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

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