Weak Convergence Methods for Nonlinear Partial Differential ...

Proof. ”⇒”: clear by the preceding theorem.
”⇐”: By linearity ϕ(x n ) → ϕ(x) ∀ϕ ∈ span A. For ψ ∈ X ′ , ε > 0, choose
ϕ ∈ span A such that ‖ϕ − ψ‖ X ′ < ε. Then
lim sup |ψ(x n ) − ψ(x)|
n→∞
≤ lim sup(|ψ(x n ) − ϕ(x n )| + |ϕ(x n ) − ϕ(x)| + |ϕ(x) − ψ(x)|)
≤ ‖ψ − ϕ‖ X ′ ‖x
} {{ } n ‖ X + ‖ψ − ϕ‖
} {{ }
X ′ ‖x‖
} {{ } X
≤ε bd.
≤ε
≤ Cε.
The analogous statement for weak*-convergence is proved similarly.
□
2.2 Weak convergence in L p spaces
We consider L p = L p (Ω), Ω ⊂ R n open, with Lebesgue-measure. Then
∫ ∫
1 ≤ p < ∞ : f k ⇀ f iff f k g → fg ∀g ∈ L q 1
,
Ω Ω
p + 1 q = 1,
∫ ∫
p = ∞ : f ∗ k ⇀ f iff f k g → fg ∀g ∈ L 1 .
Ω
Notation: In the sequel we will sometimes just write “f k
(∗)
⇀ f” meaning “f k ⇀ f”
for p < ∞ and “f k ∗ ⇀ f” for p = ∞.
As span {χ A : A ⊂ Ω measurable} is dense in L q for all q ∈ [1, ∞], by Theorem
(∗)
2.8 we deduce that for a bounded sequence (f k ) we have f k ⇀ f iff
∫ ∫
f k → f ∀A ⊂ Ω measurable,
A
A
i.e. if “local averages” of the f k converge.
In fact, it will be sufficient to choose an even smaller class of test functions:
the characteristic functions of hypercubes.
Lemma 2.9 Let (f k ) ⊂ L p (Ω) be a bounded sequence. If 1 < p ≤ ∞, then
(∗)
f k ⇀ f iff ∫ ∫
f k → f ∀A = (0, a) n + b ⊂ Ω, a, b ∈ R n .
A
A
If p = 1, then f k ⇀ f iff (f k ) is equiintegrable and
∫ ∫
f k → f ∀A = (0, a) n + b ⊂ Ω, a, b ∈ R n .
A
A
7
Ω