On the P\'olya conjecture for the Neumann problem in planar convex domains

TL;DR

This paper proves a lower bound for the Neumann eigenvalue counting function in convex planar domains, providing a partial confirmation of Pólya's conjecture with a specific constant.

Contribution

The authors establish a new lower bound for the Neumann spectrum in convex domains, advancing understanding of spectral inequalities related to Pólya's conjecture.

Findings

Proved a lower bound for convex domains involving the first zero of J0

Bound improves previous estimates for Neumann eigenvalues

Supports Pólya's conjecture in a specific convex setting

Abstract

Denote by $N_{\cal N} (\Omega,\lambda)$ the counting function of the spectrum of the Neumann problem in the domain $\Omega$ on the plane. G. P\'olya conjectured that $N_{\cal N} (\Omega,\lambda) \ge (4\pi)^{-1} |\Omega| \lambda$. We prove that for convex domains $N_{\cal N} (\Omega,\lambda) \ge (2 \sqrt 3 \,j_0^2)^{-1} |\Omega| \lambda$. Here $j_0$ is the first zero of the Bessel function $J_0$.