A mann iterative regularization method for elliptic Cauchy problems

We investigate the Cauchy problem for linear 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 manifold Gamma subset of partial derivativeOmega and our goal is to reconstruct the trace of the H-1(Omega) solution of an elliptic equation at 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.