Page 358 -

342 6. Introduction to Pseudodifferential Operators

where T(y; D) denotes the difference of the differential operators, i.e.,

T(y; D)= a 0 (y, D) − a(x, D) .

Now, the inverse of a(x, D) can formally be written in the form of Neumann’s
series
(−1)

a −1 (x, D)= a (−1) (y; D)T(y; D) a 0 (y; D) .
0
≥0
This leads us to the following successively defined sequence of kernel functions
N (0) (x, y) := a (−1) (y; D)((δ(x − y)δ jk )) p×p , (6.2.57)
0
N ( +1) (x, y) := a (−1) (y; D)T(y; D)N ( ) (x, y)for =0, 1,... .
0
The Levi function of order µ to the system of differential equations (6.2.10)
assumes the form
µ
L (µ) (x, y):= N ( ) (x, y) . (6.2.58)
=0
In the same way as for the scalar case we may seek the solution u(x)=

u 1 (x),...,u p (x) to the system (6.2.10) in the form
p
(µ)
u k (x)= L (x, y)φ (y)dy for x ∈ Ω (6.2.59)
k
=1
Ω

where φ(x)= φ 1 (x),...,φ p (x) denotes an unknown vector–valued den-
sity. By applying the differential operator A to (6.2.59) we obtain a system
of domain integral equations of the second kind,
p
(µ+1)
f j (x)= φ j (x) − T jk (y, D)N (x, y)φ m (y)dy (6.2.60)
km
k=1
Ω
by using the relation

AL (µ) (x, y)= δ(x − y) − T(x, D)N (µ+1) (x, y) . (6.2.61)

The integral operator in (6.2.60) has the kernel

T(y, D)N (µ+1) (x, y)
λ
which belongs to the class C (Ω×Ω) with λ = µ+1−n−max k=1,...,p |s k |.For
sufficiently large and compact Ω, the integral equation (6.2.60) is a classical
Fredholm integral equation of the second kind with continuous kernel in Ω.