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\begin{document}
\title{The Linear Sampling Method for Anisotropic Media: Part II.}
\author{Fioralba Cakoni}
\address{\hskip-\parindent
          Fioralba Cakoni\\
          Department of Mathematical Sciences,
          University of Delaware,\\
          Newark, Delaware 19716, U.S.A.}
\email{cakoni@math.udel.edu}
\author{Houssem Haddar}
\address{\hskip-\parindent
          Houssem Haddar\\
          Department of Mathematical Sciences,
          University of Delaware,\\
          Newark, Delaware 19716, U.S.A.}
\email{haddar@math.udel.edu}

\thanks{This paper was written while the authors were visiting the
Mathematical Sciences Research Institute at Berkeley, California.
The financial  support of MSRI is gratefully acknowledge. The
research of F.C. is also partially supported by NSF Grant
9631287.}


\begin{abstract}
We reconsider the linear sampling method for solving the inverse
scattering problem of determining the support of an anisotropic
inhomogeneous medium from a knowledge of the incident and
scattered time harmonic acoustic wave at fixed frequency. We
extend the results of a previous paper \cite{CCH01} concerning
with the same problem   to the case where the norm of the matrix
that describes the physical properties of the medium is less than
one.
\\
\\
{\bf Keywords:} Inverse scattering, anisotropic medium,
linear sampling method.
\\
{\bf MSC:}   35R30,  35J20, 35J25
\end{abstract}
\maketitle

% main text

%\label{}
\section{Introduction}
The {\it linear sampling method} has been successfully used to
solve the inverse scattering problem of determining the support
of the scatterer from the knowledge of the far field pattern at a
fixed frequency,  in the case of both acoustic and
electromagnetic waves \cite{CCH01,CCM00,CK98}. The scalar
anisotropic case was first considered in \cite{H00} and
\cite{CCH01} where the uniqueness results and the mathematical
theory of the sampling method were developed. The techniques used
in the above papers are based on an analysis of a boundary value
problem called the {\it interior transmission problem} and only
covers the case where the norm of the matrix $A$ describing the
anisotropic behaviour is greater than one. The purpose of this
paper is to extend the results of \cite{CCH01} and \cite{H00} to
the case $\|A\| <1$.
\section{The direct and inverse scattering problems for an anisotropic medium}
Let $D\subset {\mathbb R}^3$ be a nonempty, open, connected and
bounded set having a $C^2$-boundary $\partial D$ with unit
outward normal $\nu$. Let $A$ be a $3\times 3$ matrix-valued
function whose entries $a_{jk}$, $j=1,2,3$, $k=1,2,3$ are
continuously differentiable complex-valued functions in
$\overline{D}$ such that $A$ is symmetric and  satisfies
$\bar{\xi} \cdot {\mathcal Im} (A) \, \xi \, \le 0 $
 and
 $\bar{\xi} \cdot {\mathcal Re}(A) \, \xi \ge\gamma |\xi|^2$
for all $\xi \in {\mathbb C}^3$ and $x\in {\overline D}$ where
$\gamma$ is a positive constant. Note that due to the symmetry of
$A$, ${\mathcal Im}\left(\, \bar{\xi} \cdot A \, \xi \, \right) =
\bar{\xi} \cdot {\mathcal Im} (A) \, \xi $ and ${\mathcal
Re}\left(\, \bar{\xi} \cdot A \, \xi \, \right) = \bar{\xi} \cdot
{\mathcal Re} (A) \, \xi $. For a function $u \in
C^1(\overline{D})$ we define the conormal derivative by
$$
\frac{\partial u}{\partial \nu_A}(x):=\nu(x)\cdot A(x)\nabla
u(x), \qquad x\in
\partial D.
$$
We can now formulate the direct scattering problem for an
anisotropic medium. In particular, let $k>0$ be the wave number
and $n\in C(\overline{D})$ such that ${\mathcal Im} \, n\ge 0$.
Then letting $H^k$ denote the usual Sobolev space we want to find
functions $w\in H^1(D), \, u\in H^1({\mathbb R}^3\setminus
\overline{D})$ such that
\begin{eqnarray}
& (i) \quad  & \dive A \grad w + k^2 n \, w= 0 \quad  \mbox{ in }\quad D \nonumber \\
& & \nonumber\\
& (ii)  \quad &\Delta u +k^2\,u  = 0  \quad \mbox{ in } \quad
{\mathbb R}^3\setminus
\overline{D}  \nonumber \\
& & \nonumber \\
& (iii)  \quad & w -u = f  \quad \mbox{ on } \quad \partial D \label{tp} \\
& & \nonumber \\
& (iv)  \quad & \frac{\partial w}{\partial \nu_A}-\frac{\partial
u}{\partial \nu}=h
\quad  \mbox{ on } \quad \partial D \nonumber \\
& & \nonumber \\
& (v) \quad  &\lim_{r\to \infty}r\left(\frac{\partial u}{\partial
r}-iku \right)=0 \nonumber
\end{eqnarray}
where $f:=e^{ikx\cdot d}$ and $h:=\frac{\partial }{\partial
\nu}e^{ikx\cdot d}$, $d\in \Omega:=\left\{x:|x|=1\right\}$,
$r=|x|$, the boundary conditions are assumed in the sense of the
trace operator, and the radiation condition
(\ref{tp}{\it v}) holds uniformly with respect to  $\hat x=x/|x|$.  \\
More generally we consider (\ref{tp}) with $f\in
H^{\demi}(\partial D)$ and $h \in H^{-\demi}(\partial D)$
arbitrary and in the sequel will refer to this more general
problem as the {\it transmission problem} (TP). The existence of
a unique solution to (TP) has been established by H{\"a}hner in
\cite{H00}. Moreover, he has proved that this solution depends
continuously on the boundary data in the sense that the following
estimate holds
\begin{equation}\label{han}
\|w\|_{H^1(D)}+\|u\|_{H^1(B\setminus \overline{D})}\le
C\left(\|f\|_{H^{\demi}(\partial D)}+\|h\|_{H^{-\demi}(\partial
D)}\right)
\end{equation}
where $B$ is a ball containing $D$ and $C=C(B)$ is a positive constant.\\
Since $u$ satisfies the radiation condition (\ref{tp}{\it v}) we
can conclude (see \cite{CK98}) that $u$ has the asymptotic
behavior
$$
u(x)=\frac{e^{ikr}}{r}u_{\infty}(\hat
x,d)+O\left(\frac{1}{r^2}\right) \label{far}
$$
where $u_{\infty}(\hat x,d)$ is the {\it far field pattern} of
the scattered field $u$. We recall the definition of a {\it
Herglotz wave function}, which is a solution $v_g$ to
 the Helmholtz equation in ${\mathbb R}^3$ of the form
$$
v_g(x)=\int_{\Omega}e^{ikx\cdot d}g(d)\,ds(d), \qquad x\in \R^3.
$$
\\
The inverse scattering problem we are concerned with is to
determine $D$ from a knowledge of $u_{\infty}(\hat x,d)$ for
$\hat x, d\in \Omega$. Our approach for solving this inverse
scattering problem is the {\it linear sampling method} as
described in \cite{CCH01}. In particular, we pick a parameter
$y\in \R^3$ and then look for a solution $g \in L^2(\Omega)$ of
the {\it far field equation}
\begin{equation}\label{farfield}
(Fg)(\hat x):=\int_{\Omega}u_{\infty}(\hat x,
d)g(d)\,ds(d)=\Phi_{\infty}(\hat x, y)
\end{equation}
where $\Phi_{\infty}(\hat x, y):=e^{-ik\hat x \cdot y}$ is the far
field pattern of the fundamental solution to the Helmholtz
equation  $\Phi(x,y):=\frac{e^{ik|x-y|}}{|x-y|}$ and
$F:L^2(\Omega)\longrightarrow L^2(\Omega)$ is called the {\it far
field operator}. The  linear sampling method is based on an
examination of the {\it interior transmission problem} associated
to (TP) (see \cite{CCH01}), which in the sequel will be  referred
to as (ITP). The problem (ITP) is: given $f\in H^{\demi}(\partial
D)$ and $h \in H^{-\demi}(\partial D)$ find two functions $w \in
H^1(D)$ and $v\in H^1(D)$ satisfying
\begin{eqnarray}
& (i) \quad  & \dive A \grad w + k^2 n \, w= 0 \quad  \mbox{ in }\quad D \nonumber \\
& & \nonumber\\
& (ii)  \quad &\Delta u +k^2\,u  = 0  \quad \mbox{ in } \quad D \nonumber \\
& & \nonumber \\
& (iii)  \quad & w -u = f  \quad \mbox{ on } \quad \partial D \label{itp} \\
& & \nonumber \\
& (iv)  \quad & \frac{\partial w}{\partial \nu_A}-\frac{\partial
u}{\partial \nu}=h \quad  \mbox{ on } \quad \partial D. \nonumber
\end{eqnarray}
The questions of  the existence and uniqueness of (ITP) as well as
a-priori estimate for the solution are already considered  in
\cite{CCH01} where partial results are obtained under an
additional assumption on the matrix $A$. For readers convenience
we formulate these results in the following theorem.
\begin{Th} \label{existencesITP}
Assume that either ${\cal I}m \, n >0 $ or ${\cal I}m \left( \,
\bar{\xi} \cdot A \, \xi \, \right) <0 $ in a neighborhood
$B_{x_0}$ of a point $x_0 \in D$ and that there exists a constant
$\gamma > 1$ such that for almost every $x \in D$,
\begin{equation}
{\cal R}e \left( \, \bar{\xi} \cdot A(x) \, \xi \, \right) \ge
\gamma |\xi|^2 \;\;\; \forall \; \xi \in \C^3.
\end{equation}
Then {\rm (ITP)} has  a unique solution $(w, v) \in H^1(D) \times
H^1(D)$. This solution satisfies the a priori estimate
\begin{equation}
\N{w}{H^1(D)} + \N{v}{H^1(D)} \le C \left(
 \N{f}{H^\demi(\partial D)} +
\N{h}{H^{-\demi}(\partial D)}\right),
\end{equation}
where the constant $C$ is independent of $f$ and  $h$.
\end{Th}
In the case when the assertion of Theorem \ref{existencesITP}
holds one can prove by using the technique of H\"ahner in
\cite{H00} that for all incident directions $d$ and all
observation directions $\hat x$ $u_{\infty}(\hat x,d)$  uniquely
determines the support $D$ (we remind the reader that $A$ is {\it
not} uniquely determined by $u_{\infty}$ \cite{FGC}). Moreover in
this case the regularized solution to the far field equation
(\ref{farfield}) leads to a simple reconstruction algorithm for
determining the support $D$ of the inhomogeneity from the far
field pattern. In particular in \cite{CCH01} it is proved that

\begin{enumerate}
\item if $y\in D$ then for every $\epsilon>0$ there exists a solution
$g^{\epsilon}(\cdot,y)\in L^2(\Omega)$ of the inequality
$$
\|Fg_{\epsilon}(\cdot,y)- \Phi_{\infty}(\cdot,
y)\|_{L^2(\Omega)}<\epsilon
$$
such that
$$
\lim_{y \to \partial D}
\|g_{\epsilon}(\cdot,y)\|_{L^2(\Omega)}=\infty \quad \mbox{and}
\quad \lim_{y \to \partial
D}\|v_{g_{\epsilon}}(\cdot,y)\|_{H^1(D)}=\infty,
$$
where $v_{g_{\epsilon}}$ is the Herglotz wave function with kernel $g_{\epsilon}$, and \\
\item  if $y\in \R^2\setminus \overline{D}$ then for every
$\epsilon>0$ and $\delta>0$ there exists a solution
$g_{\epsilon,\delta}(\cdot,y)\in L^2(\Omega)$ of the inequality
$$
\|Fg_{\epsilon,\delta}(\cdot,y)- \Phi_{\infty}(\cdot,
y)\|_{L^2(\Omega)}<\epsilon+\delta
$$
such that
$$
\lim_{\delta \to 0} \|g_{\epsilon,
\delta}(\cdot,y)\|_{L^2(\Omega)}=\infty \quad \mbox{and} \quad
\lim_{\delta \to
0}\|v_{g_{\epsilon,\delta}}(\cdot,y)\|_{H^1(D)}=\infty,
$$
where $v_{g_{\epsilon, \delta}}$ is the Herglotz wave function
with kernel $g_{\epsilon,\delta}$.
\end{enumerate}
One can now use regularization methods to solve the far field
equation $Fg=\Phi_{\infty}(\cdot,y)$ for $y$ on an appropriate
grid containing $D$. In this way an approximation to
$g_{(\cdot,y)}$ can be obtained and hence $\partial D$ can be
determined by those points where $\|g(\cdot,y)\|_{L^2(\Omega)}$
becomes unbounded (c.f. \cite{CCM00}).

\section{The interior transmission problem}
Theorem \ref{existencesITP} addresses the case $\|A \|> 1$ and
consequently the mathematical theory of the sampling method holds
for such matrices $A$. Our aim here is to complete this result by
studying the solvability of (ITP) in the case of $\|A\|<1$. To
this end we formulate the {\it  modified interior transmission
problem} (MITP) which later will be seen as a compact
perturbation of our original interior transmission problem (ITP):
given $D$ and $A$ as above, a positive constant $m$, functions
$\ell_1 \in L^2(D)$, $\ell_2 \in L^2(D)$, $f \in H^\demi(\partial
D)$, and $h\in H^{-\demi}(\partial D)$, find $w\in H^1(D)$ and
$v\in H^1(D)$ satisfying
\begin{equation}
\label{itp:form1}
\left\{ \barr{lll}
(i) & \dive A \grad w - m \, w = \ell_1 & \mbox{ in } D \\
(ii) &\Delta v - v = \ell_2 & \mbox{ in } D \\
(iii) & w -v = f & \mbox{ on } \partial D \\
(iv) & \dna{w} - \dnu{v} = h & \mbox{ on } \partial D.
\earr
\right.
\end{equation}
Such a pair $(w,v)$ will be referred to as a strong solution to
(\ref{itp:form1}). Note that in the following for simplicity we
use the notations $ \dna:=\frac{\partial}{\partial \nu_A}$ and $
\dnu:=\frac{\partial}{\partial \nu}$.

We will now reformulate (\ref{itp:form1}) as a variational
problem. Let us introduce the Hilbert space
\begin{equation}
W(D) = \left\{ \bw \in L^2(D)^3\; : \; \dive \bw \in L^2(D)
\mbox{ and } \curl A^{-1} \bw =0 \right\}
\end{equation}
equipped with the natural norm $ \N{\bw}{W} = \left(
\N{\bw}{L^2}^2 + \N{\dive \bw}{L^2}^2 \right)^\demi$ and denote
by $\dual{\cdot}{\cdot}$ the duality pairing between
$H^\demi(\partial D)$ and $H^{-\demi}(\partial D)$. We recall the
duality identity
\begin{equation} \label{duality}
\dual{\varphi}{\bpsi \cdot \nu} = \int_D \, \varphi\;  \dive \bpsi \; dx +
\int_D \, \grad \varphi \cdot  \bpsi \; dx
\end{equation}
for $(\varphi, \bpsi) \in H^1(D) \times W(D)$ that will be of
particular interest in the sequel.
\\
We now introduce the sesquilinear form ${\cal A}$ defined on
$\{H^1(D) \times W(D)\}^2$ by
\begin{equation}
\barr{lcl} {\cal A}(U,V) &=& \dsp \int_D \grad v \cdot \grad
\bar{\varphi} \; dx + \int_D \, v \, \bar{\varphi} \; dx + \dsp
\int_D \dive \bw \, \dive
\bar{\bpsi} \; dx   \; dx \vspace{2mm} \\
&+&  \dsp m \int_D A^{-1} \bw  \cdot \bar{\bpsi} - m
\dual{v}{\bar{\bpsi} \cdot \nu} - \dual{\bar{\varphi}}{\bw\cdot
\nu} \earr
\end{equation}
where $U=(v, \bw)$ and $V=(\varphi, \bpsi)$ are in $H^1(D) \times
W(D)$.
%We introduce also the sesquilinear
%form ${\cal B}$ defined on $L^2(D)^4 \times L^2(D)^4$ by
%\begin{equation}
%{\cal B}(U,V) = (k^2 n +1) \int_D u \, \bar{\varphi} \; dx \;  + \;
%(k^2+1) \int_D \bw  \cdot \bar{\bpsi}   \; dx
%\end{equation}
%where $U=(u, \bw)$ and $V=(\phi, \bpsi)$ are in $L^2(D)^4$.
We also
introduce for $V=(\varphi, \bpsi) \in H^1(D) \times W(D)$ the
antilinear form
\begin{equation}
L(V) = \int_D ( \ell_1 \; \bar{\varphi} + \ell_2 \; \dive
\bar{\bpsi} ) \; dx -  \dual{\bar{\varphi}}{h} +m
\dual{f}{\bar{\bpsi} \cdot \nu}.
\end{equation}
The variational formulation of the problem (\ref{itp:form1}) is
\begin{equation} \label{variational:form}
\left\{ \barr{l}
\mbox{Seek } U \in H^1(D) \times W(D) \mbox{ such that }
\\
{\cal A}(U,V)  = L(V),     \;\;\;\; \forall \;
V \in H^1(D) \times W(D),
\earr
\right.
\end{equation}
and the following theorem proves the equivalence between the two formulations.
\begin{Th} \label{equivalence}
Problem (\ref{itp:form1}) has a unique strong solution $(w,v)$ if
and only if problem (\ref{variational:form}) has a unique
solution $ U \in H^1(D) \times W(D)$. Moreover if $(w,v)$ is the
unique strong solution to (\ref{itp:form1}) then $U = (v, A \grad
w)$ is the unique strong solution to (\ref{variational:form}).
Conversely, if  $U$ is the unique solution to
(\ref{variational:form}) then the unique  strong solution $(w,v)$
to (\ref{itp:form1}) is such that  $U = (v, A \grad w)$.
\end{Th}

\proof Let us prove first  the equivalence between the existence
of a strong solution $(w,v)$ to (\ref{itp:form1}) and the
existence of a solution $U$ to (\ref{variational:form}).
\vspace{2mm} \\
{\bf a)} $\exists\; (w,v) \Rightarrow \exists\; U$: Let $(w,v)$
be a strong solution to (\ref{itp:form1}). We set $\bw = A \grad w
$. From (\ref{itp:form1}{\it i}) we see that $\dive \bw = w +
\ell_1 \in L^2(D)$ then $\bw \in W(D)$. Taking the $L^2$ scalar
product of (\ref{itp:form1}{\it i}) with $\dive \bpsi$ for some
$\bpsi \in W(D)$ and using (\ref{duality}) shows that
$$
\int_D \, \dive \bw \, \dive  \bar{\bpsi} \; dx + m \int_D A^{-1}
\bw \cdot \bar{\bpsi} \; dx - m \dual{w}{\bar{\bpsi}\cdot \nu} =
\int_D \, \ell_1 \; \dive  \bar{\bpsi} \; dx.
$$
Hence by (\ref{itp:form1}{\it iii})
\begin{eqnarray}
\int_D \, \dive \bw \, \dive  \bar{\bpsi} \; dx & +& m \int_D
A^{-1}\bw \cdot
\bar{\bpsi} \; dx - m \dual{v}{\bar{\bpsi}\cdot \nu} \nonumber \\
&=&m \dual{f}{\bar{\bpsi}\cdot \nu} + \int_D \, \ell_1 \; \dive
\bar{\bpsi} \; dx  \label{vf:eq1}.
\end{eqnarray}
We now take the $L^2$ scalar product of (\ref{itp:form1}{\it ii})
with $\varphi$ in $H^1(D)$ and integrate by parts. Using the
boundary condition (\ref{itp:form1}{\it iv}), this shows that
\begin{equation}\label{vf:eq2}
\int_D \grad v \cdot \grad \bar{\varphi} \; dx + \int_D  v \,
\bar{\varphi} \; dx -\dual{\bar{\varphi}}{\bw\cdot \nu} =-
\dual{\bar{\varphi}}{h}+ \int_D \, \ell_2 \;
 \bar{\varphi} \; dx.
\end{equation}
Adding (\ref{vf:eq1}) and (\ref{vf:eq2}) together shows that
$U=(v, \bw)$ is a solution to (\ref{variational:form}).
\vspace{2mm} \\
{\bf b)} $\exists\; U  \Rightarrow \exists\; (w,v) $: We set $U =
(v, \bw) \in H^1 \times W(D)$. Since $\curl A^{-1} \bw =0$ and $D$
is simply connected, we deduce the existence of a function $w \in
H^1(D)$ such that $ \bw = A\grad w$ where $w$ is determined up to
an additive constant. As we shall see later, this constant can be
adjusted such that $(w,v)$ is a strong solution to
(\ref{itp:form1}). Obviously, if  $U$ satisfies
(\ref{variational:form}) then  $(v, \bw)$ satisfies
(\ref{vf:eq1})  and (\ref{vf:eq2}) for all $(\varphi, \bpsi) \in
H^1(D) \times W(D)$. One easily see from (\ref{vf:eq2}) that the
couple $(w,v)$ satisfies
\begin{equation} \label{res:0}
\left\{ \barr{ll}
 \Delta v - v = \ell_2 & \mbox{ in } D \\
 \dna{w} - \dnu{v} = h & \mbox{ on } \partial D.
\earr
\right.
\end{equation}
On the other hand, substituting for $\bw$ in (\ref{vf:eq1}) and using the
duality identity
(\ref{duality}) in the second integral shows that
\begin{equation} \label{vf:eq1bis}
\begin{array}{l}
\int_D \, (\dive A \grad w - m w ) \, \dive  \bar{\bpsi} \; dx  +
m \dual{w-v}{\bar{\bpsi}\cdot \nu} \\
\hspace{2cm}= m\dual{f}{\bar{\bpsi}\cdot \nu} + \int_D \, \ell_2
\; \dive \bar{\bpsi} \; dx
\end{array}
\end{equation}
for all $\bpsi$ in $W(D)$. Now consider a function $\phi \in
L^2_0(D)=\left\{\phi \in L^2(D)\; : \; \int_D \phi \; dx
=0\right\}$ and let $\chi \in H^1(D)$ be a solution to
\begin{equation} \label{prob:elem:1}
 \Delta \chi  = \bar{\phi}  \mbox{ in } D; \;\;\;
 \dnu{\chi} = 0  \mbox{ on } \partial D.
\end{equation}
Taking $\bpsi = \grad \chi$ in (\ref{vf:eq1bis}) ($\Rightarrow \dive
\bar{\bpsi} = \phi$ in $D$ and  $\bar{\bpsi}\cdot \nu = 0$ on $\partial D$) shows
that
$$
\int_D \, (\dive A \grad w \,  -  \, m w -\ell_2) \, \phi \; dx  =
0 \;\;\; \forall \; \phi \in L^2_0(D)
$$
which implies the existence of a constant $c_1$ such that
\begin{equation} \label{res:1}
\dive A \grad w - m w  -\ell_2 = c_1 \;\;\; \mbox{ in } D.
\end{equation}
We now take  $\phi \in L^2_0(\partial D)$ and let  $\chi \in H^1(D)$
be the solution to
\begin{equation} \label{prob:elem:2}
 \Delta \chi  = 0  \mbox{ in } D; \;\;\;
 \dnu{\chi} = \bar{\phi}  \mbox{ on } \partial D.
\end{equation}
Taking $\bpsi = \grad \chi$ in (\ref{vf:eq1}) ($\Rightarrow \dive
\bar{\bpsi} = 0$ in $D$ and  $\bar{\bpsi}\cdot \nu = \phi$ on $\partial D$) shows that
$$
m \int_{\partial D} \,  (w \,  -  \, v \, - \, f ) \, \phi \;
d\gamma = 0 \;\;\; \forall \; \phi \in L^2_0(\partial D)
$$
which implies the existence of a constant $c_2$ such that
\begin{equation}\label{res:2}
w \,  -  \, v \, - \, f = c_2 \;\;\; \mbox{ on }  \partial D.
\end{equation}
Substituting (\ref{res:1}) and  (\ref{res:2}) into (\ref{vf:eq1bis}) and using
(\ref{duality}) now shows that
$$
(c_1 - c_2) \int_D \; \dive \bar{\bpsi} \; dx = 0 \;\;\; \forall \; \bpsi \in
W(D)
$$
which implies $c_1 = c_2 = c$ (take $\bpsi = \grad \chi$ where
$\chi \in H^1_0(D)$ and $\Delta \chi = 1$ in $D$). Equations
(\ref{res:0}), (\ref{res:1}) and (\ref{res:2}) show that $(w-c,
v)$ is a strong solution to~(\ref{itp:form1}).
\vspace{2mm} \\
We now consider the uniqueness equivalence.
\vspace{2mm} \\
{\bf c)} Uniqueness of $(w,v) \; \Rightarrow \;$ Uniqueness of $
U $: Assume that problem (\ref{itp:form1}) has a unique strong
solution and consider two solutions $U_1 = (v_1, \bw_1)$ and $U_2
= (v_2, \bw_2)$ to (\ref{variational:form}). From step {\bf b)} we
deduce the existence of $w_1$ and $w_2$ in $H^1(D)$ such that
$\bw_1 = A \grad w_1$ and $\bw_2 = A \grad w_2$ and  $(w_1, v_1)$
and $(w_2, v_2)$ are strong solutions to (\ref{itp:form1}). Hence
$(w_1, v_1) = (w_2, v_2)$ and $(v_1, \bw_1) = (v_2, \bw_2)$.
\vspace{2mm} \\
{\bf d)} Uniqueness of $ U \; \Rightarrow \;$ Uniqueness of $
(w,v) $: Assume that problem (\ref{variational:form}) has a
unique solution and consider two strong solutions $ (w_1, v_1)$
and $(w_2, v_2)$ to (\ref{itp:form1}). We deduce from step {\bf
a)} that  $ (v_1, A \grad w_1)$ and $(v_2, A \grad w_2)$ are two
solutions to (\ref{variational:form}). Hence $ v_1 = v_2$ and  $w
= w_1 -w_2$ is a function in $H^1(D)$ that satisfies
$$
\left\{ \begin{array}{ll}
\dive A \grad w -m w =0  & \; \mbox{ in } D \\
w = \dna{w} = 0 & \; \mbox{ on } \partial D
\end{array}
\right.
$$
which implies $w=0$ (since $A$ is coercive).
\endproof


\begin{Th} \label{existence}
Assume that there exists
a constant $\gamma > 1$ such that for $x \in D$,
\begin{equation} \label{cond:A}
{\cal R}e \left( \, \bar{\xi} \cdot A^{-1}(x) \, \xi \, \right)
\ge \gamma |\xi|^2 \;\;\; \forall \; \xi \in \C^3  \; \; \mbox{
and } \;\;   \gamma^{-1} \le m  < 1.
\end{equation}
Then problem (\ref{variational:form}) has a unique solution $U =
(v, \bw) \in H^1(D) \times W(D)$. This solution satisfies the a
priori estimate
\begin{equation} \label{aprior:estimate}
\begin{array}{l}
\N{v}{H^1(D)} + \N{\bw}{W} \le \frac{4 c}{1-m}\left(
\N{\ell_1}{L^2(D)} + \N{\ell_2}{L^2(D)} \right.
\\ \hspace{6cm} +\left. \N{f}{H^\demi(\partial D)} +
\N{h}{H^{-\demi}(\partial D)}\right)
\end{array}
\end{equation}
where the constant $c$ is independent of $\ell_1$, $\ell_2$, $f$, $h$ and $\gamma$.
\end{Th}

\proof Classical trace theorems and Schwarz's inequality  insure
the continuity of the antilinear form $L$ on $H^1(D) \times W(D)$
and  the existence of a constant $c$ independent of $\ell_1$,
$\ell_2$, $f$ and $h$ such that
\begin{equation}
\N{L}{} \le c \left(\N{\ell_1}{L^2} + \N{\ell_2}{L^2} +  \N{f}{H^\demi} + \N{h}{H^{-\demi}}\right)
\end{equation}
On the other hand, if $U = (v, \bw) \in H^1(D) \times W(D)$ then,
by assumption (\ref{cond:A}),
\begin{equation}
\left| {\cal A}(U,U) \right| \ge \N{v}{H^1}^2 + \N{\bw}{W}^2 + (m
\gamma - 1) \N{\bw}{L^2}^2 - (1+m) {\cal R}e \dual{\bar v}{\bw}.
\end{equation}
According to the duality identity (\ref{duality}), one has by Schwarz's
inequality that
$$
| \dual{\bar v}{\bw} | \le \N{v}{H^1} \N{\bw}{W}
$$
and on the other hand $m\gamma -1 \ge 0$, therefore
$$
\left| {\cal A}(U,U) \right| \ge \N{v}{H^1}^2 + \N{\bw}{W}^2-
(m+1)\,\N{w}{H^1} \N{\bw}{W}
$$
Using the identity $x^2 + y^2 -(m+1)xy = \frac{m+1}{2} \left( x -
y \right)^2 + \frac{1-m}{2} (x^2 + y^2) $, we conclude that
$$
\left| {\cal A}(U,U) \right| \ge \frac{1-m}{2} \left(\N{\bw}{W}^2
+ \N{v}{H^1}^2 \right)
$$
and thus ${\cal A}$ is coercive. The continuity of ${\cal A}$ follows easily
from Schwarz's
inequality and classical trace theorems.  Theorem \ref{existence} is therefore a direct
consequence of Lax-Milgram theorem applied to (\ref{variational:form}).
\endproof
\begin{Th} \label{existence2}
Under the assumptions of Theorem  \ref{existence},  problem (\ref{itp:form1}) has a unique
strong solution $(w,v)$ that satisfies
\begin{equation} \label{aprior:estimate2}
\begin{array}{l}
\N{w}{H^1(D)} + \N{v}{H^1(D)} \le C \left( \N{\ell_1}{L^2(D)} +
\N{\ell_2}{L^2(D)} \right.
\\ \hspace{6cm} +\left. \N{f}{H^\demi(\partial D)} +
\N{h}{H^{-\demi}(\partial D)}\right)
\end{array}
\end{equation}
where the constant $C$ in independent of $\ell_1$, $\ell_2$, $f$
and $h$.
\end{Th}

\proof The existence and uniqueness of a strong solution follows
from Theorem \ref{equivalence} and Theorem \ref{existence}. The a
priori estimate (\ref{aprior:estimate2}) can be obtained directly
from (\ref{itp:form1}) but can be also deduced from
(\ref{aprior:estimate}) as follows. Theorem \ref{equivalence}
tells us that $(v, A\grad w)$ is the unique solution to
(\ref{variational:form}). Hence, according to
(\ref{aprior:estimate})
$$
\N{v}{H^1} + \N{A\grad w}{L^2} \le c_1 \left(\N{\ell_1}{L^2} +
\N{\ell_2}{L^2}+ \N{f}{H^\demi} + \N{h}{H^{-\demi}}\right).
$$
But from Poincar\'e inequality,
$$
\N{\grad w}{H^1(D)} \le c_2 \left(\N{\grad w}{L^2(D)} +
\N{w}{L^2(\partial D)} \right),
$$
and using the boundary condition (\ref{itp:form1}{\it iii}) and the trace theorem
one deduces that
$$
\N{w}{H^1(D)} \le c_2 \left( \N{\grad w}{L^2(D)} + \N{v}{H^1(D)} +
\N{f}{L^2(\partial D)} \right)
$$
for some positive constant $c_2$. Estimate
(\ref{aprior:estimate2}) follows straightforwardly.
\endproof

Now we can state the main result of this paper:
\begin{Th} \label{existenceITP}
Assume that either ${\cal I}m \, n >0 $ or ${\cal I}m \left( \, \bar{\xi}
\cdot A \, \xi \, \right) <0 $ in a neighborhood $B_{x_0}$ of a point $x_0
\in D$ and that there exists
a constant $\gamma > 1$ such that for almost every $x \in D$,
\begin{equation} \label{cond:Abis}
{\cal R}e \left( \, \bar{\xi} \cdot A(x)^{-1} \, \xi \, \right)
\ge \gamma |\xi|^2 \;\;\; \forall \; \xi \in \C^3.
\end{equation}
Then {\rm (ITP)} has  a unique solution $(w, v) \in H^1(D) \times H^1(D)$. This solution satisfies the a priori estimate
\begin{equation} \label{aprior:estimate3}
\N{w}{H^1(D)} + \N{v}{H^1(D)} \le C \left(
 \N{f}{H^\demi(\partial D)} +
\N{h}{H^{-\demi}(\partial D)}\right),
\end{equation}
where the constant $C$ is independent of $f$ and  $h$.
\end{Th}

\proof Let us set
$$
{\cal X}(D) = \left\{ (w,v) \in H^1(D) \times H^1(D) \; :\; \dive A \grad w \in
L^2(D) \mbox{ and } \Delta v \in L^2(D)\right\}
$$
and consider the operator ${\cal G}$ from ${\cal X}(D)$ into $L^2(D) \times
L^2(D) \times H^\demi(\partial D) \times  H^{-\demi}(\partial D)$ defined by
\begin{equation} \label{definition:G}
{\cal G}(w,v) = \left(\dive A \grad w - m w, \Delta v -v, (w-v)|_{\partial D}, (\dna
w - \dnu v)|_{\partial D} \right)
\end{equation}
where $m $ is a constant satisfying $\gamma^{-1} \le m < 1$.
Theorem \ref{existence2} shows that the inverse of ${\cal G}$
exists and is continuous. Since ${\cal G}$ is continuous,  we
deduce that ${\cal G}$ is a bijective operator. Now consider the
operator ${\cal T}$ from ${\cal X}(D)$ into $L^2(D) \times L^2(D)
\times H^\demi(\partial D) \times H^{-\demi}(\partial D)$ defined
by
$$
{\cal T}(w,v) = \left((k^2 \,n + m) w, \; (k^2 \, + 1) v, \; 0, \; 0 \right)
$$
By the compact embedding of $H^1(D)$ into $L^2(D)$, the operator ${\cal T}$ is
 compact. Hence ${\cal G}+{\cal T}$ is a Fredholm operator of index one.
 From Theorem 3.1 of \cite{CCH01}
we have that ${\cal G}+{\cal T}$ is injective and therefore we
deduce the existence and the continuity of $({\cal G}+{\cal
T})^{-1}$, which means in  particular the existence of a unique
solution to (ITP) satisfying the a priori estimate
(\ref{aprior:estimate3}).
\endproof

Due to a lack of uniqueness in the case where $n$ and $A$ are
real (see Theorem 3.1 of \cite{CCH01}) we cannot conclude the
solvability of (ITP). This leads to the problem of transmission
eigenvalues, i.e. the values of $k$ for which the corresponding
homogeneous (ITP) ($f=h=0$) has a non trivial solution. For these
eigenvalues the linear sampling method fails, and therefore it is
important to characterize the set of these critical values. It is
shown in \cite{CCH01} that this set is discrete under the
assumption $\left( \, \bar{\xi} \cdot A(x) \, \xi \, \right) \ge
\gamma |\xi|^2$ where $\gamma >1$. We prove here an analogous
result for the case $\left( \, \bar{\xi} \cdot A(x)^{-1} \, \xi \,
\right) \ge \gamma |\xi|^2$.

\begin{Th}
Assume that ${\cal I}m \, n = 0 $ and  ${\cal I}m \,
 A = 0 $ in $D$ and that there exists
a constant $\gamma > 1$ such that for almost every $x \in D$,
\begin{equation} \label{cond:B}
\left( \, \bar{\xi} \cdot A(x)^{-1} \, \xi \, \right) \ge \gamma
|\xi|^2 \;\;\; \forall \; \xi \in \R^3.
\end{equation}
Then the set of the values of $k \in \C$ for which {\rm (ITP)} does not have a
unique solution is discrete.
\end{Th}

\proof Consider the operator ${\cal G}$ defined by
(\ref{definition:G}) where $\gamma^{-1} \le m < 1$ and the
operator ${\cal T}_{k}$ from ${\cal X}(D)$ into $L^2(D) \times
L^2(D) \times H^\demi(\partial D) \times H^{-\demi}(\partial D)$
defined by
$$
{\cal T}_{k}(w,v) = \left((k^2 n +m) w,  (k^2+1)v, \; 0, \; 0
\right).
$$
We want to prove that the operator ${\cal G} + {\cal T}_{k}$ is
invertible for all $k \in \C \setminus S$ where $S$ is  a
discrete subset of $\C$. Since ${\cal G}$ is bijective (Theorem
\ref{existence2}), this is equivalent to showing that $(I + {\cal
G}^{-1} \, {\cal T}_k)^{-1}$ exists, where $I$ is the identity
operator from ${\cal X}(D)$ into ${\cal X}(D)$. Let us defined
the operator valued function $A$ : $\C \rightarrow {\cal L}({\cal
X}(D))$ by $A(k)= -{\cal G}^{-1} \, {\cal T}_k$. Since ${\cal
T}_k$ is compact and ${\cal G}^{-1}$ is bounded, $A(k)$ is then
compact for all $k\in \C$. Moreover, since ${\cal T}_k$ depends
polynomially on $k$ and the function $n$ is bounded, it is easy
to see that $A$ is analytic on $\C$.

We also notice that ${\cal G} + {\cal T}_i$ is injective (this
corresponds to the uniqueness of the solution to problem
(\ref{itp}) with $k=i$ (similar arguments as in the proof of
theorem 3.1 in \cite{CCH01})). Hence the operator $I - A(i) =
{\cal G}^{-1} ({\cal G} + {\cal T}_i)$ is injective and therefore
invertible. Now applying the analytic Fredholm theory (Theorem
8.26 in \cite{CK98}) shows that $(I-A(k))^{-1}$ exists for all $k
\in \C \setminus S$ where $S$ is  a discrete subset of $\C$.
\endproof
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\bibitem{CCH01}
F. Cakoni, D. Colton, H. Haddar,
\newblock{The linear sampling method for anisotropic media},
\newblock (2001) (to appear).


\bibitem{CCM00}
D. Colton, J. Coyle, P. Monk,
\newblock{Recent developments in inverse acoustic scattering theory},
\newblock{\em SIAM Review}, {\bf 42} (2000) No. 3,  369-414.


\bibitem{CK98}
D. Colton, R. Kress,
\newblock {\em Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed}, Springer Verlag, Berlin (1998).


\bibitem{FGC}
F. Gylys-Colwell,
\newblock{An inverse problem for the Helmholtz equation}, {\em Inverse
Problems} {\bf 16} (2000), 139-156.

\bibitem{H00}
P. H{\"a}hner,
\newblock{On the uniqueness of the shape of a penetrable, anisotropic obstacle}, {\em
J. Comput. Appl. Math.} {\bf 116} (2000) 167-180.


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\end{document}
