Type problem and the first eigenvalue

2024/01/22 by Chen, Bo-Yong, Xiong, Yuanpu
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2401.11803

Abstract

In this paper, we study the relationship between the type problem and the asymptotic behavior of the first eigenvalues λ₁(B_r) of ``balls'' B_r:=\ρ18.624⋯. \] Moreover, an upper bound of Λ_* in terms of volume growth ν_*:=\liminf_{r→ +∞} (log |B_r|)/(log r) is given as follows Λ_* \lesssim \begincases ν_*², amp;ν_*≫1,
ν_*log(1)/(ν_*),amp;1lt;ν_*≪1. \endcases The exponent 2 for ν_*≫1 turns out to be the best possible.

Related

• Type problem, the first eigenvalue and Hardy inequalities
• Uniform stability of the ball with respect to the first Dirichlet and Neumann ∞-eigenvalues
• Some estimates for the higher eigenvalues of sets close to the ball
• Quantitative inequality for the eigenvalue of a Schrödinger operator in the ball
• Computing the first eigenpair for problems with variable exponents