Remarks on the Monge-Ampère equation: some free boundary problems in geometry

Abstract

This paper deals with several qualitative properties of solutions of some stationary and parabolic equations associated to the Monge-Ampère operator. Mainly, we focus our attention in the occurrence of a free boundary (separating the region where the solution u is locally a hyperplane, and so were the Hessian D2u is vanishing from the rest of the domain). Among other thinfs, we take advantage of these proceedings to give a detailed version of some results already announced long time ago when dealing wiht other fully nonlinear equations (see the 1979 àèr by the authors on other parabolic equations, remark 2.25 of the 1985 monograph by the second author and the 1985 paper by the first author. In particular, our results apply to suitable formulations of the Gauss curvature flow and of the worn stones problems intensively studied in the literature.