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2.4 The Biharmonic Equation 81

where E(x, y) is the fundamental solution for the biharmonic equation
given by
1 2
E(x, y)= |x − y| log |x − y| (2.4.7)
8π
which satisfies
2
2
∆ E(x, y)= δ(x − y) in IR .
x
As in case of the Laplacian, we may rewrite u in the form
∂u

u(x)= V (Mu, Nu) − W u, (2.4.8)
∂n
where

∂E

V (Mu, Nu):= {E(x, y)Nu(y)+ (x, y) Mu(y)}ds y , (2.4.9)
∂n y
Γ
∂u
∂u
W u, := M y E(x, y) (y)+ N y E(x, y) u(y)}ds y (2.4.10)
∂n ∂n
Γ
are the simple and double layer potentials, respectively, and u| Γ , ∂u |
∂n Γ ,Mu| Γ
and Nu| Γ are the (modified) Cauchy data. This representation formula
(2.4.6) suggests two basic types of boundary conditions:
The Dirichlet boundary condition, where u| Γ and ∂u |
∂n Γ are prescribed
on Γ, and the Neumann boundary condition, where Mu| Γ and Nu| Γ
are prescribed on Γ. In thin plate theory, where u stands for the deflection
of the middle surface of the plate, the Dirichlet condition specifies the dis-
placement and the angle of rotation of the plate at the boundary, whereas
the Neumann condition provides the bending moment and shear force at
the boundary. Clearly, various linear combinations will lead to other mixed
boundary conditions, which will not be discussed here.
From the bilinear form (2.4.2), we see that
a(u, v)=0 for v ∈R := {v = c 1 x 1 + c 2 x 2 + c 3 | for all c 1 ,c 2 ,c 3 ∈ IR} .
(2.4.11)
This implies that the Neumann data need to satisfy the compatibility
condition
∂v

Mu + vNu ds = 0 for all v ∈R . (2.4.12)
∂n
Γ
We remark that looking at (2.4.2), one might think of choosing ∆u and
∂
− ∆u as the Neumann boundary conditions which correspond to the Pois-
∂n
son ratio ν = 1. This means that the compatibility condition (2.4.12) requires
that it should hold for all harmonic functions v. However, the space of har-
monic functions in Ω has infinite dimension, and this does not lead to a
regular boundary value problem in the sense of Agmon [2, p. 151].