Eigenvalue inequalities for the p-Laplacian on a Riemannian manifold and estimates for the heat kernel

Abstract

In this paper, we successfully generalize the eigenvalue comparison theorem for the Dirichlet $p$-Laplacian ($1<p<\infty$) obtained by Matei [A.-M. Matei, First eigenvalue for the $p$-Laplace operator, Nonlinear Anal. TMA 39 (8) (2000) 1051--1068] and Takeuchi [H. Takeuchi, On the first eigenvalue of the $p$-Laplacian in a Riemannian manifold, Tokyo J. Math. 21 (1998) 135--140], respectively. Moreover, we use this generalized eigenvalue comparison theorem to get estimates for the first eigenvalue of the Dirichlet $p$-Laplacian of geodesic balls on complete Riemannian manifolds with radial Ricci curvature bounded from below w.r.t. some point. In the rest of this paper, we derive an upper and lower bound for the heat kernel of geodesic balls of complete manifolds with specified curvature constraints, which can supply new ways to prove the most part of two generalized eigenvalue comparison results given by Freitas, Mao and Salavessa in [P. Freitas, J. Mao and I. Salavessa, Spherical symmetrization and the first eigenvalue of geodesic disks on manifolds, submitted (2012)].

arXiv comment27 pages, 1 figure. Minor revision to v2. A slightly different version of this paper will appear in Journal de Mathématiques Pures et Appliquées

References in corpus (2)

• Lower bounds for the first Laplacian eigenvalue of geodesic balls of spherically symmetric manifolds
• A class of rotationally symmetric quantum layers of dimension 4

Cited by in corpus (14)

• New features of the first eigenvalue on negatively curved spaces
• Extremal Domains on Hadamard manifolds
• Inverse mean curvature flow inside a cone in warped products
• Monotonicity of the first eigenvalue of the Laplace and the p-Laplace operators under a forced mean curvature flow
• Conformal upper bounds for the eigenvalues of the $p$-Laplacian
• Comparison estimates on the first eigenvalue of a quasilinear elliptic system
• The Caffarelli-Kohn-Nirenberg inequalities and manifolds with nonnegative weighted Ricci curvature
• Geometry and topology of manifolds with integral radial curvature bounds
• The Gagliardo-Nirenberg inequality on metric measure spaces
• Eigenvalue comparisons in Steklov eigenvalue problem and some other eigenvalue estimates
• Eigenvalue estimates for submanifolds in Hadamard manifolds and product manifolds $N\times \mathbb{R}$
• Prescribed Weingarten curvature equations in warped product manifolds
• Sharp eigenvalue estimates and related rigidity theorems
• On the generalized principal eigenvalue of quasilinear operators