Asymptotic estimates for the principal eigenvalue of the Laplacian in a geodesic ball
type: Item
instance of: scholarly article
title: Asymptotic estimates for the principal eigenvalue of the Laplacian in a geodesic ball
zbMATH Open document ID: 0546.35049
DOI: 10.1007/BF01448378
author: G. del Grosso
author: Federico Marchetti
published in: Applied Mathematics and Optimization
publication date: 1983-01-01T00:00:00Z
review text: Let M be a compact n-dimensional Riemannian manifold. We consider a point O of M and denote by \(B_{\epsilon}\) the geodesic ball of radius \(\epsilon\) around O and \(M_{\epsilon}=M/\bar B_{\epsilon}\). We consider the Dirichlet problem \(\Delta u=du/dt\) in \(M_{\epsilon}\), \(u|_{dM_{\epsilon}}=0\), \(u|_{t=0}=1_{M_{\epsilon}}\). We denote by \(\lambda_{\epsilon}>0\) the principal eigenvalue of the Laplace-Beltrami operator on \(M_{\epsilon}\). The aim of the paper is to give an explicit asymptotic expansion for \(\lambda_{\epsilon}\) as \(\epsilon_ 0\downarrow\) giving upper and lower bounds. The final result will be that \(\lambda_{\epsilon}\sim C/\log (1/\epsilon)\), \(n=2,\lambda_{\epsilon}\sim Ce^{n-2}\), \(n\geq 2\).
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 37A30
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 3869850
zbMATH Keywords: compact Riemannian manifold
zbMATH Keywords: geodesic ball
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: principal eigenvalue
zbMATH Keywords: Laplace-Beltrami operator
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01448378
OpenAlex ID: W2061739993
description: scientific article; zbMATH DE number 3869850