Lipschitz stability for an elliptic inverse problem with a single measurement

We consider the problem of determining the unknown boundary values of a solution of an elliptic equation outside a bounded domain B from the knowledge of the values of this solution on a boundary of an arbitrary bounded domain surrounding B. We obtain for this inverse problem Lipschitz stability under an additional hypothesis on the unknown boundary function. This result can be also interpreted as quantitative uniqueness of continuation from the Cauchy data on the boundary of the domain surrounding B. Our analysis also applies to an interior problem.