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.
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Wang, Q., Xia, C. Inequalities for eigenvalues of a clamped plate problem. Calc. Var. 40, 273–289 (2011). https://doi.org/10.1007/s00526-010-0340-4
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DOI: https://doi.org/10.1007/s00526-010-0340-4