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

Xiaoshang Jin

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.

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

Abstract

This paper investigates the first Dirichlet eigenvalue for the -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.
Paper Structure (9 sections, 14 theorems, 90 equations)

This paper contains 9 sections, 14 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Estimates for lower bound of eigenvalues via functions
  3. Some applications of Theorem \ref{['thm1.1']}
  4. The estimate of eigenvalue for domain of bounded "radius"
  5. The estimate of eigenvalue for bounded domain with bounded Ricci and mean curvature
  6. The first eigenvalue on AHE manifold
  7. Lower bound estimate
  8. Upper bound estimate
  9. The geometric property of the asymptotical behavior

Key Result

Theorem 1.1

Let $\Omega$ be a bounded smooth domain in a Riemannian manifold. Assume $p_2>p_1-1>0$ and $p=\frac{p_2}{p_2-p_1+1}.$ If $f\in W^{1,p_1}_{\rm loc}(\Omega)$ satisfies in $\Omega$ for some positive numbers $C$ and $D.$ Then

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Proposition 2.1: Theorem 2.1 in allegretto1998picone
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • ...and 17 more