Page 99 -

2.4 The Biharmonic Equation 83

A 0 = − ψ 2 ds and A 1 = (ψ 1 n + ψ 2 x)ds . (2.4.18)
Γ Γ
As a consequence of Theorem 2.4.1 one has the useful identity of Gaussian
type,
⎧
p for x ∈ Ω,
∂p
⎨
−W p, = 1 p for x ∈ Γ, for any p ∈R . (2.4.19)
∂n 2 c
0 for x ∈ Ω ,
⎩
2.4.1 Calder´on’s Projector
(See also [144].) In order to obtain the boundary integral operators as x
approaches Γ, from the simple– and double–layer potentials in the represen-
tation formulae (2.4.8) and (2.4.14), we need explicit information concerning
the kernels of the potentials. A straightforward calculation gives

∂E

V (Mu, Nu)(x) = E(x, y)Nu(y)ds y + (x, y) Mu(y)ds y
∂n y
Γ Γ
1 2
= |x − y| log |x − y|Nu(y)ds y (2.4.20)
8π
Γ
1
+ n(y) · (y − x)(2 log |x − y| +1)Mu(y)ds y
8π
Γ
∂u
∂u(y)
W u, (x) = M y E(x, y) ds y + N y E(x, y) u(y)ds y
∂n ∂n
Γ Γ
1
= (2 log |x − y| +1)+ ν(2 log |x − y| +3)
8π
Γ
((y − x) · n(y)) 2 ∂u(y)
+2(1 − ν) ds y (2.4.21)
|x − y| 2 ∂n
1 ∂ 1
+ log( )
2π ∂n y |x − y|
Γ
1 d (x − y) · t(y)(x − y) · n(y)
− (1 − ν) u(y)ds y .
2 ds y |x − y| 2

This leads to the following 16 boundary integral operators.