On the density of continuous functions in variable exponent Sobolev space
type: Item
instance of: scholarly article
title: On the density of continuous functions in variable exponent Sobolev space
zbMATH Open document ID: 1144.46031
DOI: 10.4171/RMI/492
published in: Revista Matemática Iberoamericana
publication date: 2007-08-07T00:00:00Z
full work available at URL: https://eudml.org/doc/43591
review text: Let \(W^{1,p(\cdot)}(\Omega)\) be the Sobolev space with variable exponent \(p(x)\), where \(\Omega\) is an open set in \(\mathbb{R}^n\). It is known that continuous functions are dense in such a space if either \(p(x)\) has some monotonicity properties or is log-continuous (satisfies the Dini-Lipshitz log-condition). The author gives some further results on the denseness of continuous functions. He proves two main statements. The first (Theorem 3.2) is a refinement of the above mentioned known facts. The second (Theorem 4.10) is based on a new method and is given in terms of the level sets of the variable exponent \(p(x)\). The main trick in the proof of the second result is the usage of convolutions on level sets. A lemma is proved stating that \(C(B^n)\cap W^{1,p(\cdot)}(B^n)\) is dense in \(W^{1,p(\cdot)}(B^n)\), where \(B^n\) is the unit ball, if \(p\) is a bounded function depending on \(|x|\), this statement being extended to the case of a class of domains in \(\mathbb{R}^n\). The author provides also some examples illustrating shortcomings and/or advantages of the obtained results on the density. The paper is concluded by some discussion of open questions on problems of density in Sobolev spaces with variable exponent. The author observes that when the paper was completed, two other papers on this topic appeared, by V. Zhikov and Fan, Wang and Zhao, the latter paper including a result of the type of Theorem 3.2.
reviewed by: S. G. Samko
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 26A15
zbMATH DE Number: 5177711
zbMATH Keywords: variable exponent
zbMATH Keywords: Lebesgue space
zbMATH Keywords: Sobolev space
zbMATH Keywords: density of continuous functions
author: Peter Hästö
MaRDI profile type: Publication
OpenAlex ID: W1997792773
description: scientific article; zbMATH DE number 5177711