Some inequalities between Laplacian eigenvalues on Riemannian manifolds

TL;DR

This paper derives inequalities relating the first eigenvalues of weighted p-Laplacian operators on Riemannian manifolds, using gradient estimates and comparison techniques, with applications to classical eigenvalue problems.

Contribution

It introduces new gradient estimate methods for the weighted p-Laplacian eigenfunctions and establishes novel eigenvalue comparison results on Riemannian manifolds.

Findings

Derived inequalities between weighted p-Laplacian eigenvalues.

Established eigenvalue comparison results for classical problems.

Provided gradient estimates for first eigenfunctions.

Abstract

In this paper, we study a first Dirichlet eigenfunction of the weighted $p$-Laplacian on a bounded domain in a complete weighted Riemannian manifold. By constructing gradient estimates for a first eigenfunction, we obtain some relationships between weighted $p$-Laplacian first eigenvalues. As an immediate application, we also obtain some eigenvalue comparison results between the first Dirichlet eigenvalue of the weighted Laplacian, the first clamped plate eigenvalue and the first buckling eigenvalue.