Weak Convergence Methods for Nonlinear Partial Differential ...

preserving extension L of l ◦ Ψ −1 , represented by some v ∈ L q (Ω (k) ), 1 + 1 = 1. p q
Thus,
l(u) = (l ◦ Ψ −1 )(Ψu) = ∑ ∫
v(x, α)(Ψu)(x, α) dx
α Ω
= ∑ ∫
v(x, α)∂ α u(x) dx.
α Ω
Also by choice of L and Ψ being isometric
‖l‖ (W k,p ) ′ = ‖l ◦ Ψ−1 ‖ (Ψ(W k,p (Ω))) ′ = ‖L‖ (L p (Ω (k) )) ′ = ‖v‖ L q (Ω (k) ).
If w is any element of L q (Ω (k) ) satisfying (2.1) then w, viewed as an element of
(L p (Ω (k) )) ′ , is an extension of l ◦ Ψ −1 , and so
‖w‖ L q (Ω (k) ) ≥ ‖l ◦ Ψ −1 ‖ (Ψ(W k,p )) ′ = ‖l‖ (W k,p ) ′.
Remark 2.25 This justifies saying u k ⇀ u in W k,p , 1 ≤ p < ∞, iff ∂ α u k ⇀ ∂ α u
in L p for all α with |α| ≤ k.
The elements of (W k,p (Ω (k) )) ′ are extensions of distributions:
If v satisfies (2.1), let
T = ∑
∫
(−1) α ∂ α T vα , T vα (ϕ) = v α ϕ. (2.2)
Then
|α|≤k
Tϕ = ∑
(−1) α ∂ α T vα (ϕ) = ∑ ∫
|α|≤k
= l(ϕ).
|α|≤k
v α ∂ α ϕ
□
Conversely, any distribution of the form (2.2) (with v α ∈ L q for all α) extends
to a continuous linear functional on W k,p (Ω), even uniquely to a continuous linear
functional on W k,p
0 (Ω). This follows from the fact that T is continuous with
respect to ‖ · ‖ W k,p:
∑
∫
∣∫
∣∣∣
|Tϕ| =
v α ∂ α ϕ
∣ ∣ = v Ψϕ dz
∣
Ω (k)
|α|≤k
≤ ‖v‖ L q (Ω (k) )‖Ψϕ‖ L p (Ω (k) )
= ‖v‖ L q (Ω (k) )‖ϕ‖ (W k,p ) ′ ,
and by definition D(Ω) is dense in W k,p
0 (Ω).
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