Math Conjectures

Pólya eigenvalue conjecture

OPENMajorConjectureProposed 1954 · Canonical special case

Canonical statement

Let Ω⊂Rn\Omega\subset\mathbb R^n be a bounded domain with piecewise smooth boundary and volume ∣Ω∣|\Omega|, and let ωn\omega_n be the volume of the Euclidean unit ball in Rn\mathbb R^n. Write 0<λ1≤λ2≤⋯0<\lambda_1\le\lambda_2\le\cdots for its Dirichlet Laplacian eigenvalues and 0=μ1≤μ2≤⋯0=\mu_1\le\mu_2\le\cdots for its Neumann eigenvalues, with multiplicity. For every k≥1k\ge1,
λk≥4π2(kωn∣Ω∣)2/n,μk+1≤4π2(kωn∣Ω∣)2/n. \lambda_k\ge 4\pi^2 \left(\frac{k}{\omega_n|\Omega|}\right)^{2/n}, \qquad \mu_{k+1}\le 4\pi^2 \left(\frac{k}{\omega_n|\Omega|}\right)^{2/n}.
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Let \(\Omega\subset\mathbb R^n\) be a bounded domain with piecewise smooth boundary and volume \(|\Omega|\), and let \(\omega_n\) be the volume of the Euclidean unit ball in \(\mathbb R^n\). Write \(0<\lambda_1\le\lambda_2\le\cdots\) for its Dirichlet Laplacian eigenvalues and \(0=\mu_1\le\mu_2\le\cdots\) for its Neumann eigenvalues, with multiplicity. For every \(k\ge1\), \[ \lambda_k\ge 4\pi^2 \left(\frac{k}{\omega_n|\Omega|}\right)^{2/n}, \qquad \mu_{k+1}\le 4\pi^2 \left(\frac{k}{\omega_n|\Omega|}\right)^{2/n}. \]
ANAL-FA-013:[Polya1961Eigenvalues][FilonovLevitinPolterovichSher2023Balls][Filonov2026Annuli]
Analysis and functional analysisGeometry and topologyPartial differential equations

Notes

Pólya's conjecture, posed in 1954, asserts that the eigenvalues of a bounded domain Ω⊂Rn\Omega\subset\mathbb R^n always lie on the correct side of Weyl's asymptotic law: every Dirichlet eigenvalue satisfies λk≥4π2(k/(ωn∣Ω∣))2/n\lambda_k\ge4\pi^2(k/(\omega_n|\Omega|))^{2/n}, while the Neumann eigenvalues satisfy the reversed inequality μk+1≤4π2(k/(ωn∣Ω∣))2/n\mu_{k+1}\le4\pi^2(k/(\omega_n|\Omega|))^{2/n}. Since both sequences obey the same Weyl asymptotics, the conjecture upgrades the one-term approximation from a limit statement to a bound valid for every kk [Polya1961Eigenvalues].

Pólya proved both inequalities for domains that tile Euclidean space [Polya1961Eigenvalues], and weaker versions of the bounds with dimension-dependent constants are classical. Progress beyond tiling domains is recent: Filonov, Levitin, Polterovich and Sher verified the conjecture for Euclidean balls [FilonovLevitinPolterovichSher2023Balls], and Filonov established the Dirichlet inequality for annuli [Filonov2026Annuli].

Historically the conjecture was unresolved even for basic domains such as balls until this recent work, and a full proof would require a method that does not rely on tiling or on special symmetry. Beyond these and other special families, the conjecture remains open in general.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.