Counterexamples to a higher-index Dirichlet eigenvalue-ratio conjecture

Abstract

Let ${\lambda }_k(\Omega )$ denote the $k$th Dirichlet Laplacian eigenvalue of a bounded planar domain $\Omega$, with eigenvalues counted with multiplicity. We disprove a conjectured higher-index extension of the Payne--Pólya--Weinberger inequality that was recorded as an open problem by Ashbaugh. For every integer $m\ge 3$, we construct a bounded planar domain ${\Omega }_m$ with $C^{\infty }$ boundary such that $$\frac{{\lambda }_{2m}({\Omega }_m)}{{\lambda }_m({\Omega }_m)}>\frac{13}{5}>\frac{j_{1,1}^2}{j_{0,1}^2},$$ where $j_{\nu ,1}$ is the first positive zero of $J_{\nu }$, the Bessel function of the first kind of order $\nu$. For $m=3$, the annulus $\{x\in {\mathbb{R}}^2:1/10<|x|<1\}$ already provides such a counterexample.

arXiv comment10 pages, 1 table