Inverse Boundary Value Problem by Partial data for the Neumann-to-Dirichlet-map in two dimensions

For the two dimensional Schr"odinger equation in a bounded domain, we prove uniqueness of determination of potentials in $W_p^1(\mathrm{\Omega }),p>2$ in the case where we apply all possible Neumann data supported on an arbitrarily non-empty open set $\tilde{\mathrm{\Gamma }}$ of the boundary and observe the corresponding Dirichlet data on $\tilde{\mathrm{\Gamma }}$. An immediate consequence is that one can uniquely determine a conductivity in $W_p^3(\mathrm{\Omega })$ with $p>2$ by measuring the voltage on an open subset of the boundary corresponding to current supported in the same set.