Sharp ratios for low-index Neumann eigenvalues on convex domains

Quanyu Tang; Haiqi Zhang

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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 $Ω\subset\mathbb{R}^N$ be a bounded open convex set, and let $0=μ_0(Ω)<μ_1(Ω)\le μ_2(Ω)\le\cdots$ be the Neumann eigenvalues of the Laplacian, repeated according to multiplicity. We prove the sharp bounds $$ μ_2(Ω)\le 4μ_1(Ω),\qquad μ_3(Ω)\le 9μ_1(Ω). $$ 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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