Pólya's conjecture for Dirichlet eigenvalues

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.

Problem

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.

Pólya's conjecture. For every $k \ge 1$,

$$\lambda_k(\Omega) \ge \frac{4\pi k}{|\Omega|}.$$

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.

Background

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].

What is known

• True for tiling domains (Pólya, 1961) [2].
• 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].
• 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].
• True up to a factor of 2 for every domain (Li and Yau, 1983) [3].
• The introduction of [5] surveys further partial results and the links to other problems in spectral geometry.

What would count as progress

• 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.
• 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.
• A proof of the conjecture for a new class of domains, for example all convex ones, would be progress in the other direction.

References

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
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
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
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
5. P. Freitas, J. Lagacé and J. Payette, Optimal unions of scaled copies of domains and Pólya's conjecture, 2019. arXiv:1908.08441

Source

Imported from 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.