Estimate for the first Dirichlet eigenvalue of $p-$Laplacian on non-compact manifolds
Xiaoshang Jin, Zhiwei Lü
TL;DR
This work establishes a sharp lower bound for the first Dirichlet eigenvalue of the $p$-Laplacian on bounded domains of complete non-compact manifolds with non-negative Ricci curvature: $\lambda_{1,p}(\Omega) > (p-1)\left(\frac{\pi_p}{2d}\right)^p$ where $d={\rm diam}(\Omega)$ and $\pi_p=\frac{2\pi}{p\sin(\pi/p)}$. The authors develop a Barta-type framework for the $p$-Laplacian, leveraging generalized $p$-sine functions $\sin_p$ and a Busemann function to produce a computable differential inequality, yielding the lower bound via a specialized lemma. They prove the bound is sharp by constructing warped-product manifolds that realize the limit $(p-1)\left(\frac{\pi_p}{2d}\right)^p$ as the diameter is fixed, with a detailed demonstration of sharpness in the $d=2$ case. The results extend previous linear and $p$-Laplacian estimates under non-compact Ricci bounds and connect to refined isoperimetric-type considerations and asymptotics of the $\infty$-Laplacian.
Abstract
In this paper, we establish a sharp lower bound for the first Dirichlet eigenvalue of the $p$-Laplacian on bounded domains of a complete, non-compact Riemannian manifold with non-negative Ricci curvature.