Weyl’s two-term asymptotic (Weyl’s conjecture) for the Dirichlet counting function

Establish the conjectured two-term asymptotic expansion for the Dirichlet eigenvalue counting function on domains with piecewise smooth boundary: prove that, as λ → ∞, N^D_Ω(λ) = (|Ω| ω(n) / (2π)^n) λ^{n/2} − (ℋ^{n−1}(∂Ω) / (2^{n+1} π^{(d−1)/2} Γ((n+1)/2))) λ^{(n−1)/2} + o(λ^{(n−1)/2}), where N^D_Ω(λ) := #{λ_k(Ω) : λ_k(Ω) < λ}, ℋ^{n−1}(∂Ω) is the (n−1)-dimensional Hausdorff measure of the boundary, and ω(n) is the volume of the unit ball in ℝ^n.

Background

The authors recall Weyl’s conjecture for the second term in the spectral asymptotics of the Dirichlet Laplacian. Ivrii proved the conjecture under a dynamical condition (the set of periodic geodesic billiards has measure zero), and related results hold for convex analytic domains and polygons, but the conjecture is not established in full generality for all piecewise smooth boundaries.

The paper discusses the conjecture in connection with potential improvements to Riesz mean inequalities and notes that without boundary regularity assumptions the second-term formula cannot hold in completely general settings, reinforcing the relevance of analyzing geometric dependence of spectral bounds.