PreprintCC0arXiv
Neumann eigenvalues of isosceles triangles: monotonicity and asymptotics
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↓0, with relative correction θ2/6 and remainder Ok(θ4).
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BibTeX
@misc{dai2026neumann,
title = {{Neumann eigenvalues of isosceles triangles: monotonicity and asymptotics}},
author = {Guowei Dai and Yingxin Sun and Yong Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28880},
eprint = {2609.28880},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28880},
note = {CC0},
}
RIS
TY - UNPB
TI - Neumann eigenvalues of isosceles triangles: monotonicity and asymptotics
AU - Guowei Dai
AU - Yingxin Sun
AU - Yong Zhang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.28880
UR - https://arxiv.org/abs/2609.28880
AB - 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↓0, with relative correction θ^2/6 and remainder O_k(θ^4).
ER -