# An Asymptotic Formula for Eigenvalues of the Neumann Laplacian in Domains
with a Small Star-shaped Hole

Ly Hong Hai
Department of Mathematics
University of Ostrava
30\. dubna 22
701 03 Ostrava, Czech Republic
E-mail<EMAIL_ADDRESS>
###### Abstract

This article investigates a spectral problem of the Laplace operator in a two-
dimensional bounded domain perforated by a small arbitrary star-shaped hole
and on the smooth boundary of which the Neumann boundary condition is imposed.
It is proved that the eigenvalues of this problem converge to the eigenvalues
of the Laplacian defined on the unperturbed domain as the size of the hole
approaches zero. Furthermore, our main theorem provides the rate of
convergence by showing an asymptotic expansion for all simple eigenvalues with
respect to the size and shape of the hole.

††2020 _Mathematics Subject Classification_ : Primary 35P20; Secondary
35P15.†† _Key words and phrases_ : Perforated domains, Neumann Laplacian,
eigenvalues, asymptotic formula.

##  1 Introduction

In recent years, the application of spectral theory to the study of the
effects of a small hole on the sound of a drum or bell, as well as the
behaviour of the heat equation in the study of heat distribution in objects,
has arisen great interest among scientific community. A large number of
studies have been conducted in this field, exploring topics such as the
spectral properties of Laplacian in bounded domains under Dirichlet or Neumann
conditions, the asymptotic expansion of eigenvalues for domains with small
holes, the changes in eigenvalues as the domain is varied and so on. Several
studies are briefly discussed as follows. Rauch and Taylor [21] found that the
spectrum of Laplacian in a bounded domain remains unchanged after applying
Dirichlet conditions to a compact subset of capacity zero, while Ozawa [11,
13, 14, 15, 16, 17, 18, 19, 20, 12], Besson [3] and Courtois [4] investigated
the expansion of eigenvalues in cases of small holes with Dirichlet or Neumann
boundary conditions. Kato [8], Hubert [7], as well as Ward and Keller [23]
focused on how small changes in the domain affect the eigenvalues. Gadyl’shin
and Il’in [6], Daners [5], and McGillivray [10] were considered as notable
studies in this area. In [9], Maz’ya, Nazarov, and Plamenevskij investigated
the Laplace operator on domains with obstacles and found a complete asymptotic
expansion for eigenvalues. In contrast, Barseghyan et al. [2] demonstrated the
convergence of the Laplacian spectrum on a hole-like compact set with zero
Lebesgue measure to the original domain’s Laplacian spectrum.

As far as we are concerned, much of the previous research has focused on
studying the spectral problem of the Laplace operator in domains with
specifically shaped holes, such as spherical or elliptical holes. However, the
exploration of domains with holes of general shape and smooth boundaries is
still limited, although they offer practical advantages. General-shaped holes
with smooth boundaries are commonly encountered in engineering scenarios,
including the design of heat exchangers, acoustic chambers, fluid flow control
devices, and electromagnetic waveguides. By investigating the spectral problem
in these domains, we can analyze the influence of the hole’s shape on various
physical phenomena. The smoothness of the hole’s boundary ensures the
continuity of heat flow, acoustic waves, fluid motion, or electromagnetic
fields, allowing for accurate modelling and efficient analysis.

Motivated by the observation above, we consider the spectral problem of the
Laplace operator in a bounded planar domain $\Omega$. This domain is
perforated by a small star-shaped hole $E_{\varepsilon}$ with a smooth
boundary, where $\varepsilon$ is a fixed parameter. The notion of a star-
shaped set is known as a natural generalization of a convex set. We impose the
Neumann condition on the boundary of the hole and the Dirichlet condition on
the outer boundary of the domain. Our main contribution in this paper is
constructing an asymptotic expansion with respect to the hole size for all
simple eigenvalues. To analyze the problem, we employ asymptotic techniques
and perturbation methods. More precisely, we first parameterize the small hole
by using curvilinear coordinates, which helps to develop approximations for
eigenvalues and eigenfunctions by employing the concept of Green’s kernel.
Green’s kernel has been well-known as a fundamental solution of the Laplace
equation that characterizes the behaviour of the Laplace operator in the
domain of interest. We construct Green operators by utilizing Green’s kernel,
an integral operator associated with Laplacian. Using Green operators, we
derive approximations for eigenvalues and eigenfunctions of the Laplacian with
the Neumann boundary condition on the hole. These approximations provide a
means to estimate the eigenvalues and eigenfunctions. From that, we obtain the
desired asymptotic formula of eigenvalues. The case $N=3$, the singularity of
Green function $K(x,y)$ near $x=y$ is stronger than that of the case $N=2$.
When we use the Sobolev embedding; $W^{2,p}(\Omega)\hookrightarrow
C^{2-N/p}(\bar{\Omega})$, we must take $p$ larger as $N$ increases.

The paper is organized as follows. In the next section, we state the problem
and present our main result, which is contained in Theorem 2.1. Section 3 is
devoted to preliminary results, in which we establish useful lemmas for
estimating the $L^{p}$ norm used for the rest of the paper. In Section 4,
using properties of Green functions and operators related to our problem, we
address technical issues relevant to our main result, whose proof appears in
Section 5. The last section contains conclusions and discussions.

##  2  Statement of the problem and main result

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded domain with smooth $C^{1}$
boundary $\partial\Omega$ and the origin $\tilde{w}\in\Omega$. Let
$E\subset\Omega$ be an open domain of star-shaped concerning a neighbourhood
of $\tilde{w}$ and $\partial E$ is also of class $C^{1}$. For $\varepsilon>0$,
we consider the small hole $E_{\varepsilon}=\varepsilon E$. We introduce the
curvilinear coordinates, where $x=(x_{1},x_{2})$ denotes the Cartesian
coordinates in $\mathbb{R}^{2}$ as

$\displaystyle\begin{cases}x_{1}=\alpha(\rho)\beta(\theta)\cos\theta\\\
x_{2}=\alpha(\rho)\omega(\theta)\sin\theta\end{cases}$

with $\rho\in[0,\infty)$ and $\theta\in\mathbb{R}$, where the functions
$\alpha,\beta,\omega$ are $C^{1}$-functions satisfying

  * (i)

$\beta,\omega$ are periodic of period $2\pi$ with
$\beta(0),\beta(\pi),\omega(\pi/2),\omega(3\pi/2)\neq 0$,

  * (ii)

$\alpha,\beta$ are strictly increasing and $\alpha(0)=0$,
$\lim_{\rho\to\infty}\alpha(\rho)=\infty$ and

  * (iii)

for any $\rho>0,\theta\in[0,2\pi)$

$\displaystyle\beta(\theta)\cos(\theta)(\beta(\theta)\cos(\theta))^{\prime}+\omega(\theta)\sin(\theta)(\omega(\theta)\sin(\theta))^{\prime}=0.$

The assumptions (i) and (ii) ensure that map
$(\rho,\theta)\mapsto(x_{1},x_{2})$ is a diffeomorphism from
$(0,\infty)\times[0,2\pi)$ to $\mathbb{R}\backslash\\{\tilde{w}\\}$. The
condition (iii) is for the orthogonality of the curve
$(x_{1}(\cdot,\theta_{0}),x_{2}(\cdot,\theta_{0}))$ and
$(x_{1}(\rho_{0},\cdot),x_{2}(\rho_{0},\cdot))$ for any $\rho_{0}>0$ and
$\theta_{0}\in[0,\infty)$, which is important for the sequel analysis. In the
special case, it is easy to see that choice $\alpha(\rho)=e^{\rho}-1$,
$\beta(\theta)=2+\cos(4\theta)$, and
$\omega(\theta)=2+\frac{1}{6}\frac{\sin(3\theta)}{\sin{\theta}}+\frac{1}{10}\frac{\sin(5\theta)}{\sin\theta}$
satisfies these three assumptions. Moreover, thanks to the assumption of $E$,
we can find $\rho_{0}>0$ and functions $\alpha,\beta,\omega$ satisfying
(i)–(iii) and the boundary $\partial E$ is characterized by

$\partial
E=\\{(x_{1}(\rho_{0},\theta),x_{2}(\rho_{0},\theta)):\theta\in[0,2\pi)\\}.$

Therefore, the small hole $E_{\varepsilon}$ and its boundary is characterized
by

$E_{\varepsilon}=\\{(\varepsilon x_{1}(\rho,\theta),\varepsilon
x_{2}(\rho,\theta)):\;0\leq\rho<\rho_{0},\theta\in[0,2\pi)\\}$

and

$\partial E_{\varepsilon}=\\{(\varepsilon x_{1}(\rho_{0},\theta),\varepsilon
x_{2}(\rho_{0},\theta)):\theta\in[0,2\pi)\\}.$

Define for $x\in\partial E_{\varepsilon}$ the function $f(x)=|x|^{2}$. Then,
$f$ is continuous thanks to the smoothness of $\partial E_{\varepsilon}$. Note
that there always exist two points $x_{\min},x_{\max}\in\partial
E_{\varepsilon}$ such that $f(x_{\min})=\min_{x\in\partial
E_{\varepsilon}}f(x)$ and $f(x_{\max})=\max_{x\in\partial
E_{\varepsilon}}f(x)$. It is immediate that

$\pi|x_{\min}|^{2}\leq|E_{\varepsilon}|\leq\pi|x_{\max}|^{2}.$

Since the continuous function $g(x)=f(x)-|E_{\varepsilon}|/\pi$ satisfies
$g(x_{\min})g(x_{\max})\leq 0$, there exists at least one point
$x^{*}\in\partial E_{\varepsilon}$ such that $g(x^{*})=0$, or equivalently,

(2.1)
$\displaystyle|E_{\varepsilon}|=\pi|x^{*}|^{2}=\pi(M\varepsilon)^{2}\quad\text{
where }\quad
M=\frac{|x^{*}|}{\varepsilon}=\left(\frac{|E|}{\pi}\right)^{1/2}.$

We denote $\Omega_{E_{\varepsilon}}:=\Omega\setminus\bar{E}_{\varepsilon}$ and
consider the following eigenvalue problem

(2.2) $\begin{cases}-\Delta
u(x)=\lambda(\varepsilon)u(x)&x\in\Omega_{E_{\varepsilon}},\\\
u(x)=0&x\in\partial\Omega,\\\ \frac{\partial u}{\partial\nu}(x)=0&x\in\partial
E_{\varepsilon}.\end{cases}$

Here the symbol $\partial u/\partial\nu$ is the directional derivative of $u$
along the unit outward normal vector $\nu$ to the domain
$\Omega_{E_{\varepsilon}}$, hence pointing the spectral problem mentioned
above has a countable sequence of strictly positive eigenvalues, with each
eigenvalue having a finite multiplicity [22]. These eigenvalues are
conventionally arranged in ascending order, and if an eigenvalue has a
multiplicity greater than one, it is repeated accordingly:

$0<\mu_{1}(\varepsilon)<\mu_{2}(\varepsilon)\leq\dots\leq\mu_{i}(\varepsilon)\leq\dots\to\infty.$

A crucial property is that as $\varepsilon$ approaches zero, see e.g. Rauch &
Taylor [21], each spectral problem’s eigenvalue converges to the Dirichlet
Laplacian’s corresponding eigenvalue in the original domain without a hole.
More precisely,

(2.3) $\displaystyle\mu_{i}(\varepsilon)\to\mu_{i},\text{ as }\varepsilon\to
0,$

where $\\{\mu_{i}\\}$ are the eigenvalues of the following problem

(2.4) $\begin{cases}-\Delta u(x)=\lambda u(x),&x\in\Omega\\\
u(x)=0,&x\in\partial\Omega.\end{cases}$

Let $K(x,y)$ represent the Green function of the Laplacian in the domain
$\Omega$, associated with the boundary condition mentioned in equation (2.4).
Similarly, $K_{\varepsilon}(x,y)$ denotes the Green function in the domain
$\Omega_{E_{\varepsilon}}$, associated with the boundary condition specified
in equation (2.2).

Let $\varphi_{i}(x)$ denote the eigenfunction of (2.2) corresponding to
$\mu_{i}$, which is normalized in $L^{2}(\Omega)$. To indicate the asymptotic
behaviour, we employ the notation $\mathcal{O}(\cdot)$, commonly known as the
“big-$\mathcal{O}$” Landau notation. Specifically, for any sufficiently small
positive $x$, the notation $f(x)=\mathcal{O}(x)$ implies that $\lvert
f(x)\rvert\leq Cx$, where $C$ is a constant.

We will present our main result, which provides the convergence rate (2.3).

######  Theorem 2.1.

Assume that $\mu_{i}$ is a simple eigenvalue. Then, as $\varepsilon$ tends to
zero, we have

$\mu_{i}(\varepsilon)=\mu_{i}-(2\lvert\nabla\varphi_{i}(\tilde{w})\rvert^{2}-\mu_{i}\varphi_{i}(\tilde{w})^{2})|E|\varepsilon^{2}+\mathcal{O}(\varepsilon^{3}|\log\varepsilon|^{2})$

where $\mu_{i}$ is the corresponding eigenvalue of the unperturbed problem to
$\mu_{i}(\varepsilon)$, $\tilde{w}$ is the origin and $|E|$ is the area of
$E$.

Our result extends the analysis of eigenvalue perturbations to two dimensional
domains, providing a distinct asymptotic expansion. While [9] Maz’ya’s three-
dimensional result, which shows a leading order term scaling with
$\varepsilon^{3}$, our findings indicate that in two dimensions, the leading
order term scales with $\varepsilon^{2}$. Additionally, we prove a higher-
order term $\mathcal{O}(\varepsilon^{3}|\log\varepsilon|^{2})$ for greater
precision. This highlights the different behaviours of eigenvalue
perturbations in various dimensional settings and underscores the importance
of dimensionality in asymptotic analysis.

######  Remark 2.2.

  * •

The asymptotic in Theorem 2.1 does not depend on the parameterization of $E$,
but only on the area of $E$. This also implies that the asymptotic behaviour
is invariant with respect to rotations and translations.

  * •

Our results and proofs are not applicable to eigenvalues with multiplicity
higher than one. This case is for future investigation (see e.g. [1]).

##  3 Preliminary lemmas

For any periodic function $L(\theta)$ of $\theta\in[0,2\pi]$, its Fourier
series is given by

$\displaystyle L(\theta)=S_{0}+\sum_{k=0}^{\infty}S_{k}\cos k\theta+P_{k}\sin
k\theta.$

######  Lemma 3.1.

Consider the equation system described by:

(3.1) $\displaystyle\Delta v_{\varepsilon}(x)$ $\displaystyle=0\quad\text{for
}x\in\mathbb{R}^{2}\setminus\bar{E}_{\varepsilon},$ (3.2)
$\displaystyle\frac{\partial v_{\varepsilon}}{\partial\nu}(x)$
$\displaystyle=L(\theta)\quad\text{for }x\in\partial E_{\varepsilon},$

where $L(\theta)$ is a given smooth function on $\partial E_{\varepsilon}$.
Then there exists at least one solution $v_{\varepsilon}$ that satisfies the
following condition:

(3.3) $\lvert v_{\varepsilon}(x)\rvert\leq
C\varepsilon\max_{\theta\in[0,2\pi]}\lvert
L(\theta)\rvert(1+\rho+|\log\varepsilon|),$

where $C$ is a constant independent of $\varepsilon$.

###### Proof.

Now, consider the function:

$v_{\varepsilon}(x)=A_{0}(\rho+\log\varepsilon)+\sum_{k=1}^{\infty}(-k)^{-1}e^{-k\rho}(A_{k}\cos
k\theta+B_{k}\sin k\theta).$

Thanks to the assumption (iii) of the curvilinear coordinates, we have

$\Delta
v_{\varepsilon}=\partial_{x_{1}}^{2}v_{\varepsilon}+\partial_{x_{2}}^{2}v_{\varepsilon}=\left[(\partial_{\rho}x_{1})^{2}+(\partial_{\rho}x_{2})^{2}\right]^{-1}(\partial_{\rho}^{2}v_{\varepsilon}+\partial_{\theta}^{2}v_{\varepsilon}).$

Therefore, direct computations lead to $\Delta v_{\varepsilon}(x)=0$ for
$x\in\mathbb{R}^{2}\setminus\bar{E}_{\varepsilon}$. Next, we consider the
normal derivatives $\frac{\partial}{\partial\nu}$ of $\partial
E_{\varepsilon}$. We have

$\displaystyle d\nu=\sqrt{\left(\frac{\partial
x_{1}}{\partial\rho}\right)^{2}+\left(\frac{\partial
x_{2}}{\partial\rho}\right)^{2}}d\rho$
$\displaystyle=\varepsilon|\alpha^{\prime}(\rho)|\sqrt{\beta(\theta)^{2}\cos^{2}\theta+\omega(\theta)^{2}\sin^{2}\theta}\,d\rho$
$\displaystyle=\varepsilon\tau(\rho,\theta)\,d\rho$

where
$\tau(\rho,\theta)=|\alpha^{\prime}(\rho)|\sqrt{\beta(\theta)^{2}\cos^{2}\theta+\omega(\theta)^{2}\sin^{2}\theta}\neq
0$ due to the assumptions on $\alpha,\beta,\omega$.

Differentiating the function $v_{\varepsilon}(x)$ along the exterior normal
vector at $x\in\partial E_{\varepsilon}$, we observe that

$\displaystyle\frac{\partial v_{\varepsilon}}{\partial\nu}(x)|_{x\in\partial
E_{\varepsilon}}$ $\displaystyle=$ $\displaystyle\left.\frac{\partial
v_{\varepsilon}}{\partial\rho}\frac{\partial\rho}{\partial\nu}\right\rvert_{x\in\partial
E_{\varepsilon}}$ $\displaystyle=$ $\displaystyle\left.\frac{\partial
v_{\varepsilon}}{\partial\rho}\frac{1}{\varepsilon\tau(\rho,\theta)}\right\rvert_{\rho=\rho_{0}}$
$\displaystyle=$
$\displaystyle\frac{1}{\varepsilon\tau(\rho_{0},\theta)}\left(A_{0}+\sum_{k=1}^{\infty}e^{-k\rho_{0}}(A_{k}\cos
k\theta+B_{k}\sin k\theta)\right)$ $\displaystyle=$ $\displaystyle
S_{0}+\sum_{k=1}^{\infty}(S_{k}\sin k\theta+P_{k}\cos k\theta)$
$\displaystyle=$ $\displaystyle L(\theta),$

which implies

$\displaystyle A_{0}$ $\displaystyle=$
$\displaystyle\varepsilon\tau(\rho_{0},\theta)S_{0},$ $\displaystyle A_{k}$
$\displaystyle=$ $\displaystyle\varepsilon
e^{-k\rho_{0}}\tau(\rho_{0},\theta)S_{k},$ $\displaystyle B_{k}$
$\displaystyle=$ $\displaystyle\varepsilon
e^{-k\rho_{0}}\tau(\rho_{0},\theta)P_{k},$

for $k\geq 1$. Thus, we rewrite the inequality as

(3.4) $\lvert v_{\varepsilon}(x)\rvert\leq\lvert
S_{0}\rvert\varepsilon\tau(\rho_{0},\theta)(\rho+|\log\varepsilon|)+\left(\sum_{k=1}^{\infty}\left(S_{k}^{2}+P_{k}^{2}\right)\right)^{1/2}\\\
\times\left(\sum_{k=1}^{\infty}(-k)^{-2}\varepsilon^{2}\tau(\rho_{0},\theta)^{2}e^{-2k(\rho_{0}+\rho)}\right)^{1/2}.$

By applying Schwarz’s inequality, we have

$\displaystyle\sum_{k=1}^{\infty}(-k)^{-2}\varepsilon^{2}\tau(\rho_{0},\theta)^{2}e^{-2k(\rho_{0}+\rho)}$
$\displaystyle\leq\varepsilon^{2}\tau(\rho_{0},\theta)^{2}\left(\sum_{k=1}^{\infty}\frac{1}{k^{4}}\right)^{1/2}\left(\sum_{k=1}^{\infty}e^{-4k(\rho_{0}+\rho)}\right)^{1/2}$
(3.5) $\displaystyle\leq C\tau(\rho_{0},\theta)^{2}\varepsilon^{2}.$

Combining equations (3.4) and (3), along with the inequality

$S_{0}^{2}+\sum_{k=1}^{\infty}\left(S_{k}^{2}+P_{k}^{2}\right)\leq
C\int_{0}^{2\pi}\lvert L(\theta)\rvert^{2}d\theta\leq
C^{\prime}\max_{\theta\in[0,2\pi]}\lvert L(\theta)\rvert^{2},$

we obtain the estimate

$\displaystyle\lvert v_{\varepsilon}(x)\rvert$ $\displaystyle\leq$
$\displaystyle C\varepsilon\max_{\theta\in[0,2\pi]}\lvert
L(\theta)\rvert(1+\rho+|\log\varepsilon|).$

Hence, the proof is complete. ∎

######  Lemma 3.2.

Let $N\in C^{\infty}(\partial E_{\varepsilon})$ and $g\in
H^{2}(\Omega_{E_{\varepsilon}})$ satisfy the following conditions:

(3.6) $\begin{cases}\Delta g(x)=0&x\in\Omega_{E_{\varepsilon}},\cr
g(x)=0&x\in\partial\Omega,\cr\frac{\partial
g}{\partial\nu}(x)=N(x)&x\in\partial E_{\varepsilon}.\end{cases}$

Then, the following inequality holds:

$\int_{\Omega_{E_{\varepsilon}}}\lvert\nabla g(x)\rvert^{2}dx\leq
2\pi\,\varepsilon^{2}M_{\textup{max}}\left(\max_{\partial
E_{\varepsilon}}\lvert N(x)\rvert\right)^{2}$

where $M_{\max}=\sup_{x\in\partial E}|x|$.

###### Proof.

Since $g\in H^{2}(\Omega_{E_{\varepsilon}})$, applying Green’s formula on
$\Omega_{E_{\varepsilon}}$, we get

$\int_{\Omega_{E_{\varepsilon}}}(g\Delta g+\lvert\nabla
g\rvert^{2})dx=\int_{\partial\Omega_{E_{\varepsilon}}}g\frac{\partial
g}{\partial\nu}d\sigma_{x}.$

By (3.6), it follows that

(3.7) $\displaystyle\int_{\Omega_{E_{\varepsilon}}}\lvert\nabla
g\rvert^{2}dx=\int_{\partial E_{\varepsilon}}g\frac{\partial
g}{\partial\nu}d\sigma_{x}.$

Using Schwarz’s inequality, we have

(3.8) $\displaystyle\int_{\partial E_{\varepsilon}}g(x)\frac{\partial
g}{\partial\nu}(x)d\sigma_{x}$ $\displaystyle\leq$
$\displaystyle\left(\int_{\partial
E_{\varepsilon}}g(x)^{2}d\sigma_{x}\right)^{1/2}\left(\int_{\partial
E_{\varepsilon}}\left\lvert\frac{\partial
g}{\partial\nu}(x)\right\rvert^{2}d\sigma_{x}\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle(2\pi\,\varepsilon
M_{\max})\varepsilon|\log\varepsilon|\max_{\partial E_{\varepsilon}}\lvert
N(x)\rvert\,\max_{\partial E_{\varepsilon}}\lvert N(x)\rvert$
$\displaystyle\leq$ $\displaystyle
2\pi\,M_{\text{max}}\varepsilon^{2}|\log\varepsilon|\left(\max_{\partial
E_{\varepsilon}}\lvert N(x)\rvert\right)^{2}.$

The last inequality follows from the maximum principle, $g$ is harmonic on
$\mathbb{R}^{2}\setminus{\bar{E}_{\varepsilon}}$, which satisfies the
assumption of Lemma 3.1. Then, we have $\lvert g(x)\rvert\leq
C\varepsilon|\log\varepsilon|\max_{\partial E_{\varepsilon}}\lvert N(x)\rvert$
for all $x\in\Omega_{E_{\varepsilon}}$.

From (3.7) and (3.8), we deduce

$\int_{\Omega_{E_{\varepsilon}}}\lvert\nabla g(x)\rvert^{2}dx\leq
2\pi\,\varepsilon^{2}|\log\varepsilon|M_{\text{max}}\left(\max_{\partial
E_{\varepsilon}}\lvert N(x)\rvert\right)^{2}.$

∎

######  Lemma 3.3.

Fix $M\in C^{\infty}(\bar{\Omega}_{E_{\varepsilon}})$. Let $u$ be the solution
of

(3.9) $\begin{cases}\Delta
u_{\varepsilon}(x)=0&x\in\Omega_{E_{\varepsilon}},\\\
u_{\varepsilon}(x)=0&x\in\partial\Omega,\\\
\frac{\partial}{\partial\nu}u_{\varepsilon}(x)=M(\theta)&x\in\partial
E_{\varepsilon}.\end{cases}$

Then, $u_{\varepsilon}(x)$ satisfies

$\lvert u_{\varepsilon}(x)\rvert\leq
C\varepsilon|\log\varepsilon|\max_{\theta\in[0,2\pi]}\lvert M(\theta)\rvert$

and, consequently,

$\|u_{\varepsilon}\|_{L^{2}(\Omega_{E_{\varepsilon}})}\leq
C\varepsilon|\log\varepsilon|\max_{\theta\in[0,2\pi]}\lvert M(\theta)\rvert.$

###### Proof.

Let $u_{\varepsilon}(x)$ be a solution to the problem (3.9). Initially, we
choose $L(\theta)=M(\theta)$ and obtain $v_{\varepsilon}^{(0)}(x)$ as a
solution that satisfies equations (3.1), (3.2), and (3.3). However,
$v_{\varepsilon}^{(0)}(x)$ may not satisfy $v_{\varepsilon}^{(0)}(x)=0$ for
$x\in\partial\Omega$. To address this, let $v_{\varepsilon}^{(1)}(x)$ be the
harmonic function in $\Omega$ that satisfies
$v_{\varepsilon}^{(1)}(x)=v_{\varepsilon}^{(0)}(x)$ for $x\in\partial\Omega$.
We define

$M_{\varepsilon}=\max_{\theta\in[0,2\pi]}\lvert M(\theta)\rvert.$

From (3.3) we observe that

$\max\\{\lvert
v_{\varepsilon}^{(1)}(x)\rvert,x\in\bar{\Omega}\\}\leq\hat{C}\varepsilon|\log\varepsilon|M_{\varepsilon}$

and
$\max\\{\lvert\frac{\partial}{\partial\nu}v_{\varepsilon}^{(1)}(x)\rvert,x\in\partial
E_{\varepsilon}\\}\leq\hat{C}\varepsilon|\log\varepsilon|M_{\varepsilon}$,
where $\hat{C}$ is a constant independent of $\varepsilon$. Next, we set
$L(\theta)=\frac{\partial}{\partial\nu}v_{\varepsilon}^{(1)}(x)$ for
$x\in\partial E_{\varepsilon}$ and consider $v_{\varepsilon}^{(2)}(x)$ as a
solution satisfying equations (3.1), (3.2) and (3.3). Let
$v_{\varepsilon}^{(3)}(x)$ be the harmonic function in $\Omega$ satisfying
$v_{\varepsilon}^{(3)}(x)=v_{\varepsilon}^{(2)}(x)$ for $x\in\partial\Omega$.
Then, $\max\\{\lvert
v_{\varepsilon}^{(3)}(x)\rvert,x\in\bar{\Omega}\\}\leq(\hat{C}\varepsilon|\log\varepsilon|)^{2}M_{\varepsilon}$
and
$\max\\{\lvert\frac{\partial}{\partial\nu}v_{\varepsilon}^{(3)}(x)\rvert,x\in\partial
E_{\varepsilon}\\}\leq(\hat{C}\varepsilon|\log\varepsilon|)^{2}M_{\varepsilon}$.

By repeating this procedure, we obtain the following equations for each
$n=0,1,2,\ldots$

For $v_{\varepsilon}^{(2n+1)}(x)$:

$\displaystyle\Delta v_{\varepsilon}^{(2n+1)}(x)$ $\displaystyle=$
$\displaystyle 0\quad\text{if }x\in\Omega,$ $\displaystyle
v_{\varepsilon}^{(2n+1)}(x)$ $\displaystyle=$ $\displaystyle
v_{\varepsilon}^{(2n)}(x)\quad\text{if }x\in\partial\Omega.$

For $v_{\varepsilon}^{(2n+2)}(x)$:

$\displaystyle\Delta v_{\varepsilon}^{(2n+2)}(x)$ $\displaystyle=$
$\displaystyle 0\quad\text{if
}x\in\mathbb{R}^{2}\backslash\bar{E}_{\varepsilon},$
$\displaystyle\frac{\partial}{\partial\nu}v_{\varepsilon}^{(2n+2)}(x)$
$\displaystyle=$
$\displaystyle\frac{\partial}{\partial\nu}v_{\varepsilon}^{(2n+1)}(x)\quad\text{if
}x\in\partial E_{\varepsilon}.$

By induction, we have the following inequalities:

(3.10) $\displaystyle\max_{\bar{\Omega}}\lvert
v_{\varepsilon}^{(2n+1)}(x)\rvert$ $\displaystyle\leq$
$\displaystyle(\hat{C}\varepsilon|\log\varepsilon|)^{n+1}M_{\varepsilon},$
(3.11) $\displaystyle\max_{\partial
E_{\varepsilon}}\left\lvert\frac{\partial}{\partial\nu}v_{\varepsilon}^{(2n+1)}(x)\right\rvert$
$\displaystyle\leq$
$\displaystyle(\hat{C}\varepsilon|\log\varepsilon|)^{n+1}M_{\varepsilon},$
(3.12) $\displaystyle\lvert v_{\varepsilon}^{(2n)}(x)\rvert$
$\displaystyle\leq$
$\displaystyle(\hat{C}\varepsilon|\log\varepsilon|)^{n+1}M_{\varepsilon}$

hold for $n\geq 0$. We can choose $\varepsilon$ such that
$\hat{C}\varepsilon|\log\varepsilon|\leq 1/2$. Let us define

(3.13) $\displaystyle
w_{\varepsilon}(x)=\sum_{n=0}^{\infty}(-1)^{n}v_{\varepsilon}^{(n)}(x).$

From (3.10) and (3.12), we observe that the right-hand side of (3.13)
uniformly converges on $\bar{\Omega}\backslash E_{\eta}$ for any
$\eta>\varepsilon$. Since $v_{\varepsilon}^{(n)}$ is harmonic in
$\Omega_{E_{\varepsilon}}$, it follows that $w_{\varepsilon}(x)$ is harmonic
in $\Omega_{E_{\varepsilon}}$, $w_{\varepsilon}(x)=0$ for
$x\in\partial\Omega$, and

$\displaystyle\frac{\partial w_{\varepsilon}}{\partial
x_{j}}(x)=\sum_{n=0}^{\infty}(-1)^{n}\frac{\partial
v_{\varepsilon}^{(n)}}{\partial x_{j}}(x)\quad\text{for
}x\in\Omega_{E_{\varepsilon}},\quad j=1,2.$

Next, we define

$\displaystyle
g_{\varepsilon}^{(n)}(x)=u_{\varepsilon}(x)-\sum_{i=0}^{2n+1}(-1)^{i}v_{\varepsilon}^{(i)}(x).$

Then,

(3.14) $\displaystyle\nabla
g_{\varepsilon}^{(n)}(x)\to\nabla(u_{\varepsilon}-w_{\varepsilon})(x)\qquad(n\to\infty)\text{
a.e. }x\in\Omega_{E_{\varepsilon}}.$

We have the properties that $g_{\varepsilon}^{(n)}(x)$ is harmonic in
$\Omega_{E_{\varepsilon}}$, $g_{\varepsilon}^{(n)}(x)=0$ for
$x\in\partial\Omega$, and

$\displaystyle\frac{\partial
g_{\varepsilon}^{(n)}}{\partial\nu}(x)=\frac{\partial
v_{\varepsilon}^{(2n+1)}}{\partial\nu}(x)\quad\text{for }x\in\partial
E_{\varepsilon}.$

Therefore, applying Lemma 3.2 and (3.11), we obtain

$\displaystyle\int_{\Omega_{E_{\varepsilon}}}\lvert\nabla
g_{\varepsilon}^{(n)}\rvert^{2}dx$ $\displaystyle\leq$ $\displaystyle 2\pi
M_{\text{max}}\varepsilon^{2}|\log\varepsilon|\left(\max_{\partial
E_{\varepsilon}}\left\lvert\frac{\partial
v_{\varepsilon}^{(2n+1)}}{\partial\nu}(x)\right\rvert\right)^{2}$
$\displaystyle\leq$ $\displaystyle 2\pi
M_{\text{max}}\varepsilon^{2}|\log\varepsilon|(\hat{C}\varepsilon|\log\varepsilon|)^{2n+2}M_{\varepsilon}^{2}.$

By using Fatou’s Lemma and (3.14), we deduce that

$\int_{\Omega_{E_{\varepsilon}}}\lvert\nabla(u_{\varepsilon}-w_{\varepsilon})\rvert^{2}dx\leq\liminf_{n\to\infty}\int_{\Omega_{E_{\varepsilon}}}\lvert\nabla
g^{(n)}\rvert^{2}dx\leq 0.$

Thus, $u_{\varepsilon}-w_{\varepsilon}=$ constant a.e
$\Omega_{E_{\varepsilon}}$. Since $u_{\varepsilon}(x)=w_{\varepsilon}(x)=0$
for $x\in\partial\Omega$, it follows that $u_{\varepsilon}=w_{\varepsilon}$
a.e. $\Omega_{E_{\varepsilon}}$. Therefore,

(3.15) $\displaystyle
u_{\varepsilon}(x)=\sum_{n=0}^{\infty}(-1)^{n}v_{\varepsilon}^{(n)}\qquad
x\in\Omega_{\varepsilon}.$

From (3.10) and (3.12), we have

$\displaystyle\lvert u_{\varepsilon}(x)\rvert$ $\displaystyle\leq$
$\displaystyle\sum_{n=0}^{\infty}(\lvert
v_{\varepsilon}^{(2n)}(x)\rvert+\lvert v_{\varepsilon}^{(2n+1)}(x)\rvert)$
$\displaystyle\leq$ $\displaystyle
C\varepsilon|\log\varepsilon|M_{\varepsilon}.$

Consequently, we can conclude that the function $u_{\varepsilon}$ satisfies
the following inequality:

$\|u_{\varepsilon}\|_{L^{2}(\Omega_{E_{\varepsilon}})}\leq
C\varepsilon|\log\varepsilon|\max_{\theta\in[0,2\pi]}\lvert M(\theta)\rvert,$

where $C$ is constant independent of $\varepsilon$. ∎

##  4 Some technical results

###  4.1 Green operators in perforated domains and their approximation

Consider the Green function $K(x,y)$ of the Laplacian in $\Omega$ under the
Dirichlet condition on $\partial\Omega$. Similarly, let $K_{\varepsilon}(x,y)$
be the Green function of the Laplacian in $\Omega_{E_{\varepsilon}}$
satisfying

$\displaystyle\begin{cases}-\Delta_{x}K_{\varepsilon}(x,y)=\delta(x-y),&x,y\in\Omega_{E_{\varepsilon}}\\\
K_{\varepsilon}(x,y)|_{x\in\partial\Omega}=0,&y\in\Omega_{E_{\varepsilon}}\\\
\frac{\partial}{\partial\nu}K_{\varepsilon}(x,y)|_{x\in\partial
E_{\varepsilon}}=0,&y\in\Omega_{E_{\varepsilon}}.\end{cases}$

We define the bounded linear operators $\mathbf{K}$ and
$\mathbf{K}_{\varepsilon}$ on $L^{2}(\Omega)$ and
$L^{2}(\Omega_{E_{\varepsilon}})$, respectively, as follows:

$\displaystyle(\mathbf{K}f)(x)$ $\displaystyle=$
$\displaystyle\int_{\Omega}K(x,y)f(y)dy,$
$\displaystyle(\mathbf{K_{\varepsilon}}f)(x)$ $\displaystyle=$
$\displaystyle\int_{\Omega_{E_{\varepsilon}}}K_{\varepsilon}(x,y)f(y)dy,$

By utilizing these operators, Problems (2.2) and (2.4) can be reformulated as

$\displaystyle(\mathbf{K}_{\varepsilon}u)(x)$ $\displaystyle=$
$\displaystyle\lambda(\varepsilon)^{-1}u(x),$ $\displaystyle(\mathbf{K}v)(x)$
$\displaystyle=$ $\displaystyle\lambda^{-1}v(x).$

In order to establish a connection between $\mathbf{K_{\varepsilon}}$ and
$\mathbf{K}$, we introduce the operators $\mathbf{H_{\varepsilon}}$ and
$\mathbf{\overline{H}_{\varepsilon}}$. First, we define the inner product as
follows:

$\left\langle\nabla_{w}a(x,w),\nabla_{w}b(w,y)\right\rangle=\sum_{i=1}^{2}\frac{\partial}{\partial
w_{i}}a(x,w)\frac{\partial}{\partial w_{i}}b(w,y)$

for any $a,b\in C^{1}(\Omega\times\Omega\backslash(\Omega\times\Omega)_{d})$,
where $(\Omega\times\Omega)_{d}$ represents the diagonal set of
$\Omega\times\Omega$. The inner product
$\left\langle\nabla_{w},\nabla_{w}\right\rangle$ remains invariant under any
orthogonal transformation of orthonormal coordinates $(w_{1},w_{2})$.

Similarly, by considering $a(x,w)$ and $b(x,w)$ belonging to
$C^{3}(\Omega\backslash\\{w\\})$ and satisfying
$\Delta_{w}a(x,w)=\Delta_{w}b(x,w)=0$ for $x,y\in\Omega\backslash\\{w\\}$, we
define

$\left\langle
H_{w}a(x,w),H_{w}b(w,y)\right\rangle=\sum_{i,j=1}^{2}\frac{\partial^{2}}{\partial
w_{i}\partial w_{j}}a(x,w)\frac{\partial^{2}}{\partial w_{i}\partial
w_{j}}b(w,y).$

It is noteworthy that $\left\langle
H_{w}a(x,w),H_{w}b(w,y)\right\rangle|_{x,y\in\Omega\backslash\\{w\\}}$ remains
invariant under any orthonormal transformation of the basis.

With $M$ defined by (2.1), we introduce the functions $h_{\varepsilon}(x,y)$
and $\bar{h}_{\varepsilon}(x,y)$ as follows:

$\displaystyle h_{\varepsilon}(x,y)$ $\displaystyle=$ $\displaystyle
K(x,y)+h(\varepsilon)\left\langle\nabla_{w}K(x,\tilde{w}),\nabla_{w}K(\tilde{w},y)\right\rangle$
$\displaystyle+i(\varepsilon)\left\langle
H_{w}K(x,\tilde{w}),H_{w}K(\tilde{w},y)\right\rangle,$

and

$\displaystyle\bar{h}_{\varepsilon}(x,y)$ $\displaystyle=$ $\displaystyle
K(x,y)+g(\varepsilon)K(x,w)K(w,y)$
$\displaystyle+h(\varepsilon)\left\langle\nabla_{w}K(x,\tilde{w}),\nabla_{w}K(\tilde{w},y)\right\rangle\xi_{\varepsilon}(x)\xi_{\varepsilon}(y)$
$\displaystyle+i(\varepsilon)\left\langle
H_{w}K(x,\tilde{w}),H_{w}K(\tilde{w},y)\right\rangle\xi_{\varepsilon}(x)\xi_{\varepsilon}(y),$

where $g(\varepsilon)=-\pi\mu_{i}(M\varepsilon)^{2}$,
$h(\varepsilon)=2\pi(M\varepsilon)^{2}$,
$i(\varepsilon)=\frac{\pi}{2}(M\varepsilon)^{4}$, and $\xi_{\varepsilon}(x)\in
C^{\infty}(\mathbb{R}^{2})$ satisfies $0\leq\xi_{\varepsilon}(x)\leq 1$,
$\xi_{\varepsilon}(x)=1$ for $x\in\mathbb{R}^{2}\setminus E_{\varepsilon}$ and
$\xi_{\varepsilon}(x)=0$ for $x\in E_{\varepsilon/2}$.

The operators $\mathbf{H}_{\varepsilon}$ and
$\mathbf{\overline{H}}_{\varepsilon}$ are defined as follows

$\displaystyle\begin{array}[]{llll}(\mathbf{H}_{\varepsilon}g)(x)&=&\displaystyle\int_{\Omega_{E_{\varepsilon}}}h_{\varepsilon}(x,y)g(y)dy,&x\in\Omega_{E_{\varepsilon}},\vspace*{0.15cm}\\\
(\mathbf{\overline{H}}_{\varepsilon}f)(x)&=&\displaystyle\int_{\Omega}\bar{h}_{\varepsilon}(x,y)f(y)dy,&x\in\Omega.\end{array}$

The operator $\mathbf{H}_{\varepsilon}$ serves as a very good approximation to
$\mathbf{K}_{\varepsilon}$. Comparing $\mathbf{H}_{\varepsilon}$ with
$\mathbf{\overline{H}}_{\varepsilon}$ is straightforward based on their
respective definitions. Due to the fact that
$\mathbf{\overline{H}}_{\varepsilon}$ acts on $L^{2}(\Omega)$, we can directly
compare it with $\mathbf{K}$. Consequently, we can compare
$\mathbf{K}_{\varepsilon}$ and $\mathbf{K}$.

To justify why $h_{\varepsilon}(x,y)$ provides a good approximation of
$K_{\varepsilon}(x,y)$, let us consider the difference function
$q_{\varepsilon}(x,y)=h_{\varepsilon}(x,y)-K_{\varepsilon}(x,y)$. It satisfies
the following conditions:

$\displaystyle\Delta_{x}q_{\varepsilon}(x,y)$ $\displaystyle=$ $\displaystyle
0,\qquad x,y\in\Omega_{E_{\varepsilon}},$ $\displaystyle q_{\varepsilon}(x,y)$
$\displaystyle=$ $\displaystyle 0,\qquad
x\in\partial\Omega,y\in\Omega_{E_{\varepsilon}}.$

We introduce the function

$S(x,y)=K(x,y)+(2\pi)^{-1}\log\lvert\nu-y\rvert,$

which belongs to $C^{\infty}(\Omega\times\Omega)$. Take an arbitrary
$x\in\partial E_{\varepsilon}$, and $\nu$ is the associated normal vector of
$x$. Calculating the normal derivative directly, we have

(4.1) $\displaystyle\frac{\partial}{\partial\nu}q_{\varepsilon}(x,y)$
$\displaystyle=\frac{\partial}{\partial\nu}\Bigg{(}K(x,y)-K_{\varepsilon}(x,y)+h(\varepsilon)\langle\nabla_{w}K(x,\tilde{w}),\nabla_{w}K(\tilde{w},y)\rangle+i(\varepsilon)\langle
H_{w}K(x,\tilde{w}),H_{w}K(\tilde{w},y)\rangle\Bigg{)}$
$\displaystyle=\frac{\partial}{\partial\nu}K(x,y)$
$\displaystyle\quad-h(\varepsilon)(2\pi)^{-1}\left\langle\frac{\partial}{\partial\nu}\nabla_{w}\log|\nu-\tilde{w}|,\nabla_{w}K(\tilde{w},y)\right\rangle$
$\displaystyle\quad+h(\varepsilon)\frac{\partial}{\partial\nu}\langle\nabla_{w}S(x,\tilde{w}),\nabla_{w}K(\tilde{w},y)\rangle$
$\displaystyle\quad-i(\varepsilon)(2\pi)^{-1}\left\langle\frac{\partial}{\partial\nu}H_{w}\log|\nu-\tilde{w}|,H_{w}K(\tilde{w},y)\right\rangle$
$\displaystyle\quad+i(\varepsilon)\frac{\partial}{\partial\nu}\langle
H_{w}S(x,\tilde{w}),H_{w}K(\tilde{w},y)\rangle.$

Focusing on each term separately, we start with the derivatives of the
logarithmic terms and their corresponding expressions

(4.2)
$\displaystyle\left\langle\frac{\partial}{\partial\nu}\nabla_{w}\log|\nu-\tilde{w}|,\nabla_{w}K(\tilde{w},y)\right\rangle$
$\displaystyle=\left\langle\frac{\partial}{\partial\nu}\left(\frac{-\nu_{1}}{|\nu|^{2}},\frac{-\nu_{2}}{|\nu|^{2}}\right),\nabla_{w}K(\tilde{w},y)\right\rangle$
$\displaystyle=-\left\langle\left(\nu\left(\frac{\nu_{2}^{2}-\nu_{1}^{2}}{|\nu|^{4}},\frac{-2\nu_{1}\nu_{2}}{|\nu|^{4}}\right),\nu\left(\frac{-2\nu_{1}\nu_{2}}{|\nu|^{4}},\frac{\nu_{2}^{2}-\nu_{1}^{2}}{|\nu|^{4}}\right)\right),\nabla_{w}K(\tilde{w},y)\right\rangle$
$\displaystyle=|\nu|^{-2}\left\langle\nu,\nabla_{w}K(\tilde{w},y)\right\rangle.$
(4.3)
$\displaystyle\left\langle\frac{\partial}{\partial\nu}H_{w}\log|\nu-\tilde{w}|,H_{w}K(\tilde{w},y)\right\rangle$
$\displaystyle=\left\langle\frac{\partial}{\partial\nu}\begin{pmatrix}\frac{\nu_{2}^{2}-\nu_{1}^{2}}{|\nu|^{4}}&\frac{-2\nu_{1}\nu_{2}}{|\nu|^{4}}\\\
\frac{-2\nu_{1}\nu_{2}}{|\nu|^{4}}&\frac{\nu_{1}^{2}-\nu_{2}^{2}}{|\nu|^{4}}\end{pmatrix},H_{w}K(\tilde{w},y)\right\rangle$
$\displaystyle=\left\langle\begin{pmatrix}\nu\left(\frac{-2\nu_{1}(3\nu_{2}^{2}-\nu_{1}^{2})}{|\nu|^{6}},\frac{2\nu_{2}(3\nu_{1}^{2}-\nu_{2}^{2})}{|\nu|^{6}}\right)&\nu\left(\frac{-2\nu_{2}(|\nu|^{2}-4\nu_{1}^{2})}{|\nu|^{6}},\frac{-2\nu_{1}(|\nu|^{2}-4\nu_{2}^{2})}{|\nu|^{6}}\right)\\\
\nu\left(\frac{-2\nu_{2}(|\nu|^{2}-4\nu_{1}^{2})}{|\nu|^{6}},\frac{-2\nu_{1}(|\nu|^{2}-4\nu_{2}^{2})}{|\nu|^{6}}\right)&\nu\left(\frac{2\nu_{1}(3\nu_{2}^{2}-\nu_{1}^{2})}{|\nu|^{6}},\frac{-2\nu_{2}(3\nu_{1}^{2}-\nu_{2}^{2})}{|\nu|^{6}}\right)\end{pmatrix},H_{w}K(\tilde{w},y)\right\rangle$
$\displaystyle=2|\nu|^{-4}\big{(}(\nu_{1}^{2}-\nu_{2}^{2})\frac{\partial^{2}}{\partial
w_{1}^{2}}K(\tilde{w},y)+(\nu_{2}^{2}-\nu_{1}^{2})\frac{\partial^{2}}{\partial
w_{2}^{2}}K(\tilde{w},y)+4\nu_{1}\nu_{2}\frac{\partial^{2}}{\partial
w_{1}\partial w_{2}}K(\tilde{w},y)\big{)}$
$\displaystyle=2|\nu|^{-4}\bigg{(}2\nu_{1}\left(\nu_{1}\frac{\partial^{2}}{\partial
w_{1}^{2}}K(\tilde{w},y)+\nu_{2}\frac{\partial^{2}}{\partial w_{2}\partial
w_{1}}K(\tilde{w},y)\right)$
$\displaystyle\quad+2\nu_{2}\left(\nu_{2}\frac{\partial^{2}}{\partial
w_{2}^{2}}K(\tilde{w},y)+\nu_{1}\frac{\partial^{2}}{\partial w_{1}\partial
w_{2}}K(\tilde{w},y)\right)-|\nu|^{2}\Delta_{w}K(\tilde{w},y)\bigg{)}$
$\displaystyle=4|\nu|^{-4}\nu\left(\nu_{1}\frac{\partial^{2}}{\partial
w_{1}^{2}}K(\tilde{w},y)+\nu_{2}\frac{\partial^{2}}{\partial w_{2}\partial
w_{1}}K(\tilde{w},y),\nu_{2}\frac{\partial^{2}}{\partial
w_{2}^{2}}K(\tilde{w},y)+\nu_{1}\frac{\partial^{2}}{\partial w_{1}\partial
w_{2}}K(\tilde{w},y)\right).$

By substituting (4.2) and (4.3) into (4.1) and noting that $\nu=x$, we obtain

(4.4) $\displaystyle\frac{\partial}{\partial\nu}q_{\varepsilon}(x,y)$
$\displaystyle=\nu\bigg{(}\nabla_{x}K(x,y)-h(\varepsilon)(2\pi)^{-1}|x|^{-2}\nabla_{w}K(\tilde{w},y)$
$\displaystyle\quad-i(\varepsilon)\frac{2}{\pi}|x|^{-4}\left(x_{1}\frac{\partial^{2}}{\partial
w_{1}^{2}}K(\tilde{w},y)+x_{2}\frac{\partial^{2}}{\partial w_{2}\partial
w_{1}}K(\tilde{w},y),\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\left.x_{2}\frac{\partial^{2}}{\partial
w_{2}^{2}}K(\tilde{w},y)+x_{1}\frac{\partial^{2}}{\partial w_{1}\partial
w_{2}}K(\tilde{w},y)\right)$
$\displaystyle\quad+h(\varepsilon)\nabla_{x}\left\langle\nabla_{w}S(x,\tilde{w}),\nabla_{w}K(\tilde{w},y)\right\rangle$
$\displaystyle\quad+i(\varepsilon)\nabla_{x}\left\langle
H_{w}S(x,\tilde{w}),H_{w}K(\tilde{w},y)\right\rangle\bigg{)}.$

Without loss of generality, we can assume that $x=x^{*}$. Taking into account
that $h(\varepsilon)=2\pi|x^{*}|^{2}$ and
$i(\varepsilon)=\frac{\pi}{2}|x^{*}|^{4}$, we find that (4.4) is equivalent to

(4.5)
$\displaystyle\left.\frac{\partial}{\partial\nu}q_{\varepsilon}(x,y)\right|_{x=x^{*}}$
$\displaystyle=\nu\bigg{(}\nabla_{x}K(x^{*},y)-\nabla_{w}K(\tilde{w},y)$
$\displaystyle\quad-\left(x^{*}_{1}\frac{\partial^{2}}{\partial
w_{1}^{2}}K(\tilde{w},y)+x^{*}_{2}\frac{\partial^{2}}{\partial w_{2}\partial
w_{1}}K(\tilde{w},y),\right.$
$\displaystyle\quad\quad\left.x^{*}_{2}\frac{\partial^{2}}{\partial
w_{2}^{2}}K(\tilde{w},y)+x^{*}_{1}\frac{\partial^{2}}{\partial w_{1}\partial
w_{2}}K(\tilde{w},y)\right)$
$\displaystyle\quad+\mathcal{O}(\varepsilon^{2})\nabla_{x}\left\langle\nabla_{w}S(x^{*},\tilde{w}),\nabla_{w}K(\tilde{w},y)\right\rangle$
$\displaystyle\quad+\mathcal{O}(\varepsilon^{4})\nabla_{x}\left\langle
H_{w}S(x^{*},\tilde{w}),H_{w}K(\tilde{w},y)\right\rangle\bigg{)}.$

We present two important theorems supporting the proof of Theorem 2.1. The
proofs are presented in the following subsection.

######  Theorem 4.1.

There exists a constant $C$ independent of $\varepsilon$ such that

$\|(\mathbf{K}_{\varepsilon}-\mathbf{H}_{\varepsilon})(\chi_{\varepsilon}\varphi_{i})\|_{L^{2}(\Omega_{E_{\varepsilon}})}\leq
C\varepsilon^{3}|\log\varepsilon|^{2}.$

Here $\chi_{\varepsilon}$ is the characteristic function of
$\bar{\Omega}_{E_{\varepsilon}}$.

######  Theorem 4.2.

There exists a constant $C$ independent of $\varepsilon$ such that

$\displaystyle\|(\chi_{\varepsilon}\mathbf{\overline{H}}_{\varepsilon}-\mathbf{H}_{\varepsilon}\chi_{\varepsilon})\varphi_{i}\|_{L^{2}(\Omega_{E_{\varepsilon}})}\leq
C\varepsilon^{3}|\log\varepsilon|^{2}.$

###  4.2 Proof of Theorem 4.1

We frequently use the following properties of the Green function:

(4.6) $\displaystyle\left\lvert K(x,y)\right\rvert\leq C\lvert\log\lvert
x-y\rvert\rvert,$ (4.7)
$\displaystyle\left\lvert\nabla_{x}K(x,y)\right\rvert\leq C\lvert
x-y\rvert^{-1}.$

These properties imply the following estimates:

(4.8) $\displaystyle\left\lvert(\mathbf{K}f)(x)\right\rvert\leq
C\|f\|_{L^{p}(\Omega)}\quad(p>1),$ (4.9)
$\displaystyle\left\lvert\nabla_{x}(\mathbf{K}f)(x)\right\rvert\leq
C\|f\|_{L^{p}(\Omega)}\quad(p>2).$

Now we state and prove the following lemma:

######  Lemma 4.3.

Let $p\in(2,\infty)$. Then, there exists a constant $C>0$ independent of
$\varepsilon$ such that

$\displaystyle\|\mathbf{H_{\varepsilon}}-\mathbf{K_{\varepsilon}}\|_{L^{p}(\Omega_{E_{\varepsilon}})}\leq
C\varepsilon^{2-(2/p)}\log\varepsilon.$

###### Proof.

We consider an arbitrary function $\tilde{f}\in L^{p}(\Omega)$ which is zero
on $E_{\varepsilon}$. Then, we put
$k_{\varepsilon}=(\mathbf{H_{\varepsilon}}-\mathbf{K_{\varepsilon}})\tilde{f}$
and note that $k_{\varepsilon}$ satisfies $\Delta k_{\varepsilon}(x)=0$ for
$x\in\Omega_{E_{\varepsilon}}$ and $k_{\varepsilon}(x)=0$ for
$x\in\partial\Omega$.
Taking $x=x^{*}$ and using (4.5), we obtain the expression

(4.10)
$\displaystyle\left.\frac{\partial}{\partial\nu}k_{\varepsilon}(x)\right|_{x=x^{*}}$
$\displaystyle=\nu\bigg{(}\nabla(\mathbf{K}\tilde{f})(x^{*})-\nabla(\mathbf{K}\tilde{f})(\tilde{w})$
$\displaystyle\quad-\left(x^{*}_{1}\frac{\partial^{2}}{\partial
w_{1}^{2}}(\mathbf{K}\tilde{f})(\tilde{w}),x^{*}_{2}\frac{\partial^{2}}{\partial
w_{2}^{2}}(\mathbf{K}\tilde{f})(\tilde{w})\right)$
$\displaystyle\quad-\left(x^{*}_{2}\frac{\partial^{2}}{\partial w_{2}\partial
w_{1}}(\mathbf{K}\tilde{f})(\tilde{w}),x^{*}_{1}\frac{\partial^{2}}{\partial
w_{1}\partial w_{2}}(\mathbf{K}\tilde{f})(\tilde{w})\right)$
$\displaystyle\quad+\mathcal{O}(\varepsilon^{2})\nabla_{x}\left\langle\nabla_{w}S(x^{*},\tilde{w}),\nabla(\mathbf{K}\tilde{f})(\tilde{w})\right\rangle$
$\displaystyle\quad+\mathcal{O}(\varepsilon^{4})\nabla_{x}\left\langle
H_{w}S(x^{*},\tilde{w}),H_{w}(\mathbf{K}\tilde{f})(\tilde{w})\right\rangle\bigg{)}$

By the Sobolev embedding theorem, we have

(4.11) $\displaystyle\|\mathbf{K}\tilde{f}\|_{C^{1+s}(\Omega)}\leq
C\|\tilde{f}\|_{L^{p}(\Omega)}=C\|\tilde{f}\|_{L^{p}(\Omega_{E_{\varepsilon}})}$

for $s=1-2/p$, $2<p<\infty$. Using (4.11), we have

(4.12) $\displaystyle\left\lvert\frac{\partial}{\partial
x_{n}}(\mathbf{K}\tilde{f})(x^{*})-\frac{\partial}{\partial
w_{n}}(\mathbf{K}\tilde{f})(\tilde{w})\right\rvert\leq
C\varepsilon^{s}\|\mathbf{K}\tilde{f}\|_{C^{1+s}(\Omega)}\leq
C^{\prime}\varepsilon^{s}\|\tilde{f}\|_{L^{p}(\Omega_{E_{\varepsilon}})}$

for $n=1,2$, $p>2$.

We define that $R=\sup_{x\in\partial\Omega}|x|$, and
$M_{\min}=\inf_{x\in\partial E}|x|$, then $\Omega_{E_{\varepsilon}}$ is in the
annulus with centre $\tilde{w}$:

$A(\tilde{w};\varepsilon M_{\min},R)=\\{x\in\mathbb{R}^{2}|\varepsilon
M_{\min}<|x-\tilde{w}|<R\\}.$

We obtain the estimates (4.2) and (4.14) as follows:

$\displaystyle\left\lvert\frac{\partial}{\partial
w_{n}}(\mathbf{K}\tilde{f})(\tilde{w})\right\rvert$ $\displaystyle\leq$
$\displaystyle C\left(\int_{\Omega_{E_{\varepsilon}}}\lvert
y-\tilde{w}\rvert^{-p^{\prime}}dy\right)^{1/p^{\prime}}\|\tilde{f}\|_{L^{p}(\Omega_{E_{\varepsilon}})}$
$\displaystyle\leq$ $\displaystyle C\left(\int_{\varepsilon
M_{\text{min}}}^{R}\int_{0}^{2\pi}r^{-p^{\prime}}rdrd\theta\right)^{1/p^{\prime}}\|\tilde{f}\|_{L^{p}(\Omega_{E_{\varepsilon}})}$
$\displaystyle\leq$
$\displaystyle\begin{cases}C\lvert\log\varepsilon\rvert^{1/2}\|\tilde{f}\|_{L^{2}(\Omega_{E_{\varepsilon}})}&\text{if
}p=2\\\ C\|\tilde{f}\|_{L^{p}(\Omega_{E_{\varepsilon}})}&\text{if
}p>2\end{cases}$

for $n=1,2$, $p^{\prime}$ satisfies $(1/p)+(1/p^{\prime})=1$.

Additionally, we have

(4.14) $\displaystyle\left\lvert\frac{\partial^{2}}{\partial w_{m}\partial
w_{n}}(\mathbf{K}\tilde{f})(\tilde{w})\right\rvert$ $\displaystyle\leq$
$\displaystyle C\left(\int_{\Omega_{E_{\varepsilon}}}\lvert
y-\tilde{w}\rvert^{-2p^{\prime}}dy\right)^{1/p^{\prime}}\|\tilde{f}\|_{L^{p}(\Omega_{E_{\varepsilon}})}$
$\displaystyle\leq$ $\displaystyle C\left(\int_{\varepsilon
M_{\text{min}}}^{R}\int_{0}^{2\pi}r^{-2p^{\prime}}rdrd\theta\right)^{1/p^{\prime}}\|\tilde{f}\|_{L^{p}(\Omega_{E_{\varepsilon}})}$
$\displaystyle\leq$ $\displaystyle
C\varepsilon^{-2/p}\|\tilde{f}\|_{L^{p}(\Omega_{E_{\varepsilon}})}\qquad\text{if
}p>1$

for $m\geq 1$, $n\leq 2$.

On the other hand, we see that

(4.15) $\displaystyle\frac{\partial}{\partial
x_{n}}\left\langle\nabla_{w}S(x,\tilde{w}),\nabla(\mathbf{K}\tilde{f})(\tilde{w})\right\rangle$
$\displaystyle\leq$ $\displaystyle
C\left(\int_{\Omega_{E_{\varepsilon}}}\lvert
y-\tilde{w}\rvert^{-p^{\prime}}dy\right)^{1/p^{\prime}}\|\tilde{f}\|_{L^{p}(\Omega_{E_{\varepsilon}})}$
$\displaystyle\leq$ $\displaystyle
C\|\tilde{f}\|_{L^{p}(\Omega_{E_{\varepsilon}})},$ (4.16)
$\displaystyle\frac{\partial}{\partial x_{n}}\left\langle
H_{w}S(x,\tilde{w}),H_{w}(\mathbf{K}\tilde{f})(\tilde{w})\right\rangle$
$\displaystyle\leq$ $\displaystyle
C\left(\int_{\Omega_{E_{\varepsilon}}}\lvert
y-\tilde{w}\rvert^{-2p^{\prime}}dy\right)^{1/p^{\prime}}\|\tilde{f}\|_{L^{p}(\Omega_{E_{\varepsilon}})}$
$\displaystyle\leq$ $\displaystyle
C\varepsilon^{-2/p}\|\tilde{f}\|_{L^{p}(\Omega_{E_{\varepsilon}})}\qquad\text{
for }n=1,2.$

By combining the estimates (4.12), (4.14), (4.15), and (4.16), we obtain

$\displaystyle\max_{x\in\partial
E_{\varepsilon}}\left\lvert\frac{\partial}{\partial\nu}k_{\varepsilon}(x)\right\rvert$
$\displaystyle\leq$ $\displaystyle
C\left(\varepsilon^{s}+\varepsilon^{1-2/p}+\varepsilon^{2}+\varepsilon^{4-2/p}\right)\|\tilde{f}\|_{L^{p}(\Omega_{E_{\varepsilon}})}$
$\displaystyle\leq$ $\displaystyle
C\varepsilon^{1-2/p}\|f\|_{L^{p}(\Omega_{E_{\varepsilon}})}$

for $p>2$.

Using Lemma 3.3, we can conclude that

$\displaystyle\|\mathbf{H_{\varepsilon}}-\mathbf{K_{\varepsilon}}\|_{L^{p}(\Omega_{E_{\varepsilon}})}\leq
C\varepsilon^{2-2/p}|\log\varepsilon|,$

which completes the proof of Lemma 4.3. ∎

In the following proposition, we establish the convergence rate of the
difference between operators $\mathbf{H_{\varepsilon}}$ and
$\mathbf{K_{\varepsilon}}$ in the $L^{p}$ norm, providing an estimate for
their approximation error for $p\in(1,\infty]$.

######  Proposition 4.4.

Fix $p\in(1,\infty]$. Then, for any fixed $s\in(0,1)$, the operator norm

$\|\mathbf{H_{\varepsilon}}-\mathbf{K_{\varepsilon}}\|_{L^{p}(\Omega_{E_{\varepsilon}})}=\mathcal{O}(\varepsilon^{1+s})$

as $\varepsilon$ tends to zero.

###### Proof.

Assuming $p\in(1,\infty)$, let
$\mathbf{Q_{\varepsilon}}=\mathbf{H_{\varepsilon}}-\mathbf{K_{\varepsilon}}$.
Since $\mathbf{Q_{\varepsilon}}$ is self-adjoint on
$L^{2}(\Omega_{E_{\varepsilon}})$, it follows that
$\|\mathbf{Q_{\varepsilon}}\|_{L^{q}(\Omega_{E_{\varepsilon}})}=\|\mathbf{Q_{\varepsilon}}\|_{L^{q^{\prime}}(\Omega_{E_{\varepsilon}})}$,
where $\frac{1}{q}+\frac{1}{q^{\prime}}=1$.

By applying the Riesz-Thorin interpolation theorem, we can conclude that
$\|\mathbf{Q}_{\varepsilon}\|_{L^{p}(\Omega_{E_{\varepsilon}})}\leq\|\mathbf{Q}_{\varepsilon}\|_{L^{q}(\Omega_{E_{\varepsilon}})}$
for any $p\in(q^{\prime},q)$, where $q>2$. By choosing a sufficiently large
$q$ and utilizing Lemma 4.3, we obtain Proposition 4.4 for $p\neq 1,\infty$.

Now, consider the case when $p=\infty$. Using a similar argument as in the
proof of Lemma 4.3, we can establish Proposition 4.4 with $p=\infty$.

Therefore, by combining the results for $p\in(1,\infty)$ and $p=\infty$, we
conclude the validity of Proposition 4.4 for all $p\in(1,\infty]$. ∎

Next, we aim to estimate
$\|(\mathbf{K}_{\varepsilon}-\mathbf{H}_{\varepsilon})(\chi_{\varepsilon}\varphi_{i})\|_{L^{2}(\Omega_{E_{\varepsilon}})}$.
Let us denote
$v_{\varepsilon}=(\mathbf{K}_{\varepsilon}-\mathbf{H}_{\varepsilon})(\chi_{\varepsilon}\varphi_{i})$.
Put $\hat{\chi}_{\varepsilon}=1-\chi_{\varepsilon}$. By using equation (4.10),
we obtain the following expression

(4.17)
$\displaystyle\left.\frac{\partial}{\partial\nu}v_{\varepsilon}(x)\right\rvert_{x=x^{*}}=\nu\left(I_{1}(\varepsilon)-I_{2}(\varepsilon)+I_{3}(\varepsilon)\right),$

where

$\displaystyle I_{1}(\varepsilon)$ $\displaystyle=$
$\displaystyle\nabla(\mathbf{K}\varphi_{i})(x^{*})-\nabla(\mathbf{K}\varphi_{i})(\tilde{w})-\left(x^{*}_{1}\frac{\partial^{2}}{\partial
w_{1}^{2}}(\mathbf{K}\varphi_{i})(\tilde{w}),x^{*}_{2}\frac{\partial^{2}}{\partial
w_{2}^{2}}(\mathbf{K}\varphi_{i})(\tilde{w})\right)$
$\displaystyle-\left(x^{*}_{2}\frac{\partial^{2}}{\partial w_{2}\partial
w_{1}}(\mathbf{K}\varphi_{i})(\tilde{w}),x^{*}_{1}\frac{\partial^{2}}{\partial
w_{1}\partial w_{2}}(\mathbf{K}\varphi_{i})(\tilde{w})\right)$ $\displaystyle
I_{2}(\varepsilon)$ $\displaystyle=$
$\displaystyle\nabla(\mathbf{K}\hat{\chi}_{\varepsilon}\varphi_{i})(x^{*})-\nabla(\mathbf{K}\hat{\chi}_{\varepsilon}\varphi_{i})(\tilde{w})-\left(x^{*}_{1}\frac{\partial^{2}}{\partial
w_{1}^{2}}(\mathbf{K}\hat{\chi}_{\varepsilon}\varphi_{i})(\tilde{w}),x^{*}_{2}\frac{\partial^{2}}{\partial
w_{2}^{2}}(\mathbf{K}\hat{\chi}_{\varepsilon}\varphi_{i})(\tilde{w})\right)$
$\displaystyle-\left(x^{*}_{2}\frac{\partial^{2}}{\partial w_{2}\partial
w_{1}}(\mathbf{K}\hat{\chi}_{\varepsilon}\varphi_{i})(\tilde{w}),x^{*}_{1}\frac{\partial^{2}}{\partial
w_{1}\partial
w_{2}}(\mathbf{K}\hat{\chi}_{\varepsilon}\varphi_{i})(\tilde{w})\right)$
$\displaystyle I_{3}(\varepsilon)$ $\displaystyle=$
$\displaystyle\mathcal{O}(\varepsilon^{2})\nabla_{x}\left\langle\nabla_{w}S(x,\tilde{w}),\nabla_{w}(\mathbf{K}\chi_{\varepsilon}\varphi_{i})(\tilde{w})\right\rangle$
$\displaystyle+\mathcal{O}(\varepsilon^{4})\nabla_{x}\left\langle
H_{w}S(x^{*},\tilde{w}),H_{w}(\mathbf{K}\chi_{\varepsilon}\varphi_{i})(\tilde{w})\right\rangle.$

We can express $I_{1}(\varepsilon)$ as follows, taking into account that
$\mathbf{K}\varphi_{i}(x)=\mu_{i}^{-1}\varphi_{i}(x)$:

$\displaystyle I_{1}(\varepsilon)=$
$\displaystyle\mu_{i}^{-1}\bigg{(}\nabla\varphi_{i}(x^{*})-\nabla\varphi_{i}(\tilde{w})-\left(x^{*}_{1}\frac{\partial^{2}}{\partial
w_{1}^{2}}(\varphi_{i})(\tilde{w}),x^{*}_{2}\frac{\partial^{2}}{\partial
w_{2}^{2}}(\varphi_{i})(\tilde{w})\right)$
$\displaystyle-\left(x^{*}_{2}\frac{\partial^{2}}{\partial w_{2}\partial
w_{1}}(\varphi_{i})(\tilde{w}),x^{*}_{1}\frac{\partial^{2}}{\partial
w_{1}\partial w_{2}}(\varphi_{i})(\tilde{w})\right)\bigg{)}$

By using the Taylor expansion at $\tilde{w}=(0,0)$ for
$\frac{\partial}{\partial x_{n}}\varphi_{i}(x^{*})$, $n=1,2$ we obtain:

$\displaystyle\frac{\partial}{\partial
x_{1}}\varphi_{i}(x^{*})=\frac{\partial}{\partial
w_{1}}\varphi_{i}(\tilde{w})+x^{*}_{1}\frac{\partial^{2}}{{\partial
w_{1}}^{2}}\varphi_{i}(\tilde{w})+x^{*}_{2}\frac{\partial^{2}}{\partial
w_{2}\partial w_{1}}(\varphi_{i})(\tilde{w})+\mathcal{O}(\varepsilon^{2}),$
$\displaystyle\frac{\partial}{\partial
x_{2}}\varphi_{i}(x^{*})=\frac{\partial}{\partial
w_{2}}\varphi_{i}(\tilde{w})+x^{*}_{2}\frac{\partial^{2}}{{\partial
w_{2}}^{2}}\varphi_{i}(\tilde{w})+x^{*}_{1}\frac{\partial^{2}}{\partial
w_{1}\partial w_{2}}(\varphi_{i})(\tilde{w})+\mathcal{O}(\varepsilon^{2}),$

which implies that

(4.18) $\displaystyle\lvert I_{1}(\varepsilon)\rvert\leq C\varepsilon^{2}.$

Furthermore, we have the following estimate:

(4.19) $\displaystyle\lvert I_{3}(\varepsilon)\rvert$ $\displaystyle\leq$
$\displaystyle C(\varepsilon^{2}+\varepsilon^{4}\varepsilon^{-2/p})\leq
C\varepsilon^{2}.$

To estimate $I_{2}(\varepsilon)$, we introduce the function
$L(x,y)=-(2\pi)^{-1}\log\lvert x-y\rvert$. Then, we can express
$I_{2}(\varepsilon)$ as:

(4.20) $\displaystyle
I_{2}(\varepsilon)=\left(I_{4}(\varepsilon)+I_{5}(\varepsilon)+I_{6}(\varepsilon),I^{\prime}_{4}(\varepsilon)+I^{\prime}_{5}(\varepsilon)+I^{\prime}_{6}(\varepsilon)\right),$

where

$\displaystyle I_{4}(\varepsilon)$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
x_{1}}\left(\int_{E_{\varepsilon}}L(x^{*},y)(\varphi_{i}(y)-\varphi_{i}(x^{*}))dy\right)$
$\displaystyle-\frac{\partial}{\partial
w_{1}}\int_{E_{\varepsilon}}L(w,y)(\varphi_{i}(y)-\varphi_{i}(w))dy$
$\displaystyle-(x^{*}_{1}-w_{1})\frac{\partial^{2}}{{\partial
w_{1}}^{2}}\int_{E_{\varepsilon}}L(w,y)(\varphi_{i}(y)-\varphi_{i}(w)-\sum_{n=1}^{2}(y_{n}-w_{n})\frac{\partial\varphi_{i}}{\partial
w_{n}}(w))dy$ $\displaystyle-(x^{*}_{2}-w_{2})\frac{\partial^{2}}{\partial
w_{2}\partial
w_{1}}\int_{E_{\varepsilon}}L(w,y)(\varphi_{i}(y)-\varphi_{i}(w)-\sum_{n=1}^{2}(y_{n}-w_{n})\frac{\partial\varphi_{i}}{\partial
w_{n}}(w))dy$ $\displaystyle I_{5}(\varepsilon)$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
x_{1}}(\varphi_{i}(x^{*})F(x^{*}))-\left(\frac{\partial}{\partial
w_{1}}+(x^{*}_{1}-w_{1})\frac{\partial^{2}}{{\partial
w_{1}}^{2}}\right)(\varphi_{i}(w)F(w))$
$\displaystyle-(x^{*}_{1}-w_{1})\sum_{n=1}^{2}\frac{\partial^{2}}{{\partial
w_{1}}^{2}}\left(\frac{\partial\varphi_{i}}{\partial
w_{n}}(w)K_{n}(w)\right)-(x^{*}_{2}-w_{2})\sum_{n=1}^{2}\frac{\partial^{2}}{{\partial
w_{2}\partial w_{1}}}\left(\frac{\partial\varphi_{i}}{\partial
w_{n}}(w)K_{n}(w)\right)$ $\displaystyle I_{6}(\varepsilon)$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
x_{1}}(\mathbf{S}\hat{\chi}_{\varepsilon}\varphi_{i})(x^{*})-\left(\frac{\partial}{\partial
w_{1}}+(x^{*}_{1}-w_{1})\frac{\partial^{2}}{{\partial
w_{1}}^{2}}+(x^{*}_{2}-w_{2})\frac{\partial^{2}}{\partial w_{2}\partial
w_{1}}\right)(\mathbf{S}\hat{\chi}_{\varepsilon}\varphi_{i})(w),$
$\displaystyle I^{\prime}_{4}(\varepsilon)$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
x_{2}}\left(\int_{E_{\varepsilon}}L(x^{*},y)(\varphi_{i}(y)-\varphi_{i}(x^{*}))dy\right)$
$\displaystyle-\frac{\partial}{\partial
w_{2}}\int_{E_{\varepsilon}}L(w,y)(\varphi_{i}(y)-\varphi_{i}(w))dy$
$\displaystyle-(x^{*}_{2}-w_{2})\frac{\partial^{2}}{{\partial
w_{1}}^{2}}\int_{E_{\varepsilon}}L(w,y)(\varphi_{i}(y)-\varphi_{i}(w)-\sum_{n=1}^{2}(y_{n}-w_{n})\frac{\partial\varphi_{i}}{\partial
w_{n}}(w))dy$ $\displaystyle-(x^{*}_{1}-w_{1})\frac{\partial^{2}}{\partial
w_{2}\partial
w_{1}}\int_{E_{\varepsilon}}L(w,y)(\varphi_{i}(y)-\varphi_{i}(w)-\sum_{n=1}^{2}(y_{n}-w_{n})\frac{\partial\varphi_{i}}{\partial
w_{n}}(w))dy$ $\displaystyle I^{\prime}_{5}(\varepsilon)$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
x_{2}}(\varphi_{i}(x^{*})F(x))-\left(\frac{\partial}{\partial
w_{2}}+(x^{*}_{2}-w_{2})\frac{\partial^{2}}{{\partial
w_{2}}^{2}}\right)(\varphi_{i}(w)F(w))$
$\displaystyle-(x^{*}_{2}-w_{2})\sum_{n=1}^{2}\frac{\partial^{2}}{{\partial
w_{2}}^{2}}\left(\frac{\partial\varphi_{i}}{\partial
w_{n}}(w)K_{n}(w)\right)-(x^{*}_{1}-w_{1})\sum_{n=1}^{2}\frac{\partial^{2}}{{\partial
w_{2}\partial w_{1}}}\left(\frac{\partial\varphi_{i}}{\partial
w_{n}}(w)K_{n}(w)\right)$ $\displaystyle I^{\prime}_{6}(\varepsilon)$
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial
x_{2}}(\mathbf{S}\hat{\chi}_{\varepsilon}\varphi_{i})(x^{*})-\left(\frac{\partial}{\partial
w_{2}}+(x^{*}_{2}-w_{2})\frac{\partial^{2}}{{\partial
w_{2}}^{2}}+(x^{*}_{1}-w_{1})\frac{\partial^{2}}{\partial w_{2}\partial
w_{1}}\right)(\mathbf{S}\hat{\chi}_{\varepsilon}\varphi_{i})(w),$

for $w=\tilde{w}=0$. In the given expression, the operator $\mathbf{S}$ and
functions $F(x)$ and $K_{n}(w)$ are defined as follows:

$\displaystyle(\mathbf{S}f)(x)$ $\displaystyle=$
$\displaystyle\int_{\Omega}S(x,y)f(y)dy$ $\displaystyle F(x)$ $\displaystyle=$
$\displaystyle\int_{E_{\varepsilon}}L(x,y)dy$ $\displaystyle K_{n}(w)$
$\displaystyle=$
$\displaystyle\int_{E_{\varepsilon}}L(w,y)(y_{n}-w_{n})dy\qquad(n=1,2).$

Since $(\mathbf{S}f)(x)$ is a smooth function on $\Omega$, we can estimate
$I_{6}(\varepsilon)$ as follows:

(4.21) $\displaystyle\lvert I_{6}(\varepsilon)\rvert\leq C\varepsilon^{2}.$

We evaluate the partial derivative of the integral $\frac{\partial}{\partial
x_{1}}\int_{E_{\varepsilon}}L(x,y)(\varphi_{i}(y)-\varphi_{i}(x))dy$ at the
point $x=x^{*}$. Using the properties of eigenfunctions, we can express
$\varphi_{i}(y)$ as $\varphi_{i}(y)=\varphi_{i}(x)+\mathcal{O}(\lvert
x-y\rvert)$. Substituting this into the expression, we have:

$\displaystyle\frac{\partial}{\partial
x_{1}}\int_{E_{\varepsilon}}L(x,y)(\varphi_{i}(y)-\varphi_{i}(x))dy\bigg{|}_{x=x^{*}}$
$\displaystyle=-(2\pi)^{-1}\int_{E_{\varepsilon}}\left(\frac{\partial}{\partial
x_{1}}\log\lvert x-y\rvert(\varphi_{i}(y)-\varphi_{i}(x))+\log\lvert
x-y\rvert\frac{\partial}{\partial
x_{1}}\varphi_{i}(x)\right)dy\bigg{|}_{x=x^{*}}$
$\displaystyle=-(2\pi)^{-1}\int_{E_{\varepsilon}}\left(\frac{y_{1}-x_{1}}{\lvert
x-y\rvert^{2}}\mathcal{O}(\lvert x-y\rvert)+\frac{\partial}{\partial
x_{1}}\varphi_{i}(x)\log\lvert x-y\rvert\right)dy\bigg{|}_{x=x^{*}}$
$\displaystyle\leq C\int_{E_{\varepsilon}}(1+\log\lvert
x-y\rvert)dy\bigg{|}_{x=x^{*}}$ $\displaystyle\leq
C\int_{E_{\varepsilon}}\log\lvert x-y\rvert dy\bigg{|}_{x=x^{*}}$
$\displaystyle\leq C\int_{\mathbb{B}(x,2\varepsilon M_{\text{max}})}\log\lvert
x-y\rvert dy\bigg{|}_{x=x^{*}}$ $\displaystyle\leq
C\varepsilon^{2}\lvert\log\varepsilon\rvert,$

where $C$ is a constant independent of $\varepsilon$ and
$M_{\max}=\sup_{x\in\partial E}|x|$. Note that
$E_{\varepsilon}\subset\mathbb{B}(x,2\varepsilon M_{\text{max}})$, which
allows us to replace the integral over $E_{\varepsilon}$ with the integral
over the larger ball.

Similarly, we can also establish the following inequalities:

$\displaystyle\frac{\partial}{\partial
w_{1}}\int_{E_{\varepsilon}}L(w,y)(\varphi_{i}(y)-\varphi_{i}(w))dy\leq
C\int_{E_{\varepsilon}}L(\tilde{w},y)dy,$
$\displaystyle\frac{\partial^{2}}{\partial
w_{1}^{2}}\int_{E_{\varepsilon}}L(w,y)\left(\varphi_{i}(y)-\varphi_{i}(w)-\sum_{n=1}^{2}(y_{n}-w_{n})\frac{\partial\varphi_{i}}{\partial
w_{n}}(w)\right)dy\leq C\int_{E_{\varepsilon}}L(\tilde{w},y)dy.$

Combining these estimates, we can derive the following inequality:

(4.22) $\displaystyle\lvert I_{4}(\varepsilon)\rvert$ $\displaystyle\leq
C\int_{E_{\varepsilon}}\log\lvert x-y\rvert
dy\bigg{|}_{x=x^{*}}+C\int_{E_{\varepsilon}}\log\lvert\tilde{w}-y\rvert dy$
$\displaystyle\qquad+C\varepsilon\int_{E_{\varepsilon}}\log\lvert\tilde{w}-y\rvert
dy$ $\displaystyle\leq C\varepsilon^{2}\lvert\log\varepsilon\rvert.$

After performing the necessary calculations, we arrive at the desired results.
More precisely, by the mean value theorem, there exist points $c\in
E_{\varepsilon}$, $d\in\mathbb{B}_{M\varepsilon}$, where $M$ is given by (2.1)
such that

$\displaystyle\int_{E_{\varepsilon}}L(x,y)dy=\frac{-1}{2\pi}\log|x-c||E_{\varepsilon}|,$
$\displaystyle\int_{\mathbb{B}_{M\varepsilon}}L(x,y)dy=\frac{-1}{2\pi}\log|x-d||\mathbb{B}_{M\varepsilon}|.$

Since $|\mathbb{B}_{M\varepsilon}|=|E_{\varepsilon}|$, we can deduce that

$\displaystyle
F(x)=\frac{\log|x-c|}{\log|x-d|}\int_{\mathbb{B}_{M\varepsilon}}L(x,y)dy,$

or

(4.23) $\displaystyle
F(x)=\begin{cases}\frac{\log|x-c|}{\log|x-d|}\left(-\frac{1}{2}(M\varepsilon)^{2}\lvert\log|x|\rvert\right)\quad\text{for
}x\in\mathbb{R}^{2}\backslash\overline{\mathbb{B}}_{M\varepsilon}\\\
\frac{\log|x-c|}{\log|x-d|}\left(-\frac{1}{2}(M\varepsilon)^{2}\lvert\log
M\varepsilon\rvert+\frac{(M\varepsilon)^{2}}{4}-\frac{\lvert
x\rvert^{2}}{4}\right)\quad\text{for
}x\in\mathbb{B}_{M\varepsilon},\end{cases}$

and for $K_{n}(w)$, we have:

(4.24) $\displaystyle
K_{n}(w)=\frac{\log|w-c|}{\log|w-d|}w_{n}\left(\frac{1}{2}(M\varepsilon)^{2}\lvert\log(M\varepsilon)\rvert-\frac{(M\varepsilon)^{2}}{4}+\frac{\lvert
w\rvert^{2}}{4}\right)\quad\text{for }w\in\mathbb{B}_{\varepsilon}.$

Since $x^{*}\in\partial\mathbb{B}_{M\varepsilon}$, we have

$\displaystyle F(x^{*})$
$\displaystyle=\frac{\log|x^{*}-c|}{\log|x^{*}-d|}\left(-\frac{1}{2}(M\varepsilon)^{2}\lvert\log(M\varepsilon)\rvert\right),$
(4.25) $\displaystyle\partial F(x^{*})/\partial x_{1}$
$\displaystyle=\frac{\partial}{\partial
x_{1}}\frac{\log|x^{*}-c|}{\log|x^{*}-d|}\left(\frac{-1}{2}(M\varepsilon)^{2}|\log(M\varepsilon)|\right)+\frac{\log|x^{*}-c|}{\log|x^{*}-d|}\left(-\frac{1}{2}x^{*}_{1}\right).$

For $w=(0,0)$, we obtain

(4.26) $\displaystyle\begin{split}F(w)&=\frac{\log c}{\log
d}\left(-\frac{1}{2}(M\varepsilon)^{2}\lvert\log(M\varepsilon)\rvert+\frac{(M\varepsilon)^{2}}{4}\right)\\\
\partial F(w)/\partial w_{1}&=\frac{\partial}{\partial
w_{1}}\frac{\log|w-c|}{\log|w-d|}\Big{|}_{w=(0,0)}\left(\frac{-1}{2}(M\varepsilon)^{2}|\log(M\varepsilon)|+\frac{(M\varepsilon)^{2}}{4}\right),\\\
\partial^{2}F(w)/\partial w_{1}^{2}&=\frac{\partial^{2}}{\partial
w_{1}^{2}}\frac{\log|w-c|}{\log|w-d|}\Big{|}_{w=(0,0)}\left(\frac{-1}{2}(M\varepsilon)^{2}|\log(M\varepsilon)|+\frac{(M\varepsilon)^{2}}{4}\right)-\frac{1}{2}\frac{\log
c}{\log d}.\end{split}$

Additionally, we have

(4.27) $\begin{split}K_{n}(w)&=\partial^{2}K_{n}(w)/\partial w_{1}^{2}=0,\\\
\partial K_{n}(w)/\partial w_{1}&=\delta_{1,n}\frac{\log c}{\log
d}\left(\frac{1}{2}(M\varepsilon)^{2}\lvert\log(M\varepsilon)\rvert-\frac{1}{4}(M\varepsilon)^{2}\right)\end{split}$

where $\delta_{1,n}$ is the Kronecker delta. Summing up these results, we
obtain:

(4.28) $\displaystyle I_{5}(\varepsilon)$
$\displaystyle=-\frac{1}{2}x^{*}_{1}\left(\varphi(x^{*})\frac{\log|x^{*}-c|}{\log|x^{*}-d|}-\varphi(w)\frac{\log|w-c|}{\log|w-d|}\right)+\mathcal{O}(\varepsilon^{2}\lvert\log\varepsilon\rvert)$
$\displaystyle=\mathcal{O}(\varepsilon^{2}\lvert\log\varepsilon\rvert).$

Using estimates (4.17), (4.18), (4.19), (4.20), (4.21), (4.22) and (4.28), we
conclude that:

$\left\lvert\frac{\partial}{\partial
x_{1}}v_{\varepsilon}(x)\right\rvert_{x=x^{*}}\leq
C\varepsilon^{2}\lvert\log\varepsilon\rvert.$

In the same way estimating $I_{4}(\varepsilon)$, $I_{5}(\varepsilon)$ and
$I_{6}(\varepsilon)$, we can carry out $I^{\prime}_{4}(\varepsilon)$,
$I^{\prime}_{5}(\varepsilon)$ and $I^{\prime}_{6}(\varepsilon)$ and obtain the
same results, which yields

$\left\lvert\frac{\partial}{\partial
x_{2}}v_{\varepsilon}(x)\right\rvert_{x=x^{*}}\leq
C\varepsilon^{2}\lvert\log\varepsilon\rvert.$

Combining these facts, we derive

$\left|\frac{\partial}{\partial\nu}v_{\varepsilon}(x)\right\rvert_{x=x^{*}}\leq
C\varepsilon^{2}\lvert\log\varepsilon\rvert.$

By Lemma 3.3, we can further deduce that

$\|v_{\varepsilon}\|_{L^{2}(\Omega_{E_{\varepsilon}})}\leq
C\varepsilon^{3}|\log\varepsilon|^{2}.$

Therefore, we obtain Theorem 4.1.

###  4.3 Convergence of eigenvalues

In the context of analyzing the convergence of eigenvalues, we introduce a new
kernel $\tilde{h}(x,y)$ defined as follows:

(4.29)
$\begin{split}\tilde{h}_{\varepsilon}(x,y)=K(x,y)&+h(\varepsilon)\left\langle\nabla_{w}K(x,\tilde{w}),\nabla_{w}K(\tilde{w},y)\right\rangle\chi_{\varepsilon}(x)\chi_{\varepsilon}(y)\\\
&+i(\varepsilon)\left\langle
H_{w}K(x,\tilde{w}),H_{w}K(\tilde{w},y)\right\rangle\chi_{\varepsilon}(x)\chi_{\varepsilon}(y).\end{split}$

Here, $K(x,y)$ is a given kernel, and $\nabla_{w}K(x,\tilde{w})$ and
$H_{w}K(x,\tilde{w})$ represent the gradient and Hessian of $K$ with respect
to the variable $w$, respectively. The functions $h(\varepsilon)$ and
$i(\varepsilon)$ are introduced as weight functions that depend on a small
parameter $\varepsilon$, and $\chi_{\varepsilon}(x)$ is a characteristic
function defined as $\chi_{\varepsilon}(x)=1$ if
$x\in\bar{\Omega}_{E_{\varepsilon}}$ and $\chi_{\varepsilon}(x)=0$ otherwise.

Using the kernel $\tilde{h}_{\varepsilon}(x,y)$, we define the operator
$\mathbf{\tilde{H}}_{\varepsilon}$ as:

$\displaystyle(\mathbf{\tilde{H}}_{\varepsilon}f)(x)=\int_{\Omega}\tilde{h}_{\varepsilon}(x,y)f(y)dy.$

######  Lemma 4.5.

There exists a constant $C$ independent of $\varepsilon$ such that

(4.30)
$\displaystyle\|\mathbf{\tilde{H}}_{\varepsilon}-\chi_{\varepsilon}\mathbf{\tilde{H}}_{\varepsilon}\chi_{\varepsilon}\|_{L^{2}(\Omega)}\leq
C\varepsilon$

holds.

###### Proof.

We begin by considering the operator
$\mathbf{\tilde{H}}_{\varepsilon}-\chi_{\varepsilon}\mathbf{\tilde{H}}_{\varepsilon}\chi_{\varepsilon}$.
Using the expression

$\displaystyle\mathbf{\tilde{H}}_{\varepsilon}-\chi_{\varepsilon}\mathbf{\tilde{H}}_{\varepsilon}\chi_{\varepsilon}=(1-\chi_{\varepsilon})\mathbf{\tilde{H}}_{\varepsilon}\chi_{\varepsilon}+\mathbf{\tilde{H}}_{\varepsilon}(1-\chi_{\varepsilon}),$

we can estimate its norm as follows:

(4.31)
$\displaystyle\|\mathbf{\tilde{H}}_{\varepsilon}-\chi_{\varepsilon}\mathbf{\tilde{H}}_{\varepsilon}\chi_{\varepsilon}\|_{L^{2}(\Omega)}\leq\|(1-\chi_{\varepsilon})\mathbf{\tilde{H}}_{\varepsilon}\|_{L^{2}(\Omega)}+\|\mathbf{\tilde{H}}_{\varepsilon}(1-\chi_{\varepsilon})\|_{L^{2}(\Omega)}.$

Next, we observe that in the $h(\varepsilon)$ and $i(\varepsilon)$ terms of
(4.29), the product $(1-\chi_{\varepsilon})\chi_{\varepsilon}$ equals zero.
Therefore, we have the estimate:

$\displaystyle\|(1-\chi_{\varepsilon})\mathbf{\tilde{H}}_{\varepsilon}v\|_{L^{2}(\Omega)}\leq
C\lvert E_{\varepsilon}\rvert^{1/2}\|\mathbf{K}v\|_{L^{2}(\Omega)}\leq
C\varepsilon\|v\|_{L^{2}(\Omega)},$

which holds for any $v\in L^{2}(\Omega)$. Consequently, we obtain the
following inequalities:

(4.32)
$\begin{split}&\|(1-\chi_{\varepsilon})\mathbf{\tilde{H}}_{\varepsilon}\|_{L^{2}(\Omega)}\leq
C\varepsilon\\\
&\|(1-\chi_{\varepsilon})\mathbf{\tilde{H}}_{\varepsilon}\chi_{\varepsilon}\|_{L^{2}(\Omega)}\leq
C\varepsilon.\end{split}$

Since we have a duality relation

$\displaystyle((1-\chi_{\varepsilon})\mathbf{\tilde{H}}_{\varepsilon})^{\ast}=\mathbf{\tilde{H}}_{\varepsilon}(1-\chi_{\varepsilon}),$

we can further deduce the following:

(4.33)
$\displaystyle\|\mathbf{\tilde{H}}_{\varepsilon}(1-\chi_{\varepsilon})\|_{L^{2}(\Omega)}\leq
C\varepsilon.$

We have completed the proof of the lemma by combining the estimates (4.31) and
(4.32). ∎

Next, we aim to estimate
$\|\mathbf{\tilde{H}}_{\varepsilon}-\mathbf{K}\|_{L^{2}(\Omega)}$. For this
purpose, we state and prove the following lemma:

######  Lemma 4.6.

There exists a constant $C$ independent of $\varepsilon$ such that

$\displaystyle\|\mathbf{\tilde{H}}_{\varepsilon}-\mathbf{K}\|_{L^{2}(\Omega)}\leq
C\varepsilon^{2}\lvert\log\varepsilon\rvert$

holds.

###### Proof.

We consider an arbitrary function $v\in L^{p}(\Omega)$. From the expression

$\displaystyle((\mathbf{\tilde{H}}_{\varepsilon}-\mathbf{K})v)(x)=$
$\displaystyle
h(\varepsilon)\langle\nabla_{w}K(x,w),\nabla_{w}(\mathbf{K}\xi_{\varepsilon}v)(\tilde{w})\rangle\chi_{\varepsilon}(x)$
$\displaystyle+$ $\displaystyle i(\varepsilon)\langle
H_{w}K(x,w),H_{w}(\mathbf{K}\xi_{\varepsilon}v)(\tilde{w})\rangle\chi_{\varepsilon}(x),$

we can estimate the norm as follows:

(4.34)
$\|(\mathbf{\tilde{H}}_{\varepsilon}-\mathbf{K})v\|_{L^{p}(\Omega)}\leq\lvert
h(\varepsilon)\rvert\sum_{n=1}^{2}\left(\int_{\Omega_{E_{\varepsilon}}}\left\lvert\frac{\partial}{\partial
w_{n}}K(x,\tilde{w})\right\rvert^{p}\,dx\right)^{1/p}\left\lvert\frac{\partial}{\partial
w_{n}}(\mathbf{K}\chi_{\varepsilon}v)(\tilde{w})\right\rvert\\\ +\lvert
i(\varepsilon)\rvert\sum_{m,n=1}^{2}\left(\int_{\Omega_{E_{\varepsilon}}}\left\lvert\frac{\partial^{2}}{\partial
w_{m}\partial
w_{n}}K(x,\tilde{w})\right\rvert^{p}\,dx\right)^{1/p}\left\lvert\frac{\partial^{2}}{\partial
w_{m}\partial w_{n}}(\mathbf{K}\chi_{\varepsilon}v)(\tilde{w})\right\rvert$

valid for $p<1$.

We use the following inequalities to estimate the terms in (4.33). First, for
$n=1,2$, we have

(4.35)
$\displaystyle\left(\int_{\Omega_{E_{\varepsilon}}}\left\lvert\frac{\partial}{\partial
w_{n}}K(x,\tilde{w})\right\rvert^{p}\,dx\right)^{1/p}$ $\displaystyle\leq
C\left(\int_{\Omega_{E_{\varepsilon}}}\lvert
x-\tilde{w}\rvert^{-p},dx\right)^{1/p}$
$\displaystyle\leq\begin{cases}C\lvert\log\varepsilon\rvert^{1/2}&\text{ if
}p=2,\\\ C\varepsilon^{2/p-1}&\text{ if }p>2.\end{cases}$

Next, for $1\leq m,n\leq 2$, we have

(4.36)
$\displaystyle\left(\int_{\Omega_{E_{\varepsilon}}}\left\lvert\frac{\partial^{2}}{\partial
w_{m}\partial w_{n}}K(x,\tilde{w})\right\rvert^{p}\,dx\right)^{1/p}$
$\displaystyle\leq C\left(\int_{\Omega_{E_{\varepsilon}}}\lvert
x-\tilde{w}\rvert^{-2p}\,dx\rvert\right)^{1/p}$ $\displaystyle\leq
C\varepsilon^{2/p-2}\qquad\text{ if }p>1.$

By (4.34), (4.35), (4.36) and using the estimation (4.2), (4.14) with
$\tilde{f}=\chi_{\varepsilon}v$, we can conclude that

$\displaystyle\|(\mathbf{\tilde{H}}_{\varepsilon}-\mathbf{K})v\|_{L^{2}(\Omega)}$
$\displaystyle\leq C(\lvert
h(\varepsilon)\rvert\lvert\log\varepsilon\rvert^{1/2}\lvert\log\varepsilon\rvert^{1/2}\|v\|_{L^{2}(\Omega)}$
$\displaystyle\quad+\lvert
i(\varepsilon)\rvert\varepsilon^{-1}\varepsilon^{-1}\|v\|_{L^{2}(\Omega)})$
$\displaystyle\leq
C\varepsilon^{2}\lvert\log\varepsilon\rvert\|v\|_{L^{2}(\Omega)},$

which holds for an arbitrary $v\in L^{2}(\Omega)$. Therefore, we have
completed the proof of the lemma. ∎

######  Remark 4.7.

The $i$-th eigenvalue of $\mathbf{H}_{\varepsilon}$ corresponds exactly to the
$i$-th eigenvalue of
$\chi_{\varepsilon}\mathbf{\tilde{H}}_{\varepsilon}\chi_{\varepsilon}$.

By considering Proposition 4.4, Lemma 4.5, and Lemma 4.6, we can deduce the
existence of a constant $C$ that is independent of the specific value of $i$
such that

(4.37) $\displaystyle\lvert\mu_{i}(\varepsilon)^{-1}-\mu_{i}^{-1}\rvert\leq
C(\varepsilon^{1+s}+\varepsilon+\varepsilon^{2}\lvert\log\varepsilon\rvert)\leq
C\varepsilon$

holds.

To establish Theorem 2.1, we require a more precise estimate for the left-hand
side of inequality (4.37). Utilizing the information from (4.37), we can
conclude that when the multiplicity of $\mu_{i}$ is one, the multiplicity of
$\mu_{i}(\varepsilon)$ is also one for sufficiently small $\varepsilon$.

###  4.4 Perturbation Analysis of $\mathbf{\overline{H}_{\varepsilon}}$

In this subsection, we investigate the behaviour of eigenvalues of
$\mathbf{\overline{H}}_{\varepsilon}$ as $\varepsilon$ approaches zero. We
introduce the operators $A_{0}=\mathbf{K}$ and define the operators $A_{1}$,
$A_{2}$, and $A_{3}$ as follows:

$\displaystyle(A_{1}f)(x)$
$\displaystyle=K(x,\tilde{w})(\mathbf{K}f)(\tilde{w}),$
$\displaystyle(A_{2}f)(x)$
$\displaystyle=\langle\nabla_{w}K(x,w),\nabla_{w}(\mathbf{K}\xi_{\varepsilon}f)(\tilde{w})\rangle\xi_{\varepsilon}(x)$
$\displaystyle(A_{3}f)(x)$ $\displaystyle=\langle
H_{w}K(x,w),H_{w}(\mathbf{K}\xi_{\varepsilon}f)(\tilde{w})\rangle\xi_{\varepsilon}(x).$

We express $\mathbf{\overline{H}_{\varepsilon}}$ as the sum
$\mathbf{\overline{H}_{\varepsilon}}=A_{0}+\bar{g}(\varepsilon)A_{1}+h(\varepsilon)A_{2}+i(\varepsilon)A_{3}$,
where

(4.38) $\displaystyle\bar{g}(\varepsilon)=-\pi\mu_{i}(M\varepsilon)^{2}.$

Moreover, we introduce the quantities $\lambda(\varepsilon)$ and
$\psi(\varepsilon)$, which represent approximate eigenvalues and approximate
eigenfunctions of $\mathbf{\overline{H}}_{\varepsilon}$. They are given by:

$\displaystyle\lambda(\varepsilon)$ $\displaystyle=$
$\displaystyle\lambda_{0}+\bar{g}(\varepsilon)\lambda_{1}+h(\varepsilon)\lambda_{2}+i(\varepsilon)\lambda_{3}$
$\displaystyle\psi(\varepsilon)$ $\displaystyle=$
$\displaystyle\psi_{0}+\bar{g}(\varepsilon)\psi_{1}+h(\varepsilon)\psi_{2}+i(\varepsilon)\psi_{3}.$

Let $\lambda_{0}$ be a simple eigenvalue of $A_{0}$. We begin by defining
$\psi_{0}$ as the solution to the equation:

(4.39)
$\displaystyle(A_{0}-\lambda_{0})\psi_{0}=0,\quad\|\psi_{0}\|_{L^{2}(\Omega)}=1.$

Subsequently, we proceed to solve the following equations:

(4.40)
$\displaystyle(A_{0}-\lambda_{0})\psi_{1}=(\lambda_{1}-A_{1})\psi_{0},\qquad\langle\psi_{0},\psi_{1}\rangle_{L^{2}(\Omega)}=0$
(4.41)
$\displaystyle(A_{0}-\lambda_{0})\psi_{2}=(\lambda_{2}-A_{2})\psi_{0},\qquad\langle\psi_{0},\psi_{2}\rangle_{L^{2}(\Omega)}=0$
(4.42)
$\displaystyle(A_{0}-\lambda_{0})\psi_{3}=(\lambda_{3}-A_{3})\psi_{0},\qquad\langle\psi_{0},\psi_{3}\rangle_{L^{2}(\Omega)}=0.$

According to the Fredholm alternative theory, the unique solution $\psi_{1}$,
$\psi_{2}$, $\psi_{3}$ of equations (4.40), (4.41), and (4.42) exists if and
only if the following condition holds:

(4.43) $\displaystyle\lambda_{n}=\langle
A_{n}\psi_{0},\psi_{0}\rangle_{L^{2}(\Omega)}\qquad(n=1,2,3)$

It is worth noting that $\lambda_{0}$ is chosen as $\lambda_{0}=\mu_{i}^{-1}$,
and therefore $\psi_{0}=\varphi_{i}$.

######  Lemma 4.8.

The operator norms of $A_{1}$, $A_{2}$, and $A_{3}$ satisfy the following
inequalities:

  1. 1.

For $p>1$, $\|A_{1}\|_{L^{p}(\Omega)}\leq C$.

  2. 2.

For $p=2$, $\|A_{2}\|_{L^{p}(\Omega)}\leq C\lvert\log\varepsilon\rvert$.
For $p>2$, $\|A_{2}\|_{L^{p}(\Omega)}\leq C\varepsilon^{2/p-1}$.

  3. 3.

For $p>1$, $\|A_{3}\|_{L^{p}(\Omega)}\leq C\varepsilon^{-2}$.

Here, the constant $C$ is independent of $\varepsilon$.

###### Proof.

Let $f\in L^{p}(\Omega)$ be arbitrary. We have:

$\displaystyle\|A_{1}\|_{L^{p}(\Omega)}$ $\displaystyle\leq$ $\displaystyle
C\|G(.,w)\|_{L^{p}(\Omega)}\|\mathbf{K}f\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq$ $\displaystyle C\|f\|_{L^{p}(\Omega)}\qquad(p>1).$

Applying Minkowski inequality, we obtain

$\displaystyle\|A_{2}f\|_{L^{p}(\Omega)}=\left(\int_{\Omega\backslash
E_{\varepsilon/2}}\bigg{\lvert}\sum_{n=1}^{2}\frac{\partial}{\partial
w_{n}}K(x,\tilde{w})\frac{\partial}{\partial
w_{n}}(\mathbf{K}\xi_{\varepsilon}f)(\tilde{w})\bigg{\rvert}^{p}dx\right)^{1/p}$
$\displaystyle\leq\sum_{n=1}^{2}\left(\int_{\Omega\backslash
E_{\varepsilon/2}}\left\lvert\frac{\partial}{\partial
w_{n}}K(x,\tilde{w})\right\rvert^{p}\left\lvert\frac{\partial}{\partial
w_{n}}(\mathbf{K}\xi_{\varepsilon}f)(\tilde{w})\right\rvert^{p}dx\right)^{1/p}$
(4.44) $\displaystyle\leq\sum_{n=1}^{2}\left(\int_{\Omega\backslash
E_{\varepsilon/2}}\left\lvert\frac{\partial}{\partial
w_{n}}K(x,\tilde{w})\right\rvert^{p}dx\right)^{1/p}\left\lvert\frac{\partial}{\partial
w_{n}}(\mathbf{K}\xi_{\varepsilon}f)(\tilde{w})\right\rvert.$

Furthermore, we have

(4.45) $\displaystyle\left(\int_{\Omega\backslash
E_{\varepsilon/2}}\left\lvert\frac{\partial}{\partial
w_{n}}K(x,\tilde{w})\right\rvert^{p}dx\right)^{1/p}$ $\displaystyle\leq
C\left(\int_{\Omega\backslash E_{\varepsilon/2}}\lvert
x-\tilde{w}\rvert^{-p}dx\right)^{1/p}$
$\displaystyle\leq\begin{cases}C\,\lvert\log\varepsilon\rvert\qquad\text{if
}p=2,\\\ C\varepsilon^{2/p-1}\qquad\text{if }p>2,\end{cases}$

for $n=1,2$. Using Hölder inequality, we have the estimate

(4.46) $\displaystyle\left\lvert\frac{\partial}{\partial
w_{n}}(\mathbf{K}\xi_{\varepsilon}f)(\tilde{w})\right\rvert$
$\displaystyle\leq C\left(\int_{\Omega}\lvert
y-\tilde{w}\rvert^{-q}dy\right)^{1/q}\|f\|_{L^{p}(\Omega)}$ $\displaystyle\leq
C\|f\|_{L^{p}(\Omega)}\text{ if }p>2,$

where $q$ satisfies $(1/p)+(1/q)=1$.

By combining equations (4.4), (4.45), and (4.46), we obtain:

$\displaystyle\|A_{2}\|_{L^{p}(\Omega)}$ $\displaystyle\leq$
$\displaystyle\begin{cases}C\lvert\log\varepsilon\rvert\qquad\text{if }p=2,\\\
C\varepsilon^{2/p-1}\qquad\text{if }p>2.\end{cases}$

Similarly, for $A_{3}$, we have:

$\displaystyle\|A_{3}f\|_{L^{p}(\Omega)}$ $\displaystyle\leq$
$\displaystyle\sum_{m,n=1}^{2}\left(\int_{\Omega\backslash
E_{\varepsilon/2}}\left\lvert\frac{\partial^{2}}{\partial w_{m}\partial
w_{n}}K(x,\tilde{w})\right\rvert^{p}dx\right)^{1/p}\left\lvert\frac{\partial^{2}}{\partial
w_{m}\partial w_{n}}(\mathbf{K}\xi_{\varepsilon}f)(\tilde{w})\right\rvert$
$\displaystyle\leq$ $\displaystyle
C\varepsilon^{-2}\|f\|_{L^{p}(\Omega)}\qquad\text{if }p>1,$

we complete the proof of the lemma. ∎

######  Lemma 4.9.

For a constant $C$ independent of $\varepsilon$, the following inequalities
hold:

  1. 1.

If $p>1$, then $\|\psi_{1}\|_{L^{p}(\Omega)}\leq C$.

  2. 2.

If $p=2$, then $\|\psi_{2}\|_{L^{p}(\Omega)}\leq
C\lvert\log\varepsilon\rvert$.
If $p>2$, then $\|\psi_{2}\|_{L^{p}(\Omega)}\leq C\varepsilon^{2/p-1}$.

  3. 3.

If $p>1$, then $\|\psi_{3}\|_{L^{p}(\Omega)}\leq C\varepsilon^{-2}$.

###### Proof.

We begin by observing the following inequalities:

$\displaystyle\|(\lambda_{1}-A_{1})\psi_{0}\|_{L^{2}(\Omega)}$
$\displaystyle\leq$ $\displaystyle C\|A_{1}\|_{L^{2}(\Omega)}\leq C^{\prime},$
$\displaystyle\|(\lambda_{2}-A_{2})\psi_{0}\|_{L^{2}(\Omega)}$
$\displaystyle\leq$ $\displaystyle C\|A_{2}\|_{L^{2}(\Omega)}\leq
C\lvert\log\varepsilon\rvert,$
$\displaystyle\|(\lambda_{3}-A_{3})\psi_{0}\|_{L^{2}(\Omega)}$
$\displaystyle\leq$ $\displaystyle C\|A_{3}\|_{L^{2}(\Omega)}\leq
C\varepsilon^{-2}.$

By the Fredholm theory, we obtain the following $L^{2}(\Omega)$-norm bounds
for $\psi_{1}$ and $\psi_{2}$:

$\displaystyle\|\psi_{1}\|_{L^{2}(\Omega)}$ $\displaystyle\leq$ $\displaystyle
C\|\lambda_{1}-A_{1}\|_{L^{2}(\Omega)}\|\psi_{0}\|_{L^{2}(\Omega)}\leq
C^{\prime}$ $\displaystyle\|\psi_{2}\|_{L^{2}(\Omega)}$ $\displaystyle\leq$
$\displaystyle
C\|\lambda_{2}-A_{2}\|_{L^{2}(\Omega)}\|\psi_{0}\|_{L^{2}(\Omega)}\leq
C\lvert\log\varepsilon\rvert$ $\displaystyle\|\psi_{3}\|_{L^{2}(\Omega)}$
$\displaystyle\leq$ $\displaystyle
C\|\lambda_{3}-A_{3}\|_{L^{2}(\Omega)}\|\psi_{0}\|_{L^{2}(\Omega)}\leq
C\varepsilon^{-2}.$

By similar arguments, we can obtain $L^{p}$ estimates for $\psi_{1}$,
$\psi_{2}$, and $\psi_{3}$. ∎

######  Proposition 4.10.

There exists a constant $C$ independent of $\varepsilon$ such that

(4.47)
$\displaystyle\|(\mathbf{\overline{H}}_{\varepsilon}-\lambda(\varepsilon))\psi(\varepsilon)\|_{L^{2}(\Omega)}\leq
C\varepsilon^{4}\lvert\log\varepsilon\rvert^{2}$

holds.

###### Proof.

By (4.39), (4.40), (4.41) and (4.42), we have

$\displaystyle(\mathbf{\overline{H}}_{\varepsilon}-\lambda(\varepsilon))\psi(\varepsilon)$
$\displaystyle=\bar{g}(\varepsilon)^{2}(A_{1}-\lambda_{1})\psi_{1}+h(\varepsilon)^{2}(A_{2}-\lambda_{2})\psi_{2}+i(\varepsilon)^{2}(A_{3}-\lambda_{3})\psi_{2}$
$\displaystyle+\bar{g}(\varepsilon)h(\varepsilon)((A_{1}-\lambda_{1})\psi_{2}+(A_{2}-\lambda_{2})\psi_{1})$
$\displaystyle+h(\varepsilon)i(\varepsilon)((A_{2}-\lambda_{2})\psi_{3}+(A_{3}-\lambda_{3})\psi_{2})$
(4.48)
$\displaystyle+i(\varepsilon)\bar{g}(\varepsilon)((A_{3}-\lambda_{3})\psi_{1}+(A_{1}-\lambda_{1})\psi_{3}).$

By using Lemma 4.8 and Lemma 4.9, we can estimate each term in (4.4).
Therefore, we obtain

$\|(\mathbf{\overline{H}}\varepsilon-\lambda(\varepsilon))\psi(\varepsilon)\|_{L^{2}(\Omega)}\leq
C\varepsilon^{4}\lvert\log\varepsilon\rvert^{2},$

which completes the proof of Proposition 4.10. ∎

In the following proposition, our goal is to establish an estimate for
$\|(\mathbf{K_{\varepsilon}}-\mathbf{H_{\varepsilon}})(\chi_{\varepsilon}\psi(\varepsilon))\|_{L^{2}(\Omega_{E_{\varepsilon}})}$.

######  Proposition 4.11.

There exists a constant $C$ independent of $\varepsilon$ such that

$\|(\mathbf{K_{\varepsilon}}-\mathbf{H_{\varepsilon}})(\chi_{\varepsilon}\psi(\varepsilon))\|_{L^{2}(\Omega_{E_{\varepsilon}})}\leq
C\varepsilon^{3}\lvert\log\varepsilon\rvert$

holds.

###### Proof.

We fix $\theta\in(0,1)$. Then, by Proposition 4.4, Lemma 4.9, Theorem 4.1, and
(4.38), we have

$\displaystyle\|(\mathbf{K_{\varepsilon}}-\mathbf{H_{\varepsilon}})(\chi_{\varepsilon}\psi(\varepsilon))\|_{L^{2}(\Omega_{E_{\varepsilon}})}$
$\displaystyle\leq\|(\mathbf{K_{\varepsilon}}-\mathbf{H_{\varepsilon}})(\chi_{\varepsilon}\varphi_{i})\|_{L^{2}(\Omega_{E_{\varepsilon}})}$
$\displaystyle\qquad+\|(\mathbf{K_{\varepsilon}}-\mathbf{H_{\varepsilon}})\|_{L^{2}(\Omega_{E_{\varepsilon}})}(\lvert\bar{g}(\varepsilon)\rvert\|\psi_{1}\|_{L^{2}(\Omega)}+\lvert
h(\varepsilon)\rvert\|\psi_{2}\|_{L^{2}(\Omega)}+\lvert
i(\varepsilon)\rvert\|\psi_{3}\|_{L^{2}(\Omega)})$ $\displaystyle\leq
C(\varepsilon^{3}|\log\varepsilon|^{2}+\varepsilon^{1+\theta}(\varepsilon^{2}+\varepsilon^{2}\lvert\log\varepsilon\rvert+\varepsilon^{2})$
$\displaystyle\leq C\varepsilon^{3}|\log\varepsilon|^{2}.$

The proof of Proposition 4.11 is complete. ∎

###  4.5 Proof of Theorem 4.2

In this subsection, we aim to compare the operators $\mathbf{H_{\varepsilon}}$
with $\mathbf{\overline{H}_{\varepsilon}}$ and establish an estimate for the
difference between them. To do so, we introduce the characteristic function
$\hat{\chi}_{\varepsilon}$ of the set $E_{\varepsilon}$, defined as
$\hat{\chi}_{\varepsilon}=1-\chi_{\varepsilon}$. Then, we define
$J_{\varepsilon}(x,v)=(\chi_{\varepsilon}\overline{\mathbf{H}}_{\varepsilon}v-\mathbf{H}_{\varepsilon}(\chi_{\varepsilon}v))(x)$
for $v\in L^{p}(\Omega)$. In Lemma 4.12, we establish an important estimate
for $\|J_{\varepsilon}(\cdot,v)\|_{L^{2}(\Omega_{E_{\varepsilon}})}$.

######  Lemma 4.12.

There exists a constant $C$ independent of $\varepsilon$ such that

(4.49)
$\displaystyle\|J_{\varepsilon}(\cdot,v)\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\varepsilon^{2-2/p}\|v\|_{L^{p}(\Omega)}$

holds for any $v\in L^{p}(\Omega)$ $(p>2)$.

###### Proof.

To prove this lemma, we first note that

(4.50) $\displaystyle\Delta J_{\varepsilon}(x,v)=0\qquad
x\in\Omega_{E_{\varepsilon}}$ $\displaystyle J_{\varepsilon}(x,v)=0\qquad
x\in\partial\Omega.$

By applying (4.10), we have the following expression

(4.51) $\displaystyle\frac{\partial}{\partial
x_{1}}J_{\varepsilon}(x,v)|_{x=x^{*}}=\sum_{n=1}^{4}J_{n}(\varepsilon,v),$

where

$\displaystyle J_{1}(\varepsilon,v)$ $\displaystyle=\frac{\partial}{\partial
x_{1}}(\mathbf{K}\hat{\chi}_{\varepsilon}v)(x^{*})-\left(\frac{\partial}{\partial
w_{1}}+x^{*}_{1}\frac{\partial^{2}}{{\partial
w_{1}}^{2}}+x^{*}_{2}\frac{\partial^{2}}{{\partial w_{2}\partial
w_{1}}}\right)(\mathbf{K}\xi_{\varepsilon}\hat{\chi}_{\varepsilon}v)(\tilde{w})$
$\displaystyle J_{2}(\varepsilon,v)$ $\displaystyle=2\pi(\varepsilon
M)^{2}\frac{\partial}{\partial
x_{1}}\left\langle\nabla_{w}S(x^{*},\tilde{w}),\nabla_{w}(\mathbf{K}(\xi_{\varepsilon}\hat{\chi}_{\varepsilon}v))(\tilde{w})\right\rangle$
$\displaystyle J_{3}(\varepsilon,v)$ $\displaystyle=(\pi/2)(\varepsilon
M)^{4}\frac{\partial}{\partial x_{1}}\left\langle
H_{w}S(x^{*},\tilde{w}),H_{w}(\mathbf{K}\xi_{\varepsilon}\hat{\chi}_{\varepsilon}v)(\tilde{w})\right\rangle$
$\displaystyle J_{4}(\varepsilon,v)$ $\displaystyle=-\pi\mu_{i}(\varepsilon
M)^{2}\frac{\partial}{\partial
x_{1}}K(x^{*},\tilde{w})(\mathbf{K}v)(\tilde{w}).$

Note that $E_{\varepsilon}\subset\mathbb{B}(x^{*},2\varepsilon
M_{\text{max}})$ and $(E_{\varepsilon}\setminus E_{\varepsilon/2})\subset
A(\tilde{w};\frac{\varepsilon}{2}M_{\min},\varepsilon M_{\max})$ for
$\tilde{w}=(0,0)$, we can establish the following inequalities:

(4.52) $\displaystyle\lvert J_{1}(\varepsilon,v)\rvert$ $\displaystyle\leq
C\left(\int_{\mathbb{B}(x^{*},2\varepsilon M_{\max})}\lvert
x^{*}-y\rvert^{-p^{\prime}}\,dy\right)^{1/p^{\prime}}\|v\|_{L^{p}(\Omega)}$
$\displaystyle\quad+C\left(\int_{A(\tilde{w};\frac{\varepsilon}{2}M_{\min},\varepsilon
M_{\max})}\lvert
y-w\rvert^{-p^{\prime}}\,dy\right)^{1/p^{\prime}}\|v\|_{L^{p}(\Omega)}$
$\displaystyle\quad+C\varepsilon\left(\int_{A(\tilde{w};\frac{\varepsilon}{2}M_{\min},\varepsilon
M_{\max})}\lvert
x^{*}-y\rvert^{-2p^{\prime}}\,dy\right)^{1/p^{\prime}}\|v\|_{L^{p}(\Omega)}$
$\displaystyle\leq C\varepsilon^{1-2/p}\|v\|_{L^{p}(\Omega)}\qquad(p>2),$
(4.53) $\displaystyle\lvert J_{2}(\varepsilon,v)\rvert$ $\displaystyle\leq
C\lvert
h(\varepsilon)\rvert\left(\int_{A(\tilde{w};\frac{\varepsilon}{2}M_{\min},\varepsilon
M_{\max})}\lvert
y-w\rvert^{-p^{\prime}}\,dy\right)^{1/p^{\prime}}\|v\|_{L^{p}(\Omega)}$
$\displaystyle\leq C\varepsilon^{3-2/p}\|v\|_{L^{p}(\Omega)}\qquad(p>2),$
(4.54) $\displaystyle\lvert J_{3}(\varepsilon,v)\rvert$ $\displaystyle\leq
C\lvert
i(\varepsilon)\rvert\left(\int_{A(\tilde{w};\frac{\varepsilon}{2}M_{\min},\varepsilon
M_{\max})}\lvert
y-w\rvert^{-2p^{\prime}}\,dy\right)^{1/p^{\prime}}\|v\|_{L^{p}(\Omega)}$
$\displaystyle\leq C\varepsilon^{4-2/p}\|v\|_{L^{p}(\Omega)}\qquad(p>1)$
(4.55) $\lvert J_{4}(\varepsilon,v)\rvert\leq
C\varepsilon\|v\|_{L^{p}(\Omega)}\qquad(p>1)$

where $p^{\prime}$ satisfies $(1/p)+(1/p^{\prime})=1$. Based on these
inequalities, we can summarize the results as follows:

(4.56) $\displaystyle\left\lvert\frac{\partial}{\partial
x_{1}}J_{\varepsilon}(x,v)|_{x=x^{*}}\right\rvert\leq
C\varepsilon^{1-2/p}\|v\|_{L^{p}(\Omega)}\qquad(p>2).$

Analogously, the upper estimate of $|\frac{\partial}{\partial
x_{2}}J_{\varepsilon}(x,v)|$ at $x=x^{*}$ can be obtained without any new
difficulty

(4.57) $\displaystyle\left\lvert\frac{\partial}{\partial
x_{2}}J_{\varepsilon}(x,v)|_{x=x^{*}}\right\rvert\leq
C\varepsilon^{1-2/p}\|v\|_{L^{p}(\Omega)}\qquad(p>2).$

Combining (4.56) and (4.57) yields

(4.58)
$\displaystyle\left\lvert\frac{\partial}{\partial\nu}J_{\varepsilon}(x,v)|_{x=x^{*}}\right\rvert\leq
C\varepsilon^{1-2/p}\|v\|_{L^{p}(\Omega)}\qquad(p>2).$

Using (4.58) along with Lemma 3.3, we can conclude that:

$\displaystyle\|J_{\varepsilon}(\cdot,v)\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\varepsilon^{2-2/p}|\log\varepsilon|\|v\|_{L^{p}(\Omega)}\qquad(p>2),$

we finish the proof of Lemma 4.12. ∎

Using the evaluation method similar to the proof of Theorem 4.1, we can derive
the following formula

(4.59) $\displaystyle\lvert J_{1}(\varepsilon,\varphi_{i})\rvert\leq
C\varepsilon^{2}|\log\varepsilon|.$

It is evident that

(4.60) $\displaystyle\lvert J_{4}(\varepsilon,\varphi_{i})\rvert\leq
C\varepsilon^{2}.$

Hence, we have

(4.61) $\displaystyle\lvert
J_{1}(\varepsilon,\varphi_{i})+J_{4}(\varepsilon,\varphi_{i})\rvert\leq
C\varepsilon^{2}|\log\varepsilon|.$

By summing up (4.50), (4.52), (4.53), (4.59), (4.60) and (4.61), we have

$\displaystyle\left\lvert\frac{\partial}{\partial
x_{1}}J_{\varepsilon}(x,\varphi_{i})|_{x=x^{*}}\right\rvert\leq
C\varepsilon^{2}|\log\varepsilon|.$

Similarly, we get

$\displaystyle\left\lvert\frac{\partial}{\partial
x_{2}}J_{\varepsilon}(x,\varphi_{i})|_{x=x^{*}}\right\rvert\leq
C\varepsilon^{2}|\log\varepsilon|.$

Therefore, it follows that

(4.62)
$\displaystyle\left\lvert\frac{\partial}{\partial\nu}J_{\varepsilon}(x,\varphi_{i})|_{x=x^{*}}\right\rvert\leq
C\varepsilon^{2}|\log\varepsilon|.$

Using inequality (4.62) and Lemma 3.3, we derive

(4.63)
$\displaystyle\|J_{\varepsilon}(\cdot,\varphi_{i})\|_{L^{2}(\Omega_{E_{\varepsilon}})}\leq
C\varepsilon^{3}|\log\varepsilon|^{2}.$

Consequently, we have established Theorem 4.2.

Furthermore, we aim to estimate
$\|J_{\varepsilon}(\cdot,\psi(\varepsilon))\|_{L^{2}(\Omega_{E_{\varepsilon})}}$.
By utilizing equation (4.63), Lemmas 4.9 and 4.12, we can derive the
following:

$\displaystyle\|J_{\varepsilon}(\cdot,\psi(\varepsilon))\|_{L^{2}(\Omega_{E_{\varepsilon})}}$
$\displaystyle\leq\|J_{\varepsilon}(\cdot,\varphi_{i})\|_{L^{2}(\Omega_{E_{\varepsilon})}}+\lvert
g(\varepsilon)\rvert\|J_{\varepsilon}(\cdot,\psi_{1})\|_{L^{2}(\Omega_{E_{\varepsilon})}}$
$\displaystyle\quad+\lvert
h(\varepsilon)\rvert\|J_{\varepsilon}(\cdot,\psi_{2})\|_{L^{2}(\Omega_{E_{\varepsilon})}}+\lvert
i(\varepsilon)\rvert\|J_{\varepsilon}(\cdot,\psi_{3})\|_{L^{2}(\Omega_{E_{\varepsilon})}}$
$\displaystyle\leq
C(\varepsilon^{3}|\log\varepsilon|^{2}+\varepsilon^{4-2/p}+\varepsilon^{6-2/p})$
$\displaystyle\leq C\varepsilon^{3}|\log\varepsilon|^{2}.$

As a result, we arrive at the subsequent proposition.

######  Proposition 4.13.

There exists a constant $C$ independent of $\varepsilon$ such that the
following inequality holds:

$\displaystyle\|(\mathbf{H}_{\varepsilon}\chi_{\varepsilon}-\chi_{\varepsilon}\overline{\mathbf{H}}_{\varepsilon})\psi(\varepsilon)\|_{L^{2}(\Omega_{E_{\varepsilon})}}\leq
C\varepsilon^{3}|\log\varepsilon|^{2}.$

##  5 Proof of Theorem 2.1

Armed with the estimates established in Section 4, we are now ready to
demonstrate the validity of Theorem 2.1. Using Propositions 4.10, 4.11 and
4.13, we obtain the following estimate

$\displaystyle\|(\mathbf{K}_{\varepsilon}-\lambda(\varepsilon))(\chi_{\varepsilon}\psi(\varepsilon))\|_{L^{2}(\Omega_{E_{\varepsilon}})}$
$\displaystyle\leq\|(\mathbf{K_{\varepsilon}}-\mathbf{H_{\varepsilon}})(\chi_{\varepsilon}\psi(\varepsilon))\|_{L^{2}(\Omega_{E_{\varepsilon}})}$
$\displaystyle\quad+\|\mathbf{\overline{H}}_{\varepsilon}\psi(\varepsilon)-\mathbf{H}_{\varepsilon}(\chi_{\varepsilon}\psi(\varepsilon))\|_{L^{2}(\Omega_{E_{\varepsilon}})}$
$\displaystyle\quad+\|(\mathbf{\overline{H}_{\varepsilon}}-\lambda(\varepsilon))\psi(\varepsilon)\|_{L^{2}(\Omega_{E_{\varepsilon}})}$
$\displaystyle\leq C\varepsilon^{3}|\log\varepsilon|^{2}.$

Note that $\|\psi(\varepsilon)\|_{L^{2}(\Omega_{E_{\varepsilon}})}\in(1/2,2)$
for small $\varepsilon$. As a result, there exists at least one eigenvalue
$\lambda^{\ast}(\varepsilon)$ of $\mathbf{K_{\varepsilon}}$ that satisfies the
condition

(5.1)
$\displaystyle\lvert\lambda^{\ast}(\varepsilon)-\lambda(\varepsilon)\rvert\leq
C\varepsilon^{3}|\log\varepsilon|^{2}.$

We can express $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$ as follows:

(5.2) $\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle\lvert(\mathbf{K}\psi_{0})(\tilde{w})\rvert^{2}=\mu_{i}^{-2}\varphi_{i}(\tilde{w})^{2}$
$\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\langle\nabla_{w}(\mathbf{K}\xi_{\varepsilon}\psi_{0})(\tilde{w}),\nabla_{w}(\mathbf{K}\xi_{\varepsilon}\psi_{0})(\tilde{w})\rangle$
$\displaystyle=$ $\displaystyle\sum_{n=1}^{2}\left(\frac{\partial}{\partial
w_{n}}\int_{\Omega}K(w,y)\xi_{\varepsilon}(y)\varphi_{i}(y)\,dy\right)^{2}|_{w=\tilde{w}}$
(5.4) $\displaystyle\lambda_{3}$ $\displaystyle=$ $\displaystyle\langle
H_{w}(\mathbf{K}\xi_{\varepsilon}\psi_{0})(\tilde{w}),H_{w}(\mathbf{K}\xi_{\varepsilon}\psi_{0})(\tilde{w})\rangle$
$\displaystyle=$
$\displaystyle\sum_{m,n=1}^{2}\left(\frac{\partial^{2}}{\partial w_{m}\partial
w_{n}}\int_{\Omega}K(w,y)\xi_{\varepsilon}(y)\varphi_{i}(y)\,dy\right)^{2}|_{w=\tilde{w}}.$

We observe that the second-order partial derivatives can be bounded as
follows:

$\left\lvert\frac{\partial^{2}}{\partial w_{m}\partial
w_{n}}\int_{\Omega}K(w,y)\xi_{\varepsilon}(y)\varphi_{i}(y)\,dy|_{w=\tilde{w}}\right\rvert\\\
\leq
C\int_{\Omega\setminus\mathbb{B}(w,\frac{\varepsilon}{2}M_{\text{min}})}\lvert
y-\tilde{w}\rvert^{-2}\,dy\leq C\lvert\log\varepsilon\rvert\quad(1\leq m,n\leq
2).$

Hence, we conclude that

(5.5) $\displaystyle\lambda_{3}=\mathcal{O}(\lvert\log\varepsilon\rvert^{2}).$

Similarly, we obtain the following expression:

$\displaystyle\frac{\partial}{\partial
w_{n}}\int_{\Omega}K(w,y)\xi_{\varepsilon}(y)\varphi_{i}(y)\,dy|_{w=\tilde{w}}$
(5.6) $\displaystyle=\mu_{i}^{-1}\frac{\partial}{\partial
w_{n}}\varphi_{i}(\tilde{w})+I_{1}^{(n)}(\varepsilon)+I_{2}^{(n)}(\varepsilon),$

where

$\displaystyle I_{1}^{(n)}(\varepsilon)$ $\displaystyle=$
$\displaystyle-\frac{\partial}{\partial
w_{n}}\int_{\Omega}S(w,y)(1-\xi_{\varepsilon}(y)\varphi_{i}(y)dy\lvert_{w=\tilde{w}}$
$\displaystyle I_{2}^{(n)}(\varepsilon)$ $\displaystyle=$
$\displaystyle-\frac{\partial}{\partial
w_{n}}\int_{\Omega}L(w,y)(1-\xi_{\varepsilon}(y))\varphi_{i}(y)dy|_{w=\tilde{w}}$

for $n=1,2$ and $L(w,y)=K(w,y)-S(w,y)=-(2\pi)^{-1}\log\lvert w-y\rvert$.

We can proceed with the estimation of $I_{1}^{(n)}(\varepsilon)$. Utilizing
the estimate $\lvert\frac{\partial S(w,y)}{\partial w_{n}}\rvert\leq C$ for
all $y\in E_{\varepsilon}$, which follows from the smoothness assumption on
$S(w,y)$. Since $\varphi_{i}(y)$ is an eigenfunction of the Neumann Laplacian,
it is a smooth function. Moreover, $E_{\varepsilon}$ is compact, which implies
that $\lvert\varphi_{i}(y)\rvert\leq C$ for all $y\in E_{\varepsilon}$, and
$1-\xi_{\varepsilon}(y)\leq 1$ for all $y\in E_{\varepsilon}$. Consequently,
we have

$\displaystyle\lvert I_{1}^{(n)}(\varepsilon)\rvert$
$\displaystyle\leq\int_{E_{\varepsilon}}\left\lvert\frac{\partial
S(w,y)}{\partial w_{n}}(1-\xi_{\varepsilon}(y))\varphi_{i}(y)\right\rvert dy$
$\displaystyle\leq\int_{E_{\varepsilon}}\left\lvert\frac{\partial
S(w,y)}{\partial
w_{n}}\right\rvert\lvert(1-\xi_{\varepsilon}(y))\rvert\lvert\varphi(y)\rvert
dy$ $\displaystyle\leq C\int_{E_{\varepsilon}}1dy$ (5.7) $\displaystyle\leq
C\varepsilon^{2}\qquad(n=1,2).$

Continuing our analysis, we estimate the term

(5.8) $\displaystyle\lvert I_{2}^{(n)}(\varepsilon)\rvert$
$\displaystyle=\left\lvert-\frac{\partial}{\partial
w_{n}}\int_{\Omega}L(w,y)(1-\xi_{\varepsilon}(y))\varphi_{i}(y)dy|_{w=\tilde{w}}\right\rvert$
$\displaystyle=\frac{\partial}{\partial
w_{n}}\int_{\Omega}\left((2\pi)^{-1}\log\lvert
w-y\rvert\right)(1-\xi_{\varepsilon}(y))\varphi_{i}(y)dy|_{w=\tilde{w}}$
$\displaystyle=\frac{\partial}{\partial
w_{n}}\int_{\Omega}\left((2\pi)^{-1}\log\lvert
w-y\rvert\right)(1-\xi_{\varepsilon}(y))(\varphi_{i}(y)-\varphi_{i}(w))dy|_{w=\tilde{w}}$
$\displaystyle\quad+\frac{\partial}{\partial
w_{n}}\left(\varphi_{i}(w)\int_{\Omega}\left((2\pi)^{-1}\log\lvert
w-y\rvert\right)(1-\xi_{\varepsilon}(y))dy\right)|_{w=\tilde{w}}$
$\displaystyle\leq C\varepsilon^{2}\lvert\log\varepsilon\rvert\quad(n=1,2).$

Combining the estimates (5), (5), (5) and (5.8), we obtain

(5.9)
$\displaystyle\lambda_{2}=\mu_{i}^{-2}\lvert\nabla\varphi_{i}(\tilde{w})\rvert^{2}+\mathcal{O}(\varepsilon^{2}\lvert\log\varepsilon\rvert).$

In light of equations (5.2), (5.5) and (5.9), we can deduce the following

(5.10) $\displaystyle\lambda(\varepsilon)$
$\displaystyle=\lambda_{0}+\bar{g}(\varepsilon)\lambda_{1}+h(\varepsilon)\lambda_{2}+i(\varepsilon)\lambda_{3}$
$\displaystyle=\lambda_{0}-\pi\mu_{i}(\varepsilon
M)^{2}\lambda_{1}+2\pi(\varepsilon M)^{2}\lambda_{2}+(2/\pi)(\varepsilon
M)^{4}\lambda_{3}$
$\displaystyle=\mu_{i}^{-1}-(\pi\mu_{i}^{-1}\varphi_{i}(\tilde{w})^{2}-2\pi\mu_{i}^{-2}\lvert\nabla\varphi_{i}(\tilde{w})\rvert^{2})(\varepsilon
M)^{2}+\mathcal{O}(\varepsilon^{4}\lvert\log\varepsilon\rvert^{2}).$

Summarizing the estimates with respect to (5.1), (5.10) and the result (4.37)
implies that $\lambda^{\ast}(\varepsilon)$ must be equal to
$\mu_{i}(\varepsilon)^{-1}$. Therefore, we have

$\displaystyle\lvert\mu_{i}(\varepsilon)^{-1}-(\mu_{i}^{-1}-(\pi\mu_{i}^{-1}\varphi_{i}(\tilde{w})^{2}-2\pi\mu_{i}^{-2}\lvert\nabla\varphi_{i}(\tilde{w})\rvert^{2})(\varepsilon
M)^{2})\rvert\leq C\varepsilon^{3}|\log\varepsilon|^{2}.$

From this, we deduce the asymptotic expansion for $\mu_{i}(\varepsilon)$:

$\displaystyle\mu_{i}(\varepsilon)=\mu_{i}-(2\pi\lvert\nabla\varphi_{i}(\tilde{w})\rvert^{2}-\pi\mu_{i}\varphi_{i}(\tilde{w})^{2})(\varepsilon
M)^{2}+O(\varepsilon^{3}|\log\varepsilon|^{2}).$

Equivalently,

$\mu_{i}(\varepsilon)=\mu_{i}-(2\lvert\nabla\varphi_{i}(\tilde{w})\rvert^{2}-\mu_{i}\varphi_{i}(\tilde{w})^{2})|E|\varepsilon^{2}+\mathcal{O}(\varepsilon^{3}|\log\varepsilon|^{2})$

Hence, we have completed the proof of Theorem 2.1.

##  6 Conclusions

This paper’s main contribution is developing a framework to obtain asymptotic
formulas for eigenvalues in domains with star-shaped holes, which generalise
the spherical hole studied previously. Estimates and proofs are presented in
Sections 3-5. The proposed framework can be applied in various scientific and
engineering fields. For the future, the author proposes to construct
asymptotic formulas for eigenvalues in domains with holes in compact sets with
zero Lebesgue measure.

## Acknowledgments

The author would like to thank the referees for their valuable comments which
help to improve the manuscript.

A special thanks goes to Assistant Professor Bao Quoc Tang from the Department
for Mathematics and Scientific Computing at the University of Graz for his
invaluable insights and comments.

The paper is completed during the author’s visit to the University of Graz,
and its hospitality is greatly acknowledged.

This work is supported by grant SGS01/PřF/2024.

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