Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds

By Qiaoling Wang, C. Xia. 2007

81 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/ba8c415904ff3bc6efa055bf47c460831921c554

Citations in the corpus

• A lower bound for eigenvalues of a clamped plate problem (2009, unknown): “[6,17]) 0 < 1 ≤ 2 ≤ · · · ≤ k ≤ · · · → +∞, where each i has finite multiplicity which is repeated according to its multiplicity.”