Inequalities for eigenvalues of a clamped plate problem

Abstract

In this paper we study eigenvalues of a clamped plate problem on compact domains in complete manifolds. For complete manifolds admitting special functions, we prove universal inequalities for eigenvalues of clamped plate problem independent of the domains of Payne-Pólya-Weinberger-Yang type. These manifolds include Hadamard manifolds with Ricci curvature bounded below, a class of warped product manifolds, the product of Euclidean spaces with any complete manifolds and manifolds admitting eigenmaps to a sphere. In the case of warped product manifolds, our result implies a universal inequality on hyperbolic space proved by Cheng-Yang. We also strengthen an inequality for eigenvalues of clamped plate problem on submanifolds in a Euclidean space obtained recently by Cheng, Ichikawa and Mametsuka.

Estimates of eigenvalues of a clamped problem

“…From (3.10), (3.8) and the Stokes' theorem, we have 2) for f A C y ðW; RÞ can be found in [22], and the proof there is di¤erent from ours.…”

“…that is, M is the hyperbolic space, the inequality (1.21) is the one of Wang and Xia [22] (see (1.9)).…”

Inequalities for eigenvalues of fourth-order elliptic operators in divergence form on complete Riemannian manifolds

“…Furthermore, very recently, Kristaly [18] handled the problem of clamped plates on Riemannian manifolds with negative curvature. For the eigenvalue problem of the clamped plate problem, some interesting inequalities have been established at works [5]- [7], [15], [16], [21], and [25], we present some of them in more detail below.…”

Inequalities for eigenvalues of the buckling problem of arbitrary order on bounded domains of

“…Universal inequalities of Payne–Pólya–Weinberger–Yang type for eigenvalues of Riemannian manifolds have been studied by many authors. One can find some of the interesting results about this topic, e.g., in [] and the references therein.…”