Pólya's conjecture: sharp universal lower bounds for the Dirichlet eigenvalues of general bounded Euclidean domains

For a bounded domain $\Omega\subset\mathbb{R}^d$, Weyl's asymptotic law gives $N_D(\Omega,\Lambda)\sim(2\pi)^{-d}\omega_d|\Omega|\Lambda^{d/2}$ for the Dirichlet eigenvalue counting function as $\Lambda\to\infty$, where $\omega_d$ is the volume of the unit ball. Pólya's conjecture (1961) asserts that this leading term is an upper bound at every level and for every bounded domain — equivalently, every Dirichlet eigenvalue satisfies $\lambda_k\ge C_d(k/|\Omega|)^{2/d}$ with the sharp Weyl constant $C_d=(2\pi)^2\omega_d^{-2/d}$. Pólya proved the bound for domains that tile space. Li and Yau (1983) proved the sharp-constant bound for eigenvalue sums, which yields the individual bound for every domain in every dimension but with a constant loss factor $d/(d+2)$; the sharp individual bound, Pólya's conjecture, was left open and remains open for general domains in every dimension $d\ge2$.

Background

Fix $d\ge2$ and let $\Omega\subset\mathbb{R}^d$ be a bounded open set (a "domain"). The Dirichlet Laplacian on $\Omega$ is the operator $-\Delta$ acting on square-integrable functions that vanish on the boundary of $\Omega$; it models, for example, a vibrating membrane clamped along the boundary. Its spectrum is a nondecreasing sequence of eigenvalues $0<\lambda_1(\Omega)\le\lambda_2(\Omega)\le\cdots$, each repeated according to its multiplicity, tending to infinity. The Dirichlet counting function is $N_D(\Omega,\Lambda)=\#\{k\ge1:\lambda_k(\Omega)\le\Lambda\}$. Weyl's asymptotic law — the classical leading-order description of this counting function, stated for instance in Filonov–Levitin–Polterovich–Sher (2023) — says

$$N_D(\Omega,\Lambda)\ \sim\ (2\pi)^{-d}\,\omega_d\,|\Omega|\,\Lambda^{d/2}\qquad(\Lambda\to\infty),$$

where $\omega_d$ is the volume of the Euclidean unit ball and $|\Omega|$ the volume of $\Omega$. Inverting the relation gives $\lambda_k(\Omega)\sim C_d(k/|\Omega|)^{2/d}$ with the Weyl constant $C_d=(2\pi)^2\omega_d^{-2/d}$. A domain that tiles is one whose congruent copies fill $\mathbb{R}^d$ without gaps or overlaps (a square or an equilateral triangle, for example). Pólya (1961) proved that for tiling domains the asymptotic upper bound is in fact an upper bound at every level,

$$N_D(\Omega,\Lambda)\ \le\ (2\pi)^{-d}\,\omega_d\,|\Omega|\,\Lambda^{d/2},$$

and conjectured the same for all planar domains; the extension to all dimensions $d\ge2$ is standard (the case $d=1$ is elementary). Equivalently, the conjecture states that for every bounded domain and every $k\ge1$,

$$\lambda_k(\Omega)\ \ge\ C_d\left(\frac{k}{|\Omega|}\right)^{2/d},$$

that is, the Weyl asymptotic lower bound on each individual eigenvalue holds with no loss. The companion Neumann statement — where the boundary condition of vanishing is replaced by a zero normal derivative, and the inequality reverses to $N_N(\Omega,\Lambda)\ge(2\pi)^{-d}\omega_d|\Omega|\Lambda^{d/2}$ — was also conjectured by Pólya and is likewise open in general. Li–Yau (1983) proved that for every bounded domain in every dimension the eigenvalue sums obey $\sum_{i\le k}\lambda_i(\Omega)\ge\frac{d}{d+2}C_d\,k^{1+2/d}|\Omega|^{-2/d}$, with the sharp constant $C_d$ for the sum; averaging yields the individual bound $\lambda_k\ge\frac{d}{d+2}C_d(k/|\Omega|)^{2/d}$ — the Pólya bound up to the constant factor $\frac{d}{d+2}<1$. Removing this loss is precisely Pólya's conjecture.

Problem Statement

Determine the truth of Pólya's conjecture for arbitrary bounded Euclidean domains. Let $d\ge2$ and let $\Omega\subset\mathbb{R}^d$ be a bounded open set of volume $|\Omega|$ (connectedness is no restriction, since the Dirichlet counting function is additive over connected components). Let $0<\lambda_1(\Omega)\le\lambda_2(\Omega)\le\cdots$ be the eigenvalues of the Dirichlet Laplacian on $\Omega$, listed with multiplicity, and let $N_D(\Omega,\Lambda)=\#\{k\ge1:\lambda_k(\Omega)\le\Lambda\}$ be the counting function. Let $\omega_d$ denote the volume of the unit ball in $\mathbb{R}^d$ and $C_d=(2\pi)^2\omega_d^{-2/d}$ the Weyl constant. Prove or disprove: for every $d\ge2$, every bounded open set $\Omega\subset\mathbb{R}^d$, and every $\Lambda>0$,

$$N_D(\Omega,\Lambda)\ \le\ (2\pi)^{-d}\,\omega_d\,|\Omega|\,\Lambda^{d/2},$$

equivalently, for every $k\ge1$,

$$\lambda_k(\Omega)\ \ge\ C_d\left(\frac{k}{|\Omega|}\right)^{2/d}.$$

The companion Neumann conjecture $N_N(\Omega,\Lambda)\ge(2\pi)^{-d}\omega_d|\Omega|\Lambda^{d/2}$ is likewise open in general.

References

1. G. Pólya, 'On the Eigenvalues of Vibrating Membranes', Proceedings of the London Mathematical Society (3) 11 (1961), no. 1, 419–433. DOI: 10.1112/plms/s3-11.1.419, https://doi.org/10.1112/plms/s3-11.1.419.
2. Peter Li and Shing-Tung Yau, 'On the Schrödinger equation and the eigenvalue problem', Communications in Mathematical Physics 88 (1983), 309–318. DOI: 10.1007/BF01213210, https://doi.org/10.1007/BF01213210.
3. Pedro Freitas, Isabel Salavessa, 'Families of non-tiling domains satisfying Pólya's conjecture', Journal of Mathematical Physics 64 (2023), no. 12, article 121503. arXiv:2204.08902, DOI: 10.1063/5.0161050, https://arxiv.org/abs/2204.08902.
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10. Nikolay Filonov, Michael Levitin, Iosif Polterovich, David A. Sher, 'Pólya's conjecture for Dirichlet eigenvalues of annuli', Journal of the London Mathematical Society 113 (2026), no. 2, article e70425. arXiv:2505.21737, DOI: 10.1112/jlms.70425, https://arxiv.org/abs/2505.21737.
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14. Yaojia Sun, 'Weyl's law and Pólya's conjecture for the Vladimirov-Taibleson operator', arXiv:2608.26726 (2026), https://arxiv.org/abs/2608.26726.