The Robin eigenvalue ratio at fixed volume
For $d \ge 2$, a bounded domain $\Omega \subset \mathbb{R}^d$ with smooth boundary and $\alpha > 0$, let $0 < \rho_1(\Omega; \alpha) < \rho_2(\Omega; \alpha) \le \cdots$ be the eigenvalues of $-\Delta$ on $\Omega$ with the Robin condition $\partial_\nu u + \alpha u = 0$ on $\partial\Omega$, counted with multiplicity, where $\nu$ is the outward unit normal. Unlike the Dirichlet ratio, $\rho_2/\rho_1$ is not invariant under dilations, so the volume is fixed and $\alpha$ is not rescaled. As $\alpha \to \infty$ the Robin eigenvalues tend to the Dirichlet ones, for which Ashbaugh and Benguria proved that the ball maximizes $\lambda_2/\lambda_1$. Laugesen proved that among rectangular boxes of given volume the cube maximizes $\rho_2/\rho_1$, for every $\alpha > 0$. Dai and Sun proved the inequality below for bounded connected $C^2$ domains at every $\alpha$ at which a condition on the superlevel sets of the first eigenfunction holds; for each fixed domain it holds for all sufficiently large $\alpha$. Without convexity the inequality fails: Dai and Sun also showed that for every $\alpha > 0$ the ratio is unbounded among connected domains with smooth boundary and prescribed volume.
Problem. Let $B$ be a ball with $|B| = |\Omega|$. Is $$\frac{\rho_2(\Omega; \alpha)}{\rho_1(\Omega; \alpha)} \le \frac{\rho_2(B; \alpha)}{\rho_1(B; \alpha)}$$ for every $d \ge 2$, every $\alpha > 0$ and every bounded convex domain $\Omega \subset \mathbb{R}^d$ with smooth boundary?
Source: Laugesen (2019), Conjecture G (stated for convex domains in the published version, as reported by Dai and Sun), after a question of Payne and Schaefer (2001) for planar domains; stated for convex domains with smooth boundary and α > 0 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, eigenvalue ratios, isoperimetric inequalities, Robin eigenvalues
References
@article{laugesen2019,
author = {Laugesen, Richard S.},
title = {The {R}obin {L}aplacian---spectral conjectures, rectangular theorems},
journal = {J. Math. Phys.},
volume = {60},
number = {12},
year = {2019},
pages = {121507},
doi = {10.1063/1.5116253},
eprint = {1905.07658},
archivePrefix = {arXiv},
}
@article{payne2001,
author = {Payne, L. E. and Schaefer, P. W.},
title = {Eigenvalue and eigenfunction inequalities for the elastically supported membrane},
journal = {Z. Angew. Math. Phys.},
volume = {52},
number = {5},
year = {2001},
pages = {888--895},
doi = {10.1007/PL00001580},
}
@article{ashbaugh1992,
author = {Ashbaugh, Mark S. and Benguria, Rafael D.},
title = {A sharp bound for the ratio of the first two eigenvalues of {D}irichlet {L}aplacians and extensions},
journal = {Ann. of Math. (2)},
volume = {135},
number = {3},
year = {1992},
pages = {601--628},
doi = {10.2307/2946578},
}
@misc{dai2025,
author = {Dai, Guowei and Sun, Yingxin},
title = {Upper bounds for the ratio of the first two eigenvalues of the {R}obin {L}aplacian},
year = {2025},
eprint = {2511.20988},
archivePrefix = {arXiv},
note = {Revised September 2026 (v4)},
}
@misc{aim-robin-ppw,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The Robin eigenvalue ratio at fixed volume}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/021-robin-ppw.md},
note = {Entry 021 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00306, "The Robin eigenvalue ratio at fixed volume", clopen.solutions.