Remarks on sharp boundary estimates for singular and degenerate Monge-Ampère equations

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