Sharp Neumann eigenvalue means on triangles:an exact certificate proof and stability

Guowei Dai; Yingxin Sun

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arXiv · 2609.38953

Sharp Neumann eigenvalue means on triangles:an exact certificate proof and stability

Guowei Dai·Yingxin Sun

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Abstract

We prove both inequalities in Laugesen and Siudeja's Conjecture~6.34 for the first two positive Neumann eigenvalues of triangles. The equilateral triangle uniquely maximizes their harmonic mean at fixed perimeter and their geometric mean at fixed area. Two-dimensional trial spaces reduce the proof to polynomial positivity. Projected cubic corrections repair the second-order error near equality; elementary estimates and rational Bernstein certificates cover the remaining shapes. We provide exact data and a verifier, and derive an explicit quadratic stability estimate in terms of side-length asymmetry.

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Guowei Dai, Yingxin Sun. 2026-09-30. Sharp Neumann eigenvalue means on triangles:an exact certificate proof and stability. https://arxiv.org/abs/2609.38953

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