The Neumann-to-Dirichlet map in two dimensions

For the two-dimensional Schrödinger equation in a bounded domain, we prove uniqueness of the determination of potentials in Wp1(Ω), p>2p>2 in the case where we apply all possible Neumann data supported on an arbitrarily non-empty open set Γ˜ of the boundary and observe the corresponding Dirichlet data on Γ˜. An immediate consequence is that one can uniquely determine a conductivity in Wp3(Ω) with p>2p>2 by measuring the voltage on an open subset of the boundary corresponding to a current supported in the same set.