The minimal and convex surfaces

Abstract

We consider the Monge-Ampere's right - hand side which is the sum of the integral conditional curvatures of different orders. Integrating the given equation we obtain an integral equation Applying Theorem I.Ya.Bakelmana construct nonlinear U operator mapping the cone of convex surfaces themselves. The functional - topological properties of the operator U. Obtained important assessment applying the theorem of Steiner, as well as to prove the continuity and compactness of the operator U. Applying the theorem of Krasnosel'skii prove the existence of a fixed point of this operator is the solution to this problem.