Page 28 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Continuous and Hölder continuous functions 15

It is equipped with the norm

kuk C 0,α Ω) = kuk C 0 Ω) +[u] C 0,α Ω) .
(
(
(
If there is no ambiguity we drop the dependence on the set Ω and write simply
kuk C 0,α = kuk C 0 +[u] C 0,α .
k
(iii) C k,α (Ω) is the set of u ∈ C (Ω) so that
a
[D u] < ∞
C 0,α (K)
for every compact set K ⊂ Ω and every a ∈ A k .

(iv) C k,α ¡ ¢ k ¡ ¢
Ω is the set of functions u ∈ C
Ω so that
a
[D u] C 0,α Ω) < ∞
(
for every multi-index a ∈ A k . It is equipped with the following norm
a
kuk C k,α = kuk C k +max [D u] C 0,α .
a∈A k
¡ ¢
k,α
Remark 1.8 (i) C Ω with its norm k·k C k,α is a Banach space.
k
(ii) By abuse of notations we write C (Ω)= C k,0 (Ω);or in other words,
the set of continuous functions is identified with the set of Hölder continuous
functions with exponent 0.
¡ ¢
(iii) Similarly when α =1, we see that C 0,1 Ω is in fact the set of Lipschitz
continuous functions,namely the set of functions u such that there exists a
constant γ> 0 so that
|u (x) − u (y)| ≤ γ |x − y| , ∀x, y ∈ Ω.
The best such constant is γ =[u] C 0,1.

α
Example 1.9 Let Ω =(0, 1) and u α (x)= x with α ∈ [0, 1].It is easy to see
that u α ∈ C 0,α ([0, 1]).Moreover, if 0 <α ≤ 1,then
½ α α ¾
|x − y |
[u α ] 0,α =sup =1 .
C α
x=y |x − y|
6
x,y∈[0,1]
n
Proposition 1.10 Let Ω ⊂ R be open and 0 ≤ α ≤ 1. The following properties
then hold.
¡ ¢ ¡ ¢
(i) If u, v ∈ C 0,α Ω then uv ∈ C 0,α Ω .