Eigenvalues of the Laplacian on Riemannian manifolds

Details Eigenvalues of the Laplacian on Riemannian manifolds Eigenvalues of the Laplacian on Riemannian manifolds

2011-04-26

arxiv.org/abs/1104.4847v1

Mathematics

Differential Geometry

17 pages

Scientific paper

For a bounded domain $\Omega$ with a piecewise smooth boundary in a complete Riemannian manifold $M$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of $L^2(\Omega)$ in place of the Rayleigh-Ritz formula, we obtain inequalities for eigenvalues of the Laplacian. In particular, for lower order eigenvalues, our results extend the results of Chen and Cheng \cite{CC}.

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