EQUATIONS OF ELASTIC HYPERSURFACES

102 SHELLS
In particular,
∮
∮
[L(t, D)ϕ] ⊤ ψdS =
ϕ ⊤ L ∗ (t, D)ψdS (4.6)
C
C
∑
if either L ∗ (t, D) is tangential to the boundary n νΓ kl k = 0, b = 0 on Γ, or C = S is a
closed hypersurface Γ = ∂C = ∅ (see Fig. 1).
∫
k=1
Proof: We apply the Gauß formulae (3.37) and obtain:
∫
[L(t, D)ϕ] ⊤ ψdS =
C
C
[ n∑
k=1
l k D k ϕ + bϕ] ⊤
ψdS
∮
=−
⊤ ∫
l k νΓϕ k + bϕ]
ψ ds −
[ n∑
]
ϕ ⊤ D k (l ⊤ k ψ) + b⊤ ψ dS
k=1
[ n∑
k=1
Γ
∮
=−
⊤ ∮
l k νΓϕ k + bϕ]
ψ ds −
C
ϕ ⊤ L ∗ (t, D)ψ dS
[ n∑
k=1
Γ
C
and (4.4) is proved.
Definition 4.2 The operator A(t, D) in (4.1) is called normal if
inf |detA 0 (s, ν Γ (s))| ̸= 0, s ∈ Γ , |ξ| = 1 , (4.7)
where A 0 (t, ξ) denotes the homogeneous principal symbol of A
A 0 (t, ξ) := ∑
a α (t)(−iξ) α , (t, ξ) ∈ T ∗ C . (4.8)
|α|=m
The normal derivatives ∇ k ν Γ
, k = 1, 2, . . ., where
∇ νΓ := 〈ν Γ , ∇〉 :=
n∑
ν j Γ D j =
j=1
n∑
ν j Γ ∂ j (4.9)
j=1
and ν Γ = (νΓ 1, . . . , νn Γ )⊤ is the unit outward normal vector to Γ, tangential to C , are all
normal ∇ k ν Γ
(ν Γ ) = 〈ν Γ , ν Γ 〉 k ≡ 1. The differential operator A(t, D) in (4.2) can be written
in the form
A(t, D) = ∑
|α|≤m
a α (t)D α = A 0 (t, ν Γ (t))∇ m ν Γ
+
m−1
∑
k=0
A m−k (t, D Γ )∇ k ν Γ
, (4.10)
D Γ := ( D Γ,1 , . . . , D Γ,1
) ⊤,
DΓ,k = D j − ν j Γ ∇ ν Γ
, j = 1, . . . , n ,