The first eigenvalue of the \(p\)-Laplacian on a compact Riemannian manifold.
type: Item
instance of: scholarly article
title: The first eigenvalue of the \(p\)-Laplacian on a compact Riemannian manifold.
zbMATH Open document ID: 1043.53031
DOI: 10.1016/S0362-546X(03)00209-8
author: Shigeo Kawai
author: Nobumitsu Nakauchi
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-11-16T00:00:00Z
review text: Let \(M\) be a compact Riemannian manifold of nonnegative Ricci curvature with or without boundary. In the paper under review, the authors give lower bound estimates of the first (nonlinear) eigenvalue \(\mu_{1,p}\), \(p\geq 2\), of the \(p\)-Laplacian on \(M\). More precisely, they prove that: \[ \mu_{1,p}\geq\frac{1}{p-1}\left(\frac{\pi}{4}\right)^{p} \frac{1}{d^{p}} \] where \(d\) denotes the diameter of \(M\). Moreover, if \(M\) has a boundary, they also prove similar estimates, in terms of the inscribed radius, for some nonlinear boundary value problem.
reviewed by: Salah Mehdi
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58C40
zbMATH DE Number: 2003968
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: nonlinear eigenvalue
zbMATH Keywords: estimate
Wikidata QID: Q115339190
MaRDI profile type: Publication
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full work available at URL: s0362-546x(03)00209-8
OpenAlex ID: W2021046950
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description: scientific article; zbMATH DE number 2003968