Page 32 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

p
L spaces 19

If 1 <p < ∞,we find

1
p
→ 0 in L ⇐⇒ 0 ≤ α<
u ν
p
1
p
0 in L
u ν ⇐⇒ 0 ≤ α ≤
p
(cf. Exercise 1.3.2).
Example 1.18 Let Ω =(0, 2π) and u ν (x)= sin νx,then
p
sin νx 9 0 in L , ∀ 1 ≤ p ≤∞
p
sin νx 0 in L , ∀ 1 ≤ p< ∞
and
∗
sin νx 0 in L ∞ .
These facts will be consequences of Riemann-Lebesgue Theorem (cf. Theorem
1.22).

Example 1.19 Let Ω =(0, 1), α, β ∈ R
⎧
⎨ α if x ∈ (0, 1/2)
u (x)=
⎩
β if x ∈ (1/2, 1) .
Extend u by periodicity from (0, 1) to R and define

u ν (x)= u (νx) .

Note that u ν takes only the values α and β and the sets where it takes such
values are, both, sets of measure 1/2. It is clear that {u ν } cannot be compact in
p
any L spaces; however from Riemann-Lebesgue Theorem (cf. Theorem 1.22),
we will find
α + β ∗ α + β
p
u ν in L , ∀ 1 ≤ p< ∞ and u ν in L .
∞
2 2
n
Theorem 1.20 Let Ω ⊂ R be a bounded open set. The following properties
then hold.
∗ p
(i) If u ν u in L ,then u ν u in L , ∀ 1 ≤ p< ∞.
∞
p
(ii) If u ν → u in L ,then ku ν k L p → kuk L p, 1 ≤ p ≤∞.