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A sharp isoperimetric bound for the ratio of the fifth and fourth eigenvalues of Dirichlet Laplacian
Abstract
Let Ω⊂ℝn be a bounded domain with n≥2. The well-known Payne-Pólya-Weinberger conjecture for consecutive eigenvalue ratios asserts that (λm+1(Ω))/λm(Ω)≤((jn / 2,1)/(jn / 2−1,1))2, m=1,2,3,⋯. Here, jp,k denotes the k-th positive zero of the Bessel function Jp, and λj(Ω) are the Dirichlet eigenvalues of Ω, counted with multiplicity and ordered as 0<λ1(Ω)<λ2(Ω)≤λ3(Ω)≤⋯. 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-Pólya-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 ℝn.
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BibTeX
@misc{song2026sharp,
title = {{A sharp isoperimetric bound for the ratio of the fifth and fourth eigenvalues of Dirichlet Laplacian}},
author = {Zhen Song and Ruixiao Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31088},
eprint = {2610.31088},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31088},
}
RIS
TY - UNPB
TI - A sharp isoperimetric bound for the ratio of the fifth and fourth eigenvalues of Dirichlet Laplacian
AU - Zhen Song
AU - Ruixiao Zhang
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31088
UR - https://arxiv.org/abs/2610.31088
AB - Let Ω⊂ℝ^n be a bounded domain with n≥2. The well-known Payne-Pólya-Weinberger conjecture for consecutive eigenvalue ratios asserts that (λ_m+1(Ω))/λ_m(Ω)≤((j_n / 2,1)/(j_n / 2−1,1))^2, m=1,2,3,⋯. Here, j_p,k denotes the k-th positive zero of the Bessel function J_p, and λ_j(Ω) are the Dirichlet eigenvalues of Ω, counted with multiplicity and ordered as 0<λ_1(Ω)<λ_2(Ω)≤λ_3(Ω)≤⋯. 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-Pólya-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 ℝ^n.
ER -