On the Pólya conjecture for the Neumann problem in planar convex domains

2023/09/04 by N. Filonov, Filonov, N. · 2 citations
Computer Science · Mathematics · #35P15 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2309.01432

openalex publication_date 2023/09/04 · openalex created_date 2023/09/08 · openalex updated_date 2026/07/28

Abstract

Denote by N_{\cal N} (Ω,λ) the counting function of the spectrum of the Neumann problem in the domain Ω on the plane. G. Pólya conjectured that N_{\cal N} (Ω,λ) ≥ (4π)^-1 |Ω| λ. We prove that for convex domains N_{\cal N} (Ω,λ) ≥ (2 √ 3 j₀²)^-1 |Ω| λ. Here j₀ is the first zero of the Bessel function J₀.

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• Improvement of Pólya's conjecture for balls and cylinders
• Improved Berezin-Li-Yau inequality and Kröger inequality and consequences

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