Variational problems and elliptic Monge-Ampère equations

In the present paper we study the ^-dimensional variational problems connected with the Dirichlet problem for ^-dimensional Monge-Ampere equation with zero boundary condition and prove the existence and uniqueness of an absolute minimum for this problem.This minimum is a generalized solution of the equation (*) belonging to the class of all general convex functions.The technique of convex hypersurfaces and bodies used in the geometric theory of elliptic Monge-Ampere equations turns out to be also essential for the investigations of the variational problems considered below.Therefore we also included the brief exposition of some necessary concepts and results of this theory.CONTENTS 1. Introduction.2. The functional I H {u) and its properties.3. Variational problem for the functional I H (u). 4. Dual convex hypersurfaces and Euler's equation.5. Generalizations.6.Some main concepts and the facts of the theory of elliptic solutions of the Monge-Ampere equations.