\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 134, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/134\hfil Asymptotic formula for detecting inclusions]
{Asymptotic formula for detecting inclusions via boundary measurements}

\author[Kh. Khelifi, M. Abdelwahed,  N. Chorfi,  M. Hassine \hfil EJDE-2018/134\hfilneg]
{Khalifa Khelifi, Mohamed Abdelwahed,\\
  Nejmeddine Chorfi, Maatoug Hassine}

\address{Khalifa Khelifi \newline
Department of Mathematics,
College of Sciences,
Monastir University,
Monastir, Tunisia}
\email{khalifakhelifi@hotmail.fr}

\address{Mohamed Abdelwahed (corresponding author) \newline
Department of Mathematics,
College of Sciences,
King Saud University,
Riyadh, Saudi Arabia}
\email{mabdelwahed@ksu.edu.sa}

\address{Nejmeddine Chorfi \newline
Department of Mathematics,
College of Sciences,
King Saud University,
Riyadh, Saudi Arabia}
\email{nchorfi@ksu.edu.sa}

\address{Maatoug Hassine \newline
Department of Mathematics,
College of Sciences,
Monastir University,
Monastir, Tunisia}
\email{Maatoug.Hassine@enit.rnu.tn}

\dedicatory{Communicated by Vicentiu D. Radulescu}

\thanks{Submitted September 9, 2017. Published June 28, 2018.}
\subjclass[2010]{35J15, 78M22}
\keywords{Laplace operator; asymptotic analysis; topological gradient;
\hfill\break\indent Kohn-Vogelius functional}

\begin{abstract}
 In this article, we are concerned with a geometric inverse
 problem related to the Laplace operator in a three-dimensional domain.
 The aim is to derive an asymptotic formula for detecting an inclusion
 via boundary measurement. The topological sensitivity method is applied
 to calculate a high-order topological asymptotic expansion of the semi-norm
 Kohn-Vogelius functional, when a Dirichlet perturbation is introduced
 in the initial domain.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

The detection of an object from boundary measurements is used in several
applications such as in fluid mechanics, electrical impedance tomography,
electromagnetic casting,
non-destructive testing \cite{AHM,JBOW,ACANJRF, MCDIJN}.

On the theoretical level,  these
applications correspond to  geometric inverse problems.
Among the methods to solve this type of problems, there exist a
 method based  on the
Kohn-Vogelius formulation and the topological sensitivity method
\cite{AK, SA,BHJM, ACANJRF,KEHH,Schumacher, G GP MM,PG1KS, SZ}.
The majority of works interested to this method are based on the
first-order asymptotic expansion of the Kohn-Vogelius functional
\cite{ADEK,ABB,JBGP,BCD, BHJM,ACANJRF,JC AG JM,DJCFSMV,AFMV, GH,MHKK}.
This method is sufficient in the case of small unknown object far 
from the boundary.

In general application case the size of the object to detect is finite.
For this reason, we consider high-order
terms in the asymptotic expansion of the Kohn-Vogelius functional
formula.

In this article we apply the topological sensitivity method and the
Kohn-Vogelius formulation, to derive a high-order asymptotic formula
connecting the known boundary data and the unknown inclusion
properties; its location , size and shape. More precisely in
this paper we derive a high-order topological asymptotic expansion
of the semi-norm Kohn-Vogelius functional associated to the Laplace
operator in three-dimensional domain, when a Dirichlet perturbation
is introduced in the initial domain.

The proposed approach permit to  calculate the topological gradient for
any order for the semi-norm Kohn-Vogelius functional.
We present a general approach applicable to various problems such as elasticity,
 Stokes equations, Navier-Stokes equation, Maxwell's equations, etc.

The remaining  of this paper are organized as follows. We
begin by presenting the inverse problem and the Kohn-Vogelius
formulation in section \ref{inv-prob-kohn-form}.
In section \ref{formul-prb} we present the topological sensitivity method.
In section \ref{asymp-sol}, we establish a some preliminary results,
where we derive an asymptotic formula describing the variation of
the solutions of Neumann and Dirichlet problems when a Dirichlet
perturbation is introduced in the initial domain. Section
\ref{asymp-expansion}  presents the main result of the paper.
Finally, section \ref{Proofs}  contains the proofs of the different results.
The paper ends by some concluding remarks.


\section{Inverse problem and the Kohn-Vogelius formulation}
\label{inv-prob-kohn-form}

The geometric inverse Laplace problem related to the
Laplace operator in three-dimensional domain is considered in this paper.
Let $\Omega\subset \mathbb{R}^3$ denote a bounded domain with smooth boundary
$\partial \Omega$
and satisfies $\partial \Omega=\Gamma_1\cup\Gamma_2$ with
$\Gamma_1\cap\Gamma_2=\emptyset$,
$\Gamma_2\neq\emptyset$.

We suppose that there exist a sub-domain $\mathcal{D}^*$ of $\Omega$ with a smooth
boundary $\partial\mathcal{D}^*$. The studied inverse problem can be formulated:

For regular given data $F,V$ and $\psi_m$, find the unknown domain
$\mathcal{D}^*$ such that  $\psi$ is solution of the following over determined  problem
\begin{gather*}
-\Delta \psi=F \quad \text{in }\Omega\backslash \mathcal{D}^*,\\
\nabla \psi\cdot \mathbf{n}=V \quad \text{on } \Gamma_1, \\
\psi=\psi_m \quad \text{on } \Gamma_1, \\
\psi=0 \quad \text{on } \Gamma_2,\\
\psi=0 \quad\text{on } \partial \mathcal{D}^*.
\end{gather*}
To derive an asymptotic formula connecting the boundary
measurements and the location of the unknown domain $\mathcal{D}^*$, we
propose in this work a new technique based on the Kohn-Vogelius
formulation and the topological sensitivity technique. The
Kohn-Vogelius formulation is a self regularization method which
transforms the geometric inverse problem to a shape optimization
problem. It leads to define  two problems for any given domain
$\mathcal{D}\subset \Omega$. The first one, named the Neumann problem, is associated with
 the Neumann datum $V$:
\begin{equation} \label{ePn}
\begin{gathered}
-\Delta  \psi_{n}  = F \quad \text{in } \Omega\backslash \overline{\mathcal{D}} \\
\nabla \psi_{n}\cdot \mathbf{n} =V \quad \text{on } \Gamma_1 \\
\psi_{n} =0 \quad\text{on } \Gamma_2 \\
\psi_{n} =0 \quad\text{on }\partial \mathcal{D}.
\end{gathered}
\end{equation}
The second one is associated to the measured $\psi_m$, which will be
named as the Dirichlet problem:
\begin{equation} \label{ePd}
\begin{gathered}
-\Delta \psi_{d}  =  F \quad\text{in } \Omega\backslash {\overline{\mathcal{D}}}\\
\psi_{d} = \psi_m \quad \text{on } \Gamma_1\\
\psi_{d} = 0 \quad\text{on } \Gamma_2 \\
\psi_{d} = 0 \quad\text{on } \partial \mathcal{D}.
\end{gathered}
\end{equation}
We remark that if the domains $\mathcal{D}$ and $\mathcal{D}^*$ coincide then
$\psi_{n} =\psi_{d}$.
 According to this observation,  Kohn and Vogelius \cite{RKMV} proposed
 to change the inverse problem to the minimization of a function measuring
the difference  between the Dirichlet and Neumann solutions.
  We define the Kohn-Vogelius  semi-norm function
$$
\mathcal{J}(\Omega \backslash \overline{\mathcal{D}}) =\int_{\Omega
\backslash \overline{\mathcal{D}}}\left|\nabla \psi_{n}-\nabla
\psi_{d}\right|^2d\mathbf{x},
$$
where $\psi_{n}$  (resp. $\psi_{d}$) is solution to the Neumann
 (resp. Dirichlet)  perturbed problem.

\section{Topological sensitivity method}\label{formul-prb}

To calculate a high-order topological asymptotic expansion
of the semi-norm Kohn-Vogelius functional $\mathcal{J}$, we apply
the topological sensitivity method.
 It consists in calculating the
variation of  $\mathcal{J}$
regarding to a small perturbation $B_{\mathbf{z},\epsilon}$ at the point
 $\mathbf{z}$ of the domain $\Omega$.
For $\mathbf{z}\in\Omega$ and $\epsilon>0$, we define
$B_{\mathbf{z},\epsilon}=\mathbf{z}+\epsilon B$, where  $B\subset\mathbb{R}^3$
is a bounded fixed  regular domain which contains the origin.
 We define the perturbed domain
$\Omega_{\mathbf{z},\epsilon}=\Omega\setminus\overline{B_{\mathbf{z},\epsilon}}$
Let us consider the following
overdetermined  boundary value problem
\begin{equation} \label{eq4}
\begin{gathered}
-\Delta \psi_{\epsilon}=F \quad\text{in } \Omega\backslash
\overline{B_{\mathbf{z},\epsilon}},\\
\nabla \psi_{\epsilon}\cdot \mathbf{n}=V \quad \text{on } \Gamma_1,\\
\psi_{\epsilon}=\psi_m \quad \text{on } \Gamma_1,\\
\psi_{\epsilon}=0 \quad\text{on } \Gamma_2,\\
\psi_{\epsilon}=0 \quad\text{on } \partial B_{\mathbf{z},\epsilon}.
\end{gathered}
\end{equation}
We assume here that there exists $B_{\mathbf{z}^*,\epsilon}=\mathbf{z}^*+\epsilon
B\subset\Omega$ such that there exists a solution to problem \eqref{eq4}.
Consequently,  the following geometric inverse problem is considered:
\begin{quote}
Find $B_{\mathbf{z},\epsilon}\subset\Omega$ such that the solution $\psi_{\epsilon}$
satisfies the overdetermined system \eqref{eq4}.
\end{quote}
 The Kohn-Vogelius functional for the perturbed domain is defined by
 $$
\mathcal{J}(\Omega_{\mathbf{z},\epsilon})
=\int_{\Omega_{\mathbf{z},\epsilon}}|\nabla \psi_{n,\epsilon}-\nabla
\psi_{d,\epsilon}|^2d\mathbf{x},
$$
where $\psi_{n,\epsilon}$ is the solution to the perturbed Neumann
problem
\begin{equation} \label{ePne}
\begin{gathered}
-\Delta  \psi_{n,\epsilon}  = F \quad\text{in }
\Omega\backslash {\overline{B_{\mathbf{z},\epsilon}}}, \\
\psi_{n,\epsilon} =0 \quad \text{on }\partial B_{\mathbf{z},\epsilon},\\
\nabla \psi_{n,\epsilon}{\bf n }=V \quad \text{on } \Gamma_1, \\
\psi_{n,\epsilon} =0 \quad\text{on } \Gamma_2.
\end{gathered}
\end{equation}
and $\psi_{d,\epsilon}$ is the solution to the perturbed Dirichlet
problem
\begin{equation} \label{ePde}
\begin{gathered}
-\Delta \psi_{d,\epsilon} =  F \quad \text{in }
 \Omega\backslash {\overline{B_{\mathbf{z},\epsilon}}},\\
\psi_{d,\epsilon} = 0 \quad\text{on } \partial B_{\mathbf{z},\epsilon},\\
\quad \psi_{d,\epsilon} = \psi_m \quad\text{on } \Gamma_1,\\
\psi_{d,\epsilon} =0 \quad\text{on } \Gamma_2.
\end{gathered}
\end{equation}
We remark that if $\epsilon=0$, $\Omega_{\mathbf{z},0}=\Omega$ and $\psi_0$ satisfies
\begin{gather*}
-\Delta \psi_{0}=F \quad\text{in } \Omega,\\
\nabla \psi_{0}\cdot \mathbf{n}=V \quad\text{on } \Gamma_1,\\
\psi_{0}=\psi_m \quad\text{on } \Gamma_1,\\
\psi_{0}=0 \quad\text{on } \Gamma_2,
\end{gather*}
$\psi_{n,0}$ is solution to
\begin{equation} \label{ePne2}
\begin{gathered}
-\Delta  \psi_{n,0} = F \quad\text{in } \Omega \\
\nabla \psi_{n,0}{\bf n } =V \quad\text{on } \Gamma_1 \\
\psi_{n,0} =0 \quad \text{on } \Gamma_2,
\end{gathered}
\end{equation}
and $\psi_{d,0}$ is solution to
\begin{equation} \label{epde2}
\begin{gathered}
-\Delta \psi_{d,0}  =  F \quad \text{in } \Omega\\
\quad \psi_{d,0} = \psi_m \quad\text{on } \Gamma_1\\
\psi_{d,0} =0 \quad\text{on } \Gamma_2.
\end{gathered}
\end{equation}
As mentioned in the introduction the majority of works interested to
this method are based on the first-order asymptotic expansions of
the functional $\mathcal{J}$ presented by
$$
\mathcal{J}(\Omega_{\mathbf{z},\epsilon})=\mathcal{J}(\Omega)+f(\epsilon)\delta
\mathcal{J}(\mathbf{z})+ o(f(\epsilon)),
$$
where $\delta \mathcal{J}$ is the topological
gradient and $f$ is a positive scalar function with
$\lim_{\epsilon\to 0} f(\epsilon)=0$.
Then, for small $\epsilon$ the solution of the minimization
problem
$$
\min_{B_{\mathbf{z},\epsilon}\subset \Omega}
\mathcal{J}(\Omega\setminus{\overline{\omega_{\mathbf{z},\epsilon}})},
$$
is given by $B_{{\mathbf{z}}^*,\epsilon}$, with $\mathbf{z}^*\in\Omega$ where
$\delta \mathcal{J}$ is the most negative. This is due to the fact that if
$\delta \mathcal{J}(\mathbf{z}^*)<\delta \mathcal{J}(\mathbf{z})$, we obtain
$\mathcal{J}(\Omega_{\mathbf{z}^*,\epsilon})<\mathcal{J}(\Omega_{\mathbf{z},\epsilon})$.
The purpose of this work is to obtain  an asymptotic expansion of higher
order for the Kohn-Vogelius functional $\mathcal{J}$
to detect an object of finite size and
valid when the  topological gradient $\delta \mathcal{J}$ vanishes at some
critical points inside $\Omega$, under the form:
$$
\mathcal{J}(\Omega_{\mathbf{z},\epsilon})
=\mathcal{J}(\Omega)+\sum_{i=1}^{I} f_i(\epsilon)\delta^i \mathcal{J}(\mathbf{z})+ o(f_I(\epsilon)),
$$
where
$  f_i$, $1\leq i\leq I$ are  scalar positives functions verifying
$f_{i+1}(\epsilon)= o(f_i(\epsilon))$ and vanish with $\epsilon$.
$\delta^i \mathcal{J}$ is the $i^{th}$ topological derivative of the Kohn-Vogelius
functional $\mathcal{J}$.

To derive the expected expansion, we  establish in the
next section some preliminary results. The main results of this
analysis will be presented in section \ref{asymp-expansion}.

\section{Some preliminary results}\label{asymp-sol}

The aim of this section is to present an asymptotic formula
describing the variation of the solutions $\psi_{n,\epsilon}$ and
$\psi_{d,\epsilon}$ caused by the perturbation of $\Omega$ by
$B_{\mathbf{z},\epsilon}$.

In conductivity imperfections identification context, an
asymptotic expansion describing the variation of the solutions for
$I=1$ was derived in \cite{DJCFSMV,AFMV} for the Laplace equation.
 Another application was studied using Stokes system
 \cite{MH MA 2} for the detection of obstacles in a flow via the asymptotic
expansion of the velocity filed.

In this work, to derive the desired formula, we need to find an
asymptotic expansion of the exterior problem solution for the
Laplace equation defined in $\mathbb{R}^3\setminus\overline{B}$. Let
$\Phi\in H^{1/2}(\partial B)$, denoting by $H$ the solution to
\begin{gather*}
-\Delta H =0 \quad\text{in } \mathbb{R}^3\setminus\overline{B},\\
H   \to 0 \quad\text{at } \infty,\\
H=\Phi \quad\text{on } \partial B,
\end{gather*}
Resorting to the simple layer potential representation \cite{DL,RR},
$H$ can be written as
\begin{equation} \label{rep-integ}
H(\mathbf{y})=\int_{\partial \omega}E(\mathbf{y}-t)\,q(t)ds(t), \quad
\forall \mathbf{y}\in \mathbb{R}^3\setminus\overline{B},
\end{equation}
where $E$ is the Laplace equation fundamental solution  in $\mathbb{R}^3$:
$$
E(\mathbf{y})= \frac{1}{4\pi \|\mathbf{y}\|},
$$
and $q$ is the boundary integral equation unique solution
\begin{equation} \label{eqint1}
\int_{\partial B}E(\mathbf{y}-t)q(t)ds(t)=\Phi(\mathbf{y}),\quad \forall \mathbf{y}\in
\partial B.
\end{equation}
By the change of variable $\mathbf{x}=\mathbf{z}+\epsilon \mathbf{y}$
and using the perturbation
$B_{\mathbf{z},\epsilon}$ is not close to the boundary $\partial\Omega$, we have
$$
H((\mathbf{x}-\mathbf{z})/\epsilon)=\epsilon\int_{\partial B}E(\mathbf{x}
-\mathbf{z}-\epsilon\,t)\,q(t)ds(t), \quad \forall \mathbf{x}\in
 \mathbb{R}^3\setminus\overline{B_{\mathbf{z},\epsilon}}.
$$
Denoting by $\varphi_{\mathbf{x}-\mathbf{z},t}$ the function
$$
\varphi_{\mathbf{x}-\mathbf{z},t}:\epsilon \longmapsto
\varphi_{\mathbf{x}-\mathbf{z},t}(\epsilon)
=\epsilon E((\mathbf{x}-\mathbf{z})-\epsilon t), \quad \forall \epsilon>0.
$$
Using the fact that $\varphi_{\mathbf{x}-\mathbf{z},t}$ is smooth regarding
$\epsilon$ and satisfies the following behavior
$$
\varphi_{\mathbf{x}-\mathbf{z},t}(\epsilon)
=\sum_{p=1}^{I}\frac{\epsilon^p}{p!}\varphi^{(p)}_{\mathbf{x}-\mathbf{z},t}(0)+
O(\epsilon^{I+1}),
$$
where $\varphi^{(p)}_{\mathbf{x}-\mathbf{z},t}(0)$ is the
$p^{\text{th}}$ derivative of $\varphi_{\mathbf{x}-\mathbf{z},t}$ at
 $\epsilon=0$.
Then the following lemma gives an asymptotic expansion of the
function
$$
\mathbf{x}\mapsto H(\frac{\mathbf{x}-\mathbf{z}}{\epsilon}).
$$

\begin{lemma}\label{asyp-ext}
For any $I\geq0$, we have
\begin{equation*}
H((\mathbf{x}-\mathbf{z})/\epsilon)
=\sum_{p=1}^{I}\epsilon^p\,H^{(p)}(\mathbf{x}-\mathbf{z})+O(\epsilon^{I+1}),\quad
\forall \mathbf{x}\in \mathbb{R}^3\setminus\overline{B_{\mathbf{z},\epsilon}},
\end{equation*}
where $H^{(p)}$ is the smooth function defined by
$$
H^{(p)}(\mathbf{x}-\mathbf{z})= \frac{1}{p!}\int_{\partial
B}\varphi^{(p)}_{\mathbf{x}-\mathbf{z},t}(0)q(t)ds(t).
$$
\end{lemma}

\begin{remark} \rm
The first-order asymptotic expansion of the exterior problem
solution for the Laplace equation is proved by  Guilaume and
Sid Idris  \cite[page 1049]{PG1KS}.
\end{remark}

\subsection{Asymptotic formula of the Neumann problem solution}

To present an asymptotic formula describing the variation
of the Neumann Problem solution $\psi_{n,\epsilon}$, we define the
sequences functions $(\Psi_{n,i})_{0\leq i\leq I}$ and
$(W_{n,i})_{0\leq i\leq I}$, where for all $0\leq i\leq I$,
$\Psi_{n,i}$ are smooth function defined in the initial domain
$\Omega$, obtained as the solution to a interior problem with
Neumann boundary condition on $\Gamma_1$ and $W_{n,i}$ are smooth
function defined in $\mathbb{R}^3\setminus\overline{B}$,
obtained as the solution to a exterior problems. More precisely:
\smallskip

\noindent\textbf{For $i=0$}: $\Psi_{n,0}=\psi_{n,0}$ and $W_{n,0}$ is the
solution to
\begin{equation} \label{wn0}
\begin{gathered}
-\Delta W_{n,0} =0 \quad\text{ in } \mathbb{R}^3\setminus\overline{B},\\
W_{n,0}  \to  0 \quad \text{at } \infty\\
W_{n,0}  =-\psi_{n,0}(\mathbf{z}) \quad\text{on } \partial B.
\end{gathered}
\end{equation}
\smallskip

\noindent\textbf{For $i=1$}: $\Psi_{n,1}$ is the solution to
\begin{equation} \label{e5}
\begin{gathered}
-\Delta \Psi_{n,1} =0 \quad \text{in } \Omega,\\
\nabla \Psi_{n,1}\cdot \mathbf{n}
 =-\nabla W_{n,0}^{(1)}(\mathbf{x}-\mathbf{z})\cdot \mathbf{n} \quad\text{on }
 \Gamma_1,\\
\Psi_{n,1} =-W_{n,0}^{(1)}(\mathbf{x}-\mathbf{z}) \quad\text{on } \Gamma_2,
\end{gathered}
\end{equation}
with $W_{n,0}^{(1)}$ is defined by Lemma \ref{asyp-ext} in the
particular case $i=0$, $\phi=-\psi_{n,0}(\mathbf{z})$ and $p=1$.

The function $W_{n,1}$ depends on $\Psi_{n,0}$ and $\Psi_{n,1}$, it
 is solution of  the  exterior problem
\begin{equation} \label{eqq6}
\begin{gathered}
-\Delta W_{n,1} =0 \quad\text{in } \mathbb{R}^3\setminus\overline{B},\\
W_{n,1} \to  0 \quad \text{at }  \infty\\
W_{n,1}  =-\Psi_{n,1}(\mathbf{z})-D\Psi_{n,0}(\mathbf{z})(\mathbf{y}) \quad
\text{on } \partial B.
\end{gathered}
\end{equation}
\smallskip

\noindent\textbf{For $1\leq i\leq I$}: The function $\Psi_{n,i}$ depends on $W_{n,j}$
for $0\leq j \leq i-1$
and is solution of the  interior problem
\begin{equation} \label{e7}
\begin{gathered}
-\Delta \Psi_{n,i} =0 \quad\text{in } \Omega,\\
\nabla \Psi_{n,i}\cdot \mathbf{n} =-\sum_{p=1}^{i}\nabla W_{n,i-p}^{(p)}
 (\mathbf{x}-\mathbf{z})\cdot \mathbf{n} \quad\text{on } \Gamma_1,\\
\Psi_{n,i} = -\sum_{p=1}^{i}W_{n,i-p}^{(p)}(\mathbf{x}-\mathbf{z})\quad \text{on }
\Gamma_2,
\end{gathered}
\end{equation}
with $W_{n,j}^{(p)}$ is defined by Lemma \ref{asyp-ext}.

The function $W_{n,i}$ depends on $\Psi_{n,j}$ for $0\leq j \leq i$ and
is solution of the  exterior problem
\begin{equation} %\label{eqq6}
\begin{gathered}
-\Delta W_{n,i} =0 \quad\text{in } \mathbb{R}^3\setminus\overline{B},\\
W_{n,i}  \to  0\quad \text{at }  \infty\\
W_{n,i}    =-\Psi_{n,i}(\mathbf{z})-
\sum_{p=1}^{i}\frac{1}{p!}D^p{\Psi}_{n,i-p}(\mathbf{z})(\mathbf{y}^p) \quad \text{on }
\partial B,
\end{gathered}
\end{equation}
where $D^p{\Psi}_{n,i-p}(\mathbf{z})$ is the $p^{\text{th}}$ derivative of
 ${\Psi}_{n,i-p}$ (the harmonic function) at  $\mathbf{z}\in\Omega$ and
$\mathbf{y}^p=(\mathbf{y},\dots,\mathbf{y})\in (\mathbb{R}^3)^p$.


We are now ready to present  an asymptotic formula describing the
variation of the solution $\psi_{n,\epsilon}$ raised from the perturbation
of  $\Omega$ by $B_{\mathbf{z},\epsilon}$.$\Omega$.

\begin{theorem}\label{TN0}
In the perturbed domain $\Omega_{\mathbf{z},\epsilon}$, the solution $\psi^{\epsilon}_{n}$
 of the Neumann Laplace equation  has the  asymptotic expansion
$$
\psi_{n,\epsilon} (\mathbf{x})=  \sum_{i=0}^{I} \epsilon^i [\Psi_{n,i}
(\mathbf{x})+ W_{n,i} (\frac{\mathbf{x}-\mathbf{z}}{\epsilon}) ]
+ O(\epsilon^{I+1}) \quad\text{in } \Omega_{\mathbf{z},\epsilon}.
$$
 \end{theorem}

\subsection{Asymptotic formula of the Dirichlet problem solution}

Similarly to the asymptotic of the Neumann solution, to present an
asymptotic formula describing the variation of the Dirichlet Problem
solution $\psi_{d,\epsilon}$, we define the sequences functions
$(\Psi_{d,i})_{0\leq i\leq I}$ and $(W_{d,i})_{0\leq i\leq I}$,
where for all $0\leq i\leq I$, $\Psi_{d,i}$ are smooth function
defined in the initial domain $\Omega$, obtained as the solution to
a interior problem with Dirichlet boundary condition on $\Gamma_1$
and $W_{d,i}$ are smooth function defined in
$\mathbb{R}^3\setminus\overline{B}$, obtained as the solution to a
exterior problems. More precisely, the sequences functions
$(\Psi_{d,i})_{0\leq i\leq I}$
and $(W_{d,i})_{0\leq i\leq I}$, are defined as follow:
\smallskip

\noindent\textbf{For $i=0$}: $\Psi_{d,0}=\psi_{d,0}$ and $W_{d,0}$ is the
solution to
\begin{equation} \label{w{d,0}}
\begin{gathered}
-\Delta W_{d,0} =0 \quad\text{in } \mathbb{R}^3\setminus\overline{B},\\
W_{d,0}  \to  0\quad \text{at } \infty\\
W_{d,0} =-\psi_{d,0}(\mathbf{z}) \quad\text{on } \partial B.
\end{gathered}
\end{equation}
\smallskip

\noindent\textbf{For $i=1$}: $\Psi_{d,1}$ is the solution to
\begin{equation} \label{e10}
\begin{gathered}
-\Delta \Psi_{d,1} =0 \quad\text{in } \Omega,\\
 \Psi_{d,1} =- W_{d,0}^{(1)}(\mathbf{x}-\mathbf{z}) \quad\text{on } \partial\Omega,
\end{gathered}
\end{equation}
with $W_{d,0}^{(1)}$ is defined by Lemma \ref{asyp-ext} in the
particular case $i=0$, $\phi=-\psi_{d,0}(\mathbf{z})$ and $p=1$.

The function $W_{d,1}$ depends on $\Psi_{d,0}$ and $\Psi_{d,1}$, and
is solution of the  exterior problem
\begin{equation} %\label{eqq6}
\begin{gathered}
-\Delta W_{d,1} =0 \quad\text{in } \mathbb{R}^3\setminus\overline{B},\\
W_{d,1}  \to  0\quad \text{at }  \infty\\
W_{d,1}  =-\Psi_{d,1}(\mathbf{z})-D\Psi_{d,0}(\mathbf{z})(\mathbf{y}) \quad
\text{on } \partial B.
\end{gathered}
\end{equation}
\smallskip

\noindent\textbf{For $1\leq i\leq I$}:
The function $\Psi_{d,i}$ depends on $W_{d,j}$ for $0\leq j \leq i-1$
and is solution of the interior problem
\begin{equation} \label{e12}
\begin{gathered}
-\Delta \Psi_{d,i} =0 \quad\text{in } \Omega,\\
\Psi_{d,i} = -\sum_{p=1}^{i}W_{d,i-p}^{(p)}(\mathbf{x}-\mathbf{z})\quad
\text{on } \partial\Omega,
\end{gathered}
\end{equation}
with $W_{d,j}^{(p)}$  defined in Lemma \ref{asyp-ext}.

The function $W_{d,i}$ depends on $\Psi_{d,j}$ for $0\leq j \leq i$ and
is solution of the  exterior problem
\begin{equation} %\label{eqq6}
\begin{gathered}
-\Delta W_{d,i} =0 \quad\text{in } \mathbb{R}^3\setminus\overline{B},\\
W_{d,i}  \to  0\quad \text{at }  \infty\\
W_{d,i} =-\Psi_{d,i}(\mathbf{z})-
\sum_{p=1}^{i}\frac{1}{p!}D^p{\Psi}_{d,i-p}(\mathbf{z})(\mathbf{y}^p) \quad
\text{on }\partial B.
\end{gathered}
\end{equation}

We are now ready to present  an asymptotic formula giving the
variation of  $\psi_{d,\epsilon}$ raised from the perturbation
of  $\Omega$ by $B_{\mathbf{z},\epsilon}$.

\begin{theorem}\label{TD0}
In the perturbed domain $\Omega_{\mathbf{z},\epsilon}$, the solution
 $\psi_{d,\epsilon}$ of the Dirichlet Laplace equation has the
asymptotic expansion
$$
\psi_{d,\epsilon} (\mathbf{x})=  \sum_{i=0}^{I} \epsilon^i [\Psi_{d,i}
(\mathbf{x})+ W_{d,i} (\frac{\mathbf{x}-\mathbf{z}}{\epsilon}) ]
+ O(\epsilon^{I+1}) \quad\text{in } \Omega_{\mathbf{z},\epsilon}.
$$
\end{theorem}

\section{Asymptotic formula}\label{asymp-expansion}

The main result is presented in this section. A high-order
topological asymptotic expansion is derived for the semi-norm
Kohn-Vogelius functional $\mathcal{J}$, when a Dirichlet
perturbation is introduced in the initial domain.
The functional $\mathcal{J}$ can be decomposed as
\begin{align*}
\mathcal{J} (\Omega_{\mathbf{z},\epsilon})
&= \int_{\Omega_{\mathbf{z},\epsilon}} | \nabla
\psi_{d,\epsilon} |^2 \text{dx}
+ \int_{\Omega_{\mathbf{z},\epsilon}}  | \nabla \psi_{n,\epsilon} |^2 \text{dx}
-2 \int_{\Omega_{\mathbf{z},\epsilon}} \nabla \psi_{d,\epsilon}.
 \nabla \psi_{n,\epsilon} \text{dx}\\
&=\mathcal{J}_d (\Omega_{\mathbf{z},\epsilon})+\mathcal{J}_n (\Omega_{\mathbf{z},\epsilon})+\mathcal{J}_{dn}
 (\Omega_{\mathbf{z},\epsilon}),
\end{align*}
where
\begin{gather*}
\mathcal{J}_d (\Omega_{\mathbf{z},\epsilon})=\int_{\Omega_{\mathbf{z},\epsilon}} | \nabla
\psi_{d,\epsilon}|^2 \text{dx},\\
\mathcal{J}_n (\Omega_{\mathbf{z},\epsilon})=\int_{\Omega_{\mathbf{z},\epsilon}} |
\nabla \psi_{n,\epsilon} |^2 \text{dx},\\
\mathcal{J}_{d,n} (\Omega_{\mathbf{z},\epsilon})=-2 \int_{\Omega_{\mathbf{z},\epsilon}} \nabla
\psi_{d,\epsilon} . \nabla \psi_{n,\epsilon} \text{dx}.
\end{gather*}
The Dirichlet term $\mathcal{J}_d$ has the following variation
\begin{align*}
\mathcal{J}_d (\Omega_{\mathbf{z},\epsilon})-\mathcal{J}_d (\Omega)
&=\int_{\Omega_{\mathbf{z},\epsilon}} |
\nabla \psi_{d,\epsilon} |^2 \text{dx}-\int_{\Omega} | \nabla \psi_{d,0}
|^2 \text{dx}\\
&=\int_{\Omega_{\mathbf{z},\epsilon}}\nabla
(\psi_{d,\epsilon}+\psi_{d,0})\cdot\nabla
(\psi_{d,\epsilon}-\psi_{d,0})\text{dx}
-\int_{B_{\mathbf{z},\epsilon}} | \nabla \psi_{d,0} |^2 \text{dx}.
\end{align*}
By the Green formula, from the problems \eqref{ePde} and \eqref{ePde}
with $\epsilon=0$, we deduce
\begin{align*}
&\int_{\Omega_{\mathbf{z},\epsilon}}\nabla (\psi_{d,\epsilon}+u_{d,0})\cdot\nabla
(\psi_{d,\epsilon}-\psi_{d,0})\text{dx}\\
&=-\int_{\partial B_{\mathbf{z},\epsilon}}\nabla (\psi_{d,\epsilon}+u_{d,0})
\cdot \mathbf{n} \psi_{d,0}\text{ds}
+2\int_{\Omega_{\mathbf{z},\epsilon}}
F(\psi_{d,\epsilon}-\psi_{d,0})\text{dx}.
\end{align*}
From  problem \eqref{ePde}
with $\epsilon=0$ we derive
\begin{equation}\label{gren-int}
\int_{B_{\mathbf{z},\epsilon}} | \nabla \psi_{d,0} |^2 \text{dx}
=-\int_{\partial B_{\mathbf{z},\epsilon}}\nabla \psi_{d,0} \cdot \mathbf{n}
\psi_{d,0}\text{ ds}
+\int_{B_{\mathbf{z},\epsilon}} F\,\psi_{d,0}\text{ dx },
\end{equation}
then, we obtain
\begin{align*}
&\mathcal{J}_d (\Omega_{\mathbf{z},\epsilon})-\mathcal{J}_d(\Omega) \\
&=-\int_{\partial B_{\mathbf{z},\epsilon}}\nabla \psi_{d,\epsilon}\cdot \mathbf{n}
\psi_{d,0}\text{ ds }
 +2\int_{\Omega_{\mathbf{z},\epsilon}} F\,(\psi_{d,\epsilon}-\psi_{d,0})\text{ dx }
 -\int_{B_{\mathbf{z},\epsilon}} F\psi_{d,0}\text{ dx }\\
&=-\int_{\partial B_{\mathbf{z},\epsilon}}\nabla
(\psi_{d,\epsilon}-\psi_{d,0})\cdot \mathbf{n} \psi_{d,0}\text{ ds}
-\int_{\partial B_{\mathbf{z},\epsilon}}\nabla
\psi_{d,0}\cdot \mathbf{n} \psi_{d,0}\text{ ds }\\
&\quad +2\int_{\Omega_{\mathbf{z},\epsilon}}
F(\psi_{d,\epsilon}-\psi_{d,0})\text{ dx }-\int_{B_{\mathbf{z},\epsilon}}
F\psi_{d,0}\text{ dx }.
\end{align*}
Then, from \eqref{gren-int} it follows that
\begin{align*}
\mathcal{J}_d (\Omega_{\mathbf{z},\epsilon})-\mathcal{J}_d (\Omega)
&= - \int_{\partial B_{\mathbf{z},\epsilon}} (\nabla \psi_{d,\epsilon}
 -\nabla \psi_{d,0}).{\mathbf{n}} \psi_{d,0} \text{ ds }
 + \int_{B_{\mathbf{z},\epsilon}}|\nabla \psi_{d,0} |^2  \text{ dx} \\
&\quad + 2 \int_{\Omega_{\mathbf{z},\epsilon}}
F\,(\psi_{d,\epsilon}-\psi_{d,0})\text{ dx }- 2\int_{ B_{\mathbf{z},\epsilon}}
F\, \psi_{d,0} \text{ dx }.
\end{align*}
Similarly, the Neumann term $\mathcal{J}_n$ has the  variation
\begin{align*}
\mathcal{J}_n (\Omega_{\mathbf{z},\epsilon})-\mathcal{J}_n (\Omega)
&=\int_{\Omega_{\mathbf{z},\epsilon}}
|\nabla \psi_{n,\epsilon}|^2 \text{dx} -\int_{\Omega}
|\nabla \psi_{n,0} |^2  \text{dx}\\
&= \int_{\partial B_{\mathbf{z},\epsilon}} (\nabla \psi_{n,\epsilon}
-\nabla \psi_{n,0}).{\mathbf{n}} \,\psi_{n,0} \text{ ds } -
\int_{B_{\mathbf{z},\epsilon}} |\nabla \psi_{n,0} |^2  \text{ dx}.
\end{align*}
The Dirichlet/Neumann term $\mathcal{J}_{d,n}$ has the  variation
\begin{align*}
\mathcal{J}_{d,n} (\Omega_{\mathbf{z},\epsilon})-\mathcal{J}_{d,n}(\Omega)
&=\int_{\Omega_{\mathbf{z},\epsilon}}  \nabla
\psi_{d,\epsilon}.\nabla \psi_{n,\epsilon}  \text{ dx }
-\int_{\Omega} \nabla \psi_{d,0} . \nabla \psi_{n,0}
\text{ dx}\\
&= \int_{\Omega_{\mathbf{z},\epsilon}}  F (\psi_{d,\epsilon} -
\psi_{d,0}) \text{ dx }- \int_{B_{\mathbf{z},\epsilon}} F\,
\psi_{d,0} \text{ dx }.
\end{align*}
Then the functional $\mathcal{J}$ has the  variation
\begin{align*}
\mathcal{J}(\Omega_{\mathbf{z},\epsilon})-\mathcal{J}(\Omega)
&=\int_{B_{\mathbf{z},\epsilon}}|\nabla \psi_{d,0} |^2d\mathbf{x} -
\int_{B_{\mathbf{z},\epsilon}}|\nabla \psi_{n,0} |^2d\mathbf{x}  \\
&\quad - \int_{\partial B_{\mathbf{z},\epsilon}}(\nabla \psi_{d,\epsilon}
-\nabla \psi_{d,0}).{\mathbf{n}} \,\psi_{d,0}ds \\
&\quad + \int_{\partial
B_{\mathbf{z},\epsilon}}(\nabla \psi_{n,\epsilon} -\nabla
\psi_{n,0}).{\mathbf{n}} \psi_{n,0}ds  .
\end{align*}
From Theorem \ref{TN0}, we have
\begin{align*}
&\int_{\partial B_{\mathbf{z},\epsilon}}(\nabla \psi_{n,\epsilon}
-\nabla \psi_{n,0}).{\mathbf{n}}\psi_{n,0}ds \\
&=\sum_{i=1}^{I} \epsilon^i
\int_{\partial B_{\mathbf{z},\epsilon}}\nabla \Psi_{n,i}(\mathbf{x}).
\mathbf{n}(\mathbf{x})\,\psi_{n,0}(\mathbf{x})ds \\
&\quad + \sum_{i=0}^{I} \epsilon^i \int_{\partial
B_{\mathbf{z},\epsilon}}\nabla_{\mathbf{x}} W_{n,i}
 ((\mathbf{x}-\mathbf{z})/\epsilon))\cdot \mathbf{n}\,\psi_{n,0}ds + O(\epsilon^{I+1}),
\end{align*}
and using Theorem \ref{TD0}, we have
\begin{align*}
&\int_{\partial B_{\mathbf{z},\epsilon}}(\nabla \psi_{d,\epsilon}
-\nabla \psi_{d,0}).{\mathbf{n}} \,\psi_{d,0}ds\\
&= \sum_{i=1}^{I} \epsilon^i
\int_{\partial B_{\mathbf{z},\epsilon}}\nabla \Psi_{d,i}(\mathbf{x}).
\mathbf{n}(\mathbf{x})\,\psi_{d,0}(\mathbf{x})ds \\
&\quad +   \sum_{i=0}^{I} \epsilon^i \int_{\partial
B_{\mathbf{z},\epsilon}}\nabla_{\mathbf{x}} W_{d,i}
 (\frac{\mathbf{x}-\mathbf{z}}{\epsilon})\cdot
\mathbf{n}\psi_{d,0}ds + O(\epsilon^{I+1}).
\end{align*}
Consequently, the functional $\mathcal{J}$ has the following variation
\begin{equation}\label{varJ}
\begin{aligned}
\mathcal{J}(\Omega_{\mathbf{z},\epsilon})-\mathcal{J}(\Omega)
&=\sum_{i=0}^{I} \epsilon^i\int_{\partial
B_{\mathbf{z},\epsilon}} \nabla_{\mathbf{x}} W_{n,i} 
 (\frac{\mathbf{x}-\mathbf{z}}{\epsilon})\cdot \mathbf{n}\,\psi_{n,0}ds \\
&\quad -\sum_{i=0}^{I} \epsilon^i\int_{\partial
 B_{\mathbf{z},\epsilon}} \nabla_{\mathbf{x}} W_{d,i}
 (\frac{\mathbf{x}-\mathbf{z}}{\epsilon})\cdot \mathbf{n}\,\psi_{d,0}ds\\
&\quad +\sum_{i=1}^{I} \epsilon^i \int_{\partial B_{\mathbf{z},\epsilon}}
\nabla \Psi_{n,i}(\mathbf{x})\cdot \mathbf{n}(\mathbf{x})\,\psi_{n,0}(\mathbf{x})ds\\
&\quad -\sum_{i=1}^{I} \epsilon^i \int_{\partial
B_{\mathbf{z},\epsilon}}\nabla \Psi_{d,i}(\mathbf{x})\cdot
\mathbf{n}(\mathbf{x})\,\psi_{d,0}(\mathbf{x})ds    \\
&\quad+\int_{B_{\mathbf{z},\epsilon}}|\nabla \psi_{d,0} |^2d\mathbf{x} -
\int_{B_{\mathbf{z},\epsilon}}|\nabla \psi_{n,0} |^2d\mathbf{x}+O(\epsilon^{I+1}).
\end{aligned}
\end{equation}
To present the desired asymptotic expansion of the
Kohn-Vogelius functional $\mathcal{J}$, for all $\mathbf{z}\in\Omega$ we consider the
following notation:
\begin{gather*}
\mathcal{T}_{n,1}^{i}(\mathbf{z})=\sum_{p=0}^{i}\frac{1}{p!}
\int_{\partial B}\hspace{-0.25cm}\nabla_\mathbf{y} W_{n,i-p}(\mathbf{y})\cdot
\mathbf{n}(\mathbf{y})[\nabla^{(p)} \psi_{n,0} (\mathbf{z})(\mathbf{y}^{p})]ds
(\mathbf{y}),
\\
\mathcal{T}_{d,1}^{i}(\mathbf{z})
= -\sum_{p=0}^{i}\frac{1}{p!} \int_{\partial
B}\hspace{-0.25cm}\nabla_\mathbf{y} W_{d,i-p}(\mathbf{y})\cdot
\mathbf{n}(\mathbf{y})[\nabla^{(p)} \psi_{d,0} 
(\mathbf{z})(\mathbf{y}^{p})]ds(\mathbf{y}),
\\
\begin{aligned}
\mathcal{T}_{n,2}^{i}(\mathbf{z})
&= \sum_{p=0}^{i} \sum_{q=0}^{p}\frac{1}{q!(p-q)!} \int_{\partial
B} [ \nabla^{(q+1)} \Psi_{n,i-p+1}
(\mathbf{z})(\mathbf{y}^q)] \cdot \mathbf{n}(\mathbf{y}) \\
&\quad\times [\nabla^{(p-q)} \psi_{n,0} (\mathbf{z})(\mathbf{y}^{p-q})]ds(\mathbf{y}),
\end{aligned}\\
\begin{aligned}
\mathcal{T}_{d,2}^{i}(\mathbf{z})
&= -\sum_{p=0}^{i} \sum_{q=0}^{p}\frac{1}{q!(p-q)!} \int_{\partial
B} [ \nabla^{(q+1)} \Psi_{d,i-p+1}
(\mathbf{z})(\mathbf{y}^q)] \cdot \mathbf{n}(\mathbf{y}) \\
&\quad\times [\nabla^{(p-q)} \psi_{d,0} 
(\mathbf{z})(\mathbf{y}^{p-q})]ds(\mathbf{y}),
\end{aligned}\\
\mathcal{T}_{d,3}^{i}(\mathbf{z})
= \sum_{p=0}^{i}\frac{1}{p!(i-p)!} \int_{B} \nabla^{(p+1)} \psi_{d,0}
(\mathbf{z})(\mathbf{y}^p)\cdot \nabla^{(i-p+1)}
\psi_{d,0} (\mathbf{z})(\mathbf{y}^{i-p})d\mathbf{y},
\\
\mathcal{T}_{n,3}^{i}(\mathbf{z})
=-\sum_{p=0}^{i}\frac{1}{p!(i-p)!} \int_{B} \nabla^{(p+1)} \psi_{n,0}
(\mathbf{z})(\mathbf{y}^p)\cdot \nabla^{(i-p+1)} \psi_{n,0} 
(\mathbf{z})(\mathbf{y}^{i-p})d\mathbf{y}.
\end{gather*}
We can know derive the topological asymptotic expansion of
the Kohn-Vogelius cost functional $\mathcal{J}$ by giving
the variation $\mathcal{J}(\Omega_{\mathbf{z},\epsilon})-\mathcal{J}(\Omega)$ regarding 
to the geometric perturbation  of the domain at any
point. The main result is described by the following
theorem.

\begin{theorem}\label{T}
The topological asymptotic expansion of the Kohn-Vogelius functional 
$\mathcal{J}$ is given by
$$
\mathcal{J}(\Omega_{\mathbf{z},\epsilon})-\mathcal{J}(\Omega)
=\sum_{i=1}^{I}\epsilon^i\delta^i \mathcal{J}(\mathbf{z})+O(\epsilon^{I+1}),
$$
where 
\begin{equation*} 
\delta^i \mathcal{J}(\mathbf{z})=
\begin{cases}
\mathcal{T}_{n,1}^{i-1}(\mathbf{z})- T_{d,1}^{i-1}(\mathbf{z})
 &\text{if } i\leq2,\\[4pt]
\mathcal{T}_{n,1}^{i-1}(\mathbf{z})-T_{d,1}^{i-1}(\mathbf{z})
+T_{n,2}^{i-3}(\mathbf{z})-T_{d,2}^{i-3}+T_{n,3}^{i-3}-T_{d,3}^{i-3}(\mathbf{z})
&\text{if } 3\leq i\leq I.
\end{cases}
\end{equation*}
\end{theorem}

\section{Proofs}\label{Proofs}

The aim of this section is to prove Theorems \ref{TN0} and \ref{TD0},
and the main result described in Theorem \ref{T}.

\subsection*{Proofs of Theorems \ref{TN0} and \ref{TD0}}
To prove Theorem \ref{TN0}, 
 in $\Omega_{\mathbf{z},\epsilon}$ we define the function
\begin{align*}
R^{\epsilon}_{n,I}
&=\Psi_{n,0} (\mathbf{x})+ W_{n,0}
(\frac{\mathbf{x}-\mathbf{z}}{\epsilon})+\epsilon\,(\Psi_{n,1} (\mathbf{x})
 + W_{n,1} (\frac{\mathbf{x}-\mathbf{z}}{\epsilon}) \\
&\quad + \dots + \epsilon^N(\Psi_{n,I} (\mathbf{x})+ W_{n,I} 
(\frac{\mathbf{x}-\mathbf{z}}{\epsilon})-\psi^{\epsilon}_n (\mathbf{x}).
\end{align*}
We can easily show that  $R^{\epsilon}_{n,I}$ is harmonic in
$\Omega_{\mathbf{z},\epsilon}$.

On $\partial B_{\mathbf{z},\epsilon}$ we have
\begin{equation}
\begin{aligned}
R^{\epsilon}_{n,I}(\mathbf{x})
&= \Psi_{n,0} (\mathbf{x})+ W_{n,0} (\frac{\mathbf{x}-\mathbf{z}}{\epsilon}) 
 +\sum_{i=1}^{I}\epsilon^i [\Psi_{n,i} (\mathbf{x})
 + W_{n,i} (\frac{\mathbf{x}-\mathbf{z}}{\epsilon})]\\
&= \sum_{i=0}^{I}\epsilon^i \Psi_{n,i} (\mathbf{x}) 
 - \sum_{i=0}^{I} \epsilon^i\Big[ \sum_{p=0}^{i}
 \frac{1}{p!}D^p{\Psi}_{n,i-p}(\mathbf{z})((\frac{\mathbf{x}
 -\mathbf{z}}{\epsilon})^p)\Big].
\end{aligned}
\end{equation}
From the multilinearity of $D^p{\Psi}_{n,i-p}(\mathbf{z})$, it follows that
\begin{align*}
\sum_{i=1}^{I} \epsilon^i\Big[
\sum_{p=0}^{i}\frac{1}{p!}D^p{\Psi}_{n,i-p}(\mathbf{z})
 ((\frac{\mathbf{x}-\mathbf{z}}{\epsilon})^p)\Big]
&= \sum_{i=0}^{I}\sum_{p=0}^{i}\frac{\epsilon^{i-p}}{p!}D^p{\Psi}_{n,i-p}
 (\mathbf{z})((\mathbf{x}-\mathbf{z})^p)\\
&= \sum_{i=0}^{I}\epsilon^i
\sum_{p=0}^{N-i}\frac{1}{p!}D^p{\Psi}_{n,i}(\mathbf{z})((\mathbf{x}-\mathbf{z})^p).
\end{align*}
Then, one can deduce
\begin{equation}
 R^{\epsilon}_{n,I}=\sum_{i=0}^{I}\epsilon^i
\Big[\Psi_{n,i}(\mathbf{x})-\sum_{p=0}^{I-i}\frac{1}{p!}D^p{\Psi}_{n,i}
(\mathbf{z})((\mathbf{x}-\mathbf{z})^p)\Big].
\end{equation}
Using that $\|\mathbf{x}-\mathbf{z}\| =
O(\epsilon)$ on $\partial B_{\mathbf{z},\epsilon}$ and Taylor's Theorem  \cite{R}, we have 
$$ 
R^{\epsilon}_{n,I}(\mathbf{x})= O(\epsilon^{I+1}), \quad \text{on }
\partial B_{\mathbf{z},\epsilon}.
$$


On $\Gamma_2$ we have
\begin{align*} 
R^{\epsilon}_{n,I}(\mathbf{x})
&= \sum_{i=0}^{I}\epsilon^i W_{n,i}(\frac{\mathbf{x}-\mathbf{z}}{\epsilon}) 
 - \sum_{i=1}^{I}\epsilon^i \big[\sum_{p=1}^{i}W_{n,i-p}^{(p)}(\mathbf{x}-\mathbf{z})\big]\\
&= \sum_{i=0}^{I}\epsilon^i W_{n,i}(\frac{\mathbf{x}-\mathbf{z}}{\epsilon}) 
 - \sum_{i=0}^{I-1}\epsilon^i
\big[\sum_{p=1}^{I-i}\epsilon^p W_{n,i}^{(p)}(\mathbf{x}-\mathbf{z})\big].
\end{align*}
This equality can be written as
$$
R^{\epsilon}_{n,I}(\mathbf{x})=\epsilon^I W_{n,I}(\frac{\mathbf{x}-\mathbf{z}}{\epsilon}) 
+ \sum_{i=0}^{I-1}\epsilon^i \big[W_{n,i}(\frac{\mathbf{x}-\mathbf{z}}{\epsilon})
 - \sum_{p=1}^{I-i} \epsilon^p W_{n,i}^{(p)}(\mathbf{x}-\mathbf{z})\big].
$$
Then, by Lemma \ref{asyp-ext} we obtain
$$
R^{\epsilon}_{n,I}= O(\epsilon^{I+1}) \quad \text{on } \Gamma_2.
$$

On $\Gamma_n$, using the same analysis we obtain
    $$
\nabla R^{\epsilon}_{n,I}\cdot \mathbf{n}= O(\epsilon^{I+1}) \quad \text{ on } \Gamma_1.
$$

Similarly, to prove Theorem \ref{TD0}, we define the
function $R^{\epsilon}_{d,I}$ in $\Omega_{\mathbf{z},\epsilon}$ by
\begin{align*}
R^{\epsilon}_{d,I}
&=\Psi_{d,0} (\mathbf{x})+ W_{d,0}
(\frac{\mathbf{x}-\mathbf{z}}{\epsilon})+\epsilon\,(\Psi_{d,1} 
(\mathbf{x})+ W_{d,1} (\frac{\mathbf{x}-\mathbf{z}}{\epsilon}) \\
&\quad + \dots + \epsilon^I (\Psi_{d,I} (\mathbf{x})+ W_{d,I}
(\frac{\mathbf{x}-\mathbf{z}}{\epsilon})-\psi^{\epsilon}_d (\mathbf{x})
\end{align*}
and using the same analysis in the proof of Theorem \ref{TN0} we
derive $R^{\epsilon}_{d,I}= O(\epsilon^{I+1})$.

\subsection{Proofs of the main results in Theorem \ref{T}}

To prove Theorem \ref{T}, we have to estimate  each
term of the equality \eqref{varJ}.
\smallskip

\noindent\textbf{Estimate for the first and the second terms.}
By changing  $\mathbf{x}=\mathbf{z}+\epsilon \mathbf{y}$, we have
\[
\int_{\partial B_{\mathbf{z},\epsilon}}\nabla_{\mathbf{x}} W_{n,i}
 (\frac{\mathbf{x}-\mathbf{z}}{\epsilon})\cdot
\mathbf{n}(\mathbf{x})\psi_{n,0}(\mathbf{x})ds
=\epsilon \int_{\partial B}\nabla_\mathbf{y} W_{n,i}(\mathbf{y})\cdot
\mathbf{n}(\mathbf{y})\,\psi_{n,0}(\mathbf{z}+\epsilon \mathbf{y})ds(\mathbf{y}).
\]
Since $\psi_{n,0}$ is smooth in  a neighborhood of
$\mathbf{z}$, one obtains
\begin{align*} 
\psi_{n,0}(\mathbf{z}+\epsilon \mathbf{y})
&= \psi_{n,0}(\mathbf{z}) + \sum_{p=1}^{I-1}\frac{\epsilon^p}{p!} 
 \nabla^{(p)} \psi_{n,0} (\mathbf{z})(\mathbf{y}^p)+O(\epsilon^{I})\\
&= \sum_{p=0}^{I-1}\frac{\epsilon^p}{p!} \nabla^{(p)} \psi_{n,0}
(\mathbf{z})(\mathbf{y}^p)+O(\epsilon^{I}).
\end{align*}
Then, we have
\begin{align*}
&\int_{\partial B_{\mathbf{z},\epsilon}}
\nabla_{\mathbf{x}} W_{n,i} (\frac{\mathbf{x}-\mathbf{z}}{\epsilon})\cdot 
\mathbf{n}(\mathbf{x})\psi_{n,0}(\mathbf{x})ds\\
&= \sum_{p=0}^{I-1}\frac{\epsilon^{p+1}}{p!}
\int_{\partial B}\nabla_\mathbf{y} W_{n,i}(\mathbf{y})\cdot 
\mathbf{n}(\mathbf{y})[\nabla^{(p)}
\psi_{n,0} (\mathbf{z})(\mathbf{y}^{p})]ds(\mathbf{y})+O(\epsilon^{I+1}).
\end{align*}
Consequently, 
\begin{align*} 
&\sum_{i=0}^{I} \epsilon^i \int_{\partial B_{\mathbf{z},\epsilon}} \nabla_{\mathbf{x}}
W_{n,i} ((\mathbf{x}-\mathbf{z})/\epsilon))\cdot \mathbf{n}\,\psi_{n,0}ds 
\\
& = \sum_{i=0}^{I} \epsilon^i\sum_{p=0}^{I-1}\frac{\epsilon^{p+1}}{p!}  
\int_{\partial B}\hspace{-0.25cm}\nabla_\mathbf{y} W_{n,i}(\mathbf{y})
\cdot \mathbf{n}(\mathbf{y})[\nabla^{(p)} \psi_{n,0} (\mathbf{z})
(\mathbf{y}^{p})]ds(\mathbf{y})+O(\epsilon^{I+1})\\
& = \sum_{i=1}^{I}
\epsilon^{i}\sum_{p=0}^{i-1}\frac{1}{p!}
\int_{\partial\omega}\hspace{-0.25cm}\nabla_\mathbf{y} W_{n,i-p-1}(\mathbf{y})\cdot
\mathbf{n}(\mathbf{y})[\nabla^{(p)} \psi_{n,0} (\mathbf{z})(\mathbf{y}^{p})]ds
(\mathbf{y})+O(\epsilon^{I+1})\\
&=\sum_{i=1}^{I} \epsilon^{i}
\mathcal{T}_{n,1}^{i-1}(\mathbf{z})+ O(\epsilon^{I+1}),
\end{align*}
Similarly, we obtain
\[
 \sum_{i=0}^{I} \epsilon^i \int_{\partial
B_{\mathbf{z},\epsilon}} \nabla_{\mathbf{x}} W_{d,i} 
((\mathbf{x}-\mathbf{z})/\epsilon))\cdot \mathbf{n}\,\psi_{d,0}ds  
= -\sum_{i=1}^{I}
\epsilon^{i} \mathcal{T}_{d,1}^{i-1}(\mathbf{z})+ O(\epsilon^{I+1}).
\]
\smallskip

\noindent\textbf{Estimate for the third  and the fourth terms.}
By changing  $\mathbf{x}=\mathbf{z}+\epsilon \mathbf{y}$, we have
\[
\int_{\partial B_{\mathbf{z},\epsilon}}  \nabla \Psi_{n,i}(\mathbf{x})\cdot
\mathbf{n}(\mathbf{x})\psi_{n,0}(\mathbf{x})\,ds 
=\epsilon^2 \int_{\partial B} \nabla
\Psi_{n,i}(\mathbf{z}+\epsilon \mathbf{y})\cdot \mathbf{n}(\mathbf{z}
+\epsilon \mathbf{y}) \,u_{n,0}(\mathbf{z}+\epsilon \mathbf{y})ds(\mathbf{y}).
\]
Since $\psi_{n,0}$ is smooth in  a neighborhood of
$\mathbf{z}$, one obtains
\begin{align*}
\psi_{n,0}(\mathbf{z}+\epsilon \mathbf{y})
&= \psi_{n,0}(\mathbf{z}) + \sum_{p=1}^{I-1}\frac{\epsilon^p}{p!} 
 \nabla^{(p)} \psi_{n,0} (\mathbf{z})(\mathbf{y}^p)+O(\epsilon^{I})\\
&= \sum_{p=0}^{I-1}\frac{\epsilon^p}{p!} \nabla^{(p)} \psi_{n,0}
(\mathbf{z})(\mathbf{y}^p)+O(\epsilon^{I}).
\end{align*}
Similarly, $\Psi_i$ is smooth in  a neighborhood of $\mathbf{z}$, then
\[
\nabla \Psi_{n,i}(\mathbf{z}+\epsilon
\mathbf{y})=\sum_{q=0}^{I-1}\frac{\epsilon^q}{q!} \nabla^{(q+1)}
\Psi_{n,i} (\mathbf{z})(\mathbf{y}^q)+O(\epsilon^{I}).
\]
Then
\begin{align*} 
 &\int_{\partial B_{\mathbf{z},\epsilon}}
\nabla U_{n,i}(\mathbf{x})\cdot \mathbf{n}(\mathbf{x})u_{n,0}(\mathbf{x})ds \\
& =\epsilon^2 \int_{\partial B}
[\sum_{q=0}^{I-1}\frac{\epsilon^q}{q!} \nabla^{(q+1)} U_{n,i}
(\mathbf{z})(\mathbf{y}^q)] \cdot \mathbf{n}(\mathbf{y}) [\sum_{p=0}^{I-1}\frac{\epsilon^p}{p!}
\nabla^{(p)} u_{n,0} (\mathbf{z})(\mathbf{y}^p)]ds(\mathbf{y})
+ O(\epsilon^{I+1}).
\end{align*}
Using the Cauchy product formula, we derive
\begin{align*} 
&\int_{\partial B_{\mathbf{z},\epsilon}}
\nabla \Psi_{n,i}(\mathbf{x})\cdot \mathbf{n}(\mathbf{x})\psi_{n,0}(\mathbf{x})ds\\
&=\sum_{p=0}^{I-2}\epsilon^{p+2}
\sum_{q=0}^{p}\frac{1}{q!(p-q)!}\hspace{-0.1cm}\int_{\partial
B}[ \nabla^{(q+1)} \Psi_{n,i} (\mathbf{z})(\mathbf{y}^q)] \cdot \mathbf{n}(\mathbf{y})\\
&\quad \times [\nabla^{(p-q)} \psi_{n,0} (\mathbf{z})(\mathbf{y}^{p-q})]ds(\mathbf{y})
+O(\epsilon^{I+1}).
\end{align*}
Consequently,
\begin{align*}
&\sum_{i=1}^{I} \epsilon^i \int_{\partial B_{\mathbf{z},\epsilon}}  
\nabla \Psi_{n,i}(\mathbf{x})\cdot \mathbf{n}(\mathbf{x})
 \psi_{n,0}(\mathbf{x})ds \\
&=\sum_{i=1}^{I}\sum_{p=0}^{I-2}\epsilon^{i+p+2} \sum_{q=0}^{p}\frac{1}{q!(p-q)!}\\
&\quad\times\int_{\partial B} [ \nabla^{(q+1)} \Psi_{n,i} (\mathbf{z})
 (\mathbf{y}^q)] \cdot \mathbf{n}(\mathbf{y}) [\nabla^{(p-q)} \psi_{n,0} 
 (\mathbf{z})(\mathbf{y}^{p-q})]ds(\mathbf{y})+O(\epsilon^{I+1})\\
&=\sum_{i=3}^{I} \epsilon^{i} \sum_{p=0}^{i-3}
\sum_{q=0}^{p}\frac{1}{q!(p-q)!}\\
&\quad\times \int_{\partial B} [
\nabla^{(q+1)} \Psi_{n,i-p-2} (\mathbf{z})(\mathbf{y}^q)] 
\cdot \mathbf{n}(\mathbf{y}) [\nabla^{p-q} \psi_{n,0}
(\mathbf{z})(\mathbf{y}^{(p-q)})]ds(\mathbf{y})+O(\epsilon^{I+1})\\
&= \sum_{i=3}^{I} \epsilon^{i} \mathcal{T}_{n,2}^{i-3}(\mathbf{z}) 
+ O(\epsilon^{I+1}).
\end{align*}
Similarly,
\begin{align*}
&\sum_{i=1}^{I} \epsilon^i \int_{\partial B_{\mathbf{z},\epsilon}}  
 \nabla \Psi_{d,i}(\mathbf{x})\cdot \mathbf{n}(\mathbf{x})
 \psi_{d,0}(\mathbf{x})ds \\
&=\sum_{i=3}^{I} \epsilon^{i} \sum_{p=0}^{i-3} \sum_{q=0}^{p}\frac{1}{q!(p-q)!}\\
 &\quad\times \int_{\partial B} [
\nabla^{(q+1)} \Psi_{d,i-p-2} (\mathbf{z})(\mathbf{y}^q)] \cdot 
\mathbf{n}(\mathbf{y}) [\nabla^{p-q} \psi_{d,0}
(\mathbf{z})(\mathbf{y}^{(p-q)})]ds(\mathbf{y})+O(\epsilon^{I+1})\\
&=-\sum_{i=3}^{I} \epsilon^{i} \mathcal{T}_{d,2}^{i-3}(\mathbf{z}) 
+ O(\epsilon^{I+1}).
\end{align*}
\smallskip

\noindent\textbf{Estimate for the fifth and the sixth terms.}
Since $\psi_{d,0}$ and $\psi_{n,0}$ are sufficiently regular in
$B_{\mathbf{z},\epsilon}$, we have
\begin{gather*} 
\nabla \psi_{d,0}(\mathbf{z}+\epsilon \mathbf{y})
 =\nabla \psi_{d,0}(\mathbf{z}) 
 + \sum_{i=1}^{I-1}\frac{\epsilon^i}{i!} \nabla^{(i+1)} \psi_{d,0} 
 (\mathbf{z})(\mathbf{y}^i)+O(\epsilon^{I})\\
\nabla \psi_{n,0}(\mathbf{z}+\epsilon \mathbf{y})=\nabla \psi_{n,0}(\mathbf{z}) 
 +\sum_{i=1}^{I-1}\frac{\epsilon^i}{i!} \nabla^{(i+1)} \psi_{n,0}
(\mathbf{z})(\mathbf{y}^i)+O(\epsilon^{I}).
\end{gather*}
By the change of variable $\mathbf{x}=\mathbf{z}+\epsilon \mathbf{y}$, 
we obtain
\begin{align*} 
\int_{B_{\mathbf{z},\epsilon}}|\nabla \psi_{d,0}|^2d\mathbf{x} 
&= \epsilon^3 \int_{B}|\nabla \psi_{d,0}(\mathbf{z}+\epsilon \mathbf{y})|^2d\mathbf{y}\\
&= \epsilon^3 \int_{B}
\Big(\sum_{i=0}^{I-1}\frac{\epsilon^i}{i!} |\nabla^{(i+1)}
\psi_{d,0} (\mathbf{z})(\mathbf{y}^i)\Big)|^2d\mathbf{y} +O(\epsilon^{I+1}).
\end{align*}
Using the Cauchy product formula, we obtain
\begin{align*}
 &\int_{B_{\mathbf{z},\epsilon}}|\nabla \psi_{d,0}|^2d\mathbf{x}\\
 &= \sum_{i=0}^{I-3}\epsilon^{i+3}\Big
  (\sum_{p=0}^{i}\frac{1}{p!(i-p)!}
 \int_{B}  \nabla^{(p+1)} \psi_{d,0}
(\mathbf{z})(\mathbf{y}^p)\cdot  \nabla^{(i-p+1)} \psi_{d,0}
 (\mathbf{z})(\mathbf{y}^{i-p})d\mathbf{y} \Big) \\
&\quad +O(\epsilon^{I+1})\\
 &=\sum_{i=3}^{I}\epsilon^i \mathcal{T}_{d,3}^{i-3}(\mathbf{z})+ O(\epsilon^{I+1}).
\end{align*}
Similarly, 
\begin{align*} 
 &\int_{B_{\mathbf{z},\epsilon}}|\nabla \psi_{n,0}|^2d\mathbf{x}\\
 & = \sum_{i=0}^{I-3}\epsilon^{i+3}\Big
(\sum_{p=0}^{i}\frac{1}{p!(i-p)!}
  \int_{B}  \nabla^{(p+1)} \psi_{n,0}
(\mathbf{z})(\mathbf{y}^p)\cdot  \nabla^{(i-p+1)} \psi_{n,0} 
(\mathbf{z})(\mathbf{y}^{i-p})d\mathbf{y} \Big) \\
&\quad +O(\epsilon^{I+1})\\
 &=-\sum_{i=3}^{I}\epsilon^i \,\mathcal{T}_{n,3}^{i-3}(\mathbf{z})
+ O(\epsilon^{I+1}).
\end{align*}
Finally, the desired result is obtained  by using the above estimates.

\subsection*{Concluding remarks}\label{conclusion}
This work is concerned with a geometric inverse problem related to
the Laplace operator in three-dimensional domain. More precisely, 
the topological sensitivity method is applied  to calculate
a high-order topological asymptotic expansion of the semi-norm
Kohn-Vogelius functional, when a Dirichlet perturbation is
introduced in the initial domain.

The obtained expansion of the  semi-norm Kohn-Vogelius functional is 
of higher interest and improves the detection of objects with any size 
of perturbation.  The other advantage is when the topological derivative 
of order one is equal to zero for some critical points in the initial domain.


\subsection*{Acknowledgments} 
The authors would like to extend their sincere appreciation to the 
Deanship of Scientific Research at King Saud University for funding 
this Research group No (RG-1435-026).

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\end{document}