eta})\geq c\varepsilon^{-1}\eta^{\frac{d}{p}-1}$

However, (9.22) also yields $B_{p}(\Omega_{\varepsilon,\eta})\geq
cD_{p}(\Omega_{\varepsilon,\eta})\geq c$. As a result, we obtain
$B_{p}(\Omega_{\varepsilon,\eta})\geq
c(1+\varepsilon^{-1}\eta^{\frac{d}{p}-1})$. ∎

Finally, we establish lower bounds for $A_{p}(\Omega_{\varepsilon,\eta})$.

######  Theorem 9.6.

Let $2<p<\infty$. Let $\Omega_{\varepsilon,\eta}$ be given by (9.7). Then, if
$0<\sigma_{\varepsilon}\leq 1$,

$A_{p}(\Omega_{\varepsilon,\eta})\geq\begin{cases}\begin{aligned}
&c\eta^{-d|\frac{1}{2}-\frac{1}{p}|}\qquad&&\text{if}\ d\geq 3,\\\
&c\eta^{-2|\frac{1}{2}-\frac{1}{p}|}|\ln\eta|^{-\frac{1}{2}}\qquad&&\text{if}\
d=2.\end{aligned}\end{cases}$ (9.23)

If $\sigma_{\epsilon}\geq 1$ we have

$A_{p}(\Omega_{\varepsilon,\eta})\geq\begin{cases}\begin{aligned}
&c(1+\varepsilon^{-1}\eta^{\frac{d}{p}-1})\qquad&&\text{if}\ d\geq 3\text{ and
}2<p<d,\\\ &c\varepsilon^{-1}\eta^{\frac{d}{p}-1}\qquad&&\text{if}\ d\geq
3\text{ and }d\leq p<\infty,\\\
&c\varepsilon^{-1}\eta^{\frac{2}{p}-1}|\ln\eta|^{-1}\qquad&&\text{if}\
d=2\text{ and }2<p<\infty.\end{aligned}\end{cases}$ (9.24)

The constants $c>0$ only depend on $d$, $p$, $Y^{s}$ and $\Omega$.

###### Proof.

Let $u_{\varepsilon}$ be a solution to $-\Delta u_{\varepsilon}=\text{\rm
div}(f)$ in $\Omega_{\varepsilon,\eta}$ with $u_{\varepsilon}=0$ on
$\partial\Omega_{\varepsilon,\eta}$. By a Sobolev imbedding, we have

$\|u_{\varepsilon}\|_{L^{q^{\prime}}(\Omega_{\varepsilon,\eta})}\leq C\|\nabla
u_{\varepsilon}\|_{L^{p^{\prime}}(\Omega_{\varepsilon,\eta})}\leq
CA_{p}(\Omega_{\varepsilon,\eta})\|f\|_{L^{p^{\prime}}(\Omega_{\varepsilon,\eta})},$

where $\frac{1}{q^{\prime}}=\frac{1}{p^{\prime}}-\frac{1}{d}$ and
$1<p^{\prime}<d.$ By duality this implies that if $-\Delta v_{\varepsilon}=G$
in $\Omega_{\varepsilon,\eta}$ and $v_{\varepsilon}=0$ on
$\partial\Omega_{\varepsilon,\eta},$ then

$\|\nabla v_{\varepsilon}\|_{L^{p}(\Omega_{\varepsilon,\eta})}\leq
CA_{p}(\Omega_{\varepsilon,\eta})\|G\|_{L^{q}(\Omega_{\varepsilon,\eta})}.$

Thus, if $-\Delta u_{\varepsilon}=F+\text{\rm div}(f)$ in
$\Omega_{\varepsilon,\eta}$ with $u_{\varepsilon}=0$ on
$\partial\Omega_{\varepsilon,\eta}$, then

$\|\nabla u_{\varepsilon}\|_{L^{p}(\Omega_{\varepsilon,\eta})}\leq
CA_{p}(\Omega_{\varepsilon,\eta})\left\\{\|F\|_{L^{q}(\Omega_{\varepsilon,\eta})}+\|f\|_{L^{p}(\Omega_{\varepsilon,\eta})}\right\\},$
(9.25)

where $d^{\prime}<p<\infty$ and $\frac{1}{q}=\frac{1}{p}+\frac{1}{d}$.

Let $\phi$ and $\psi_{\varepsilon,\eta}$ be the same as in the proof of
Theorem 9.3. Note that

$-\Delta(\psi_{\varepsilon,\eta}\phi)=\varepsilon^{-2}\eta^{d-2}\phi-2\text{\rm
div}(\psi_{\varepsilon,\eta}\nabla\phi)+\psi_{\varepsilon,\eta}\Delta\phi$

in $\Omega_{\varepsilon,\eta}$ and $\psi_{\varepsilon,\eta}\phi=0$ on
$\partial\Omega_{\varepsilon,\eta}$. It follows from (9.25) that

$\displaystyle
R^{\frac{d}{p}}\|\nabla\psi_{\varepsilon,\eta}\|_{L^{p}(Q(0,\varepsilon))}\leq
C\|\nabla(\psi_{\varepsilon,\eta}\phi)\|_{L^{p}(\Omega_{\varepsilon,\eta})}$
$\displaystyle\leq
CA_{p}(\Omega_{\varepsilon,\eta})\left\\{\varepsilon^{\frac{d}{q}-2}\eta^{d-2}R^{\frac{d}{q}}+\varepsilon^{-1}R^{\frac{d}{p}-1}\|\psi_{\varepsilon,\eta}\|_{L^{p}(Q(0,\varepsilon))}+\varepsilon^{-2}R^{\frac{d}{q}-2}\|\psi_{\varepsilon,\eta}\|_{L^{q}(Q(0,\varepsilon))}\right\\},$

where we have used the periodicity of $\psi_{\varepsilon,\eta}$. Hence,

$\displaystyle\|\nabla\psi_{\varepsilon,\eta}\|_{L^{p}(Q(0,\varepsilon))}$
(9.26) $\displaystyle\leq
CA_{p}(\Omega_{\varepsilon,\eta})\left\\{\varepsilon^{\frac{d}{p}-1}\eta^{d-2}R+\varepsilon^{-1}R^{-1}\|\psi_{\varepsilon,\eta}\|_{L^{p}(Q(0,\varepsilon))}+\varepsilon^{-2}R^{-1}\|\psi_{\varepsilon,\eta}\|_{L^{q}(Q(0,\varepsilon))}\right\\}.$

Case 1: $d\geq 3$ and $0<\sigma_{\varepsilon}\leq 1.$

Applying Lemma 9.2 to (9.26) yields

$\eta^{\frac{d}{p}-1}\leq
CA_{p}(\Omega_{\varepsilon,\eta})\left\\{\eta^{d-2}R+R^{-1}\right\\}.$ (9.27)

Since $\sigma_{\varepsilon}\leq 1$, we may pick
$R\approx\eta^{-\frac{d-2}{2}}$ in (9.27) and obtain

$A_{p}(\Omega_{\varepsilon,\eta})\geq
C\eta^{\frac{d}{p}-\frac{d}{2}}=C\eta^{-d|\frac{1}{p}-\frac{1}{2}|}.$

Case 2: $d=2$ and $0<\sigma_{\varepsilon}\leq 1.$

Applying Lemma 9.2 to (9.26) now yields

$\eta^{\frac{2}{p}-1}\leq
CA_{p}(\Omega_{\varepsilon,\eta})\left\\{R+|\ln\eta|R^{-1}\right\\}.$ (9.28)

Picking $R\approx|\ln\eta|^{\frac{1}{2}}$ in (9.28) gives

$\eta^{\frac{2}{p}-1}\leq
CA_{p}(\Omega_{\varepsilon,\eta})|\ln\eta|^{\frac{1}{2}},$

which leads to

$A_{p}(\Omega_{\varepsilon,\eta})\geq
c\eta^{-2|\frac{1}{2}-\frac{1}{p}|}|\ln\eta|^{-\frac{1}{2}}.$

Case 3: $d\geq 3$ and $\sigma_{\varepsilon}\geq 1.$

Choosing $R\approx\varepsilon^{-1}$ in (9.27) yields

$\displaystyle\eta^{\frac{d}{p}-1}$ $\displaystyle\leq
CA_{p}(\Omega_{\varepsilon,\eta})\left\\{\eta^{d-2}\varepsilon^{-1}+\varepsilon\right\\}$
$\displaystyle\leq C\varepsilon A_{p}(\Omega_{\varepsilon,\eta}).$

Thus,

$A_{p}(\Omega_{\varepsilon,\eta})\geq
c\,\varepsilon^{-1}\eta^{\frac{d}{p}-1}.$

This gives the lower bound in (9.24) for the case $d\leq p<\infty$. Note that
if $2<p<d$, by the Poincaré inequality
$\|u_{\varepsilon}\|_{L^{p}(\Omega_{\varepsilon,\eta})}\leq C\|\nabla
u_{\varepsilon}\|_{L^{p}(\Omega_{\varepsilon,\eta})}$, we have

$A_{p}(\Omega_{\varepsilon,\eta})\geq
c\,C_{p}(\Omega_{\varepsilon,\eta})=c\,B_{p^{\prime}}(\Omega_{\varepsilon,\eta})\geq
c.$

Hence,

$A_{p}(\Omega_{\varepsilon,\eta})\geq
c(1+\varepsilon^{-1}\eta^{\frac{d}{p}-1}).$

Case 4: $d=2$ and $\sigma_{\varepsilon}\geq 1.$

Now we take $R\approx\varepsilon^{-1}$ in (9.28). This gives

$\displaystyle\eta^{\frac{2}{p}-1}$ $\displaystyle\leq
CA_{p}(\Omega_{\varepsilon,\eta})\left\\{\varepsilon^{-1}+\varepsilon|\ln\eta|\right\\}$
$\displaystyle\leq C\varepsilon A_{p}(\Omega_{\varepsilon,\eta})|\ln\eta|,$

which leads to

$A_{p}(\Omega_{\varepsilon,\eta})\geq
C\varepsilon^{-1}\eta^{\frac{2}{p}-1}|\ln\eta|^{-1}.$

This completes the proof. ∎

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Robert Righi, Department of Mathematics, University of Kentucky, Lexington,
Kentucky 40506, USA.

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

E-mail<EMAIL_ADDRESS>