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2.5 Remarks 91

2.5 Remarks

Very often in applications, on different parts of the boundary, different bound-
ary conditions are required or, as in classical crack mechanics (Cruse [58]),
the boundaries are given as transmission conditions on some bounded mani-
fold, the crack surface, in the interior of the domain. A similar situation can
be found for screen problems (see also Costabel and Dauge [52] and Stephan
[293]).
As an example of mixed boundary conditions let us consider the Lam´e
system with given Dirichlet data on Γ D ⊂ Γ and given Neumann data on
Γ N ⊂ Γ where Γ = Γ D ∪ Γ N ∪ γ with the set of collision points γ of the two
boundary conditions (which might also be empty if Γ D and Γ N are separated
components of Γ) (see e.g. Fichera [76], [145], Kohr et al [164], Maz‘ya [202],
Stephan [295]) where meas (Γ D ) > 0.
For n = 2, where Γ is a closed curve, we assume that either γ = ∅ or
consists of finitely many points; for n = 3, the set γ is either empty or a
closed curve and as smooth as Γ. For the Lam´e system (2.2.1) the classical
mixed boundary value problem reads:
2
α
Find u ∈ C (Ω) ∩ C (Ω) , 0 <α< 1, satisfying
−∆ u = f in Ω with
∗
(2.5.1)
γ 0 u = ϕ D on Γ D and Tu = ψ N on Γ N .
For reformulating this problem with boundary integral equations we first ex-
tend ϕ D from Γ D and ψ N from Γ N onto the complete boundary Γ such that

ϕ D = ϕ| Γ D and ψ N = ψ| Γ N (2.5.2)
α
with ϕ ∈ C (Γ) , 0 <α< 1 and appropriate ψ. Then
γ 0 u = ϕ + ϕ ,Tu = ψ + ψ (2.5.3)

where now
α α
ϕ ∈ C (Γ N )= {ϕ ∈ C (Γ) | supp ϕ ⊂ Γ N } (2.5.4)
0
and ψ with supp ψ ⊂ Γ D are the yet unknown Cauchy data to be determined.

With (2.5.3), the representation formula (2.2.4) reads

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

+ E(x, y)ψ(y)ds y − T y E(x, y) ϕ(y)ds y (2.5.5)

Γ Γ

+ E(x, y)f(y)dy for x ∈ Ω.
Ω