A localization theorem and boundary regularity for a class of degenerate Monge Ampere equations

TL;DR

This paper establishes boundary regularity and a localization theorem for a class of degenerate Monge-Ampère equations where the right-hand side degenerates near the boundary, extending previous results to higher dimensions.

Contribution

It provides new $C^2$ boundary estimates for degenerate Monge-Ampère equations with boundary degeneracy, generalizing prior two-dimensional results to higher dimensions.

Findings

Established $C^2$ boundary regularity estimates.

Proved a localization theorem for degenerate Monge-Ampère equations.

Extended boundary regularity results from 2D to higher dimensions.

Abstract

We consider degenerate Monge-Ampere equations of the type $$\det D^2 u= f \quad \{in $\Om$}, \quad \quad f \sim \, d_{\p \Om}^\alpha \quad \{near $\p \Om$,}$$ where $d_{\p \Om}$ represents the distance to the boundary of the domain $\Om$ and $\alpha>0$ is a positive power. We obtain $C^2$ estimates at the boundary under natural conditions on the boundary data and the right hand side. Similar estimates in two dimensions were obtained by J.X. Hong, G. Huang and W. Wang.