Domain Identification for Semilinear Elliptic Equations in the Plane: the Zero Flux Case

Abstract

We consider the problem of identifying the domain $\Omega \subset {\mathbb{R}}^n$ of a semilinear elliptic equation subject to given Cauchy data on part of the known outer boundary $\Gamma$ and to the zero flux condition on the unknown inner boundary $\gamma$, where it is assumed that $\Gamma$ is a piecewise $C^1$ curve and that $\gamma$ is the boundary of a finite disjoint union of simply connected domains, each bounded by a piecewise $C^1$ Jordan curve. It is shown that, under appropriate smoothness conditions, the domain $\Omega$ is uniquely determined. The problem of existence of solution for given data is not considered since it is usually of lesser importance in view of measurement errors giving data for which no solution exists. Keywords: Domain identification, semilinear elliptic