Pólya’s conjecture for (0,1) × aΩ in the small-a regime

Determine whether Pólya’s conjecture holds for the Laplace eigenvalues on product domains of the form (0,1) × aΩ ⊂ ℝ^{d+1} for sufficiently small a > 0, for any bounded Euclidean domain Ω ⊂ ℝ^d with suitable boundary regularity; that is, prove that the Dirichlet eigenvalues satisfy the Pólya lower bound and the positive Neumann eigenvalues satisfy the Pólya upper bound with the volume |(0,1) × aΩ| entering the Weyl-type term.

Background

Although the authors establish Pólya’s conjecture for thin products of the form (0,a) × Ω (with the interval as the thin factor), they explicitly state that they are unable to prove the corresponding result when the thinness is in the Ω-factor, i.e., for (0,1) × aΩ with small a.

Because Pólya’s inequalities are scale-invariant, resolving the small-a case for (0,1) × aΩ would have implications for the general product (0,1) × Ω, underscoring the significance of this unresolved question.