Weak Convergence Methods for Nonlinear Partial Differential ...

Using (2.10) (with η = ξ ) the second term can be estimated by
|ξ|
∫
{|ξ|>1}
∫
Re ˜f(ŵ(ν) (ξ)) ≥ −ε
∣
∣ŵ(ν) (ξ)
{|ξ|>1}
∫
∣ 2 dξ − C ε
{|ξ|>1}
Noting that 1 ≤ 2
t
(1+t 2 ) 1 2
for t ≥ 1, by (2.8) we get
∫
lim inf
∫
Re ˜f(ŵ(ν) (ξ)) dξ ≥ −ε |ŵ(ν) (ξ)| 2 dξ
{|ξ|>1}
{|ξ|>1}
Since ε > 0 was arbitrary, we indeed obtain (2.9).
≥ −ε‖w (ν) ‖ L 2 ≥ −Cε.
∑
∑
a
∣ ijk ŵ (ν) ξ k
j
|ξ| ∣
Remark 2.62 Up to extracting subsequences, the condition that f(u (ν) ) converges
in D ′ (even weakly* in M) is always satisfied.
Corollary 2.63 Let E = span Λ, d = dim E. Choose coordinates so that
E = {y ∈ R m : y d+1 = . . . = y m = 0}.
If u (ν) ⇀ ∗ u in L ∞ (Ω; R m ), the sequence Au (ν) is compact in W −1,2
loc
and
i
j,k
□
2
.
f : R m−d → R
is continuous,
then
strongly in L p (Ω) ∀ p < ∞.
Proof. The mapping
f(u (ν)
d+1 , . . .,u(ν) m ) → f(u d+1, . . .,u m )
vanishes on Λ and so
y = (y 1 , . . .,y m ) ↦→ y 2 d+1 + . . . + y 2 m
(u (ν)
d+1 )2 + . . . + (u (ν)
m ) 2 → u 2 d+1 + . . . + u 2 m in D ′ .
By boundedness in L ∞ this convergence holds also w*-L ∞ . But then
∫
∫
(u (ν)
d+1 )2 + . . . + (u (ν)
m ) 2 → u 2 d+1 + . . . + u 2 m,
which shows that
‖u (ν)
j ‖ 2 L 2 → ‖u j‖ 2 L 2 ∀ j ≥ d + 1
48