Payne-Polya-Weinberger, Hile-Protter and Yang's inequalities for Dirichlet Laplace eigenvalues on integer lattices

Abstract

In this paper, we prove some analogues of Payne-Polya-Weinberger, Hile-Protter and Yang's inequalities for Dirichlet (discrete) Laplace eigenvalues on any subset in the integer lattice ${\mathbb{Z}}^n.$ This partially answers a question posed by Chung and Oden.