\documentclass[twoside]{article} \usepackage{amssymb} % used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Uniqueness for a Boundary Identification Problem \hfil}% {\hfil Kurt Bryan \& Lester F. Caudill, Jr. \hfil} \begin{document} \setcounter{page}{23} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Differential Equations and Computational Simulations III}\newline J. Graef, R. Shivaji, B. Soni J. \& Zhu (Editors)\newline Electronic Journal of Differential Equations, Conference~01, 1997, pp. 23--39.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp 147.26.103.110 or 129.120.3.113 (login: ftp)} \vspace{\bigskipamount} \\ Uniqueness for a Boundary Identification Problem in Thermal Imaging \thanks{ {\em 1991 Mathematics Subject Classifications:} 35A40, 35J25, 35R30. \hfil\break\indent {\em Key words and phrases:} Inverse problems, non-destructive testing, thermal imaging. \hfil\break\indent \copyright 1998 Southwest Texas State University and University of North Texas. \hfil\break\indent Published November 12, 1998. \hfil\break\indent Partially supported by NSF Grant DMS-9623279. } } \date{} \author{Kurt Bryan \& Lester F. Caudill, Jr.} \maketitle \begin{abstract} An inverse problem for an initial-boundary value problem is considered. The goal is to determine an unknown portion of the boundary of a region in ${\mathbb R}^n$ from measurements of Cauchy data on a known portion of the boundary. The dynamics in the interior of the region are governed by a differential operator of parabolic type. Utilizing a unique continuation result for evolution operators, along with the method of eigenfunction expansions, it is shown that uniqueness holds for a large and physically reasonable class of Cauchy data pairs. \end{abstract} \newtheorem{thm}{Theorem}[section] \newtheorem{lemma}{Lemma}[section] \def\rf#1{(\ref{#1})} \def\pderiv#1#2{\frac{\partial #1}{\partial #2}} \section{Introduction} \label{sec.intro} The goal of non-destructive evaluation is to gather information about the interior or other inaccessible portion of some material object from exterior measurements. Thermal imaging is one approach to this problem; a prescribed heat flux is applied to a portion of the surface of the object and the resulting surface temperature response is measured. From this information one attempts to determine the internal thermal properties of the object, or the shape of some unknown, inaccessible portion of the boundary. Thermal imaging holds promise as a tool for corrosion detection in aircraft, and has found utility in industrial applications. The interested reader is referred to \cite{BC}, and the references therein, for a discussion of some of these applications. Thermal imaging methods have also found application in the medical field. For example, infrared thermography has been used to investigate the distribution and structure of skin blood vessels; this has implications regarding potential recovery from burn injuries (\cite{CSCJ}), and also bears on the selection of donor sites for skin grafts (\cite{IA}). We are interested in the use of thermal imaging for the detection of so-called back surface'' corrosion and damage. The most elementary model of such a process is simple material loss which leads to a change in the surface profile of the object's boundary. This is the model we have chosen for this paper. Our long-term goal is to develop a reliable method for determining the presence and extent of corrosion. Of course, any such method requires a sound theoretical foundation. To this end, our focus in this work is on the issue of uniqueness---under what conditions do the proposed data measurements provide sufficient information from which to determine the shape of an unknown portion of the object's boundary? This problem may be formulated mathematically as an inverse problem for the heat equation. More precisely, let $\Omega\subseteq{\mathbb R}^n$ represent the object to be imaged. We assume that the surface $\partial\Omega$ of $\Omega$ is piecewise $C^2$. We use $\Gamma$ to denote the known,'' accessible portion of $\partial\Omega$, and we assume that both $\Gamma$ and $\partial\Omega\setminus\Gamma$ have nonzero surface measure as subsets of $\partial\Omega$. Let $S_0$ denote some open portion of $\Gamma$ with positive measure and let the applied heat flux $g(t,x)$ be defined for each $(t,x)\in{\mathbb R}^+; \partial\Omega$ with support in $S_0$. With some rescaling we model the propagation of heat through $\Omega$ with an initial-boundary value problem for the heat equation, \begin{eqnarray} \label{eqn.bvp1a} u_t(t,x)-\Delta_x u(t,x) &=& 0\,,\quad {\rm for} \,\,t\in{\mathbb R}^+,\,x\in\Omega\,,\\ \label{eqn.bvp1b} \pderiv{u}{\eta}(t,x) &=& g(t,x)\,,\quad {\rm for}\,\,t\in{\mathbb R}^+,\,x\in\partial\Omega\,, \\ \label{eqn.bvp1c} u(0,x) &=& u_0(x)\,,\quad {\rm for}\,\,x\in\Omega. \end{eqnarray} Here $u(t,x)$ denotes the temperature in the domain $\Omega$ at the point $x$ at time $t$, $u_t$ the derivative of $u$ with respect to $t$, and $\eta$ an outward unit normal vector field on $\partial\Omega$. Throughout this paper, we will refer to \rf{eqn.bvp1a}-\rf{eqn.bvp1c} collectively as (IBVP). Let $S_1\subset\Gamma$ denote the portion of the boundary on which we take temperature measurements. We consider the following inverse problem: Does knowledge of $u(t,x)$ on $S_1$ for some time period $t_0 < t < t_1$ uniquely determine $\partial\Omega\setminus\Gamma$? Specifically, suppose $\Omega_1\subseteq{\mathbb R}^n$ and $\Omega_2\subseteq{\mathbb R}^n$ with $\Gamma$ contained in $\partial\Omega_1 \cap\partial\Omega_2$. For $j=1,\,2$, let $u_j(t,x)$ be the solution of \rf{eqn.bvp1a}-\rf{eqn.bvp1c} with $\Omega$ replaced by $\Omega_j$. If, $u_1=u_2$ on $(t_0,t_1)\times S_1$, must it be true that $\partial\Omega_1\setminus\Gamma=\partial\Omega_2\setminus\Gamma$? \paragraph{Remark.} Implicit in the formulation of (IBVP) is the assumption that on the unknown part of the boundary the condition $\pderiv{u}{\eta}=0$ holds, so that the back surface acts as a perfect insulator. This is only a first approximation in most situations. In Section \ref{sec.newtonbcs} we discuss other boundary conditions in which the back surface loses heat to the ambient environment. \medskip The answer to the uniqueness problem posed in the present paper will be seen to depend on certain properties of the domain $\Omega$, the initial condition $u_0$, and the flux $g$. We will show that uniqueness holds for constant $u_0$ and any non-zero flux $g$. For non-constant $u_0$ one can impose reasonable conditions on the flux $g$ to ensure uniqueness, provided $\Omega$ is bounded. The case in which the flux $g$ is time-periodic was analyzed in \cite{BC}. This paper is organized as follows. The case of constant initial condition and non-constant flux is analyzed in \S \ref{sec.constantIC}, where uniqueness is proved. In \S \ref{sec.efn.expansion} we derive a useful eigenfunction representation and associated estimates for solutions of (IBVP), which are used in \S \ref{sec.second.unique} to prove a uniqueness result for bounded domains. In \S \ref{sec.newtonbcs}, we extend our results to include other possibilities for the boundary conditions on $\Gamma$. The fact that uniqueness for the inverse problem fails without additional hypotheses on the ingredients in \rf{eqn.bvp1a}-\rf{eqn.bvp1c} may be illustrated by a simple example in ${\mathbb R}^2$. Let $\Omega_1$ be the rectangle defined by $00$, then $u_1=u_2$ on $(0,T)\times \Omega'$. \end{lemma} In proving this lemma, we will make use of the following unique continuation result for parabolic equations. Its proof is based on the derivation of inequalities of Carleman type, and is omitted here. The interested reader is referred to the work of Saut and Scheurer \cite{SS}. \begin{lemma} \label{lemma.unique.continuation} Let $\Omega$ be a connected open set in ${\mathbb R}^n$ and $Q=(-T,T)\times\Omega$. Let ${u\in L^2\left((-T,T);\,H^2_{loc}\left(\Omega\right)\right)}$ be a solution of $u_t-\Delta u=0$ which vanishes in some open subset ${\cal O}$ of $Q$. Then $u$ vanishes in the horizontal component of ${\cal O}$. \end{lemma} \paragraph{Note:} Following Nirenberg \cite{Ni}, we define the {\it horizontal component} ${\cal O}_h$ of ${\cal O}$ to be the union of all open hyperplanes of the form $t=constant$ in $Q$ which have nonempty intersection with ${\cal O}$. \paragraph{Proof of Lemma \ref{lemma.agree.both.defined}.} The function $w\equiv u_1-u_2$ obeys \begin{eqnarray*} w_t-\Delta w &=& 0\,,\quad {\rm on}\,\,(0,T)\times\Omega', \\ w=\pderiv{w}{\eta} &=& 0\,,\quad {\rm on}\,\,(0,T)\times S_1, \\ w(x,0) &=& 0\,,\quad {\rm on}\,\,\Omega'. \end{eqnarray*} We can choose some open connected subset $I$ with $\bar{I}\subset S_1$ and open ball $B\subset{\mathbb R}^n$ such that $B\cap\partial\Omega'=I$. Let $\Omega_B=B\setminus \Omega'$ set ${\tilde\Omega}\equiv\Omega'\cup\Omega_B$. Define the function $${\tilde w}\equiv\cases{w,&(0,T)\times\Omega';\cr 0,&(0,T)\times\Omega_B;\cr 0,&(-T,0]\times {\tilde\Omega}.\cr}$$ For a smooth test function $\phi$, $$\int_0^T\int_{{\tilde\Omega}}{\tilde w}\left[\phi_t+\Delta\phi\right]dxdt=0\,,$$ so that ${\tilde w}$ satisfies \rf{eqn.bvp1a} on ${\tilde\Omega}$. Using standard parabolic regularity arguments (see, e.g., \cite{LSU}), one can show that ${{\tilde w}\in H^1\left((-T,T);\,H^2\left({\tilde\Omega}\right)\right)}$. Make the identifications $Q\equiv(-T,T)\times{\tilde\Omega}$ and ${\cal O}\equiv(-T,T)\times{\rm int}(\Omega_B)$ (in this case, the horizontal component of ${\cal O}$ is $Q$) to see that ${\tilde w}$ satisfies the hypotheses of Lemma \ref{lemma.unique.continuation}. We conclude that ${\tilde w}$ vanishes on $Q$ and so $u_1=u_2$ on $(0,T)\times\Omega'$. \hfill$\Box$\medskip We now present the main result of this section. \begin{thm} \label{thm.first.unique} Let $(u_1,\Omega_1)$ and $(u_2,\Omega_2)$ be solutions of \rf{eqn.bvp1a}-\rf{eqn.bvp1c}, with $(S_0\cup S_1)\subseteq (\partial\Omega_1\cap\partial\Omega_2)$. Suppose $u_0(x)=u_0$, a constant, and suppose that there is some time $T>0$ for which the applied flux $g(t,x)$ is not identically zero on $(0,T)\times S_0$. If $u_1=u_2$ on $(0,T)\times S_1$ then $\Omega_1=\Omega_2$ and $u_1=u_2$ on $\Omega_1=\Omega_2$. \end{thm} \paragraph{proof} By replacing $u_j$ with $u_j-u_0$, for $j=1,2$, if necessary, it suffices to consider the case $u_0=0$. Suppose that $\Omega_1\ne \Omega_2$, and let $\Omega'$ be as above. Then there exists some nonempty connected component $D$, sharing a portion of its boundary with $\partial\Omega'$, of either $\Omega_1\setminus\Omega_2$ or $\Omega_2\setminus\Omega_1$. Let us suppose the latter, so that $u_2$ is defined and satisfies \rf{eqn.bvp1a} on $D$. The boundary $\partial D$ of $D$ is comprised of a portion $\Gamma_1$ of $\partial\Omega_1$ and $\Gamma_2$ of $\partial\Omega_2$. On $\Gamma_2$ we know that the normal derivative of $u_2$ is identically zero; on $\Gamma_1$, we know that the normal derivative (from inside $\Omega_1$) of $u_1$ is zero, and since $u_2\equiv u_1$ on $\Omega'$ (by Lemma \ref{lemma.agree.both.defined}) and $u_2$ is smooth across $\Gamma_1$, we conclude that the normal derivative of $u_2$ vanishes on the boundary of $D$. Since $u_2$ satisfies equation \rf{eqn.bvp1a} with zero initial data on $D$, this forces $u_2\equiv 0$ on $(0,T)\times D$. Finally, by extending $u_2$ to be zero on $(-T,0]\times(\Omega'\cup D)$, we may appeal to Lemma \ref{lemma.unique.continuation} to conclude that $u_2\equiv 0$ on $(0,T)\times(\Omega'\cup D)$. This in turn implies that the flux $g$ is identically zero on $(0,T)\times S_0$, a contradiction, and we must conclude that $\Omega_1=\Omega_2$, as asserted. \hfill$\Box$ \section{Eigenfunction Expansion} \label{sec.efn.expansion} In this section we record a useful eigenfunction expansion for the function $u(t,x)$ which satisfies the parabolic initial-boundary value problem (IBVP) (\ref{eqn.bvp1a})-(\ref{eqn.bvp1c}). The technique, as well as the derivation of the accompanying estimates, are standard, and we have spared the reader the details. Instead, we defer to \cite{St}, or virtually any text on classical PDE. We assume that the initial condition $u_0$ belongs to $L^2(\Omega)$ and that for all $t>0$ the applied flux $g(t,x)$ belongs to $C^1((0,T); L^2(\partial\Omega))$, the space of continuously differentiable functions from $(0,T)$ to $L^2(\partial\Omega)$. We seek a solution $u(t,x)$ to (IBVP) in the space $C((0,T); L^2(\Omega))$; for such a solution the derivatives of $u$ with respect to $t$ and $x$ are not well-defined, and so we cast (IBVP) into a weak form. Multiply equation (\ref{eqn.bvp1a}) by a smooth test function $\phi(t,x)$ with $\phi(T,x)\equiv 0$ and $\frac{\partial\phi}{\partial\eta}=0$ on $\partial\Omega$, and then integrate over $(0,T)\times\Omega$. Integrate the term involving $\phi u_t$ by parts in $t$ use Green's second identity on the term involving $\phi\bigtriangleup u$ to obtain $$\label{weak} \int_{\Omega} u_0(x)\phi(0,x)\,dx + \int_0^T\int_{\Omega} u\left (\phi_t + \Delta \phi\right )\,dx\,dt + \int_0^T\int_{\partial\Omega} \phi g\,dS_x\,dt = 0.$$ The restriction of the $L^2(\Omega)$ function $u$ to $\partial\Omega$ is not well-defined, but since $\pderiv{\phi}{\eta}=0$ on $\partial\Omega$ the boundary integral involving $u \pderiv{\phi}{\eta}$ vanishes. This is our weak form of (\ref{eqn.bvp1a}) - (\ref{eqn.bvp1c}). In preparation for the eigenfunction expansion, let $\{\lambda_k,\,\psi_k(x)\}$, $k=0,1,\ldots$ be an eigensystem for $-\Delta$ on $\Omega$ with homogeneous Neumann boundary conditions, so that \begin{eqnarray*} \Delta\psi_k+\lambda_k\psi_k & = & 0\;\;{\rm in}\;\;\Omega,\\ \pderiv{\psi_k}{\eta} & = & 0 \;\;{\rm on}\;\;\partial\Omega .. \end{eqnarray*} The eigenvalues $\lambda_k$ are non-negative; order them by magnitude, so $\lambda_k \leq \lambda_{k+1}$. With the boundary condition $\pderiv{\psi_k}{\eta}=0$, the first eigenvalue $\lambda_0=0$, is simple, and has a constant eigenfunction. We normalize the eigenfunctions so that $\|\psi_k\|_{L^2(\Omega)}=1$ for all $k$, and so obtain an orthonormal basis for $L^2(\Omega)$. The function $\psi_0(x)$ is constant and $\psi_0(x)=1/\sqrt{|\Omega |}$. Orthogonality of the eigenfunctions then implies that $$\int_{\Omega}\psi_k(x)\,dx=0\,,\quad k\ge 1\,.$$ In later sections we will make use of the following standard estimate for solutions to Poisson's equation with Neumann boundary conditions. \begin{lemma} \label{lemma.v.bound} Let $f_1\in L^2(\Omega)$, $f_2\in L^2(\partial\Omega)$, and let $\psi(x)\in H^1(\Omega)$ satisfy \begin{eqnarray*} \bigtriangleup \psi & = & f_1\;\;\mbox{in}\;\;\Omega,\\ \pderiv{\psi}{\eta} & = & f_2\;\;\mbox{on}\;\;\partial\Omega, \\ \int_{\Omega}\psi(x)\,dx & = & 0,\nonumber \end{eqnarray*} Then $$\|\psi\|_{H^1(\Omega)} \leq C (\|f_2\|_{L^2(\partial\Omega)} + \|f_1\|_{L^2(\Omega)})$$ where $C$ depends on the domain $\Omega$. \end{lemma} The main result of this section is \begin{lemma} \label{replemma} The solution $u(t,x)$ to (\ref{weak}) is unique in $C({\mathbb R}^+; L^2(\Omega))$, and can be expanded as $$\label{ueqn} u(t,x) = v(t,x) + \frac{d_0}{\sqrt{|\Omega |}} + \frac{1}{|\Omega |} \int_0^t G(s)\,ds + \sum_{k=1}^{\infty} T_k(t)\psi_k(x)$$ where $v(t,x)$ defined on ${\mathbb R}^+\times\Omega$ denotes the unique function which satisfies the family of elliptic problems (indexed by $t$) \begin{eqnarray} \bigtriangleup_x v & = & \frac{1}{|\Omega|} G(t)\;\;\mbox{in}\;\;\Omega, \label{veqn}\\ \pderiv{v}{\eta} & = & g(t,x)\;\;\mbox{on}\;\;\partial\Omega, \nonumber\\ \int_{\Omega}v(t,x)\,dx & = & 0,\nonumber \end{eqnarray} and \begin{eqnarray} G(t) & = & \int_{\partial\Omega} g(t,x)\,dS_x, \label{Geqn}\\ d_k & = & \int_{\Omega}(u_0(x)-v(0,x))\psi_k(x)\,dx,\, \label{eqn.dk}\\ c_k(t)& = & -\int_{\Omega} v_t\psi_k(x)\,dx\,,\quad k>0\,,\label{ceqn}\\ c_0(t) & = & \frac{G(t)}{\sqrt{|\Omega |}},\nonumber\\ T_k(t) & = & d_ke^{-\lambda_k t} + \int_0^t c_k(s)e^{-\lambda_k (t-s)}\,ds, \;\;k>0, \label{Teqn} \end{eqnarray} where $|\Omega |$ denotes the measure of $\Omega$, $dS_x$ denotes surface measure on $\partial\Omega$, and $v_t$ is the derivative of $v(t,x)$ with respect to $t$ (which exists with the given hypotheses). We also have the estimate $$\label{growthest} \sum_{k=1}^{\infty} T_k^2(t) \leq C\left (e^{-2\lambda_1 t}\|u_0\|_{L^2(\Omega)} + \frac{t\|g_t(t,\cdot)\|^2_{L^2(\partial\Omega)}}{2\lambda_1}\right )$$ where $C$ is a constant which depends on $\Omega$. \end{lemma} \section{Uniqueness for Bounded Regions} \label{sec.second.unique} We now consider the more general case in which the initial condition $u_0$ need not be constant. Here we will assume that $\Omega$ is a bounded region. The essential idea in this section is simple. We note from the proof of Theorem \ref{thm.first.unique} that if uniqueness fails then there must be some insulated'' region $D$ inside $\Omega$. Within such a region, heat neither enters nor leaves, so that the average temperature of $D$ cannot increase with time. This is the basis of the argument that follows: intuitively, if the applied flux $g$ pumps enough heat into $\Omega$ over a long enough period then no region $D$ can remain at the same average temperature, and so a uniqueness result must hold. We make this physical argument precise below. \begin{thm} \label{thm.second.unique} Let $g(t,x)$ denote a flux in the class $C^1({\mathbb R};L^2(\partial\Omega))$ supported for $x\in S_0$ with $\|g(t,\cdot)\|_{L^2(\partial\Omega)} \leq M_0$ for all $t>0$ and $\|g_t(t,\cdot)\|_{L^2(\partial\Omega)}\leq M_1$ for all $t>0$. Suppose also that $G(t)$ defined by equation (\ref{Geqn}) satisfies $G(t)\geq G_0>0$ for all $t$. Let $u(t,x)$ be the solution to (\ref{eqn.bvp1a})-(\ref{eqn.bvp1c}) or its weak form (\ref{weak}) (the initial condition $u_0$ is not considered known). Then knowledge of $u(t,x)$ for $00$ such that measurements of $u_1$ and $u_2$ on $(0,T)\times S_1$ must differ. We proceed by contradiction. Assume that $u_1\equiv u_2$ on $(0,\infty)\times S_1$, and as before, let $\Omega'$ denote the connected component of $\Omega_1\cap\Omega_2$ for which $\Gamma\subseteq\partial\Omega'$. Let $w=u_1-u_2$. We will show that $w(t,x)\equiv 0$ on $(0,\infty)\times\Omega'$. To see this note that the function $w$ satisfies $$\label{weqn} \frac{\partial w}{\partial t} - \bigtriangleup w = 0 \,,\;\;\mbox{in}\;\;\Omega'\times (0,\infty)\,,$$ with $w=\frac{\partial w}{\partial\eta} =0$ on $S_1 \times (0,\infty)$. Let $p$ be a point in $S_1$ and $B$ a ball centered at $p$ such that $B\cap\partial \Omega'\subset S_1$. Let $B_0$ denote that portion of $B$ which lies outside $\Omega'$. Define $\tilde{w}(t,x)$ on $\Omega'\cup B_0$ as $$\tilde{w}(t,x) = \left \{ \begin{array}{ll} w(t,x), & x\in\Omega'\\ 0 & x\in B_0 \end{array} \right .$$ Standard regularity results (see \cite{LSU}) show that $w\in L^2((0,T); H^2(\Omega'))$ for any $T>0$. Since $w=\frac{\partial w}{\partial\eta}\equiv 0$ on $S_1$, it is easy to check that $\tilde{w}\in L^2((0,T); H^2(\Omega'\cup B_0))$. The function $\tilde{w}$ vanishes on $B_0\times (0,\infty)$ and we conclude from Lemma \ref{lemma.unique.continuation} (with the minor alteration $-T\rightarrow 0$) that $\tilde{w}$ vanishes on $\Omega'\times (0,\infty)$. This shows that $u_1\equiv u_2$ on $\Omega'\times (0,\infty)$. Also, since (\ref{weqn}) has a unique solution for given initial and boundary conditions, we conclude that $u_0=\tilde{u}_0$ on $\Omega'$. If $\Omega_1\neq \Omega_2$ then either $\Omega_1\setminus\Omega_2$ or $\Omega_2\setminus\Omega_1$ contains a nonempty connected component $D$ for which $\partial D\cap\partial\Omega'$ has positive surface measure. For specificity, we assume that $D\subset (\Omega_2\setminus\Omega_1)$. The boundary of $D$ consists of portions of $\partial\Omega_1\setminus (S_0\cup S_1)$ and $\partial \Omega_2\setminus (S_0\cup S_1)$. On these portions of the boundary the applied flux $g$ is identically zero. Standard regularity results then show that $u_2$ is a classical solution to the heat equation and smooth on $\bar{D}$, and we therefore have $\frac{\partial u_2}{\partial\eta}\equiv 0$ on $\partial D$. Since $D$ is bounded and $u_2$ is smooth, $$\frac{d}{dt}\int_{D}u_2(t,x)\,dx = \int_D \frac{\partial u_2}{\partial t}\,dx = \int_D \bigtriangleup u_2\,dx = \int_{\partial D}\frac{\partial u_2}{\partial\eta} \,dS_x = 0.$$ The integral $\int_{D}u_2(t,x)\,dx$ on the left is just the total thermal energy inside $D$. We have thus shown that the assumption that $u_1=u_2$ on $(0,\infty) \times S_1$ and $\Omega_1\neq\Omega_2$ force the existence of an insulated region $D$ for which $\int_D u_2(t,x)\,dx$ is constant. We will show that this is impossible for an applied flux $g(t,x)$ of the form specified in the statement of the theorem. Let $u_2(t,x)$ be expressed via an eigenfunction expansion as in equation (\ref{ueqn}). Integrating over $D$ shows that $$\label{eqnu2} \int_D u_2(t,x)\,dx = \int_D v(t,x)\,dx + \frac{d_0|D|}{\sqrt{|\Omega_2 |}} + \frac{|D|}{|\Omega_2 |} \int_0^t G(s)\,ds + \int_D \sum_{k=1}^{\infty} T_k(t)\psi_k(x)\,dx$$ where $T_k(t)$ is defined by equation (\ref{Teqn}), $d_0$ by equation (\ref{eqn.dk}), and $v$ satisfies (\ref{veqn}) with $\Omega$ replaced by $\Omega_2$. Since $G(t)\geq G_0$ for all $t$, the integral $\int_0^t G(s)\,ds$ grows at least as fast as $G_0t$; however, the other terms in the equation can be shown to be $o(t)$ as $t\to\infty$, and this will show that $\int_D u_2(t,x)\,dx$ cannot be constant. The first integral on the right side of equation \rf{eqnu2} is bounded in $t$, which can be proved by noting that \begin{eqnarray*} \left|\int_D v(t,x)\,dx \right|& \leq & \sqrt{|D|}\|v(t,\cdot)\|_{L^2(D)}\\ & \leq & \sqrt{|D|}\|v(t,\cdot)\|_{L^2(\Omega)\,,} \end{eqnarray*} and applying Lemma \ref{lemma.v.bound} with the fact that $\|g(t,\cdot)\|_{L^2(\partial\Omega)} \leq M_0$. The second term in equation (\ref{eqnu2}) is constant and, therefore, bounded in $t$. The last term can be estimated by noting that \begin{eqnarray} \left|\int_D \sum_{k=1}^{\infty} T_k(t)\psi_k(x)\,dx \right| & \leq & \sqrt{|D|} \left\|\sum_{k=1}^{\infty} T_k(t)\psi_k(x)\,dx\right\|_{L^2(D)},\nonumber\\ & \leq & \sqrt{|D|} \left\|\sum_{k=1}^{\infty} T_k(t)\psi_k(x)\,dx\right\|_{L^2(\Omega)}, \nonumber\\ & = & \sqrt{|D|}\sqrt{\sum_{k=1}^{\infty} T^2_k(t)}\label{Tbound}. \end{eqnarray} Combining (\ref{Tbound}) with the estimate (\ref{growthest}) in Lemma \ref{replemma} shows that $$\label{T2bound} \int_D \sum_{k=1}^{\infty} T_k(t)\psi_k(x)\,dx \leq C\sqrt{|D|}\left (e^{-2\lambda_1 t} \|u_0\|^2_{L^2(\Omega)} + \frac{M_1t}{2\lambda_1}\right)^{1/2}.$$ The quantity on the right side of (\ref{T2bound}) is clearly $o(t)$, and so grows more slowly than $\int_0^t G(s)\,ds$. Equation (\ref{eqnu2}) then shows that for sufficiently large $t$ the integral $\int_D u_2(t,x)\,dx$ must increase, a contradiction that proves Theorem \ref{thm.second.unique}. \hfill$\Box$\medskip If in addition to the conditions above $g$ is analytic in $t$ (for example, if $g$ is independent of $t$, so $g=g(x)$) then we can do better. Suppose that $g(t,x) \in C^{\omega}((0,T); L^2(\partial\Omega))$, i.e. for each $t_0 > 0$ there is some $\delta >0$ such that $g(t,x)$ can be written as $$g(t,x) = \sum_{k=0}^{\infty} \frac{(t-t_0)^k}{k!}g_k(x)$$ for all $t$ with $|t-t_0|<\delta$, where $g_k\in L^2(\partial\Omega)$. In this case the solution to (\ref{eqn.bvp1a})-(\ref{eqn.bvp1c}) is analytic in $t$, $$u(t,x) = \sum_{k=0}^{\infty} \frac{(t-t_0)^k}{k!}u_k(x)$$ where $u_k\in L^2(\Omega)$. Suppose that two domains $\Omega_1$ and $\Omega_2$ give rise to the same temperature measurements on $(t_1,t_2)\times S_1$ with $t_10$ and $\|g_t(t,\cdot)\|_{L^2(\partial\Omega)} \leq M_1$ for $t>0$. Suppose also that $G(t)$ defined by equation (\ref{Geqn}) satisfies $G(t)\geq G_0>0$ for all $t$. Let $u(t,x)$ be the solution to the IBVP (\ref{eqn.bvp1a})-(\ref{eqn.bvp1c}) or its weak form (\ref{weak}) (the initial condition $u_0$ is not considered known). Then knowledge of $u(t,x)$ for any open time interval $0 0$ and $g$ supported for $x\in S_0\subset\partial\Omega$. The Robin boundary condition $\frac{\partial u}{\partial n} + \alpha u = 0$ corresponds to a Newton-cooling type of heat loss on the boundary with ambient temperature scaled to zero; note that we have assumed that the loss term $-\alpha u$ applies even on $S_0$, where the flux $g$ is applied. The solution $u$ to the initial-boundary value problem \rf{newtoncool}-\rf{newtoncool.3} can be represented with an eigenfunction expansion, as $$\label{newtonrep} u(t,x) = v(t,x) + \sum_{k=0}^{\infty} T_k(t)\psi_k(x)$$ where $v(t,x)$ satisfies the family of elliptic problems (indexed by $t$) \begin{eqnarray} \bigtriangleup_x v & = & 0 \;\;\mbox{in}\;\;\Omega, \label{newveqn}\\ \frac{\partial v}{\partial n} + \alpha v & = & g\;\;{\rm on}\;\;\partial\Omega, \label{newveqn.2} \end{eqnarray} $T_k(t)$ is defined by $$\label{newTeqn} T_k(t) = d_ke^{-\lambda_k t} + e^{-\lambda_k t}\int_0^t c_k(s)e^{\lambda_k s} \,ds,$$ and \begin{eqnarray} c_k(t) & = & -\int_{\Omega} v_t \psi_k(x)\,dx, \label{newceqn}\\ d_k & = & \int_{\Omega}(u_0(x)-v(0,x))\psi_k(x)\,dx\nonumber \end{eqnarray} and finally, $\{\lambda_k,\psi_k(x)\}$ is an orthonormal eigensystem for $-\bigtriangleup$ with the Robin boundary conditions, so for each $k$, \begin{eqnarray*} \bigtriangleup\psi_k+\lambda_k\psi_k & = & 0\;\;{\rm in}\;\;\Omega,\\ \frac{\partial \psi_k}{\partial n} + \alpha \psi_k & = & 0,\;\;{\rm on}\;\;\partial\Omega\,. \end{eqnarray*} We order the eigenvalues by magnitude. It is easy to check that all eigenvalues are strictly positive. The next result gives sufficient conditions on the induced flux $g$ which guarantee uniqueness. As before, we assume that we have measurements of $u(t,x)$ for $x\in S_1\subset\partial\Omega$. Loosely speaking, we require the flux to be nonnegative and decaying in time, but not too quickly. More precisely, we have \begin{thm} \label{newtonunique} Let $(u_1,\Omega_1)$ and $(u_2,\Omega_2)$ be solutions of \rf{newtoncool}-\rf{newtoncool.3} with $(S_0\cup S_1)\subseteq (\partial\Omega_1\cap\partial\Omega_2)$. Suppose that the applied flux $g(t,x) \in C^1({\mathbb R};L^2(\partial\Omega))$ and is supported in $S_0$ for each $t$. Also, assume that \begin{enumerate} \item $g(t,x)$ is not identically zero. \item $g(t,x), \frac{\partial g}{\partial t}(t,x) \geq 0$ for all $x$ and $t$. \item $\left\|g_t(t,\cdot)\right\|_{L^2(\partial\Omega)} \rightarrow 0$ as $t\rightarrow\infty$ in such a way that \newline ${\sup_{s>t}\|g_t(s,\cdot)\|^2_{L^2(\partial\Omega)} = o\left(\frac{1}{\ln t}\right)}$ as $t\to\infty$. \end{enumerate} Then $u_1\equiv u_2$ on ${\mathbb R}^+\times S_1$ implies that $\Omega_1=\Omega_2$. \end{thm} In the case where $\Omega$ belongs to ${\mathbb R}^2$ or ${\mathbb R}^3$, one may relax the decay condition 3. in this result. More precisely, \begin{thm} \label{newtonuniqueprime} In space dimension 2 or 3, Theorem \ref{newtonunique} holds, with hypothesis 3. relaxed to \begin{itemize} \item[$3'.$] $\left\|g_t(t,\cdot)\right\|_{L^2(\partial\Omega)} \rightarrow 0$ as $t\rightarrow\infty$. \end{itemize} \end{thm} We shall prove Theorem \ref{newtonunique} first. Afterward, we will indicate the changes necessary to establish Theorem \ref{newtonuniqueprime}. \paragraph{Proof of Theorem \ref{newtonunique}.} Our proof proceeds by contradiction. Suppose $\Omega_1\neq \Omega_2$. The same reasoning as in the proof of Theorem \ref{thm.first.unique} shows that we must have some nonempty region $D\subset\Omega$ (where $\Omega$ is $\Omega_1$ or $\Omega_2$), with $\partial D\cap\partial\Omega'$ having positive surface measure, on which \begin{eqnarray*} \frac{\partial u}{\partial t} - \bigtriangleup u & = & 0\;\;{\rm on}\;\; {\mathbb R}^+ \times D\\ \frac{\partial u}{\partial n} + \alpha u & = & 0,\;\;{\rm on}\;\; {\mathbb R}^+\times \partial D\\ u(0,x) & = & u_0(x)\;\;{\rm on}\;\; D \end{eqnarray*} (where $u$ is either $u_1$ or $u_2$). (This is the same $\Omega'$ as in the proof of Theorem \ref{thm.second.unique}.) We first observe that the integral $\int_D u(t,x)\,dx$ must tend exponentially rapidly to zero as $t\to\infty$. To see this, note that $u(t,x)$ can be expanded on $D$ in terms of eigenfunctions $$\label{newueqn} u(t,x) = \sum_{k=0}^{\infty} d_k e^{-\tilde{\lambda}_k t}\tilde{\psi}_k(x)$$ with $$d_k = \int_{D}u_0(x)\tilde{\psi}_k(x)\,dx.$$ and $\{\tilde{\lambda}_k,\tilde{\psi}_k(x)\}$ is an eigensystem for $-\bigtriangleup$ on $D$ with boundary conditions \mbox{$\frac{\partial \tilde{\psi}_k}{\partial n} + \alpha \tilde{\psi}_k = 0$} on $\partial D$. Again, the eigenvalues are strictly positive. From the representation (\ref{newueqn}) \begin{eqnarray} \left | \int_D u(t,x)\,dx \right | & \leq & \sqrt{|D|}\|u\|_{L^2(D)}\nonumber\\ & = & O(e^{-\tilde{\lambda}_0 t}) \label{newthing} \end{eqnarray} where ${\tilde{\lambda}_0>0}$ is the smallest eigenvalue for the above eigensystem. We shall complete the proof of Theorem \ref{newtonunique} by contradicting relation \rf{newthing} in the following way: We shall show that, under the hypotheses on $g$, \begin{eqnarray} & & \lim_{t\rightarrow\infty} \int_D u(t,x)\,dx\longrightarrow \int_D v(t,x)\,dx \label{assert1}.\\ & & \int_D v(t,x)\,dx\;\;{\rm is\; bounded\; away\; from\; zero,\;\; uniformly\; in}\;\; t \label{assert2}. \end{eqnarray} From these two facts, it is clear that ${\int_D u(t,x)\,dx}$ must be bounded away from zero, uniformly in $t$, the desired contradiction to \rf{newthing}. To establish (\ref{assert1}), we first show that ${\|u-v\|_{L^2(\Omega)}\to 0}$ as $t\to\infty$. Note that \rf{newtonrep} and \rf{newTeqn} imply \begin{eqnarray} \|u-v\|^2_{L^2(\Omega)} & = & \sum_{k=0}^{\infty}T^2_k(t) \nonumber\\ & \le & 2\sum_{k=0}^{\infty}\left(\int_0^tc_k(s)e^{-\lambda_k (t-s)}ds\right)^2 +o(1) \label{uvdiff} \end{eqnarray} where the last equality follows from the fact that $\lambda_k>0$ for each $k$. The integral appearing inside the sum on the right can be bounded as \begin{eqnarray} \left(\int_0^t c_k(s)e^{-\lambda_k (t-s)}ds\right )^2 & = & \left(\int_0^{\beta} c_k(s)e^{-\lambda_k (t-s)}ds + \int_{\beta}^t c_k(s)e^{-\lambda_k (t-s)}ds \right )^2\nonumber\\ & \le & \left (\frac{e^{-2\lambda_k (t-\beta)} - e^{-2\lambda_k t}}{\lambda_k}\right ) \int_0^{\beta} c^2_k(s)\,ds\nonumber\\ & & + \left (\frac{1 - e^{-2\lambda_k (t-\beta)}}{\lambda_k}\right ) \int_{\beta}^t c^2_k(s)\,ds\label{messy} \end{eqnarray} where $\beta=\beta(t)\in (0,t)$ is to be specified in a moment. From the bounds (\ref{uvdiff}) and (\ref{messy}) we conclude that \begin{eqnarray} \|u-v\|^2_{L^2(\Omega)} & \leq & C \left( \left(\frac{e^{-2\lambda_0 (t-\beta)} - e^{-2\lambda_0 t}}{\lambda_0}\right) \int_0^{\beta}\sum_{k=0}^{\infty} c_k^2(s)\,ds\right . \nonumber\\ & & +\left . \left(\frac{1 - e^{-2\lambda_0 (t-\beta)}}{\lambda_0}\right)\int_{\beta}^t \sum_{k=0}^{\infty} c_k^2(s)\,ds\right) \label{uvbound2} \end{eqnarray} for some constant $C$, where we have interchanged the summation and integral for the convergent series and used that fact that ${\frac{e^{-\lambda_k t} - e^{-2\lambda_k t}}{\lambda_k}}$ and ${\frac{1-e^{-\lambda_k t}}{\lambda_k}}$ are decreasing functions of $\lambda_k$ for $\lambda_k,t>0$. Next, note that from equation (\ref{newceqn}) we have $\sum_{k=0}^{\infty} c_k(t)^2 = \|v_t(t,\cdot)\|^2_{L^2(\Omega)}$, where $v_t$ satisfies \begin{eqnarray} \bigtriangleup v_t & = & 0\;\;{\rm in}\;\;\Omega,\label{newvteqn}\\ \frac{\partial v_t}{\partial\eta} + \alpha v_t & = & g_t\;\;{\rm on}\;\;\partial\Omega \label{newvteqn.2} \end{eqnarray} Standard estimates similar to Lemma \ref{lemma.v.bound} show that we can bound $$\label{othervbound} \|v_t\|_{L^2(\Omega)} \leq C \|g_t\|_{L^2(\partial\Omega)}.$$ It then follows from (\ref{uvbound2}) and (\ref{othervbound}) that \begin{eqnarray} \|u-v\|^2_{L^2(\Omega)} & \leq & C \left ( \right . (e^{-2\lambda_0 (t-\beta)} - e^{-2\lambda_0 t}) \int_0^{\beta} \|g_t(s,\cdot)\|^2_{L^2(\partial\Omega)}\,ds \nonumber\\ & & + (1 - e^{-2\lambda_0 (t-\beta)}) \int_{\beta}^t \|g_t(s,\cdot)\|^2_{L^2(\partial\Omega)}\,ds \left . \right ) \nonumber\\ &\leq & C \left ( \right . (e^{-2\lambda_0 (t-\beta)}- e^{-2\lambda_0 t}) \beta\cdot\sup_{0r} \|g_t(s,\cdot)\|^2_{L^2(\partial\Omega)}$, and observe that$\epsilon(r)$has the following properties: \begin{itemize} \item For each$r$,$\epsilon(r)\ge 0$and$\epsilon'(r)\le 0$. \item$\epsilon(0)<\infty$and$\epsilon(r)\to 0$as$r\to\infty$. \end{itemize} From equation \rf{funstuff}, we have, in terms of$\epsilon(r)$, \begin{eqnarray} \|u-v\|^2_{L^2(\Omega)} & \leq & C \left ( \right . (e^{-2\lambda_0 (t-\beta)}- e^{-2\lambda_0 t}) \beta\cdot\epsilon(0) \nonumber\\ & & + (1 - e^{-2\lambda_0 (t-\beta)})(t-\beta)\epsilon(\beta)\left . \right ) \label{morefunstuff} \end{eqnarray} We now specify, for each fixed$t>0$, a corresponding value of$\beta$implicitly by the relation $$\beta + K\ln\beta = t\,,$$ where$K\in{\mathbb R}^+$satisfies$K>\frac{1}{2\lambda_0}$. It is simple to check that this relation defines$\beta$uniquely as a function of$t$, and that$\beta(t)\to\infty$as$t\to\infty$. Furthermore, $$t-\beta = K\ln\beta\to\infty$$ as$t\to\infty$. With$\beta(t)$so defined, we note that $$\label{eqn.goaway1} (t-\beta)\epsilon(\beta) = K\ln(\beta)\epsilon(\beta)\to 0$$ as$t$(and hence$\beta$)$\to\infty$, by virtue of the decay condition on$\epsilon$. Also, $$\label{eqn.goaway2} \beta e^{-2\lambda_0(t-\beta)} = \beta e^{-2\lambda_0K\ln\beta} = \beta^{1-2\lambda_0K}\to 0$$ as$t\to\infty$, since$2\lambda_0 K>1$. In light of \rf{eqn.goaway1} and \rf{eqn.goaway2}, we see that the right-hand side of \rf{morefunstuff} decays to$0$as$t\to\infty$. From this, we conclude that${\|u-v\|_{L^2(\Omega)}\to 0}$as$t\to\infty$. In fact,${\|u-v\|_{L^2(D)}\to 0}$for any$D\subset\Omega$, so that $$\left|\int_D (u-v)\,dx\right| \leq \|u-v\|_{L^1(D)} \leq \sqrt{|D|}\|u-v\|_{L^2(D)} \rightarrow 0\,,$$ from which we conclude $$\label{eqn.utov} \int_D u(t,x)\,dx \rightarrow \int_D v(t,x)\,dx$$ as$t\rightarrow \infty$, which is (\ref{assert1}). It remains only to establish (\ref{assert2}). To this end, let us first consider the case in which$g(t,x)\in C^1({\mathbb R}; C^2(\partial\Omega))$. Then the function$v(t,\cdot)\in C^2(\bar{\Omega})$for all$t$. We have by the maximum principle that the minimum value of$v(t,x)$on$\bar{\Omega}$occurs at a point on$\partial\Omega$at which$\frac{\partial v}{\partial\eta} \leq 0$. At such a point,$\alpha v = g - \frac{\partial v}{\partial\eta} \geq 0,$from which we conclude that$v(t,x)\geq 0$for$x\in\Omega$. In particular, for any$D\subset\Omega$we have$\int_D v(t,x)\,dx \geq 0$, with equality if and only if$v(t,x)\equiv 0$. Since$v\equiv 0$if and only if$g\equiv 0$(from \rf{newvteqn}-\rf{newvteqn.2}), the hypotheses on$g$imply that${\int_D v(t,x)\,dx > 0}$for each$t$. (In particular,${\int_D v(0,x)\,dx > 0}$.) Furthermore, since$g_t\geq 0$, the same reasoning shows that$\int_D v_t(t,x)\,dx \geq 0$for any$D\subset\Omega$and all$t>0$. Consequently, for each$t>0$, \begin{eqnarray*} \int_D v(t,x)\,dx &=& \int_D v(0,x)\,dx + \int_0^t \pderiv{}{s}\int_D v(s,x)\,dxds \\ &=& \int_D v(0,x)\,dx + \int_0^t \int_D v_t(s,x)\,dxds \\ &\geq& \int_D v(0,x)\,dx > 0\,, \end{eqnarray*} where we have used the fact that$D$is bounded and smooth enough to interchange the order of integration and differentiation. This establishes (\ref{assert2}) for$g(t,x)\in C^1({\mathbb R}; C^2(\partial\Omega))$. Finally, we note that the same conclusion holds if$g(t,\cdot)$is merely$L^2(\partial\Omega)$, rather than$C^2(\partial\Omega)$, for we can approximate any non-negative$g\in L^2(\partial\Omega)$arbitrarily closely (in$L^2(\partial\Omega)$) with a non-negative function$\tilde{g}(t,\cdot) \in C^2 (\partial\Omega)$. From the standard estimate $$\|v - \tilde{v}\|_{L^2(\Omega)} \leq C \|g - \tilde{g}\|_{L^2(\partial\Omega)}$$ where$\tilde{v}$satisfies the boundary value problem \rf{newveqn}-\rf{newveqn.2} with$g$replaced by$\tilde{g}$, we conclude that$\|v - \tilde{v}\|_{L^2(D)}$can be made arbitrarily small. Since $$\left|\int_D v(t,x)\,dx - \int_D \tilde{v}(t,x)\,dx \right| \leq \sqrt{|D|} \|v - \tilde{v}\|_{L^2(D)} \leq C \|g - \tilde{g}\|_{L^2(\partial\Omega)}\,,$$ and since$\int_D \tilde{v}(t,x)\,dx>0$uniformly in$t$, we conclude that$\int_D v(t,x)\,dx>0$uniformly in$t$also. This establishes (\ref{assert2}), and completes the proof. \hfill$\Box$\paragraph{Proof of Theorem \ref{newtonuniqueprime}.} Noting that the decay condition${o\left(\frac{1}{\ln t}\right)}$was used only to establish \rf{eqn.utov}, the preceding proof also works for Theorem \ref{newtonuniqueprime}, provided we show that this relation still holds. Let$\Omega\subseteq{\mathbb R}^n$with$n=2$or$n=3$, and set $$\phi(t)\equiv\sum_{k=0}^{\infty}\left(\int_0^tc_k(s)e^{-\lambda_k(t-s)}ds\right)^2\,.$$ In light of \rf{uvdiff}, it suffices to show that $$\lim_{t\to\infty}\phi(t) = 0\,.$$ To this end, set $$M(t)\equiv \sup_{tt and all k. We can estimate \begin{eqnarray} \nonumber \phi(t) & = & \sum_{k=0}^{\infty} \left (\int_0^{t/2} c_k(s)e^{-\lambda_k (t-s)}\,ds + \int_{t/2}^{t} c_k(s)e^{-\lambda_k (t-s)}\,ds\right )^2 \\ & \leq & 2M^2(0)e^{-\lambda_0 t/2}\sum_{k=0}^{\infty}\frac{1}{\lambda^2_k} + 2M^2(t/2)\sum_{k=0}^{\infty}\frac{1}{\lambda^2_k} \label{eqn.new} \end{eqnarray} In {\mathbb R}^2 or {\mathbb R}^3, we have {\sum_{k=0}^{\infty}\frac{1}{\lambda^2_k}<\infty\,,} so that \rf{eqn.new} yields$$\phi(t) \le C_1e^{-\lambda_0 t/2} + C_2M^2(t/2)\,.$$By hypothesis$3'.$,$M\left(\frac{t}{2}\right)\to 0$as$t\to\infty$, so that \rf{eqn.utov} holds. \hfill$\Box$\section{Concluding Remarks} We have examined a variety of settings in which the Cauchy data for the heat equation uniquely determines the shape of the region on which the heat equation is defined. Specifically, if the initial condition is constant over the region of interest, then the Cauchy data---temperature and heat flux---on any open portion of the boundary of the region over any time interval determines the shape of the region. In the more general case in which the initial condition is not necessarily constant, the Cauchy data for the time interval$(0, \infty)$uniquely determines the shape of a bounded region$\Omega$, provided that the data satisfy certain reasonable conditions. For insulate boundary conditions$\frac{\partial u}{\partial \eta} = 0$on the unknown portion$B$of the boundary, the prescribed flux$g(t,x)$on$\partial\Omega\setminus B$must provide a net positive flux at all times and be bounded away from zero, and$g_t$must be bounded. For the Robin boundary condition$\frac{\partial u}{\partial\eta} + \alpha u = 0\$, we require the flux to be positive at all points and times, and obey a certain decay property in time. The techniques employed in this analysis can be used to investigate other types of boundary conditions. The choices presented here reflect sensible conditions within the context of the particular physical situation---remote corrosion detection---which motivates our study of this model. We also note that these techniques may be extended in a straightforward fashion to include more general parabolic equations. For example, one could incorporate a spatially-varying thermal conductivity, and conduct the preceding analysis within the framework of appropriately-weighted Hilbert spaces, with similar results. While a uniqueness result holds, this inverse problem is most certainly ill-posed; the shape of the region will not be a continuous function of the measured data in any reasonable norm. A next logical step would be to examine and quantify the nature of the ill-posedness and identify the features of the boundary which can be stably estimated from the Cauchy data. This should give insight into useful and practical reconstruction algorithms. Such an algorithm might be based on the ideas in \cite{BC}---linearize the forward problem and examine the linearized map from the boundary shape'' space to the measured temperature data. The forward map will be given as an integral operator with smooth kernel, and will have an unbounded inverse. We are currently investigating such an approach to gain an understanding of stability and reconstruction possibilities. \begin{thebibliography}{9} \bibitem{Adams} R.A.\ Adams, {\sl Sobolev Spaces}, Academic Press, New York, 1975. \bibitem{BC} K.\ Bryan and L.F.\ Caudill,\ Jr., {\sl An inverse problem in thermal imaging}, SIAM \ J.\ Appl.\ Math., 56 (1996), pp. 715--735. \bibitem{CSCJ} R.\ Cole, P.\ Shakespeare, H.\ Chissell, and S.\ Jones, {\sl Thermographic assessment of burns using a nonpermeable membrane as wound covering}, Burns 17 (1991), pp. 117-122. \bibitem{GT} D.\ Gilbarg and N.\ Trudinger, {\sl Elliptic Partial Differential Equations of Second Order} (Second Edition), Springer-Verlag, Berlin, 1983. \bibitem{IA} Y.\ Itoh and K.\ Arai, {\sl Use of recovery-enhanced thermography to localize cutaneous perforators}, Annals of Plastic Surgery 34 (1995), pp. 507-511. \bibitem{LSU} O.\ Ladyzhenskaya, V.\ Solonnikov, and N.\ Uraltseva, {\sl Linear and Quasilinear Equations of Parabolic Type} (Translations of Mathematical Monographs, v.\ 23), American Mathematical Society, Providence, RI, 1968. \bibitem{Ni} L.\ Nirenberg, {\sl Uniqueness in Cauchy problems for differential equations with constant leading coefficients}, Comm.\ Pure\ Appl.\ Math.\ 10 (1957), pp. 89-105. \bibitem{SS} J.C.\ Saut and B.\ Scheurer, {\sl Unique continuation for some evolution equations}, J.\ Differential\ Equations\ 66 (1987), 118-139. \bibitem{St} I.\ Stakgold, {\sl Green's Functions and Boundary Value Problems}, John Wiley \& Sons, New York, 1979. \end{thebibliography} \newpage {\sc Kurt Bryan} \\ Department of Mathematics\\ Rose-Hulman Institute of Technology\\ Terre Haute, IN 47803, USA.\\ E-mail: kurt.bryan@rose-hulman.edu \\ \medskip {\sc Lester F. Caudill, Jr.} \\ Department of Mathematics and Computer Science\\ University of Richmond\\ Richmond, VA 23173, USA\\ E-mail: lcaudill@richmond.edu \end{document}