Regularity of free boundary for the Monge–Ampère obstacle problem

Abstract

We prove the regularity of the free boundary in the Monge–Ampère obstacle problem $det{D}^{2}v=f(y){\mathit{\chi }}_{\{v>0\}}$. 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 $det{D}^{2}u=1∕f(Du)+{\mathit{\delta }}_0$ 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.