8 2. NOTATION AND PRELIMINARIES
we fix 3 p ∞. In connection with the domain Ω, we have already fixed R0 0
so that Ω ⊂⊂ BR0 . We further fix the reals α0 and α1 so that α1 α0 R0.
2.2. Preliminaries
The deformation of an elastic body plays a central role in our analysis. If
the map Φ : Ω R3 is such a deformation, it will be crucial that we can obtain
a corresponding deformation of the exterior domain E := R3 \ Ω, that is, a map
Ψ : E R3 \ Φ(Ω). The following lemma yields such a mapping in the setting
of function spaces we shall be using. The lemma is not new, see for example [19,
Lemma 1], but due to the importance of the lemma in our analysis, we present here
a complete and elementary proof.
Lemma 2.1. There exists a bounded linear operator
F : W
2,p (Ω)3
W
2,p (R3)3
with the properties
F(u)|Ω = u|Ω, (2.11)
supp F(u) BR0 , (2.12)









∃K0 0 ∀u W
2,p(Ω)3
: u
2,p
K0
χu := Id +F(u) maps E
C1-diffeomorphically
onto
R3
\ [Id +u](Ω)
and det ∇χu(x)
1
2
for all x
R3.
(2.13)
Proof. Since Ω is a C2-smooth domain, there is a bounded linear extension
operator
Ext : W
2,p(Ω)
W
2,p(R3),
see for example [1, Theorem 4.26]. Let ψ Cc
∞(R3;
R) be a “cut-off” function with
ψ = 1 in Ω and ψ = 0 in
R3
\ BR0 . Define
F : W
2,p(Ω)3
W
2,p(R3)3,
F(u) := ψ ·
(
Ext(u1), Ext(u2), Ext(u3)
)
.
Then F is a bounded linear operator satisfying (2.11) and (2.12). In order to
show (2.13), we first recall that p 3 whence W 2,p(R3) is continuously embedded
in
C1(R3).
Employing this embedding, we immediately obtain that χu is a
C1-
mapping of
R3
into itself. Next, observe that
det ∇χu(x) = det
(
I + ∇F(u)(x)
)
= 1 + H
(
∇F(u)(x)
)
(2.14)
with H
(
∇F(u)(x)
)
a sum of 12 higher order terms with respect to the entries of
∇F(u)(x). Denote the Sobolev constant of the embedding W
1,p(R3)

C0(R3),
that is, the norm of the embedding, by C4. Recall (2.10) and choose
K0 := min{1/(24C4 F ), 1/( F C3)}.
Then u
2,p
K0 implies
H(∇F(u))

C4 H(∇F(u))
1,p,R3
12C4∇F(u)
1,p,R3

1
2
,
and consequently det ∇χu(x) 1/2 for all x
R3.
Clearly, lim|x|→∞|χu(x)| = ∞,
whence χu :
R3

R3
is proper, that is, χu
−1(K)
is compact for all compact
K
R3.
From the global inverse function theorem of Hadamard, see for example
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