Weak Convergence Methods for Nonlinear Partial Differential ...

Suppose (u (ν) ), u satisfy (H) and, in addition,
Then
lim inf
ν→∞
∫
Ω
(u (ν) ) − u ∈ ker(A) ∀ν.
∫
f(u (ν) (x))dx ≥
Ω
f(u(x)) ∀Ω ⊂ R n open.
Proof. Approximate Ω by unions of hypercubes of side-length 1 k :
• H k =
⋃1≤i≤I ˙
k
D ki , D ki ⊂ Ω translates of (0, 1 k )n
• |Ω\H k | → 0 k → ∞.
If x ∈ H k , then set
Then
∫
∫
u k (x) = − u(y)dy for x ∈ D ki .
D ki
H k
|u(x) − u k (x)|dx = ∑ i
≤ ∑ i
∫ ∫
|u(x) − − u(y) dy| dx
D ki D ki
∫ ∫
1
|u(x) − u(y)| dy dx.
D ki
|D ki | D ki
If u is uniformly continuous, then this expression converges to zero as k → ∞.
But also for general u ∈ L 1 (Ω) we have
∫
|u k | ≤ ∑ ∫ ∫
|D ki |
H k i
∫D −1 |u(y)| dy = |u|,
ki D ki H k
So that u ↦→ u k is a contraction on L 1 (H k ). Let ε > 0. Then approximate u by
a uniformly continuous v such that ‖u − v‖ L 1 < ε. This implies
‖u − u k ‖ L 1 ≤ ‖u − v‖ + ‖v − v k ‖ + ‖u k − v k ‖ ≤ ‖v − v k ‖ + 2‖u − v‖ ≤ 3ε
for k large. So
∫
H k
|u(x) − u k (x)|dx → 0 (2.4)
as k → ∞.
Note also that if f is continuous and bounded, then for given ε > 0 there
exists a constant C = C(ε) such that f(y 1 ) −f(y 2 )| ≤
ε +C|y 4|Ω| 1 −y 2 | holds true
for all y 1 , y 2 ∈ R m . It follows that
|f(u + (u
∫H (ν) − u)) − f(u k + (u (ν) − u))| ≤ ε ∫
k
4 + C |u − u k | < ε (2.5)
H k
2
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