P\'olya conjecture for the Neumann eigenvalues

TL;DR

This paper proves the Pólya conjecture for Neumann eigenvalues of the Laplacian on bounded domains with smooth boundaries, establishing an upper bound for the eigenvalues that aligns with the conjecture's prediction.

Contribution

It provides a proof of the Pólya conjecture specifically for Neumann eigenvalues on smooth bounded domains, a longstanding open problem.

Findings

Established the inequality or all eigenvalues k.

Confirmed the conjecture's bound matches the eigenvalue growth.

Applicable to domains with ^1-smooth boundary.

Abstract

For a given bounded domain $\Omega\subset {\Bbb R}^n$ with $C^1$-smooth boundary, we prove the P\'olya conjecture for the Neumann eigenvalues. In other words, we prove that \begin{eqnarray*} \mu_{k+1}\le \frac{(2\pi)^2k^{2/n}}{(\omega_n \cdot \mbox{vol}\, (\Omega))^{2/n}} \quad \;\; \mbox{for all} \;\; k=0,1,2,3,\cdots,\end{eqnarray*} where $\mu_k$ is the $k$-th Neumann eigenvalue of the Laplacian for $\Omega$.