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156 3. Representation Formulae

Find u satisfying the differential equation

Pu = f in Ω ∪ Ω c
and the transmission conditions

Rγu = ϕ − and [Rγu] Γ =(Rγ c u − Rγu)=[ϕ]on Γ (3.9.37)
n
where f ∈ C (IR ) ,ϕ , [ϕ] ∈ C (Γ) are given functions and where u
−
∞
∞
0
satisfies the radiation condition (3.6.15).
We begin with some properties of the modified Calder´on projectors. Ob-
viously, they enjoy the same properties as the ordinary Calder´on projectors,
namely
2
C = C Ω , C Ω + C Ω c = I, C 2 (3.9.38)
Ω c = C Ω c .
Ω

In terms of the boundary potentials and boundary operators R and S,the
explicit forms of C Ω and C Ω c are given by

ϕ RQ Ω PN 1 ,RQ Ω PN 2 ϕ
C Ω = − ,

λ SQ Ω PN 1 ,SQ Ω PN 2 λ
(3.9.39)

ϕ RQ Ω cPN 1 ,RQ Ω cPN 2 ϕ
C Ω c = .

λ SQ Ω cPN 1 ,SQ Ω cPN 2 λ
If we denote the jump of the boundary potentials u across Γ by
[QPN j ]= Q Ω cPN j −Q Ω PN j
then, as a consequence of (3.8.27), we have the following jump relations for
any ϕ, λ ∈ C (Γ) across Γ:
∞
[RQPN 1 ϕ]= ϕ, [RQPN 2 λ] = 0 ,
(3.9.40)
[SQPN 1 ϕ]= 0 , [SQPN 2 λ] = λ.
From the representation formulae (3.7.10), (3.7.11) and (3.7.13), (3.7.14)
we arrive at the representation formula for the transmission problem,

u(x) = E(x, y)f(y)dy + M(x; u)
IR n
2m−1 2m− −1
p
∂
+ E(x, y) P p+ +1 γ c u(y) − γ u(y) ds y
∂n y
=0 p=0
Γ

= E(x, y)f(y)dy + M(x; u) (3.9.41)
IR n
2m−1 2m− −1

+ K pP p+ +1{N 1[ϕ]+ N 2[λ]}
=0 p=0
c
for x ∈ Ω ∪ Ω ,x ∈ Γ.