1,373 research outputs found

    Eigenvalues of the Laplacian on Riemannian manifolds

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    For a bounded domain Ω\Omega with a piecewise smooth boundary in a complete Riemannian manifold MM, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of L2(Ω)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}.Comment: 17 page

    Bounds for eigenvalue ratios of the Laplacian

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    For a bounded domain Ω\Omega with a piecewise smooth boundary in an nn-dimensional Euclidean space Rn\mathbf{R}^{n}, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. First we give a general inequality for eigenvalues of the Laplacian. As an application, we study lower order eigenvalues of the Laplacian and derive the ratios of lower order eigenvalues of the Laplacian.Comment: 14 page
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