Boundary regularity for Monge-Ampère equations with unbounded right hand side

Abstract

We consider Monge-Ampère equations with right hand side $f$ that degenerate to $\infty$ near the boundary of a convex domain $\Omega$, which are of the type $$\det D^{2}u=f\mathrm{in}\Omega ,f\sim d_{\partial \Omega }^{-\alpha }\mathrm{near}\partial \Omega ,$$ where $d_{\partial \Omega }$ represents the distance to $\partial \Omega$ and $-\alpha$ is a negative power with $\alpha \in (0,2)$. We study the boundary regularity of the solutions and establish a localization theorem for boundary sections.