EQUATIONS OF ELASTIC HYPERSURFACES

118 SHELLS
which are present in the second line of formula (4.74), have order less or equal µ. On the
other hand, h ∈ W j+1/p
p
(Γ) is arbitrary and from (4.75) by the duality we conclude the
′
existence of traces (∇ µ+j
ν Γ
ϕ) + ∈ W −j−1/p
p (Γ), k = 0, . . . , µ − 1. The first µ traces
(∇ j ν Γ
ϕ) + ∈ H µ−j−1/p
p (Γ) exist for all k = 0, . . . , µ − 1 . (4.77)
exist due to the Theorem 4.18 on traces and the assumption (4.73).
For a boundary operator B(t, D) we apply a representation (4.10):
γ−1
∑
B(t, D) = B j0 (t, ν Γ (t))∇ γ ν Γ
+ B k (t, D Γ )∇ k ν Γ
, (4.78)
where B k (t, D Γ ), k = 0, . . . , γ − 1, are the tangential operator to the boundary Γ of order
γ − k. Since the traces (∇ k ν Γ
ϕ) + ∈ W µ−k−1/p
p (Γ) exist for all k = 1, . . . , 2µ − 1 and
k=0
B k (t, D Γ )ϕ + = ( B k (t, D Γ )ϕ ) +
provided ( B k (t, D Γ )ϕ ) +
exists, the operators B(t, D) can be applied to ϕ ∈ W
µ
p (Γ) and the
trace on the boundary exists (B(t, D)ϕ) + ∈ W µ−γ−1/p
p (Γ).
−µ
Remark 4.21 The same Lemma 4.20 holds if we replace f ∈ p (C ) by f ∈ ˜W p (C ) and
ϕ ∈ H µ p(C ) by ϕ ∈ W µ p(C ); the conclusion then is like: ( B(t, D)ϕ ) + µ−γ−1/p ∈ W p (Γ).
˜H
−µ
p
Moreover, the condition f ∈ (C ) in the foregoing Lemma 4.20 can not be relaxed
(C ), because for some elliptic equations A(t, D)ϕ = f the set of solutions
up to f ∈ H −µ
p
for all f ∈ H −µ
p
only for k = 0, . . . , µ − 1.
˜H
−µ
(C ) covers the space H µ p(C ) completely and the traces ( ∇ k ν(t, D Γ )ϕ ) +
exist
4.4 ON PSEUDODIFFERENTIAL OPERATORS
A partial differential operator (abbreviation-PDO)
can also be written as follows
P(x, D x ) = ∑
|α|≤m
a α (x)D α x = ∑
|α|≤m
a α (x)(i∂) α (4.79)
P(x, D x ) = F −1
ξ↦→x P (x, ξ)F y↦→ξ , (4.80)
where
P (x, ξ) := ∑
a α (x)ξ α (4.81)
|α|≤m