Abstract
This chapter deals with the delicate question of when every function in a Sobolev space can be approximated by a more regular function, such as a smooth or Lipschitz continuous function. For the Lebesgue space, this question was solved in Theorem 3.4.12. An important fact is that log-Hölder continuity is sufficient for density of smooth functions. This is shown in Sect. 9.1. However, for the density question log-Hölder continuity is by no means necessary. Despite the contributions of many researchers, there remain substantial gaps in our understanding of this question. Indeed, it is fair to say that the results are in a transitory state and will hopefully be improved and unified in the future.
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© 2011 Springer-Verlag Berlin Heidelberg
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Diening, L., Harjulehto, P., Hästö, P., Růžička, M. (2011). Density of Regular Functions. In: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics(), vol 2017. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18363-8_9
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DOI: https://doi.org/10.1007/978-3-642-18363-8_9
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-18362-1
Online ISBN: 978-3-642-18363-8
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