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.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.10663
 Bibcode:
 2020arXiv200610663F
 Keywords:

 Mathematics  Spectral Theory;
 35P15