Pólya’s conjecture for the product domain (0,1) × Ω

Determine whether Pólya’s conjecture holds for the Laplace eigenvalues on the Cartesian product domain (0,1) × Ω ⊂ ℝ^{d+1}, for an arbitrary bounded Euclidean domain Ω ⊂ ℝ^d with suitable boundary regularity. Specifically, establish whether, for every k ≥ 1, the Dirichlet eigenvalues satisfy λ_k((0,1) × Ω) ≥ 4π^2 / ((ω_{d+1} |(0,1) × Ω|)^{2/(d+1)} k^{2/(d+1)}) and the positive Neumann eigenvalues satisfy μ_k((0,1) × Ω) ≤ 4π^2 / ((ω_{d+1} |(0,1) × Ω|)^{2/(d+1)} k^{2/(d+1)}).

Background

The paper proves Pólya’s conjecture for several classes of “thin” product domains, including (0,a) × Ω for sufficiently small a, and aΩ × Ω₂ when one factor is thin. Laptev’s approach via Riesz mean inequalities requires γ ≥ 1, which does not cover the case when the first factor has dimension d₁ = 1 (i.e., an interval).

Despite tiling properties of the interval, the authors note that the general case of the product domain (0,1) × Ω remains unresolved. This motivates their separate paper of thin products (0,a) × Ω, for which they obtain positive results, while explicitly stating that the case (0,1) × Ω is still open.