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.

翻译：我们用固定的 $\\\\\\ subset R ⁇ 2$ 来调查使用 $C\\\ infty$- covaltics 的椭圆形操作者的问题。 这里描述的方法是概括Maz'ya et al. [Ma] 为Laplace 操作者开发的算法,该操作者提出了一种方法,其依据是用给定的 $Gamma \ subset\ repart\\ omega$ 来解决相继存在的混合边界价值问题。 我们的目标是重建 $H1 (\\\\ omega) 的 $H1 (\ \ \ \ \ \ \ \ \ \ \ \ omega) 解决方案的痕量。 我们用 $C\ \ inty \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \