Inequalities between Dirichlet, Neumann and buckling eigenvalues on Riemannian manifolds

Abstract

In this paper, we discuss various comparison results between Laplacian (or bilaplacian) eigenvalues for a mean-convex bounded domain in a Riemannian manifold of non-negative Ricci curvature, under different boundary conditions, namely Dirichlet, Neumann, clamped and buckling conditions. We first obtain a new integral gradient estimate for a first Dirichlet eigenfunction, and deduce therefrom that the first buckling eigenvalue of the bi-Laplacian is lower than four times the first eigenvalue of the Dirichlet Laplacian. An immediate consequence is a new comparison result between Dirichlet and Neumann Laplacian first eigenvalues. These inequalities as well as the intermediate key results are given in the general context of weighted Riemannian manifolds and under various curvature conditions.