高维Laplace特征值的和--旋转与紧标架
Sums of Laplace eigenvalues --- rotations and tight frames in higher dimensions
摘要：
拉普拉斯算子的第一个$n \geq 1$特征值之和在正则单纯形（三维的正四面体）的单形中是最大的，在超立方体的平行六面体中是最大的，在球的椭球体中是最大的，只要“逆”体的体积和惯性矩被适当地归一化。这个结果适用于Dirichlet，Robin和Neumann特征值。此外，三次环面被证明是最大的平面环面。证明的方法涉及到由极值域的旋转群的轨道生成的欧几里得空间的紧框架。在一般凸区域中，球被构造成最大化Neumann特征值之和，条件是该区域的极对偶的体积和转动惯量被适当地归一化。
英文摘要：
The sum of the first $n \geq 1$ eigenvalues of the Laplacian is shown to be maximal among simplexes for the regular simplex (the regular tetrahedron, in three dimensions), maximal among parallelepipeds for the hypercube, and maximal among ellipsoids for the ball, provided the volume and moment of inertia of an "inverse" body are suitably normalized. This result holds for Dirichlet, Robin and Neumann eigenvalues. Additionally, the cubical torus is shown to be maximal among flat tori. The method of proof involves tight frames for euclidean space generated by the orbits of the rotation group of the extremal domain. The ball is conjectured to maximize sums of Neumann eigenvalues among general convex domains, provided the volume and moment of inertia of the polar dual of the domain are suitably normalized.
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