Sharp ratios for low-index Neumann eigenvalues on convex domains

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.

Disclosure

“SHARP NEUMANN EIGENVALUE RATIOS 15 Acknowledgments and AI disclosure During the development and preparation of this work, the authors used a generative AI tool for preliminary, non-authoritative exploratory assistance, including organizational discussion and the consideration of possible approaches. The AI-generated outputs were not treated as mathematical sources, and no argument was includ”

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Classification

Brainstorming or outlining

Multiplier

2

Verified

2026-08-24T00:34:09.244Z

Structural counts