Variational Problems and Generalized Elliptic Solutions of Monge-Ampere Equations

Abstract

There is a profound connection between n-dimensional variational problems and the Dirichlet problem for n-dimensional Monge-Ampere equations. The absolute minimum of these variational problems turns out to be a generalized solution of the corresponding Monge-Ampere equations. In this chapter we study explicitly the main variational problem connected with the Monge- Ampere equation

$$ \det \left( {{u_{{ij}}}} \right) = f\left( {{x_{1}},{x_{2}}, \ldots ,{x_{n}}} \right) $$

(7.1)

and also consider generalizations of it.

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