The sum of the first n ≥ 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 auxiliary 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 proof involves euclidean tight frames generated by orbits of the rotation group of the extremal domain. Among general convex domains, the ball is conjectured to maximize sums of Neumann eigenvalues, provided the volume and moment of inertia of the polar dual are suitably normalized.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics