On the higher eigenvalues for the \(\infty\)-eigenvalue problem
type: Item
instance of: scholarly article
title: On the higher eigenvalues for the \(\infty\)-eigenvalue problem
zbMATH Open document ID: 1080.35057
author: Petri Juutinen
author: Peter Lindqvist
published in: Calculus of Variations and Partial Differential Equations
publication date: 2005-05-12T00:00:00Z
review text: The authors consider a nonlinear eigenvalue problem associated with a limiting version of the \(p\)-Laplacian for \(p=\infty\). Namely, if \(\Omega\) is an open subset of \(\mathbb R^n\), \(S_{n\times n}\) is the set of \(n\times n\) real symmetric matrices with real entries, the authors consider the nonlinear problem \( F_{\Lambda}(u,Du,D^2u)=0\) in \(\Omega\), \(u_{| \partial\Omega}=0\) on \(\partial\Omega\), in the viscosity sense in the unknowns \(\Lambda\in \mathbb R\) (a \(\infty\)-eigenvalue) and \(u\in C^0(\overline{\Omega})\) (an eigenfunction), where \(F_{\Lambda}\) is the function of \(\mathbb R\times\mathbb R^n\times S_{n\times n}\) to \(\mathbb R\) defined by \(F(s,\xi,X)=\min\{| \xi| -\Lambda s, -X\xi\cdot\xi\}\) if \(s>0\), \(F(s,\xi,X)=-X\xi\cdot\xi\) if \(s=0\), \(F(s,\xi,X)=\max\{-\Lambda s-| \xi| ,-X\xi\cdot\xi\}\) if \(s<0\). The authors continue the analysis of their joint paper with \textit{J. J.~Manfredi} [Arch. Ration. Mech. Anal. 148, No. 2, 89--105 (1999; Zbl 0947.35104)] and show that the second \(\infty\)-eigenvalue has a very explicit geometric characterization, and that it is the limit of the \((1/p)\)-power of the second eigenvalue of the \(p\)-Laplacian as \(p\) tends to infinity and that it admits a variational characterization. Then the authors show that contrary to what happens for the \(p\)-Laplacian, the second \(\infty\)-eigenvalue may coincide with the first \(\infty\)-eigenvalue. Then they turn to analyze the higher eigenvalues and show that if \(\Omega\) is bounded, then the set of the \(\infty\)-eigenvalues is unbounded and closed. Moreover, the authors give a bound from above of the number of nodal domains for an \(\infty\)-eigenvalue and show the validity of certain properties which are known to hold for the \(2\)-Laplacian, i.e., the ordinary Laplacian, but which seem to be unknown for the \(p\)-Laplacian with \(p\in ]1,+\infty[\setminus\{2\}\). Namely, they show the validity of the Payne-Pólya-Weinberger conjecture conce…
reviewed by: Massimo Lanza de Cristoforis
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 49R50
Mathematics Subject Classification ID: 35P15
zbMATH DE Number: 2167754
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: \(\infty\)-Laplacian
zbMATH Keywords: eigenvalues
zbMATH Keywords: Payne-Pólya-Weinberger conjecture
zbMATH Keywords: Pólya conjecture
zbMATH Keywords: nodal domains
zbMATH Keywords: eigenfunctions
MaRDI profile type: Publication
full work available at URL: s00526-004-0295-4
OpenAlex ID: W1971327226
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DOI: 10.1007/S00526-004-0295-4
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description: scientific article; zbMATH DE number 2167754