A Denseness Theorem with an Application to a Two-Dimensional Inverse Potential Refraction Problem

Abstract

In this paper we will be concerned with the problem of reconstructing a region $D\subset\Omega\subset\R^2$, from the knowledge of the boundary potential $u|_{\partial\Omega}$, where u satisfies {\rm div}\,((\chi_{\clos{\Omega}\setminus D} + a\chi_D)\nabla u)=0\quad \text{in $\Omega$}\qquad\text{and}\qquad \frac{\partial u}{\partial\nu}= I\quad\text{on $\partial\Omega$},with a a real, positive constant. We will show that the domain derivative of the corresponding forward mapping is injective. This is done by proving the denseness of a certain subspace of $L^1(\partial D)$. The novelty of the result is that our proof is valid without any restriction on the Neumann boundary data I.