A sharp isoperimetric bound for the ratio of the fifth and fourth eigenvalues of Dirichlet Laplacian

Abstract

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with $n\geq2$. The well-known Payne-P\'olya-Weinberger conjecture for consecutive eigenvalue ratios asserts that $$ \frac{\lambda_{m+1}(\Omega)}{\lambda_m(\Omega)}\leq\left(\frac{j_{n / 2,1}}{j_{n / 2-1,1}}\right)^2,\quad m=1,2,3,\cdots. $$ Here, $j_{p,k}$ denotes the $k$-th positive zero of the Bessel function $J_p$, and $\lambda_j(\Omega)$ are the Dirichlet eigenvalues of $\Omega$, counted with multiplicity and ordered as $$ 0<\lambda_1(\Omega)<\lambda_2(\Omega)\leq\lambda_3(\Omega)\leq\cdots. $$ The cases $m=1,2,3$ were established by Ashbaugh and Benguria in their seminal works. In this paper, we give an affirmative answer to the Payne-P\'olya-Weinberger conjecture for $m=4$ by combining a quadratic-form identity with a Brouwer degree argument. We also extend the corresponding spectral inequality to arbitrary bounded open sets in $\mathbb{R}^n$.