Solution of the Monge-Ampere equation in a ring domain

Abstract

The problem of recovering surfaces by the total or extrinsic curvature is related to the solution of the nonlinear elliptic equation of the Monge-Ampere type. Using the geometric method, the existence and uniqueness of a solution to the Monge-Ampere equation  is shown in the problem of recovering a surface by its total curvature in isotropic space. In this article, an exact solution to the Dirichlet problem for a ring domain is found if the total curvature function is given exact form.  In this, isotropic space geometry is used.