Weak Convergence Methods for Nonlinear Partial Differential ...

Proof. Clearly (i) =⇒ (ii), so we only prove (ii). ”⇒”: Suppose
lim inf
n→∞ F(u n) ≥ F(u) ∀(u n ) s.t. u n ∗ ⇀ u.
In particular, for a, b ∈ R m , λ ∈ [0, 1], let f : [0, 1] n → R
{
a, x 1 ≤ λ,
u(x) =
b, x 1 > λ,
extended to R n by periodicity and define u k (x) = u(kx)χ Ω (x).
As u k ∗ ⇀ u = ∫ [0,1] n u = λa + (1 − λ)b in L ∞ (R n ), also u| Ω ∗ ⇀ u in L ∞ (Ω).
Similarly, f(u n ) ⇀ ∗ λf(a) + (1 − λ)f(b) and consequently
∫ ∫
F(u n ) = f(u n ) n→∞
→ λf(a) + (1 − λ)f(b) = |Ω|[λf(a) + (1 − λ)f(b)].
But
Ω
Ω
∫
F(u) =
f(u) = |Ω|f(λa + (1 − λ)b),
and so f is convex.
”⇐:” As f is convex there are affine functions l 1 , l 2 , ... such that f = sup i l i .
For fixed n let
E j := {x ∈ R m : l j (x) > l i (x) for 1 ≤ i < j and l j (x) ≥ l i (x) for j < i ≤ n} ,
so that E 1 ˙∪ · · · ˙∪E n = R m and max 1≤i≤n l i (x) = l j (x) on E j . Suppose u ∗ k ⇀ u.
Then also χ A u ∗ k ⇀ χ A u for all measurable A. But then
∫
lim inf F(u k) = lim inf f(u k )
k→∞
k→∞
Ω
n∑
∫
≥ lim inf l j (u k )
k→∞
If l j (x) = a j x + b j , this reads
n∑
∫
lim inf F(u k ) ≥ lim inf
=
=
∫
=
j=1
n∑
∫
j=1
n∑
∫
j=1
Ω
u −1 (E j )
u −1 (E j )
max l i(u).
1≤i≤n
10
j=1
u −1 (E j )
a j χ u −1 (E j )u k + χ u −1 (E j )b j
l j (u)
max l i(u)
1≤i≤n