论文标题
加强的Alexandrov最大原理或蒙格解决方案的统一Hölder连续性 - 右侧有限的Ampère方程
A Strengthened Alexandrov Maximum Principle or Uniform Hölder Continuity for Solutions of the Monge--Ampère Equation with Bounded Right-Hand Side
论文作者
论文摘要
本文是关于Monge的凸解决方案$ u $ - Ampère方程至少在带有Dirichlet边界数据和非负界右侧的二维开放式凸面域上。 对于具有零边界数据的凸功能，Alexandrov最大原理$ | u（x）| \ leq c \ operatatorName {dist}（x，\partialΩ）^α$等于（均匀）Hölder连续性，具有相同的常数和指数。 convex $α$-Hölder连续功能为$ w^{1，p} $，对于$ p <1/（1 { - }α）$。我们证明了hölder的连续性，指数$α= 2/n $ for $ n \ geq 3 $，以及$ n = 2 $的任何$α\ in（0,1）$ in（0,1）$，前提是边界数据满足此Hölder的连续性，并表明这些指数的界限是锋利的。唯一的手段是绑定Hessian在$ n $维圆柱上的特定显式功能的决定因素，并使用Princple比较。
This article is about the convex solution $u$ of the Monge--Ampère equation on an at least 2-dimensional open bounded convex domain with Dirichlet boundary data and nonnegative bounded right-hand side. For convex functions with zero boundary data, an Alexandrov maximum principle $|u(x)| \leq C \operatorname{dist}(x,\partialΩ)^α$ is equivalent to (uniform) Hölder continuity with the same constant and exponent. Convex $α$-Hölder continuous functions are $W^{1,p}$ for $p < 1/(1{-}α)$. We prove Hölder continuity with the exponent $α=2/n$ for $n \geq 3$ and any $α\in (0,1)$ for $n=2$, provided that the boundary data satisfy this Hölder continuity, and show that these bounds for the exponent are sharp. The only means is to bound the Hessian determinant of a certain explicit function on an $n$-dimensional cylinder and to use the comparison princple.
