Page 44 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Sobolev spaces 31

Since a similar argument holds for h< 0, we deduce from (1.8) and (1.9) that,
if 1 <p ≤∞,
¯ ¯
Z b
¯ ¯
¯ ¯
u (x)[ϕ (x − h) − ϕ (x)] dx¯ ≤ c |h|kϕk L p 0 .
¯
¯ a ¯
Letting |h| tend to zero, we get
¯ ¯
Z b
¯ ¯
¯ 0 ¯ ∞
uϕ dx¯ ≤ c kϕk L p 0 , ∀ϕ ∈ C (a, b)
¯ 0
¯ a ¯
which is exactly (ii).
(i) ⇒ (iii). From Lemma 1.38 below, we have for every x ∈ ω
Z x+h Z 1
u (x + h) − u (x)= u (t) dt = h u (x + sh) ds
0
0
x 0
and hence
Z 1
0
|u (x + h) − u (x)| ≤ |h| |u (x + sh)| ds .
0
Let 1 <p< ∞ (the conclusion is obvious if p = ∞), we have from Hölder
inequality that
1
Z
p p p
0
|u (x + h) − u (x)| ≤ |h| |u (x + sh)| ds
0
and hence after integration
Z Z Z 1
p p p
0
|u (x + h) − u (x)| dx ≤ |h| |u (x + sh)| dsdx
ω ω 0
Z 1 Z
p p
0
= |h| |u (x + sh)| dxds .
0 ω
Since ω + sh ⊂ (a, b),we find
Z Z
p p 0 p
0
|u (x + sh)| dx = |u (y)| dy ≤ ku k L p
0
ω ω+sh
and hence
¶ 1
µZ
p
p
0
|u (x + h) − u (x)| dx ≤ ku k p |h|
L
ω
which is the claim.
In the proof of Theorem 1.36, we have used a result that, roughly speaking,
p
says that functions in W 1,p are continuous and are primitives of functions in L .