Weak Convergence Methods for Nonlinear Partial Differential ...

Note that if Au (ν) is bounded in L 2 , then it is compact in W −1,2 by Rellich’s
theorem, and hence in W −1,2
loc
. (If u (νk) ⇀ u in W −1,2 , then also ϕu (νk) → ϕu in
W −1,2 ∀ ϕ ∈ Cc ∞ (Ω).)
Proof. We only need to prove the first statement. First note that l ∈ M because
f(u (ν) ) is bounded in L 1 and so f(u (ν) ) ⇀ ∗ l in M.
1. Let v (ν) = u (ν) − u. Then v (ν) satisfies ( ˜H) with u = 0, f(u) = 0. If we
succeed to prove the theorem for v (ν) , then
Thus
0 ≤ lim f(v (ν) ) = lim(u (ν) − u) T M(u (ν) − u)
= lim f(u (ν) ) + f(u) + lim(u (ν) ) T Mu − lim u T Mu (ν)
= lim f(u (ν) ) − f(u).
lim f(u (ν) ) ≥ f(u).
2. For ϕ ∈ Cc ∞ (Ω) let w (ν) = ϕv (ν) . Then
⎧
⎪⎨
w (ν) ⇀ 0 in L 2 ,
∑
j,k
⎪⎩
a ∂w (ν)
j
ijk ∂x k
→ 0 in W −1,2 (R n ), i ∈ {1, ..., q},
supp w (ν) ⊂ K ⊂⊂ Ω
for some suitable K. To see this, note that
and so
∂w (ν)
j
∂x k
= ϕ ∂v(ν) j ∂ϕ
+ v (ν)
j
∂x k ∂x
} {{ k
}
bounded in L 2
Aw (ν) = ϕAv (ν) + something bounded in L 2 .
} {{ }
compact in W −1,2
Now w (ν) ⇀ 0 in L 2 implies Aw (ν) ⇀ 0 in W −1,2 . By compactness this
gives
Aw (ν) → 0 in W −1,2 .
We need to prove that
lim inf
ν→∞
∫
R n (w (ν) ) T Mw (ν) ≥ 0.
If this is done, the proof is finished: ∀ ϕ ∈ D(Ω)
∫
∫
0 ≤ lim inf
ϕ 2
ν→∞
and so
(w (ν) ) T Mw (ν) = lim inf
R n ν→∞
l ≥ 0
45
}{{}
∈D
as measures.
(v (ν) ) T Mv (ν) = l(ϕ 2 )
} {{ }
→l in D ′