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

TL;DR

This paper proves the regularity of the free boundary in the Monge-Ampère obstacle problem by analyzing the asymptotic behavior of solutions and transforming the problem into a singular elliptic equation.

Contribution

It introduces a novel approach using duality and a partial Legendre transform to establish free boundary regularity in the Monge-Ampère obstacle problem.

Findings

Established asymptotic estimates for solutions near singular points

Proved regularity of solutions to the transformed singular elliptic equation

Demonstrated the equivalence of free boundary regularity to asymptotic cone regularity

Abstract

In this paper, we prove the regularity of the free boundary in the Monge-Amp\`ere 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-Amp\`ere 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-Amp\`ere equation to a singular, fully nonlinear elliptic equation, and establish the regularity of solutions to the singular elliptic equation.