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3.2. Elliptic Boundary Value Problems 111

3.2.2 An Example of a Boundary Value Problem of Fourth
Order

The Dirichlet problem for the biharmonic equation reads as follows:
1 ¯
4
Find u ∈ C (Ω) ∩ C (Ω) such that
$ 2
∆ u = f in Ω ,
(3.36)
∂ ν u = u =0 on ∂Ω ,
where
d
2 2
2
∆ u := ∆ (∆u)= ∂ i ∂ u .
j
i,j=1
2
In the case d = 1 this collapses to ∆ u = u (4) .
2
For u, v ∈ H (Ω) it follows from Corollary 3.9 that

(u ∆v − ∆uv) dx = {u∂ ν v − ∂ ν uv}dσ
Ω ∂Ω
4
2
and hence for u ∈ H (Ω),v ∈ H (Ω) (by replacing u with ∆u in the above
equation),

2
∆u ∆vdx = ∆ uv dx − ∂ ν ∆uv dσ + ∆u∂ ν vdσ.
Ω Ω ∂Ω ∂Ω
For a Lipschitz domain Ω we define

2
2
H (Ω) := v ∈ H (Ω) v = ∂ ν v =0 on ∂Ω

0
2
and obtain the variational formulation of (3.36) in the space V := H (Ω):
0
Find u ∈ V , such that

a(u, v):= ∆u ∆vdx = b(v):= fv dx for all v ∈ V.
Ω Ω
More general, for a boundary value problem of order 2m in conservative
m
m
form, we obtain a variational formulation in H (Ω) or H (Ω).
0
3.2.3 Regularity of Boundary Value Problems
In Section 3.2.1 we stated conditions under which the linear elliptic bound-
ary value problem admits a unique solution u (˜u, respectively) in some
1
subspace V of H (Ω). In many cases, for instance for the interpolation of
the solution or in the context of error estimates (also in norms other than
the · V norm)itis notsufficientthat u (˜u, respectively) have only first
2
weak derivatives in L (Ω).
Therefore, within the framework of the so-called regularity theory, the
question of the assumptions under which the weak solution belongs to
2
H (Ω), for instance, has to be answered. These additional conditions
contain conditions about