Page 31 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

18 Preliminaries

p
We now turn our attention to the notions of convergence in L spaces. The
natural notion, called strong convergence, is the one induced by the k·k L p norm.
We will often need a weaker notion of convergence known as weak convergence.
We now define these notions.
n
Definition 1.15 Let Ω ⊂ R be an open set and 1 ≤ p ≤∞.
p
(i) A sequence u ν is said to (strongly) converge to u if u ν ,u ∈ L and if

lim ku ν − uk L p =0 .
ν→∞
p
We will denote this convergence by: u ν → u in L .
(ii) If 1 ≤ p< ∞,we say that thesequence u ν weakly converges to u if u ν ,
p
u ∈ L and if
Z
0
p
lim [u ν (x) − u (x)] ϕ (x) dx =0, ∀ϕ ∈ L (Ω) .
ν→∞
Ω
p
This convergence will be denoted by: u ν u in L .
(iii) If p = ∞,the sequence u ν is said to weak ∗ converge to u if u ν ,u ∈ L ∞
and if
Z
1
lim [u ν (x) − u (x)] ϕ (x) dx =0, ∀ϕ ∈ L (Ω)
ν→∞
Ω
∗
and will be denoted by: u ν u in L .
∞
Remark 1.16 (i) We speak of weak ∗ convergence in L ∞ instead of weak con-
1
vergence, because as seen above the dual of L ∞ is strictly larger than L .For-
p
mally, however, weak convergence in L and weak ∗ convergence in L ∞ take the
same form.
(ii) The limit (weak or strong) is unique.
(iii) It is obvious that
⎧ p
⎨ u ν u in L if 1 ≤ p< ∞
p
u ν → u in L ⇒
⎩ ∗
u ν u in L ∞ if p = ∞ .
Example 1.17 Let Ω =(0, 1), α> 0 and
⎧ α
⎨ ν if x ∈ (0, 1/ν)
u ν (x)=
⎩
0 if x ∈ (1/ν, 1) .