  We prove for a two dimensional bounded domain that the Cauchy data for the
Schroedinger equation measured on an arbitrary open subset of the boundary
determines uniquely the potential. This implies, for the conductivity equation,
that if we measure the current fluxes at the boundary on an arbitrary open
subset of the boundary produced by voltage potentials supported in the same
subset, we can determine uniquely the conductivity. We use Carleman estimates
with degenerate weight functions to construct appropriate complex geometrical
optics solutions to prove the results.
