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2.4 The Biharmonic Equation 87

As a consequence, both, interior and exterior Dirichlet problems lead to
the same uniquely solvable system (2.4.28) where only the right–hand sides
are different and, for the interior Dirichlet problem, ω =0.
Clearly, the solution of the Dirichlet problems can also be treated by using
the boundary integral equations of the second kind. To illustrate the idea we
∂n Γ = ϕ 2 .
consider again the interior Dirichlet problem where u| Γ = ϕ 1 and ∂u |
From the representation formula (2.4.6) we obtain the following system for
the unknown σ =(Mu, Nu) on Γ:
1
& '& ' & '& '
2 I + K 33 −V 34 σ 1 D 31 D 32 ϕ 1
1 = =: Dϕ . (2.4.30)
−D 43 I − K 44 σ 2 D 41 D 42 ϕ 2
2
This system (2.4.30) of integral equations has a unique solution. As we shall
see in Chapter 10, for 0 ≤ ν< 1 the Fredholm alternative is still valid for
α
these integral equations and σ ∈ C 1+α (Γ) × C (Γ). So, uniqueness implies
existence.
◦ α 1+α
Lemma 2.4.4. Let σ ∈ C (Γ) × C (Γ) be the solution of the homoge-
neous system
1
◦
◦
( I + K 33 )σ 1 − V 34 σ 2 =0
2
(2.4.31)
◦ 1 ◦
−D 34 σ 1 +( I − K 44 )σ 2 =0 on Γ.
2
◦
Then σ = 0.
Proof: For the proof we consider the simple layer potential
◦
u 0 (x)= V σ
which is a solution of (2.4.1) for x ∈ Γ. Then for x ∈ Ω we obtain with
(2.4.31):
− 1 ◦ ◦ ◦
Mu | Γ = ( I − K 33 )σ 1 + V 34 σ 2 = σ 1 ,
0
2
− 1 ◦ ◦ ◦
Nu | Γ = ( I + K 44 )σ 2 + D 43 σ 1 = σ 2 .
0
2
Then the Green formula (2.4.2) implies
◦ ∂v

◦
σ 1 + σ 2 v ds = 0 for all v ∈ . (2.4.32)
Γ ∂u
+
+
c
For x ∈ Ω , we find Mu | Γ =0 and Nu | Γ = 0 due to (2.4.31). Then
0 0
Theorem 2.4.1 implies with (2.4.32) that
c
u 0 (x)= p(x)for x ∈ Ω ∪ Γ with some p ∈ .
But u 0 (x) is continuously differentiable across Γ and satisfies (2.4.1) in Ω
− ∂p
−
with boundary conditions u | Γ = p| Γ and ∂u 0 | Γ = ∂n Γ . Hence, with
|
0
∂n
Theorem 2.4.1 we find