A Mann iterative regularization method for elliptic Cauchy problems

We investigate the Cauchy problem for linear elliptic operators with $C^{\mathrm{\infty }}$-coefficients at a regular set $\mathrm{\Omega }\subset R^2$, which is a classical example of an ill-posed problem. The Cauchy data are given at the manifold $\mathrm{\Gamma }\subset \mathrm{\partial }\mathrm{\Omega }$ and our goal is to reconstruct the trace of the $H^1(\mathrm{\Omega })$ 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.