发表机构
Shing-Tung Yau Center and School of Mathematics, Southeast University（东南大学数学学院及苏步青数学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对含d维平坦边界段的有界凸域上三类Monge-Ampère方程的凸解，确定了其在平坦边界段处斜率为无穷的指数阈值，证明了部分方程的无解阈值，并在特定外锥条件下验证了阈值的尖锐性。
AI 中文摘要
对于任意有界凸域Ω⊂ℝⁿ，其边界∂Ω包含一个d维平坦边界段F，我们考虑满足零边界条件的凸解，对应三类Monge-Ampère方程：detD²u=dist(·,Aff(F))^α、detD²u=dist(·,∂Ω)^α、detD²u=(-u)^α，其中dist表示欧氏距离，Aff(F)是F的仿射包。我们证明，使得u在F处斜率为无穷的指数α∈ℝ的阈值为α≤2d−n；对于前两类方程，还证明当α≤2d−2n时不存在解。此外，当Ω在F处满足“带d维脊的外锥条件”时，第一类方程的两个阈值均是尖锐的，第三类方程的无穷斜率阈值也是尖锐的。
英文摘要
On any bounded convex domain $Ω\subset\mathbb{R}^n$ with a $d$-dimensional flat boundary piece $F\subset\partialΩ$, we consider convex solutions with zero boundary values to the Monge-Ampère equations $$ \det\mathsf{D}^2 u= \mathrm{dist}(\,\cdot\,,\mathrm{Aff}(F))^α,\quad\det\mathsf{D}^2 u=\mathrm{dist}(\,\cdot\,,\partialΩ)^α,\quad \det\mathsf{D}^2 u=(-u)^α, $$ where $\mathrm{dist}$ stands for Euclidean distance and $\mathrm{Aff}(F)$ is the affine hull of $F$. We show that the threshold of the exponent $α\in\mathbb{R}$ for $u$ to have infinite slope at $F$ is $α\leq 2d-n$. For the first two equations, we also show that no solution exists when $α\leq 2d-2n$. Moreover, when $Ω$ satisfies the ``exterior cone condition with a $d$-dimensional ridge'' at $F$, both thresholds are sharp for the first equation, and the infinite-slope threshold is sharp for the last equation.
Comments26 pages, 4 figures
