Weak Convergence Methods for Nonlinear Partial Differential ...

Recall Schauder’s theorem from functional analysis: A linear operator T : X → Y
between Banach spaces X and Y is compact if and only if the adjoint operator
T ′ : Y ′ → X ′ is compact. Applying this to the compact embeddings W 1,p
0 ֒→
L p , 1 ≤ p < ∞, yields:
Theorem 2.31 Suppose Ω ⊂ R n is open and bounded. The embeddings L q (Ω) ֒→
W −1,q (Ω), 1 < q ≤ ∞ are compact.
Proof. Let T : W 1,p
0 → L p , Tu = u be the compact embedding operator. Then
T ′ : (L p ) ′ → (W 1,p
0 ) ′
∫
T ′ ϕ(u) = ϕ(Tu) = ϕu ∀ϕ ∈ (L p ) ′ , u ∈ W 1,p
0 ,
i.e. T ′ : L q → W −1,q is the natural embedding from L q into W −1,q and compact,
too.
□
For p = 1 we have argued earlier that it is convenient to embed L 1 in the
larger space M of Radon measures. We therefore prove a compact embedding
result directly for M.
Theorem 2.32 Suppose Ω ⊂ R n is open and bounded. The embeddings M(Ω) ֒→
W −1,q (Ω), q < n , are compact.
n−1
Proof. For p > n, the embedding W 1,p
0 ֒→ C 0 is compact. Similarly as before it
follows that then also
M ∼ = (C 0 ) ′ ֒→ (W 1,p
0 ) ′ ∼ = W −1,q
is compact for q = p
n
. It remains to note that p > n is equivalent to q < .
p−1 n−1
□
2.5 A-quasiconvexity
We now resume our disccussion of Section 2.2 on the (semi-)continuity properties
of nonlinear functionals. Now, however, under additional differential constraints.
More precisely, we will consider sequences (u (ν) ) ⊂ L ∞ (Ω; R m ) such that
⎧
⎪⎨
(H)
⎪⎩
u (ν) ∗ ⇀ u L ∞
( ) ∑
Au (ν) =
j,k a ∂u (ν)
j
ijk ∂x k
i=1,...,q
f(u (ν) ) ⇀ ∗ l L ∞ ,
bd. in L 2 (Ω; R q ),
where the a ijk ∈ R are constants and so A is a linear partial differential operator
with constant coefficients.
If (H) holds and additionally Au (ν) = 0 ∀ν we say that (H 0 ) holds.
Examples:
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