An iterative method for solving elliptic Cauchy problems

We investigate the Cauchy problem for elliptic operators with C-infinity- coefficients at a regular set Omega subset of R-2, which is a classical example of an ill-posed problem. The Cauchy data are given at the subset Gamma subset of partial derivative Omega and our objective is to reconstruct the trace of the H-1(Omega) solution of an elliptic equation at partial derivative Omega/Gamma. The method described here is a generalization of the algorithm developed by Maz'ya et al. [Ma] for the Laplace operator, who proposed a method based on solving successive well-posed mixed boundary value problems (BVP) using the given Cauchy data as part of the boundary data. We give an alternative convergence proof for the algorithm in the case we have a linear elliptic operator with C-infinity- coefficients. We also present some numerical experiments for a special non linear problem and the obtained results are very promisive.