Determination of second-order elliptic operators in two dimensions from partial Cauchy data.

We consider the inverse boundary value problem in two dimensions of determining the coefficients of a general second-order elliptic operator from the Cauchy data measured on a nonempty arbitrary relatively open subset of the boundary. We give a complete characterization of the set of coefficients yielding the same partial Cauchy data. As a corollary we prove several uniqueness results in determining coefficients from partial Cauchy data for the isotropic conductivity equation, the Schrödinger equation, the convection-diffusion equation, the anisotropic conductivity equation modulo a group of diffeomorphisms that are the identity at the boundary, and the magnetic Schrödinger equations modulo gauge transformations. The key step is the construction of novel complex geometrical optics solutions using Carleman estimates.