24.11. SOME EMBEDDING THEOREMS 695
Let n be this large. Then let{
ukXGε∗ψn
}r
k=1be a η
8 net for Aεn in Lp(Gε ;W
).
Then consider the balls B(
ukXGε, η
4
)in Lp
(Gε ;W
). If wXGε
is in Aε , is it in some
B(
ukXGε, η
2
)? By what was just shown, there is k such that∥∥∥wXGε
∗ψn−ukXGε∗ψn
∥∥∥Lp(Gε ;W)
<η
8
and also ∥∥∥wXGε−wXGε
∗ψn
∥∥∥Lp(Gε ;W)
<η
8+
η
25∥∥∥ukXGε−ukXGε
∗ψn
∥∥∥Lp(Gε ;W)
<η
8+
η
25
Thus, ∥∥∥wXGε−ukXGε
∥∥∥Lp(Gε ;W )
≤∥∥∥wXGε
−wXGε∗ψn
∥∥∥Lp(Gε ;W)
+∥∥∥wXGε
∗ψn−ukXGε∗ψn
∥∥∥Lp(Gε ;W)
+∥∥∥ukXGε
∗ψn−ukXGε
∥∥∥Lp(Gε ;W)
<3η
8+
2η
25<
η
2
It follows that{
ukXGε
}r
k=1is a η/2 net for Lp
(Gε ;W
)contrary to the construction. Thus
A has an η net after all. ■In case Ω is a closed interval, there are several versions of these sorts of embeddings
which are enormously useful in the study of nonlinear evolution equations or inclusions.The following theorem is an infinite dimensional version of the Ascoli Arzela theorem.
It is like a well known result due to Simon [52]. It is an appropriate generalization whenyou do not necessarily have weak derivatives. I am giving another proof although Theorem24.11.3 given above is actually more general.
Theorem 24.11.6 Let q > 1 and let E ⊆W ⊆ X where the injection map is contin-uous from W to X and compact from E to W. Let S be defined by{
u such that ∥u(t)∥E ≤ R for all t ∈ [a,b] , and ∥u(s)−u(t)∥X ≤ R |t− s|1/q}.
Thus S is bounded in L∞ (a,b,E) and in addition, the functions are uniformly Holder contin-uous into X . Then S⊆C ([a,b] ;W ) and if {un}⊆ S, there exists a subsequence,
{unk
}which
converges to a function u ∈C ([a,b] ;W ) in the following way: limk→∞
∥∥unk −u∥∥
∞,W = 0.
Proof: First consider the issue of S being a subset of C ([a,b] ;W ) . Let ε > 0 be given.Then by Lemma 24.11.1, there exists a constant, Cε such that for all u ∈W
∥u∥W ≤ε
6R∥u∥E +Cε ∥u∥X .
Therefore, for all u ∈ S,
∥u(t)−u(s)∥W ≤ε
6R∥u(t)−u(s)∥E +Cε ∥u(t)−u(s)∥X