Kenneth Kuttler

21.10. SOME EMBEDDING THEOREMS 697

and also ∥∥∥wXGε−wXGε

∗ψn

∥∥∥Lp(Gε ;W)

<η

25∥∥∥ukXGε−ukXGε

∗ψn

∥∥∥Lp(Gε ;W)

<η

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

2η

25<

It follows that{

ukXGε

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 [117]. It is an appropriate generalization whenyou do not necessarily have weak derivatives.

Theorem 21.10.6 Let q > 1 and let E ⊆W ⊆ X where the injection map is continuousfrom 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 con-tinuous 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 21.10.1, 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

≤ ε

6R(∥u(t)∥E +∥u(s)∥E)+Cε ∥u(t)−u(s)∥X

≤ ε

3+Cε R |t− s|1/q . (21.10.47)

21.10. SOME EMBEDDING THEOREMS 697and also n 7| w2G,—w2ees Wr L(Gew) < 8 + 5Nn 1Ds — up Uz. 4| %ee— ue Pact We L(Gaw) 8 | 25Thus,|y%ee— Pee L?(GeW) S [VPage —weee Vn L?(GeW)sfexcen- mel aiu a %ee Yn ~ 42 Celln(Gew)3n 2n 71< 3495 <2,k=@ has an 7 net after all. JIn case Q is a closed interval, there are several versions of these sorts of embeddingswhich 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 [117]. It is an appropriate generalization whenyou do not necessarily have weak derivatives.It follows that {u XS ; is a 1/2 net for L? (Ge; W) contrary to the construction. ThusTheorem 21.10.6 Let g > 1 and let ECW CX where the injection map is continuousfrom W to X and compact from E to W. Let S be defined by{u such that ||u(t)||,~ <R for allt € [a,b], and ||u(s) —u(t)|ly <Rir—s|"/4}.Thus S is bounded in L® (a,b,E) and in addition, the functions are uniformly Holder con-tinuous into X. Then S C C(a,b];W) and if {un} C S, there exists a subsequence, {un, }which converges to a function u € C (|a,b];W) in the following way.Him |hn, ~ Wl =0Proof: First consider the issue of S being a subset of C([a,b];W). Let € > 0 be given.Then by Lemma 21.10.1, there exists a constant, Cg such that for all u € WElel S Ge llelle + Ce [lull -Therefore, for allu € S,le (0) —we(s)lly Selle (0) — wes) +Ce lel) —(8)lly< F(lle(lle + lle) le) +Ce le) — (9) hy< 5 + CeR |r 5)!" (21.10.47)