On the P\'olya conjecture for the Neumann problem in tiling sets

Abstract

In 1954, G. P\'olya 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, P\'olya 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 P\'olya conjecture in the Neumann case for all tiling sets.