Analytic properties of conditional curvatures of convex hypersurfaces and the Dirichlet problem for the Monge-Ampere equation

The existence and uniqueness of a sur face with given geometric characteristics is one of the important topical problems of global differential geometry. By stating this problem in terms of analysis, we arrive at second-order elliptic and parabolic partial differential equations. In the present paper we consider generalized solutions of the Monge-Ampere equation \\z(ij)\\ = phi(x, z, p) in A(n), where z = z(x(1),..., x(n)) is a convex function, p = (p(1),..., p(n)) = (partial derivative z/partial derivative x(1),..., partial derivative(z)/partial derivative x(n)), and z(ij) = partial derivative(2)z/partial derivative x(i)partial derivative x(j). We consider the Cayley-Klein model of the space A(n) and use a method based on fixed point principle for Banach spaces.