A MANN ITERATIVE REGULARIZATION METHOD FOR ELLIPTIC CAUCHY PROBLEMS

We investigate the Cauchy problem for linear elliptic operators with C ^∞–coefficients at a regular set Ω ? R ², which is a classical example of an ill-posed problem. The Cauchy data are given at the manifold Γ ? ?Ω and our goal is to reconstruct the trace of the H ¹(Ω) solution of an elliptic equation at ?Ω/Γ. 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.