Weak Convergence Methods for Nonlinear Partial Differential ...

Finally sending n → ∞, by monotone convergence we find
∫
lim inf F(u k) ≥ f(u) = F(u).
k→∞
As L 1 is not reflexive, bounded sequences need not admit weakly convergent
subsequences. This can be seen already in the following simple one-dimensional
situation.
Example: The sequence nχ (0,
1
n ) ⊂ L1 is bounded, but does not admit a convergent
subsequence. (If so, then nχ n ⇀ 0, but ∫ nχ n ≡ 1 ↛ 0.)
This lack of compactness can be cured by embedding L 1 in a larger space of
measures: Naturally L 1 (Ω) ⊂ M(Ω), the set of finite (signed) Radon measures.
By the Riesz theorem, this space can be identified with the dual of C 0 (Ω) :=
C c (Ω) ‖.‖∞ :
Ω
C 0 (Ω) ′ = M(Ω).
So naturally we have the notion of weak*-convergence on M(Ω):
∫ ∫
µ ∗ k ⇀ µ M(Ω) :⇔ fdµ k → fdµ ∀f ∈ C 0 (Ω).
Since ‖f‖ L 1 = ‖fdx‖ M , every L 1 -bounded sequence has a weak*-convergent
subsequence in M.
Note: If Ω is compact, then C(Ω) = C 0 (Ω) = C c (Ω).
Example: uχ ∗ [0,
1
n ] ⇀ δ 0 in M(−1, 1).
For easy reference we finally recall the definition and some characterizations
of equiintegrability from general measure theory.
Definition 2.13 A family/set F ⊂ L 1 (Ω, µ) in a measure space (Ω, µ) is called
equintegrable if
□
(i) ∀ε > 0
(ii) ∀ε > 0
∃ A measurable with µ(A) < ∞ such that
∫
|f|dµ < ε ∀ f ∈ F.
Ω\A
∃ δ > 0 such that ∀E measurable with
∫
µ(E) < δ =⇒ |f|dµ < ε ∀ f ∈ F.
E
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