Kodai Mathematical Journal

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

Jianghai Shi

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 $\Omega$ is a bounded domain in an $n$-dimensional complete Riemannian manifold $M^n$, $A$ is a positive semidefinite symmetric (1,1)-tensor on $M^n$, $V$ is a non-negative continuous function on $\Omega$, $v$ denotes the outwards unit normal vector field of $\partial \Omega$ and $\rho$ is a weight function which is positive and continuous on $\Omega$. 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.

Article information

Source
Kodai Math. J., Volume 40, Number 2 (2017), 229-253.

Dates
First available in Project Euclid: 12 July 2017

https://projecteuclid.org/euclid.kmj/1499846596

Digital Object Identifier
doi:10.2996/kmj/1499846596

Mathematical Reviews number (MathSciNet)
MR3680560

Zentralblatt MATH identifier
1377.53052

Citation

Shi, Jianghai. Universal inequalities for eigenvalues of a class of operators on Riemannian manifold. Kodai Math. J. 40 (2017), no. 2, 229--253. doi:10.2996/kmj/1499846596. https://projecteuclid.org/euclid.kmj/1499846596