Page 95 -

2.4 The Biharmonic Equation 79

The equations (2.3.53) or (2.3.54) are uniquely solvable for every c ∈
[0, ∞) and so, the general elastic solution u e for almost incompressible ma-
terial has the form
M

u e (x)= V e ψ(x) − W e u 0 (x)+ α j m j (x)
j=1
M
1 2c
= u st + V ∆ ψ − c(W ∆ u 0 + L 1 u 0 (x)+ α j m j (x)
1+ c µ
j=1
for x ∈ Ω with arbitrary α j ∈ IR and where u st is the solution of the Stokes
problem with given boundary tractions ψ,and L 1 is defined in (2.3.43). For
c → 0 we see that for the elastic Neumann problem
u e = u st + O(c)

up to rigid motions, i.e., a regular perturbation with respect to the Stokes
solution.

2.4 The Biharmonic Equation

In both problems, plane elasticity and plane Stokes flow, the systems of partial
differential equations can be reduced to a single scalar 4th–order equation,

2
2
c
∆ u =0 in Ω (or Ω ) ⊂ IR , (2.4.1)
kwown as the biharmonic equation. In the elasticity case, u is the Airy stress
function, whereas in the Stokes flow u is the stream function of the flow. The
Airy function W(x) is defined in terms of the stress tensor σ ij (u) for the
displacement field u as
2
2
2
∂ W ∂ W ∂ W
σ 11 (u)= 2 , σ 12 (u)= − , σ 22 (u)= 2 ,
∂x 2 ∂x 1 ∂x 2 ∂x 1
which satisfies the equilibrium equation divσ(u) = 0 automatically for any
smooth function W. Then from the stress-strain relation in the form of
Hooke’s law, it follows that
∆W = σ 11 (u)+ σ 22 (u)=2(λ + µ)divu ;

and thus, W satisfies (2.4.1) since ∆(div u)= 0 from the Lam´e system. On
the other hand, the stream function u is defined in terms of the velocity u in
the form
⊥
u =(∇u) .
Here ⊥ indicates the operation of rotating a vector counter–clockwise by
a right angle. From the definition, the continuity equation for the velocity