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.