PROC
Parabolic Monge-Ampere equations giving rise to a free boundary: The worn stone model
Gregorio Díaz Jesús Ildefonso Díaz
This paper deals with several qualitative properties of solutions of some parabolic equations associated to the Monge--Ampère operator arising in suitable formulations of the Gauss curvature flow and the worn stone problems.
keywords: Gauss curvature surfaces free boundary problem. Monge-Ampère equation
DCDS
On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions
Gregorio Díaz Jesús Ildefonso Díaz
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.
keywords: Monge--Ampère equation Gauss curvatures surfaces free boundary problem.

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