Universal inequalities for the eigenvalues of the biharmonic operator on submanifolds

Saïd Ilias and Ola Makhoul

Vol. 251 (2011), No. 2, 315–329

DOI: 10.2140/pjm.2011.251.315

Abstract

We establish universal inequalities for the eigenvalues of the clamped plate problem on compact submanifolds of Euclidean space, of spheres and of real, complex and quaternionic projective spaces. We prove similar results for the biharmonic operator on domains of Riemannian manifolds that admit spherical eigenmaps (this includes compact homogeneous Riemannian spaces) and finally on domains of hyperbolic space.

Keywords

eigenvalue, biharmonic operator, universal inequality, submanifold, eigenmap

Mathematical Subject Classification 2000

Primary: 35P15, 58A10, 58C40, 58J50

Milestones

Received: 28 April 2010

Accepted: 22 September 2010

Published: 3 June 2011

Proposed: Paul Yang