21,612 research outputs found
Bounds for eigenvalue ratios of the Laplacian
For a bounded domain with a piecewise smooth boundary in an
-dimensional Euclidean space , 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
Eigenvalues of the Laplacian on Riemannian manifolds
For a bounded domain with a piecewise smooth boundary in a complete
Riemannian manifold , we study eigenvalues of the Dirichlet eigenvalue
problem of the Laplacian. By making use of a fact that eigenfunctions form an
orthonormal basis of 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
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