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Properties of Sobolev Spaces

  • Ulrich Wilbrandt
Chapter
Part of the Advances in Mathematical Fluid Mechanics book series (AMFM)

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 Ws, 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

  1. [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
  2. [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
  3. [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

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ulrich Wilbrandt
    • 1
  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany

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