Families of non-tiling domains satisfying Pólya's conjecture

Abstract

We show the existence of classes of non-tiling domains satisfying Pólya's conjecture in any dimension, in both the Euclidean and non-Euclidean cases. This is a consequence of a more general observation asserting that if a domain satisfies Pólya's conjecture eventually, that is, for a sufficiently large order of the eigenvalues, and may be partitioned into $p$ non-overlapping isometric sub-domains, with $p$ arbitrarily large, then there exists an order $p_0$ such that for $p$ larger than $p_0$ all such sub-domains satisfy Pólya's conjecture. In particular, this allows us to show that families of sectors of domains of revolution with analytic boundary, and thin cylinders satisfy Pólya's conjecture, for instance. We also improve upon the Li-Yau constant for general cylinders in the Dirichlet case.

This version 9 pages. Corrected some misprints and added some more examples

References in corpus (2)

• Extremal Eigenvalues of the Laplacian on Euclidean domains and closed surfaces
• Pólya's conjecture for Euclidean balls

Cited by in corpus (3)

• Maximization of Neumann eigenvalues
• Pólya's conjecture for Dirichlet eigenvalues of annuli
• Pólya-type inequalities on spheres and hemispheres