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Stable determination of a crack from boundary measurements

Published online by Cambridge University Press:  14 November 2011

Giovanni Alessandrini
Affiliation:
Dipartimento di Scienze Matematiche, Università degli Studi di Trieste, Italy

Synopsis

We treat the problem of determining a crack inside a conductor when two pairs of current and voltage boundary measurements are given. We prove a theorem of continuous dependence from the data.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1993

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References

1Alessandrini, G.. Determining conductivity by boundary measurements: the stability issue. In Applied and Industrial Mathematics, ed. Spigler, R. (Dordrecht: Kluwer, 1991).Google Scholar
2Beretta, E. and Vessella, S.. Stability results for an inverse problem in potential theory. Ann. Mat. Pura Appl. (4) 156 (1990), 381404.CrossRefGoogle Scholar
3Bryan, K. and Vogelius, M.. A uniqueness result concerning the identification of a collection of cracks from finitely many elecrostatic boundary measurements (preprint).Google Scholar
4Duren, P. L. and Schiffer, M. M.. Robin functions and energy functinals of multiply connected domains. Pacific J. Math. (2) 148 (1991), 251273.CrossRefGoogle Scholar
5Friedman, A. and Vogelius, M.. Determining cracks by boundary measurements. Indiana Univ. Math. J. 38 (1989), 527556.CrossRefGoogle Scholar
6Hadamard, J.. Lectures on Cauchy's Problem (New York; Dover, 1952).Google Scholar
7Isakov, V.. Inverse Source Problems (Providence R.I.; American Mathematical Society, 1990).CrossRefGoogle Scholar
8Lavrent, M. M.'ev, Romanov, V. G. and Sisatskii, S. P.. Problemi Non Ben Posti in Fisica Matematica e Analisi (Firenze: Pubblicazioni dell'Istituto di Analisi Globale e Applicazioni Serie “Problemi non ben posti ed inversi”, 1983; Italian translation).Google Scholar
9Nachman, A.. Reconstruction from boundary measurements. Ann. of Math. 128 (1988), 531576.CrossRefGoogle Scholar
10Nehari, Z.. Conformal Mapping (New York: McGraw Hill, 1952).Google Scholar
11Sylvester, J. and Uhlmann, G.. A uniqueness theorem for an inverse boundary value problem. Ann. of Math. 125 (1987), 153169.CrossRefGoogle Scholar
12Santosa, F. and Vogelius, M.. A computational algorithm to determine cracks from electrostatic boundary measurements. Internal. J. Engrg. Sci. 29 (1991), 917937.CrossRefGoogle Scholar
13Walsh, J. L.. The Location of Critical Points of Analytic and Harmonic Functions (New York: American Mathemaical Society, 1950).CrossRefGoogle Scholar