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 D2v = 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 det D2u = 1=f (Du) + ı0 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. Show more