Bulletin of the
Korean Mathematical Society BKMS

In this paper, we firstly prove some general inequalities for the Neumann eigenvalues for domains contained in a Euclidean $n$-space $\R^n$. Using the general inequalities, we can derive some new Neumann eigenvalues estimates which include an upper bound for the $(k+1)^{th}$ eigenvalue and a new estimate for the gap of the consecutive eigenvalues. Moreover, we give sharp lower bound for the first eigenvalue of two kinds of eigenvalue problems of the biharmonic operator with Navier boundary condition on compact Riemannian manifolds with boundary and positive Ricci curvature.

Keywords: Eigenvalues, Neumann problem, Navier problem, upper bound, lower bound

MSC numbers: Primary 35P15, 53C40

Supported by: http://bkms.kms.or.kr/journal/view.html?doi=10.4134/BKMS.b200170