Page 109 -

2.5 Remarks 93

−∆ u = 0 in Ω c , (2.5.8)
∗
α
in C (Ω c ∪ Γ) and up to Γ c from either side with possibly different
traces at Γ c ,
± ± ± ±
c
0
γ u| Γ c = ϕ and T u = ψ (2.5.9)
where we have the transmission properties
+ − + − α
0
0
0
[γ 0 u]| Γ c := (γ u − γ u)| Γ c =[ϕ]| Γ c =(ϕ − ϕ )| Γ c ∈ C (Γ c ) (2.5.10)
and
+ − + − α
c c 1
[T c u]| Γ c := (T u − T u)| Γ c =[ψ]| Γ c := (ψ − ψ )| Γ c ∈ C (Γ c ) (2.5.11)
with
α
+
α
C (Γ c ):= {v ∈ C (Γ c ) | (γ v − γ v)| γ = 0} , (2.5.12)
−
0 0 0
and
α
α − 1 2 ψ (x) | ψ ∈ C (Γ c )} .
C (Γ c ):= {ψ = {dist (x − γ)} 1 1 (2.5.13)
1
For the classical insertion problem with Dirichlet conditions γ 0 u = ϕ ∈
α
α
C (Γ)on Γ the functions ϕ ∈ C (Γ c ) are given. The unknown field u then
±
0
has to satisfy the boundary conditions
+
−
γ 0 u| Γ = ϕ on Γ and with (ϕ − ϕ )| γ = 0 ,
(2.5.14)
+ + − −
0 0
γ u| Γ c = ϕ and γ u| Γ c = ϕ on Γ c
as well as the transmission conditions (2.5.10) and (2.5.11).
By extending Γ c up to the boundary Γ ficticiously and applying the Green
formula to the two ficticiously separated subdomains of Ω one finds the rep-
resentation formula

u(x)= E(x, y)ψ(y)ds y − T y E(x, y) ϕ(y)ds y
Γ Γ
(2.5.15)

c
− E(x, y)[ψ]| Γ c (y)ds y + T E(x, y) [ϕ]| Γ c (y)ds y
y
y∈Γ c Γ c
for x ∈ Ω c .
By taking the traces of (2.5.15) at Γ and at Γ c one obtains the following
system of equations on Γ and on Γ c :