Pólya’s Conjecture for Dirichlet and Neumann Laplacians

Establish Pólya’s Conjecture by proving that for any bounded Euclidean domain Ω in R^n, the eigenvalue counting function N_D(λ) for the Dirichlet Laplacian satisfies N_D(λ) ≤ (ω_n/(2π)^n) Vol(Ω) λ^{n/2} and that the eigenvalue counting function N_N(λ) for the Neumann Laplacian satisfies N_N(λ) ≥ (ω_n/(2π)^n) Vol(Ω) λ^{n/2}, thereby matching the leading term in Weyl’s law with the correct inequality direction in each case.

Background

The paper studies upper bounds for the counting function of Steklov eigenvalues and frames the main result as a weakened analogue of Pólya’s Conjecture in the Steklov setting. As context, it recalls the classical Pólya’s Conjecture for Laplacian eigenvalues on Euclidean domains, originally posed for the Dirichlet problem and with a corresponding Neumann version.

The authors note that while Pólya proved the conjecture for plane-covering domains and there are weakened results by Li and Yau, the general form of Pólya’s Conjecture (both Dirichlet upper bound and Neumann lower bound) remains unresolved, with a recent breakthrough confirming it for Euclidean balls.