Page 51 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

38 Preliminaries

Since u (a)= 0,wehave
¯Z ¯ Z
x b
¯ ¯
0
0
|u (x)| = |u (x) − u (a)| = ¯ u (t) dt ≤ |u (t)| dt = ku k L 1 .
0
¯
¯ ¯
a a
From this inequality we immediately get that (1.13) is true for p = ∞.When
p =1, we have after integration that
Z b
0
kuk 1 = |u (x)| dx ≤ (b − a) ku k 1 .
L L
a
So it remains to prove (1.13) when 1 <p< ∞. Applying Hölder inequality, we
obtain
à ! 1 à ! 1
Z Z
b p 0 b p 1
0 p
0
|u (x)| ≤ 1 p 0 |u | =(b − a) p 0 ku k
L p
a a
and hence
à ! 1
b p
Z
p
kuk L p = |u| dx
a
à ! 1
Z b p
p 0 p
0
≤ (b − a) p 0 ku k dx =(b − a) ku k L p .
L p
a
This concludes the proof of the theorem when n =1.
1.4.1 Exercises
n
Exercise 1.4.1 Let 1 ≤ p< ∞, R> 0 and B R = {x ∈ R : |x| <R}.Let for
f ∈ C ∞ (0, +∞) and for x ∈ B R
u (x)= f (|x|) .
p
(i) Show that u ∈ L (B R ) if and only if
R
Z
p
r n−1 |f (r)| dr < ∞ .
0
(ii) Assume that
£ n−1 ¤
lim r |f (r)| =0 .
r→0
p
Prove that u ∈ W 1,p (B R ) if and only if u ∈ L (B R ) and
Z R
p
0
r n−1 |f (r)| dr < ∞ .
0
(iii) Discuss all the cases of Example 1.33.