Weak Convergence Methods for Nonlinear Partial Differential ...

Proof. The detailed proof is left as an excercise. For p > 1 it suffices to show
that span {χ A : A = (0, a) n + b ⊂ Ω measurable} is dense in L q for 1 ≤ q < ∞,
which follows from the fact that C 0 (Ω) is dense in L q . For p = 1 one induces
from the Dunford-Pettis theorem (Theorem 2.14) that (f k ) is relatively weakly
sequentially compact if and only if (f k ) is equiintegrable. It then only remains to
identify weak limit points uniquely.
□
One of our main tasks is to investigate nonlinear operations on weakly convergent
sequences.
Example: Suppose f n ⇀ f in L p (0, 1). Let ψ : R → R. Question: When can we
guarantee ψ(f n ) ⇀ ψ(f)? Surely, if ψ is affine. But indeed the converse is true,
too:
Proposition 2.10 If ψ(f n ) ⇀ ψ(f) for any sequence (f n ) with f n ⇀ f, then ψ
is affine.
Before we prove this, we consider one of the most important examples of
weakly but not strongly convergent sequences.
Lemma 2.11 Suppose D = (0, 1) n (or a general hypercube in R n ). Let f ∈
L p (D) and extend f : R n → R periodically. Let f k ∈ L p (Ω), Ω ⊂ R n bounded and
open,
f k (x) := f(kx) “a highly oscillatory function”.
Then
(∗)
f k ⇀ 1 ∫ ∫
f(x) dx = − f(x) dx on L p (Ω).
|D| D
D
∫
(Here the right hand side is the constant function taking the value −
mean value of f over the periodic unit cell.)
D
f, i.e., the
Proof. Choose k 0 ∈ N such that k 0 D ⊃ Ω. If p = ∞, then clearly (f k ) is bounded.
For p < ∞ this follows from
∫
∫
‖f k ‖ p L = |f(kx)| p dx = k −n |f(y)| p dy
p
Ω
∫
kΩ
∫
≤ k −n |f(y)| p dy = k −n k n k0
n |f(y)| p dy = ‖f‖ p L
k n p 0 .
kk 0 D
Similarly for any subhypercube Q ⊂ Ω:
∫ ∫
f k = k −n
Q
kQ
D
f(y).
Now consider a partitioning of R n by translates of D. For large k, the number
of translates of D completely contained in kQ is k n |Q|
+ |D| O(kn−1 ), the number of
translates of D hitting ∂Ω is O(k n−1 ). It follows that
∫
f k = |Q| ∫
f + O(k −1 ).
|D|
Q
D
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