Polya's conjecture for Dirichlet and Neumann Laplace spectra

Pólya proposed in 1954 that the Dirichlet and Neumann eigenvalues of every Euclidean domain obey global bounds given by the leading Weyl term. The full assertion remains unresolved in dimensions $n\ge 2$.

Known results

• Pólya (1954) proved the conjecture for planar tiling domains; the method extends to higher dimensions, with the Neumann tiling result later generalized by Kellner.
• The conjecture holds in dimension $n=1$ by explicit eigenvalue computation.
• Filonov, Levitin, Polterovich, and Sher (2023) proved the disk case and the Dirichlet case for balls in every dimension; related work covers sectors and annuli.
• Thin-product results establish Dirichlet inequalities, and under smoother boundaries Neumann inequalities, when one factor is sufficiently small.

2025--2026 special-case advances

A 2025 paper proved an $\epsilon$-loss Dirichlet version for bounded Lipschitz domains and identified broad classes satisfying the exact Dirichlet inequality. Later work improved ball and cylinder estimates, including the Neumann conjecture for cylinders in $\mathbb{R}^{3}$; another result shows the conjectured inequality for infinitely many eigenvalues in several dimensions. None settles every eigenvalue for every domain.

Current status (as of August 2026): The conjecture is settled in dimension $n=1$ and in several special geometric settings, but the general Dirichlet and Neumann assertions for $n\ge 2$ remain open.