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On the density of continuous functions in variable exponent Sobolev space. (English) Zbl 1144.46031
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.

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
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