On the Pólya conjecture for the Neumann problem in planar convex domains

Abstract

Denote by $N_{\mathcal{N}}(\Omega ,\lambda )$ the counting function of the spectrum of the Neumann problem in the domain $\Omega$ on the plane. G. Pólya conjectured that $N_{\mathcal{N}}(\Omega ,\lambda )\ge (4\pi {)}^{-1}|\Omega |\lambda$. We prove that for convex domains $N_{\mathcal{N}}(\Omega ,\lambda )\ge (2\sqrt{3}{j}_0^2{)}^{-1}|\Omega |\lambda$. Here $j_0$ is the first zero of the Bessel function $J_0$.