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2.4 The Biharmonic Equation 85

Some more explanations are needed here. In order to maintain consistency
with our notations for the Laplacian, we have adopted the notations V ij ,K ij
and D ij for the weakly and hypersingular boundary integral operators ac-
cording to our terminology. These boundary integral operators are obtained
by taking limits of the operations ∇ z (•)·n x ,M z ,N z , respectively on the cor-
responding potentials V and W as Ω
z → x ∈ Γ. As in the case of the
Laplacian, for any solution of (2.4.1), the Cauchy data (u, ∂u ,Mu,Nu) Γ on Γ
∂n
are reproduced by the matrix operators in (2.4.23), and C Ω is the Calder´on
projector corresponding to the bi–Laplacian. In the classical H¨older function
spaces, we have the following lemma.

Lemma 2.4.2. Let Γ ∈ C 2,α , 0 <α< 1.Then C Ω maps % 3 C 3+α−k (Γ)
k=0
into itself continuously. Moreover,
2
C = C Ω . (2.4.24)
Ω
As a consequence of this lemma, one finds the following specific identities:
1
2
V 12 D 21 + V 13 D 31 + V 14 D 41 = ( I − K ),
11
4
2
1
D 21 V 12 + V 23 D 32 + V 24 D 42 = ( I − K ),
22
4
1
2
D 31 V 13 + D 32 V 23 + V 34 D 43 = ( I − K ),
33
4
2
1
D 41 V 14 + D 42 V 24 + D 43 V 34 = ( I − K ).
44
4
Clearly, from (2.4.24) one finds 12 more identities between these operators.
In the same manner as in the case for the Laplacian, for any solution u
c
of (2.4.1) in Ω with p = 0, we may introduce the Calder´on projection C Ω c
for the exterior domain for the biharmonic equation. Then clearly, we have
C Ω c = I− C Ω ,
where I denotes the identity matrix operator. This relation then provides the
corresponding boundary integral equations for exterior boundary value prob-
lems. As will be seen, the boundary integral operators in C Ω are pseudodif-
ferential operators on Γ and their orders are summarized systematically in
the following:
⎛ ⎞
0 −1 −3 −3
+1 0 −1 −3
⎜ ⎟
Ord(C Ω ):= ⎜ ⎟ (2.4.25)
⎝ +1 +1 0 −1 ⎠
+3 +1 +1 0
The orders of these operators can be calculated from their symbols and
provide the mapping properties in the Sobolev spaces to be discussed in
Chapter 10.
2.4.2 Boundary Value Problems and Boundary Integral Equations
We begin with the boundary integral equations for the Dirichlet problems.
For the integral equations of the first kind we employ the second and the