28,152 research outputs found

    Eigenvalues of the Laplacian on Riemannian manifolds

    Full text link
    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