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154 3. Representation Formulae

In comparison to equations (3.9.23) and (3.9.32), we see that A Ω and A Ω c
are the same for interior and exterior problems. Hence, we simply denote this
operator in the sequel by

A := A Ω = A Ω c = −RDPN 2 .

It is not difficult to see that under the same assumptions as for the interior
problem, the equivalence Theorem 3.9.1 remains valid for the exterior
c
problem (3.9.27), (3.9.28) with the radiation condition provided f ∈ C (Ω )
∞
n
and f has compact support in IR .
The following theorem shows the relation between the solutions of the ho-
mogeneous boundary value problems and the homogeneous boundary integral
equations of the first kind.
Theorem 3.9.2. Under the previous assumptions Γ ∈ C ∞ and a α ∈ C ,
∞
we have for the eigenspaces of the interior and exterior boundary value prob-
lems and for the eigenspace of the boundary integral operator A on Γ, respec-
tively, the relation:

kerA = X (3.9.34)
where

∞
kerA = {λ ∈ C (Γ) Aλ =0 on Γ}
and

∞
X = span {Sγu u ∈ C (Ω) ∧ Pu =0 in Ω and Rγu =0 on Γ}

c
∞
c
∪{Sγ c u c u c ∈ C (Ω ) ∧ Pu c =0 in Ω ,Rγ c u c =0 on Γ

and the radiation condition M(x; u c )=0}
Proof:
i.) Suppose λ ∈ kerA, i.e. Aλ =0 on Γ. Then define the potential
2m−1 2m− −1

u(x):= K p P p+ +1 (N 2 λ)
=0 p=0
where (ψ) denotes the –th component of the vector ψ.Now, u is a solution
of
c
Pu =0 in Ω and in Ω .
Then we find in Ω
Rγu = RQ Ω PN 2 λ = −Aλ =0 and
(3.9.35)
Sγu = SQ Ω PN 2 λ.