Year: 2026
Type: preprint
Abstract: We prove explicit positive corrections to the Dirichlet and Neumann Pólya inequalities at every positive eigenvalue index of every annular sector. The coefficients depend only on the radii and are uniform in the aperture. Real-order angular integration preserves the exact area term, while directional rounding estimates and radial trials control all frequencies. Optimized bounds attain the sharp circular-arc coefficient for the continuous count. We identify the full sector perimeter asymptotic and a low-index obstruction to using its coefficient throughout the spectrum. The strict Neumann inequality also passes to products with bounded Lipschitz domains.
Source: arXiv (Cornell University)
Authors: Dai, Guowei, Sun, Yingxin
Institution: Dalian University of Technology
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G4772512679
OpenAlex ID: W7222160575