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

Published online by Cambridge University Press:  15 September 2002

B. Canuto ,
Edi Rosset  and
S. Vessella
B. Canuto
Affiliation:
Laboratoire de Mathématiques Appliquées, UMR 7641 du CNRS, Université de Versailles, 45 avenue des États-Unis, 78035 Versailles Cedex, France; [email protected].
Edi Rosset
Affiliation:
Dipartimento di Scienze Matematiche, Università degli Studi di Trieste, Via Valerio 12/1, 34100 Trieste, Italy; [email protected].
S. Vessella
Affiliation:
DIMAD, Università degli Studi di Firenze, Via C. Lombroso 6/17, 50134 Firenze, Italy; [email protected].
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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 ${\mathbb R}^n$, n ≥ 2, from a single pair of boundary measurements of temperature and thermal flux.

Keywords

Parabolic equationsstrong unique continuationstabilityinverse problems.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 7 , 2002 , pp. 521 - 565
Copyright
© EDP Sciences, SMAI, 2002

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References

V. Adolfsson and L. Escauriaza, C 1,α domains and unique continuation at the boundary. Comm. Pure Appl. Math. L (1997) 935-969.
Alessandrini, G. and Rosset, E., The inverse conductivity problem with one measurement: Bounds on the size of the unknown object. SIAM J. Appl. Math. 58 (1998) 1060-1071.
Alessandrini, G. and Rondi, L., Optimal stability for the inverse problem of multiple cavities. J. Differential Equations 176 (2001) 356-386. CrossRef
G. Alessandrini, E. Beretta, E. Rosset and S. Vessella, Optimal stability for inverse elliptic boundary value problems with unknown boundaries. Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) XXIX (2000) 755-806.
Bryan, K. and Candill Jr, L.F.., An inverse problem in thermal imaging. SIAM J. Appl. Math. 56 (1996) 715-735. CrossRef
K. Bryan and L.F. Candill Jr., Uniqueness for boundary identification problem in thermal imaging, in Differential Equations and Computational Simulations III, edited by J. Graef, R. Shivaji, B. Soni and J. Zhu.
Bryan, K. and Candill Jr, L.F.., Stability and reconstruction for an inverse problem for the heat equation. Inverse Problems 14 (1998) 1429-1453. CrossRef
Canuto, B., Rosset, E. and Vessella, S., Quantitative estimates of unique continuation for parabolic equations and inverse initial-boundary value problems with unknown boundaries. Trans. AMS 354 (2002) 491-535. CrossRef
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1. Wiley, New York (1953).
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer, New York (1983).
V. Isakov, Inverse problems for partial differential equations. Springer, New York (1998).
Ito, S. and Yamabe, H., A unique continuation theorem for solutions of a parabolic differential equation. J. Math. Soc. Japan 10 (1958) 314-321. CrossRef
O.A. Ladyzhenskaja, V.A. Solonnikov and N.N. Ural'ceva, Linear and quasilinear equations of parabolic type. Amer. Math. Soc., Providende, Math. Monographs 23 (1968).
Landis, E.M. and Oleinik, O.A., Generalized analyticity and some related properties of solutions of elliptic and parabolic equations. Russ. Math. Surveys 29 (1974) 195-212. CrossRef
F.H. Lin, A uniqueness theorem for parabolic equations. Comm. Pure Appl. Math. XLIII (1990) 127-136.
J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications II. Springer, New York (1972).
A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer, New York (1983).
Vessella, S., Stability estimates in an inverse problem for a three-dimensional heat equation. SIAM J. Math. Anal. 28 (1997) 1354-1370. CrossRef