EQUATIONS OF ELASTIC HYPERSURFACES

74 SHELLS
Theorem 3.12 For any real-valued function ϕ ∈ C 1 (S ) and any 1 ≤ j < k ≤ n, there
hold:
∫
∮
[ ]
M jk ϕ dS = νj νΓ k − ν k ν j Γ ϕ ds , (3.30)
S
Γ
where ν Γ (ξ) = ( ν 1 Γ (ξ), . . . , νn Γ (ξ)) ⊤
is the unit tangential vector to S at the boundary point
ξ ∈ Γ := ∂S and outward (unit) normal vector to the boundary Γ = ∂S .
Proof: With formula (2.174) at hand the integrand in (3.30) can be represented as a total
differential
(M jk ϕ) dS = (∂ k ϕ)ω j
∣
∣S − (∂ j ϕ)ω k
∣
∣S = d [ ϕω jk
]∣
∣S .
Applying the Stoke’s formula (2.176) and formula (2.175) we get:
∫
∫
M jk ϕ dS = d [ ∫ ∫
]∣ ∣ [ ]
ϕω ∣S jk = ϕω ∣Γ jk = νj νΓ k − ν k ν j Γ ϕ ds
S
and (3.30) is proved.
The formal adjoint (in R n ) to P is defined by
S
Γ
Γ
P ∗ u = − ∑ j
∂ j a ⊤ j u + b ⊤ u (3.31)
If Ω ⊂ R n is a smooth, bounded domain, and if P is a first-order operator, weakly
tangential to ∂Ω, then, applying (2.121) (cf. § 2.5), P can be integrated by parts over Ω
without boundary terms, i.e.
∫
∫
(P u, v) Ω := 〈P u, v〉 dx = 〈u, P ∗ v〉 dx =: (u, P ∗ v) Ω (3.32)
Ω
Ω
for all vector-valued sections of vector fields
u, v ∈ C 1 (¯Ω).
For a weakly tangential differential operator Q on a closed hypersurface S let Q ∗ S
denote the “surface” adjoint:
∮
∮
(Q S u, v) S := 〈Qϕ, ψ〉 dS = 〈ϕ, Q ∗ S ψ〉 dS = (u, Q ∗ S v) S (3.33)
S
S
for all vector-valued sections of vector fields ϕ, ψ ∈ C 1 (¯Ω).
Corollary 3.13 The surface-adjoint operator PS ∗ to the weakly tangential differential operator
P in (3.20) is equal to the formally adjoint one
n∑
PS ∗ ϕ = P ∗ ϕ = − ∂ j a ⊤ j u + b ⊤ u . (3.34)
j=1