Lower bound for the first eigenvalue of $p-$Laplacian and applications in asymptotically hyperbolic Einstein manifolds

TL;DR

This paper derives new lower bounds for the first Dirichlet eigenvalue of the $p$-Laplacian on Riemannian manifolds, with applications to large geodesic balls in asymptotically hyperbolic Einstein manifolds.

Contribution

It introduces improved lower bounds for the $p$-Laplacian eigenvalue based on divergence, gradient, and distance functions, extending understanding in geometric analysis.

Findings

Established a lower bound involving specific functions satisfying divergence and gradient conditions.

Provided an enhanced lower bound linked to the domain's distance function.

Applied bounds to estimate eigenvalues of large geodesic balls in asymptotically hyperbolic Einstein manifolds.

Abstract

This paper investigates the first Dirichlet eigenvalue for the $p$-Laplacian in Riemannian manifolds. Firstly, we establish a lower bound for this eigenvalue under the condition that the domain includes a specific function which fulfills certain criteria related to divergence and gradient conditions. In the subsequent section, we introduce an enhanced lower bound for the eigenvalue, which is linked to the distance function defined in the domain. As a practical application, we provide an estimation for the first Dirichlet eigenvalue of geodesic balls with large radius in asymptotically hyperbolic Einstein manifolds.