Remarks on sharp boundary estimates for singular and degenerate Monge-Amp\`ere equations

TL;DR

This paper establishes sharp boundary estimates for solutions to singular and degenerate Monge-Ampère equations, confirming the optimality of known regularity results and exploring boundary behavior and applications to related equations.

Contribution

It constructs supersolutions to derive boundary bounds, proving optimality of regularity results and analyzing gradient blow-up in solutions.

Findings

Sharp lower bounds near boundary for solutions

Optimality of global Hölder regularity results

Gradient blow-up near flat boundary parts for certain q

Abstract

By constructing appropriate smooth, possibly non-convex supersolutions, we establish sharp lower bounds near the boundary for the modulus of nontrivial solutions to singular and degenerate Monge-Amp\`ere equations of the form $\det D^2 u =|u|^q$ with zero boundary condition on a bounded domain in $\mathbb{R}^n$. These bounds imply that currently known global H\"older regularity results for these equations are optimal for all $q$ negative, and almost optimal for $0\leq q\leq n-2$. Our study also establishes the optimality of global $C^{\frac{1}{n}}$ regularity for convex solutions to the Monge-Amp\`ere equation with finite total Monge-Amp\`ere measure. Moreover, when $0\leq q<n-2$, the unique solution has its gradient blowing up near any flat part of the boundary. The case of $q$ being $0$ is related to surface tensions in dimer models. We also obtain new global log-Lipschitz estimates, and apply them to the Abreu's equation with degenerate boundary data.