Regularity and classification of the free boundary for a Monge-Ampère obstacle problem

Abstract

We study convex solutions to the Monge-Ampère obstacle problem \[ \operatorname{det} D^2 v=g v^qχ_{\{v>0\}}, \quad v \geq 0, \] where $q\in [0,n)$ is a constant and $g$ is a bounded positive function. This problem emerges from the $L_p$ Minkowski problem. We establish $C^{1,\alpha }$ regularity for the strictly convex part of the free boundary $\partial \{v=0\}$. Furthermore, when $g\in C^{\alpha }$, we prove a Schauder-type estimate. As a consequence, when $g\equiv 1$, we obtain a Liouville theorem for entire solutions with unbounded coincidence sets $\{v=0\}$. Combined with existing results, this provides a complete classification of entire solutions for the case $q=0$.

48 pages