The next theorem is very useful in getting estimates in partial differential equations. It is
called Erling’s lemma.
Definition 32.7.1Let 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 32.7.2Let 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 ε||u||
W E X
Proof:Suppose not. Then there exists ε > 0 and for each n ∈ ℕ, u_{n} such
that
||un||W > ε||un ||E + n ||un||X
Now let v_{n} = u_{n}∕
||un||
_{E}. Therefore,
||vn||
_{E} = 1 and
||vn||W > ε+ n ||vn||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||
_{W}> ε and passing to the limit yields 0 > ε, a
contradiction. ■
Definition 32.7.3Define 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.
It is like a well known result due to Simon [?]. It is an appropriate generalization when
you do not have weak derivatives.
Theorem 32.7.4Let 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 ≤ 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 Holdercontinuous into X. Then S ⊆ C
([a,b];W )
and if
{un}
⊆ S, there exists asubsequence,
{un}
k
which converges to a function u ∈ C
([a,b];W )
in the followingway.
lim ||unk − u||∞,W = 0.
k→∞
Proof:First consider the issue of S being a subset of C
([a,b];W )
. Let ε > 0 be
given. Then by Theorem 32.7.2 there exists a constant, C_{ε} such that for all
u ∈ W
||u||W ≤ -ε-||u||E + Cε||u ||X .
6R
Therefore, for all u ∈ S,
||u (t)− u (s)||W ≤ -ε-||u (t)− u (s)||E + Cε||u (t)− u (s)||X
6Rε
≤ ---(∥u(t)∥E + ∥u (s)∥E)+ C ε∥u(t) − u(s)∥X
6εR 1∕q
≤ 3 + CεR |t− s| . (32.7.56)
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 =
{t}
n
_{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)
}
(t )
1
converges to a point in W. Now take a subsequence of this, called
(n,2)
such that as n →∞,u_{(n,2)
}
(t)
2
converges to a point in W. It follows that u_{(n,2)
}
(t)
1
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}
(t )
j
converges for all t_{j}∈ D.
Claim:Let
{uk}
be as just defined, converging at every point of D ≡
∑k 1 ∫ ti ∑k 1 ∫ ti
= t−-t--- un(t)dsX [ti−1,ti)(t)− t-−-t-- un (s)dsX [ti−1,ti) (t)
i=1 i i−1 ti−1 i=1 i i−1 ti−1
∫
∑k ---1--- ti
= ti − ti−1 ti−1 (un(t)− un(s))dsX[ti−1,ti)(t).
i=1
It follows from Jensen’s inequality that
||un (t)− un (t)||pW
k |||| ∫ ti ||||p
= ∑ ||||---1--- (un (t) − un (s))ds|||| X[t ,t)(t)
i=1||ti − ti−1 ti−1 ||W i−1 i
∑k ∫ ti
≤ ---1--- ||un(t) − un (s)||pW dsX [ti−1,ti)(t)
i=1 ti − ti−1 ti−1