Page 29 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

16 Preliminaries

(ii) If 0 ≤ α ≤ β ≤ 1 and k ≥ 0 is an integer, then

Ω ⊃ C
Ω .
Ω ⊃ C
Ω ⊃ C
C k ¡ ¢ k,α ¡ ¢ k,β ¡ ¢ k,1 ¡ ¢
(iii) If, in addition, Ω is bounded and convex, then
Ω ⊃ C
C k,1 ¡ ¢ k+1 ¡ ¢
Ω .
1.2.1 Exercises
Exercise 1.2.1 Show Proposition 1.10.

p
1.3 L spaces
n
Definition 1.11 Let Ω ⊂ R be an open set and 1 ≤ p ≤∞. We say that a
p
measurable function u : Ω → R belongs to L (Ω) if
⎧ ¶ 1
µZ
p
⎪ p
⎪
⎨ |u (x)| dx if 1 ≤ p< ∞
kuk L p = Ω
⎪
⎪
⎩
inf {α : |u (x)| ≤ α a.e. in Ω} if p = ∞
¡
N
1
i
p
is finite. As above if u : Ω → R , u = u , ..., u N ¢ ,is such that u ∈ L (Ω),for
every i =1, ..., N, we write u ∈ L p ¡ Ω; R N ¢ .
Remark 1.12 The abbreviation “a.e.” means that a property holds almost every-
where. For example, the function
⎧
⎨ 1 if x ∈ Q
χ (x)=
Q
⎩
0 if x/∈ Q
where Q is the set of rational numbers, is such that χ =0 a.e.
Q
In the next theorem we summarize the most important properties of L p
spaces that we will need. We however will not recall Fatou lemma, the dominated
convergence theorem and other basic theorems of Lebesgue integral.
n
Theorem 1.13 Let Ω ⊂ R be open and 1 ≤ p ≤∞.
p
(i) k·k L p is a norm and L (Ω), equippedwiththisnorm,isaBanachspace.
2
The space L (Ω) is a Hilbert space with scalar product given by
Z
hu; vi = u (x) v (x) dx .
Ω