ITERATIVE METHODS FOR k-HESSIAN EQUATIONS

On a domain of the n-dimensional Euclidean space, and for an integer k = 1,..., n, the k-Hessian equations are fully nonlinear elliptic equations for k > 1 and consist of the Poisson equation for k = 1 and the Monge-Ampere equation for k = n. We analyze for smooth non degenerate solutions a 9-point finite difference scheme. We prove that the discrete scheme has a locally unique solution with a quadratic convergence rate. In addition we propose new iterative methods which are numerically shown to work for non smooth solutions. A connection of the latter with a popular Gauss-Seidel method for the Monge-Ampere equation is established and new Gauss-Seidel type iterative methods for 2-Hessian equations are introduced.