ON A LOGARITHMIC STABILITY ESTIMATE FOR AN INVERSE HEAT CONDUCTION PROBLEM

. We are concerned with an inverse problem arising in thermal imaging in a bounded domain Ω ⊂ R n , n = 2 , 3. This inverse problem consists in the determination of the heat exchange coeﬃcient q ( x ) appearing in the boundary of a heat equation with Robin boundary condition.

1. Introduction. This paper investigates an inverse boundary coefficient problem in thermal imaging. The inverse problem under consideration consists in recovering the unknown heat exchange (heat loss) coefficient q(x) appearing in the heat equation with Robin boundary condition. This recovery may be obtained from boundary temperature measurements. In practice, this kind of inverse problem can be used to model the damage localization or corrosion detection in an inaccessible portion of some material object [23,28] and heat loss as well [8,17,32,34,33].
In this paper we consider a C 3 -smooth bounded domain Ω of R n (n = 2, 3) with boundary Γ := ∂Ω. We assume that there exist two subsets of Γ disjoint, Γ a and Γ i , with nonzero surface measure such that Γ := Γ a ∪ Γ i ;Γ a = ∅, andΓ i = ∅, (1.1) where Γ a denotes the "known" accessible portion of ∂Ω. We impose on the portion Γ i (may be inaccessible) a condition modeling the heat loss. The support of the applied heat flux g (stationary or time dependent) is contained in Γ a . The temperature distribution, denoted by u, satisfies the following initial-boundary value problem where ∂ ν denotes the derivative in the direction of the exterior unit normal vector ν to Γ.
In engineering applications, the stationary heat flux g corresponds to an uniform heating of the outer surface. Typically, this is the case when heat or flash lamps are used to provide the input flux g. In this paper, we study separately the two cases: stationary and time-dependent heat flux. Here, it is worth noticing that the time-periodic heat flux problem was studied in [6].
Besides, for an initial condition u 0 ∈ L 2 (Ω), one can prove that the problem (1.2) has a unique solution u ∈ C([0, ∞[; L 2 (Ω)) ∩ C(]0, ∞[; H 2 (Ω)) ∩ L ∞ ((0, ∞); H 1 (Ω)) and to be able to get some estimations of the solution, we assume that for a fixed positive constant R 0 , we have In this paper, we restrict our attention to the Robin boundary condition ∂ ν u + q(x)u = 0 which, according to [7], corresponds to a Newton-cooling type of heat loss on the boundary with ambient temperature scaled to zero. Now, using the same notations as in [14], we introduce the vector space where S (R n ) is the space of temperate distributions on R n ,ŷ is the Fourier transform of the function y and B s,r (R n ) is a Besov space (see Chapter 10 in [22]). The Besov spaces B s,r (R n ) play an important role in generalizing many classic functional spaces. Moreover, the space B s,2 (R n ) is the Sobolev space H s (R n ). In addition, if s ∈ (0, 1) and r = ∞, we have B s,∞ (R n ) = C s (R n ) where C s is the Hölder space (see the Appendix of [21]). Using local charts and partition of unity, B s,r (∂Ω) is defined from B s,r (R n−1 ) in the same way as H s (R n ) is built from H s (R n−1 ). In our current study, some smoothness properties of the solution to the problem (1.2) are needed. In order to give sufficient conditions on data guaranteeing these smoothness properties, we introduce the following sets of boundary coefficients: where M > 0 is a given constant. Let us recall that the function q(x) (heat exchange coefficient) in (1.2) is known as the Robin coefficient with a support in Γ i . So, the introduction of the space D 0 M is suitable and it will be useful in the rest of the paper. The inverse Robin problem has been investigated, theoretically and numerically by several authors (see e.g. [1,3,4,5,6,7,9,10,11,12,13,18,19,23,25,26,27,29,32,34]) where various type of stability estimates are given.
In this work, we firstly established a double-logarithmic stability estimate for recovering the boundary coefficient. Our result is considerably different from those already established in [3,12] in two or three dimensional spaces where Ω is of class C ∞ . In this paper, Ω is just of class C 3 . Moreover in [3,12] the error q −q is only estimated in a compact subset of {x ∈ Γ i ; u q (x) = 0} by a simple logarithmic stability estimates. In this paper, the error q −q is obtained in the whole Γ i by a double-logarithmic stability. This means that the stability decreases less rapidly near the points where the solution to the problem (1.2) vanishes. To the best of our knowledge, our results generalize the majority of previous works.
The paper is organized as follows. In Section 2, we first describe the considered inverse problem and we state the logarithmic stability estimate results. Then, we prove rigorously and separately some important results by considering some hypothesis introduced in Section 1. Finally, we give a stability result for the case of a time dependent heat flux.
2. Stability of the determination of Robin coefficient. In this section, we establish a double logarithmic stability estimate for the determination of a boundary coefficient appearing in a boundary value problem for the heat equation with Robin boundary condition. Firstly, we consider the case where the heat flux g is stationary (g := g(x)). Then, the time-dependent case (g := h(x, t)) can be deduced using the same techniques. Now, we introduce γ × (0, ∞) as a subset of the accessible sub-boundary Γ a × (0, ∞). We assume that γ does not meet supp(g) and the following condition holds true: The inverse problem associated to the problem (1.2) can be formulated as follows: Inverse problem. Determine q, supported on Γ i , from the boundary measurements where Γ i is assumed to be a priori known and u q is the solution to the problem (1.2) with coefficient q.
The uniqueness results of the determination of Γ i in (1.2)-(2.2) can be inspired from [7] and [6]. More results for the stationary case can be found in the literature, where different kinds of methods are used to establish uniqueness, stability and numerical algorithms ( see e.g, [1,2,9,12,13,15] and references therein).
2.1. Double logarithmic stability estimate. In this subsection, we establish a double logarithmic stability estimate for the inverse problem in thermal imaging described above. Here, we require that the heat flux on a part of the accessible boundary Γ a remains the same at every time. Moreover, we assume that the boundary function g introduced in the problem (1.2) only depends on the space variable x. A similar result can be deduced for the case of a time dependent heat flux. This latter case will be detailed later. Now, let us consider the following assumption: where k ≥ 0 is an integer and χ Γa is the characteristic function related to Γ a . Note that, g is not identically equal to zero. Thus, let us assume that we have (1.1), (2.2), (2.1) and (A 1 ). By applying stationary heat flux, we get the following result.
The proof of Theorem 2.1 is postponed to subsection 2.2. So, following [15] and [14], we modulate the problem of detecting corrosion damage by electric measurements. To this end, we consider the following boundary value problem Using Theorem 2.3 of [16] and the fact that B n−1/2,1 (Γ) is continuously embedded in B n−3/2,1 (Γ), we obtain that (for any q ∈ D) the problem (2.3) has a unique solution v q ∈ H n (Ω) (n = 2, 3). Moreover, we have where R 1 = R 1 (Ω, g, M ) denotes a positive constant. In addition, we introduce the auxiliary initial-boundary value problem (2.5) Next, let us denote by u q the solution to the problem (1.2) and let us decompose this solution into the sum where v q is the solution to (2.3) and w q is the solution to (2.5).

2.2.
Proof of the stability estimate. To prove Theorem 2.1, we need the following technical lemmas. The first lemma, was proved in [3] and is based on the classical theory of analytic semigroups ( see e.g. [30,31] where C depending on Ω, Γ, R 0 , and R 1 but independent of q. Note that by virtue of Lemma 4.2 of [4], there exist δ * > 0 and 0 < r * ≤ diam(∂Ω) such that for any 0 < δ < δ * andx ∈ ∂Ω, we have where v q is the solution to (2.3) and B(x, r * ) means the ball, in R n , with centerx and with radius r * .
To state the second technical lemma, we first consider an arbitrary function f ∈ C α (Γ) such that where M 0 > 0 is a given constant and [f ] α denotes the infimum over all constants L such that is well satisfied. Let us recall that f ∈ C α (Γ) if there exists L ≥ 0 such that (2.9) holds. We have the following result.

10)
where v q is the solution to (2.3).
Since |v q | ≤ δ in B(x, r) ∩ Γ, one can use Corollary 2.6 of [14] to justify the existence of a constant c 1 > 0 such that Applying the function ln to the previous inequality, we obtain where c is a positive constant depending on c 1 and Γ. Thus, (2.12) implies that , for all 0 < δ < δ * .
Now, we are ready to give the main result of the current subsection. Proof of the Theorem 2.1. Let γ ⊂ Γ a ⊂ Γ be defined as in (2.1). Let u q (resp. uq) be the solution to the problem (1.2) with the coefficient q (resp. with the coefficientq). Analogously, we can define the functions v q and vq . Using relation (2.6), Lemma 2.1 and the trace Theorem, we can find a constant λ > 0 such that where the positive constants C and µ are given in Lemma 2.1. Letting t tends to infinity, we get v q − vq L 2 (γ) ≤ u q − uq L ∞ ((0,∞);L 2 (γ)) . (2.14) Besides, let us defineṽ = v q − vq satisfying ∆ṽ = 0. From the interpolation inequality and (2.4), we get: where C 1 , C 2 , C 3 are positive constants. Hence, by the trace Theorem, we get where C 4 is a positive constant. Now, let γ 0 ⊂⊂ γ and P v = ∆ṽ = 0. From Theorem 2.2, we have As done previously, we use the interpolation inequality, to obtain . By the trace Theorem, relations (2.4) and (2.14), we obtain L ∞ ((0,∞);L 2 (γ)) + τ ṽ H 2 (Ω) . Using relation (2.4) once again, we infer that The function f is non increasing with f (0 + ) = +∞ and f (+∞) = 0, so that the above equation has a unique solution min .
-If 0 > min , and by choosing min = in (2.18) we have where min is sufficiently small satisfying for some c > c, ≤ e c / min .