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arXiv · 2607.05388

Sharp ratios for low-index Neumann eigenvalues on convex domains

Quanyu Tang·Haiqi Zhang
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Abstract

Let $\Omega\subset\mathbb{R}^N$ be a bounded open convex set, and let $0=\mu_0(\Omega)<\mu_1(\Omega)\le \mu_2(\Omega)\le\cdots$ be the Neumann eigenvalues of the Laplacian, repeated according to multiplicity. We prove the sharp bounds $$ \mu_2(\Omega)\le 4\mu_1(\Omega),\qquad \mu_3(\Omega)\le 9\mu_1(\Omega). $$ The first estimate resolves a problem attributed to Henrot, while the second gives the next sharp case predicted by the one-dimensional model. The constants are optimal in every dimension.

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Quanyu Tang, Haiqi Zhang. 2026-07-06. Sharp ratios for low-index Neumann eigenvalues on convex domains. https://arxiv.org/abs/2607.05388

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