Eigenvalues of the Laplacian on Riemannian manifolds

2011/04/26 by Qing-Ming Cheng, Cheng, Qing-Ming, Xuerong Qi +1
Computer Science · Mathematics · #35P15 #58C40 #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics #math.DG #msc:35P15 #msc:58C40

paper · pdf · doi:10.48550/arxiv.1104.4847

17 pages

arxiv created 2011/04/26 · openalex publication_date 2011/04/26 · arxiv updated 2011/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a bounded domain Ω 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 L²(Ω) 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 \citeCC.

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