PreprintCC0arXiv
Robin eigenvalue ratios: monotonicity on boxes and counterexamples
Abstract
We disprove Laugesen's conjecture that the ratio of the first two Robin eigenvalues decreases with the positive boundary parameter. On any prescribed compact interval of positive parameters, the ratio is strictly increasing on a suitable connected polyhedral domain in every dimension at least two; in the plane, the domain can be a simply connected polygon. In contrast, we prove strict decrease on every rectangular box. Both results follow from a comparison of the relative growth of interval eigenvalues.
Related
- A self-contained proof of the variational characterization of the principal eigenvalue of −Δ+q under general Robin boundary conditions of non-classical type
- Optimization of the second Robin eigenvalue on a planar exterior domain under fixed area constraint
- Non-resonant angles in a Robin eigenvalue problem
- On the Isoperimetric Inequality for the Third Robin Eigenvalue on Surfaces
- Counterexamples to a higher-index Dirichlet eigenvalue-ratio conjecture
BibTeX
@misc{dai2026robin,
title = {{Robin eigenvalue ratios: monotonicity on boxes and counterexamples}},
author = {Guowei Dai and Yingxin Sun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23544},
eprint = {2609.23544},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23544},
note = {CC0},
}
RIS
TY - UNPB
TI - Robin eigenvalue ratios: monotonicity on boxes and counterexamples
AU - Guowei Dai
AU - Yingxin Sun
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23544
UR - https://arxiv.org/abs/2609.23544
AB - We disprove Laugesen's conjecture that the ratio of the first two Robin eigenvalues decreases with the positive boundary parameter. On any prescribed compact interval of positive parameters, the ratio is strictly increasing on a suitable connected polyhedral domain in every dimension at least two; in the plane, the domain can be a simply connected polygon. In contrast, we prove strict decrease on every rectangular box. Both results follow from a comparison of the relative growth of interval eigenvalues.
ER -