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arXiv · 2609.28880

Neumann eigenvalues of isosceles triangles: monotonicity and asymptotics

Guowei Dai·Yingxin Sun·Yong Zhang
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Abstract

We prove Laugesen and Siudeja's conjecture on Neumann eigenvalues of isosceles triangles. After multiplication by the squared diameter, the first positive symmetric eigenvalue increases strictly with the apex angle $θ$, while the first antisymmetric eigenvalue decreases strictly up to the equilateral triangle and increases strictly thereafter. \rev{The proof combines a slice-average estimate of directional energies with the explicit equilateral eigenfunction.} A quadratic correction across slices also gives a two-term expansion for every fixed positive spectral index $k$ as $θ\downarrow0$, with relative correction $θ^2/6$ and remainder $O_k(θ^4)$.

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Guowei Dai, Yingxin Sun, Yong Zhang. 2026-09-24. Neumann eigenvalues of isosceles triangles: monotonicity and asymptotics. https://arxiv.org/abs/2609.28880

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