EQUATIONS OF ELASTIC HYPERSURFACES

3. CALCULUS OF DIFFERENTIAL OPERATORS ON HYPERSURFACES 97
for some constant M > 0 or, what is equivalent, the mapping
[ ∥∥U ∣
U ↦→ ∣L2 (S ) ∥ 2 + ∥ DefS (U) ∣ L2 (S ) ∥ 2] 1/2
(3.142)
is an equivalent norm on the space W 1 (S ).
Proof: The proof is essentially a modification of the one given by P. Ciarlet in [Ci3] for
curvilinear coordinates and covariant derivatives.
Consider the space
{
Ŵ 1 (S ) := U = ( ) }
⊤
U 1 . . . , U n : Uj , D jk (U) ∈ L 2 (S ) for all j, k = 1, . . . n (3.143)
endowed with the norm (cf. (3.141) and (3.142)):
∥ U
∣ ∣Ŵ 1 (S ) ∥ ∥ :=
[ ∥∥U ∣ ∣L2 (S ) ∥ ∥ 2 + ∥ ∥ DefS (U) ∣ ∣ L2 (S ) ∥ ∥ 2] 1/2
. (3.144)
The derivatives here are understood in the sense of distributions: D jk (U) ∈ L 2 (S ) means
that there exists a function in L 2 (S ) denoted by D jk (U) such that
∫ [
(D jk (U), V ) S := U j (X )Dk ∗V (X ) + U 1,k(X )Dj ∗V (X )
+
S
n∑
V (X )U m (X )Dm( ∗ νj (X )ν k (X ) )] dS ∀ ψ ∈ L 2 (S ) ,
m=1
(cf. (3.36) for the formal dual D ∗ m).
It is sufficient, obviously, to prove that the spaces W 1 (S ) and Ŵ1 (S ) are identical.
The inclusion W 1 (S ) ⊂ Ŵ1 (S ) is trivial and we concentrate on the proof of the inverse
inclusion Ŵ1 (S ) ⊂ W 1 (S ).
To this end take U ∈ Ŵ1 (S ) and note that the inclusions U ∈ L 2 (S ), Def S (U) ∈
L 2 (S ) (i.e., D jk U ∈ L 2 (S ) for all j, k = 1, . . . , n) imply
˜D jk (U) = 1 ]
[D k U j + D j U k = D jk (U) − 1 n∑ ( )
∂ r νj ν k Ur ∈ L 2 (S ) (3.145)
2
2
Then
r=1
for all j, k = 1, . . . , n .
D j U k ∈ W −1 (S ) for all j, k, m = 1, . . . , n ,
[
Dj , D k
]
Um =D j D k U m − D k D j U m =
n∑
(M jk ν r )D r U m ∈ W −1 (S ) (see Lemma 3.11.x)
r=1
for all j, k, m = 1, . . . , n ,
D j D k U m =D j ˜Dkm (U) + D k ˜Djm (U) − D m ˜Djk (U) − [ D j , D k
]
Um − [ D j , D m
]
U1,k
− [ D k , D m
]
Uj ∈ W −1 (S ) for all j, k, m = 1, . . . , n ,