  We show in two dimensions that measuring Dirichlet data for the conductivity
equation on an open subset of the boundary and, roughly speaking, Neumann data
in slightly larger set than the complement uniquely determines the conductivity
on a simply connected domain. The proof is reduced to show a similar result for
the Schr\"odinger equation. Using Carleman estimates with degenerate weights we
construct appropriate complex geometrical optics solutions to prove the
results.
