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# Properties of Sobolev Spaces

Chapter

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## Abstract

This chapter collects a number of important properties of Sobolev spaces. Almost every claim is provided together with a proof. The main result is the density of \(\mathcal D(\overline \varOmega )\) in *W*^{s, p}(*Ω*) and proved in Theorem 3.4.5. The necessary tools to establish these proofs are introduced and intermediate results are presented in the following subsections. This entire chapter can be viewed as a preparation for subsequent ones on traces, Chap. 4, and to meaningfully define the weak forms of some partial differential equations in Chap. 5.

## References

- [AF03]Robert A. Adams and John J. F. Fournier.
*Sobolev spaces*, volume 140 of*Pure and Applied Mathematics (Amsterdam)*. Elsevier/Academic Press, Amsterdam, second edition, 2003.Google Scholar - [KO88]N. Kikuchi and J. T. Oden.
*Contact problems in elasticity: a study of variational inequalities and finite element methods*, volume 8 of*SIAM Studies in Applied Mathematics*. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988.Google Scholar - [MM07]Irina Mitrea and Marius Mitrea. The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains.
*Trans. Amer. Math. Soc.*, 359(9):4143–4182 (electronic), 2007.MathSciNetCrossRefGoogle Scholar

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