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2.1. Variational Formulation 49

∞
• C (Ω) is dense in V with respect to · 0. (2.9)
0
The first and second statements are obvious. The two others require a
certain technical effort. A more general statement will be formulated in
Theorem 3.7.
With that, we obtain from (2.5) the following result:
1 ¯
Lemma 2.1 Let u be a classical solution of (2.1), (2.2) and let u ∈ C (Ω).
Then

a(u, v)= f, v for all v ∈ V. (2.10)
0
Equation (2.10) is also called a variational equation.

∞
Proof: Let v ∈ V .Then v n ∈ C (Ω) exist with v n → v with respect
0
to · 0 and also to · a . Therefore, it follows from the continuity of the
bilinear form with respect to · a (see (A4.22)) and the continuity of the
functional defined by the right-hand side v → f, v 0 with respect to · 0
2
(because of the Cauchy–Schwarz inequality in L (Ω)) that
and a(u, v n ) → a(u, v) for n →∞ .
f, v n 0 → f, v 0
Since a(u, v n )= f, v n 0 ,we get a(u, v)= f, v 0 .

The space V in the identity (2.10) can be further enlarged as long as (2.8)
and (2.9) will remain valid. This fact will be used later to give a correct
definition.
Definition 2.2 A function u ∈ V is called a weak (or variational) solution
of (2.1), (2.2) if the following variational equation holds:
a(u, v)= f, v for all v ∈ V.
0
If u models e.g. the displacement of a membrane, this relation is called
the principle of virtual work.
Lemma 2.1 guarantees that a classical solution u is a weak solution.
The weak formulation has the following properties:
• It requires less smoothness: ∂ i u has to be only piecewise continuous.

• The validity of the boundary condition is guaranteed by the definition
of the function space V .

We now show that the variational equation (2.10) has exactly the same
solution(s) as a minimization problem:
Lemma 2.3 The variational equation (2.10) has the same solutions u ∈ V
as the minimization problem
F(v) → min for all v ∈ V, (2.11)