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

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.