Eigenvalue bounds for the poly-harmonic operators

Abstract

We study some estimates for the sums of eigenvalues of the Dirichlet poly-harmonic operator $(-\Delta)^{l}$ restricted to a bounded domain $\Omega\subset\mathbb{R}^{d}$ with $d\ge2$, $l\ge1$. Our approach yields estimates sharper than the estimates recently obtained by Q.-M. Cheng, X. Qi and G. Wei (Pacific J. Math. 262 (2013) 35–47) and G. Wei and L. Zeng (Estimates for eigenvalues of poly-harmonic operators, preprint). Another central object of study in this paper is to establish some certain estimates for the sums of powers of the eigenvalues of the poly-harmonic operator $(-\Delta)^{l}|_{\Omega}$.