An iterative method for solving elliptic Cauchy problems

Abstract: We investigate the Cauchy problem for elliptic operators with $C^\infty$-coefficients at a regular set $\Omega \subset R^2$, which is a classical example of an ill-posed problem. The Cauchy data are given at the subset $\Gamma \subset \partial\Omega$ and our objective is to reconstruct the trace of the $H^1(\Omega)$ solution of an elliptic equation at $\partial \Omega / \Gamma$. 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 $C^\infty$-coefficients. We also present some numerical experiments for a special non linear problem and the obtained results are very promisive.