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Exercises 99

1
Here it is important that the components of K∇v belong to H (Ω), using
2
∞
the fact that for v ∈ L (Ω), k ∈ L (Ω),
2
kv ∈ L (Ω) and kv 0 ≤ k ∞ v 0 .
d
Theorem 3.10 Suppose Ω ⊂ R is a bounded Lipschitz domain.
If k> d/2,then
¯
k
H (Ω) ⊂ C(Ω) ,
and the embedding is continuous.

Proof: See, for example, [37].

For dimension d = 2 this requires k> 1, and for dimension d =3we
3
need k> . Therefore, in both cases k = 2 satisfies the assumption of the
2
above theorem.

Exercises

3.1 Prove the Lax–Milgram Theorem in the following way:

(a) Show, by using the Riesz representation theorem, the equivalence of
(3.5) with the operator equation

A¯u = f
for A ∈ L[V, V ]and f ∈ V .

(b) Show, for T ε ∈ L[V, V ], T ε v := v − ε(Av − f)and ε> 0, that for
some ε> 0, the operator T ε is a contraction on V . Then conclude
the assertion by Banach’s fixed-point theorem (in the Banach space
setting, cf. Remark 8.5).

1
3.2 Prove estimate (3.9) by showing that even for v ∈ H (a, b),
1/2
|v(x) − v(y)|≤ |v| 1 |x − y| for x, y ∈ (a, b) .
2
3.3 Suppose Ω ⊂ R is the open disk with radius 1 and centre 0. Prove

α 2 1
that for the function u(x):= ln |x| , x ∈ Ω \{0}, α ∈ (0, )we have

2
1
u ∈ H (Ω), but u cannot be extended continuously to x =0.
1
2
3.4 Suppose Ω ⊂ R is the open unit disk. Prove that each u ∈ H (Ω)
√
4
has a trace u| ∈ L 2 (∂Ω) satisfying u 0,∂Ω ≤ 8 u 1,Ω .
∂Ω