A stability result in the localization of cavities in a thermic conducting medium
ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 521-565.

We prove a logarithmic stability estimate for a parabolic inverse problem concerning the localization of unknown cavities in a thermic conducting medium $\Omega$ in ${ℝ}^{n}$, $n\ge 2$, from a single pair of boundary measurements of temperature and thermal flux.

DOI : https://doi.org/10.1051/cocv:2002066
Classification : 35R30,  35R25,  35R35
Mots clés : parabolic equations, strong unique continuation, stability, inverse problems
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author = {Canuto, B. and Rosset, Edi and Vessella, S.},
title = {A stability result in the localization of cavities in a thermic conducting medium},
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Canuto, B.; Rosset, Edi; Vessella, S. A stability result in the localization of cavities in a thermic conducting medium. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 521-565. doi : 10.1051/cocv:2002066. http://www.numdam.org/articles/10.1051/cocv:2002066/

[1] V. Adolfsson and L. Escauriaza, ${C}^{1,\alpha }$ domains and unique continuation at the boundary. Comm. Pure Appl. Math. L (1997) 935-969. | MR 1466583 | Zbl 0899.31004

[2] G. Alessandrini and E. Rosset, The inverse conductivity problem with one measurement: Bounds on the size of the unknown object. SIAM J. Appl. Math. 58 (1998) 1060-1071. | MR 1620386 | Zbl 0953.35141

[3] G. Alessandrini and L. Rondi, Optimal stability for the inverse problem of multiple cavities. J. Differential Equations 176 (2001) 356-386. | MR 1866280 | Zbl 0988.35163

[4] 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. | Numdam | MR 1822407 | Zbl 1034.35148

[5] K. Bryan and L.F. Candill Jr., An inverse problem in thermal imaging. SIAM J. Appl. Math. 56 (1996) 715-735. | MR 1389750 | Zbl 0854.35125

[6] 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.

[7] K. Bryan and L.F. Candill Jr., Stability and reconstruction for an inverse problem for the heat equation. Inverse Problems 14 (1998) 1429-1453. | MR 1662444 | Zbl 0914.35152

[8] B. Canuto, E. Rosset and S. Vessella, Quantitative estimates of unique continuation for parabolic equations and inverse initial-boundary value problems with unknown boundaries. Trans. AMS 354 (2002) 491-535. | MR 1862557 | Zbl 0992.35112

[9] R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1. Wiley, New York (1953). | Zbl 0051.28802

[10] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer, New York (1983). | MR 737190 | Zbl 0562.35001

[11] V. Isakov, Inverse problems for partial differential equations. Springer, New York (1998). | MR 1482521 | Zbl 0908.35134

[12] S. Ito and H. Yamabe, A unique continuation theorem for solutions of a parabolic differential equation. J. Math. Soc. Japan 10 (1958) 314-321. | MR 99519 | Zbl 0088.30403

[13] 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). | Zbl 0174.15403

[14] E.M. Landis and O.A. Oleinik, Generalized analyticity and some related properties of solutions of elliptic and parabolic equations. Russ. Math. Surveys 29 (1974) 195-212. | MR 402268 | Zbl 0305.35014

[15] F.H. Lin, A uniqueness theorem for parabolic equations. Comm. Pure Appl. Math. XLIII (1990) 127-136. | MR 1024191 | Zbl 0727.35063

[16] J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications II. Springer, New York (1972). | Zbl 0227.35001

[17] A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer, New York (1983). | MR 710486 | Zbl 0516.47023

[18] S. Vessella, Stability estimates in an inverse problem for a three-dimensional heat equation. SIAM J. Math. Anal. 28 (1997) 1354-1370. | MR 1474218 | Zbl 0888.35130

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