Differentiability of monotone Sobolev functions.
type: Item
instance of: scholarly article
title: Differentiability of monotone Sobolev functions.
zbMATH Open document ID: 1055.46021
author: Jani Onninen
published in: Real Analysis Exchange
publication date: 2003-01-05T00:00:00Z
review text: The author of this interesting paper studies the question of what are the minimal integrability conditions on the partial derivatives of a~homeomorphic Sobolev mapping from \(W^{1,1}\) that would guarantee differentiability almost everywhere. A~known sufficient condition, due to \textit{E. M. Stein} [Ann.\ Math. (2) 113, 383--385 (1981; Zbl 0531.46021)], is that the gradient \(| \nabla f| \) belongs to the Lorentz space \(L^{n,1}\). A still weaker sufficient condition, due to Väisälä, is that \(| \nabla f| \) is \(p\)-integrable with some \(p>n-1\) (with \(p=n-1\), this does not suffice when \(n>2\)). The author shows that a~natural sufficient condition is \(| \nabla f| \in L^{n-1,1}\), and that this condition implies differentiability almost everywhere for a~fairly large class of mappings. Moreover, he shows that \(| \nabla f| \in L^{n-1,q}\) with any \(q>1\) is no longer sufficient; in such cases, there always exists a~weakly monotone function \(u\) satisfying this condition but not differentiable anywhere.
reviewed by: Luboš Pick
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 26B35
zbMATH DE Number: 1848876
zbMATH Keywords: differentiability
zbMATH Keywords: weak monotonicity
zbMATH Keywords: Sobolev functions
zbMATH Keywords: Lorentz spaces
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1848876