Mathematical Sciences: Isoperimetric Inequalities for Eigenvalue Ratios

数学科学：特征值比的等周不等式

• 批准号: 9114162
• 负责人: Mark Ashbaugh
• 金额: $ 3.74万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-03-01 至 1995-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9114162&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Isoperimetric; Inequalities; Eigenvalue

Work supported by this award concentrates on geometric questions related to partial differential operators defined on surface-like regions known as manifolds. One can visualize the general thrust of the work by considering a vibrating surface or membrane and asking for analytic quantities which capture the geometric properties of the surface. One such class of quantities consists of eigenvalues of the Laplace operator - a partial differential operator - defined on the surface with certain boundary conditions imposed on its boundary. It had long been conjectured that the ratio of the first two Dirichlet eigenvalues of the Laplacian was maximized for surfaces which were flat discs. This (the Payne-Polya-Weinberger conjecture) has recently been verified by the principal investigator. This project will seek to describe surfaces which maximize other eigenvalue ratios. At the same time, efforts will be made to extend results obtained in Euclidean space to manifolds. Since much of what is known is confined to two-dimensional surfaces, extensions to surfaces and manifolds of higher dimension will also be addressed.

该奖项支持的工作集中在与被称为歧管的表面样区域定义的部分差分运算符有关的几何问题。 可以通过考虑振动表面或膜并要求捕获表面的几何特性的分析量来可视化工作的总体推力。 这样的数量类别由在表面上定义的拉普拉斯操作员的特征值（一种部分差分运算符）组成，其边界上施加了某些边界条件。 长期以来，人们一直认为，在平盘的表面最大化的Laplacian的前两个Dirichlet特征值的比率是最大化的。 该（Payne-Polya-Weinberger的猜想）最近已由首席研究员进行了验证。 该项目将寻求描述最大化其他特征值比率的表面。 同时，将努力将在欧几里得空间中获得的结果扩展到歧管。 由于已知的许多内容仅限于二维表面，因此也将解决对较高维度的表面和歧管的扩展。