Mean value iterations for nonlinear elliptic Cauchy problems

We investigate the Cauchy problem for a class of nonlinear elliptic operators with $C^{\mathrm{\infty }}$-coefficients at a regular set $\mathrm{\Omega }\subset R^n$. The Cauchy data are given at a manifold $\mathrm{\Gamma }\subset \mathrm{\partial }\mathrm{\Omega }$ and our goal is to reconstruct the trace of the $H^1(\mathrm{\Omega })$ solution of a nonlinear elliptic equation at $\mathrm{\partial }\mathrm{\Omega }\mathrm{/}\mathrm{\Gamma }$. We propose two iterative methods based on the segmenting Mann iteration applied to fixed point equations, which are closely related to the original problem. The first approach consists in obtaining a corresponding linear Cauchy problem and analyzing a linear fixed point equation; a convergence proof is given and convergence rates are obtained. On the second approach a nonlinear fixed point equation is considered and a fully nonlinear iterative method is investigated; some preliminary convergence results are proven and a numerical analysis is provided.