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92 2. Boundary Integral Equations

α
As is well known, even if ψ ∈ C (Γ) then ψ will have singularities at γ which

1 α
2 ψ with ψ ∈ C (Γ D )
need to be taken into account either by {dist (x, γ)} −
1 1
or by adding singular functions at γ.
Taking the trace and the traction of (2.5.5) on Γ leads with (2.5.3) to the
system of boundary integral equations
1
V ψ(x) − K ϕ(x)= ϕ(x)+ Kϕ(x) − V ψ(x) − Nf(x) for x ∈ Γ D ,

2
1

K ψ(x)+ D ϕ(x)= ψ(x) − K ψ(x) − Dϕ(x) − T x Nf(x)for x ∈ Γ N .

2
(2.5.6)
As will be seen in Chapter 5, the system (2.5.6) is uniquely solvable for
α 1
ϕ ∈ C (Γ N )and ψ either in the space with the weight {dist (x, γ)} − 2 or
0

in an augmented space according to the asymptotic behaviour of the solu-
tion and involving the stress intensity factors (Stephan et al [297] provided
meas(Γ D ) > 0.
In a similar manner one might also use the system of integral equations
of the second kind
1 ϕ(x)+ K ϕ(x) − V ψ(x)= V ψ(x) − ϕ(x) − Kϕ(x)+ Nf(x)
1

2 2
for x ∈ Γ N ,
1

ψ(x) − K ψ(x) − D ϕ(x)= − ψ(x)+ K ψ(x)+ Dϕ(x)+ T x Nf(x)
1

2 2
for x ∈ Γ D . (2.5.7)
For the Laplacian and the Helmholtz equation and mixed boundary value
problems as well as for the Stokes system one may proceed in the same man-
ner. As will be seen in Chapter 5, the variational formulation for the mixed
boundary conditions provides us with the right analytical tools for show-
ing the well–posedness of the formulation (2.5.6) (see e.g., Kohr et al [164],
Sauter and Schwab [266] and Steinbach [290]). In the engineering literature,
usually the system (2.5.7) is used for discretization and then the equations
corresponding to (2.5.7) are obtained by assembling the discrete given and
unknown Cauchy data appropriately (see e.g., Bonnet [18], Brebbia et al
[23, 24] and Gaul et al [94]).
For crack and insertion problems let us again consider just the example
of classical linear theory without volume forces. Let us consider a bounded
n
open domain Ω ⊂ IR with n = 2 or 3 enclosing a given bounded crack
α
or insertion surface as an oriented piece of a curve Γ c ∈ C ,if n =2 or,
α
if n = 3, as an open piece of an oriented surface Γ c ∈ C , with a simple,
α
closed boundary curve ∂Γ c = γ ∈ C . Further the crack should not reach
the boundary ∂Ω = Γ of Ω, i.e., Γ c ⊂ Ω. The annulus Ω c := Ω \ F c is not a
Lipschitz domain anymore but if we distinguish the two sides of Γ c assigning
with + the points near Γ c on the side of the normal vector n c given due to
the orientation of Γ c and the points of the opposite side with −, the traces
from either side are still defined. For the crack or insertion problem, an elastic
2
field u ∈ C (Ω c ) is sought which satisfies the homogeneous Lam´esystem