EQUATIONS OF ELASTIC HYPERSURFACES

92 SHELLS
∑
since n ν j D j = 0 on S .
j=1
Next we concentrate on the first integrand in (3.116): we employ the commutator identity
from Lemma 3.11.x plus that the fields U and V are tangential to write
∫ n∑
∫ [ n∑
n∑
−2 V k D j D k U j dS = −2 V k D k D j U j + V k [D j , D k ]U j
]dS
S
j,k=1
on S , because
∫
= −2
S
S
[
〈∇ S Div S U, V 〉 +
∫ [
= −2 〈∇ S Div S U, V 〉 +
S
n∑
ν j U j =
j=1
j,k=1
n∑
j,k,l=1
n∑
j,k,l=1
j,k=1
[
Vk ν j D k ν l − ν k V k D j ν l
]
Dl U j
]
dS
V k (D k ν l ) [ D l (ν j U j ) − (D l ν j )U j
] ] dS
∫ [
n∑
]
= −2 〈∇ S Div S U, V 〉 − (∂ k ν l )(∂ l ν j )U j V k dS
S
j,k,l=1
∫
∫
= −2 〈∇ S Div S U, V 〉 dS − 2 〈WS 2 U, V 〉 dS (3.117)
S
n∑
ν k V k = 0.
k=1
For the second integrand in (3.116) we use Lemma 3.3.i and that the field U is tangential:
n∑
∫
∫
n∑ [ ( ) ( ) ]
2 HS 0 ν j (D k U j )V k dS = 2 HS
0 V k Dk νj U j − Dk ν j Uj dS
j,k=1
S
S
j,k=1
At this point, we may therefore conclude that
∫ n∑
(D j U k + D k U j )(D j V k + D k V j ) dS
S
j,k=1
S
∫
= −2 HS 0 〈W S U, V 〉 dS. (3.118)
S
∫
= 2 〈−∆ S U − ∇ S Div S U + WS 2 U − HS 0 W S U, V 〉 dS. (3.119)
S
We now proceed to analyze the remaining terms in (3.115). More precisely, we still
have to take into account the terms containing either ∂ U (ν j ν k ) or ∂ V (ν j ν k ). We start with the
identity
n∑
n∑
n∑
(D k U j )D V (ν j ν k )= ν k (D k U j )D V ν j + (D V ν k ) [ ( ) ]
D k νj U j − Uj D k ν j
j,k=1
=−
j,k=1
j,k=1
n∑
(D V ν k )(D U ν k ) = −〈WS 2 U, V 〉, (3.120)
k,j=1