Weak Convergence Methods for Nonlinear Partial Differential ...

Definition 2.26 Let 1 < p ≤ ∞.
{
W −k,p := T ∈ D ′ (Ω) : ∃v α ∈ L p (Ω) : T = ∑ α
∂ α v α
}
with norm
‖T ‖ W −k,p := min
{
‖v‖ L p (Ω (k) ) : T = ∑ α
∂ α v α
}.
As a corollary to our previous considerations we obtain:
Theorem 2.27 If 1 ≤ p < ∞, 1 p + 1 q
= 1, then
(W k,p
0 (Ω)) ′ ∼ = W −k,q (Ω).
Recall the Rellich-Kondrachov Theorem:
Theorem 2.28 If Ω ⊂ R n is open and bounded, then the embeddings
• W 1,p
0 ֒→ L q for 1 ≤ p < n, 1 ≤ q < p ∗ = pn
n−p and
• W 1,p
0 ֒→ C 0 (Ω) for p > n
are compact. If in addition Ω has a Lipschitz boundary ∂Ω, then the embeddings
• W 1,p ֒→ L q for 1 ≤ p < n, 1 ≤ q < p ∗ = pn
n−p and
• W 1,p ֒→ C(Ω) for p > n
are compact, too.
Corollary 2.29 Under the assumptions of Theorem 2.28, the embedding W 1,p
0 (Ω)
(resp. W 1,p (Ω)) ֒→ L p (Ω) is compact for any 1 ≤ p ≤ ∞. In particular, any
bounded sequence in W 1,p (Ω) has an L p -strongly convergent subsequence and
(∗)
u n ⇀ u in W 1,p implies u n → u in L p .
Remark 2.30 Compactness theorems are important to handle nonlinear expressions.
E.g., suppose u ε is W 1,α -bounded sequence of solutions of the quasilinear
PDE
a(x, u ε ) · ∇u ε = b(x, u ε ).
Then for a subsequence u ∗ ε ⇀ u in W 1,∞ and, in particular, u ε → u uniformly.
If a and b are continuous, we find that u also solves
a(x, u) · ∇u = b(x, u).
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