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inverse problems, Dirichlet-to-Neumann map
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\begin{document}
\begin{titlepage}
\begin{center}
{\bf SPECTRAL ASYMPTOTICS OF THE DIRICHLET-TO-NEUMANN MAP \\
ON MULTIPLY CONNECTED DOMAINS IN $\R^d$ }
\vspace{1 cm}
\setcounter{footnote}{0}
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{\bf P.\ D.\ Hislop \footnote{Supported in part by NSF grant DMS-9707049.}
} \\
\vspace{0.3 cm}
{\ten Department of Mathematics \\
University of Kentucky \\
Lexington, KY 40506--0027 USA}
\vspace{0.3 cm}
{\bf C.\ V.\ Lutzer}
\vspace{0.3 cm}
{\ten Department of Mathematics and Statistics \\
Rochester Institute of Technology \\
Rochester, NY 14623-5603 USA}
\end{center}
\vspace{0.5 cm}
\begin{center}
{\bf Abstract}
\end{center}
\noindent
We study the spectral asymptotics of the Dirichlet-to-Neumann operator
$\Lambda_\gamma$ on a multiply-connected, bounded, domain in $\R^d$,
$d \geq 3$, associated
with the uniformly elliptic operator $L_\gamma = - \sum_{i,j=1}^d
\partial_i \gamma_{ij} \partial_j$, where $\gamma$
is a smooth, positive-definite, symmetric matrix-valued function
on $\Omega$.
We prove that the operator is approximately diagonal
in the sense that $\Lambda_\gamma = D_\gamma + R_\gamma$, where $D_\gamma$
is a direct sum of operators, each of which acts on one boundary
component only, and $R_\gamma$ is a smoothing operator.
This representation follows from the
fact that the $\gamma$-harmonic
extensions of eigenfunctions of $\Lambda_\gamma$ vanish
rapidly away from the boundary.
Using this representation, we study the inverse problem of
determining the number of holes in the body, that is,
the number of the connected
components of the boundary, by using the high-energy spectral
asymptotics of $\Lambda_\gamma$.
\vspace{0.3 cm}
\noindent
\today
\end{titlepage}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction and Main Results}\label{S.1}
We study the spectral asymptotics of the Dirichlet-to-Neumann (DN)
operator $\Lambda_\gamma$ associated with a uniformly elliptic,
second-order differential operator $L_\gamma$, on a bounded,
multiply-connected region $\Omega \subset \R^d$, $d \geq 3$, with smooth boundary.
The elliptic operator $L_\gamma$ has the form
\beq
L_\gamma \equiv - \Sum_{j,k = 1}^d \partial_j \; \gamma_{jk} \; \partial_k ,
\eeq
where we assume that the $d \times d$-matrix-valued function $\gamma (x)
= [ \gamma_{jk} (x) ]$ satisfies
the following hypotheses:
\begin{verse}
H1. The real coefficients satisfy $\gamma_{jk} (x) = \gamma_{kj} ( x)
\in C^{\infty} ( \overline{ \Omega } ) $. \\
H2. There exist constants $0 < \lambda_0 \leq \lambda_1 < \infty$ such that
for all $\xi \in \R^d$,
we have
$$
\lambda_0 \| \xi \|^2 \leq \Sum_{j,k = 1}^d
\xi_j \xi_k \gamma_{jk} ( x) \leq \lambda_1 \|
\xi \|^2 .
$$
\end{verse}
The DN map $\Lambda_\gamma$ is defined as follows. Let $f \in C^{ \infty
} ( \partial \Omega )$, and denote by $u_f (x)$ the unique solution of the
Dirichlet problem
\bea
L_\gamma u (x) = 0 , & ~~~x \in \Omega \nonumber \\
u \: | \: \partial \Omega = f & ~~~x \in \partial \Omega .
\eea
We then define
\beq
\Lambda_\gamma f \equiv \sum_{l, m =1}^d
\left. \left\{ \nu_l \gamma_{lm} \frac{ \partial u_f }{ \partial x_m } \right\}
\: \right| \: \partial \Omega ,
\eeq
where $ \nu $ denotes the outward normal vector on
$\partial \Omega$.
The DN operator $\Lambda_\gamma$
extends to a bounded map $\Lambda_\gamma : H^{1/2} ( \partial
\Omega ) \rightarrow H^{-1/2} ( \partial \Omega )$.
Furthermore, the DN operator $\Lambda_\gamma$ is an unbounded, self-adjoint
operator on $L^2 ( \partial \Omega )$
with a compact resolvent (cf. \cite{[Taylor2],[Uhlmann2]}).
Consequently, the
$L^2$-spectrum of $\Lambda_\gamma$ is discrete with no finite accumulation
point.
Let $\{ \lambda_j \: | \: \lambda_1 = 0 , \; \lambda_j \leq \lambda_{j+1},
j = 1 , 2 , \ldots \}$
denote the eigenvalues of the DN operator $\Lambda_\gamma$
listed in nondecreasing order, including multiplicity.
Let $\Lambda_{\gamma , \Omega} ( x , \xi )$, for $(x, \xi) \in T^*
\partial \Omega $, be the symbol of
the first-order, elliptic
pseudodifferential operator $\Lambda_\gamma$. The
eigenvalues $\lambda_j$ of $\Lambda_\gamma$ satisfy classical
Weyl asymptotics (cf.\ \cite{[Hormander]}, chapter XXIX, or
\cite{[Taylor]}, chapter XII):
$$
\lambda_j \sim ( j /C( \partial \Omega,
\Lambda_{ \gamma , \partial \Omega}))^{ 1 / ( d - 1
) } ,
$$
where
\begin{eqnarray*}
C( \partial \Omega , \Lambda_{ \gamma, \partial \Omega} ) &=& (2
\pi )^{- (d-1)} \; \mbox{Vol} \; \{ (x , \xi ) \in T^*\partial
\Omega \backslash \{0\} \; | \; \Lambda_{\gamma , \partial \Omega} ( x ,
\xi ) \leq 1 \} \\
&=& (2\pi )^{- (d-1)} \int_{
\Lambda_{\gamma , \partial \Omega }( x , \xi ) \leq 1} \; dx ~d \xi .
\end{eqnarray*}
The unique solution $u_f$ of the Dirichlet problem (1.2) with
boundary datum $f$ is called the {\it $\gamma$-harmonic extension of
$f$}.
Our first result concerns the localization of the $\gamma$-harmonic
extension of an eigenfunction of $\Lambda_\gamma$ near the boundary.
We say that a function $g$ {\it decays rapidly}, written
$g(m) = O(m^{-\infty})$, if
$\lim_{m\rightarrow\infty} m^{k}g(m) = 0$, for every $k\in \N$.
\begin{theorem}\hspace{-0.2cm}{\bf .}
\label{rapid_localization}
Let $\phi_m$ be an eigenfunction of $\Lambda_\gamma$ satisfying
$\Lambda_\gamma \phi_m = \lambda_m \phi_m$,
with $\|\phi_m\|_{\ltwos{\pom}} = 1$, and let $u_m$ be the
$\gamma$-harmonic extension of $\phi_m$ to $\Omega $. For any compact
$K\subset \Omega$,
$||u_m||_{H^1(K)} = O(m^{-\infty})$.
\end{theorem}
This result reflects the fact that the eigenfunctions of the DN operator
$\Lambda_\gamma$ become highly oscillatory as the eigenvalue increases,
and hence the $\gamma$-harmonic extensions decay rapidly away from the boundary.
We believe that the decay is actually of order $e^{- \mbox{dist} ( K ,
\partial \Omega ) |m| }$ in the case of an analytic boundary and analytic
coefficients, but we have not been able to prove this.
The localization result in Theorem 1.1 is the basis of our other results.
We are interested in the situation when $\partial \Omega$ consists of
$k$ mutually disconnected components, $\partial \Omega_j$, with each
boundary component
$\partial \Omega_j$ a smooth, connected, compact surface: $\pom =
\cup_{j=1}^{k} \pom_j$. We label the boundary components so that
$\pom_1$ is the boundary of the
unbounded component of $\R^d \backslash \Omega$.
We note that
the regions $\mbox{Int} ( \pom_j )$, bounded by the other boundary components
$\pom_j$, are disjoint. They are contained in the bounded region interior to
$\partial \Omega_1$.
It follows that $\ltwos{\pom} = \oplus_{j=1}^{k}\ltwos{\pom_j}$.
We write
$\phi\in\ltwos{\pom}$ as the $k$-tuple
$\phi=(\phi_1,\phi_2,\ldots ,\phi_k)$, where
$\phi_j = \phi \: | \: {\pom_j}$. The DN operator $\Lambda_\gamma$ is a map
between $k$-tuples of functions
defined on the boundary.
For each boundary component, we define the restriction operator
$R_j:\ltwos{\pom}\rightarrow\ltwos{\pom_j}$ by $R_j\phi = \phi_j$, and
the extension operator $E_j:\ltwos{\pom_j}\rightarrow\ltwos{\pom}$
by $E_j\phi = \psi$, where
$\psi_i = 0$, when $i\not = j$, and $\psi_j = \phi$.
It is easy to check that $C_j \equiv E_j R_j$ is an
orthogonal projection on $\ltwos{\pom}$.
The family of operators $\{ C_j \; | \; j = 1 ,
\ldots, k \}$ satisfies $C_j C_l = \delta_{jl} C_j$, and $Id = \sum_{j = 1}^k C_j$,
on $\ltwos{\pom}$. A pseudodifferential operator $P$ on a smooth manifold $X$
is said to be {\it smoothing}
if $P : H^{-t} ( X ) \rightarrow H^s (X)$, for all $s, t \in \R^+$.
\begin{theorem}\hspace{-0.2cm}{\bf .}
\label{diagonal_plus_remainder}
The DN operator $\Lambda_\gamma$
admits a decomposition into two self-adjoint operators
$\Lambda_\gamma = D_\gamma + R_\gamma$,
where $D_\gamma = \sum_{j=1}^k C_j \Lambda_\gamma C_j$, and $R_\gamma$
is a smoothing operator.
\end{theorem}
Viewed as a map between $k$-tuples of functions on $\partial \Omega$, the
DN operator $\Lambda_\gamma$ can be represented as a $k \times k$-matrix.
Theorem \ref{diagonal_plus_remainder}
indicates that, in this representation, the DN
map $\Lambda_\gamma$ is diagonal up to a
smoothing error.
The fact that $\Lambda_\gamma$ and $D_\gamma$ differ by a smoothing operator
indicates that their spectra have the same asymptotics.
\begin{theorem}\hspace{-0.2cm}{\bf .}
\label{spectral_asymptotics_1}
Let $\{ \mu_l \; | \; l \in \N \} $ be the eigenvalues of
$D_\gamma$, written in nondecreasing order, including multiplicity,
and let $\{ \lambda_l \; | \; l \in \N \}$ be the eigenvalues of
$\Lambda_\gamma$ written similarly. Then, we have
$\lambda_l=\mu_l+O(l^{-\infty})$.
\end{theorem}
We can make precise the nature of the operators $C_j \Lambda_\gamma C_j$
composing the diagonal operator $D_\gamma$. We define operators
$\Lambda_j$ on the boundary component $\Omega_j$ by $\Lambda_j
\equiv R_j \Lambda_\gamma E_j : L^2 ( \partial \Omega_j ) \rightarrow L^2
( \partial \Omega_j )$. Since $R_j E_j = 1$ on $L^2 ( \partial \Omega_j)$,
it is clear that the eigenvalues of $C_j
\Lambda_\gamma C_j$ coincide with those of $\Lambda_j$. Although the operator
$\Lambda_j$ involves the other boundaries through
$\Lambda_\gamma$, we will show that the effect of the other boundary
components is small in the high-energy limit.
One result in this direction is the following (see section 5 for
more details). We extend
$\gamma_{ij} \in C^\infty (\overline{\Omega})$
to be a smooth function on all of $\R^d$. We assume
that the extended matrix of functions $\gamma = [ \gamma_{ij} ]$
remains symmetric
and uniformly elliptic so that H2 remains valid for some
constants
$0 < \lambda_0 \leq \lambda_1 < \infty$.
Furthermore, we assume
\begin{verse}
H3. There exists $0 < R < \infty$ so that $\Omega \subset B_R
(0)$, and $\gamma_{ij} ( x ) = \delta_{ij}$, for $\|x\| > R$.
\end{verse}
It follows from section 5 that the difference of two DN operators
associated with various connected components of $\pom$,
and constructed with two different extensions of $\gamma$, is a smoothing
operator.
We introduce operators $\lsh_j$ that involve only the
$j^{th}$-boundary component as follows.
For $\pom_1$, we denote by $\Omega_1^\#$ the bounded component of
$\R^d\backslash \pom_1$.
For the other boundary components with $1 < j \leq k$,
let $\Omega_j^\#$ be the unbounded
component of $\R^d\backslash \pom_j$.
We define operators
$\lsh_j:\ltwos{\pom_j}\rightarrow\ltwos{\pom_j} $,
acting on a single boundary component, by
$\lsh_j f = \nu \cdot \gamma \nabla u_f^\#| {\pom_j}$, where $u_f^\#$ is the
unique $\gamma$-harmonic extension of $f$ to $\Omega_j^\#$,
that decays at infinity for $1 < j \leq k$ (see the appendix, section 8), and
$\nu$ is the outward normal to that region.
\begin{theorem}\hspace{-0.2cm}{\bf .}
\label{smoothing_argument}
The difference $\Lambda_j - \Lambda_j^\#$,
on $L^2 ( \pom_j )$, is a smoothing operator.
Consequently, the difference of the $m^{th}$ eigenvalues of these two operators
vanishes like $O ( m^{- \infty} )$.
\end{theorem}
We now discuss the application of these results to the inverse problem of
determining the connectivity of a body $\Omega$ from the high-energy
asymptotics of the DN operator $\Lambda_\gamma$.
By the {\it weighted measure} of the boundary component $\pom_j$, we mean
the constant, $C( \partial \Omega_j ,
\Lambda_{ \gamma, \partial \Omega} )$, similar to the one
appearing in the Weyl eigenvalue asymptotics, given by
$C( \partial \Omega_j ,
\Lambda_{ \gamma, \partial \Omega} ) = (2\pi )^{- (d-1)}
\; \mbox{Vol} \; \{ (x , \xi ) \in T^*\partial
\Omega_j \backslash \{0\} \; | \; \Lambda_{\gamma , \partial
\Omega} ( x , \xi ) \leq 1 \}$,
where $\Lambda_{\gamma, \partial \Omega } ( x , \xi )$ is the symbol of
$\Lambda_\gamma$. In the case that $\gamma_{ij} = \delta_{ij}$,
we have $C( \partial \Omega_j , \Lambda_{ 1, \partial
\Omega} ) = | \partial \Omega_j | \; ( \Gamma ( (d-1)/2 + 1 ) (4 \pi
)^{(d-1)/2} )^{-1}$.
\begin{theorem}\hspace{-0.2cm}{\bf .}
\label{spectrum_geometry}
The high-energy spectrum of $\Lambda_\gamma $ determines a lower bound
on the number of connected components of $R^d\backslash\Omega $, and
also determines the weighted measure of each boundary component (not counting
multiplicities).
\end{theorem}
The phrase ``not counting multiplicities''
means that the asymptotics of $\sigma(\Lambda_\gamma )$ may not
indicate the existence of more than one boundary component with
the same weighted measure.
This result is outside the domain of
nondestructive evaluation since in order to determine the asymptotics of
the eigenvalues of $\Lambda_\gamma$, boundary data on the interior
surfaces $\pom_j , j \neq 1$, must be specified.
In order to model a more realistic situation for which the methods of
nondestructive evaluation can be applied, we can modify the problem as follows.
We assume that the voids $\mbox{Int}( \pom_j)$, for $j \neq 1$,
are filled with perfectly insulating material.
This implies that the normal component of the current
across each interior
boundary component $ \pom_j$, for $j \neq 1$,
satisfies $\nu \cdot \gamma \nabla u \: | \: \pom_j = 0$.
We further suppose that boundary data
on the outer surface $\partial \Omega_1$ is
specified. We can then define a
corresponding DN operator for this mixed-problem on $L^2 (
\partial \Omega_1 )$, and ask if the
high-energy asymptotics of the spectrum of this operator allow us to
determine the number of interior components.
However, we prove that a result similar to Theorem 1.1 holds in this case also.
That is, we prove that the $\gamma$-harmonic extensions of the
eigenfunctions of the DN operator for the
mixed-problem localize near $\partial \Omega_1$.
Consequently, the number of connected components of $\pom$
cannot be determined from
the high-energy asymptotics of the spectrum of this DN operator.
The results of this initial investigation for nondestructive evaluation
are negative in that the high-energy asymptotics of the spectrum,
obtained by measurements external to the body, are not
sufficient to determine the connectivity of a body. It might be possible,
however, to use all the eigenvalues in order to determine the
connectivity. One result in this direction is due to J.\ Edward
\cite{[Edward]}. Using the zeta function associated with the
eigenvalues of the DN operator, Edward
proved that the disk in $\R^2$ of radius $1$
is determined by the spectrum of the DN operator in the sense
that any other simply-connected, bounded region in the plane with
boundary measure $2 \pi$ is isomorphic to the disk under Euclidean motions.
(In fact, a stronger result is known in this two-dimensional
case. The first eigenvalue
of the DN operator determines the disk \cite{[Weinstock]} among all simply
connected regions with the same boundary measure).
Quite recently, another approach has been taken by Lassas and Uhlmann
\cite{[LassasUhlmann]} who
proved that the DN map, restricted to a nonempty, open,
real-analytic subset of the boundary, determines a compact, connected,
real-analytic Riemannian manifold
in dimensions $d \geq 3$, and the conformal class of a smooth,
connected, compact Riemannian surface.
This result was refined by Lassas, Taylor, and Uhlmann \cite{[LTU]}
who proved that for $d \geq 3$, the DN operator, restricted to
a nonempty, open, subset of the boundary determines the complete,
connected, real-analytic Riemannian manifold (not necessarily
compact, but with compact, nonempty boundary).
Consequently, the DN map restricted to Dirichlet data supported
on a piece of the
boundary determines the connectivity in the real analytic case.
The methods of these papers, however, do not indicate how to
determine the connectivity from the DN map acting on functions
supported on a piece of the boundary.
In this paper, we show that the spectral asymptotics of the DN map
(employing Dirichlet data on the entire boundary)
do determine the connectivity, and that a lower bound on the
connectivity can be calculated from these asymptotics.
We mention some related works concerning the location of discontinuities
within a body by measurements on the exterior surface. Isakov
\cite{[Isakov]} proved that one can locate a discontinuity
(supported on an open set) in the scalar
conductivity within a body using the DN map associated with the
exterior surface. There is some similarity between the contents
of section 7 and the work of Friedman and Vogelius \cite{[FV]}.
These authors consider, for the case of scalar conductivity,
the question of locating small
inhomogeneities of extreme conductivity in a conducting body
using the DN operator.
The classic inverse problem of determining the scalar
conductivity from the DN operator has been studied extensively,
see, for example \cite{[KohnVogelius],[Nachman],[SylvesterUhlmann]}.
For a general discussion of inverse problems for isotropic and
anisotropic materials, we refer the reader to the lecture notes of
Uhlmann \cite{[Uhlmann],[Uhlmann2]}.
\vspace{.1in}
\noindent
{\bf Acknowledgments.} We thank R.\ M.\ Brown, P.\ A.\ Perry, Z.\ Shen,
and G.\ Uhlmann for many valuable remarks.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\thechapter}{\arabic{chapter}}
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\section{Preliminaries}\label{S.2}
We review some basic material needed in the proofs of the main theorems.
The results on solutions to the Dirichlet problem can be found in
many texts, cf.\ \cite{[Evans]}. A nice
account of the Dirichlet-to-Neumann operator can be found in
\cite{[Uhlmann2]}.
We always assume that the boundary components are smooth and that the
matrix $\gamma$ satisfies hypotheses H1 and H2. We are concerned with the
Dirichlet problem (1.2) for $L_\gamma$ and $\Omega$.
The trace onto the boundary of $\Omega$ plays a key role.
\begin{lemma}\hspace{-0.2cm}{\bf .}
\label{trace_Lemma}
Suppose $s> 1/2$. The restriction map $Tu=u|_{\pom}$, for $u\in
H^s(\Omega )\cap C(\overline{\Omega})$, extends to a bounded linear map
$T:H^s(\Omega )\rightarrow H^{s-1/2}(\pom )$.
The kernel of $T$ is exactly $H_0^s(\Omega )$.
\end{lemma}
We next need an estimate on the $H^1$-bound of a function on $\Omega$ in
terms of its boundary value and the $L^2$-norm of its derivatives.
%used in existence of solns and in section 7 on perfect insulators
\begin{lemma}\hspace{-0.2cm}{\bf .}
\label{L2_bnding_Lemma}
Suppose $\Gamma\subset\pom$ is a set of positive
($d$-$1$)-dimensional measure. Then there exists a
constant $C$, dependent upon $\Gamma$, so that for any
$u \in H^1(\Omega )$,
\begin{equation}
\label{poincare_like_bnd}
||u||_{H^1(\Omega )}^2 \leq C \left \{
||Tu||_{\ltwos{\Gamma} }^2 +
\sum_{j=1}^d \left | \left | \pard{u}{x_j} \right | \right
|_{\ltwos{\Omega}}^2 \right \} .
\end{equation}
\end{lemma}
We recall the main theorem on the solvability of the Dirichlet problem,
which we write in its nonhomogeneous version. Let $f \in H^{1/2} ( \pom
)$, and $F \in H^{-1} ( \Omega)$. We say
that $u_f$ is a solution to the Dirichlet problem in the weak
sense if for any $\phi \in C_0^{\infty} ( \Omega)$, we have,
\bea
\label{gamma_harmonic}
\Sum_{j, k=1}^d \Int_\Omega \partial_j \phi \gamma_{jk} \partial_k u_f
& = & \Int_\Omega F \phi, \\
T u_f &=& f .
\eea
\begin{theorem}\hspace{-0.2cm}{\bf .}
\label{mapping}
The mapping
${\cal F}_\gamma :H^1(\Omega)\rightarrow
H^{-1}(\Omega )\times H^{1/2}(\partial \Omega )$
defined by ${\cal F}_\gamma u = ( L_\gamma u, Tu )$ is an isomorphism.
Further, if ${\cal F}_\gamma u = (F,f)$, the function $u$ satisfies the estimate
\beq
||u||_{H^1(\Omega)}
\leq C \left \{ ||F||_{H^{-1}(\Omega)} + ||f||_{H^{1/2}(\partial \Omega)} \right \}
.
\eeq
\end{theorem}
The unique function $u_f$ that solves (\ref{gamma_harmonic})
with $F = 0$ will be called
the {\it $\gamma$-harmonic extension} of $f$ into $\Omega$.
Under the smoothness assumptions, the $\gamma$-harmonic
extension of $f \in H^{1/2} ( \pom )$ is actually in $C^{\infty} ( {\overline
\Omega } )$.
Given Theorem \ref{mapping}, the DN operator can be defined as follows.
Initially,
we define
$\Lambda_\gamma$ on $H^{3/2} ( \pom )$, to insure that the trace of the
outward normal derivative of the $\gamma$-harmonic extension of $f$ exists.
We have
\begin{equation}
\Lambda_\gamma f = \sum_{l, m = 1 }^d \nu_l \gamma_{lm} \partial_m u_f \: | \: {\pom},
\label{dtn_def}
\end{equation}
where $ u_f $ is the $\gamma$-harmonic extension of $f$.
To simplify the notation, we will write
\beq
\nu \cdot \gamma \nabla \equiv \sum_{l, m = 1 }^d \nu_l \gamma_{lm} \partial_m.
\eeq
The domain of $\Lambda_\gamma$ can be extend through a duality argument.
\begin{theorem}\hspace{-0.2cm}{\bf .}
\label{extend domain of Lambda}
The linear operator $\Lambda_\gamma $ defined by (\ref{dtn_def}) extends to a
bounded map $\Lambda_\gamma : H^{1/2}(\pom) \rightarrow H^{-1/2}(\pom)$.
\end{theorem}
As an operator on the Hilbert space $L^2 ( \pom )$, the DN
operator is a nonnegative, self-ajoint operator with compact
resolvent. Consequently, the spectrum of $\Lambda_\gamma$, as an
operator on $L^2 ( \pom )$, is discrete, and consists of
eigenvalues $\lambda_j$, with $\lambda_j \rightarrow \infty$.
We mention that in the simple case $\gamma_{ij} =
\delta_{ij}$, the DN operator is, roughly, the operator
$ \sqrt{-\Delta_{\pom}}$, where $- \Delta_{\pom}$ is the
Laplace-Beltrami operator on
$\pom$, with the induced metric.
We refer the reader to \cite{[Taylor2]} and \cite{[LeeUhlmann]}
for additional information about this representation.
Extensions of boundary data into unbounded
sets containing $\Omega$ will be of interest. We extend
$\gamma_{ij} \in C^\infty (\overline{\Omega})$
to be a smooth function on all of $\R^d$. We assume
that the extended matrix of functions $\gamma = [ \gamma_{ij} ]$
remains symmetric
and uniformly elliptic so that H2 remains valid for some constants
$0 < \lambda_0 \leq \lambda_1 < \infty$.
Furthermore, we assume
\begin{verse}
H3. There exists $0 < R < \infty$ so that $\Omega \subset B_R
(0)$, and $\gamma_{ij} ( x ) = \delta_{ij}$, for $\|x\| > R$.
\end{verse}
We will always assume this extension has been taken in later sections
when we consider $\gamma$-harmonic functions
outside of, and into the interior of, the region $\Omega$.
It follows from the analysis in section 5
that the main results of this paper are independent of the choice of this
extension in the following sense.
Suppose that $\gamma_1$ and
$\gamma_2$ are two smooth extensions of $\gamma$ satisfying H1,
H2, and H3. Then, the differences of the DN operators,
$\Lambda_j^\# (\gamma_1) - \Lambda_j^\# (\gamma_2)$, and
$\Lambda_{\pom_j}(\gamma_1) - \Lambda_{\pom_j} (\gamma_2)$,
associated with $\pom_j$, and defined with $\gamma$-harmonic
extensions to the exterior, respectively, interior, regions (see
section 5), are smoothing operators.
Hence, each pair of DN operators has the same high energy
spectral asymptotics.
Representation formulae for solutions of Dirichlet problems will play
a central role in this analysis. For this, we need information on
the Green's function $G_\gamma$, corresponding to the extended, elliptic
operator $L_\gamma$ on $\R^d$, and on the Dirichlet Green's function
$G_{\gamma, \Omega}$, corresponding to the (extended) elliptic
operator $L_\gamma$ on an open domain (bounded or unbounded)
$\Omega \subset \R^d, d\geq 3$, with $\partial \Omega \neq
\emptyset$.
Consequently, we state the following theorem for a
general, open region $\Omega
\subset \R^d$, for $d \geq 3$. The proof of parts of this theorem for
real, symmetric $\gamma_{ij}$ is contained in the paper of
Littman, Stampacchia, and Weinberger \cite{[LSW]}.
Their results hold for $\gamma_{ij} \in L^\infty ( \R^d)$, although we will
state them here only under conditions H1--H3.
The results of \cite{[LSW]} were generalized to not necessarily
symmetric $\gamma_{ij}$ by Gr\"uter and Widman \cite{[GruterWidman]}.
In their paper,
Gr\"uter and Widman construct the Dirichlet Green's function for
$L^\infty$-coefficients on bounded, open domains.
In a separate note \cite{[HislopLutzer]}, we
show how this proof can be extended
to general open regions (including $\Omega = \R^d, d \geq 3$) with,
or without, smooth boundary.
In the following theorem, the local Sobolev space
$H^{1,s}_{loc} ( \R^d)$ consists of those measurable functions $f$
so that for any bounded, open region $\Omega \subset \R^d$, we have
$f \in H^{1,s} ( \Omega)$. The local Sobolev space
$H^{1,s,0}_{loc} ( \Omega )$, for an unbounded region $\Omega \subset
\R^d$, with $\partial \Omega \in C^1$, consists of those functions $g \in
H_{loc}^{1,s} ( \Omega )$ for which $Tg = 0$, where $T : H_{loc}^{1,s} (\Omega)
\rightarrow L^s ( \partial \Omega )$ is the trace map.
\begin{theorem}\hspace{-0.2cm}{\bf .}
\label{Green's function}
Let $\Omega \subset \R^d$ be an open subset of $\R^d, d \geq 3$, with smooth
boundary. We assume that the coefficients of the uniformly
elliptic operator $L_\gamma$
satisfy hypotheses H1, H2, and H3.
There is a nonnegative function
$G_{\gamma, \Omega}: \Omega \times \Omega \rightarrow \R^+ \cup \{ \infty \}$,
such that, for any $s \in [1 , d / (d-1) )$,
\begin{enumerate}
\item If $\Omega = \R^d$, we have
$G_{\gamma}(x , y) \in H^{1,s}_{loc} ( \R^d)$, for each fixed $y
\in \R^d$;
\item If $\partial \Omega \neq \emptyset$,
with Dirichlet boundary conditions,
we have $G_{\gamma, \Omega}(x,y) \in H^{1,s,0}_{loc} (
\Omega )$, for each fixed $y \in \Omega$.
\end{enumerate}
For all $\phi\in C_0^1(\Omega )$, this function satisfies,
\begin{equation}
\label{eq:G_0_is_a_weak_delta}
\int_{\Omega} \grad\phi \cdot \gamma \grad G_{\gamma, \Omega}
(x,\cdot)~dy=\phi(x).
\end{equation}
Furthermore, $G_{\gamma, \Omega}$ has the following properties:
\begin{enumerate}
\item[{\bf (a.)}]
When $x\not = y$, $G_{\gamma, \Omega} (x,y) = G_{\gamma, \Omega} (y,x)$.
\item[{\bf (b.)}]
The function $G_{\gamma, \Omega} (x,y)$ is smooth for $x\not = y$.
\item[{\bf (c.)}]
There is a finite constant $K > 0$ such that
\beq
K^{-1} \| x - y \|^{2-d} \leq G_{\gamma, \Omega} (x,y) \leq K \| x-y \|^{2-d}.
\eeq
\item[{\bf (d.)}] There is a finite constant $K_1 > 0$ such that
\beq
|\grad G_{\gamma, \Omega} (x,y)| \leq K_1 \| x-y\|^{1-d}.
\eeq
\item[{\bf (e.)}]
For the case when $\partial \Omega \neq \emptyset$, we
have the Dirichlet boundary conditions:
$G_{\gamma, \Omega}( \omega , y ) = 0$, for $\omega \in \partial
\Omega$ and $y \in \Omega$.
\end{enumerate}
\end{theorem}
In the following, we will write $dy$ for the volume measure, $d
\sigma$ for the induced surface measure, and $d \omega$ for the surface
measure on the unit sphere $S^{d-1}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\setcounter{chapter}{3}
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\section{Rapid Decay of the $\gamma$-Harmonic Extensions}\label{S.3}
The Green's function $G_\gamma$ for $\R^d, d \geq 3,$ described in Theorem 2.3
allows us to write a
representation formula for the solution to the Dirichlet problem (1.2)
with boundary data $f \in H^{1/2} ( \partial \Omega)$.
This representation formula will be the basis of the proof of Theorem
1.1.
\begin{prop}\hspace{-0.2cm}{\bf .}
\label{w_f representation Lemma}
Suppose $w_f$ is the $\gamma$-harmonic extension of $f \in H^{1/2} (
\partial \Omega)$ into
$\Omega$, and $G_\gamma$ is the function from
Theorem 2.3 for $\R^d, d \geq 3$. Then
\begin{equation}
\label{gamma harmonic representation formula}
w_f(x) = \int_{\pom}
\left\{ G_\gamma(x, \cdot ) \Lambda_\gamma f -
f \nu \cdot \gamma \nabla G_\gamma(x,\cdot) \right \} ~d\sigma .
\end{equation}
\end{prop}
{\sc Proof:}
Fix $x\in\Omega$ and choose a function $\eta\in C_0^\infty(\R^d)$
that is identically one in a neighborhood of $x$ and vanishes near
$\pom$. Then $w_f = \eta w_f + (1-\eta)w_f$. Because $w_f$ lies in
the kernel of an elliptic operator with smooth coefficients, it is
smooth away from $\pom$, so
$\eta w_f\in C^\infty_0$. We use Theorem 2.3 to write
\begin{eqnarray}
\label{eq:w_f_summand1}
w_f(x) = (\eta w_f)(x)
& = &
\int_\Omega \grad G_\gamma(x,\cdot)\cdot \gamma \grad (\eta w_f)~dy
\nonumber \\
& = &
\int_\Omega \left\{ \mbox{div}\{ G_\gamma(x,\cdot)\gamma\grad (\eta w_f)\}
- G_\gamma(x , \cdot)\Lg (\eta w_f) \right \}~dy
\nonumber \\
& = &
-\int_\Omega G_\gamma(x , \cdot)\Lg (\eta w_f)~dy
\end{eqnarray}
according to the Divergence Theorem, since $\grad (\eta w_f)$ is
compactly supported away from $\pom$. We also know, because of
(\ref{eq:G_0_is_a_weak_delta}) and part (b.) of Theorem
2.3 that
\begin{eqnarray}
\label{eq:w_f_summand2}
\lefteqn{ -\int_\Omega G_\gamma(x,\cdot)\Lg (1-\eta)w_f~dy}
\nonumber \\
& = & \int_\Omega \left \{ (1-\eta )w_f\Lg G_\gamma (x,\cdot ) -
G_\gamma (x,\cdot) \Lg (1-\eta)w_f \right \}~dy \nonumber \\
& = & \int_{\pom} \left\{ (1-\eta )w_f \nu \cdot \gamma \nabla
G_\gamma (x,\cdot )
- G_\gamma (x,\cdot) \nu \cdot \gamma \nabla (1-\eta)w_f \right \}~d\sigma
\nonumber \\
& = & \int_{\pom} \left\{ f \nu \cdot \gamma \nabla G_\gamma (x,\cdot)-
G_\gamma (x,\cdot) \Lambda_\gamma f \right \} ~d\sigma ,
\end{eqnarray}
since $\eta$ vanishes near $\pom$. Adding equations
(\ref{eq:w_f_summand1}) and (\ref{eq:w_f_summand2}), we obtain
\begin{eqnarray*}
w_f(x) & = & -\int_\Omega G_\gamma (x , \cdot)\Lg (\eta w_f)~dy
-\int_\Omega G_\gamma (x,\cdot)\Lg (1-\eta)w_f~dy \\
&& +
\int_{\pom} \left\{ G_\gamma (x , \cdot)\Lambda_\gamma f-
f \nu \cdot \gamma \nabla G_\gamma (x,\cdot) \right \} ~d\sigma\\
& = & \int_{\pom} \left\{ G_\gamma (x , \cdot)\Lambda_\gamma f-
f \nu \cdot \gamma \nabla G_\gamma (x,\cdot) \right \} ~d\sigma, \\
\end{eqnarray*}
since $w_f$ is $\gamma$-harmonic in $\Omega$.\bx
We can now prove Theorem 1.1.
Let us recall that the eigenvalues $\{ \lambda_j \: | \: \lambda_1 = 0
, \lambda_j \leq \lambda_{j+1},
j = 1 , 2 , \ldots \}$ of $\Lambda_\gamma$ form a
nondecreasing sequence of nonnegative real numbers, and that
$\lambda_j \sim j^{1 / (d-1)}$.
\vspace{.1in}
\noindent
{\bf Theorem 1.1}. {\it Let $\phi_m$ be an eigenfunction of
$\Lambda_\gamma$ satisfying
$\Lambda_\gamma \phi_m = \lambda_m \phi_m$,
with $||\phi_m||_{\ltwos{\pom}} = 1$, and let $u_m$ be the
$\gamma$-harmonic extension of $\phi_m$ to $\Omega $. For any compact
$K\subset \Omega$, $\|u_m\|_{H^1(K)} = O(m^{-\infty})$.
}
\vspace{.1in}
\noindent
{\sc Proof:} Because $\pom\in C^\infty$, its outward normal
vector is smooth. This fact, in conjunction with the
regularity of $G_\gamma (x,\cdot )|_{\pom}$ for $x\in K$, and the
self-adjointness of $\Lambda_\gamma$, allows us
to use Proposition \ref{w_f representation Lemma} to write
\begin{eqnarray}
\label{multiply by one trick}
u_n(x)
& = &
\int_{\pom} \left \{ G_\gamma (x,\cdot ) \Lambda_\gamma \phi_m-
\phi_m \nu \cdot \gamma \nabla G_\gamma (x,\cdot) \right \} ~d\sigma
\nonumber \\
& = &
\frac{1}{\lambda_m^p} \int_{\pom} \{
G_\gamma (x,\cdot ) \Lambda_\gamma^{p+1} \phi_m -
\Lambda_\gamma^p \phi_m \nu \cdot \gamma \nabla G_\gamma (x,\cdot)
\} ~d\sigma \nonumber \\
& = & \frac{1}{\lambda_m^p} \int_{\pom} \{
\phi_m \Lambda_\gamma^{p+1} G_\gamma (x,\cdot )
- \phi_m \Lambda_\gamma^p \nu \cdot \gamma \nabla G_\gamma (x,\cdot)
\} ~d\sigma .
\end{eqnarray}
We now use H\"older's inequality to write
(\ref{multiply by one trick}) as
\bea
\label{after Holder's inequality}
|u_m(x)| & \leq & \frac{1}{\lambda_m^p} \left \{
\| \Lambda_\gamma^{p+1} G_\gamma (x,\cdot ) \|_{\ltwos{\pom}}
+ \left\|\Lambda_\gamma^p \nu \cdot \gamma \nabla G_\gamma
(x,\cdot)\right\|_{\ltwos{\pom}}
\right \} \nonumber \\
& = &
\frac{C(x;p)}{\lambda_m^p} .
\eea
Because the singularity of $G_\gamma
(x,\cdot)$ depends only upon the distance
from $x$ to $\pom$, there is a finite constant $C_p > 0$ so that
$|u_m(x)|\leq C_p \lambda_m^{-p} $, for every $x\in K$. A similar
inequality for $\|Du_n\|_{\ltwos{K}}$ is obtained by passing
differentiation through the integral in Proposition
\ref{w_f representation Lemma}. The classical spectral asymptotics,
$\lambda_m \sim m^{p / (d-1)}$, and the fact that $p$ was chosen
arbitrarily, complete the proof.\bx
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\setcounter{equation}{0}
\section{Approximate Diagonalization of $\Lambda_\gamma$}\label{S.4}
The rapid decay asserted by Theorem 1.1,
in conjunction with the
representation formula developed in
Proposition \ref{w_f representation Lemma}, allows us
to prove the decomposition theorem, Theorem 1.2.
We recall that $R_j$ is the restriction operator from $L^2 (
\pom )$ onto $L^2 ( \pom_j )$, and that $E_j$ is the extension operator
from $L^2 ( \pom_j )$ into $L^2 (\pom )$. The operator $C_j \equiv E_j R_j$ is
an orthogonal projector on $L^2 ( \pom )$.
\vspace{.1in}
\noindent
{\bf Theorem 1.2.} {\it
The DN map $\Lambda_\gamma$
admits a decomposition into two self-adjoint operators,
$\Lambda_\gamma = D_\gamma +R_\gamma$, where
$D_\gamma =\sum_{j=1}^k C_j \Lambda_\gamma C_j$, and
$R_\gamma$ is a smoothing operator.}
\vspace{.1in}
\noindent
{\sc Proof:} The decomposition follows from the orthogonality and
completeness properties of the operators $C_j$. For any function
$f\in D (\Lambda_\gamma )$,
$$
\Lambda_\gamma f
= \sum_{j=1}^k \Lambda_\gamma C_jf
= \sum_{j, l=1}^k C_j \Lambda_\gamma C_lf
= D_\gamma f + R_\gamma f ,
$$
where
$$
D_\gamma f = \sum_{j=1}^k C_j\Lambda_\gamma C_j f ,
$$
and
$$
R_\gamma f = \sum_{j,l=1 : l \neq j}^k C_j \Lambda_\gamma C_l f.
$$
Since the projections $C_j$ preserve the domain of $\Lambda_\gamma$, it
is clear that both operators $D_\gamma$ and $R_\gamma$ are self-adjoint.
In order to prove that $R_\gamma$ is smoothing,
we prove the following result in Lemma 4.1.
Let $\phi_n$ be an eigenfunction of $\Lambda_\gamma$ satisfying
$\Lambda_\gamma \phi_n = \lambda_n \phi_n$. The rapid decay of the
$\gamma$-harmonic extension of $\phi_n$ away from the boundary implies
that for all $j , l = 1 , \ldots , k$,
\beq
R_l \Lambda_\gamma E_j R_j \phi_n = \lambda_n \delta_{lj} R_j \phi_n + O (
n^{- \infty} ).
\eeq
We prove in Lemma 4.2 ahead that this result implies that both
$\Lambda_\gamma^p R_\gamma$ and $R_\gamma
\Lambda_\gamma^p$ extend to bounded operators on
$L^2 ( \pom )$, for any $p \in \Z$.
This immediately proves that $R_\gamma$ is a bounded operator.
To show that $R_\gamma$ is smoothing, we note that
\begin{equation}
R_\gamma = \bfrac{1}{(1+\Lambda_\gamma )^s}
\left( (1+\Lambda_\gamma )^s R_\gamma \right ) :
\ltwos{\pom}\rightarrow H^s (\pom ) ,
\end{equation}
is bounded for any $s > 0$, and that
\beq
R_\gamma = ( R_\gamma (1+\Lambda_\gamma )^t ) \bfrac{1}{(1+\Lambda_\gamma )^t}
: H^{-t} ( \pom ) \rightarrow \ltwos{\pom} ,
\eeq
is bounded for any $t > 0 $, so $R_\gamma$ is smoothing.
\bx \\
\begin{lemma}\hspace{-0.2cm}{\bf .}
\label{localization}
Let $\phi_n$ be an eigenfunction of $\Lambda_\gamma$ satisfying
$\Lambda_\gamma \phi_n = \lambda_n \phi_n$ and $\| \phi_n \|_{
L^2 ( \partial \Omega )} = 1.$.
For all $j , l = 1 , \ldots , k$, we have
\beq
R_l \Lambda_\gamma E_j R_j \phi_n = \lambda_n \delta_{lj} R_j \phi_n
+ O ( n^{- \infty} ).
\eeq
\end{lemma}
{\sc Proof:} We need to compute the image of $E_j R_j \phi_n$ under
the DN map $\Lambda_\gamma$.
Let $u_n$ be the $\gamma$-harmonic extension of $\phi_n$ to $\Omega$.
We denote by $w_n$ the $\gamma$-harmonic extension of $E_j R_j \phi_n$
to $\Omega$. This is computed from $u_n$ as follows.
Let $\xi_j \in C^\infty ({\overline \Omega})$,
with $\xi_j \equiv 1$ in a neighborhood
of $\pom_j$, and $\xi_j \equiv 0$ in a neighborhood of
$\pom\backslash\pom_j$. The $\gamma$-harmonic extension
$w_n$ of $E_j R_j\phi_n$ to $\Omega$ can be written as
\begin{equation}
\label{eta_representation}
w_n = \xi_j u_n - \ho (L_\gamma \xi_j u_n) ,
\end{equation}
where $\ho $ denotes the solution operator of the
inhomogeneous Dirichlet problem with Dirichlet boundary data.
That is, $\ho (F)$ is the function which solves
\begin{eqnarray}
\label{inhom_with_Dirichlet_bndry_data}
L_\gamma u & = & F\mbox{ in } \Omega \\
u & = & 0 \mbox{ on } \pom \nonumber .
\end{eqnarray}
One can easily check that $w_n$ is $\gamma$-harmonic and satisfies the proper
boundary conditions. This representation of $w_n$ greatly simplifies
our calculation of the outward normal derivative at the boundary.
Because $\xi_j \equiv 1$ near $\pom_j$,
\begin{eqnarray}
\label{booger_derivative}
\nu \cdot \gamma \nabla w_n \: | \: {\pom_j} & = &
[ \nu \cdot \gamma \nabla u_n - \nu\cdot\gamma \nabla \ho (L_\gamma \xi_j u_n)
] \: | \: {\pom_j} \nonumber \\
& = & \lambda_n R_j \phi_n - [ \nu\cdot\gamma \nabla \ho (L_\gamma \xi_j u_n)
] \: | \: {\pom_j} .
\end{eqnarray}
Using the Green's function $G_{\gamma, \Omega}$ of Theorem 2.3 to solve
(\ref{inhom_with_Dirichlet_bndry_data}), we may write
\bea
\ho (L_\gamma \xi_j u_n)(x)& = & \int_\Omega L_\gamma (\xi_j
u_n)G_{\gamma,\Omega}(x,\cdot )~dy
\nonumber \\
&= & \int_{\mbox{\small{supp}}(\grad\xi_j )}
u_n (L_\gamma \xi_j) G_{\gamma,\Omega}(x,\cdot )~dy \nonumber \\
& & - 2 \: \sum_{l,m=1}^d \int_{\mbox{\small{supp}}(\grad\xi_j )}
( \partial_l \xi_j ) \gamma_{lm} (\partial_m u_n )
G_{\gamma,\Omega}(x,\cdot )~dy . \nonumber \\
& &
\eea
Thus, we rewrite (\ref{booger_derivative}) as
\bea
\label{deriv_a}
\lefteqn{ \nu \cdot \gamma \nabla w_n \: | \: {\pom_j} } \\
& = & \lambda_n R_j \phi_n -
\left[ \int_{\mbox{\small{supp}}(\grad\xi_j )} u_n ( L_\gamma \xi_j )
\: \nu\cdot \gamma \nabla_x G_{\gamma,\Omega}
(x,\cdot ) ~dy \right] \: | \: {\pom_j} \nonumber \\
& & + 2 \: \left. \left[ \sum_{l,m=1}^d \int_{\mbox{\small{supp}}(\grad\xi_j )}
( \partial_l \xi_j ) \gamma_{ml} (\partial_m u_n )
\: \nu\cdot \gamma \nabla_x G_{\gamma,\Omega} (x,\cdot )~dy
\right] \: \right| \: {\pom_j}. \nonumber \\
& &
\eea
Recall that $\xi_j $ is constant in a neighborhood of $\pom_j $ so, for
$x$ near $\pom_j$, the local Green's function
$G_{\gamma,\Omega}(x,y)$ is smooth as $y$ ranges over
$\mbox{supp}(\grad\xi_j )$.
Since both $\xi_j $ and $\grad_x G_{\gamma,\Omega}(x,\cdot )$ are smooth
functions on $\mbox{supp}(\grad \xi_j )$, and
$\mbox{supp}(\grad \xi_j )$ is a compact subset of $\Omega $,
there is a constant $C(\xi_j ; \gamma ; \Omega)$ such that
$$
\left| \int_{\mbox{\small{supp}}(\grad\xi_j )}
[ u_n ( L_\gamma \xi_j) - 2 \: \sum_{l,m=1}^d ( \partial_l \xi_j)
\gamma_{lm} (\partial_m u_n ) ]
\: \nu\cdot \gamma \nabla_x G_{\gamma,\Omega}(x,\cdot )~dy \right|
$$
\beq
\leq C(\xi_j ; \gamma ; \Omega )
\int_{\mbox{\small{supp}}(\grad\xi_j )}
(|u_n| + \|\grad u_n\|)~dy .
\eeq
Since $\mbox{supp} ( \grad \xi_j ) \subset \Omega$,
we can apply Theorem 1.1, to conclude that the right-hand side
of (4.11) is $O(n^{-\infty} ),$
that is, $\Lambda_j R_j\phi_n = \lambda_n R_j\phi_n + O(n^{-\infty})$.
On the other hand, when we restrict to $\pom_i$, $i\not =j$,
equations (4.5), (4.7) and (4.10) imply
\bea
\label{row_of_R}
\lefteqn{ \nu (x) \cdot \nabla w_n(x) | {\pom_i}} & & \nonumber \\
&=& - \int_{\mbox{\small{supp}}(\grad\xi_j )}
\left[ u_n ( L_\gamma \xi_j) - 2 \: \sum_{l,m=1}^d
( \partial_l \xi_j ) \gamma_{lm} ( \partial_m u_n)
\right] \nu\cdot \gamma \nabla_x G_{\gamma,\Omega}(x,\cdot ) ~dy ~|
\pom_i . \nonumber \\
& &
\eea
Since $x$ near $\pom_i$ is disjoint from
$\mbox{supp}(\grad \xi_j )$, the second summand of
(\ref{row_of_R}) is $O(n^{-\infty})$. \bx \\
\begin{lemma}\hspace{-0.2cm}{\bf .}
\label{R_Lambda_alpha}
For any $p \in \N$, the operators $R_\gamma \Lambda_\gamma^p $ and
$\Lambda_\gamma^p R_\gamma$ can be
extended to bounded operators on $\ltwos{\pom}$.
\end{lemma}
{\sc Proof:} Suppose that $\psi $ is
in the domain of $\Lambda_\gamma^p $, with
$\|\psi \|_{\ltwos{\pom}}=1$. Because $\{ \phi_n \}$ is an
orthonormal basis of eigenfunctions of $\Lambda_\gamma$
for $\ltwos{\pom}$, we can write
$\psi = \sum_n \beta_n \phi_n$, where
$\beta_n = \langle \psi ,\phi_n\rangle_{\ltwos{\pom}}$. Using the
Cauchy-Schwarz
inequality for sequences in $\ell^2(\N)$ and the linearity of
$\Lambda_\gamma^p$ and $R_\gamma$, we see that
\bea
\|R_\gamma \Lambda_\gamma^p \psi \|_{\ltwos{\pom}} & \leq &
\sum_n | \beta_n | \lambda_n^p \|R_\gamma \phi_n \|_{\ltwos{\pom}}
\nonumber \\
& \leq &
\left( \sum_n \lambda_n^{2p} \|R_\gamma \phi_n\|_{\ltwos{\pom}}
^2 \right)^{1/2} \equiv C(p ).
\eea
The constant $C(p )$ is finite since, by Lemma 4.1,
$\|R_\gamma \phi_n\|_{\ltwos{\pom}} = O(\lambda_n^{-k})$, for every $k\in\N$.
This bound, and the fact that the domain of $\Lambda^p_\gamma $ is dense
in $\ltwos{\pom}$, allow us to extend $R_\gamma \Lambda_\gamma^p $ to a unique
bounded operator on all of $\ltwos{\pom}$.
Turning to the second operator, $\Lambda_\gamma^p R_\gamma$,
for any $\psi \in D ( \Lambda_\gamma^p )$, and
for any $\xi\in\ltwos{\pom}$, we have
\begin{equation}
|\langle \Lambda_\gamma^p\psi, R_\gamma \xi\rangle |=
|\langle R_\gamma \Lambda_\gamma^p \psi, \xi\rangle |\leq
C(p ) \: \|\psi \|_{\ltwos{\pom}} \: \|\xi \|_{\ltwos{\pom}} .
\end{equation}
We conclude that $R_\gamma \xi$ is in the domain of
$\Lambda^p_\gamma$.
Since the domain of $\Lambda^p_\gamma $ is a dense set in
$\ltwos{\pom}$,
it follows that $\Lambda_\gamma^p R_\gamma
:\ltwos{\pom}\rightarrow\ltwos{\pom}$
can be extended to a bounded operator (see \cite{[HislopSigal]}).
\bx \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Spectral Asymptotics}\label{S.5}
In this section, we show that the
approximate diagonalization formula for $\Lambda_\gamma$ of Theorem
1.2 allows us to approximate the spectrum of $\Lambda_\gamma$ by that of
the diagonal operator $D_\gamma = \Sum_{j=1}^k C_j \Lambda_\gamma C_j$.
We then study the operator $C_j \Lambda_\gamma C_j$ in detail and prove
that its spectrum is asymptotically close
to the spectrum of a DN operator $\Lambda_j$ associated
only with the $j^{th}$-boundary component.
\vspace{.1in}
\noindent
{\bf Theorem 1.3.} {\it Let $\{ \mu_l \; | \; l \in \N \} $ be the
eigenvalues of
$D_\gamma$ listed in nondecreasing order, including multiplicity,
and let $\{ \lambda_l \; | \; l \in \N \}$ be the eigenvalues of
$\Lambda_\gamma$, listed similarly. Then, we have
$\lambda_l=\mu_l+O(l^{-\infty})$.
}
\vspace{.1in}
\noindent
{\sc Proof:} Let $\{ \theta_j , j = 1 , \ldots \}$ be an orthonormal
basis of eigenvectors of $D_\gamma$.
We denote by $M_j$ the eigenspace spanned
by the first $j$ eigenvectors of $D_\gamma$.
We use a variational formula (cf.\ \cite{[ReedSimon4]}, section
XIII.1) to calculate $\lambda_{j+1}$:
\begin{eqnarray}
\label{Rx_is_small}
\lambda_{j+1}
& = &
\max_{\mbox{\scriptsize{dim}}(V^\perp )=j} \left(
\min_{f\in V \cap D (\Lambda_\gamma ) }
\{ \langle \Lambda_\gamma f,f\rangle_{\ltwos{\pom}} \; | \;
\|f\|_{\ltwos{\pom}}=1 \} \right) \nonumber \\
& = &
\max_{\mbox{\scriptsize{dim}}(V^\perp )=j} \left(
\min_{f\in V \cap D( \Lambda_\gamma) }
\{ \langle D_\gamma f,f\rangle_{\ltwos{\pom}} +
\langle R_\gamma f,f\rangle_{\ltwos{\pom}} \; | \;
\|f\|_{\ltwos{\pom}}=1 \} \right) \nonumber \\
& \geq & \max_{\mbox{\scriptsize{dim}}(V^\perp )=j} \left(
\min_{\shortstack{\scriptsize{$f\in V \cap D(\Lambda_\gamma) $} \\
\scriptsize{$\|f\|=1$}}}
\{ \langle D_\gamma f,f\rangle_{\ltwos{\pom}} \} -
\max_{\shortstack{\scriptsize{$f\in V \cap D(\Lambda_\gamma)$} \\
\scriptsize{$\|f\|=1$}}}
\{ |\langle R_\gamma f,f\rangle_{\ltwos{\pom}}| \} \right) \nonumber \\
& \geq &
\min_{\shortstack{\scriptsize{$f\in M_j^\perp \cap
D(\Lambda_\gamma)$} \\
\scriptsize{$\|f\|=1$}}}
\{ \langle D_\gamma f,f\rangle_{\ltwos{\pom}}\} -
\max_{\shortstack{\scriptsize{$f\in M_j^\perp \cap
D(\Lambda_\gamma)$} \\
\scriptsize{$\|f\|=1$}}}
\{ |\langle R_\gamma f,f\rangle_{\ltwos{\pom}}| \}
\nonumber \\
& = &
\mu_{j+1} -
\max_{f\in M_j^\perp \cap D(\Lambda_\gamma)}
\{ |\langle R_\gamma f,f\rangle_{\ltwos{\pom}}| \; | \;
\|f\|_{\ltwos{\pom}}=1\} .
\end{eqnarray}
Note that, according to Lemma \ref{R_Lambda_alpha},
$R_\gamma D_\gamma^q = R_\gamma (\Lambda_\gamma - R_\gamma )^q$
is a bounded operator for any $q\in\N$. It follows that,
for any eigenfunction
$\theta_j$ of $D_\gamma$, there is a constant $C(q ) \equiv \|
R_\gamma D_\gamma^q \|$, independent of the eigenvalue index $j$, such that
$\| R_\gamma \theta_j \|_{\ltwos{\pom}} \leq C(q ) \mu_j^{-q}$.
For any $p \in \N$, we choose $q = q(p) \in \N$
sufficiently large so that, recalling the classical eigenvalue
asymptotics, we have
$\{\mu_j^{p-q}\}\in\ell^2(\N )$. It follows from this and Lemma 4.2 that
\beq
\| R_\gamma D_\gamma^p \theta_j \|_{\ltwos{\pom}} =
\mu_j^p \|R_\gamma \theta_j\|_{\ltwos{\pom}} \leq
C(q) \mu_j^{p-q},
\eeq
where the constant $C (q)$, because of
the way in which we choose $q$, depends on $p$, but is
independent of $j$. Thus,
the sequence, indexed by $j$,
$\{ ||R_\gamma D_\gamma^p\theta_j ||_{\ltwos{\pom}} \}\in \ell^2(\N )$, for any
natural number $p$.
Now we can estimate the second summand on the right of (5.1). We may
write any vector $f\in M_j^\perp$, $\|f\| = 1$,
in terms of its Fourier coefficients,
$f = \sum_{k>j} f_k \theta_k$ so that,
\begin{eqnarray}
\label{R_has_correct_order}
\|R_\gamma f\|_{\ltwos{\pom}}
& \leq &
\sum_{k>j} \left | \frac{f_k}{\mu_k^p} \right |~
\|R_\gamma D_\gamma^p\theta_k\|_{\ltwos{\pom}} \nonumber \\
&\leq &
\frac{1}{\mu_{j+1}^p}
\sum_{k>j} \left | \left (\frac{\mu_{j+1}}{\mu_k}\right )^p f_k\right |
~||R_\gamma D_\gamma^p\theta_k||_{\ltwos{\pom}} \nonumber \\
& \leq &
\frac{1}{\mu_{j+1}^p}
\left\{ \sum_{k>j} ||R_\gamma D_\gamma^p \theta_k||_{\ltwos{\pom}}^2
\right\}^{1/2} \nonumber \\
&\leq &
\frac{1}{\mu_{j+1}^p}
\left\{ \sum_{k=1}^\infty ||R_\gamma D_\gamma^p \theta_k||_{\ltwos{\pom}}^2
\right\}^{1/2} \nonumber \\
& =& \frac{c(p )}{\mu_{j+1}^p} ,
\end{eqnarray}
where we used the Cauchy-Schwarz
inequality for sequences in $\ell^2(\N )$, the fact that
$\| f \| =1$, and the nondecreasing nature of the
sequence $\{\mu_j \}$. Inequality (\ref{R_has_correct_order}) allows us
to rewrite (5.1) as
$$
\lambda_{j+1} \geq \mu_{j+1} - O(j^{-\infty}) .
$$
Similarly, by reversing the role of $\Lambda_\gamma$ and
$D_\gamma$, we find that $\mu_{j+1}\geq\lambda_{j+1}- O(j^{-\infty})$
and the result follows.\bx \\
Having shown that the asymptotic behavior of the
spectrum of $\Lambda_\gamma$ is determined by $D_\gamma$,
we next describe the nature of the spectrum of the operator $D_\gamma =
\sum_{j = 1}^k \; C_j \Lambda_\gamma C_j$.
The operator $C_j \Lambda_\gamma C_j$ on $L^2 ( \partial \Omega )$
depends on data on the other boundary components through
$\Lambda_\gamma$. Due to the localization
of the $\gamma$-harmonic extensions of the eigenfunctions, though, this
dependence is very weak at high-energy.
Let us note an obvious fact. We define $\Lambda_j \equiv R_j
\Lambda_\gamma E_j : L^2 ( \partial \Omega_j ) \rightarrow L^2 ( \partial
\Omega_j )$. Then, it is clear that the spectrum of $C_j \Lambda_\gamma
C_j$ and $\Lambda_j$ coincide. Given a boundary component $\partial
\Omega_j$, we want to define a DN map associated solely with this
surface. There are two natural ways to do this since the surface $\partial
\Omega_j$ partitions $\R^d$ into two regions. We can consider
either the $\gamma$-harmonic extension to the bounded
region {\it interior} to the surface $\partial \Omega_j$, or
the extension to the {\it exterior}, unbounded region.
In order to maintain the sense of the normal derivative used in
the definition of $\Lambda_\gamma$, we distinguish the boundary
$\partial \Omega_1$ from the other boundary components. For
$\pom_1$, we denote by $\Omega_1^\#$ the bounded component of
$\R^d\backslash \pom_1$.
For $1 < j\leq k$, let $\Omega_j^\#$ be the unbounded component of
$\R^d\backslash \pom_j$. We define
$\lsh_j:\ltwos{\pom_j}\rightarrow\ltwos{\pom_j} $ by
$\lsh_j f = \nu \cdot \gamma \nabla u_f^\#|_{\pom_j}$, where $u_f^\#$ is the
$\gamma$-harmonic extension of $f$ to $\Omega_j^\#$ decaying at
infinity.
We assume hypothesis H3
so that we have chosen a smooth extension of $\gamma$ to $\R^d$
that coincides with the identity matrix outside a sufficiently large
ball. The existence of
such a $\gamma$-harmonic extension
for the unbounded regions $\pom_j^\#, ~1 < j \leq k$
is verified in the appendix, section 8. The vector
$\nu$ is the outward normal to the region.
Consequently, the outward normal is the same as was used in the
definition of $\Lambda_\gamma$ on each surface
$\pom_j, ~1 \leq j \leq k$.
This allows us to understand
one $\gamma$-harmonic extension as an approximation of the other.
\begin{lemma}\hspace{-0.2cm}{\bf .}
\label{exterior_likeness}
Suppose $\phi_n^{(j)}$ is a normalized eigenfunction of
$\lsh_j$ satisfying $\lsh_j\phi_n^{(j)} =\lambda_n\phi_n^{(j)}$.
We then have,
$$
\langle \phi_m^{(j)}, \Lambda_j \phi_n^{(j)} \rangle_{\ltwos{\pom_j}}
= \lambda_n\delta_{nm} +
O(\lambda_n^{-\infty})\cdot O(\lambda_m^{-\infty}) ,
$$
where the symbol on the right means rapid decay in each eigenvalue
separately.
\end{lemma}
{\sc Proof:} Consider the inner product,
$$
\langle \phi_m^{(j)},\Lambda_j \phi_n^{(j)} \rangle_{\ltwos{\pom_j}}
= \lambda_n\delta_{nm} +
\langle \phi_m^{(j)},(\Lambda_j-\lsh_j) \phi_n^{(j)} \rangle_{\ltwos{\pom_j}} .
$$
Using the Divergence Theorem, with $u_n$ as the $\gamma$-harmonic extension of
$E_j\phi_n^{(j)}$ to $\Omega$, we have
\begin{eqnarray*}
\langle \phi_m^{(j)},\Lambda_j \phi_n^{(j)} \rangle_{\ltwos{\pom_j}} & = &
\int_{\pom_j} \phi_m^{(j)} \nu \cdot \gamma \nabla u_n ~d\sigma =
\int_{\pom} (E_j\phi_m^{(j)}) \nu \cdot \gamma \nabla u_n ~d\sigma \\
~ & = & \int_{\pom} u_m \nu \cdot \gamma \nabla u_n ~d\sigma =
\int_\Omega \grad u_m \cdot \gamma \grad u_n ~dx .
\end{eqnarray*}
Let $u_n^\#$ denote the $\gamma$-harmonic extension of the
eigenfunction $\phi_n^{(j)}$ to $\Omega_j^\#$.
Because of the decay estimates on the $\gamma$-harmonic
solutions to the exterior region $\Omega_j^\#$, given in
Theorem 8.1, we may use the Divergence Theorem
with the functions $u_n^\#$ on the unbounded region $\Omega_j^\#$ to
write
$$
\langle \phi_m^{(j)} ,\lsh_j \phi_n^{(j)} \rangle =
\int_{\pom_j} u_m^\# \nu \cdot \gamma \nabla u_n^\# ~d\sigma =
\int_{\Omega_j^\#}
\grad u_m^\# \cdot \gamma \grad u_n^\# ~dx .
$$
Therefore, combining these two equations, we obtain,
\begin{eqnarray}
\label{chaba5}
\langle \phi_m^{(j)},(\Lambda_j-\lsh_j) \phi_n^{(j)} \rangle
& = &
\int_\Omega \left( \grad u_m\cdot \gamma \grad u_n - \grad u_m^\#\cdot \gamma
\grad u_n^\# \right) ~dx
\nonumber \\
&& -\int_{\Omega_j^\# \backslash \Omega}
\grad u_m^\# \cdot \gamma \grad u_n^\# ~dx .
\end{eqnarray}
Using the Divergence Theorem again, we see that the second
integral of (\ref{chaba5}) can be rewritten as
\begin{equation}
\label{chaba6}
\int_{\partial(\Omega_j^\#\backslash\Omega )}
u_m^\# \nu \cdot \gamma \nabla u_n^\# ~d\sigma .
\end{equation}
In Corollary 8.1, we derive the following representation
formula for $u_n^\#$,
$$
u_n^\# (x)=\int_{\pom_j}
\phi_n^{(j)} (y) {\cal G}_{\Omega_j^\#} (x,y)~d\sigma(y)
$$
where ${\cal G}_{\Omega_j^\#}(\cdot ,\cdot )$ is a smooth function away
from the diagonal. Consequently, we can apply
the techniques of Lemma \ref{rapid_localization} to conclude
that $\|u_n^\#\|_{H^1(K)} = O(\lambda_n^{-\infty} )$, for any
compact set $K\subset \Omega_j^\#$. In particular,
since $\partial (\Omega_j^\#\backslash\Omega)$ is removed from
$\pom_j$, we can conclude that the integral in (\ref{chaba6})
is of order $O(\lambda_m^{-\infty})\cdot O(\lambda_n^{-\infty})$.
To address the first integral of (\ref{chaba5}), we
add zero:
\begin{eqnarray}
\label{chaba7}
\int_\Omega \left( \grad u_m\cdot \gamma \grad u_n-\grad u_m^\#\cdot \gamma
\grad u_n^\# \right) ~dx
& ~ & \nonumber \\
& \hspace{-6cm} = & \hspace{-3cm}
\int_\Omega \left( \grad u_m^\# \cdot \gamma (\grad u_n - \grad u_n^\# ) +
(\grad u_m -\grad u_m^\# )\cdot \gamma \grad u_n \right) ~dx \nonumber \\
& \hspace{-6cm} = & \hspace{-3cm}
\int_{\Omega} \left( \mbox{div}((u_m-u_m^\#) \gamma \grad u_n) +
\mbox{div}((u_n-u_n^\#) \gamma \grad u_m^\# ) \right) ~dx \nonumber \\
& \hspace{-6cm} = & \hspace{-3cm}
-\int_{\pom\backslash\partial\Omega_j}
\left( u_m^\# \nu \cdot \gamma \nabla u_n + u_n^\#
\nu \cdot \gamma \grad u_m^\# \right) ~d\sigma,
\end{eqnarray}
since $(u_n-u_n^\#)$ vanishes on $\pom_j$. Because
$\pom\backslash\pom_j$ is a compact subset of $\Omega_j^\#$ that
is removed from $\pom_j^\#$, as in the analysis of
(\ref{chaba6}), we have that $\| u_m^\# \|_{H^1 (K) } = O(
m^{- \infty} )$. To estimate $\Lambda_\gamma u_n$
restricted to $\partial \Omega \backslash \partial \Omega_j$,
we use the method of proof of Lemma 4.1.
Recall that $u_n$ is the
$\gamma$-harmonic extension of $E_j \phi_n^{(j)}$ to $\Omega$.
As in the (4.5), we have the representation
\beq
u_n = \xi_j u_n^\# - H_\Omega ( L_\gamma \xi_j u_n^\# ) ,
\eeq
where $\xi_j$ is $1$ in a neighborhood of $\partial \Omega_j$, and $H_\Omega$
is the solution operator for (4.6). For $l \neq j$, we compute
\beq
\Lambda_\gamma u_n \; | \; \partial \Omega_l = - \nu \cdot \gamma \nabla
H_\Omega ( L_\gamma \xi_j u_n^\# ) \; | \; \partial \Omega_l .
\eeq
We express the solution operator $H_\Omega$ in terms of the Green's
function as in (4.8) to arrive at the analog of (4.12) with $w_n$ there replaced
by $u_n$, and $u_n$ there replaced by $u_n^\# $. Since $\mbox{supp} ( \nabla
\xi_j )$ is disjoint from $\partial \Omega_l$, for $l \neq j$,
the decay for the $\gamma$-harmonic function $u_n^\# $, away
from $\partial \Omega_j$, establishes that the right side of (5.8)
is $ O ( \lambda_n^{- \infty} )$. As a consequence, the
last term in (5.6)
is $O(\lambda_n^{-\infty})\cdot O(\lambda_m^{-\infty})$.
This completes the proof.\bx
Lemma \ref{exterior_likeness} tells us the action of $\Lambda_\gamma$ on
functions $f\in D (\Lambda_\gamma )$, with $\mbox{supp} \; f \subset\pom_j$, is
very similar to the action on $R_jf$ of the Dirichlet-to-Neumann
operator defined by way of a $\gamma$-harmonic extension to the {\it exterior}
region $\Omega_j^\#, ~1 < j \leq k$, or to the {\it interior}
region $\Omega_1^\#$.
We make that statement somewhat more precise with the following lemma.
\begin{lemma}\hspace{-0.2cm}{\bf .}
\label{smoothing_argument}
The operator difference $A_j \equiv \Lambda_j - \Lambda_j^\#$,
is a smoothing operator.
\end{lemma}
{\sc Proof:} Let $\{ \phi_n^{(j)} \}$ be the eigenfunctions of
$\lsh_j$ with associated eigenvalues $\{ \lambda_n \}$. Let
$\psi\in\ltwos{\pom_j}$ be in the domain of $(\lsh_j)^p$, for some
arbitrary but fixed $p$, and $||\psi ||_{\ltwos{\pom_j}} = 1$. We
expand $\psi$ as
$$
\psi = \sum_{n\geq 0} \beta_n \phi_n^{(j)},
\mbox{ where } \beta_n = \langle \psi , \phi_n^{(n)} \rangle_{\ltwos{\pom_j}} .
$$
Using the linearity of $A_j$ and $(\lsh_j)^p$, we see that
\begin{equation}
\label{smoothing_step2}
|| A_j(\lsh_j)^p\psi ||_{\ltwos{\pom_j}}
\leq
\sum_{n\geq 0} \beta_n \lambda_n^p ||A_j\phi_n^{(j)} ||_{\ltwos{\pom_j}}
\leq
\left( \sum_{n\geq 0} \lambda_n^{2p}
\|A_j\phi_n^{(j)}\|_{\ltwos{\pom_j}}^2 \right)^{1/2},
\end{equation}
where the final inequality of (\ref{smoothing_step2}) arises from the
Cauchy-Schwarz inequality for sequences in $\ell^2(\N)$, and the fact that
$||\psi|| = 1$. Let us attend to $||A_j\phi_n^{(j)}||_{\ltwos{\pom_j}}^2$:
$$
||A_j\phi_n^{(j)}||_{\ltwos{\pom_j}}^2
= \sum_{m\geq 0} |\langle A_j\phi_m^{(j)} ,\phi_n^{(j)} \rangle |^2
$$
and, according to Lemma \ref{exterior_likeness},
$ |\langle A_j\phi_m^{(j)} ,\phi_n^{(j)} \rangle |^2 =
O(\lambda_n^{-\infty})\cdot O(\lambda_m^{-\infty})$. Thus,
$$
||A_j\phi_n^{(j)}||_{\ltwos{\pom_j}}^2
= \sum_{m\geq 0} O(\lambda_n^{-\infty})\cdot O(\lambda_m^{-\infty})
= O(\lambda_n^{-\infty}) ,
$$
whence the right-hand side of (\ref{smoothing_step2}) converges,
independently of $\psi$. That is, $A_j(\lsh_j)^p$ is a bounded operator
on the domain of $(\lsh_j)^p$, which is dense in $\ltwos{\pom_j}$.
We extend the operator to all of $\ltwos{\pom_j}$ and denote this extension,
also, by $A_j(\lsh_j)^p$. As a bounded operator, we know
$A_j(\lsh_j)^p$ has a bounded adjoint,
$(A_j(\lsh_j)^p)^* = (\lsh_j)^pA_j$. It follows that
$((\lsh_j)^p + 1)A_j:\ltwos{\pom_j}\rightarrow\ltwos{\pom_j}$
is a bounded operator and, thus,
$$
A_j = \frac{1}{(\lsh_j)^p + 1} ((\lsh_j)^p + 1)A_j :
\ltwos{\pom_j}\rightarrow H^p(\pom_j) .
$$
The fact that $p$ is arbitrary concludes the proof.\bx \\
One might naturally ask if the spectral asymptotics depends on
the choice of the extension to the exterior of the regions
bounded by $\pom_j$, for $1 < j \leq k$, and the interior of
$\pom_1$. To answer this,
we now consider extensions to the interior of the region bounded by
$\pom_j$, for $1 < j \leq k$, and, similarly, the extension to
the exterior region bounded by $\pom_1$.
We denote by $\Lambda_{\pom_j}$ the Dirichlet-to-Neumann operator on
$\ltwos{\pom_j}$ that is defined using a $\gamma$-harmonic extension to
the bounded region enclosed by $\pom_j, ~1 < j \leq k$, and to
the unbounded region exterior to $\pom_1$.
We will show that, up to smoothing, it does not matter which
operators, $\Lambda_j^\#$ or $\Lambda_{\pom_j}$,
we choose to model the boundary.
The next result constitutes the final step of this section.
It will be used in the next section
in order to extract geometric information concerning $\pom$
from the DN operator.
\begin{lemma}\hspace{-0.2cm}{\bf .}
\label{symbol_from_metric}
For any $1 \leq j \leq k$, the operator
$\lsh_j-\Lambda_{\pom_j}$, on $L^2 ( \pom_j)$, is a
smoothing operator.
\end{lemma}
{\sc Proof:} In \cite{[LeeUhlmann]}, Lee and Uhlmann express the
ambient Riemannian metric of $\R^d$ in boundary local coordinates, and
proceed to calculate the full symbol of the DN
operator in terms of the induced metric tensor on $\pom$.
Let us suppose that the
outward normal vector to $\R^d\backslash\Omega_j^\#$ (the
bounded region enclosed by $\pom_j$), written in boundary
normal coordinates, is $\nu = -\pard{~}{x_d}$. To choose
$\nu = \pard{~}{x_d}$, instead, is to define the DN
operator by way of a $\gamma$-harmonic extension to $\Omega_j^\#$
and, as one might expect, a factor of negative one is introduced
into the symbol. However, we cannot use the same alternating
$(d$-$1)$-form in our representation of the DN
operator as defined with an extension into $\Omega_j^\#$
as was used when our extension was into $\R^d\backslash\Omega_j^\#$
because this form induces an orientation on
the manifold $\pom_j^\#$ under which it is {\it not} the
boundary of $\Omega_j^\#$. To properly orient
$\pom_j^\#$, we commute the differentials of the alternating form and,
thus, the negative sign that arose from our choice of outward
normal is absorbed. The conclusion follows from
fact that the symbols of $\lsh_j$ and
$\Lambda_{\pom_j}$ agree to all orders.\bx
\begin{corollary}\hspace{-0.2cm}{\bf .}
\label{sharp_to_pom_j_evalues}
Suppose the spectrum of $\Lambda_{\pom_j}$ is denoted by
$\{ \lambda_n \}$, and the spectrum of $\Lambda_j$ is written as
$\{ \mu_n \}$. Then, $\lambda_n = \mu_n + O(n^{-\infty})$.
\end{corollary}
{\sc Proof:} We can write
\bea
\label{smoothing_step3}
\Lambda_{\pom_j} & =&
\{ (\Lambda_{\pom_j} - \lsh_j) + (\lsh_j - \Lambda_j) \} +
\Lambda_j \nonumber \\
& = & \Lambda_j + R_j ,
\eea
where $R_j$ is a sum of smoothing operators, and so is a smoothing operator.
Consequently, if we write eigenfunctions of $\Lambda_{\pom_j}$ as
$\phi_n^{(j)}$, we have
\beq
\| R_j \phi_n^{(j)} \| \; = \; \| R_j \Lambda_{\partial \Omega_j
}^p \Lambda_{\partial \Omega_j}^{-p} \phi_n^{(j)} \| \; \leq C(p)
\lambda_n^{-p} .
\eeq
That is, we have
$||(\Lambda_{\pom_j} - \Lambda_j )\phi_n^{(j)}||_{\ltwos{\pom}}
= O(n^{-\infty})$. We have a similar estimate if
$\phi_n^{(j)}$ is replaced by an eigenfunction of $\Lambda_j$. The proof of
Theorem \ref{spectral_asymptotics_1} can now be applied, and the
result follows.\bx
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\thechapter}{\arabic{chapter}}
\renewcommand{\thesection}{\thechapter}
\setcounter{chapter}{6}
\setcounter{equation}{0}
\section{Splitting of $\sigma(\Lambda_\gamma )$
and Geometric Interpretation}\label{S.6}
Let us summarize of our analysis up to this point. The
Dirichlet-to-Neumann operator $\Lambda_\gamma$
was first approximated by a diagonal
operator, $D_\gamma = \sum_{j = 1}^k C_j \Lambda_\gamma C_j$.
While the operators $C_j \Lambda_\gamma C_j$ depend only on the
boundary data on $\pom_j$, they still depend on the other
boundary components $\pom_k , k \neq j$, through
$\Lambda_\gamma$.
We first replaced $C_j \Lambda_\gamma C_j$, acting on $L^2 ( \pom
)$, by
$\Lambda_j$, acting on $L^2 ( \pom_j)$. The operator $\Lambda_j$
has the same spectrum as $C_j \Lambda_\gamma C_j$,
but still depends on the other boundary components.
However, each $\Lambda_j $ is
closely approximated by a similar operator $\Lambda_j^\#$ that is
defined by way of a $\gamma$-harmonic extension to a region with only one
boundary component. These operators $\lsh_j$, or,
equivalently, the operators $\Lambda_{\pom_j}$, have the same
spectral asymptotics as the $C_j \Lambda_\gamma C_j$, and are independent of
$\pom_i$, for $i\not = j$. In each step, we have
moved toward a simpler approximation of $\Lambda_\gamma$ while maintaining
the asymptotics of its spectrum. It is this sequence of approximations
that lead us to the following assertion.
\begin{theorem}\hspace{-0.2cm}{\bf .}
\label{spectrum_geometry}
The spectrum of $\Lambda_\gamma $ determines a lower bound
on the number of components of $R^d\backslash\Omega $ and, further,
also determines the weighted measure of each boundary component (not counting
multiplicities).
\end{theorem}
In the statement of this theorem, the phrase
``not counting multiplicities''
is intended to say that the asymptotics of $\sigma(\Lambda_\gamma )$ may not
indicate the existence of more than one boundary component with
the same weighted measure.\\
\noindent
{\sc Proof:} Theorem \ref{spectral_asymptotics_1} tells us the
asymptotics of $\sigma(\Lambda_\gamma )$ are determined by $\sigma(D_\gamma)$,
and we know $\sigma(D_\gamma)= \bigcup_{j=1}^k\sigma(C_j\Lambda_\gamma C_j)=
\bigcup_{j=1}^k\sigma(\Lambda_j )$. Because
$\lsh_j = \Lambda_j - A_j$, where
$| \langle \phi_n^{(j)}, A_j \phi_n^{(j)} \rangle_{\ltwos{\pom_j}}|
= (\lambda_n^{-\infty})$, for eigenfunctions $\phi_n^{(j)}$
of $\lsh_j$, the proof of
Theorem \ref{spectral_asymptotics_1} gives us
that $\sigma(\lsh_j) \sim \sigma(\Lambda_j)$. It follows from
Corollary \ref{sharp_to_pom_j_evalues} that
$\sigma(\lsh_j)\sim \sigma(\Lambda_{\pom_j} )$.
Let $\sigma(\Lambda_{\pom_j} ) = \{\lambda_m^{j} \}$, listed in
nondecreasing order. The classical eigenvalue asymptotics give us
$$
\lambda_m^j \sim
\left ( \bfrac{m}{C(\pom_j, \Lambda_{\gamma, \partial \Omega }
)}\right )^{1/(d-1)}
,
$$
where,
$$
C(\pom_j, \Lambda_{\gamma, \partial \Omega } )=
(2 \pi )^{- (d-1)} \; \mbox{Vol} \; \{ (x , \xi ) \in T^*\partial
\Omega_j \backslash \{0\} \; | \; \Lambda_{\gamma , \partial \Omega} ( x ,
\xi ) \leq 1 \} .
$$
Thus, we have the following equivalence:
$$
\sigma(\Lambda_\gamma )\sim\sigma(D_\gamma )= \Cup_{j=1}^k \sigma(\Lambda_j )
~~\mbox{ and }~~
\sigma(\Lambda_j ) \sim \sigma(\lsh_j ) \sim \sigma (\Lambda_{\pom_j}).
$$
That is to say, asymptotically, $\sigma(\Lambda_\gamma )$ splits into at
most $k$ disjoint sets, each of which is asymptotic to
$( m/C(\pom_j, \Lambda_{\gamma, \partial \Omega } ))^{1/(d-1)}$,
for some $j$, $1\leq j\leq k$. From this
behavior, we can determine whether there {\it are} voids in $\Omega $
and, if so, bound the volume of each from above. However,
if $i\not = j$, and $\pom_i$ and $\pom_j$ have the same
$(d-1)$-dimensional weighted measure, the asymptotics of
$\sigma(\Lambda_i)$ and $\sigma(\Lambda_j)$ are the same.
Because of this, $\sigma(\Lambda_\gamma )$ may not
indicate the existence of {\it both} holes, but only that there is
{\it some} part of $\pom$ with the appropriate weighted measure.\bx \\
Let us remark that this theorem implies that the high energy
asymptotics of the spectrum of the DN map $\Lambda_\gamma$ clearly
indicate if the region is simply- or multiply-connected.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\thechapter}{\arabic{chapter}}
\renewcommand{\thesection}{\thechapter}
\setcounter{chapter}{7}
\setcounter{equation}{0}
\section{Perfect Insulators}\label{S.7}
The problem of the previous chapter was geometric in nature,
because the DN map $\Lambda_\gamma$ is defined by the
voltage to current mapping on the {\it entire} boundary,
including the interior boundaries.
Clearly, if our goal is to detect
imperfections in the interior of $\Omega$,
that is to say, if we do not already
know such defects exist, then we surely cannot take measurements
to help us calculate the spectrum based on fixing potentials on the
inaccessible, interior boundaries.
The following section reformulates our question so that only
measurements on the outer boundary, $\pom_1$, are required for
calculation.
This is more in keeping with the methods of nondestructive evaluation.
Suppose the body $\Omega$ is interrupted not by voids but by
a finite collection of smoothly bounded perfect insulators.
As in the previous section, $\pom_1$ is understood to be the
boundary of the reference region $\Omega_1^\#$, the bounded
component of $\R^d \backslash \partial \Omega_1$, and $\pom_j$,
$1 1/2$,
we define $K^s_f(\Omega) =
\{ w\in H^s(\Omega ) \; | \; w=f\mbox{ on }\pom_1\mbox{ in the trace
sense}\}$.
\end{definition}
\begin{theorem}\hspace{-0.2cm}{\bf .}
\label{simple_mixed_soln}
The mixed problem (\ref{mixed_problem}) has a unique solution in the
weak sense for any $f\in H^{1/2}(\pom_1)$.
\end{theorem}
{\sc Proof:} The hypothesis that $f\in H^{1/2}(\pom)$ allows us to
assert that $K_f^1(\Omega )$ is not empty, since the Dirichlet
data $E_1f$ on $\partial \Omega$ has a unique $\gamma$-harmonic extension.
It is not immediately obvious that the
energy functional $L_\gamma$ achieves a minimum on
$K_f^1(\Omega )$. However, it is bounded from below by H2 so an
infimum exists. Let $\lambda$ be this infimum. Then there is a
sequence of functions
$\{ v_j \}\subset K_f^1(\Omega )$ such that $L_\gamma [v_j]\geq
L_\gamma [v_{j+1}]$ and
$\limit{j}{\infty} L_\gamma [v_j] = \lambda$.
It is easy to check that
\beq
L_\gamma \left [ \frac{u+w}{2} \right ] +
L_\gamma \left [ \frac{u-w}{2} \right ] = \frac{1}{2} \{ L_\gamma [u] +
L_\gamma [w] \},
\eeq
whence
\beq
L_\gamma \left [ \frac{v_j-v_k}{2} \right ]
= \frac{1}{2}\{ L_\gamma [v_j]+ L_\gamma [v_k]\} -
L_\gamma \left [ \frac{v_j+v_k}{2} \right ] .
\eeq
Because $(v_j+v_k)/2 \in K_f^1(\Omega)$,
we know that
$L_\gamma \left [ \frac{v_j+v_k}{2} \right ] \geq \lambda$. Consequently,
(7.4) implies that
\begin{equation}
\label{converging}
L_\gamma \left [ \frac{v_j-v_k}{2} \right ] \leq \frac{1}{2}\{ L_\gamma [v_j]
+ L_\gamma [v_k]\} - \lambda .
\end{equation}
Clearly, the right-hand side of (\ref{converging}) tends to zero as
$j$ and $k$ tend to infinity. It follows from
Lemma \ref{L2_bnding_Lemma} that $\{ v_j \}$ is a Cauchy sequence
in $H^1(\Omega )$ so there is a function $v \in H^1(\Omega )$ to which
it converges. The continuity of the trace operator ensures that
$Tv=f$, as desired and, for any
$\phi\in K^1_0(\Omega)$, the functional
$L_\gamma [v + t\phi]$ takes its minimum at
$t=0$. Therefore,
\begin{equation}
\label{minimizer_weak_harmonic}
0 = \left.\frac{d}{dt} \right |_{t=0} \int_\Omega
\grad (v+t\phi )\cdot \gamma \grad (v+t\phi)~dx =
2\int_\Omega \grad v \cdot \gamma \grad\phi ~dx ,
\end{equation}
so $v$ is $\gamma$-harmonic in the weak (or distributional) sense. According
to elliptic regularity (cf.\ \cite{[Evans]}),
the weak solution $v$ is
smooth in $\Omega$, and $L_\gamma v=0$ in the classical sense. Therefore,
for any $\phi\in K^1_0(\Omega)$,
\begin{equation}
\label{getting_outward_normal_d}
\int_\Omega \mbox{div}(\phi \gamma \cdot \grad v)~dx =
\int_\Omega \left( \grad\phi \cdot \gamma \grad v + \phi L_\gamma v
\right) ~dx .
\end{equation}
The first summand of the right-hand side of
(\ref{getting_outward_normal_d}) is zero according to
(\ref{minimizer_weak_harmonic}) and the second vanishes because
$L_\gamma v \equiv 0$ in $\Omega$. So we write, formally,
$$
\int_{\pom\backslash\pom_1} (T\phi) \nu \cdot \gamma \grad v ~d\sigma =
\int_{\pom} (T\phi) \nu \cdot \gamma \grad v ~d\sigma =
\int_\Omega \mbox{div}(\phi \cdot \gamma \grad v)~dx = 0.
$$
Because the range of the trace map, $T$, is dense in $\ltwos{\pom}$,
we conclude that the outward normal derivative of $v$ on $\pom_j$,
$1 R$.
The function $F$ has compact support in the interior
of $B_{2R} (0) \backslash B_R (0)$. From Theorem 2.3,
there exists a Green's function $G_{\gamma , \Omega}
(x,y)$ for $L_\gamma$
on the region $\Omega$, with Dirichlet boundary conditions on the
boundary $\partial \Omega$, and decaying at infinity.
Using this Green's function, we construct a function
\beq
z(x ) \equiv \int_{ B_{2R} (0) \backslash B_R (0) }
\; G_{\gamma, \Omega} ( x , y ) F(y ) ~dy .
\eeq
This function
satisfies
\beq
L_\gamma z = F, ~~\mbox{in} \; \Omega,
~~\mbox{and} \; z \; | \; \partial \Omega = 0 .
\eeq
Consequently, the function $u \equiv \psi - z$ is a solution
to the boundary-value problem (8.1).
The manner in which $u$ is constructed, and Theorem 2.3,
allow us to compute the
decay rate of the solution $u$. For $\|x\| > 2R$, the solution is
\beq
u(x) = v(x) - z(x) .
\eeq
It is clear from parts (c.) and (d.) of Theorem 2.3, and the
properties of $G_R$, that (8.4) holds. It remains only
to prove that this function is unique among all
those that vanish at infinity. To the contrary, suppose there
is another such function, $v$, solving (8.1) and vanishing at
infinity.
Then $\phi=(u-v)$ is $\gamma$-harmonic in $\Omega$, satisfies
$\phi \; | \; \partial \Omega = 0$, and vanishes at infinity.
It follows from Moser's form of Liouville's Theorem \cite{[Moser]}
that $\phi=0$. \bx \\
Given the existence of a unique solution to (8.1) decaying at infinity
as in (8.2), we can now prove a representation formula
for the solution.
\begin{corollary}\hspace{-0.2cm}{\bf .}
\label{exterior_solvability}
Suppose $\Omega \subset \R^d, d\geq3$ is an open region in $\R^d$
with smooth,
nonempty boundary whose complement is a compact set.
Then, there is a function
${\cal G}_{\gamma, \Omega}: \Omega\times\pom \rightarrow \R$ such that
$$
u(x) = \int_{\pom} {\cal G}_{\gamma, \Omega } (x, \omega )
f(\omega)~d\sigma(\omega)
$$
is the unique solution to
\begin{equation}
\label{exterior_Dirichlet_problem}
\begin{array}{rcl}
L_\gamma u & = & 0 \mbox{ in } \Omega \\
u & = & f \mbox{ on }\pom,
\end{array}
\end{equation}
for $f \in C ( \partial \Omega)$,
among all functions that tend to zero at infinity. Furthermore,
the solution satisfies the decay estimate (8.2).
\end{corollary}
{\sc Proof:}
Let $u_f$ be the unique solution to (8.1) constructed in Theorem
8.1.
Using the Dirichlet Green's function for $\Omega$, as given in
Theorem 2.3, we derive a representation formula
for the solution to (8.1). Let $R >> 0$ be chosen. An application
of the Divergence Theorem to the region $B_R (0) \backslash
\Omega^c$ yields
\bea
u_f (x) & = & \int_{\partial \Omega } \; f(\omega) \nu \cdot \gamma
(\omega) \grad G_{\gamma, \Omega} ( \omega , x ) ~d \sigma (\omega)
\nonumber \\
& & + \int_{\partial B_R (0)} \; \left[ u (R \omega) \nu \cdot \grad
G_{ \gamma, \Omega }( R \omega , x ) - G_{\gamma, \Omega}( R \omega , x )
\nu \cdot \grad u (R \omega ) \right ] ~R^{d-1} ~d \omega , \nonumber \\
& &
\eea
where $d\omega$ is the measure on the sphere $S^{d-1}$,
and $\gamma = 1$ for $R$ large enough.
We now use the decay estimates on $G_{\gamma , \Omega}$ and
$\grad G_{\gamma , \Omega}$, given in Theorem 2.3, parts (c.) and
(d.), and the decay estimate on $u_f$ and $\grad u_f$ in (8.2), to prove that
the integral over $\partial B_R (0)$ vanishes as $R \rightarrow \infty$.
As a result, we obtain
\beq
u_f (x) = \int_{\partial \Omega } \; f(\omega) \nu \cdot \gamma
(\omega) \grad G_{\gamma, \Omega} ( \omega , x ) ~d \sigma
(\omega) ,
\eeq
so that ${\cal G}_\Omega (x, \omega ) =
\nu \cdot \gamma (\omega) \grad G_{\gamma, \Omega} ( x, \omega ).
$ \bx \\
We can now define the Dirichlet-to-Neumann operator
for $\partial \Omega$
by way of $\gamma$-harmonic extensions to the unbounded region
$\Omega$.
\begin{prop}\hspace{-0.2cm}{\bf .}
Assume $\Omega$ and $L_\gamma$ satisfy the same properties
as in Theorem 8.1, so that $\Omega^c$ is compact.
If $u_f$ and $u_g$ are the $\gamma$-harmonic extensions of
$f,g \in C ( \partial \Omega)$ to $\Omega$, as in Theorem 8.1, and
$\nu$ is the outward normal to $\Omega$, then
$$
\left \langle g, \Lambda_\gamma f \right \rangle_{\ltwos{\pom}}
= \langle g, \nu \cdot \gamma \grad u_f \rangle_{\ltwos{\pom}} =
\int_\Omega \grad u_g \cdot \gamma \grad u_f ~dx .
$$
\end{prop}
{\sc Proof:} Choose $R > >0$ so that $R^d\backslash\Omega^c \subset
B_R (0)$, and write $\Omega_R = \Omega\cap B_R (0)$.
We use the Divergence Theorem on this smoothly bounded region to write
\begin{equation}
\label{r_ball_and_Omega}
\int_{\Omega_R} \grad u_g \cdot \gamma \grad u_f ~dx =
\int_{\pom} g \nu \cdot \gamma \grad u_f ~d\sigma +
\int_{\partial B_R (0)} u_g \nu \cdot \gamma \grad u_f ~d\sigma ,
\end{equation}
where $\nu$ is the outward normal to $\Omega_R$. Properties
(8.2) of the $\gamma$-harmonic extensions $u_f$ and $\grad u_g$
show that the second integral in (8.16) vanishes as $R \rightarrow
\infty$. \bx
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}
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