On the free boundary associated to 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.