Variational problems of Monge-Ampère type

Abstract

We consider the existence and regularity of maximizers (or minimizers) to two Monge-Ampere type functionals, one is the affine volume functional and the other arises in the study of Calabi’s extremal metrics on toric Kahler manifolds. The Euler equations of both functionals are fourth order partial differential equations, which are elliptic when the solution is a convex function. Both equations can be written as a system of a Monge-Ampere equation and a linearized Monge-Ampere equation. We discuss recent development on the regularity of maximizers to these two functionals Show more