Optimal Stability for Inverse Elliptic Boundary Value Problems with Unknown Boundaries

In this paper we study a class of inverse problems associated to elliptic boundary value problems. More precisely, those inverse problems in which the role of the unknown is played by an inaccessible part of the boundary and the role of the data is played by overdetermined boundary data for the elliptic equation assigned on the remaining, accessible, part of the boundary. We treat the case of arbitrary space dimension n ≥ 2. Such problems arise in applied contexts of nondestructive testing of materials for either electric or thermal conductors, and are known to be ill-posed. In this paper we obtain essentially best possible stability estimates. Here, in the context of ill-posed problems, stability means the continuous dependence of the unknown upon the data when additional a priori information on the unknown boundary (such as its regularity) is available.