Type problem, the first eigenvalue and Hardy inequalities

2024/03/28 by Gilles Carron, Carron, Gilles, Bo-Yong Chen +3
Computer Science · Mathematics · #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

paper · pdf · doi:10.48550/arxiv.2403.19086

openalex publication_date 2024/03/28 · openalex created_date 2025/02/19 · openalex updated_date 2026/08/01

Abstract

In this paper, we study the relationship between the type problem and the asymptotic behaviour of the first (Dirichlet) eigenvalues λ₁(B_r) of ``balls'' B_r:=\ρr₀ r² λ₁(B_r)≥ γgt;0, we obtain a sharp estimate of the volume growth: |B_r|≥ cr^μ(γ). Moreover when γ>j₀²≈ 5.784, where j₀ denotes the first positive zero of the Bessel function J₀, then M is hyperbolic and we have a Hardy type inequality. In the case where r₀=0, a sharp Hardy type inequality holds. These spectral conditions are satisfied if one assumes that Δρ²≥2μ(γ)>0. In particular, when inf_MΔρ²>4, M is hyperbolic and we get a sharp Hardy type inequality. Related results for finite volume case are also studied.

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