P\'olya's conjecture for thin products

Let Ω ⊂ Rᵈ be a bounded Euclidean domain. According to the famous Weyl law, both its Dirichlet eigenvalue λₖ(Ω) and its Neumann eigenvalue μₖ(Ω) have the same leading asymptotics wₖ(Ω)=C(d,Ω)k2/d as k → ∞. G. P\'olya conjectured in 1954 that each Dirichlet eigenvalue λₖ(Ω) is greater than wₖ(Ω), while each Neumann eigenvalue μₖ(Ω) is no more than wₖ(Ω). In this paper we prove P\'olya's conjecture for thin products, i.e. domains of the form (aΩ₁) × Ω₂, where Ω₁, Ω₂ are Euclidean domains, and a is small enough. We also prove that the same inequalities hold if Ω₂ is replaced by a Riemannian manifold, and thus get P\'olya's conjecture for a class of ``thin" Riemannian manifolds with boundary.