EQUATIONS OF ELASTIC HYPERSURFACES

3. CALCULUS OF DIFFERENTIAL OPERATORS ON HYPERSURFACES 91
Theorem 3.29 The following identities hold for the deformation tensor and the Lamé operator
on S :
Def S (U) := [ D jk (U) ] , n×n Djk (U) = 1 [
( ) ]
D k U j + D j U k + D U νj ν k , (3.111)
2
[Def S (U)] ⊤ = Def S (U) and Def S (U)ν = 0 , (3.112)
L S =µ π S ∇ ∗ S ∇ S + (λ + µ) ∇ S ∇ ∗ S − µ H 0
S W S
=−µ ∆ S − (λ + µ) π S ∇ S Div S − µ H 0
S W S . (3.113)
Proof: Given the local nature of the identities we seek to prove, it suffices to work locally,
in a small open subset O of S , where an orthonormal frame T 1 , . . . , T n−1 to TS has been
fixed. As before, we set T n := ν so that {T j } 1≤j≤n is an ortho-normal frame for R n , at points
in O.
For a tangential field U to S with supp U ⊂ O by the definition of the deformation
tensor we have:
〈Def S (U)V, W 〉 := Def S (U)(π S V, π S W ), ∀ V, W ∈ R n . (3.114)
The equalities (3.112) are easy to check and we proceed with the proof of (3.113). Applying
(3.68) and (2.225) we eventually obtain (3.111):
D jk (U)= 1 [ ] 1
Uj;k + U k;j =
2
2
= 1 2
[
D S k U j + D S j U k +
[
D k U j + D j U k + D U
(
νj ν k
) ] .
n∑
U r (D r ν k )ν j +
r=1
n∑ ]
U r (D r ν j )ν k
Now if V is also a smooth vector field, tangential to S , applying (3.111) we get
∫
∫
〈Def ∗ S Def S (U), V 〉 dS = 〈Def S (U), Def S (V )〉 dS
S
=
j,k=1
S
n∑
∫
1
[
][
]
D k U j + D j U k + D U (ν j ν k ) D k V j + D j V k + D V (ν j ν k ) dS . (3.115)
4 S
To proceed, we first consider
∫
n∑
S
j,k=1
∫
= 2
∫
(D j U k + D k U j )(D j V k + D k V j ) dS = 2
n∑
S
j,k=1
[
−V k D 2 j U k − V k D j D k U j − H 0
∫
= − 〈∆ S U, V 〉 dS − 2
S
∫
n∑
S
j,k=1
S
j,k=1
r=1
n∑
Dj ∗ (D j U k + D k U j )V k dS
]
S ν j (D j U k )V k − HS 0 ν j (D k U j )V k dS
[
V k D j D k U j + H 0
S ν j (D k U j )V k
]
dS , (3.116)