American Institute of Mathematical Sciences

April  2015, 35(4): 1447-1468. doi: 10.3934/dcds.2015.35.1447

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

Received  June 2013 Revised  October 2013 Published  November 2014

This paper deals with several qualitative properties of solutions of some stationary equations associated to the Monge--Ampère 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.
Citation: Gregorio Díaz, Jesús Ildefonso Díaz. On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1447-1468. doi: 10.3934/dcds.2015.35.1447
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