EQUATIONS OF ELASTIC HYPERSURFACES

4. BOUNDARY VALUE PROBLEMS: GENERAL RESULTS 101
4 BOUNDARY VALUE PROBLEMS: GENERAL RESULTS
4.1 THE GREEN FORMULA
Let S ⊂ R n be a closed smooth hypersurface of co-dimension 1 and C ⊂ S be a subsurface
with the smooth boundary Γ = ∂C (see Fig. 2). Consider the following boundary value
problem on the open hypersurface C
{
A(t, D)ϕ = f, on C ,
(B j (s, D)ϕ) + = g j , j = 0, . . . , µ − 1, on Γ = ∂C ,
(4.1)
where A(t, D) is the ”basic” tangential differential operator and B j (t, D) are the ”boundary”
tangential partial differential operators:
[ ]
∑
A(t, D)=[A jk (t 0 , D)] N×N
= a jkα (t 0 )D α = ∑ a α (t)D α ,
|α|≤m
|α|≤m
N×N
a α := [a jkα ] N×N
∈ C ∞ (C ) ,
[
∑
B j (t, D)=[B jjk (D)] N×N
= b jjkα D α =
|α|≤m j
]N×N
∑
(4.2)
b jα (t)D α ,
|α|≤m j
b jα := [b jjkα ] N×N
∈ C ∞ (C Γ ) ,
where C Γ ⊂ S denotes some neighborhood of Γ ⊂ C .
Lemma 4.1 For a strongly tangential differential operator of order one
L(D) :=
n∑
n∑
l k ∂ k + b = l k D k + b = L(t, D) , (4.3)
k=1
k=1
n∑
ν k l k ≡ 0 , l j , b ∈ (C 1 ) N×N (C )
k=1
(cf. (4.5)) the following rule for integration by parts is valid:
∮
∮
[L(t, D)ϕ] ⊤ ψ dS = −
⊤ ∮
l k νΓϕ k + bϕ]
ψ ds −
ϕ ⊤ L ∗ (t, D)ψ dS . (4.4)
[ n∑
k=1
C
Γ
C
Here
L ∗ (t, D)ψ :=
n∑
D k l ⊤ k ψ + b ⊤ ψ (4.5)
k=1
is the formal dual operator. ν Γ = (ν 1 Γ , . . . , νn Γ )⊤ is the unit outward normal vector to Γ at
the point s ∈ Γ, which is tangential to the hypersurface 〈ν(s), ν Γ (s)〉 = 0 for all s ∈ Γ.