Pólya's conjecture for Dirichlet eigenvalues of the Laplacian on bounded Euclidean domains, and its poly-Laplacian generalization

Pólya's conjecture (1961) asserts that for every bounded domain $\Omega\subset\mathbb{R}^n$ with $n\ge 2$ each Dirichlet eigenvalue of the Laplacian is at least its Weyl asymptotic value: $\lambda_k(\Omega)\ge 4\pi^2 k^{2/n}/(\omega_n V(\Omega))^{2/n}$ for all $k$. After six decades it remains open for general domains — even convex ones. Confirmed cases are limited to tiling domains, Euclidean balls, planar sectors, annuli, thin products, and strip-tiling domains (hence all triangles); very recently the $\varepsilon$-loss version for every bounded Lipschitz domain was reduced to a finite computation above an explicit eigenvalue threshold. Ji–Luo (arXiv:2604.20367, 2026) prove optimal averaged-sum (Berezin–Li–Yau-type) lower bounds and explicitly do not settle the conjecture. This record states, as the problem, the two named conjectures that paper leaves open: the original Pólya conjecture and the generalized Pólya conjecture of Ku–Ku–Tang (1992) for the clamped eigenvalues $\Lambda_k$ of the poly-Laplacian, $\Lambda_k\ge (2\pi)^{2l}k^{2l/n}/(\omega_n V(\Omega))^{2l/n}$. The task is to prove or refute each.

Background

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain (a connected open set of finite volume $V(\Omega)$) and consider the Dirichlet eigenvalue problem $-\Delta u=\lambda u$ in $\Omega$ with $u=0$ on the boundary $\partial\Omega$. Its spectrum $0<\lambda_1\le\lambda_2\le\cdots\to\infty$ gives the squared vibration frequencies of a membrane fixed along the boundary. Weyl's law states that $\lambda_k\sim 4\pi^2 k^{2/n}/(\omega_n V(\Omega))^{2/n}$ as $k\to\infty$, where $\omega_n$ denotes the volume of the unit ball in $\mathbb{R}^n$; equivalently, the eigenvalue counting function $N_\Omega(\lambda)=\#\{j:\lambda_j\le\lambda\}$ satisfies $N_\Omega(\lambda)\sim \omega_n V(\Omega)\lambda^{n/2}/(2\pi)^n$. In 1961, Pólya proved by a packing (exhaustion) argument that for planar domains which tile the plane every eigenvalue is at least its Weyl value, and he conjectured the same for all bounded domains:

$\lambda_k(\Omega)\ \ge\ \frac{4\pi^2}{(\omega_n V(\Omega))^{2/n}}\,k^{2/n}\quad\text{for all } k\ge 1,$

equivalently $N_\Omega(\lambda)\le \omega_n V(\Omega)\lambda^{n/2}/(2\pi)^n$ for all $\lambda\ge 0$. More than six decades later this remains one of the central open problems of spectral geometry.

The strongest general partial results concern averages of eigenvalues. The Berezin–Li–Yau inequality (proved by Li and Yau in 1983 by Fourier-transform methods) gives $(1/k)\sum_{i\le k}\lambda_i\ge \frac{n}{n+2}\cdot\frac{4\pi^2}{(\omega_n V(\Omega))^{2/n}}k^{2/n}$, which yields Pólya's bound only with the defect factor $n/(n+2)$. Melas (2003) sharpened it by an additive term involving the moment of inertia of $\Omega$, and the 2025–2026 line of work by Ji–Xu and by Ji–Luo (arXiv:2604.20367) pushed this averaged-sum strategy to what is optimal along this line, via exact expansions of certain binary polynomials that control the radially decreasing rearrangement of the eigenvalue sums. None of it touches the individual-eigenvalue conjecture.

For the conjecture itself, all confirmed cases exploit special structure. Laptev (1997) settled tiling domains in arbitrary dimension. Filonov–Levitin–Polterovich–Sher proved the conjecture for Euclidean balls in every dimension and for planar sectors (2023) — the first non-tiling domains, via a rigorous computer-assisted combination of variational bounds, Bessel phase estimates and lattice-point counting — and subsequently for annuli (2026). He–Wang proved it for thin products $(a\Omega_1)\times\Omega_2$ with $a$ small enough (2026). Jiang–Lin (2025) proved it for strip-tiling domains — which include every triangle — and for classes of quite irregular domains, and showed that for every bounded Lipschitz domain and every $\varepsilon\in(0,1)$ the conjectured bound holds up to a factor $(1+\varepsilon)$ for all eigenvalues above an explicitly computable threshold, reducing that $\varepsilon$-loss version to a finite computation. Frank–Larson (2025) analyze the consequences of a possible failure of the conjecture for convex sets, so it is open even for convex domains. The problem is genuinely delicate: the analogous statement for the fractional Laplacian $(-\Delta)^{\alpha/2}$ with $0<\alpha<2$ is false (Kwaśnicki–Laugesen–Siudeja 2016), so any proof must use the local second-order structure of the Laplacian.

The poly-Laplacian generalization. For an integer $l\ge 2$, the Dirichlet problem for the poly-Laplacian (the $l$-th iterate of the Laplacian) is $(-\Delta)^l u=\Lambda u$ with clamped boundary conditions $u=\partial_\nu u=\cdots=\partial_\nu^{l-1}u=0$ on $\partial\Omega$, where $\partial_\nu$ denotes the normal derivative; for $l=2$ this is the vibrating clamped plate. Its eigenvalues $\Lambda_k$ obey $\Lambda_k\sim (2\pi)^{2l}k^{2l/n}/(\omega_n V(\Omega))^{2l/n}$, and Ku–Ku–Tang (1992), who posed their generalized Pólya conjecture for nonnegative combinations of powers of elliptic operators, conjectured in particular

$\Lambda_k\ \ge\ \frac{(2\pi)^{2l}}{(\omega_n V(\Omega))^{2l/n}}\,k^{2l/n}\quad\text{for all } k\ge 1.$

This too is open; recent work (Ji–Xu 2025; Ji–Luo 2026) proves Berezin–Li–Yau-type lower bounds for averaged sums of these eigenvalues but not the conjecture. A further direction flagged in Ji–Luo (arXiv:2604.20367) — uniform lower bounds carrying the second Weyl term, i.e. the boundary correction of order $k^{1/n}$ proportional to $|\partial\Omega|$, known under suitable boundary regularity and realized as a lower bound so far only for special geometries such as polygons (Kovářík–Vugalter–Weidl 2009) — also remains open in the literature.

Problem Statement

Decide Pólya's conjecture: prove or refute that, for every bounded domain $\Omega\subset\mathbb{R}^n$ with $n\ge 2$ and every $k\ge 1$, the $k$-th Dirichlet eigenvalue of the Laplacian on $\Omega$ satisfies

$\lambda_k(\Omega)\ \ge\ \frac{4\pi^2}{(\omega_n V(\Omega))^{2/n}}\,k^{2/n},$

equivalently $N_\Omega(\lambda)\le \omega_n V(\Omega)\lambda^{n/2}/(2\pi)^n$ for all $\lambda\ge 0$. As the companion generalization singled out in Ji–Luo (arXiv:2604.20367), prove or refute that for every integer $l\ge 2$ the clamped (Dirichlet) eigenvalues of the poly-Laplacian satisfy

$\Lambda_k(\Omega)\ \ge\ \frac{(2\pi)^{2l}}{(\omega_n V(\Omega))^{2l/n}}\,k^{2l/n}\quad\text{for all } k\ge 1.$

The common objective is to determine whether the Weyl asymptotic value is a universal lower bound for individual eigenvalues on arbitrary bounded domains — for the Laplacian and for the clamped poly-Laplacian. The two statements are not logically independent: the clamped poly-Laplacian eigenvalues dominate the $l$-th powers of the Dirichlet Laplacian eigenvalues, $\Lambda_k(\Omega)\ \ge\ \lambda_k(\Omega)^l$ for all $k\ge 1$, so the poly-Laplacian inequality holds whenever Pólya's conjecture does, and a counterexample to the poly-Laplacian inequality would also refute Pólya's conjecture. The poly-Laplacian conjecture of Ku–Ku–Tang is the companion generalization of Pólya's conjecture, and the two are posed as separate components, each to be proved or refuted on its own.

References

1. Zhengchao Ji, Yong Luo, Improved lower bounds for Dirichlet eigenvalues of the Laplacian and poly-Laplacian on bounded Euclidean domains, 2026, arXiv:2604.20367, https://arxiv.org/abs/2604.20367
2. G. Pólya, On the eigenvalues of vibrating membranes, 1961, Proc. Lond. Math. Soc. (3) 11, 419-433, doi:10.1112/plms/s3-11.1.419
3. Ari Laptev, Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, 1997, J. Funct. Anal. 151, 531-545, doi:10.1006/jfan.1997.3155
4. Peter Li, Shing-Tung Yau, On the Schrödinger equation and the eigenvalue problem, 1983, Comm. Math. Phys. 88, 309-318, doi:10.1007/BF01213210
5. A. D. Melas, A lower bound for sums of eigenvalues of the Laplacian, 2003, Proc. Amer. Math. Soc. 131, 631-636, doi:10.1090/S0002-9939-02-06834-x
6. Hsu-Tung Ku, Mei-Chin Ku, Ding-Yi Tang, Inequalities for eigenvalues of elliptic equations and the generalized Pólya conjecture, 1992, J. Differential Equations 97, 127-139, doi:10.1016/0022-0396(92)90087-4
7. Nikolay Filonov, Michael Levitin, Iosif Polterovich, David A. Sher, Pólya's conjecture for Euclidean balls, 2023, Invent. Math. 234, 129-169, doi:10.1007/s00222-023-01198-1, https://arxiv.org/abs/2203.07696
8. Nikolay Filonov, Michael Levitin, Iosif Polterovich, David A. Sher, Pólya's conjecture for Dirichlet eigenvalues of annuli, 2026, J. Lond. Math. Soc. 113, doi:10.1112/jlms.70425, https://arxiv.org/abs/2505.21737
9. Xiang He, Zuoqin Wang, Pólya's conjecture for thin products, 2026, Int. Math. Res. Not., doi:10.1093/imrn/rnag182, https://arxiv.org/abs/2402.12093
10. Renjin Jiang, Fanghua Lin, Pólya's conjecture up to epsilon-loss and quantitative estimates for the remainder of Weyl's law, 2025, arXiv:2507.04307, to appear in Comm. Pure Appl. Math., https://arxiv.org/abs/2507.04307
11. Rupert L. Frank, Simon Larson, Semiclassical inequalities for Dirichlet and Neumann Laplacians on convex domains, 2025, Comm. Pure Appl. Math. 79, 762-822, doi:10.1002/cpa.70019, https://arxiv.org/abs/2410.04769
12. Mateusz Kwaśnicki, Richard S. Laugesen, Bartłomiej A. Siudeja, Pólya's conjecture fails for the fractional Laplacian, 2016, arXiv:1608.06905, https://arxiv.org/abs/1608.06905
13. Zhengchao Ji, Hongwei Xu, Deep estimates for higher eigenvalues of the poly-Laplacian, 2025, arXiv:2508.04069, https://arxiv.org/abs/2508.04069
14. Hynek Kovařík, Semjon Vugalter, Timo Weidl, Two-dimensional Berezin-Li-Yau inequalities with a correction term, 2009, Comm. Math. Phys. 287, 959-981, doi:10.1007/s00220-008-0692-1
15. Yue He, Proof of the Pólya conjecture, 2014, arXiv:1411.1135, https://arxiv.org/abs/1411.1135 (withdrawn by the author)
16. Genqian Liu, The generalised Pólya conjecture for the Dirichlet eigenvalues, 2014, arXiv:1411.2400, https://arxiv.org/abs/1411.2400 (withdrawn by the author)
17. Jingwei Guo, Changxing Miao, Weiwei Wang, Guoqing Zhan, Improvement of Pólya's conjecture for balls and cylinders, 2025, arXiv:2511.17050, https://arxiv.org/abs/2511.17050
18. Pedro Freitas, Isabel Salavessa, Spherical n-lunes: billiards and eigenvalues, 2026, arXiv:2608.09557, https://arxiv.org/abs/2608.09557