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.