Global $C^{1,\beta}$ and $W^{2, p}$ regularity for some singular Monge-Amp\`ere equations

TL;DR

This paper proves global regularity results in $C^{1,eta}$ and $W^{2,p}$ spaces for certain singular Monge-Ampère equations, extending understanding of solutions' smoothness under specific boundary and domain conditions.

Contribution

It establishes new global regularity results for singular Monge-Ampère equations with boundary singularities, particularly for equations involving the distance to the boundary and power-type singularities.

Findings

Solutions are globally $C^{1,eta}$ regular for the considered equations.

Solutions belong to $W^{2,p}$ for all $p<1/\alpha$ under the given conditions.

The results apply to convex Aleksandrov solutions with boundary data on smooth, uniformly convex domains.

Abstract

We establish global $C^{1,\beta}$ and $W^{2, p}$ regularity for singular Monge-Amp\`ere equations of the form \[\det D^2 u \sim \text{dist}^{-\alpha}(\cdot,\partial\Omega),\quad \alpha\in (0, 1),\] under suitable conditions on the boundary data and domains. Our results imply that the convex Aleksandrov solution to the singular Monge-Amp\`ere equation \[\det D^2 u=|u|^{-\alpha}\quad \text{in}\quad\Omega,\quad u=0\quad \text{in}\quad \partial\Omega, \quad \alpha\in (0, 1),\] where $\Omega$ is a $C^3$, bounded, and uniformly convex domain, is globally $C^{1,\beta}$ and belongs to $W^{2, p}$ for all $p<1/\alpha$.