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

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For the two dimensional Schrödinger equation in a bounded domain, we prove uniqueness of determination of potentials in W¹ₚ(Ω), p>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 W³ₚ(Ω) with p>2 by measuring the voltage on an open subset of the boundary corresponding to current supported in the same set.