On the Kuratowski measure of noncompactness for duality mappings
type: Item
instance of: scholarly article
title: On the Kuratowski measure of noncompactness for duality mappings
zbMATH Open document ID: 1290.47052
published in: Topological Methods in Nonlinear Analysis
publication date: 2013-04-26T00:00:00Z
review text: In a real Banach space \(X\) having a Fréchet differentiable norm, the duality mapping on \(X\) is considered, namely \(J_\varphi:X\rightarrow X^*\), with \(\varphi:\mathbb{R}_+\rightarrow \mathbb{R}_+\) being a gauge function. The main result of the paper is that, for the Kuratowski measure on noncompactness of \(J_\varphi\), one has \[ \alpha(J_\varphi)\geq \sup\left\{\frac{\varphi(r)}{r}: r>0\right\} \text{ if } \dim X=\infty\;, \] whereas \[ \alpha(J_\varphi)=0 \;\text{ if and only if } \dim X<\infty\;. \] \smallskip As an application, the minus \(p\)-Laplacian \(-\Delta_p:W^{1,p}_0(\Omega)\rightarrow W^{-1,p'}(\Omega)\), \(1<p<\infty\), \(1/p+1/p'=1\), is viewed as the duality mapping on \(W^{1,p}_0(\Omega)\) corresponding to the gauge function \(\varphi(t)=t^{p-1}\). It is deduced that the Kuratowski measure of noncompactness for the \(p\)-Laplacian satisfies \[ \alpha(-\Delta_p)=1 \text{ for } p=2 \] and \[ \alpha(-\Delta_p)=\infty \text{ for } p\in (1,2)\cup (2,\infty)\;. \]
reviewed by: Paola Rubbioni
Mathematics Subject Classification ID: 47H08
Mathematics Subject Classification ID: 35J70
zbMATH DE Number: 6158521
zbMATH Keywords: Kuratowski measure of noncompactness
zbMATH Keywords: smooth Banach spaces
zbMATH Keywords: duality mappings
zbMATH Keywords: \(p\)-Laplacian
author: George Dinca
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6158521