A parabolic inverse problem with mixed boundary data. Stability estimates for the unknown boundary and impedance

V. Bacchelli, M. Di Cristo, E. Sincich, S. Vessella
2014 Transactions of the American Mathematical Society  
We consider the problem of determining an unaccessible part of the boundary of a conductor by means of thermal measurements. We study a problem of corrosion where a Robin type condition is prescribed on the damaged part and we prove logarithmic stability estimate. We remark that the mathematical model we consider here is a simplified version of the more general one in which the Robin condition is assumed on the whole boundary ∂Ω, [5, 6, 13] . In fact, our mixed boundary conditions correspond to
more » ... the case in which the Biot number (a simple index of the ratio of heat transfer resistance inside of and at the surface of the body; see [25] Sect. 4.3.2, page 250) is very small (see also [5] ). Nevertheless, it is rather simple to adapt our mathematical treatment to such a general situation (see Remark 4.6). At the same time the mixed condition permits us to simplify the exposition of the proof of our main result of the paper. Under the assumption that γ ≡ 0 (i.e. the Neumann condition on an inaccessible portion I) the analysis of stability has been carried out both in the framework of elliptic and parabolic equations. In [2, 4] , in which the problem is studied in the elliptic case, the authors show that, keeping as minimal as possible the a priori assumptions, the unknown boundary depends continuously on the known boundary measurements with a rate of continuity of logarithmic type, which is the best possible as shown in [9] . In the framework of parabolic equations, the stability issue has been studied in [8, 10, 23, 24] . Also, in such papers logarithmic stability estimates have been obtained. A refined analysis of the problem has been proposed in [10] . In such a paper the authors give a detailed study about the optimality of the logarithmic rate of continuity in the parabolic case. Under the assumption that γ ≡ 0 the problem of determining the portion I of the boundary and γ has been considered in [3] , where a uniqueness result in the stationary case is proved, provided two suitable measurements are performed. In view of the example given in [7], the number of measurements turns out to be optimal. The corresponding stability issue has been addressed in [22] , where logarithmic stability estimates have been achieved. This result is optimal as well (see [9] ). Let us finally mention [21] , where a uniqueness result under weaker regularity assumptions on the boundary has been obtained. In the parabolic case the problem is considered in [14] . In such a paper a uniqueness result, both for the Robin coefficient and the unknown portion I of the boundary, is proved under the additional assumptions that γ and g do not depend on t (more precisely, it is required that g is time independent on an interval (T 1 , T ), T 1 < T ). The main novelty of the present paper is to consider the case in which the Robin coefficient on the unknown boundary depends on x and t. Such an assumption can be considered as a first step in studying the case in which γ also depends on the temperature u. In this paper we show that I depends on thermal boundary measurements on the accessible portion of the boundary with a rate of continuity of logarithmic type. Although the precise results are written in Theorem 2.5 below, here we would like to anticipate something about them and briefly discuss the approach we follow to prove them. As in the elliptic case, we perform two boundary measurements. More precisely we prescribe two different heat fluxes, g andg, and we read the corresponding temperatures, u andũ, on a portion Σ of the accessible part of the boundary. For i = 1, 2, let u i be the solution to (1.1) when Ω = Ω i , γ = γ i , g = g and letũ i be the solution to (1.1) when Ω = Ω i , γ = γ i , g =g. We assume that g andg are linearly independent and one of them with sign. We proved that if for a given ε > 0 we have (1.2) u 1 − u 2 L 2 (Σ×(0,T )) ≤ ε, ũ 1 −ũ 2 L 2 (Σ×(0,T )) ≤ ε,
doi:10.1090/s0002-9947-2014-05807-8 fatcat:qrcb2yorqve6zatsaprtssueue