论文标题
关于单数和堕落的蒙格 - 安培方程的尖锐边界估计的评论
Remarks on sharp boundary estimates for singular and degenerate Monge-Ampère equations
论文作者
论文摘要
通过构建适当的平滑，可能是非凸的超填充，我们在边界附近建立了尖锐的下限，以实现$ \ det d^2 u = | u |^q $的单数和退化的monge-ampère方程，以$ \ mathbbbb的边界域的零边界条件。这些界限意味着这些方程式目前已知的全球Hölder规律性结果对于所有$ Q $ nater来说都是最佳的，几乎最佳的是$ 0 \ leq q \ leq n-2 $。我们的研究还确定了全局$ c^{\ frac {1} {n}} $规律性的最佳性，以凸出对Monge-ampère方程，并具有有限的总Monge-Ampère度量。此外，当$ 0 \ leq Q <n-2 $时，独特的解决方案在边界的任何平坦部分附近都会爆炸。 $ Q $的情况为$ 0 $与二聚体模型中的表面紧张关系有关。我们还获得了新的全局log-lipschitz估计值，并将其应用于使用退化边界数据的Abreu方程。
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-Ampère 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 Hölder 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-Ampère equation with finite total Monge-Ampère 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.
