The next theorem is very useful in getting estimates in partial differential equations. It is
called Erling’s lemma.
Definition 42.5.1Let E,W be Banach spaces such that E ⊆ Wand theinjectionmap from E into W is continuous. The injection map is said to be compact if everybounded set in E has compact closure in W. In other words, if a sequence is boundedin E it has a convergent subsequence converging in W. This is also referred to bysaying that bounded sets in E are precompactin W.
Theorem 42.5.2Let E ⊆ W ⊆ X where the injection map is continuous from W to Xand compact from E to W. Then for every ε > 0 there exists a constant, C_{ε}such that forall u ∈ E,
||u|| ≤ ε||u|| + C ||u||
W E ε X
Proof:Suppose not. Then there exists ε > 0 and for each n ∈ ℕ, u_{n} such
that
||u || > ε||u || + n ||u ||
n W n E n X
Now let v_{n} = u_{n}∕
||un||
_{E}. Therefore,
||vn||
_{E} = 1 and
||vn|| > ε+ n ||vn||
W X
It follows there exists a subsequence, still denoted by v_{n} such that v_{n} converges to v in
W. However, the above inequality shows that
||vn||
_{X}→ 0. Therefore, v = 0. But then the
above inequality would imply that
||vn||
> ε and passing to the limit yields 0 > ε, a
contradiction.
Definition 42.5.3Define C
([a,b];X )
the space of functions continuous at everypoint of
[a,b]
having values in X.
You should verify that this is a Banach space with norm
||u||∞,X = max {||unk (t) − u (t)||X : t ∈ [a,b]}.
The following theorem is an infinite dimensional version of the Ascoli Arzela theorem.
[?].
Theorem 42.5.4Let 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 + ||u′||Lq([a,b];X) ≤ R for all t ∈ [a,b] .
Then S ⊆ C
([a,b];W )
and if
{un}
⊆ S, there exists a subsequence,
{unk}
whichconverges to a function u ∈ C
([a,b];W )
in the following way.
lim ||unk − u||∞,W = 0.
k→∞
Proof:First consider the issue of S being a subset of C
([a,b];W )
. By Theorem
42.1.9 on Page 4829 the following holds in X for u ∈ S.
∫ t
u (t)− u(s) = u′(r)dr.
s
Thus S ⊆ C
([a,b];X )
. Let ε > 0 be given. Then by Theorem 42.5.2 there exists a
constant, C_{ε} such that for all u ∈ W
||u||W ≤ -ε-||u||E + Cε||u ||X .
4R
Therefore, for all u ∈ S,
ε
||u(t)− u(s)||W ≤ 6R-||u(t)− u(s)||E + C ε||u(t)− u(s)||X
||||∫ t ||||
≤ ε + Cε|||| u′(r)dr||||
3 ∫ s X
ε t ′ ε 1∕q
≤ 3 + Cε s ||u (r)||X dr ≤ 3 + C εR |t− s| (.42.5.22)
Since ε is arbitrary, it follows u ∈ C
([a,b];W )
.
Let D = ℚ ∩
[a,b]
so D is a countable dense subset of
[a,b]
. Let D =
{tn}
_{n=1}^{∞}. By
compactness of the embedding of E into W, there exists a subsequence u_{(n,1)
} such that as
n →∞, u_{(n,1)
}
(t1)
converges to a point in W. Now take a subsequence of this, called
(n,2)
such that as n →∞,u_{(n,2)
}
(t2)
converges to a point in W. It follows that u_{(n,2)
}
(t1)
also converges to a point of W. Continue this way. Now consider the diagonal sequence,
u_{k}≡ u_{(k,k)
} This sequence is a subsequence of u_{(n,l)
} whenever k > l. Therefore, u_{k}
(tj)
converges for all t_{j}∈ D.
Claim:Let
{uk}
be as just defined, converging at every point of D ≡ℚ∩
and so it follows that if δ is sufficiently small and s ∈ B
(t,δ)
, then when n,m > N_{t}
||u (s)− u (s)|| < ε.
n m
Since
[a,b]
is compact, there are finitely many of these balls,
{B (ti,δ)}
_{i=1}^{p}, such that for
s ∈ B
(ti,δ)
and n,m > N_{ti}, the above inequality holds. Let N > max
{Nt ,⋅⋅⋅,Nt }
1 p
.
Then if m,n > N and s ∈
[a,b]
is arbitrary, it follows the above inequality must hold.
Therefore, this has shown the following claim.
Claim:Let ε > 0 be given. Then there exists N such that if m,n > N, then
||un − um ||
_{∞,W}< ε.
Now let u
(t)
= lim_{k→∞}u_{k}
(t)
.
||u (t)− u (s)|| ≤ ||u (t)− un (t)|| + ||un (t)− un (s)|| + ||un (s)− u (s)||
W W W (42.5W.23)
(42.5.23)
Let N be in the above claim and fix n > N. Then
||u (t)− un (t)||W = lim ||um (t)− un (t)||W ≤ ε
m→ ∞
and similarly,
||un (s)− u(s)||
_{W}≤ ε. Then if
|t− s|
is small enough, 42.5.22 shows
the middle term in 42.5.23 is also smaller than ε. Therefore, if
|t− s|
is small
enough,
||u(t)− u(s)||W < 3ε.
Thus u is continuous. Finally, let N be as in the above claim. Then letting m,n > N, it
follows that for all t ∈
[a,b]
,
||um (t)− un(t)|| < ε.
Therefore, letting m →∞, it follows that for all t ∈
[a,b]
,
||u(t)− u (t)|| ≤ ε.
n
and so
||u− un||
_{∞,W}≤ ε. Since ε is arbitrary, this proves the theorem.
The next theorem is another such imbedding theorem found in [?]. It is often used in
partial differential equations.
Theorem 42.5.5Let 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) : u′ ∈ Lq([a,b];X)
and ||u|| p + ||u ′||q ≤ R }.
L ([a,b];E) L ([a,b];X)
Then S is precompact in L^{p}
([a,b];W )
. This means that if
{un}
_{n=1}^{∞}⊆ S, it has asubsequence
{unk}
which converges in L^{p}
([a,b];W )
.
Proof:By Proposition 6.2.5 on Page 410 it suffices to show S has an η net in
L^{p}
([a,b];W )
for each η > 0.
If not, there exists η > 0 and a sequence
{un}
⊆ S, such that
||u − u || ≥ η (42.5.24)
n m
(42.5.24)
for all n≠m and the norm refers to L^{p}
([a,b];W )
. Let
a = t0 < t1 < ⋅⋅⋅ < tk = b,ti − ti−1 = (b− a)∕k.
Now define
∫
-- ∑k -- -- ---1--- ti
un (t) ≡ uniX [ti−1,ti)(t),uni ≡ ti − ti−1 ti−1 un (s)ds.
i=1
The idea is to show that u_{n} approximates u_{n} well and then to argue that a subsequence
of the
--
{un}
is a Cauchy sequence yielding a contradiction to 42.5.24.
Therefore,
-- ∑k 1 ∫ ti --
un (t)− un(t) = ti −-ti−-1 (un(t)− un(s))dsX[ti−1,ti)(t).
i=1 ti−1
It follows from Jensen’s inequality that
-- p
||un (t)− un (t)||W
∑k |||| 1 ∫ ti ||||p
= ||||------- (un (t) − un (s))ds|||| X[ti−1,ti)(t)
i=1||ti − ti−1 ti−1 ||W
∑k 1 ∫ ti p
≤ t−-t--- ||un(t) − un (s)||W dsX [ti−1,ti)(t)
i=1 i i−1 ti−1
and so
∫
b -- p
a ||(un(t)− un(s))||W ds
∫ b∑k ∫ ti
≤ ---1--- ||un (t)− un (s)||pW dsX[ti−1,ti)(t)dt
a i=1ti − ti− 1 ti−1
∑k ∫ ti ∫ ti
= ---1--- ||un(t)− un(s)||pW dsdt. (42.5.25)
i=1ti − ti− 1 ti−1 ti−1
From Theorems 42.5.2 and 42.1.9, if ε > 0, there exists C_{ε} such that
||un (t)− un(s)||p ≤ ε||un (t) − un (s)||p + Cε||un (t)− un (s)||p
W E X
||∫ ||
p−1 p p |||| t ′ ||||p
≤ 2 ε (||un (t)|| + ||un (s)||)+ Cε || s un(r)dr||X
(∫ t )p
≤ 2p−1ε (||un (t)||p + ||un (s)||p)+ Cε ||u′n(r)||X dr
s
≤ 2p−1ε (||un (t)||p + ||un (s)||p)
((∫ t )1∕q ′)p
+Cε ||u′n (r)||qX dr |t− s|1∕q
s
= 2p−1ε (||un (t)||p + ||un (s)||p)+ CεRp ∕q|t− s|p∕q′ .
∑k 1 ∫ ti ∫ ti ( p−1 p p
ti −-ti−1 2 ε(||un(t)|| + ||un(s)||)
i=1 ti− 1′ti)−1
+C εRp ∕q |t− s|p∕q dsdt
∫ ∫ ∫
∑k p ti p p∕q---1--- ti ti p∕q′
= 2ε ti−1 ||un (t)||W + CεR ti − ti−1 ti−1 ti−1 |t− s| dsdt
i=1 ∫ k ∫ ∫
= 2pε b||u (t)||p dt+ C Rp∕q∑ ----1----(t − t )p∕q′ ti tidsdt
a n ε i=1(ti − ti−1) i i−1 ti−1 ti−1
∫ b k
= 2pε ||u (t)||p dt+ C Rp∕q∑ ----1----(t − t )p∕q′ (t− t )2
a n ε i=1(ti − ti−1) i i−1 i i− 1
k ( )1+p∕q′
≤ 2pεRp + CεRp∕q∑ (ti − ti−1)1+p∕q′ = 2pεRp + CεRp∕qk b−-a .
i=1 k
Taking ε so small that 2^{p}εR^{p}< η^{p}∕8^{p} and then choosing k sufficiently large, it
follows
||un − un||Lp([a,b];W ) < η.
4
Now use compactness of the embedding of E into W to obtain a subsequence such
that
--
{un}
is Cauchy in L^{p}
(a,b;W )
and use this to contradict 42.5.24. The details
follow.
Suppose u_{n}
(t)
= ∑_{i=1}^{k}u_{i}^{n}X_{[ti−1,ti)}
(t)
. Thus
k
||u-(t)|| = ∑ ||un|| X (t)
n E i=1 i E [ti−1,ti)