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.
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Notes
- 1.
In this definition Ω may be \(\mathbb {R}^d\), then Ω ε = Ω for all ε.
- 2.
For simplicity here the norm on \(\mathbb {R}^d\) is the 1-norm, i.e.,
. - 3.
In contrast to the definition of the weak derivative, the test function φ is not in \(\mathcal D(\mathbb {R}^d_{+})\) but in \(\mathcal D(\mathbb {R}^d)\).
- 4.
- 5.
Of course this is only true for K ≥ 2, for K = 1 there is no integration over a second component y 2. Similarly in the following steps.
- 6.
The set (0, ∞)K + {y} has to be understood element-wise, i.e., it is defined as
, see also Footnote 3 on page 15.
.
, see also Footnote References
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.
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.
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.
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Wilbrandt, U. (2019). Properties of Sobolev Spaces. In: Stokes–Darcy Equations. Advances in Mathematical Fluid Mechanics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-02904-3_3
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DOI: https://doi.org/10.1007/978-3-030-02904-3_3
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