2.1. NOTATION 7
and recall that one can identify
D0,2(E) 1
= {u
D1,2(E)

L6(E)
| u|∂E = 0}.
Furthermore, we set
Wloc
m,q(E)
:= {u Lloc(E)
1
| ∀R 0 : u W
m,q(ER)}.
We recall, see for example [17, Chapter II.5], that
Dm,q(E)
Wloc
m,q(E).
For 1 t
3
2
we introduce the spaces
(2.5)
D2,t(E)
:= {u
D2,t(E)
| u
3t
3−2t
+ |u|1,
3t
3−t
∞},
D1,t(E)
:= {u
D1,t(E)
| u
3t
3−t
∞}.
Finally, we shall need the space
W2,p(Ω)
:= {u W
2,p(Ω)3
|
Ω
u dx = 0 and
Ω
(
∇u
∇uT
)
dx = 0}
of proper deformation vector fields of Ω.
For general properties of the homogeneous Sobolev spaces we refer the reader
to [17]. Here, we just recall the Sobolev inequality:
(2.6) ∀u
D1,2(E)

L6(E)
: u
6

2

3
|u|1,2.
Concerning classical Sobolev spaces, we recall the boundedness of the trace opera-
tor, 1 p ∞,
(2.7) TrR : W
1,p(ER)
W
1−1/p,p(∂Ω),
and that the operator norm of TrR is independent of R. Moreover, when p 3
the Sobolev space W
1,p(Ω)
possesses an algebraic structure, see for example [1,
Theorem 5.25], and we have
(2.8) ∀u, v W
1,p(Ω)
: uv
1,p
C1 u
1,p
v 1,p.
One can further show for 1 s p and p 3 that
(2.9) ∀(u, v) W
1,p(ER)
× W
1,s(ER)
: uv
1,s,ER
C2 u
1,p,ER
v
1,s,ER
,
with C2 = C2(R). The algebraic property also holds for unbounded domains, in
particular we have
(2.10) ∀u, v W
1,p(R3)
: uv
1,p
C3 u
1,p
v
1,p
for the whole space
R3.
We use X Y to denote a continuous embedding of a Banach space X into a
Banach space Y . We use xj x to denote convergence of a sequence {xj}j=1

X
with respect to the weak topology of X.
We will make use of the Landau symbols, that is, the Big-O and Small-o
notation. Recall that f(x) = O(g(x)) iff |f(x)| C|g(x)| as |x| ∞, and
f(x) = o(g(x)) iff |f(x)|/|g(x)| 0 as |x| ∞.
We use small letters (c0,c1,...) to denote constants appearing only within a
single proof, and capital letters (C0,C1,...) to denote constants appearing globally.
For the sake of convenience, we shall fix at this point a number of constants.
Throughout the paper, the real number p is used as a Sobolev space exponent, and
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