Stability for an inverse problem in potential theory
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- by Hamid Bellout, Avner Friedman and Victor Isakov PDF
- Trans. Amer. Math. Soc. 332 (1992), 271-296 Request permission
Abstract:
Let $D$ be a subdomain of a bounded domain $\Omega$ in ${\mathbb {R}^n}$ . The conductivity coefficient of $D$ is a positive constant $k \ne 1$ and the conductivity of $\Omega \backslash D$ is equal to $1$. For a given current density $g$ on $\partial \Omega$ , we compute the resulting potential $u$ and denote by $f$ the value of $u$ on $\partial \Omega$. The general inverse problem is to estimate the location of $D$ from the known measurements of the voltage $f$. If ${D_h}$ is a family of domains for which the Hausdorff distance $d(D,{D_h})$ equal to $O(h)$ ($h$ small), then the corresponding measurements ${f_h}$ are $O(h)$ close to $f$. This paper is concerned with proving the inverse, that is, $d(D,{D_h}) \leq \frac {1}{c}\left \| {f_h} - f\right \|$ , $c > 0$ ; the domains $D$ and ${D_h}$ are assumed to be piecewise smooth. If $n \geq 3$ , we assume in proving the above result, that ${D_h} \supset D$ (or ${D_h} \subset D$) for all small $h$ . For $n = 2$ this monotonicity condition is dropped, provided $g$ is appropriately chosen. The above stability estimate provides quantitative information on the location of ${D_h}$ by means of ${f_h}$ .References
- Hamid Bellout and Avner Friedman, Identification problems in potential theory, Arch. Rational Mech. Anal. 101 (1988), no. 2, 143–160. MR 921936, DOI 10.1007/BF00251458
- L. Escauriaza, E. B. Fabes, and G. Verchota, On a regularity theorem for weak solutions to transmission problems with internal Lipschitz boundaries, Proc. Amer. Math. Soc. 115 (1992), no. 4, 1069–1076. MR 1092919, DOI 10.1090/S0002-9939-1992-1092919-1
- Gregory Eskin, Index formulas for elliptic boundary value problems in plane domains with corners, Trans. Amer. Math. Soc. 314 (1989), no. 1, 283–348. MR 961621, DOI 10.1090/S0002-9947-1989-0961621-0
- Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
- Avner Friedman and Björn Gustafsson, Identification of the conductivity coefficient in an elliptic equation, SIAM J. Math. Anal. 18 (1987), no. 3, 777–787. MR 883568, DOI 10.1137/0518059
- 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, DOI 10.1512/iumj.1989.38.38027
- P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
- 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 G. Stampacchia, Équations elliptiques du second ordre à coefficients discontinues, Lecture Notes, Univ. de Montreal, 1965.
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 332 (1992), 271-296
- MSC: Primary 31B20; Secondary 31B35, 35J25, 35R30
- DOI: https://doi.org/10.1090/S0002-9947-1992-1069743-3
- MathSciNet review: 1069743