Boundary regularity for Monge-Amp\`ere equations with unbounded right hand side

TL;DR

This paper investigates the boundary regularity of solutions to Monge-Ampère equations with unbounded right-hand sides near the boundary, establishing a localization theorem for boundary sections.

Contribution

It introduces new boundary regularity results for Monge-Ampère equations with degenerate right-hand sides and proves a localization theorem for boundary sections.

Findings

Established boundary regularity for solutions with unbounded RHS.

Proved a localization theorem for boundary sections.

Analyzed equations with RHS behaving like a negative power of boundary distance.

Abstract

We consider Monge-Amp\`ere equations with right hand side $f$ that degenerate to $\infty$ near the boundary of a convex domain $\Omega$, which are of the type $$\mathrm{det}\;D^2 u=f\quad\mathrm{in}\;\Omega,\quad\quad f\sim d^{-\alpha}_{\partial\Omega}\quad\mathrm{near}\;\partial\Omega,$$ 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.