Page 25 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

12 Preliminaries

to the subject. The monograph of Gilbarg-Trudinger [49] can also be of great
help. The book of Adams [1] is surely one of the most complete in this field, but
its reading is harder than the three others.
Finally in Section 1.5 we will gather some important properties of convex
functions such as, Jensen inequality, the Legendre transform and Carathéodory
theorem. The book of Rockafellar [87] is classical in this field. One can also
consult Hörmander [60] or Webster [96], see also [31].

1.2 Continuous and Hölder continuous functions

n
Definition 1.1 Let Ω ⊂ R be an open set and define

0
(i) C (Ω)= C (Ω) is the set of continuous functions u : Ω → R. Similarly
¡
N
we let C 0 ¡ Ω; R N ¢ = C Ω; R N ¢ be the set of continuous maps u : Ω → R .
¡ ¢ ¡ ¢
(ii) C 0 Ω = C Ω is the set of continuous functions u : Ω → R,which can
N
be continuously extended to Ω. When we are dealing with maps, u : Ω → R ,
¡ ¢ ¡ ¢
we will write, similarly as above, C 0 Ω; R N = C Ω; R N .
(iii) The support of a function u : Ω → R is defined as

supp u = {x ∈ Ω : u (x) 6=0} .

(iv) C 0 (Ω)= {u ∈ C (Ω) : supp u ⊂ Ω is compact}.

¡ ¢
(v) We define the norm over C Ω ,by
kuk 0 =sup |u (x)| .
C
x∈Ω
¡ ¢
Remark 1.2 C Ω equipped with the norm k·k C 0 is aBanachspace.
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Theorem 1.3 (Ascoli-Arzela Theorem) Let Ω ⊂ R be a bounded domain.
¡ ¢
Let K ⊂ C Ω be bounded and such that the following property of equicontinuity
holds: for every > 0 there exists δ> 0 so that
|x − y| <δ ⇒ |u (x) − u (y)| <ε, ∀x, y ∈ Ω and ∀u ∈ K,

then K is compact.

We will also use the following notations.