24.11. SOME EMBEDDING THEOREMS 699
k
∑i=1
1ti− ti−1
∫ ti
ti−1
∫ ti
ti−1
(2p−1
ε (∥un (t)∥p +∥un (s)∥p)+Cε |t− s|p/q)
dsdt
=k
∑i=1
2pε
∫ ti
ti−1
∥un (t)∥pW +
Cε
ti− ti−1
∫ ti
ti−1
∫ ti
ti−1
|t− s|p/q dsdt
≤ 2pε
∫ b
a∥un (t)∥p dt +Cε
k
∑i=1
1(ti− ti−1)
(ti− ti−1)p/q∫ ti
ti−1
∫ ti
ti−1
dsdt
= 2pε
∫ b
a∥un (t)∥p dt +Cε
k
∑i=1
1(ti− ti−1)
(ti− ti−1)p/q (ti− ti−1)
2
≤ 2pεRp +Cε
k
∑i=1
(ti− ti−1)1+p/q = 2p
εRp +Cε k(
b−ak
)1+p/q
.
Taking ε so small that 2pεRp < η p/8p and then choosing k sufficiently large, it followsthat ||un−un||Lp([a,b];W ) <
η
4 .Thus k is fixed and un at a step function with k steps having values in E. Now use
compactness of the embedding of E into W to obtain a subsequence such that {un} isCauchy in Lp (a,b;W ) and use this to contradict 24.49. The details follow.
Suppose un (t) = ∑ki=1 un
i X[ti−1,ti) (t) . Thus ∥un (t)∥E = ∑ki=1 ∥un
i ∥E X[ti−1,ti) (t) and soR≥
∫ ba ∥un (t)∥p
E dt = Tk ∑
ki=1 ∥un
i ∥pE . Therefore, the {un
i } are all bounded. It follows that af-ter taking subsequences k times there exists a subsequence
{unk
}such that unk is a Cauchy
sequence in Lp (a,b;W ) . You simply get a subsequence such that unki is a Cauchy sequence
in W for each i. Then denoting this subsequence by n,
∥un−um∥Lp(a,b;W ) ≤ ∥un−un∥Lp(a,b;W )+∥un−um∥Lp(a,b;W )+∥um−um∥Lp(a,b;W )
≤ η
4+∥un−um∥Lp(a,b;W )+
η
4< η
provided m,n are large enough, contradicting 24.49. ■You can give a different version of the above to include the case where there is, instead
of a Holder condition, a bound on u′ for u∈ S. It is stated next. We are assuming a situationin which
∫ ba u′ (t)dt = u(b)−u(a) . This happens, for example, if u′ is the weak derivative.
This is discussed in the exercises. These kind of theorems are in [52].
Corollary 24.11.9 Let E ⊆W ⊆ X where the injection map is continuous from W to Xand compact from E to W. Let p≥ 1, let q > 1, and define
S≡ {u ∈ Lp ([a,b] ;E) : for some C, ∥u(t)−u(s)∥X ≤C |t− s|1/q and ∥u∥Lp([a,b];E) ≤ R}.
Thus S is bounded in Lp ([a,b] ;E) and Holder continuous into X. Then S is precompact inLp ([a,b] ;W ). This means that if {un}∞
n=1 ⊆ S, it has a subsequence{
unk
}which converges
in Lp ([a,b] ;W ) . The same conclusion can be drawn if it is known instead of the Holdercondition that ∥u′∥L1([a,b];X) is bounded.
Proof: The first part is Theorem 24.11.8. Therefore, we just prove the new stuff whichinvolves a bound on the L1 norm of the derivative. It suffices to show S has an η net inLp ([a,b] ;W ) for each η > 0.
If not, there exists η > 0 and a sequence {un} ⊆ S, such that
∥un−um∥ ≥ η (24.51)