Identification of small inhomogeneities of extreme conductivity by boundary measurements: A theorem on continuous dependence
The authors consider the inverse problem to determine perturbations of the coefficient \(\gamma\) (x) in the differential equation \(\nabla \cdot (\gamma \nabla u)=0\) in \(\Omega <{\mathbb{R}}^ n\), \(n\geq 2\), from measurements of u and \(\gamma\) (\(\partial u/\partial \nu)\) on the boundary \(\partial \Omega\). They treat the cases where the perturbed coefficients are either infinite or zero on a finite (but unknown) number of small (order \(\epsilon)\) subregions of \(\Omega\). In the first part they prove an asymptotic formula of the field \(u_{\epsilon}\) with respect to \(\epsilon\). In the second part this result is used to prove a Lipschitz-continuous dependence estimate of the function which maps - for fixed Neumann data - the Dirichlet data onto the parameters describing \(\gamma\).
scientific article; zbMATH DE number 4120698
scientific article