Regularity of free boundary for the Monge-Ampère obstacle problem

Abstract

In this paper, we prove the regularity of the free boundary in the Monge-Ampère obstacle problem $\det D^{2}v=f(y){\chi }_{\{v>0\}}.$ By duality, the regularity of the free boundary is equivalent to that of the asymptotic cone of the solution to the singular Monge-Ampère equation $\det D^{2}u=1/f(Du)+{\delta }_0$ at the origin. We first establish an asymptotic estimate for the solution $u$ near the singular point, then use a partial Legendre transform to change the Monge-Ampère equation to a singular, fully nonlinear elliptic equation, and establish the regularity of solutions to the singular elliptic equation.