8 2. NOTATION AND PRELIMINARIES we fix 3 p ∞. In connection with the domain Ω, we have already fixed R0 0 so that Ω ⊂⊂ BR 0 . 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) BR 0 , (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 \ BR 0 . 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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