EQUATIONS OF ELASTIC HYPERSURFACES

178 SHELLS
The Neumann problem (5.165), (5.20) has a solution only for those right-hand sides
h ∈ W s−1− 1 p
p (Γ) which are orthogonal to all killing vector fields:
∮
h(s)w(s)ds = 0 ∀w ∈ K (C ) . (5.166)
Γ
If condition (5.166) holds, the Neumann problem has a solution u 0 ∈ W s p(C ) and a
general solution reads u = u 0 + w.
Proof: First we prove the uniqueness. Taking g = 0 (h = 0) we should prove that the
corresponding homogeneous Dirichlet BVP has only a trivial solution u = 0 while solutions
w to the homogeneous Neumann BVP are the killing vectors D S (t, D) = 0 only.
In fact, due to Lemma 5.32, these solutions are infinitely smooth and by taking v = u =
w in the Green formula (5.161) we get
E ( w(t), w(t)) ≡ 0 ⇐⇒ D j,k (t) = 0 ∀j, k = 1, . . . , n ⇐⇒ w ∈ K (C ) .(5.167)
(cf. (5.152)). For the Neumann BVP (5.165) the result is proved. For the Dirichlet BVP
(5.164) we have in addition u(s) = 0 ∀s ∈ Γ and this yields u(t) = 0 ∀t ∈ C (see (5.153)).
For the proof of existence we recall that the BVPs (5.164) and (5.165), with conditions
(5.20), are Fredholm (see Theorem 4.34) and their kernels coincide with cokernels. Therefore,
by the part proved already, the Dirichlet BVP (5.164) is solvable uniquely, while for
solvability of the Neumann problem (5.165) there must hold the orthogonality condition
(5.166) for the data with the solution of the homogeneous equation (cf. (5.166)).
Let us choose the function B ∈ C ∞ (R n ) to be supported in the complemented domain
supp B ⊂ C c := S \C and consider the fundamental solution K A (t, t−τ) to the perturbed
elasticity operator A S (t, D)−BI (see Theorem 5.33). Any solution of BVPs (5.164), (5.20)
and (5.165), (5.20) is represented by the formulae (cf. (5.26))
u(t) = (N C f)(t) + (W Γ u + )(t) − (V Γ (T C (ν Γ , D) u) + )(t) , t ∈ C , (5.168)
where the corresponding potential operators are defined as follows (cf. (5.27))
∮
(N C f)(t) u(t) := K A (t, t − τ)f(τ) dS ,
∮
(W Γ v)(t) :=
C
[(T C (ν Γ (s), D s )K A )(t, s − t)] ⊤ v + (s) ds , (5.169)
Γ
∮
(V Γ v)(t) :=
K A (t, t − s) v + (s) ds , t ∈ C .
Γ
The potentials N C , W Γ and V Γ have the the same mapping properties and for them are
valid the Plemelji formulae as listed in Corollary 5.8.
The equivalent boundary ΨDOs to BVPs (5.164), (5.20) and (5.165), (5.20) are defined
by (5.33) and (5.35) with two important differences: