Weak Convergence Methods for Nonlinear Partial Differential ...

and thus
Consequently,
u (ν)
j → u j in L 2 ∀ j ≥ d + 1.
f(u (ν)
d+1
, ..., u(ν) m ) → f(u d+1 , ..., u m )
strongly in L 1 while being bounded in L ∞ . Hence,
f(u (ν)
d+1
, ..., u(ν) m ) → Lp
f(u d+1 , ..., u m ) ∀ p < ∞.
Exercise: Show that f n → f in L 1 and (f n ) bounded in L q implies f n → f in L p
for all p < q.
Corollary 2.64 (The ‘div-curl lemma’) Suppose v, w ∈ L 2 (Ω; R n ) are vector
fields such that
⎧
⎪⎨ div v (ν) = ∑ n
(
⎪⎩ curl w (ν) =
j=1
∂w (ν)
j
∂x k
∂v (ν)
j
(or just compact in W −1,2
loc
(Ω), resp.).
Then v (ν) ⇀ v, w (ν) ⇀ w in L 2 (Ω; R n ) implies
∂x j
is bounded in L 2 ,
)
− ∂w(ν) k
∂x j
j,k
v (ν) · w (ν) D ′
→ v · w.
is bounded in L 2
□
Proof.
{
}
m∑
Λ = (λ, µ) : ∃ ξ ≠ 0 : λ j ξ j = 0, µ j ξ k − µ k ξ j = 0 ∀ j, k
j=1
= {(λ, µ) : ∃ ξ ≠ 0 : λ · ξ = 0, µ ‖ ξ}
= {(λ, µ) : λ · µ = 0}.
u (ν) = (u (ν)
1 , u(ν) 2 ) := (v(ν) , w (ν) ) satisfies ( ˜H) with
f(y 1 , y 2 ) = y 1 · y 2
vanishing on Λ. The assertion thus follows.
□
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