Comparison Theorems for Eigenvalues of Elliptic Operators and the Generalized Pólya Conjecture

摘要

We establish comparison theorems for eigenvalues between higher order elliptic equations on compact manifolds with boundary. As an application, it follows that if the Pólya conjecture is true then so is the generalized Pólya conjecture proposed by Ku et al. (J Differ Equ 97:127-139, 1992). We also obtain new lower bound for the eigenvalues of higher order elliptic equations on bounded domains in a Euclidean space.