Pólya's conjecture for Dirichlet eigenvalues

Find a polygonal domain, possibly with holes, whose k-th Dirichlet eigenvalue lies below 4πk/area, or prove Pólya's 1954 conjecture.

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1	# Pólya's conjecture for Dirichlet eigenvalues
2
3	The frequencies of a vibrating drum are the Dirichlet eigenvalues of its shape, and in 1954 George Pólya conjectured that the $k$-th eigenvalue never falls below the leading term of Weyl's law. It is proved for shapes that tile the plane, for disks and for annuli. The task is to find a polygonal drum, possibly with holes, that violates it, or to prove the conjecture.
4
5	## Problem
6
7	Let $\Omega \subset \mathbb{R}^2$ be a bounded open set with Lipschitz boundary, and let $0 < \lambda_1(\Omega) \le \lambda_2(\Omega) \le \cdots$ be the eigenvalues of $-\Delta$ on $\Omega$ with Dirichlet boundary conditions, counted with multiplicity. Write $\|\Omega\|$ for the area.
8
9	Pólya's conjecture. For every $k \ge 1$,
10	$$\lambda_k(\Omega) \ge \frac{4\pi k}{\|\Omega\|}.$$
11
12	Task. Find a polygon $P$ with rational vertices, possibly with polygonal holes, and an index $k$ such that $\|P\|\, \lambda_k(P) < 4\pi k$, with a rigorous margin; or prove the conjecture.
13
14	## Background
15
16	Weyl's law gives $\lambda_k(\Omega) \sim 4\pi k / \|\Omega\|$ as $k \to \infty$. Pólya conjectured in 1954 that the Dirichlet eigenvalues always lie on one side of this leading term (and the Neumann eigenvalues on the other) [1]. He proved it in 1961 for domains that tile the plane [2]. For general domains, Li and Yau proved the bound up to a factor of 2, $\lambda_k(\Omega) \ge 2\pi k/\|\Omega\|$ [3]. The disk resisted for decades even though its eigenvalues are explicit zeros of Bessel functions, and was settled only in 2023 [1].
17
18	## What is known
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20	- True for tiling domains (Pólya, 1961) [2].
21	- True for the disk, the first non-tiling planar domain to be settled, and for balls in every dimension in the Dirichlet case (Filonov, Levitin, Polterovich and Sher, 2023) [1]; true for annuli (the same authors, 2026) [4].
22	- True for $k = 1$ and $k = 2$ on every domain, by the Faber–Krahn and Krahn–Szegő inequalities; these are the only eigenvalues for which it is known in general [1].
23	- True up to a factor of 2 for every domain (Li and Yau, 1983) [3].
24	- The introduction of [5] surveys further partial results and the links to other problems in spectral geometry.
25
26	## What would count as progress
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28	- A polygonal domain (outer polygon and holes, rational coordinates), an index $k$ and a margin $\varepsilon > 0$, with a rigorous upper bound showing $\|P\|\, \lambda_k(P) < 4\pi k - \varepsilon$. Upper bounds on eigenvalues follow from the min–max principle with explicit trial functions, so they can be certified.
29	- Epoch scores submissions with its own verifier: it computes a rigorous bound on $\lambda_k(P)$ at the requested mesh refinement and accepts if the inequality holds with the stated margin.
30	- A proof of the conjecture for a new class of domains, for example all convex ones, would be progress in the other direction.
31
32	## References
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34	1. N. Filonov, M. Levitin, I. Polterovich and D. A. Sher, Pólya's conjecture for Euclidean balls, Invent. Math. 234 (2023), 129–169. [doi:10.1007/s00222-023-01198-1](https://doi.org/10.1007/s00222-023-01198-1)
35	2. G. Pólya, On the eigenvalues of vibrating membranes, Proc. London Math. Soc. (3) 11 (1961), 419–433. [doi:10.1112/plms/s3-11.1.419](https://doi.org/10.1112/plms/s3-11.1.419)
36	3. P. Li and S.-T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (1983), 309–318. [doi:10.1007/BF01213210](https://doi.org/10.1007/BF01213210)
37	4. N. Filonov, M. Levitin, I. Polterovich and D. A. Sher, Pólya's conjecture for Dirichlet eigenvalues of annuli, J. London Math. Soc. 113 (2026), e70425. [doi:10.1112/jlms.70425](https://doi.org/10.1112/jlms.70425)
38	5. P. Freitas, J. Lagacé and J. Payette, Optimal unions of scaled copies of domains and Pólya's conjecture, 2019. [arXiv:1908.08441](https://arxiv.org/abs/1908.08441)
39
40	## Source
41
42	Imported from [FrontierMath: Open Problems, polya-laplacian-spectrum](https://epoch.ai/frontiermath/open-problems/polya-laplacian-spectrum), curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.