6 2. NOTATION AND PRELIMINARIES
for any differentiable vector field Φ :
R3

R3.
As a consequence,
(2.2) div(U Φ cof ∇Φ) = (det ∇Φ) (div U) Φ
holds for any differentiable U :
R3

R3×3.
For a differentiable vector field
u :
R3

R3
it holds that
div(cof
∇ΦT
u Φ) = (det ∇Φ) (div u) Φ. (2.3)
Moreover, we have the relation
(2.4) Φ =
1
|(cof ∇Φ) · n|
(cof ∇Φ) · n
between the outer normal on ∂Φ(Ω) and n on Ω.
For a domain D Rn we denote by Lq(D) the usual Lebesgue space, and by
·q,D the associated norm. We use W m,q(D) to denote the classical Sobolev space
equipped with the usual norm
u
m,q,D
:=
|α|≤m
D
|Dαu|q
dx
1/q
.
Here, α N0
3
denotes a multi-index,

:= ∂x1
α1
∂x2
α2
∂x3
α3
, and |α| = α1 + α2 + α3.
When no confusion can arise, we simply write ·q and ·m,q for the norms ·q,D
and ·m,q,D, respectively. Depending on the context, function spaces may consist
of tensor- and vector-valued functions, which we indicate by, for example,
Lq(D)3×3
and
Lq(D)3.
The notation
Cm(D)
is used to denote the space of functions u
Cm(D)
for
which Dαu is bounded and uniformly continuous for all |α| m. Cm(D) is a
Banach space when equipped with the norm
u
Cm(D)
:= max
|α|≤m
sup|Dαu(x)|.
x∈D
We use Cc
∞(D)
to denote the space of all smooth functions with compact support
in D. Moreover, we denote by
D(D) := Cc
(D)3
| div ϕ = 0}
the space of all smooth three-dimensional solenoidal (divergence free) vector fields
with compact support.
For a bounded domain D
R3
we let
L0(D)
q
:= {u
Lq(D)
|
D
u dx = 0}.
For the exterior domain E R3 we introduce the homogeneous Sobolev spaces
Dm,q(E)
:= {u Lloc(E)
1
|
Dαu

Lq(E),
|α| = m}
and associated semi-norms
|u|m,q :=
|α|=m
E
|Dαu|q
dx
1/q
.
We put
D0,2(E) 1
:= Cc ∞(E)
|·|1,2
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