Page 27 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

14 Preliminaries

(v) C 0 ∞ (Ω)= D (Ω)= C ∞ (Ω) ∩ C 0 (Ω).

N
(vi) When dealing with maps u : Ω → R , we will write, for example,
¡ ¢
C k Ω; R N , and similarly for the other cases.
¡ ¢
Remark 1.5 C k Ω with its norm k·k C k isaBanachspace.
We will also need to define the set of piecewise continuous functions.

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Definition 1.6 Let Ω ⊂ R be an open set.
(i) Define C 0 ¡ ¢ ¡ ¢
Ω = C piec Ω to be the set of piecewise continuous func-
piec
tions u : Ω → R. This means that there exists a finite (or more generally a
countable) partition of Ω into open sets Ω i ⊂ Ω, i =1, ..., I,sothat
I
Ω = ∪ Ω i , Ω i ∩ Ω j = ∅, if i 6= j, 1 ≤ i, j ≤ I
i=1
and u| is continuous.
Ω i
k
(ii) Similarly C piec ¡ ¢ k−1 ¡ ¢
Ω ,whose
Ω , k ≥ 1,is the set of functions u ∈ C
¡ ¢
partial derivatives of order k are in C 0 Ω .
piec
We now turn to the notion of Hölder continuous functions.
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Definition 1.7 Let D ⊂ R , u : D → R and 0 <α ≤ 1.We let
½ ¾
|u (x) − u (y)|
[u] =sup .
C 0,α (D) α
x,y∈D |x − y|
x=y
6
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Let Ω ⊂ R be open, k ≥ 0 be an integer. We define the different spaces of
Hölder continuous functions in the following way.
(i) C 0,α (Ω) is the set of u ∈ C (Ω) so that
½ ¾
|u (x) − u (y)|
[u] =sup < ∞
C 0,α (K) α
x,y∈K |x − y|
x=y
6
for every compact set K ⊂ Ω.
¡ ¢ ¡ ¢
(ii) C 0,α Ω is the set of functions u ∈ C Ω so that
[u] C 0,α Ω) < ∞ .
(