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94 2. Boundary Integral Equations

E(x, y)ψ(y)ds y − E(x, y)[ψ](y)ds y
y∈Γ y∈Γ c

1 c
= ϕ(x)+ Kϕ(x) − T E(x, y) [ϕ]ds y for x ∈ Γ,
2 y
y∈Γ c
(2.5.16)

− E(x, y)ψ(y)ds y + E(x, y)[ψ](y)ds y
y∈Γ Γ c

1 + − c
= − ϕ (x)+ ϕ (x) + T E(x, y) [ψ]ds y for x ∈ Γ c .
2 y
Γ c
α
α
This is a coupled system for ψ ∈ C (Γ)on Γ and [ψ] ∈ C (Γ c )on Γ c , which,
1
+
in fact, is uniquely solvable for any given triple (ϕ, ϕ , ϕ ) with the required
−
properties.
For the classical crack problem with Dirichlet conditions on Γ, e.g. as
α
α
−
ψ + ∈ C (Γ c )and ψ − ∈ C (Γ c ) are given with (ψ + − ψ )| γ = 0;the
desired fields u has to satisfy (2.5.8) and the boundary conditions
γ 0 u| Γ = ϕ on Γ + + − = ψ − (2.5.17)
c
c = ψ ,T u| Γ c on Γ c
and T u| Γ c
as well as the transmission conditions (2.5.10), (2.5.11).
Again from the representation formula (2.5.15) we now obtain the coupled
system

c
E(x, y)ψ(y)ds y + T E(x, y) [ϕ](y)ds y
y
y∈Γ y∈Γ c

1
= ϕ(x)+ Kϕ(x)+ E(x, y)[ψ]| Γ c (y)ds y for x ∈ Γ,
2
y∈Γ c

c (2.5.18)
D c [ϕ](x) − T E(x, y)ψ(y)ds y
x
y∈Γ
= 1 + − c )(x)
2 ψ (x)+ ψ (x) − K | ([ψ]| Γ c

− T x c T y E(x.y) ϕ(y)ds y for x ∈ Γ c
y∈Γ
α
α
for the unknowns ψ ∈ C (Γ) and [ϕ] ∈ C (Γ c ). As it turns out, this sys-
0
+
tem always has a unique solution for any given triple (ϕ, ψ , ψ ) with the
−
required properties.
The desired displacement field in Ω c is in both cases given by (2.5.15).