Mathematical Sciences: Isoperimetric Inequalities for Eigenvalue Ratios

该奖项支持的工作集中在几何问题上 与定义在类曲面上的偏微分算子有关 称为流形的区域。 人们可以想象出 通过考虑振动的表面或膜， 要求解析量， 表面的性质。 这样一类量包括 的特征值的拉普拉斯算子-偏微分 算子-定义在具有一定边界的曲面上 边界上的条件。 它早已被 的前两个Dirichlet特征值之比， 拉普拉斯算子最大化的表面是平坦的圆盘。 这 (the Payne-Polya-Weinberger猜想）最近被 由主要研究者验证。 该项目将寻求描述最大限度地提高其他 特征值比 与此同时，将努力 将欧氏空间中的结果推广到流形上。 以来 许多已知的都局限于二维表面， 高维曲面和流形的扩展将 也要解决。

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.