Abstract
One important property of certain Sobolev spaces is the fact that there is a well defined restriction onto the boundary, even though the boundary has measure zero. Such restrictions are known as traces and allow for prescribed boundary data of solutions of partial differential equations.
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Notes
- 1.
The index i in the definition of w n is omitted for better readability.
- 2.
As in Sect. 2.3 it is \((\cdot ,\cdot )_0 = (\cdot ,\cdot )_{L^2(\varOmega )}\) and \((\cdot ,\cdot )_{0,M} = (\cdot ,\cdot )_{L^2(M)}\) for sets M.
- 3.
A domain \(\varOmega \subset \mathbb {R}^d\) is called star-shaped with respect to B ⊂ Ω, if for all x ∈ Ω and all y ∈ B the connecting line segment
is a subset of Ω.
is a subset of Ω.References
Chérif Amrouche, Philippe G. Ciarlet, and Cristinel Mardare. Remarks on a lemma by Jacques-Louis Lions. C. R. Math. Acad. Sci. Paris, 352(9):691–695, 2014.
Chérif Amrouche, Philippe G. Ciarlet, and Cristinel Mardare. On a lemma of Jacques-Louis Lions and its relation to other fundamental results. J. Math. Pures Appl. (9), 104(2):207–226, 2015.
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Wilbrandt, U. (2019). Traces. In: Stokes–Darcy Equations. Advances in Mathematical Fluid Mechanics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-02904-3_4
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DOI: https://doi.org/10.1007/978-3-030-02904-3_4
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