Differential Equations and Computational Simulations III J. Graef, R. Shivaji, B. Soni J. & Zhu (Editors) Electronic Journal of Differential Equations, Conference 01, 1997, pp. 23–39. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp 147.26.103.110 or 129.120.3.113 (login: ftp)

Uniqueness for a Boundary Identification Problem in Thermal Imaging ^{∗}

Kurt Bryan & Lester F. Caudill, Jr.

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 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.

1 Introduction

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 [2], 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 ([3]), and also bears on the selection of donor sites for skin grafts ([5]).

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 Ω ⊆ ℝ^{n} represent the object to be imaged. We assume that the surface ∂Ω of Ω is piecewise C^{2}. We use Γ to denote the “known,” accessible portion of ∂Ω, and we assume that both Γ and ∂Ω \ Γ have nonzero surface measure as subsets of ∂Ω. Let S_{0} denote some open portion of Γ with positive measure and let the applied heat flux g(t, x) be defined for each (t, x) ∈ ℝ^{+}; ∂Ω with support in S_{0}. With some rescaling we model the propagation of heat through Ω with an initial-boundary value problem for the heat equation,

∗1991 Mathematics Subject Classifications: 35A40, 35J25, 35R30.

Key words and phrases: Inverse problems, non-destructive testing, thermal imaging.

\begin{matrix}c \\ ◯ \end{matrix}1998 Southwest Texas State University and University of North Texas.

Published November 12, 1998.

Partially supported by NSF Grant DMS-9623279.

u_{t}(t, x) − Δ_{x}u(t, x) = 0 , for t ∈ ℝ^{+}, x ∈ Ω ,  (1)
\frac{∂u}{∂η}(t, x) = g(t, x) , for t ∈ ℝ^{+}, x ∈ ∂Ω ,  (2)
u(0, x) = u_{0}(x) , for x ∈ Ω.  (3)

Here u(t, x) denotes the temperature in the domain Ω at the point x at time t, u_{t} the derivative of u with respect to t, and η an outward unit normal vector field on ∂Ω. Throughout this paper, we will refer to (1)-(3) collectively as (IBVP). Let S_{1} ⊂ Γ 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 ∂Ω \ Γ? Specifically, suppose Ω_{1} ⊆ ℝ^{n} and Ω_{2} ⊆ ℝ^{n} with Γ contained in ∂Ω_{1} ∩ ∂Ω_{2}. For j = 1, 2, let u_{j}(t, x) be the solution of (1)-(3) with Ω replaced by Ω_{j}. If, u_{1} = u_{2} on (t_{0}, t_{1}) × S_{1}, must it be true that ∂Ω_{1} \ Γ = ∂Ω_{2} \ Γ?

Remark. Implicit in the formulation of (IBVP) is the assumption that on the unknown part of the boundary the condition \frac{∂u}{∂η} = 0 holds, so that the back surface acts as a perfect insulator. This is only a first approximation in most situations. In Section 5 we discuss other boundary conditions in which the back surface loses heat to the ambient environment.

The answer to the uniqueness problem posed in the present paper will be seen to depend on certain properties of the domain Ω, 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 Ω is bounded. The case in which the flux g is time-periodic was analyzed in [2].

This paper is organized as follows. The case of constant initial condition and non-constant flux is analyzed in §2, where uniqueness is proved. In §3 we derive a useful eigenfunction representation and associated estimates for solutions of (IBVP), which are used in §4 to prove a uniqueness result for bounded domains.

In §5, we extend our results to include other possibilities for the boundary conditions on Γ.

The fact that uniqueness for the inverse problem fails without additional hypotheses on the ingredients in (1)-(3) may be illustrated by a simple example in ℝ^{2}. Let Ω_{1} be the rectangle defined by 0 < x < 2π, 0 < y < π. Let Ω_{2} be Ω_{1} minus the rectangle \frac{2}{3}π < x < \frac{4}{3}π, 0 < y < \frac{2}{3}π, so that Ω_{1} and Ω_{2} share the “known” top boundary Γ = {(x, y) : 0 < x < 2π, y = π}. Let u(t, x, y) be the function

u(t, x, y) = e^{− \frac{9}{2} t} cos(\frac{3}{2}x) cos(\frac{3}{2}y) ,

and set u_{1} ≡ u|_{Ω_{1}} and u_{2} ≡ u|_{Ω_{2}}. One may verify directly that, for j = 1, 2,

\frac{∂u_{j}}{∂η} = e^{− \frac{9}{2} t} cos(\frac{3}{2}x) on Γ while \frac{∂u_{j}}{∂η} = 0 everywhere else on ∂Ω_{j}. Both u_{1} and u_{2} satisfy (1) with the same initial condition, with the same Cauchy data on Γ, and so in this case uniqueness fails. Analogous counter-examples can be constructed in other dimensions.

2 Constant Initial Condition

In this section we develop a uniqueness result for the case in which the initial condition u_{0}(x) is constant. The only condition on the applied flux g(t, x) is that it be regular enough for (IBVP) to possess a unique solution, e.g., g ∈ C(ℝ; L^{2}(∂Ω)).

In what follows, we will require the following lemma. Let Ω_{1} and Ω_{2} represent two objects which share the “known” boundary portion Γ. One can see that there exists a connected component of Ω_{1} ∩ Ω_{2} which has Γ as part of its boundary. We shall denote this component by Ω′.

Lemma 2.1 Let (u_{1}, Ω_{1}) and (u_{2}, Ω_{2}) each satisfy (1)-(3). If u_{1} = u_{2} on (0, T) × S_{1} for some time T > 0, then u_{1} = u_{2} on (0, T) × Ω′.

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 [8].

Lemma 2.2 Let Ω be a connected open set in ℝ^{n} and Q = (−T, T) × Ω. Let u ∈ L^{2} ((−T, T); H_{loc}^{2} (Ω)) be a solution of u_{t} − Δu = 0 which vanishes in some open subset O of Q. Then u vanishes in the horizontal component of O.

Note: Following Nirenberg [7], we define the horizontal component O_{h} of O to be the union of all open hyperplanes of the form t = constant in Q which have nonempty intersection with O.

Proof of Lemma 2.1. The function w ≡ u_{1} − u_{2} obeys

w_{t} − Δw = 0 , on (0, T) × Ω′,
w = \frac{∂w}{∂η} = 0 , on (0, T) × S_{1},
w(x, 0) = 0 , on Ω′.

We can choose some open connected subset I with Ī ⊂ S_{1} and open ball B ⊂ ℝ^{n} such that B ∩ ∂Ω′ = I. Let Ω_{B} = B \ Ω′ set Ω̃ ≡ Ω′ ∪ Ω_{B}. Define the function

w̃ ≡ \{ \begin{matrix}w, & (0, T) × Ω′; \\ 0, & (0, T) × Ω_{B}; \\ 0, & (−T, 0] × Ω̃. \end{matrix}

For a smooth test function ϕ,

∫_{0}^{T} ∫Ω̃ w̃ [ϕ_{t} + Δϕ] dxdt = 0 ,

so that w̃ satisfies (1) on Ω̃. Using standard parabolic regularity arguments (see, e.g., [6]), one can show that w̃ ∈ H^{1} ((−T, T); H^{2} \binom{}{Ω̃}).

Make the identifications Q ≡ (−T, T)×Ω̃ and O ≡ (−T, T)×int(Ω_{B}) (in this case, the horizontal component of O is Q) to see that w̃ satisfies the hypotheses of Lemma 2.2. We conclude that w̃ vanishes on Q and so u_{1} = u_{2} on (0, T)×Ω′. □

We now present the main result of this section.

Theorem 2.1 Let (u_{1}, Ω_{1}) and (u_{2}, Ω_{2}) be solutions of (1)-(3), with (S_{0}∪S_{1}) ⊆ (∂Ω_{1} ∩ ∂Ω_{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)×S_{0}. If u_{1} = u_{2} on (0, T) × S_{1} then Ω_{1} = Ω_{2} and u_{1} = u_{2} on Ω_{1} = Ω_{2}.

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 Ω_{1} ≠ Ω_{2}, and let Ω′ be as above. Then there exists some nonempty connected component D, sharing a portion of its boundary with ∂Ω′, of either Ω_{1} \ Ω_{2} or Ω_{2} \ Ω_{1}. Let us suppose the latter, so that u_{2} is defined and satisfies (1) on D. The boundary ∂D of D is comprised of a portion Γ_{1} of ∂Ω_{1} and Γ_{2} of ∂Ω_{2}. On Γ_{2} we know that the normal derivative of u_{2} is identically zero; on Γ_{1}, we know that the normal derivative (from inside Ω_{1}) of u_{1} is zero, and since u_{2} ≡ u_{1} on Ω′ (by Lemma 2.1) and u_{2} is smooth across Γ_{1}, we conclude that the normal derivative of u_{2} vanishes on the boundary of D. Since u_{2} satisfies equation (1) with zero initial data on D, this forces u_{2} ≡ 0 on (0, T) × D. Finally, by extending u_{2} to be zero on (−T, 0] × (Ω′ ∪ D), we may appeal to Lemma 2.2 to conclude that u_{2} ≡ 0 on (0, T) × (Ω′ ∪ D). This in turn implies that the flux g is identically zero on (0, T) × S_{0}, a contradiction, and we must conclude that Ω_{1} = Ω_{2}, as asserted. □

3 Eigenfunction 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) (1)-(3). 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 [9], or virtually any text on classical PDE.

We assume that the initial condition u_{0} belongs to L^{2}(Ω) and that for all t > 0 the applied flux g(t, x) belongs to C^{1}((0, T); L^{2}(∂Ω)), the space of continuously differentiable functions from (0, T) to L^{2}(∂Ω). We seek a solution u(t, x) to (IBVP) in the space C((0, T); L^{2}(Ω)); 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 (1) by a smooth test function ϕ(t, x) with ϕ(T, x) ≡ 0 and \frac{∂ϕ}{∂η} = 0 on ∂Ω, and then integrate over (0, T) × Ω. Integrate the term involving ϕu_{t} by parts in t use Green’s second identity on the term involving ϕ △ u to obtain

∫_{Ω} u_{0}(x)ϕ(0, x) dx + ∫_{0}^{T} ∫_{Ω} u (ϕ_{t} + Δϕ) dx dt + ∫_{0}^{T} ∫_{∂Ω} ϕg dS_{x} dt = 0. (4)

The restriction of the L^{2}(Ω) function u to ∂Ω is not well-defined, but since \frac{∂ϕ}{∂η} = 0 on ∂Ω the boundary integral involving u\frac{∂ϕ}{∂η} vanishes. This is our weak form of (1) - (3).

In preparation for the eigenfunction expansion, let {λ_{k}, ψ_{k}(x)}, k = 0, 1, … be an eigensystem for −Δ on Ω with homogeneous Neumann boundary conditions, so that

Δψ_{k} + λ_{k}ψ_{k} = 0 in Ω,
\frac{∂ψ_{k}}{∂η} = 0 on ∂Ω..

The eigenvalues λ_{k} are non-negative; order them by magnitude, so λ_{k} ≤ λ_{k+1}. With the boundary condition \frac{∂ψ_{k}}{∂η} = 0, the first eigenvalue λ_{0} = 0, is simple, and has a constant eigenfunction. We normalize the eigenfunctions so that ∥ψ_{k}∥_{L^{2}(Ω)} = 1 for all k, and so obtain an orthonormal basis for L^{2}(Ω). The function ψ_{0}(x) is constant and ψ_{0}(x) = 1/\sqrt{|Ω|}. Orthogonality of the eigenfunctions then implies that

∫_{Ω} ψ_{k}(x) dx = 0 , k ≥ 1 .

In later sections we will make use of the following standard estimate for solutions to Poisson’s equation with Neumann boundary conditions.

Lemma 3.1 Let f_{1} ∈ L^{2}(Ω), f_{2} ∈ L^{2}(∂Ω), and let ψ(x) ∈ H^{1}(Ω) satisfy

△ψ = f_{1} in Ω,
\frac{∂ψ}{∂η} = f_{2} on ∂Ω,
∫_{Ω} ψ(x) dx = 0,

Then

∥ψ∥_{H^{1}(Ω)} ≤ C(∥f_{2}∥_{L^{2}(∂Ω)} + ∥f_{1}∥_{L^{2}(Ω)})

where C depends on the domain Ω.

The main result of this section is

Lemma 3.2 The solution u(t, x) to (4) is unique in C(ℝ^{+}; L^{2}(Ω)), and can be expanded as

u(t, x) = v(t, x) + \frac{d_{0}}{\sqrt{|Ω|}} + \frac{1}{|Ω|} ∫_{0}^{t} G(s) ds + ∑_{k=1}^{∞} T_{k}(t)ψ_{k}(x)  (5)

where v(t, x) defined on ℝ^{+} × Ω denotes the unique function which satisfies the family of elliptic problems (indexed by t)

△_{x}v = \frac{1}{|Ω|}G(t) in Ω,  (6)
\frac{∂v}{∂η} = g(t, x) on ∂Ω,
∫_{Ω} v(t, x) dx = 0,

and

G(t) = ∫_{∂Ω} g(t, x) dS_{x},  (7)
d_{k} = ∫_{Ω}(u_{0}(x) − v(0, x))ψ_{k}(x) dx,  (8)
c_{k}(t) = − ∫_{Ω} v_{t}ψ_{k}(x) dx , k > 0 ,  (9)
c_{0}(t) = \frac{G(t)}{\sqrt{|Ω|}},

T_{k}(t) = d_{k}e^{−λ_{k}t} + ∫_{0}^{t} c_{k}(s)e^{−λ_{k}(t−s)} ds, k > 0,  (10)

where |Ω| denotes the measure of Ω, dS_{x} denotes surface measure on ∂Ω, 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

∑_{k=1}^{∞} T_{k}^{2}(t) ≤ C (e^{−2λ_{1}t}∥u_{0}∥_{L^{2}(Ω)} + \frac{t∥g_{t}(t, ·)∥_{L^{2}(∂Ω)}^{2}}{2λ_{1}} )  (11)

where C is a constant which depends on Ω.

4 Uniqueness for Bounded Regions

We now consider the more general case in which the initial condition u_{0} need not be constant. Here we will assume that Ω is a bounded region. The essential idea in this section is simple. We note from the proof of Theorem 2.1 that if uniqueness fails then there must be some “insulated” region D inside Ω. 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 Ω 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.

Theorem 4.1 Let g(t, x) denote a flux in the class C^{1}(ℝ; L^{2}(∂Ω)) supported for x ∈ S_{0} with ∥g(t, ·)∥_{L^{2}(∂Ω)} ≤ M_{0} for all t > 0 and ∥g_{t}(t, ·)∥_{L^{2}(∂Ω)} ≤ M_{1} for all t > 0. Suppose also that G(t) defined by equation (7) satisfies G(t) ≥ G_{0} > 0 for all t. Let u(t, x) be the solution to (1)-(3) or its weak form (4) (the initial condition u_{0} is not considered known). Then knowledge of u(t, x) for 0 < t < ∞ and x ∈ S_{1} uniquely determines the region Ω and the initial condition u_{0}.

Proof of Theorem 4.1. Suppose that u_{1} and u_{2} are solutions to the weak form (4) of (IBVP) on domains Ω_{1} and Ω_{2}, respectively, with initial conditions u_{1}(0, x) = u_{0}(x) and u_{2}(0, x) = ũ_{0}(x). We assume that temperature measurements are taken on an open subset S_{1} ⊂ (∂Ω_{1} ∩ ∂Ω_{2}) and the same flux g(t, x) applied on an open subset S_{0} ⊂ (∂Ω_{1} ∩ ∂Ω_{2}). We will show that there is some time T > 0 such that measurements of u_{1} and u_{2} on (0, T) × S_{1} must differ.

We proceed by contradiction. Assume that u_{1} ≡ u_{2} on (0, ∞) × S_{1}, and as before, let Ω′ denote the connected component of Ω_{1} ∩ Ω_{2} for which Γ ⊆ ∂Ω′. Let w = u_{1} − u_{2}. We will show that w(t, x) ≡ 0 on (0, ∞) × Ω′. To see this note that the function w satisfies

\frac{∂w}{∂t} − △w = 0 , in Ω′ × (0, ∞) ,  (12)

with w = \frac{∂w}{∂η} = 0 on S_{1} × (0, ∞). Let p be a point in S_{1} and B a ball centered at p such that B ∩∂Ω′ ⊂ S_{1}. Let B_{0} denote that portion of B which lies outside Ω′. Define w̃(t, x) on Ω′ ∪ B_{0} as

w̃(t, x) = \{ \begin{matrix}w(t, x), & x ∈ Ω′ \\ 0 & x ∈ B_{0} \end{matrix}

Standard regularity results (see [6]) show that w ∈ L^{2}((0, T); H^{2}(Ω′)) for any T > 0. Since w = \frac{∂w}{∂η} ≡ 0 on S_{1}, it is easy to check that w̃ ∈ L^{2}((0, T); H^{2}(Ω′ ∪ B_{0})). The function w̃ vanishes on B_{0} × (0, ∞) and we conclude from Lemma 2.2 (with the minor alteration −T → 0) that w̃ vanishes on Ω′ × (0, ∞). This shows that u_{1} ≡ u_{2} on Ω′ × (0, ∞). Also, since (12) has a unique solution for given initial and boundary conditions, we conclude that u_{0} = ũ_{0} on Ω′.

If Ω_{1} ≠ Ω_{2} then either Ω_{1} \ Ω_{2} or Ω_{2} \ Ω_{1} contains a nonempty connected component D for which ∂D ∩ ∂Ω′ has positive surface measure. For specificity, we assume that D ⊂ (Ω_{2} \ Ω_{1}). The boundary of D consists of portions of ∂Ω_{1} \ (S_{0} ∪ S_{1}) and ∂Ω_{2} \ (S_{0} ∪ 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 D̄, and we therefore have \frac{∂u_{2}}{∂η} ≡ 0 on ∂D. Since D is bounded and u_{2} is smooth,

\frac{d}{dt} ∫_{D} u_{2}(t, x) dx = ∫_{D} \frac{∂u_{2}}{∂t} dx = ∫_{D} △u_{2} dx = ∫_{∂D} \frac{∂u_{2}}{∂η} dS_{x} = 0.

The integral ∫_{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, ∞) × S_{1} and Ω_{1} ≠ Ω_{2} force the existence of an insulated region D for which ∫_{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 (5). Integrating over D shows that

∫_{D} u_{2}(t, x) dx = ∫_{D} v(t, x) dx+ \frac{d_{0}|D|}{\sqrt{|Ω_{2}|}} + \frac{|D|}{|Ω_{2}|} ∫_{0}^{t} G(s) ds+∫_{D} ∑_{k=1}^{∞} T_{k}(t)ψ_{k}(x) dx

(13)

where T_{k}(t) is defined by equation (10), d_{0} by equation (8), and v satisfies (6) with Ω replaced by Ω_{2}. Since G(t) ≥ G_{0} for all t, the integral ∫_{0}^{t} G(s) ds grows at least as fast as G_{0}t; however, the other terms in the equation can be shown to be o(t) as t → ∞, and this will show that ∫_{D} u_{2}(t, x) dx cannot be constant.

The first integral on the right side of equation (13) is bounded in t, which can be proved by noting that

|∫_{D} v(t, x) dx| ≤ \sqrt{|D|}∥v(t, ·)∥_{L^{2}(D)}
≤ \sqrt{|D|}∥v(t, ·)∥_{L^{2}(Ω) ,}

and applying Lemma 3.1 with the fact that ∥g(t, ·)∥_{L^{2}(∂Ω)} ≤ M_{0}.

The second term in equation (13) is constant and, therefore, bounded in t. The last term can be estimated by noting that

|∫_{D} ∑_{k=1}^{∞} T_{k}(t)ψ_{k}(x) dx| ≤ \sqrt{|D|} ∥∑_{k=1}^{∞} T_{k}(t)ψ_{k}(x) dx∥_{L^{2}(D)} ,

≤ \sqrt{|D|} ∥∑_{k=1}^{∞} T_{k}(t)ψ_{k}(x) dx∥_{L^{2}(Ω)} ,

= \sqrt{|D|}\sqrt{∑_{k=1}^{∞} T_{k}^{2}(t)}.  (14)

Combining (14) with the estimate (11) in Lemma 3.2 shows that

∫_{D} ∑_{k=1}^{∞} T_{k}(t)ψ_{k}(x) dx ≤ C\sqrt{|D|} (e^{−2λ_{1}t}∥u_{0}∥_{L^{2}(Ω)}^{2} + \frac{M_{1}t}{2λ_{1}})^{1/2} . (15)

The quantity on the right side of (15) is clearly o(t), and so grows more slowly than ∫_{0}^{t} G(s) ds. Equation (13) then shows that for sufficiently large t the integral ∫_{D} u_{2}(t, x) dx must increase, a contradiction that proves Theorem 4.1. □

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) ∈ C^{ω}((0, T); L^{2}(∂Ω)), i.e. for each t_{0} > 0 there is some δ > 0 such that g(t, x) can be written as

g(t, x) = ∑_{k=0}^{∞} \frac{(t − t_{0})^{k}}{k!} g_{k}(x)

for all t with |t − t_{0}| < δ, where g_{k} ∈ L^{2}(∂Ω). In this case the solution to (1)-(3) is analytic in t,

u(t, x) = ∑_{k=0}^{∞} \frac{(t − t_{0})^{k}}{k!} u_{k}(x)

where u_{k} ∈ L^{2}(Ω). Suppose that two domains Ω_{1} and Ω_{2} give rise to the same temperature measurements on (t_{1}, t_{2})×S_{1} with t_{1} < t_{2}. Arguing as in the proof of Theorem 4.1 we find that u_{1} ≡ u_{2} on (t_{1}, t_{2}) × Ω′, but since u_{1} and u_{2} are analytic in t we have u_{1} ≡ u_{2} on (0, ∞) × Ω′. The rest of the proof of Theorem 4.1 remains unchanged and we have

Theorem 4.2 Let g(t, x) denote a flux in the class C^{ω}(ℝ; L^{2}(∂Ω)) supported for x ∈ S_{0} with ∥g(t, ·)∥_{L^{2}(∂Ω)} ≤ M_{0} for all t > 0 and ∥g_{t}(t, ·)∥_{L^{2}(∂Ω)} ≤ M_{1} for t > 0. Suppose also that G(t) defined by equation (7) satisfies G(t) ≥ G_{0} > 0 for all t. Let u(t, x) be the solution to the IBVP (1)-(3) or its weak form (4) (the initial condition u_{0} is not considered known). Then knowledge of u(t, x) for any open time interval 0 < t_{1} < t < t_{2} and x ∈ S_{1} uniquely determines the region Ω and the initial condition u_{0}.

5 Other Boundary Conditions

The results of the previous section show that we can uniquely identify the unknown portion of the surface of Ω if we pump in enough heat for a long enough time. However, in this situation the insulated boundary condition \frac{∂u}{∂n} = 0 on ∂Ω becomes physically less realistic. Those portions of the boundary on which a nonzero flux is not applied will tend to lose heat to the surrounding environment. In this section we consider uniqueness results under boundary conditions which model this heat loss. The proofs are quite similar to those of the previous section.

Suppose that u(t, x) satisfies the initial-boundary value problem

\frac{∂u}{∂t} − △u = 0 on ℝ^{+} × Ω  (16)
\frac{∂u}{∂n} + αu = g(t, x), on ℝ^{+} × ∂Ω  (17)
u(0, x) = u_{0}(x) on Ω  (18)

with α > 0 and g supported for x ∈ S_{0} ⊂ ∂Ω. The Robin boundary condition \frac{∂u}{∂n} +αu = 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 −αu applies even on S_{0}, where the flux g is applied.

The solution u to the initial-boundary value problem (16)-(18) can be represented with an eigenfunction expansion, as

u(t, x) = v(t, x) + ∑_{k=0}^{∞} T_{k}(t)ψ_{k}(x)  (19)

where v(t, x) satisfies the family of elliptic problems (indexed by t)

△_{x}v = 0 in Ω,  (20)
\frac{∂v}{∂n} + αv = g on ∂Ω,  (21)

T_{k}(t) is defined by

T_{k}(t) = d_{k}e^{−λ_{k}t} + e^{−λ_{k}t} ∫_{0}^{t} c_{k}(s)e^{λ_{k}s} ds,  (22)

and

c_{k}(t) = − ∫_{Ω} v_{t}ψ_{k}(x) dx,  (23)
d_{k} = ∫_{Ω}(u_{0}(x) − v(0, x))ψ_{k}(x) dx

and finally, {λ_{k}, ψ_{k}(x)} is an orthonormal eigensystem for −△ with the Robin boundary conditions, so for each k,

△ψ_{k} + λ_{k}ψ_{k} = 0 in Ω,
\frac{∂ψ_{k}}{∂n} + αψ_{k} = 0, on ∂Ω .

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 ∈ S_{1} ⊂ ∂Ω. Loosely speaking, we require the flux to be nonnegative and decaying in time, but not too quickly. More precisely, we have

Theorem 5.1 Let (u_{1}, Ω_{1}) and (u_{2}, Ω_{2}) be solutions of (16)-(18) with (S_{0} ∪ S_{1}) ⊆ (∂Ω_{1} ∩ ∂Ω_{2}). Suppose that the applied flux g(t, x) ∈ C^{1}(ℝ; L^{2}(∂Ω)) and is supported in S_{0} for each t. Also, assume that

1. g(t, x) is not identically zero.

2. g(t, x), \frac{∂g}{∂t} (t, x) ≥ 0 for all x and t.

3. ∥g_{t}(t, ·)∥_{L^{2}(∂Ω)} → 0 as t → ∞ in such a way that sup_{s>t} ∥g_{t}(s, ·)∥_{L^{2}(∂Ω)}^{2} = o (\frac{1}{ln t}) as t → ∞.

Then u_{1} ≡ u_{2} on ℝ^{+} × S_{1} implies that Ω_{1} = Ω_{2}.

In the case where Ω belongs to ℝ^{2} or ℝ^{3}, one may relax the decay condition 3. in this result. More precisely,

Theorem 5.2 In space dimension 2 or 3, Theorem 5.1 holds, with hypothesis 3. relaxed to

3′. ∥g_{t}(t, ·)∥_{L^{2}(∂Ω)} → 0 as t → ∞.

We shall prove Theorem 5.1 first. Afterward, we will indicate the changes necessary to establish Theorem 5.2.

Proof of Theorem 5.1. Our proof proceeds by contradiction. Suppose Ω_{1} ≠ Ω_{2}. The same reasoning as in the proof of Theorem 2.1 shows that we must have some nonempty region D ⊂ Ω (where Ω is Ω_{1} or Ω_{2}), with ∂D∩∂Ω′ having positive surface measure, on which

\frac{∂u}{∂t} − △u = 0 on ℝ^{+} × D
\frac{∂u}{∂n} + αu = 0, on ℝ^{+} × ∂D
u(0, x) = u_{0}(x) on D

(where u is either u_{1} or u_{2}). (This is the same Ω′ as in the proof of Theorem 4.1.) We first observe that the integral ∫_{D} u(t, x) dx must tend exponentially rapidly to zero as t → ∞. To see this, note that u(t, x) can be expanded on D in terms of eigenfunctions

u(t, x) = ∑_{k=0}^{∞} d_{k}e^{−λ̃_{k}t}ψ̃_{k}(x)  (24)

with

d_{k} = ∫_{D} u_{0}(x)ψ̃_{k}(x) dx.

and {λ̃_{k}, ψ̃_{k}(x)} is an eigensystem for −△ on D with boundary conditions \frac{∂ψ̃_{k}}{∂n} + αψ̃_{k} = 0 on ∂D. Again, the eigenvalues are strictly positive. From the representation (24)

|∫_{D} u(t, x) dx| ≤ \sqrt{|D|}∥u∥_{L^{2}(D)}
= O(e^{−λ̃_{0}t})  (25)

where λ̃_{0} > 0 is the smallest eigenvalue for the above eigensystem.

We shall complete the proof of Theorem 5.1 by contradicting relation (25) in the following way: We shall show that, under the hypotheses on g,

lim_{t→∞} ∫_{D} u(t, x) dx ⟶ ∫_{D} v(t, x) dx.  (26)
∫_{D} v(t, x) dx is bounded away from zero, uniformly in t. (27)

From these two facts, it is clear that ∫_{D} u(t, x) dx must be bounded away from zero, uniformly in t, the desired contradiction to (25).

To establish (26), we first show that ∥u − v∥_{L^{2}(Ω)} → 0 as t → ∞. Note that (19) and (22) imply

∥u − v∥_{L^{2}(Ω)}^{2} = ∑_{k=0}^{∞} T_{k}^{2}(t)

≤ 2 ∑_{k=0}^{∞} (∫_{0}^{t} c_{k}(s)e^{−λ_{k}(t−s)}ds)^{2} + o(1)  (28)

where the last equality follows from the fact that λ_{k} > 0 for each k. The integral appearing inside the sum on the right can be bounded as

(∫_{0}^{t} c_{k}(s)e^{−λ_{k}(t−s)}ds)^{2} = (∫_{0}^{β} c_{k}(s)e^{−λ_{k}(t−s)}ds + ∫_{β}^{t} c_{k}(s)e^{−λ_{k}(t−s)}ds)^{2}

≤ (\frac{e^{−2λ_{k}(t−β)} − e^{−2λ_{k}t}}{λ_{k}} ) ∫_{0}^{β} c_{k}^{2}(s) ds
+ (\frac{1 − e^{−2λ_{k}(t−β)}}{λ_{k}} ) ∫_{β}^{t} c_{k}^{2}(s) ds  (29)

where β = β(t) ∈ (0, t) is to be specified in a moment. From the bounds (28) and (29) we conclude that

∥u − v∥_{L^{2}(Ω)}^{2} ≤ C ((\frac{e^{−2λ_{0}(t−β)} − e^{−2λ_{0}t}}{λ_{0}} ) ∫_{0}^{β} ∑_{k=0}^{∞} c_{k}^{2}(s) ds

+ (\frac{1 − e^{−2λ_{0}(t−β)}}{λ_{0}} ) ∫_{β}^{t} ∑_{k=0}^{∞} c_{k}^{2}(s) ds)  (30)

for some constant C, where we have interchanged the summation and integral for the convergent series and used that fact that \frac{e^{−λ_{k}t}−e^{−2λ_{k}t}}{λ_{k}} and \frac{1−e^{−λ_{k}t}}{λ_{k}} are decreasing functions of λ_{k} for λ_{k}, t > 0.

Next, note that from equation (23) we have ∑_{k=0}^{∞} c_{k}(t)^{2} = ∥v_{t}(t, ·)∥_{L^{2}(Ω)}^{2}, where v_{t} satisfies

△v_{t} = 0 in Ω,  (31)
\frac{∂v_{t}}{∂η} + αv_{t} = g_{t} on ∂Ω  (32)

Standard estimates similar to Lemma 3.1 show that we can bound

∥v_{t}∥_{L^{2}(Ω)} ≤ C∥g_{t}∥_{L^{2}(∂Ω)}.  (33)

It then follows from (30) and (33) that

∥u − v∥_{L^{2}(Ω)}^{2} ≤ C ( (e^{−2λ_{0}(t−β)} − e^{−2λ_{0}t}) ∫_{0}^{β} ∥g_{t}(s, ·)∥_{L^{2}(∂Ω)}^{2} ds
+(1 − e^{−2λ_{0}(t−β)}) ∫_{β}^{t} ∥g_{t}(s, ·)∥_{L^{2}(∂Ω)}^{2} ds )
≤ C ( (e^{−2λ_{0}(t−β)} − e^{−2λ_{0}t})β · sup_{0<s<β} ∥g_{t}(s, ·)∥_{L^{2}(∂Ω)}^{2}
+(1 − e^{−2λ_{0}(t−β)})(t − β) sup_{β<s<t} ∥g_{t}(s, ·)∥_{L^{2}(∂Ω)}^{2} ) (34)

(the 1/λ_{0} factor has been absorbed into the constant C). For r ≥ 0, define ϵ(r) = sup_{s>r} ∥g_{t}(s, ·)∥_{L^{2}(∂Ω)}^{2}, and observe that ϵ(r) has the following properties:

• For each r, ϵ(r) ≥ 0 and ϵ′(r) ≤ 0.

• ϵ(0) < ∞ and ϵ(r) → 0 as r → ∞.

From equation (34), we have, in terms of ϵ(r),

∥u − v∥_{L^{2}(Ω)}^{2} ≤ C ( (e^{−2λ_{0}(t−β)} − e^{−2λ_{0}t})β · ϵ(0)
+(1 − e^{−2λ_{0}(t−β)})(t − β)ϵ(β) )  (35)

We now specify, for each fixed t > 0, a corresponding value of β implicitly by the relation

β + K ln β = t ,

where K ∈ ℝ^{+} satisfies K > \frac{1}{2λ_{0}} . It is simple to check that this relation defines β uniquely as a function of t, and that β(t) → ∞ as t → ∞. Furthermore,

t − β = K ln β → ∞

as t → ∞. With β(t) so defined, we note that

(t − β)ϵ(β) = K ln(β)ϵ(β) → 0  (36)

as t (and hence β) → ∞, by virtue of the decay condition on ϵ. Also,

βe^{−2λ_{0}(t−β)} = βe^{−2λ_{0}K ln β} = β^{1−2λ_{0}K} → 0  (37)

as t → ∞, since 2λ_{0}K > 1. In light of (36) and (37), we see that the right-hand side of (35) decays to 0 as t → ∞. From this, we conclude that ∥u − v∥_{L^{2}(Ω)} → 0 as t → ∞. In fact, ∥u − v∥_{L^{2}(D)} → 0 for any D ⊂ Ω, so that

|∫_{D}(u − v) dx| ≤ ∥u − v∥_{L^{1}(D)} ≤ \sqrt{|D|}∥u − v∥_{L^{2}(D)} → 0 ,

from which we conclude

∫_{D} u(t, x) dx → ∫_{D} v(t, x) dx  (38)

as t → ∞, which is (26).

It remains only to establish (27). To this end, let us first consider the case in which g(t, x) ∈ C^{1}(ℝ; C^{2}(∂Ω)). Then the function v(t, ·) ∈ C^{2}(Ω̄) for all t. We have by the maximum principle that the minimum value of v(t, x) on Ω̄ occurs at a point on ∂Ω at which \frac{∂v}{∂η} ≤ 0. At such a point, αv = g − \frac{∂v}{∂η} ≥ 0, from which we conclude that v(t, x) ≥ 0 for x ∈ Ω. In particular, for any D ⊂ Ω we have ∫_{D} v(t, x) dx ≥ 0, with equality if and only if v(t, x) ≡ 0. Since v ≡ 0 if and only if g ≡ 0 (from (31)-(32)), the hypotheses on g imply that ∫_{D} v(t, x) dx > 0 for each t. (In particular, ∫_{D} v(0, x) dx > 0.) Furthermore, since g_{t} ≥ 0, the same reasoning shows that ∫_{D} v_{t}(t, x) dx ≥ 0 for any D ⊂ Ω and all t > 0. Consequently, for each t > 0,

∫_{D} v(t, x) dx = ∫_{D} v(0, x) dx + ∫_{0}^{t} \frac{∂}{∂s} ∫_{D} v(s, x) dxds
= ∫_{D} v(0, x) dx + ∫_{0}^{t} ∫_{D} v_{t}(s, x) dxds
≥ ∫_{D} v(0, x) dx > 0 ,

where we have used the fact that D is bounded and smooth enough to interchange the order of integration and differentiation. This establishes (27) for g(t, x) ∈ C^{1}(ℝ; C^{2}(∂Ω)).

Finally, we note that the same conclusion holds if g(t, ·) is merely L^{2}(∂Ω), rather than C^{2}(∂Ω), for we can approximate any non-negative g ∈ L^{2}(∂Ω) arbitrarily closely (in L^{2}(∂Ω)) with a non-negative function g̃(t, ·) ∈ C^{2}(∂Ω). From the standard estimate

∥v − ṽ∥_{L^{2}(Ω)} ≤ C∥g − g̃∥_{L^{2}(∂Ω)}

where ṽ satisfies the boundary value problem (20)-(21) with g replaced by g̃, we conclude that ∥v − ṽ∥_{L^{2}(D)} can be made arbitrarily small. Since

|∫_{D} v(t, x) dx − ∫_{D} ṽ(t, x) dx| ≤ \sqrt{|D|}∥v − ṽ∥_{L^{2}(D)} ≤ C∥g − g̃∥_{L^{2}(∂Ω)} ,

and since ∫_{D} ṽ(t, x) dx > 0 uniformly in t, we conclude that ∫_{D} v(t, x) dx > 0 uniformly in t also. This establishes (27), and completes the proof. □

Proof of Theorem 5.2. Noting that the decay condition o (\frac{1}{ln t}) was used only to establish (38), the preceding proof also works for Theorem 5.2, provided we show that this relation still holds. Let Ω ⊆ ℝ^{n} with n = 2 or n = 3, and set

ϕ(t) ≡ ∑_{k=0}^{∞} (∫_{0}^{t} c_{k}(s)e^{−λ_{k}(t−s)}ds)^{2} .

In light of (28), it suffices to show that

lim_{t→∞} ϕ(t) = 0 .

To this end, set

M(t) ≡ sup_{t<s<∞} ∥{c_{k}(s)}∥_{l^{2}(k)} .

(Recall that ∥{c_{k}(s)}∥_{l^{2}(k)} is of the same order as ∥g_{t}(t, ·)∥_{L^{2}(∂Ω)} as t → ∞.) Note that |c_{k}(s)| ≤ M(t) for s > t and all k. We can estimate

ϕ(t) = ∑_{k=0}^{∞} (∫_{0}^{t/2} c_{k}(s)e^{−λ_{k}(t−s)} ds + ∫_{t/2}^{t} c_{k}(s)e^{−λ_{k}(t−s)} ds)^{2}

≤ 2M^{2}(0)e^{−λ_{0}t/2} ∑_{k=0}^{∞} \frac{1}{λ_{k}^{2}} + 2M^{2}(t/2) ∑_{k=0}^{∞} \frac{1}{λ_{k}^{2}}  (39)

In ℝ^{2} or ℝ^{3}, we have ∑_{k=0}^{∞} \frac{1}{λ_{k}^{2}} < ∞ , so that (39) yields

ϕ(t) ≤ C_{1}e^{−λ_{0}t/2} + C_{2}M^{2}(t/2) .

By hypothesis 3′., M (\frac{t}{2}) → 0 as t → ∞, so that (38) holds. □

6 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, ∞) uniquely determines the shape of a bounded region Ω, provided that the data satisfy certain reasonable conditions. For insulate boundary conditions \frac{∂u}{∂η} = 0 on the unknown portion B of the boundary, the prescribed flux g(t, x) on ∂Ω \ 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{∂u}{∂η} + αu = 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 illposed; 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 [2]—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.

References

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[2] K. Bryan and L.F. Caudill, Jr., An inverse problem in thermal imaging, SIAM J. Appl. Math., 56 (1996), pp. 715–735.

[3] R. Cole, P. Shakespeare, H. Chissell, and S. Jones, Thermographic assessment of burns using a nonpermeable membrane as wound covering, Burns 17 (1991), pp. 117-122.

[4] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order (Second Edition), Springer-Verlag, Berlin, 1983.

[5] Y. Itoh and K. Arai, Use of recovery-enhanced thermography to localize cutaneous perforators, Annals of Plastic Surgery 34 (1995), pp. 507-511.

[6] O. Ladyzhenskaya, V. Solonnikov, and N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type (Translations of Mathematical Monographs, v. 23), American Mathematical Society, Providence, RI, 1968.

[7] L. Nirenberg, Uniqueness in Cauchy problems for differential equations with constant leading coefficients, Comm. Pure Appl. Math. 10 (1957), pp. 89-105.

[8] J.C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations 66 (1987), 118-139.

[9] I. Stakgold, Green’s Functions and Boundary Value Problems, John Wiley & Sons, New York, 1979.

Kurt Bryan Department of Mathematics Rose-Hulman Institute of Technology Terre Haute, IN 47803, USA. E-mail: kurt.bryan@rose-hulman.edu

Lester F. Caudill, Jr. Department of Mathematics and Computer Science University of Richmond Richmond, VA 23173, USA E-mail: lcaudill@richmond.edu