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86 2. Boundary Integral Equations

first row of C Ω which leads to the following system for the interior Dirichlet
problem,
1
& '& ' & '& '
V 23 V 24 σ 1 −D 21 2 I − K 22 ϕ 1
Vσ := = =: f .
1 i
V 13 V 14 σ 2 I + K 11 −V 12 ϕ 2
2
(2.4.26)
The solution of the interior Dirichlet problem has associated Cauchy data σ
which satisfy the three compatibility conditions:

(σ 1 n + σ 2 x)ds x = 0 , − σ 2 ds =0 . (2.4.27)
Γ Γ
As we shall see in Chapter 10 , V is known as a strongly elliptic operator
for which the classical Fredholm alternative holds. Hence uniqueness will
α
imply the existence of exactly one solution σ ∈ C 1+α (Γ) × C (Γ).
For the exterior Dirichlet problem, by using C Ω c and the representation
(2.4.13) we obtain the system with integral equations of the first kind,

1
& '& '
D 21 2 I + K 22 ϕ 1
Vσ + Rω = − =: f ,
e
1
2 I − K 11 V 12 ϕ 2

(2.4.28)
(σ 1 n + σ 2 x)= A 1 , − σ 2 ds = A 0
Γ Γ
where

0 n 1 n 2 3

R(x)= − and ω =(ω 0 ,ω 1 ,ω 2 ) ∈ IR .
1 x 1 x 2
Lemma 2.4.3. The homogeneous system corresponding to (2.4.28) has only
3
α
the trivial solution in C 1+α (Γ) × C (Γ) × IR .
Proof: Let σ 0 , ω 0 be any solution of
∂v

Vσ 0 + Rω 0 =0 on Γ, ,v σ 0 ds = 0 for all v ∈ (2.4.29)
∂n
Γ
and consider the solution of (2.4.1),
◦ ◦ ◦ c
u 0 (x):= V σ 0 (x)+ p 0 (x) with p 0 (x)= ω 0 + ω 1 x 1 + ω 2 x 2 for x ∈ Ω .
Then A 0 =0 , A 1 = 0 because of (2.4.29) and (2.4.18), hence u 0 = O(|x|)at
c
infinity due to (2.4.13) which implies u 0 (x)= p 0 (x) for all x ∈ Ω ∪ Γ.On
the other hand, u 0 (x) is also a solution of (2.4.1) in Ω and is continuous
across Γ where u 0 | Γ = 0. Hence, due to Theorem 2.4.1, u 0 (x) = 0 for all x
±
±
in Ω. Consequently, Mu | Γ =0 and Nu | Γ = 0. Then the jump relations
0
0
corresponding to C Ω c −C Ω imply σ 0 =([Mu]| Γ , [Nu]| Γ ) = 0 on Γ and
+
◦
−
0= u | Γ = u | Γ = p 0 implies p 0 (x) = 0 for all x, i.e., ω =0.
0
0