On the existence of solution for an inverse problem

Abstract

We consider a boundary detection problem. We present physical motivations. We formulate the problem as a shape optimization problem by introducing the Neumann condition of the accessible part in a cost functional to be minimized, which complicates the study of continuity state that requires more regularity of the free boundary. We show the existence of the optimal solution of the problem by the J. Haslinger and P. Neittaanm¨aki principle.