EQUATIONS OF ELASTIC HYPERSURFACES

142 SHELLS
Calderón-Zygmund singular integral operator and the integral in (4.179) is understood in the
Cauchy principal value sense:
∮
[
⊤
V jj (t, D)ϕ(t) := lim B j (t, D t ) C j (τ, D τ )KA ⊤ (t, τ)]
ϕ(τ) ds . (4.182)
ε→0
Γ\γ(t,ε)
Here γ(t, ε) := S n−2 (t, ε)∩Γ is the part of the boundary surface Γ inside the sphere S n−2 (t, ε)
with radius ε centered at t ∈ Γ. Then V j,j is continuous in the spaces X θ p and Z θ (S ) as stated
in (4.180) and in (4.181).
Theorem 4.40 Let the BVP (4.13) be formally adjoint to (4.1) and
suppose that the Green formula (4.15) holds. Then for the traces ( ) +
B j V
−
k
following Plemelji formulae:
we have the
(
Bj (t, D)V k (t, D)ϕ ) −
(t) =
(
Bj (t, D)V k (t, D)ϕ ) +
(t) = Vjk (t, D)ϕ(t) (4.183a)
for k ≠ j ,
(
Bj (t, D)V j (t, D)ϕ ) − + (t) = −
+ 1 2 ϕ(t) + V jj(t, D)ϕ(t) , t ∈ Γ , (4.183b)
k, j = 0, . . . , 2µ − 1 , ϕ ∈ H θ p(Γ) .
Proof: For the detailed proof we quote [Du2, § 6.4]. QED
Assumed that the conditions of Theorem 4.4 hold and the Green formula (4.15) is valid.
Let
{
X θ, − +
(Bj
p (A, B j , Γ) := ϕ ) }
− + : ϕ ∈ X θ+m j+ 1 p
p (C −+ ) , A(t, D)ϕ = 0 (4.184)
for j = 0, . . . , 2µ − 1 ,
θ ∈ R , 1 < p < ∞ , 1 ≤ q ≤ ∞, where u −+ denote the traces.
Theorem 4.41 The decompositions
X θ p(Γ) = X θ,−
p (A, B j , Γ) ⊕ X θ,+
p (A, B j , Γ)
X θ,−
p
(A, B j , Γ) ∩ X θ,+ (A, B j , Γ) = ∅
p
(4.185)
hold and the corresponding Calderón projections
are defined as follows
P −+ A,j : Xθ p(Γ) −→ X θ, − +
p (A, B j , Γ) (4.186)
P −+ A,j ϕ = − +( B j V j ϕ ) − + for j = 0, . . . , 2µ − 1 . (4.187)
A similar results hold for the Zygmund spaces Z θ (Γ) = W θ ∞(Γ),