Optimal boundary regularity for some singular Monge-Amp\`ere equations on bounded convex domains

TL;DR

This paper establishes the optimal boundary regularity for certain singular Monge-Ampère equations on convex domains by constructing explicit supersolutions, revealing their degenerate nature and extending understanding of affine geometric equations.

Contribution

It introduces a method to determine optimal boundary regularity for singular Monge-Ampère equations using explicit supersolutions, applicable to affine hyperbolic and hyperspheres.

Findings

Achieved optimal global Hölder regularity results.

Identified the degenerate nature of specific singular equations.

Extended regularity theory to broader classes of convex domains.

Abstract

By constructing explicit supersolutions, we obtain the optimal global H\"older regularity for several singular Monge-Amp\`ere equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as $ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $ with zero boundary data, have unexpected degenerate nature.