Inequalities for the Navier and Dirichlet eigenvalues of elliptic operators

Qiaoling Wang and Changyu Xia

Vol. 251 (2011), No. 1, 219–237

DOI: 10.2140/pjm.2011.251.219

Abstract

This paper studies eigenvalues of elliptic operators on a bounded domain in a Euclidean space. We obtain lower bounds for the eigenvalues of elliptic operators of higher orders with Navier boundary condition. We also prove lower bounds and universal inequalities of Payne–Pólya–Weinberger–Yang type for the eigenvalues of second order elliptic equations in divergence form with Dirichlet boundary condition.

Keywords

eigenvalue, lower bound, universal inequality, elliptic operator, Navier boundary condition

Mathematical Subject Classification 2000

Primary: 35P15, 53C20, 53C42

Secondary: 35P15, 53C42

Milestones

Received: 22 April 2009

Revised: 20 October 2009

Accepted: 16 January 2010

Published: 22 April 2011