Inverse boundary value problem by measuring Dirichlet data and Neumann data on disjoint sets

We discuss the inverse boundary value problem of determining the conductivity in two dimensions from the pair of all input Dirichlet data supported on an open subset + and all the corresponding Neumann data measured on an open subset -.We prove the global uniqueness under some additional geometric condition, in the case where + ∩ -= ∅, and we prove also the uniqueness for a similar inverse problem for the stationary Schrödinger equation.The key of the proof is the construction of appropriate complex geometrical optics solutions using Carleman estimates with a singular weight.