2.2. PRELIMINARIES 9
[33, Corollary 4.3], it therefore follows that χu :
R3

R3
is a
C1-diffeomorphism.
Since χu(Ω) = [Id +u](Ω), we deduce that χu maps E
C1-diffeomorphically
onto
R3
\ [Id +u](Ω).
Lemma 2.2. Let K0 be the constant from Lemma 2.1. Let {un}n=1

W
2,p(Ω)3
with un
2,p
K0 and un u in W 2,p(Ω) as n ∞. Then χun χu in C1(R3)
as n ∞. Moreover, cof ∇χun cof ∇χu, (cof ∇χun )−1 (cof ∇χu)−1, and
∇χun −1 ∇χu −1 in C0(R3) as n ∞.
Proof. Since F from Lemma 2.1 is a bounded linear operator, it is weakly
continuous. Consequently F(un) F(u) in W 2,p(R3). Thus, χun χu in C1(R3)
simply follows from the compactness of the embedding W
2,p(R3)

C1(BR0
) and
the fact that χu = χun in
R3\BR0
. It now follows trivially that cof ∇χun cof ∇χu
in
C0(R3).
Finally, since
∇χun
−1
= (det ∇χun
)−1
cof(∇χun
)T
,
we also find, recalling the representation (2.14), ∇χun −1 ∇χu −1 and, subsequently,
(cof ∇χun )−1 (cof ∇χu)−1 in C0(R3) as n ∞.
Lemma 2.3. Let K0 be the constant from Lemma 2.1. Let u W 2,p(Ω)3 with
u
2,p
K0, R R0, and ϕ W0
1,p
(ER)3
with div
(
cof ∇χu
T
ϕ
)
= 0. Then there is
a sequence {ϕn}n=1

Cc
1(ER)3
with div(cof ∇χu
T
ϕn) = 0 and ϕn ϕ in W
1,p(ER)
as n ∞.
Proof. By Lemma 2.1 we see that χu : ER χu(ER) is a
C1-diffeomorphism.
Thus, χu(ER) is of class
C1.
Moreover, ϕ χu
−1
W0
1,p
(
χu(ER)
)3
. From the Piola
identity (2.3) it follows that div(ϕ χu −1) = 0. It is well-known, see for example
[17, Section III.4.1], that there exists a sequence {ηn}n=1 Cc
(
χu(ER)
)3
with
div ηn = 0 and ηn ϕ χu −1 in W 1,p
(
χu(ER)
)
as n ∞. The lemma now follows
with ϕn := ηn χu.
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