Abstract
In this chapter we study fine properties of Sobolev functions. By definition, Sobolev functions are defined only up to Lebesgue measure zero and thus it is not always clear how to use their point-wise properties. We pick a good representative from every equivalence class of Sobolev functions and show that this representative, called quasicontinuous, has many good properties. Our main tools are the capacities studied in Chap. 10. Our results general- ize classical ones to the variable exponent case. In Sect. 11.1 we show that each Sobolev function has a quasicontinuous representatives under natural conditions on the exponent p and define a capacity based on quasicontinuous functions.
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© 2011 Springer-Verlag Berlin Heidelberg
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Diening, L., Harjulehto, P., Hästö, P., Růžička, M. (2011). Fine Properties of Sobolev 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_11
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DOI: https://doi.org/10.1007/978-3-642-18363-8_11
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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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