Abstract
In this chapter most of the notation used in this monograph is introduced; in particular, Lipschitz domains on which the so-called Lebesgue and Sobolev spaces are defined, together with a few basic inequalities. Furthermore, the important theorem of Lax–Milgram is shown. To begin with, some definitions and results from functional analysis are stated.
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Notes
- 1.
Sometimes also called elliptic; if the space is not clear it is often explicitly mentioned as X-coercivity or X-ellipticity.
- 2.
Also called adjoint.
- 3.
Here and in what follows B(x 0, α) ⊂ X denotes the ball around x 0 with radius α, i.e.,
. Furthermore, for sets M, N ⊂ X and a real number a the product aM and the sum M + N have to be understood element-wise: 
, 
. - 4.
Note that
. - 5.
In this work it is always assumed that infima or suprema are taken with respect to a set for which the following expression is defined. In this case, this means x ∈ X and x≠0.
- 6.
In the case p = q = 2 the Hölder inequality is the Cauchy–Schwarz inequality (2.1).
- 7.
The derivative D α v is defined as
. - 8.
Some generalization on Rademacher’s theorem can be found in [EG92].
. Furthermore, for sets M, N ⊂ X and a real number a the product aM and the sum M + N have to be understood element-wise: 
, 
.
.
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Wilbrandt, U. (2019). Notation and Preliminary Results. In: Stokes–Darcy Equations. Advances in Mathematical Fluid Mechanics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-02904-3_2
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