# A stability result in the localization of cavities in a thermic conducting medium

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cocvVol7-22.dvi ESAIM: Control, Optimisation and Calculus of Variations August 2002, Vol. 7, 521{565 URL: http://www.emath.fr/cocv/ DOI: 10.1051/cocv:2002066 A STABILITY RESULT IN THE LOCALIZATION OF CAVITIES IN A THERMIC CONDUCTING MEDIUM B. Canuto1, Edi Rosset2 and S. Vessella3 Abstract. We prove a logarithmic stability estimate for a parabolic inverse problem concerning the localization of unknown cavities in a thermic conducting medium Ω in Rn , n � 2, from a single pair of boundary measurements of temperature and thermal flux. Mathematics Subject Classi�cation. 35R30, 35R25, 35R35. Received September 27, 2001. Revised May 28, 2002. 1. Introduction and the main result In the present paper we are concerned with the study of a problem in thermal imaging. This is a technique used to determine some physical and geometrical proprieties of a thermic conducting medium via boundary measurements of temperature and thermal flux. More precisely we denote by Ω a thermic conducting medium, i.e. a su�ciently smooth, bounded domain in Rn, n � 2, and by D a cavity in Ω (i.e. D is a domain compactly contained in Ω), of which neither the form nor the position is known. On the other hand we can measure the temperature f and the thermal flux g on the boundary of the medium @Ω. The goal is then to identify the cavity D via the boundary data f , g. This problem can occur in nondestructive tests of materials, for example in detecting the corrosion parts of an aircraft which are inaccessible to direct inspections (see Bryan and Caudill [5{7], and their references). We denote by u(t; x) the temperature at the time t and at the point x 2 ΩnD, u0 the initial temperature in ΩnD, f the temperature on (0; T )� @Ω, and k(x) the anisotropic thermal di�usion coe�cient, that is k is an n� n symmetric matrix-valued function in Ω satisfying the following conditions: (i) there exists a constant � � 1, such that for all x 2 Ω, and for all � 2 Rn, �−1j�j2 � k(x)� � � � �j�j2 (ellipticity); (1.1) Keywords and phra

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