\documentclass[11pt]{article} \usepackage{m300,epsfig} \psfigdriver{dvips} \def\Del{\mbox{$\Delta$}} \def\dt{\,{\mbox{\rm d}t}} \def\lam{\lambda} \def\Lam{\Lambda} \def\sig{\sigma} \def\iint{\mbox{$\dis{\int\!\!\!\int}$}} \def\D{\mbox{$\cal D$}} \textwidth 5.5 in \newtheorem{theorem}{Theorem} \newtheorem{definition}{Definition} \begin{document} \def\endproof{\rule{1ex}{1ex}} \def\proof{{\em Proof.\ }} \def\uni{ \mbox{{$ \raisebox{-.8ex}{$\rightarrow$} \atop \raisebox{.8ex}{$\rightarrow$}$}}} % The next command shows how to use \mbox'es. \def\B{\mbox{$\cal B$}} \newcommand{\ClassN}{\mbox{${\mb{C}(N,\sig^*)}$}} \newcommand{\Cprime}{\mbox{${\mb{C}'(h,H,A,N,\partial D)}$}} \newcommand{\Clip}{\mbox{${\mb{C}^{l}(h,H,P,\partial D)}$}} \def\deru{\left(\Frac{\partial u_k^*}{\partial \vec{\nu}}\right)^2} % Now we see how to use bold face symbols in Math mode. \newcommand{\Class}{\mbox{${\mb{C}(h,H,A,N,\partial D)}$}} \bibliographystyle{siam} \pagestyle{myheadings} \markboth{DAVID C. BARNES and ROGER KNOBEL}{THE FINITE INVERSE PROBLEM} \begin{center}{\large\bf ON THE RECOVERY OF IMPEDANCE BOUNDARY CONDITIONS FROM SPECTRAL DATA\\ DAVID C. BARNES\\ Department of Pure and Applied Mathematics\\ Washington State University\\ Pullman, Washington, 99164-3113\\ Email to, BARNES@WSUMATH.BITNET.} \end{center} \centerline{THIS IS VERSION \today} % This will center the text and print a % date \vspace{1em} % Add a little vertical space. \begin{center} \epsfig{file=odd11.ps} \end{center} \begin{abstract} Let $D$\ be a domain in $\Re^2$\ having a piece-wise smooth boundary and ******* eigenvalues are denoted by $\lam_n(\sig,q)$. ******* considered the problem of reconstructing the coefficient function $q(x,y)$\ using a finite ******* for solving for $\sig$\ along with some examples. \end{abstract} \section{Formulation of the problem} % Start a new section with a proper % number. Let $\D$\ be a given domain in the $x$-$y$ plane having a piece-wise smooth boundary and consider the eigenvalue problem % Next we have a displayed equation. \begin{equation} u_{xx} + u_{yy} + (\lam - q(x,y))u =0 \txt{in} D \txt{and} \partial u/\partial{\vec{\nu}}+\sig(s)u=0 \txt{on} \partial D. \Label{one}\end{equation} % The command \Label is defined in barnes.sty as is \txt{}. The coefficient $\sig(s)$\ is defined on $\partial D$\ and $s$\ stands for a ****** $\mb{\Lam_N}=(\Lam_1,\,\Lam_2,\,\cdots ,\Lam_N)$\ denote the eigenvalues of (\ref{one}) % A reference to the above equation. corresponding to some unknown function $\sig^*(s)$\ so that ******* Although, uniqueness theorems for this problem are scarce, of particular note is the following \cite{gm}.\\ % The \\ command starts a new line. % This theorem illustrates the use of \ref{} and Label{} \begin{theorem}\Label{gm} Let $D$\ be an elliptical region and suppose that $\sig^*(s)$\ is invariant under the group of symmetries of $D$. If $q(x)\equiv 0$, then the spectrum of (\ref{one}) uniquely determines the function $\sig^*(s)$.\\ \end{theorem} Another related result has been given by Zayed cite. He used an asymptotic ******\\ % The definition environment is defined in the file siam.sty. You may also use % the command \newtheorem to define your custom environments. \begin{definition} The sequence of functions $\sig_N(s)$\ is said to % \mb{#1} is defined by % \def\mb#1{{\mbox{\boldmath${#1}$\unboldmath}}} interpolate to the data $\mb{\Lam_N}$\ if \begin{equation} \max_{1\leq i\leq N}\left|\lam_i (\sig_N,q(x,y)) - \Lam_i \right| \ar 0 \txt{as} N\ar \infty . \Label{B}\end{equation} \end{definition} A method for constructing such an interpolating sequence will be given below. Having constructed a good interpolating sequence one may then ask: % The next structure is a list of items. We also have \enumerate and we may use % the \item[...] command to control the labels on the list. \begin{itemize} \item As $N\ar\infty$, does the interpolating sequence $\sig_N$\ converge (and if so, in which topology) to the unknown function $\sig^*$. \item If the inverse problem does have a well-posed solution, then how much spectral data is necessary in order to obtain good approximations to the unknown function? \item How sensitive is any such solution to a perturbation in the spectral data $\mb{\Lam}$?\\ \end{itemize} % The \noi command is defined in barnes.sty to mean do not indent the % paragraph. \noi This work will give partial but essential steps toward answering these questions. ******* for any function $\sig(s)$\ that solves the equations \begin{equation} \lam_n(\sig,q)=\Lam_n \txt{for} n=1,2,\cdots, N. \Label{iso}\end{equation} In general, such an inverse problem will not have a unique solution and, in ***** \section{Continuity of the eigenvalues} % A new section with the next number. The continuity properties of the eigenvalues play a major role in understanding ******* ***** Sturm-Liouville equation \begin{equation} y'' + (\lam - q(x) ) y = 0, \qquad \begin{array}{c} % The \array command will allow line a_1 y(0)+a_2y'(0)+a_3y(1)+a_4y'(1)=0\\ % breaking inside equations. b_1 y(0)+b_2y'(0)+b_3y(1)+b_4y'(1)=0 % It must be used in math-mode! \end{array}. % y'' is the second derivative. \Label{sl}\end{equation} It was shown that the topology induced on the functions $q(x)$\ by the $1Max$\ ******* depending only on the domain and the bounds $H,h$\ on $\sig$, such that if $u_i$\ has $L_2(D)$\ norm 1 then % The command \max is interesting. There are also others like \sin, \log, etc. $\max_{(x,y)\in D)}|u_i(x,y)|^2\leq C \lam_i^2$. Thus, there is a constant $A$\ such that \begin{equation} \max_{(x,y\in D)}|u_i(x,y)|\leq A \txt{for}\ (x,y)\in D \txt{and} i=1,2,\cdots,N. \Label{second}\end{equation} ****** however, there can be little doubt that \Cprime\ is still a very large subset of \Class. Using these two classes, the proof of continuity in the $L_{\infty}$\ % \infty prints the infinity symbol. norm given by Courant and Hilbert \cite{CH} may be generalized as follows.\\ \begin{theorem}\Label{main} A) When endowed with the $L_1[\partial D]$\ topology the eigenvalues are continuous on \Class\ as functions of $\sig(s)$.\\[1em] B) When given the weak topology the eigenvalues are continuous on \Cprime.\\ \end{theorem} \proof The Rayleigh quotient for the eigenvalue problem is given by \[ % A displayed equation without a number attached to it and a \Frac command. \lam_n(\sig,q)=\max_{V_i}\min_{u} \Frac{\iint_{D}u_x^2+u_y^2 + u^2 \d A + \int_{\partial D}\sig(s)u^2}{\iint_{D}u^2} \d s \] where the minimum is taken over all functions $u\in H^1(D)$, that are ****** $|\lam_n(\sig_1)-\lam_n(\sig_2)|\leq A^2\|\Del \sig\|_{L_1}$. % The command \| prints the double absolute value symbol. The following relation will be used to study the weak continuity in \Cprime. % Notice the \bar,\d commands in the following out of barnes.sty \begin{equation} \int_{\partial D}\Del \sig(s)u^2(s)\d s =u^2(s)\int_0^s\Del \sig(t)\dt\bar_0^{\ell} - \int_0^{\ell}\left(\int_0^s\Del \sig(t)\dt\right) 2u(s)\Frac{\d u}{\d s}\d s \Label{parts}\end{equation} \begin{theorem}\Label{unique} Suppose that $\sig^*(s)\in\Cprime$\ and that there is a unique function $\sig^*$\ that satisfies the equations $\lam_n(\sig^*,q)=\Lam_n$\ for all $n$. If $\sig_N(s)\in\Cprime$\ is any interpolating sequence then, it follows that $\sig_N(s)$\ converges weakly to $\sig^*(s)$. \end{theorem} \proof % Prints Proof in italics. Let $\sig_N(s)$\ be an interpolating sequence. Since the class \Class\ ****** to \Cprime, at least it belongs to \Class\ % Note the \ after the \Class % command. ******* convergent subsequence must converge to the same function $\sig^*(s)$, so the whole sequence $\sig_N(s)$\ must converge weakly. \endproof % Prints a box. $\pi^2$. It is easy to see that $\|\sig^*-\sig_k\|\not\ar 0$\ as $k\ar\infty$\ and the following table gives some numerical results illustrating the weak convergence as $k$\ increases. \\[.5em] % The command \\[dim] makes a new line with extra dim space down. % The tabular command is complex but.... \begin{center} \begin{tabular}{||c|c|c||} \hline k & \dis{\max_{1\leq j\leq 15} |\lam_j(\sig^*) - \lam_j(\sig_k)|} & $\|\sig^*-\sig_k\|_{1Max}$ \\ \hline 2 & .0520 &.0500 \\ \hline 4 & .0350 &.0250 \\ \hline 6 & .0320 &.0166 \\ \hline 8 & .0005 &.0125 \\ \hline 10 & .0003 &.0100 \\ \hline \end{tabular} \end{center} \vspace{1em} \noi Thus, the first 15 eigenvalues can not distinguish between $\sig^*(s)$\ and % A numbered list of items. \begin{enumerate} \item $h\leq \sig(s) \leq H$\ $\forall s\in[0,\ell]$. \item There exist $P+1$\ points $0=x_0