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