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82 2. Boundary Integral Equations

As for the exterior boundary value problems, in order to ensure the
c
uniqueness of the solution of (2.4.1) in Ω , we need to augment (2.4.1) with
an appropriate radiation condition (see (2.3.18)). We require that
A 1 · x

u(x)= A 0 r + r log r + O(r) as r = |x|→∞ (2.4.13)
|x|
for given constant A 0 and constant vector A 1 . Under the condition (2.4.9),
c
we then have the representation formula for the solution of (2.4.1) in Ω ,
∂u

u(x)= −V (Mu, Nu)+ W u, + p(x), (2.4.14)
∂n
where p ∈R is a polynomial of degree less than or equal to one.
Before we formulate the boundary integral equations we first summarize
some classical basic results.
Theorem 2.4.1. (Gakhov [90], Mikhlin [208, 209, 211] and Muskhelishvili
[223]). Let Γ ∈ C 2,α , 0 <α< 1.
i) Let
∂u 3+α 2+α

ϕ =(ϕ 1 ,ϕ 2 ) = u| Γ , | Γ ∈ C (Γ) × Γ (Γ)
∂n
4
be given. Then there exists a unique solution u ∈ C 3+α (Ω) ∩ C (Ω) of the
interior Dirichlet problem satisfying the Dirichlet conditions
∂u
u| Γ = ϕ 1 and | Γ = ϕ 2 . (2.4.15)
∂n
2
For given A 0 ∈ IR and A 1 ∈ IR there also exists a unique solution
4
c
c
u ∈ C 3+α (Ω ∪ Γ) ∩ C (Ω ) of the exterior Dirichlet problem which behaves
at infinity as in (2.4.13).
ii) For given
α

ψ =(ψ 1 ,ψ 2 ) ∈ C 1+α (Γ) × C (Γ)
satisfying the compatibility conditions (2.4.12), i.e.,
∂v

ψ 1 + ψ 2 v ds =0 for all v ∈R , (2.4.16)
∂n
Γ
the interior Neumann problem consisting of (2.4.1) and the Neumann condi-
tions
(2.4.17)
Mu| Γ = ψ 1 and Nu| Γ = ψ 2
4
has a solution u ∈ C 3+α (Ω) ∩ C (Ω) which is unique up to a linear function
p ∈R.
c
If, for the exterior Neumann problem in Ω , in addition to ψ the linear
c
function p ∈R is given, then it has a unique solution u ∈ C 3+α (Ω ∪ Γ) ∩
4
C (Ω) with the behaviour (2.4.13), (2.4.14) where