37.1 Embedding Theorems For Wm,p
Recall Theorem 34.5.1 which is listed here for convenience.
Theorem 37.1.1 Suppose u,u,i ∈ Llocp
,n and p > n. Then u has a
representative, still denoted by u, such that for all x,y ∈ℝn,
This amazing result shows that every u ∈ Wm,p
has a representative which is
p > n.
Using the above inequality, one can give an important embedding theorem.
Definition 37.1.2 Let X,Y be two Banach spaces and let f : X → Y be a
function. Then f is a compact map if whenever S is a bounded set in X, it follows
is precompact in Y .
Theorem 37.1.3 Let U be a bounded open set and for u a function defined on
ℝn, let rUu
for x ∈ U. Then if p > n, rU
continuous and compact.
Proof: First suppose uk → 0 in W1,p
Then if rUuk
does not converge to 0,
follows there exists a sequence, still denoted by k
and ε >
0 such that uk →
Selecting a further subsequence which is still denoted by k,
you can also assume uk
0 a.e. Pick such an x0 ∈ U
where this convergence takes
place. Then from 37.1.13
, for all x ∈U,
showing that uk converges uniformly to 0 on U contrary to
is continuous as claimed.
Next let S be a bounded subset of W1,p
1,p < M
for all u ∈ S.
u ∈ S
Now choosing r large enough, Mp∕rp < mn
and so, for such
there exists xu ∈ U
Therefore from 37.1.13
, whenever x ∈ U,
is uniformly bounded. But also, for
is equicontinuous. By the Ascoli Arzela theorem, it follows
is precompact and so
Definition 37.1.4 Let α ∈ (0,1] and K a compact subset of ℝn
is a complete normed linear space called a Holder space.
The verification that this is a complete normed linear space is routine and is left for
you. More generally, one considers the following class of Holder spaces.
Definition 37.1.5 Let K be a compact subset of ℝn and let λ ∈ (0,1]. Cm,λ
the set of functions, u which are restrictions of functions defined on ℝn to Ksuch that for
. The norm of a function in Cm,λ
is given by
Lemma 37.1.6 Let m be a positive integer, K a compact subset of ℝn, and let
0 < β < λ ≤ 1. Then the identity map from Cm,λ
Proof: First note that the containment is obvious because for any function, f,
Suppose the identity map, id, is not compact. Then there exists ε > 0 and a sequence,
k=1∞ ⊆ Cm,λ
m,λ < M
for all k
β ≥ ε
By the Ascoli Arzela theorem, there exists a subsequence of this, still denoted by fk
such that ∑
∞ < δ
for all k≠l.
It follows that there exist pairs
of points and a multi index, α
and so considering the ends of the above inequality,
Now also, since ∑
∞ < δ,
it follows from the first inequality in
Since δ < ε∕2, this implies
contrary to 37.1.14. This proves the lemma.
Corollary 37.1.7 Let p > n,U and rU be as in Theorem 37.1.3 and let m be a
nonnegative integer. Then rU : Wm+1,p
is continuous as a map
for all λ ∈
and rU is compact if λ <
Proof: Suppose uk → 0 in Wm+1,p
Then from 37.1.13
, if λ ≤
From Theorem 37.1.3
it follows that for
0. This proves the claim about continuity. The claim about
compactness for λ <
follows from Lemma
(Bounded in Wm,p
It is just as important to consider the case where p < n. To do this case the following
lemma due to Gagliardo [?] will be of interest. See also [?].
Lemma 37.1.8 Suppose n ≥ 2 and wj does not depend on the jth component of x, xj.
In this inequality, assume all the functions are continuous so there can be no
Proof: First note that for n = 2 the inequality reduces to the statement
which is obviously true. Suppose then that the inequality is valid for some n. Using
Fubini’s theorem, Holder’s inequality, and the induction hypothesis,
This proves the lemma.
Lemma 37.1.9 If ϕ ∈ Cc∞
and n ≥
Proof: The case where n = 1 is obvious if n∕
is interpreted as
then that n >
1 and note that for ai ≥
In fact, the term on the left is one of many terms of the expression on the right.
Therefore, taking nth roots
Then observe that for each j = 1,2,
and from Lemma 37.1.8 this is dominated by
and this proves the lemma.
The above lemma is due to Gagliardo and Nirenberg.
With this lemma, it is possible to prove a major embedding theorem which
Theorem 37.1.10 Let 1 ≤ p < n and
. Then if f ∈ W1,p
Proof: From the definition of W1,p