Authors’ abstract: “Let \(D\) be a subdomain of a bounded domain \(\Omega\) in \(\mathbb{R}^ n\). The conductivity coefficient of \(D\) is a positive constant \(k \neq 1\) 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 \(d(D,D_ h)\) is 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 \[ d(D,D_ h) \leq{1 \over c} \| f_ h-f \|,\;c>0; \] 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 \(f_ h\)”.