Partial Data for the Neumann-to-Dirichlet Map

Abstract

We show that measurements of the Neumann-to-Dirichlet map, roughly speaking, on a certain part of the boundary of a smooth domain in dimension 3 or higher, for inputs with support restricted to the other part, determine an electric potential on that domain. Given a convexity condition on the domain, either the set on which measurements are taken, or the set on which input functions are supported, can be made to be arbitrarily small. The result is analogous to the result by Kenig, Sj\"{o}strand, and Uhlmann for the Dirichlet-to-Neumann map. The main new ingredient in the proof is a Carleman estimate for the Schr\"{o}dinger operator with appropriate boundary conditions.