A Mann iterative regularization method for elliptic Cauchy problems

Abstract

We investigate the Cauchy problem for linear 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 manifold $\Gamma \subset \partial\Omega$ and our goal is to reconstruct the trace of the $H^1(\Omega)$ solution of an elliptic equation at $\partial \Omega / \Gamma$. The method proposed here composes the segmenting Mann iteration with a fixed point equation associated with the elliptic Cauchy problem. Our algorithm generalizes the iterative method developed by Maz'ya et al., who proposed a method based on solving successive well-posed mixed boundary value problems. We analyze the regularizing and convergence properties both theoretically and numerically.