Pólya's eigenvalue inequalities
Let $d \ge 2$ and let $\Omega \subset \mathbb{R}^d$ be a bounded connected Lipschitz domain. Let $\lambda_1 \le \lambda_2 \le \cdots$ be the Dirichlet eigenvalues and $0 = \mu_0 < \mu_1 \le \mu_2 \le \cdots$ the Neumann eigenvalues of $-\Delta$ on $\Omega$, counted with multiplicity, and let $\omega_d$ be the volume of the unit ball in $\mathbb{R}^d$. The middle term below is the leading term of Weyl's law. Pólya proved both inequalities for tiling domains, in the Neumann case under an extra assumption that Kellner removed, and they hold for every domain when $k = 1, 2$. Filonov, Levitin, Polterovich and Sher proved them for the disk and, in the Dirichlet case, for balls in every dimension (2023) and for planar annuli (2026). Two 2026 preprints, theirs and Li's, prove the Neumann case for balls of every dimension $d \ge 3$.
Problem. Is $$\mu_k(\Omega) \le 4\pi^2 \left( \frac{k}{\omega_d |\Omega|} \right)^{2/d} \le \lambda_k(\Omega)$$ for every $k \ge 1$, every $d \ge 2$ and every such $\Omega$?
Source: Pólya (1954), for planar domains; in this form, Filonov, Levitin, Polterovich and Sher (2023), (1.6) and Remark 1.1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, Dirichlet eigenvalues, Neumann eigenvalues, Weyl's law
References
@book{polya1954,
author = {P{\'o}lya, George},
title = {Mathematics and Plausible Reasoning},
note = {In two volumes},
publisher = {Princeton University Press},
address = {Princeton, NJ},
year = {1954},
}
@article{polya1961,
author = {P{\'o}lya, George},
title = {On the eigenvalues of vibrating membranes},
journal = {Proc. London Math. Soc. (3)},
volume = {11},
year = {1961},
pages = {419--433},
doi = {10.1112/plms/s3-11.1.419},
}
@article{kellner1966,
author = {Kellner, Robert},
title = {On a theorem of {P}olya},
journal = {Amer. Math. Monthly},
volume = {73},
number = {8},
year = {1966},
pages = {856--858},
doi = {10.2307/2314181},
}
@article{filonov2023,
author = {Filonov, Nikolay and Levitin, Michael and Polterovich, Iosif and Sher, David A.},
title = {P{\'o}lya's conjecture for {E}uclidean balls},
journal = {Invent. Math.},
volume = {234},
number = {1},
year = {2023},
pages = {129--169},
doi = {10.1007/s00222-023-01198-1},
eprint = {2203.07696},
archivePrefix = {arXiv},
}
@article{filonov2026annuli,
author = {Filonov, Nikolay and Levitin, Michael and Polterovich, Iosif and Sher, David A.},
title = {P{\'o}lya's conjecture for {D}irichlet eigenvalues of annuli},
journal = {J. Lond. Math. Soc. (2)},
volume = {113},
number = {2},
year = {2026},
pages = {e70425},
doi = {10.1112/jlms.70425},
eprint = {2505.21737},
archivePrefix = {arXiv},
}
@misc{filonov2026neumann,
author = {Filonov, Nikolay and Levitin, Michael and Polterovich, Iosif and Sher, David A.},
title = {P{\'o}lya's conjecture for higher-dimensional {N}eumann balls},
year = {2026},
eprint = {2607.29305},
archivePrefix = {arXiv},
}
@misc{li2026,
author = {Li, Yutian},
title = {P{\'o}lya's Conjecture for the {N}eumann Eigenvalues on {E}uclidean Balls},
year = {2026},
eprint = {2607.25958},
archivePrefix = {arXiv},
}
@misc{aim-polya-eigenvalue-bound,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Pólya's eigenvalue inequalities}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/c9929805cece044705abaaebe1cebddac1ba11a6/problems/001-polya-eigenvalue-bound.md},
note = {Entry 001 at commit c992980, 2026-10-05},
}
Resolution claims
We establish strict Dirichlet Pólya inequalities for spherical shells in every dimension $d\ge2$ and for every radius ratio. The inclusive eigenvalue counting function lies strictly below the leading Weyl term at every positive energy; consequently, each eigenvalue strictly exceeds its Weyl prediction, with multiplicities included. We also prove the same strict inequality for the Aharonov--Bohm operator on planar annuli at …
We introduce the Enhanced Neumann Pólya inequality, which is stronger than Pólya's conjecture, and verify it for many classes of domains, including strip-tiling domains, Euclidean balls, thin product domains, and certain higher-dimensional tiling domains. We develop a geometric removal approach for the Enhanced Neumann Pólya inequality. In particular, our results provide new classes of domains satisfying the Neumann part of …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-23 | Open → Partially resolved | The registry's curators | New Qualitative and Quantitative Estimates for Neumann Eigenvalues toward Pólya's Conjecture v1 recorded from arXiv:2610.19455v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00270, "Pólya's eigenvalue inequalities", clopen.solutions.