Variational problems and elliptic Monge-Ampère equations (Q792670)

The author studies the minimization problem for the functional \[ I_ n(u)=-\int_{G}[\det(\partial^ 2u/\partial x_ i\partial x_ j)- (n+1)f(x)]udx, \] where G is a convex bounded domain in \(R^ n\), f is given, and u is a convex function vanishing on the boundary of G. The expression in brackets in the integrand can be naturally defined for continuous convex functions. Conditions are established for existence and uniqueness of the absolute minimum. The results are generalizations of earlier results of the author for \(n=2\).