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), 3031-3041

MSC:
Primary 35R30; Secondary 31A25, 86A22

DOI:
https://doi.org/10.1090/S0002-9947-1995-1303113-8

MathSciNet review:
1303113

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 .

**[AM]**G. Alessandrini and R. Magnanini,*Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions*, SIAM J. Math. Anal.**25**(1994), no. 5, 1259–1268. MR**1289138**, https://doi.org/10.1137/S0036141093249080**[BFI]**Hamid Bellout, Avner Friedman, and Victor Isakov,*Stability for an inverse problem in potential theory*, Trans. Amer. Math. Soc.**332**(1992), no. 1, 271–296. MR**1069743**, https://doi.org/10.1090/S0002-9947-1992-1069743-3**[Be]**Lipman Bers,*Local behavior of solutions of general linear elliptic equations*, Comm. Pure Appl. Math.**8**(1955), 473–496. MR**0075416**, https://doi.org/10.1002/cpa.3160080404**[C]**V. G. Cherednichenko,*A problem on the conjugation of harmonic functions and its inverse*, Differentsial′nye Uravneniya**18**(1982), no. 4, 682–689, 734 (Russian). MR**658435****[CV]**V. G. Cherednichenko and G. V. Veryovkina,*Inverse conductivity problem in the two-dimensional case*, Ill-posed problems in natural sciences (Moscow, 1991) VSP, Utrecht, 1992, pp. 270–276. MR**1219987****[DEF]**Emmanuele DiBenedetto, Charles M. Elliott, and Avner Friedman,*The free boundary of a flow in a porous body heated from its boundary*, Nonlinear Anal.**10**(1986), no. 9, 879–900. MR**856872**, https://doi.org/10.1016/0362-546X(86)90076-3**[E]**A. Erdelyi et al,*Higher transcendental functions*, vol. II, McGraw-Hill, New York, 1953.**[FI]**Avner Friedman and Victor Isakov,*On the uniqueness in the inverse conductivity problem with one measurement*, Indiana Univ. Math. J.**38**(1989), no. 3, 563–579. MR**1017325**, https://doi.org/10.1512/iumj.1989.38.38027**[I]**Victor Isakov,*Inverse source problems*, Mathematical Surveys and Monographs, vol. 34, American Mathematical Society, Providence, RI, 1990. MR**1071181****[IP]**Victor Isakov and Jeffrey Powell,*On the inverse conductivity problem with one measurement*, Inverse Problems**6**(1990), no. 2, 311–318. MR**1046169****[M]**L. G. Michailov,*A new class of singular integral equations and its application to differential equations with singular coefficients*, Translated from the Russian by M. D. Friedman, Wolters-Noordhoff Publishing, Groningen, 1970. MR**0264216****[Mu]**N. I. Muskhelishvili,*Singular integral equations*, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR**0355494****[P]**Jeffrey Powell,*On a small perturbation in the two-dimensional inverse conductivity problem*, J. Math. Anal. Appl.**175**(1993), no. 1, 292–304. MR**1216762**, https://doi.org/10.1006/jmaa.1993.1169**[V]**I. N. Vekua,*Generalized analytic functions*, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962. MR**0150320**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35R30,
31A25,
86A22

Retrieve articles in all journals with MSC: 35R30, 31A25, 86A22

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1303113-8

Article copyright:
© Copyright 1995
American Mathematical Society