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Pólya’s conjecture

Let Ω⊂Rd\Omega\subset\mathbb{R}^{d} be a bounded domain, and let λk(Ω)\lambda_k(\Omega) and μk(Ω)\mu_k(\Omega) denote its kk-th Dirichlet and Neumann Laplacian eigenvalues, respectively. If ωd\omega_d is the volume of the unit ball in Rd\mathbb{R}^{d}, then for every k≥1k\ge 1, Pólya's conjecture asserts that λk(Ω)≥4π2(ωd∣Ω∣)2/dk2/d≥μk(Ω)\lambda_k(\Omega)\ge \frac{4\pi^2}{(\omega_d\lvert\Omega\rvert)^{2/d}}k^{2/d}\ge \mu_k(\Omega), where ∣Ω∣\lvert\Omega\rvert denotes the volume of Ω\Omega.

Sources & referencesView supporting material

Primary source

International Mathematics Research Notices

Additional references

partial-differential-equationsspectral-geometry

Progress summary

Refreshed
Partially solved

The conjecture remains open for arbitrary shapes, but it is now proved for several important families, including every Euclidean ball, and holds approximately for all sufficiently large eigenvalues.

Proposed by Pólya in 1954, the conjecture gives matching pointwise lower and upper bounds for Dirichlet and Neumann Laplacian eigenvalues using only a domain’s volume. It remains unresolved for general domains.

Known results

  • Pólya (1954, 1961): both inequalities for domains tiling Euclidean space; Kellner later removed the Neumann regularity assumption.
  • Li and Yau: averaged Dirichlet bounds, but not the conjectured pointwise bounds.
  • The inequalities hold in every dimension for the first two eigenvalues; the planar Neumann case is due to Bucur and Henrot.
  • Filonov, Levitin, Polterovich, and Sher (2023): both inequalities for disks and, more generally, Euclidean balls; planar sectors are also covered.

2025–2026 advances

A 2025 paper proves an explicit epsilon-loss version for every bounded Lipschitz domain at sufficiently large eigenvalues and reduces the remaining verification to computation. Two 2026 arXiv papers establish the Neumann inequality for balls in dimensions at least three, completing the ball case in every dimension; no general proof or counterexample is reported. A 2014 claimed proof was withdrawn for a crucial error.

Current status (as of August 2026): The conjecture is proved for several domain classes and approximately for general bounded Lipschitz domains at high energy, but remains open for general domains.

Sources

Solutions 0

No solutions have been posted yet.