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arXiv · 2208.03463

On the P\'olya conjecture for circular sectors and for balls

N. Filonov
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Abstract

In 1954, G. Polya conjectured that the counting function $N(\Omega,\Lambda)$ of the eigenvalues of the Laplace operator of the Dirichlet (resp. Neumann) boundary value problem in a bounded set $\Omega\subset R^d$ is lesser (resp. greater) than $(2\pi)^{-d} \omega_d |\Omega| \Lambda^{d/2}$. Here $\Lambda$ is the spectral parameter, and $\omega_d$ is the volume of the unit ball. We prove this conjecture for both Dirichlet and Neumann boundary problems for any circular sector, and for the Dirichlet problem for a ball of arbitrary dimension. We heavily use the ideas from \cite{LPS}.

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N. Filonov. 2022-08-06. On the P\'olya conjecture for circular sectors and for balls. https://arxiv.org/abs/2208.03463

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