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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Stability for an inverse problem in potential theory


Authors: Hamid Bellout, Avner Friedman and Victor Isakov
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
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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\Vert {f_h} - f\right\Vert $ , $ 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}$ .


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DOI: https://doi.org/10.1090/S0002-9947-1992-1069743-3
Article copyright: © Copyright 1992 American Mathematical Society

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