Local uniqueness in the inverse conductivity problem with one measurement
Authors:
G. Alessandrini, V. Isakov and J. Powell
Journal:
Trans. Amer. Math. Soc. 347 (1995), 30313041
MSC:
Primary 35R30; Secondary 31A25, 86A22
MathSciNet review:
1303113
Fulltext PDF Free Access
Abstract 
References 
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Abstract: We prove local uniqueness of a domain entering the conductivity equation in a bounded planar domain given the Cauchy data for on a part of . The main assumption is that has zero index on which is easy to guarantee by choosing special boundary data for . To achieve our goals we study index of critical points of on .
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 [BFI]
 H. Bellout, A. Friedman, and V. Isakov, Stability for an inverse problem in potential theory, Trans. Amer. Math. Soc. 332 (1992), 271296. MR 1069743 (92j:31010)
 [Be]
 L. Bers, Local behavior of solutions of general linear elliptic equations, Comm. Pure Appl. Math. 8 (1955), 473496. MR 0075416 (17:743a)
 [C]
 V. G. Cherednichenko, A problem in the conjugation of harmonic functions and its inverse, Differential Equations 18 (1982), 503509. MR 658435 (83h:31002)
 [CV]
 V. G. Cherednichenko and G. V. Veryovkina, Inverse conductivity problem in twodimensional case, IllPosed Problems in Natural Sciences (A. N. Tikhonov, ed.), VSP, Utrecht, 1992, pp. 270276. MR 1219987 (94b:35294)
 [DEF]
 E. DiBenedetto, C. M. Elliot, and A. Friedman, The free boundary of a flow in a porous body heated from its boundary, Nonlinear Anal. 10 (1986), 879900. MR 856872 (87j:76083)
 [E]
 A. Erdelyi et al, Higher transcendental functions, vol. II, McGrawHill, New York, 1953.
 [FI]
 A. Friedman and V. Isakov, On the uniqueness in the inverse conductivity problem with one measurement, Indiana Univ. Math. J. 38 (1989), 563579. MR 1017325 (91a:35164)
 [I]
 V. Isakov, Inverse source problems, Math. Surveys Monographs, vol. 34, Amer. Math. Soc., Providence, RI, 1990. MR 1071181 (92g:35230)
 [IP]
 V. Isakov and J. Powell, On the inverse conductivity problem with one measurement, Inverse Problems 6 (1990), 31318. MR 1046169 (91e:35212a)
 [M]
 L. G. Mikhailov, A new class of singular integral equations and its applications to differential equations with singular coefficients, WoltersNoordhoff, Groningen, 1970. MR 0264216 (41:8812)
 [Mu]
 N. I. Mushelishvili, Singular integral equations, Noordhoff, Groningen, 1953. MR 0355494 (50:7968)
 [P]
 J. Powell, On a small perturbation in the twodimensional inverse conductivity problem, J. Math. Anal. Appl. 175 (1993), 292304. MR 1216762 (94c:35169)
 [V]
 I. N. Vekua, Generalized analytic functions, Pergamon Press, 1962. MR 0150320 (27:321)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199513031138
PII:
S 00029947(1995)13031138
Article copyright:
© Copyright 1995
American Mathematical Society
