Pólya's conjecture (1954) asserts that the counting function of the Dirichlet eigenvalues of a bounded planar domain Ω\Omega satisfies NΩDir(λ)<Area(Ω)λ2/(4π)N^{Dir}_\Omega(\lambda)<\mathrm{Area}(\Omega)\lambda^2/(4\pi) for all λ>0\lambda>0, equivalently every eigenvalue obeys λk(Ω)≥4πk/Area(Ω)\lambda_k(\Omega)\ge 4\pi k/\mathrm{Area}(\Omega). It is proved for tiling domains, disks and balls, thin products, and — in a 2026 work of Filonov, Levitin, Polterovich and Sher — for all concentric circular annuli, the first non-simply connected examples. The open problem addressed here is Pólya's conjecture for bounded planar domains that are not simply connected and not concentric annuli, with the eccentric (non-concentric circular) annulus as the canonical first open case and general multiply connected domains as the full target: a resolution must prove the conjecture, or refute it by a certified counterexample. Every known proof exploits either a tiling/packing structure or an explicit spectrum from separation of variables (solving the eigenvalue problem as a sum of product modes), both of which fail off concentricity, and domain monotonicity gives no inheritance because removing a hole raises eigenvalues while lowering area.

Background

For a bounded domain Ω⊂R2\Omega\subset\mathbb{R}^2 with Dirichlet boundary conditions (the eigenfunctions vanish on the boundary), the eigenvalue problem −Δu=λu-\Delta u=\lambda u, u∣∂Ω=0u|_{\partial\Omega}=0 has a discrete spectrum 0<λ1(Ω)≤λ2(Ω)≤⋯0<\lambda_1(\Omega)\le\lambda_2(\Omega)\le\cdots. The eigenvalue counting function NΩDir(λ)=#{k:λk(Ω)≤λ2}N^{Dir}_\Omega(\lambda)=\#\{k:\lambda_k(\Omega)\le\lambda^2\} obeys Weyl's law NΩDir(λ)∼Area(Ω)λ2/(4π)N^{Dir}_\Omega(\lambda)\sim\mathrm{Area}(\Omega)\lambda^2/(4\pi) as λ→∞\lambda\to\infty. Pólya (1954, Mathematics and Plausible Reasoning) conjectured that the leading term is in fact a one-sided bound for all λ\lambda: NΩDir(λ)<Area(Ω)λ2/(4π)N^{Dir}_\Omega(\lambda)<\mathrm{Area}(\Omega)\lambda^2/(4\pi), equivalently λk(Ω)≥4πk/Area(Ω)\lambda_k(\Omega)\ge 4\pi k/\mathrm{Area}(\Omega) for every kk. He later proved it for plane-tiling domains by a packing argument, in his 1961 paper on the eigenvalues of vibrating membranes. The best general result remains the averaged (Li–Yau) bound, whose constant is smaller than Pólya's by a factor depending on dimension, so the sharp pointwise bound is open for arbitrary domains. Positive classes beyond tiling domains are known: Euclidean balls in all dimensions (Filonov–Levitin–Polterovich–Sher, Invent. Math. 2023, via Faber–Krahn-type variational arguments and lattice-point counting), finite sectors and revolution-type non-tiling families (Freitas–Salavessa 2023), and thin products (aΩ1)×Ω2(a\Omega_1)\times\Omega_2 with aa small (He–Wang). Recently Filonov–Levitin–Polterovich–Sher (J. London Math. Soc. 2026) proved the conjecture for every concentric circular annulus Ar={x:r<∣x∣<1}A_r=\{x:r<|x|<1\}, 0<r<10<r<1 — the first verification on a non-simply connected planar domain. Their proof splits the parameter space into regions handled by the Faber–Krahn and Krahn–Szegő inequalities (the facts that the disk and two disjoint half-area disks minimize the first and second Dirichlet eigenvalues, respectively, among sets of given area), a flat-cylinder comparison, bounds on phase differences of Bessel functions (the special functions arising in the radial separation of variables for the Laplacian, i.e. solving the eigenvalue problem in product modes), refined trapezoidal floor-sum lattice-point counting, and a rigorous computer-assisted check of a bounded residual parameter region using interval arithmetic (arithmetic on machine-representable intervals with outward rounding, which turns floating-point computation into a proof), with public scripts. Every ingredient of their proof is anchored in the concentric radial geometry. Beyond concentric annuli nothing is known: an eccentric annulus has no separation-of-variables spectrum, and domain monotonicity for Dirichlet eigenvalues (shrinking the domain raises each λk\lambda_k) does not transfer the result, because deleting the hole from an enclosing disk both raises eigenvalues and reduces area, making the required bound strictly stronger than the disk bound inherited by continuity arguments.

Problem Statement

Prove, or refute by a certified counterexample, Pólya's conjecture for the Dirichlet eigenvalues of bounded multiply connected planar domains other than concentric circular annuli: for every bounded domain Ω⊂R2\Omega\subset\mathbb{R}^2 whose complement in the plane is not connected and which is not a concentric circular annulus (a domain bounded by two circles sharing a center), is it true that NΩDir(λ)<Area(Ω)4πλ2for all λ>0,N^{Dir}_\Omega(\lambda)<\frac{\mathrm{Area}(\Omega)}{4\pi}\lambda^2\quad\text{for all }\lambda>0, equivalently λk(Ω)≥4πk/Area(Ω)\lambda_k(\Omega)\ge 4\pi k/\mathrm{Area}(\Omega) for every k≥1k\ge 1? The canonical first open case is the eccentric annulus, the region between two non-concentric circles; the general question covers arbitrary doubly connected domains (holes of non-circular shape) and domains with three or more boundary components. A resolution may proceed by any method — variational, geometric, or rigorous computer-assisted — but must establish the stated inequality for the whole targeted class of domains. A refutation consists of an explicit domain and a rigorously certified eigenvalue bound violating the inequality. No finite-size, symmetry, or regularity restriction may be added to the class beyond boundedness of the domain.

References

  1. Nikolay Filonov, Michael Levitin, Iosif Polterovich, David A. Sher, Pólya's conjecture for Dirichlet eigenvalues of annuli, Journal of the London Mathematical Society 113(2) (2026) e70425, DOI 10.1112/jlms.70425; arXiv:2505.21737, https://arxiv.org/abs/2505.21737
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