Stability for an inverse problem in potential theory

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 ${f_h}$ .