SIAM J. MATH. ANAL. Vol. 27, No. 2, pp. 361-375, March 1996 () 1996 Society for Industrial and Applied Mathematics OO3 DETERMINING LINEAR CRACKS BY BOUNDARY MEASUREMENTS: LIPSCHITZ STABILITY* GIOVANNI ALESSANDRINIt, ELENA BERETTA$, AND SERGIO VESSELLA Abstract. We consider the inverse boundary value problem of crack detection in a two- dimensional electrical conductor. We prove an estimate of Lipschitz type on the continuous de- pendence of an unknown linear crack from the boundary measurements. Key words, inverse boundary value problem, stability, crack AMS subject classifications. 35R30, 31A25 1. Introduction. The inverse problem of. detecting a crack in an electrically conducting body can be modeled as the determination of a curve a in a planar domain t from boundary measurements of solutions u (potentials) to the problem (1.1a) An--0 (1.1b) u const. CU (1.1c) O = in f\a, on 0", on 0, when various profiles (currents) are assigned. Friedman and Vogelius have proved that a crack a is uniquely determined when boundary measurements corresponding to two appropriate profiles are known; see IF-V]. They also observe, by a duality argument, that a similar result holds when (1.1b) is replaced with (1.1b’) 0 0 on a. When (1.1b) holds, the crack a is said to be perfectly conducting, whereas in (1.1b’), it is said to be perfectly insulating. They also address the stability issue, discuss the relevance for the actual reconstruction of the crack, and give some initial results in this direction. A stability estimate for perfectly conducting cracks has been obtained in [A1] and generalized by Diaz Valenzuela to the perfectly insulating case [DV]. Unfortu- nately, such estimates are of logarithmic type; see [A2] for a partial improvement. In [A1], [DV], and [A2], bounds on the smoothness and the size of the unknown crack are assumed, but they involve only a finite number of derivatives of the crack parametriza- tion. One can expect that better stability estimates might be obtained when stronger Received by the editors February 4, 1994; accepted for publication (in revised form) August 2, 1994. This research was partially supported by the Fondi Ministero dell’ University4 della Ricerca Scientifica e Tecnologica and the Consiglio Nazionale della Richerche. Dipartimento di Scienze Matematiche, Universit di Trieste, 34100, Trieste, Italy. The research of this author was partially supported by the Ministero Ricerca Scientifica e Tecnologica. Istituto per le Applicazioni del Calcolo "Mauro Picone," Viale del Policlinico 137, 00161 Rome, Italy. The research of this author was supported by the Consiglio Nazionale delle Ricerche. Dipartimento di Matematica per le Decisioni Economiche, Finanziarie, Attuariali, e Sociali, Universit di Firenze, 6/17, 50134 Firenze, Italy. 361 Downloaded 11/02/12 to 150.217.1.25. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php