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

Gregorio Díaz; Jesús Ildefonso Díaz

doi:10.3934/dcds.2015.35.1447

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2015, Volume 35, Issue 4: 1447-1468. Doi: 10.3934/dcds.2015.35.1447

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

Gregorio Díaz^1, and
Jesús Ildefonso Díaz^2,

1.
Instituto Matemático Interdisciplinar and Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de las Ciencias 3, 28040-Madrid
2.
Instituto de Matemática Interdiciplinar, Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de las Ciencias, 3, 28040 Madrid

Received: May 31, 2013

Revised: September 30, 2013

Published: May 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J96, 35K96, 35R35; Secondary: 53C35.

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