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2.1. Variational Formulation 51

1
where F(v):= a(v, v) − b(v) .
2
Lemma 2.5 The weak solution according to (2.10) (or (2.11)) is unique.

Proof: Let u 1 ,u 2 be two weak solutions, i.e.,
a(u 1 ,v)= f, v ,
0
for all v ∈ V.
a(u 2 ,v)= f, v ,
0
By subtraction, it follows that
a(u 1 − u 2 ,v) = 0 for all v ∈ V.

Choosing v = u 1 − u 2 implies a(u 1 − u 2,u 1 − u 2 ) = 0 and consequently
u 1 = u 2 , because a is definite.
Remark 2.6 Lemma 2.5 is generally valid if a is a definite bilinear form
and b is a linear form.
So far, we have defined two different norms on V : · a and · 0 .The
difference between these norms is essential because they are not equivalent
on the vector space V defined by (2.7), and consequently, they generate
different convergence concepts, as will be shown by the following example:
Example 2.7 Let Ω = (0, 1), i.e.
1
a(u, v):= u v dx ,

0
and let v n :Ω → R for n ≥ 2 be defined by (cf. Figure 2.1)
 1
nx , for 0 ≤ x ≤ ,
 n

v n (x)= 1 , for 1 ≤ x ≤ 1 − 1 ,
n n

 1
n − nx , for 1 − ≤ x ≤ 1 .
n
1 v n

1 n-1
n n 1
Figure 2.1. The function v n.

Then
1
1/2
v n 0 ≤ 1 dx =1 ,
0