EQUATIONS OF ELASTIC HYPERSURFACES

166 SHELLS
respectively. The operator W ∗ 0(t, D) is indeed the adjoint to W 0 (s, D)). For these operators
we have the standard Plemelji formulae, similar to ones listed in Corollary 5.8 (cf. Theorem
4.40):
(
WΓ ϕ ) − + (s) = −
+ 1 2 ϕ(s) + W 0(s, D)ϕ(s) ,
(
TC (ν Γ , D)V Γ ϕ ) − + (s) = + −1 2 ϕ(s) + W∗ 0(s, D)ϕ(s) ,
(
VΓ ϕ ) −
(s) = (VΓ ϕ) + (s) = V −1 (s, D)ϕ(s) ,
(5.104)
(
TC (ν Γ , D)W Γ ϕ ) −
(s) = (TC (ν Γ , D)W Γ ϕ) + (s) = W +1 (s, D)ϕ(s) , s ∈ Γ .
Lemma 5.24 The homogeneous equation
V −1 (s, D)ϕ(s) = 0 , ϕ ∈ H θ p(Γ) , s ∈ Γ (5.105)
has only a trivial solution ϕ = 0 for arbitrary s > 0 and 1 < p < ∞.
Proof (cf. Lemma 4.44): Let ϕ be a non-trivial solution of (5.105). Then, due to (4.167) (also
see (5.38)), the function u := V Γ ϕ is a solution to the homogeneous equation ∆ C u(t) = 0 in
both open complemented surfaces t ∈ C and in t ∈ C c . Moreover, u −+ (s) = V −1 ϕ(s) = 0
on the boundary Γ. Therefore, u(t) is a non-trivial solution to the homogeneous Dirichlet
BVPs (5.16) and a similar one for the complemented surface C c with the vanishing data
f(t) ≡ 0 and g(s) ≡ 0. Due to Theorem 5.7 u(t) = V Γ ϕ 0 (t) ≡ 0 in C and in C c . Then due
to (5.31)
and the proof is completed.
Theorem 5.25 The operator
is self adjoint and positive definite
ϕ(s) = (D νΓ V Γ ϕ) − (s) − (D νΓ V Γ ϕ) + (s) ≡ 0
for some M > 0.
The operator (cf Corollary 5.8 for the spaces)
is invertible for arbitrary θ ∈ R and 1 < p < ∞.
−V −1 : H −1/2 (Γ) → H 1/2 (Γ) (5.106)
−(V −1 w, w) Γ ≥ M‖w ∣ ∣ H −1/2 (Γ)‖ 2 (5.107)
V −1 : X θ p(Γ) −→ X θ+1
p (Γ) , (5.108)
Proof: Let us insert ϕ = ψ := V Γ w, w ∈ H −1/2 (S ) into the first Green formula (5.19a) for
the surface S and into a similar one
(∆ C (t, D)ϕ, ψ) C c + (∇ C ϕ, ∇ C ψ) C c = (D νΓ ϕ − , ψ − ) Γ (5.109)