Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds

In this paper we consider eigenvalues of the Dirichlet biharmonic operator on compact Riemannian manifolds with boundary (possibly empty) and prove a general inequality for them. By using this inequality, we study eigenvalues of the Dirichlet biharmonic operator on compact domains in a Euclidean space or a minimal submanifold of it and a unit sphere. We obtain universal bounds on the (k + 1)th eigenvalue on such objects in terms of the first k eigenvalues independent of the domains. The estimate for the (k + 1)th eigenvalue of bounded domains in a Euclidean space improves an important inequality obtained recently by Cheng and Yang. © 2006 Elsevier Inc. All rights reserved