Pólya’s conjecture for Dirichlet eigenvalues on arbitrary bounded domains

Prove that for every bounded open domain Ω ⊂ ℝ^n (n ≥ 2) and every k ∈ ℕ, the k-th Dirichlet eigenvalue λ_k(Ω) satisfies the inequality k ≤ (|Ω| ω(n)/(2π)^n) λ_k(Ω)^{n/2}.

Background

The paper recalls the classical problem posed by Pólya in 1954 concerning Dirichlet eigenvalues of the Laplacian on bounded domains. Pólya’s inequality is known to hold for certain special classes of domains (e.g., tiling domains, balls and some product domains), but it remains unsettled in general.

The authors position their results as achieving an ε-loss version of Pólya’s conjecture and provide quantitative estimates that reduce verification to computational checks for finitely many low eigenvalues. Nevertheless, the full conjecture for arbitrary bounded open domains is still widely open.

References

In 1954, P olya conjectured that, it should hold on arbitrary bounded open domain that $$ k\le \frac{|\Omega|\omega(n)}{(2\pi)n}\lambda_k{n/2},\ \forall\, k\in.$$ The P olya conjecture has been wide open since then.

— Pólya's conjecture up to $ε$-loss and quantitative estimates for the remainder of Weyl's law  (2507.04307 - Jiang et al., 6 Jul 2025) in Section 1.1 (Introduction: Weyl’s law and Pólya’s conjecture)

Pólya's conjecture asks whether the Weyl term is a termwise lower bound, \begin{equation}\label{eq:polya}\lambda_k\geq C_nV{-2/n}k{2/n}, \end{equation} for every $k$. The conjecture remains open for general domains.

eq:polya:

λk≥CnV−2/nk2/n,\lambda_k\geq C_nV^{-2/n}k^{2/n},

— Toward Pólya's Conjecture: Improving the Individual Li-Yau Bound via Energy Orthogonality  (2609.17219 - Wang et al., 15 Sep 2026) in Section 1, Introduction