Regularity of free boundary for the Monge-Amp\`ere obstacle problem

In this paper, we prove the regularity of the free boundary in the MongeAmpère obstacle problem detDv = 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 detDu = 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.

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