Pólya conjecture for the Neumann eigenvalues

2014/11/08 by Genqian Liu, Liu, Genqian
Computer Science · Mathematics · #35P15 #58J50 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1411.2094

openalex publication_date 2014/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a given bounded domain Ω⊂ \Bbb Rⁿ with C¹-smooth boundary, we prove the Pólya conjecture for the Neumann eigenvalues. In other words, we prove that μ_k+1≤ \frac(2π)²k^2/n(ω_n ⋅ vol (Ω))^2/n for all k=0,1,2,3,⋯, where μ_k is the k-th Neumann eigenvalue of the Laplacian for Ω.

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