EQUATIONS OF ELASTIC HYPERSURFACES

144 SHELLS
Remark 4.43 The operators
V (0,µ) (t, D) : X θ−m µ
(Γ) × · · · × X θ−m 2µ−1
(Γ)
−→ X θ−m 0
(Γ) × · · · × X θ−m µ−1
(Γ) (4.193)
V (µ,0) (t, D) : X θ−m 0
(Γ) × · · · × X θ−m µ−1
(Γ)
−→ X θ−mµ (Γ) × · · · × X θ−m 2µ−1
(Γ) (4.194)
are continuous for θ ∈ R, since m j + γ j = 2µ − 1 (cf. (4.180)) and, therefore, (cf. (4.16))
ord V jk (t, D) = m j + γ k − 2µ + 1 = m j − m k , j, k = 0, . . . , 2µ − 1 . (4.195)
Similarly, the operators
V (0,0) (t, D) : X θ−m 0
(Γ) × · · · × X θ−m µ−1
(Γ)
−→ X θ−m 0
(Γ) × · · · × X θ−m µ−1
(Γ) (4.196)
V (µ,µ) (t, D) : X θ−mµ (Γ) × · · · × X θ−m 2µ−1
(Γ)
are continuous for θ ∈ R.
−→ X θ−mµ (Γ) × · · · × X θ−m 2µ−1
(Γ) (4.197)
Lemma 4.44 Let the basic operator A(t, D) in (4.2) be defined on the entire closed hypersurface
S = C − ∪ Γ ∪ C + , where C = C + , C c = C − are complemented open surfaces
with the common boundary Γ = ∂C −+ . Let the kernels Ker −
+A of the BVP (4.1) with the
constraints (4.141) (equivalently-of the map (4.142)) for the complemented domains C −+ be
trivial Ker + −
A = {0} (contain only constants Ker −
+A = {const}).
Then the homogeneous pseudodifferential equations
(
Vr (t, D)ϕ ) −
(t) = 0 ,
(
Vr (t, D)ϕ ) +
(t) = 0 , ϕ ∈ W
θ
p (Γ) , t ∈ Γ
(4.198)
which are ΨDO of order γ r (cf. (4.180)) have only trivial solutions ϕ = 0 (respectively, have
only constant solutions ϕ = const) for all r = µ, . . . 2µ − 1.
Proof: Let ϕ be a non-trivial solution one of equation in (4.198). Then, due to (4.167),
A(t, D)V r ϕ(t) = 0 in both complemented surfaces C and in C c and
V jr (t, D)ϕ(t) = ( B j (t, D)V r ϕ(t) = 0 on Γ ∀ j = 0, . . . , µ − 1
since j < r (cf. (4.183a)). Therefore, ψ(t) is a non-trivial solution to the homogeneous BVP
(4.1) and a similar one for the complemented surface C c with the vanishing data f(t) ≡ 0
and g 0 (s) ≡ · · · ≡ g µ−1 (s) ≡ 0. Due to the assumption ψ(t) = V Γ ϕ(t) ≡ 0 in C −+ . Then,
due to (4.183b),
ϕ(t) = ( B j (t, D)V j ϕ ) +
(t) −
(
Bj (t, D)V j ϕ ) −
(s) ≡ 0 .