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Pólya’s conjecture on Dirichlet eigenvalues

For every integer d≥1d\ge 1, every bounded open set Ω⊂Rd\Omega\subset\mathbb{R}^d of finite positive volume ∣Ω∣|\Omega|, and every k∈Nk\in\mathbb{N}, if λk(Ω)\lambda_k(\Omega) denotes the kk-th eigenvalue of the Dirichlet Laplacian on Ω\Omega and ωd\omega_d is the volume of the unit ball in Rd\mathbb{R}^d, then λk(Ω)≥4π2(kωd∣Ω∣)2/d\lambda_k(\Omega)\ge 4\pi^2\left(\frac{k}{\omega_d|\Omega|}\right)^{2/d}. In the planar case d=2d=2, this is λk(Ω)≥4πk∣Ω∣\lambda_k(\Omega)\ge \frac{4\pi k}{|\Omega|}.

References

Primary source

arXiv

Additional references

partial-differential-equationsspectral-geometry
Posted · edited

Progress summary

Refreshed
Claimed progress

A new unrefereed result reaches two thirds of the predicted lower bound in the plane, while the full conjecture remains open for general domains.

Pólya formulated the conjecture in 1954. It predicts a sharp lower bound for every Dirichlet eigenvalue of a bounded domain, but no proof is known for arbitrary domains.

Known results

  • Pólya, 1954: proved the conjecture for tiling domains, including higher-dimensional tiling domains.
  • Filonov, Levitin, Polterovich, and Sher: proved it for the disk and, in the Dirichlet case, balls in every dimension.
  • A 2024 universal inequality improves the Berezin--Li--Yau bound but does not reach Pólya's constant.
  • A 2025 result establishes an epsilon-loss form for sufficiently large eigenvalues on bounded Lipschitz domains.

October 5, 2026 two-thirds planar bound

Romain Speciel reported a trace-identity argument giving, for every eigenvalue of a bounded planar domain of area AA, a lower bound equal to two thirds of the conjectured Pólya bound. This is quantitative progress, not a proof of the full conjecture, and the preprint is unrefereed.

Current status (as of October 2026): The conjecture is proved for several special classes of domains and has new quantitative partial bounds, but remains open for general bounded domains; the latest two-thirds claim is unverified.

Sources

Solutions 0

No solutions have been posted yet.