Abstract
We provide several inequalities between eigenvalues of some classical eigenvalue problems on compact Riemannian manifolds with C^2 boundary. A key tool in the proof is the generalized Rellich identity on a Riemannian manifold. Our results in particular extend some inequalities due to Kuttler and Sigillito from subsets of mathbb {R}^2 to the manifold setting.
Highlights
The objective of this manuscript is to establish several inequalities between eigenvalues of the classical eigenvalue problems mentioned below
The Dirichlet eigenvalues describe the fundamental modes of vibration of an idealized drum, and for n = 2, the Neumann eigenvalues appear naturally in the study of the vibrations of a free membrane; see e.g. [3,6]
We extend Kuttler–Sigillito’s results in two ways
Summary
The objective of this manuscript is to establish several inequalities between eigenvalues of the classical eigenvalue problems mentioned below. In the following theorem we provide several inequalities for eigenvalues of (1.2)–(1.5) on star shaped manifolds under the assumption of bounded sectional curvature. Apart from the Laplace and Hessian comparison theorems, and the variational characterization of the eigenvalues, the key tool in the proof of Theorems 1.3 and 1.4 is a generalization of the classical Rellich identity to the manifold setting. This is the content of the theorem. A special case of Theorem 3.1, called the generalized Pohozaev identity, was proved in [21,25] in order to get some spectral inequalities between the Steklov and Laplace eigenvalues.
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