Regularity of free boundary for the Monge-Ampère obstacle problem

2021/11/20 by Huang, Genggeng, Tang, Lan, Wang, Xu-Jia · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2111.10575

Abstract

In this paper, we prove the regularity of the free boundary in the Monge-Ampère obstacle problem det D² v= 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 D² u = 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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• Ancient Gauss Curvature Flows of Bounded Width
• On a general class of free boundary Monge-Ampère equations

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