Principal eigenvalue of a very badly degenerate operator and applications
type: Item
instance of: scholarly article
title: Principal eigenvalue of a very badly degenerate operator and applications
zbMATH Open document ID: 1132.35066
author: Petri Juutinen
published in: Journal of Differential Equations
publication date: 2007-06-04T00:00:00Z
review text: The author considers the eigenvalue problem \[ -\Delta_{\infty}u=\lambda u\;{\text{in}} \;\Omega,\qquad u=0\;{\text{on}} \;\partial\Omega\,, \] where \(\Omega\) is a bounded open subset of \(\mathbb R^{n}\), \(u\) is a function of \(\Omega\) to \(\mathbb R\), \(\lambda\) plays the role of an eigenvalue, and \[ \Delta_{\infty}u\equiv \biggl(D^{2}u \frac{Du}{| Du| }\biggr) \frac{Du}{| Du| }. \] It is proved that the least eigenvalue is positive and is delivered by the formula \[ \lambda_{1}=\sup\{\lambda:\exists v>0\text{ in }\Omega,\text{ such that } -\Delta_{\infty}v\geq\lambda v\}. \] The author also proves that \(\lambda_{1}\) admits a positive eigenfunction and that it can be characterized as the supremum of the values \(\lambda\) for which \(\Delta_{\infty}+\lambda I\) satisfies the maximum principle. Then some applications are presented to the analysis of the problem \[ -\Delta_{\infty}\varphi= \lambda\varphi +f \quad \text{in }\Omega, \] with respect to unknown \(\varphi\) for some positive function \(f\) of \(\Omega\) to \(\mathbb R\).
reviewed by: Massimo Lanza de Cristoforis
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 47J10
zbMATH DE Number: 5161141
zbMATH Keywords: Infinity Laplacian
zbMATH Keywords: Principal eigenvalue
zbMATH Keywords: Maximum principle
MaRDI profile type: Publication
full work available at URL: j.jde.2007.01.020
OpenAlex ID: W1991369319
Wikidata QID: Q109406490
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DOI: 10.1016/J.JDE.2007.01.020
description: scientific article; zbMATH DE number 5161141