Partial Cauchy Data for General Second Order Elliptic Operators in Two Dimensions

We consider the problem of determining the coefficients of a first-order perturbation of the Laplacian in two dimensions by measuring the corresponding Cauchy data on an arbitrary open subset of the boundary. From this information we obtain a coupled system of ∂z̄ and ∂z which the coefficients satisfy. As a corollary we show that for a simply connected domain we can determine uniquely the coefficients up to the natural obstruction. Another consequence of our result is that the magnetic field and the electric potential are uniquely determined by measuring the partial Cauchy data associated to the magnetic Schrödinger equation measured on an arbitrary open subset of the boundary. of the boundary. We also show that the coefficients of any real vector field perturbation of the Laplacian, the convection terms, are uniquely determined by their partial Cauchy data.