Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds

1949/06/01 by S. Minakshisundaram, Å. Pleijel, Åke Pleijel · 750 citations
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Algebraic and Geometric Analysis #Boundary (topology) #Boundary value problem #Curvature #Differential operator #Eigenfunction #Eigenvalues and eigenvectors #Elasticity and Wave Propagation #Geometry #Laplace operator #Laplace's equation #Laplace–Beltrami operator #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Metric (unit) #Operator (biology) #Pseudo-Riemannian manifold #Pure mathematics #Real line #Ricci curvature #Riemannian manifold #p-Laplacian

paper · pdf · doi:10.4153/cjm-1949-021-5

published in Canadian Journal of Mathematics 1(3), 242-256 (Cambridge University Press)

openalex publication_date 1949/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

Let V be a connected, compact, differentiable Riemannian manifold. If V is not closed we denote its boundary by S . In terms of local coordinates ( x i ), i = 1, 2, … Ν, the line-element dr is given by where gik (x 1 , x 2 , … x N ) are the components of the metric tensor on V We denote by Δ the Beltrami-Laplace-Operator and we consider on V the differential equation (1) Δu + λu = 0.

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