On the Pólya conjecture for the Neumann problem in tiling sets

Abstract

In 1954, G. Pólya conjectured that the counting function of the eigenvalues of the Laplace operator of Dirichlet (resp. Neumann) boundary value problem in a bounded set $\Omega \subset {\mathbb{R}}^d$ is lesser (resp. greater) than $C_W|\Omega |{\lambda }^{d/2}$. Here $\lambda$ is the spectral parameter, and $C_W$ is the constant in the Weyl asymptotics. In 1961, Pólya proved this conjecture for tiling sets in the Dirichlet case, and for tiling sets under some additional restrictions for the Neumann case. We prove the Pólya conjecture in the Neumann case for all tiling sets.

References in corpus (1)

• An aperiodic hexagonal tile