PreprintCC0arXiv
Sharp Neumann eigenvalue means on triangles
Abstract
We prove two conjectures of Laugesen and Siudeja: the equilateral triangle uniquely maximizes the harmonic mean of the first two positive Neumann eigenvalues at fixed perimeter and their geometric mean at fixed area. The proof uses the two equilateral eigenfunctions and four projected polynomial corrections. Triangle symmetry reduces the associated variational matrices to two-by-two operations and two scalar invariants. Angular comparison and horizontal trial functions then yield the global inequalities. The same argument gives explicit quadratic stability bounds in terms of normalized side-length asymmetry.
Related
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BibTeX
@misc{dai2026sharp,
title = {{Sharp Neumann eigenvalue means on triangles}},
author = {Guowei Dai and Yingxin Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.38953},
eprint = {2609.38953},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38953},
note = {CC0},
}
RIS
TY - UNPB
TI - Sharp Neumann eigenvalue means on triangles
AU - Guowei Dai
AU - Yingxin Sun
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2609.38953
UR - https://arxiv.org/abs/2609.38953
AB - We prove two conjectures of Laugesen and Siudeja: the equilateral triangle uniquely maximizes the harmonic mean of the first two positive Neumann eigenvalues at fixed perimeter and their geometric mean at fixed area. The proof uses the two equilateral eigenfunctions and four projected polynomial corrections. Triangle symmetry reduces the associated variational matrices to two-by-two operations and two scalar invariants. Angular comparison and horizontal trial functions then yield the global inequalities. The same argument gives explicit quadratic stability bounds in terms of normalized side-length asymmetry.
ER -