EQUATIONS OF ELASTIC HYPERSURFACES

4. BOUNDARY VALUE PROBLEMS: GENERAL RESULTS 139
Note that layer potentials V j ψ, j = 1, . . . , 2µ − 1, ψ ∈ H θ p(Γ), θ ∈ R, 1 < p < ∞,
represent solutions of the homogeneous equation A(t, D)ψ ≡ 0. In fact, from (4.162) and
(4.166) follows that
A(t, D)V j ψ(t)= ∑ ∮
∂τ α A(t, D)K A (t, τ)c ⊤ jα (τ)ϕ(τ)ds
|α|≤γ j
Γ
= ∑
|α|≤γ j
∮
Γ
on Γ since ∂ α τ δ(t − τ) = 0 for all t ∉ Γ and all τ ∈ Γ.
∂τ α δ(t − τ)ψ(τ)c ⊤ jα (τ)ϕ(τ) ds ≡ 0 j = 1, . . . , 2µ − 1 (4.167)
The integrals in (4.164)-(4.167) are understood as functionals K A (t, ·), (∂t α K (β)
A
(t, ·)
etc.) with a parameter t ∈ S applied to the test function ϕ(τ) (to c ⊤ α (τ)ϕ(τ)).
Summing up (4.164) for the domains C −+ we get
u(t) = F A f(t) +
2µ−1
∑
j=0
[v](s) := v + (s) − v − (s) , s ∈ Γ ,
where f is the right-hand side in BVP (4.1) and
∫
F A ϕ(t) := N S −v(t) + N S +v(t) =
V j [B j u](t) , t ∈ S \ Γ = S + ∪ S − , (4.168)
S
K A (t, τ)ϕ(τ) dS (4.169)
is the Newton potential (the inverse, the fundamental solution) for A(t, D).
Pseudodifferential operators a(t, D) on the hypersurface S and b(x, D) on the Euclidean
space R n−1 are called locally quasi-equivalent at (t 0 , x 0 ), t 0 ∈ S , x 0 ∈ R n−1 if
inf
χ
‖χ [ κ −1
j,∗ a(·, D)κ j,∗ − b(·, D) ] ∣ ∣H θ p(R n )‖ = 0 (4.170)
for the appropriate local chart {κ j , X j , Y j }, x 0 ∈ X j , t 0 = κ j (x 0 ) (cf. Definition 4.23 for
κ j,∗ ). where the infimum is taken over the set of all smooth functions χ ∈ C0 ∞ (R n ) which are
equal to the identity, χ(t) ≡ 1, in some neighborhood of x 0 . Usually, the local equivalence
at (t 0 , x 0 ) is abbreviated as
a(t, D) (t 0,x 0 )
∼ b(x, D)
and we refer to [Du1] for elementary properties of the local quasi-equivalence.
Remark 4.35 The “basic” partial differential operator A(t, D) in (4.1), (4.1) is locally
quasi-equivalent
A(t, D) (t 0,0)
∼ A(t 0 , D) (4.171)
to the operator with constant “frozen” coefficients
⎡
⎤
A(t 0 , D) = [A jk (t 0 , D)] N×N
:= ⎣ ∑
a jkα (t 0 )∂ α ⎦ = ∑
a α (t 0 )∂ α , (4.172)
|α|≤m
N×N
|α|≤m
a α (t) := [a jkα (t)] N×N
, a α (t 0 ) = const .