凸多边形边数众多时的Pólya--Szegő猜想
The Pólya--Szegő conjecture for convex polygons with many sides
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中文总结 AI 辅助
该文证明对足够大的N，正N边形在给定面积且边数至多N的凸多边形中唯一最小化第一Dirichlet特征值，通过逆三次Fourier不等式及精确约束坐标实现。
中文摘要 AI 辅助
对于每个足够大的N，我们证明在给定面积且边数至多为N的凸多边形中，正N边形在刚体运动意义下唯一地最小化第一Dirichlet特征值。定量的Faber--Krahn稳定性将极小元局部化在圆盘附近，但未确定其离散几何。我们通过中心化的外角测度来编码该几何，并证明一个逆三次Fourier不等式，其等号情形为均匀根配置。该不等式的亏损给出了正多边形上二次谱超出的下界。循环相消和公共物质坐标比较使得非线性差异相对于同一亏损为小量。精确约束坐标将此比较推广到任意小的正亏损情形。极小性迫使亏损为零，这刻画了正多边形。
英文摘要
For every sufficiently large N, we prove that the regular N-gon uniquely minimizes the first Dirichlet eigenvalue among convex polygons of prescribed area with at most N sides, up to rigid motions. Quantitative Faber--Krahn stability localizes minimizers near the disk but leaves their discrete geometry undetermined. We encode that geometry by a centered measure of exterior angles and prove an inverse-cubic Fourier inequality whose equality case is the uniform root configuration. Its defect gives a lower bound for the quadratic spectral excess over the regular polygon. Cyclic cancellation and a common material-coordinate comparison make the nonlinear difference small relative to the same defect. Exact constraint coordinates extend this comparison to arbitrarily small positive defects. Minimality then forces zero defect, which characterizes the regular polygon.
发表机构
- Central South University（中南大学）
- University of Macau（澳门大学）
- Zhuhai UM Science and Technology Research Institute（珠海UM科学技术研究院）
- Hunan University（湖南大学）
机构由 AI 辅助整理，请以论文原文为准。
