EQUATIONS OF ELASTIC HYPERSURFACES

134 SHELLS
I step: We apply a local principle (see e.g. [Du1]). It is rather straightforward to prove that
the operator A = A(ω, D) in (4.142) is locally quasi-equivalent to operators with constant
coefficients, frozen at ω 0 ∈ C = C ∪ Γ:
A(ω, D) ω 0
∼
{
Apr (ω 0 , ∇ ′ ) = ∑
|α|=m
a α (ω 0 )(∇ ′ ) α if ω 0 ∈ C ,
A (s0 )(∇ ′ ) if ω 0 = s 0 ∈ Γ ,
(
A pr ω0 , ∇ ′ ) : H µ (R n−1 ) −→ H −µ (R n−1 ) if ω ∈ C , (4.146)
(
A (s0 ) ∇ ′′ d ) ( (
= A (s0 ) ∇ ′′ , d ) (
, r + B 0,(s0 ) ∇ ′′ , d )
(
, . . . , r + B µ−1,(s0 ) ∇ ′′ , d ))
dt
dt
dt
dt
µ−1
∏
: H µ (R n−1
+ ) −→ H −µ (R n−1
+ ) × H µ−mj−1/2 (R n−2 ) if ω 0 = s 0 ∈ Γ , (4.147)
j=0
(
where A (s0 ) ∇ ′′ , d ) (
, B j,(s0 ) ∇ ′′ , d )
are the differential operators with the symbols
dt
dt
described in (4.125) and ∇ ′ = ( ∇ ′′ , ∂ n−1 ), ∇ ′′ = ( ∂ 1 , . . . , ∂ n−2 ) and (r + ϕ)(x 1 , . . . , x n−1 ) =
ϕ(x 1 , . . . , x n−2 , 0) is the trace operator, restricting function to the boundary ∂R n−1
+ = R n−2 .
According the local principle the operator A(ω, D) in (4.142) is Fredholm if and only if
the local representatives in (4.146) and (4.147) ( are locally invertible for all ω 0 ∈ C and all
s 0 ∈ Γ. Moreover, since A pr (ω 0 , ∇ ′ ), A (s0 ) ∇ ′′ , d )
(
and r + B j,(s0 ) ∇ ′′ , d )
are homogeneous,
they are locally invertible if and only if they are globally invertible (cf. [Du1,
dt
dt
DS1]).
Thus, we need to only establish the invertibility of operators in (4.147) for all values of
ω 0 ∈ C and s 0 ∈ Γ.
II step: Let F p (R n−1 ) := { F ϕ : ϕ ∈ L p (R n−1 ) } denote the Fourier transformed space
endowed with the naturally induced norm. Let further
F 〈µ〉
p (R n−1 ) := { 〈λ〉 µ ϕ(λ) : ϕ ∈ F p (R n−1 ) }
be the corresponding weighted space. Clearly,
F 〈µ〉
p (R n−1 ) = F H µ p(R n−1 )
and after the Fourier transform F ξ ′ →η ′ we get the operator
A pr (ω 0 , η ′ )I : F 〈µ〉
p (R n−2 ) −→ F 〈−µ〉
p (R n−2 ) (4.148)
equivalent to A pr (ω 0 , ∇ ′ ) in (4.147). Here A pr (ω 0 , η ′ )I is the multiplication operator by the
symbol. Since the symbol is elliptic, the multiplication operator is obviously invertible and
the case of operator A pr (ω 0 , ∇ ′ ), ω 0 ∈ C is completed.