Mathematical Sciences: Isoperimetric Inequalities for Eigenvalue Ratios

该奖项支持的工作集中在与在被称为流形的类曲面区域上定义的偏微分算子相关的几何问题上。人们可以通过考虑一个振动的表面或膜，并要求捕捉表面几何特性的解析量，来可视化工作的总体推力。一类这样的量由拉普拉斯算子的特征值组成——一个偏微分算子——定义在表面上，在其边界上施加一定的边界条件。长期以来，人们一直推测，对于平面圆盘曲面，拉普拉斯函数的前两个狄利克雷特征值之比是最大的。这个猜想（佩恩-波利亚-温伯格猜想）最近得到了首席研究员的证实。这个项目将寻求描述最大化其他特征值比率的表面。同时，将在欧几里德空间中得到的结果推广到流形。由于已知的大部分内容都局限于二维表面，因此也将讨论对更高维度表面和流形的扩展。

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.