Deep estimates for higher eigenvalues of the poly-Laplacian

Abstract

We investigate the lower bound for higher eigenvalues $\lambda_i$ of the poly-Laplace operator on a bounded domain and improve the famous Li-Yau inequality and its related results. Firstly, we consider the low dimensional cases for the P\'{o}lya conjecture, the clamped plate problem and the eigenvalue problem of the poly-Laplacian and deliver a series of deep eigenvalue inequalities for these problems respectively. Secondly, we establish a sharp lower bound for the eigenvalues of the poly-Laplacia in arbitrary dimension under some certain restrictive conditions. Finally, we provide an improved inequality for $\lambda_i$ in arbitrary dimension without any restrictive conditions. Our results also yield the improvement of the lower bounds for the Stokes eigenvalue problems and the Generalized P\'{o}lya conjecture.