The Pólya conjecture for Dirichlet eigenvalues of the Laplacian on arbitrary bounded Euclidean domains

For a bounded domain $\Omega\subset\mathbb{R}^n$, let $0<\lambda_1(\Omega)\le\lambda_2(\Omega)\le\cdots$ be the eigenvalues of the Dirichlet Laplacian. Weyl's law gives $\lambda_k\sim 4\pi^2(\omega_n|\Omega|)^{-2/n}k^{2/n}$ as $k\to\infty$, where $\omega_n$ is the volume of the unit ball, and Pólya conjectured (1961, for domains that tile space, where he proved it) that this asymptotic value is a universal lower bound: $\lambda_k(\Omega)\ge(2\pi)^2\bigl(k/(\omega_n|\Omega|)\bigr)^{2/n}$ for every $k$, equivalently the eigenvalue counting function satisfies $N(\Omega,\Lambda)\le(2\pi)^{-n}\omega_n|\Omega|\Lambda^{n/2}$ for all $\Lambda>0$. Wang and Xia (2010) prove comparison theorems implying that a positive answer would also settle the generalized Pólya conjecture of Ku–Ku–Tang (1992) for polynomial elliptic operators. Despite deep partial results — tiling domains, balls, disks, sectors and annuli, an averaged (Li–Yau) bound with the factor $n/(n+2)$, and a recent $\varepsilon$-loss version for bounded Lipschitz domains — the conjecture for arbitrary bounded domains remains open in every dimension $n\ge2$. The problem is to prove or refute it.

Background

The Dirichlet Laplacian on a bounded open set $\Omega\subset\mathbb{R}^n$ is the operator $-\Delta$ acting on functions that vanish on $\partial\Omega$; its eigenvalues $0<\lambda_1(\Omega)\le\lambda_2(\Omega)\le\cdots\to\infty$ form the fundamental frequency spectrum of a fixed membrane occupying $\Omega$. Equivalently, they are characterized variationally by the Rayleigh–Ritz (min–max) principle: $\lambda_k$ is the infimum, over all $k$-dimensional spaces of compactly supported smooth functions on $\Omega$, of the largest Rayleigh quotient $\int_\Omega|\nabla u|^2/\int_\Omega u^2$ on that space. The eigenvalue counting function $N(\Omega,\Lambda)$ is the number of eigenvalues not exceeding $\Lambda$.

Weyl's law states that $N(\Omega,\Lambda)\sim(2\pi)^{-n}\omega_n|\Omega|\Lambda^{n/2}$ as $\Lambda\to\infty$, where $\omega_n$ is the volume of the unit ball in $\mathbb{R}^n$; equivalently $\lambda_k(\Omega)\sim 4\pi^2(\omega_n|\Omega|)^{-2/n}k^{2/n}$. G. Pólya (1961) proved that for domains that tile $\mathbb{R}^n$ (domains whose congruent copies pack the space without gaps or overlaps, such as parallelepipeds, certain triangles and hexagons) the leading Weyl term is already a one-sided bound for every finite $\Lambda$: $N(\Omega,\Lambda)\le(2\pi)^{-n}\omega_n|\Omega|\Lambda^{n/2}$, via a transplant/packing argument (eigenfunctions on $\Omega$ are transplanted onto the packed congruent copies of $\Omega$, and their mutual orthogonality caps how many eigenvalues can lie below any $\Lambda$), and he conjectured that this holds for all bounded domains. In eigenvalue form the conjecture reads $\lambda_k(\Omega)\ge(2\pi)^2\bigl(k/(\omega_n|\Omega|)\bigr)^{2/n}$ for all $k$; it is an equality in dimension $n=1$ (where $\omega_1=2$) and is open in every dimension $n\ge2$. The averaged version is known: Li and Yau (1983) proved $\frac{1}{k}\sum_{i=1}^k\lambda_i(\Omega)\ge\frac{n}{n+2}(2\pi)^2(\omega_n|\Omega|)^{-2/n}k^{2/n}$ for all $k$, and Melas (2003) sharpened this with an additive correction term proportional to $|\Omega|/I(\Omega)$, where $I(\Omega)$ is the moment of inertia of $\Omega$; but the factor $n/(n+2)$ separating the averaged result from the pointwise conjecture has never been improved, and the pointwise bound itself is the content of the conjecture.

A higher-order generalization: Ku, Ku and Tang (1992) proposed the generalized Pólya conjecture for Dirichlet eigenvalues of polynomial elliptic operators $\mathfrak{L}=\sum_{m=r+1}^{t}a_m(-\Delta)^m$ with nonnegative coefficients (the case of a single power $(-\Delta)^m$ is the poly-Laplacian; $m=2$ with clamped boundary conditions, meaning that both the function and its normal derivative vanish on $\partial\Omega$, is the clamped plate problem), namely $\lambda_{k,r}(\mathfrak{L})\ge\sum_{m=r+1}^{t}a_m\bigl(4\pi^2/\omega_n^{2/n}\bigr)^m(k/|\Omega|)^{2m/n}$, i.e. the full Weyl-polynomial value as a pointwise lower bound. Wang and Xia (2010) established comparison theorems for eigenvalues of higher-order elliptic equations on compact manifolds with boundary, showing that if the ordinary Pólya conjecture is true for the Laplacian, then this generalized conjecture follows; they also proved new lower bounds for such higher-order eigenvalue problems (with constants denoted $b_{n,m}$, $b_{n,p}$, whose optimality they leave open).

For general domains the state of the art is: the conjecture holds for tiling domains (Pólya), for disks, planar sectors and Dirichlet balls of any dimension (Filonov–Levitin–Polterovich–Sher 2023, via Bessel-function phase estimates and lattice-point counting), for annuli (the same authors 2026), and for further special classes (thin products, unions of scaled copies, certain non-tiling families); Frank–Loss–Weidl (2009) proved the magnetic analogue with a constant field on arbitrary domains; Jiang–Lin (2026) proved the $\varepsilon$-loss version (the bound with factor $1+\varepsilon$ above an explicit spectral threshold) on all bounded Lipschitz domains. No proof or counterexample for the exact original statement on arbitrary bounded domains is known, and cautionary analogues exist: the Neumann counterpart of the conjecture and the fractional-Laplacian analogue (Kwaśnicki–Laugesen–Siudeja) are known to fail in general.

Problem Statement

For every integer $n\ge2$ and every bounded open set $\Omega\subset\mathbb{R}^n$ (no regularity of $\partial\Omega$ assumed beyond boundedness), is it true that the Dirichlet eigenvalues of the Laplacian on $\Omega$ satisfy

$$\lambda_k(\Omega)\;\ge\;(2\pi)^2\left(\frac{k}{\omega_n\,|\Omega|}\right)^{2/n}\qquad\text{for every }k=1,2,\dots,$$

equivalently, that the eigenvalue counting function satisfies

$$N(\Omega,\Lambda)\;\le\;(2\pi)^{-n}\,\omega_n\,|\Omega|\,\Lambda^{n/2}\qquad\text{for all }\Lambda>0?$$

Prove the inequality for all bounded domains in all dimensions $n\ge2$, or exhibit a bounded open set $\Omega\subset\mathbb{R}^n$ ($n\ge2$) and an index $k$ for which it fails. The statement is sharp as stated: the constant $(2\pi)^2\omega_n^{-2/n}$ cannot be increased (it is attained asymptotically by Weyl's law, and with equality in $n=1$), and by Wang–Xia's comparison theorem a positive answer would also settle the generalized Pólya conjecture of Ku–Ku–Tang for Dirichlet eigenvalues of polynomial elliptic operators $\sum_m a_m(-\Delta)^m$. The two further questions raised in Wang–Xia (2010) — extending the Melas correction-term inequality to unbounded finite-measure domains, and the optimality of Wang–Xia's constants $b_{n,m}$, $b_{n,p}$ — are auxiliary refinements that are neither implied by nor needed for this resolution and are not part of its completion boundary.

References

1. G. Pólya, On the eigenvalues of vibrating membranes, Proc. London Math. Soc. s3-11 (1961) 419–433, DOI 10.1112/plms/s3-11.1.419, https://doi.org/10.1112/plms/s3-11.1.419
2. P. Li, 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
3. A. D. Melas, A lower bound for sums of eigenvalues of the Laplacian, Proc. Amer. Math. Soc. 131 (2003) 631–636, DOI 10.1090/S0002-9939-02-06834-X, https://doi.org/10.1090/S0002-9939-02-06834-X
4. Hsu-Tung Ku, Mei-Chin Ku, Ding-Yi Tang, Inequalities for eigenvalues of elliptic equations and the generalized Pólya conjecture, J. Differential Equations 97 (1992) 127–139, DOI 10.1016/0022-0396(92)90087-4, https://doi.org/10.1016/0022-0396(92)90087-4
5. Qiaoling Wang, Changyu Xia, Comparison theorems for eigenvalues of elliptic operators and the generalized Pólya conjecture, Math. Phys. Anal. Geom. 13 (2010) 235–253, DOI 10.1007/s11040-010-9077-8, https://doi.org/10.1007/s11040-010-9077-8
6. Rupert L. Frank, Michael Loss, Timo Weidl, Pólya's conjecture in the presence of a constant magnetic field, J. Eur. Math. Soc. 11 (2009) 1365–1383, DOI 10.4171/JEMS/184, https://doi.org/10.4171/JEMS/184
7. Nikolay Filonov, Michael Levitin, Iosif Polterovich, David 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
8. Nikolay Filonov, Michael Levitin, Iosif Polterovich, David A. Sher, Pólya's conjecture for Dirichlet eigenvalues of annuli, J. Lond. Math. Soc. 113 (2026), DOI 10.1112/jlms.70425, https://doi.org/10.1112/jlms.70425
9. Nikolay Filonov, Michael Levitin, Iosif Polterovich, David A. Sher, Pólya's conjecture for higher-dimensional Neumann balls, arXiv:2607.29305 (2026), https://arxiv.org/abs/2607.29305
10. Renjin Jiang, Fanghua Lin, Pólya's conjecture up to ε-loss and quantitative estimates for the remainder of Weyl's law, Comm. Pure Appl. Math. (2026, published online), DOI 10.1002/cpa.70058, https://doi.org/10.1002/cpa.70058
11. Rupert L. Frank, Simon Larson, Semiclassical inequalities for Dirichlet and Neumann Laplacians on convex domains, Comm. Pure Appl. Math. 79 (2025) 762–822, DOI 10.1002/cpa.70019, https://doi.org/10.1002/cpa.70019
12. Pedro Freitas, Isabel Salavessa, Families of non-tiling domains satisfying Pólya's conjecture, J. Math. Phys. 64 (2023), DOI 10.1063/5.0161050, https://doi.org/10.1063/5.0161050
13. Mateusz Kwaśnicki, Richard S. Laugesen, Bartłomiej A. Siudeja, Pólya's conjecture fails for the fractional Laplacian, J. Spectr. Theory 9 (2018) 127–135, DOI 10.4171/JST/242, https://doi.org/10.4171/JST/242
14. Nikolay Filonov, On the Pólya conjecture for the Neumann problem in planar convex domains, arXiv:2309.01432 (2023), https://arxiv.org/abs/2309.01432
15. Zhengchao Ji, Hongwei Xu, Deep estimates for higher eigenvalues of the poly-Laplacian, arXiv:2508.04069 (2025), https://arxiv.org/abs/2508.04069
16. Zhengchao Ji, Yong Luo, Improved lower bounds for Dirichlet eigenvalues of the Laplacian and poly-Laplacian on bounded Euclidean domains, arXiv:2604.20367 (2026), https://arxiv.org/abs/2604.20367
17. Qing-Ming Cheng, He-Jun Sun, Guoxin Wei, Lingzhong Zeng, Estimates for lower bounds of eigenvalues of the poly-Laplacian and quadratic polynomial operator of the Laplacian, arXiv:1112.2781 (2011), https://arxiv.org/abs/1112.2781
18. Hynek Kovařík, Semjon Vugalter, Timo Weidl, Two-dimensional Berezin-Li-Yau inequalities with a correction term, arXiv:0802.2792 (2008), https://arxiv.org/abs/0802.2792
19. Yue He, Proof of the Pólya conjecture, arXiv:1411.1135 (2014, withdrawn by the author due to a crucial error), https://arxiv.org/abs/1411.1135
20. Genqian Liu, The generalised Pólya conjecture for the Dirichlet eigenvalues, arXiv:1411.2400 (2014, withdrawn by the author due to an error), https://arxiv.org/abs/1411.2400
21. Genqian Liu, Pólya conjecture for the Neumann eigenvalues, arXiv:1411.2094 (2014, withdrawn by the author due to an error), https://arxiv.org/abs/1411.2094
22. Zhiyuan Xu, Hongwei Xu, A generalization of Pólya conjecture and Li-Yau inequalities for higher eigenvalues, Calc. Var. Partial Differential Equations 59 (2020), DOI 10.1007/s00526-020-01839-w, https://doi.org/10.1007/s00526-020-01839-w
23. Fanghua Lin, Extremum problems of Laplacian eigenvalues and generalized Pólya conjecture, Chinese Ann. Math. Ser. B 38 (2017) 497–512, DOI 10.1007/s11401-017-1080-y, https://doi.org/10.1007/s11401-017-1080-y
24. Xiang He, Zuoqin Wang, Pólya's conjecture for thin products, arXiv:2402.12093 (2024), https://arxiv.org/abs/2402.12093
25. Jingwei Guo, Changxing Miao, Weiwei Wang, Guoqing Zhan, Improvement of Pólya's conjecture for balls and cylinders, arXiv:2511.17050 (2025), https://arxiv.org/abs/2511.17050
26. Anderson L. A. Araujo, On the asymptotic behavior for eigenvalues of p-Laplacian and the Pólya conjecture for Dirichlet eigenvalues in certain annular domains, Proc. Amer. Math. Soc. (2016), DOI 10.1090/proc/13548, https://doi.org/10.1090/proc/13548