Regularity of free boundary for the Monge–Ampère obstacle problem

We prove the regularity of the free boundary in the Monge–Ampère obstacle problem detD2v=f(y)χ{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 detD2u=1∕f(Du)+δ0 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.