Spectrum of one-dimensional \(p\)-Laplacian with an indefinite integrable weight
type: Item
instance of: scholarly article
title: Spectrum of one-dimensional \(p\)-Laplacian with an indefinite integrable weight
zbMATH Open document ID: 1202.34147
author: Gang Meng
author: Ping Yan
author: Meirong Zhang
published in: Mediterranean Journal of Mathematics
publication date: 2010-06-29T00:00:00Z
review text: For \(1 < p < \infty\), the authors study the spectrum of the one-imensional \(p\)-Laplacian \[ (\varphi_p (x'))' + \mu \omega (t) \varphi (x) = 0\text{ on }[0,1] \] with Dirichlet boundary conditions \(x(0) = x(1) = 0\), where \(\varphi_p (x) := |x|^{p-2}x\) and \(\omega\) is a (generally) indefinite weight in \(L^\gamma([0,1], \mathbb{R})\) \((=: \mathcal{L}^\gamma)\) with \(1 \leq \gamma \leq \infty\). If \(\omega\) is positive, then it is well known that the spectrum consists of an increasing sequence of positive eigenvalues accumulating at \(\infty\). If \(\omega\) allows a sign change but is still strictly positive on a set of non-zero measure, then it is shown by means of Prüfer transformations that this property remains true for the non-negative part of the spectrum. The set \(W_+^\gamma\) of all such weights is an open subset of \(\mathcal{L}^\gamma\) with respect to the usual norm \(\|\cdot\|_\gamma\). The two main results of the paper deal with the dependence of the \(m\)th positive eigenvalue \(\mu_m^+(\omega) \; (m \in \mathbb{N})\) on the weight, i.e., with the mapping \(\omega \rightarrow \mu_m^+(\omega)\): 1. As a mapping \(W_+^\gamma \rightarrow (0,\infty)\), it is continuously differentiable with respect to \(\|\cdot\|_\gamma\) and the differential can be expressed in terms of eigenfunctions. 2. With respect to the weak topology in \(\mathcal{L}^\gamma\), the same mapping is continuous. Moreover, similar (but more complicated) results are also obtained for Neumann (instead of Dirichlet) boundary conditions. Finally, the results are applied to extremal problems of eigenvalues, i.e., to estimates of \(\inf \mu_m^+(\omega)\) and \(\sup \mu_m^+(\omega)\), where \(\omega\) varies over certain subsets of \(\mathcal{L}^\gamma\).
reviewed by: Andreas Fleige
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 47J10
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34B09
Mathematics Subject Classification ID: 34L05
zbMATH DE Number: 5728337
zbMATH Keywords: spectrum
zbMATH Keywords: eigenvalue
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: indefinite weight
zbMATH Keywords: continuity
zbMATH Keywords: weak topology
zbMATH Keywords: differentiability
MaRDI profile type: Publication
full work available at URL: s00009-010-0040-5
OpenAlex ID: W2020044911
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DOI: 10.1007/S00009-010-0040-5
description: scientific article; zbMATH DE number 5728337