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

We study convex solutions to the Monge-Ampère obstacle problem $$\det D^{2}v=g{v}^q{\chi }_{\{v>0\}},v\ge 0,$$ 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 $\mathrm{\partial }\{v=0\}$. 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 $q=0$.