Universal inequalities for eigenvalues of a class of operators on Riemannian manifold

Abstract

In this paper, we investigate the boundary value problem of the following operator
$$\left\{\begin{array}{l}\Delta^{2}u-a\rm div\it A\nabla{u}+Vu=\rho\lambda{u}\ \ \hbox{in} \ \rm\Omega, \\ u|_{\partial \Omega}=\frac{\partial u}{\partial v}|_{\partial \Omega}=0,\\ \end{array}\right.$$
where Ω is a bounded domain in an n-dimensional complete Riemannian manifold Mⁿ, A is a positive semidefinite symmetric (1,1)-tensor on Mⁿ, V is a non-negative continuous function on Ω, v denotes the outwards unit normal vector field of ∂Ω and ρ is a weight function which is positive and continuous on Ω. By the Rayleigh-Ritz inequality, we obtain universal inequalities for the eigenvalues of these operators on bounded domain of complete manifolds isometrically immersed in a Euclidean space, and of complete manifolds admitting special functions which include the Hadamard manifolds with Ricci curvature bounded below, a class of warped product manifolds, the product of Euclidean spaces with any complete manifold and manifolds admitting eigenmaps to a sphere.