On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions

Abstract

This paper deals with several qualitative properties of solutions of some stationary equations associated to the Monge--Ampere operator on the set of convex functions which are not necessarily understood in a strict sense. Mainly, we focus our attention on the occurrence of a free boundary (separating the region where the solution $u$ is locally a hyperplane, thus, the Hessian $D^{2}u$ is vanishing, from the rest of the domain). In particular, our results apply to suitable formulations of the Gauss curvature flow and of the worn stones problems intensively studied in the literature.