  For the two dimensional Schr\"odinger 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.
