The boundary inverse problem for the Laplace equation in two dimensions

We have studied the problem of determining part of the boundary of a domain Omega where a potential satisfies the Laplace equation. The potential and its normal derivative have prescribed values on the known part of the boundary that encloses Omega while its normal derivative must vanish on the remaining part. We establish a sufficient condition for the potential to be monotonic along the unknown boundary. This allows us to use the potential to parametrize the boundary. Two methods are presented that solve the problem under this assumption. The first one solves the problem in a closed form and it can be used to define a parameter that will describe the ill-posedness of the problem. The effect of this parameter on the second method presented has been determined for a particular numerical example.