Then Λ is one to one because one of the multi indices is 0. Also
Λ(Xm,p (U ))
is a closed subspace of L^{p}
(U)
^{w}. To see this, suppose
(u ,D α2u ,⋅⋅⋅,Dαw u ) → (f ,f,⋅⋅⋅,f )
k k k 1 2 w
in L^{p}
(U)
^{w}. Then u_{k}→ f_{1} in L^{p}
(U)
and D^{αj}u_{k}→ f_{j} in L^{p}
(U)
. Therefore, letting
ϕ ∈ C_{c}^{∞}
(U)
and letting k →∞,
∫ α |α|∫ α
U (D juk)ϕdx = (− 1) U ukD jϕdx
∫ ↓ |α|∫ ↓
U fjϕdx (− 1) U f1D αjϕdx ≡ D αj (f1)(ϕ)
It follows D^{αj}
(f1)
= f_{j} and so Λ
(Xm,p (U ))
is closed as claimed. This is clearly also a
subspace of L^{p}
(U)
^{w} and so it follows that Λ
(Xm,p (U))
is a reflexive Banach space. This
is because L^{p}
(U )
^{w}, being the product of reflexive Banach spaces, is reflexive and any
closed subspace of a reflexive Banach space is reflexive. Now Λ is an isometry of
X^{m,p}
(U)
and Λ
(Xm,p(U ))
which shows that X^{m,p}
(U )
is a reflexive Banach space.
Finally, W^{m,p}
(U)
is a closed subspace of the reflexive Banach space, X^{m,p}
(U)
and so it is also reflexive. To see X^{m,p}
(U)
is separable, note that L^{p}
(U )
^{w} is
separable because it is the finite product of the separable hence completely
separable metric space, L^{p}
(U )
and Λ
(Xm,p (U))
is a subset of L^{p}
(U )
^{w}. Therefore,
Λ
(Xm,p(U ))
is separable and since Λ is an isometry, it follows X^{m,p}
(U )
is
separable also. Now W^{m,p}
(U )
must also be separable because it is a subset of
X^{m,p}
(U)
.
The following theorem is obvious but is worth noting because it says that
if a function has a weak derivative in L^{p}
(U )
on a large open set, U then the
restriction of this weak derivative is also the weak derivative for any smaller open
set.
Theorem 37.0.4Suppose U is an open set and U_{0}⊆ U is another open set. Supposealso D^{α}u ∈ L^{p}
(U)
. Then for all ψ ∈ C_{c}^{∞}
(U0)
,
∫ ∫
(D αu)ψdx = (− 1)|α| u (Dαψ ).
U0 U0
The following theorem is a fundamental approximation result for functions in
X^{m,p}
(U)
.
Theorem 37.0.5Let U be an open set and let U_{0}be an open subset of U with theproperty thatdist
(U-,U C)
0
> 0. Then if u ∈ X^{m,p}
(U )
and
^u
denotes the zero extentionof u off U,
lim ||^u∗ ϕ − u|| = 0.
l→ ∞ l Xm,p(U0)
Proof:Always assume l is large enough that 1∕l <dist
(U-,UC )
0
. Thus for
x ∈ U_{0},
∫
^u ∗ϕl(x) = u(x − y )ϕl(y )dy. (37.0.1)
B (0,1l)
(37.0.1)
The theorem is proved if it can be shown that D^{α}
|||| ||||
||Dα (^u∗ϕl)− D αu||Lp(U0) = || ^Dαu ∗ϕl − D αu||p
|||| ||L|| (U0)
≤ || ^Dαu ∗ϕl − D^αu||p n → 0.
L (ℝ )
This proves the theorem.
As part of the proof of the theorem, the following corollary was established.
Corollary 37.0.6Let U_{0}and U be as in the above theorem. Then for all l large enoughand ϕ_{l}a mollifier,
( )
D α(^u ∗ϕ ) = D^αu∗ ϕ (37.0.2)
l l
(37.0.2)
as distributions on C_{c}^{∞}
(U0)
.
Definition 37.0.7Let U be an open set. C^{∞}
(U )
denotes the set of functions whichare defined and infinitely differentiable on U.
Note that f
(x)
=
1x
is a function in C^{∞}
(0,1)
. However, it is not equal to the
restriction to
(0,1)
of some function which is in C_{c}^{∞}
(ℝ)
. This illustrates the distinction
between C^{∞}
(U)
and C^{∞}
(-)
U
. The set, C^{∞}
(--)
U
is a subset of C^{∞}
(U)
. The following
theorem is known as the Meyer Serrin theorem.
Theorem 37.0.8(Meyer Serrin) Let U be an open subset of ℝ^{n}. Then if δ > 0
and u ∈ X^{m,p}
(U)
, there exists J ∈ C^{∞}
(U )
such that
||J − u ||
_{m,p,U}< δ.
Proof:Let
⋅⋅⋅
U_{k}⊆U_{k}⊆ U_{k+1}
⋅⋅⋅
be a sequence of open subsets of U whose union
equals U such that U_{k} is compact for all k. Also let U_{−3} = U_{−2} = U_{−1} = U_{0} = ∅.
Now define V_{k}≡ U_{k+1}∖U_{k−1}. Thus
{Vk}
_{k=1}^{∞} is an open cover of U. Note
the open cover is locally finite and therefore, there exists a partition of unity
subordinate to this open cover,
{ηk}
_{k=1}^{∞} such that each spt
(ηk)
∈ C_{c}
(Vk)
.
Let ψ_{m} denote the sum of all the η_{k} which are non zero at some point of V_{m}.
Thus
----- ∞ ∑∞
spt(ψm ) ⊆ Um+2 ∖ Um−2,ψm ∈ C c (U ), ψm (x) = 1 (37.0.3)
m=1
(37.0.3)
for all x ∈ U, and ψ_{m}u ∈ W^{m,p}
(U )
m+2
.
Now let ϕ_{l} be a mollifier and consider
∑∞
J ≡ uψm ∗ϕlm (37.0.4)
m=0
(37.0.4)
where l_{m} is chosen large enough that the following two conditions hold:
Now apply the monotone convergence theorem to conclude that
||J − u||
_{m,p,U}≤ δ. This
proves the theorem.
Note that J = 0 on ∂U. Later on, you will see that this is pathological.
In the study of partial differential equations it is the space W^{m,p}
(U )
which is of the
most use, not the space X^{m,p}
(U )
. This is because of the density of C^{∞}
(-)
U
.
Nevertheless, for reasonable open sets, U, the two spaces coincide.
Definition 37.0.9An open set, U ⊆ ℝ^{n}is said to satisfy the segment condition iffor all z ∈U, there exists an open set U_{z}containing z and a vector a suchthat
-- --
U ∩ Uz + ta ⊆ U
for all t ∈
(0,1)
.
PICT
You can imagine open sets which do not satisfy the segment condition. For example, a
pair of circles which are tangent at their boundaries. The condition in the above
definition breaks down at their point of tangency.
Here is a simple lemma which will be used in the proof of the following
theorem.
Lemma 37.0.10If u ∈ W^{m,p}
(U )
and ψ ∈ C_{c}^{∞}
n
(ℝ )
, then uψ ∈ W^{m,p}
(U)
.
Proof: Let
|α|
≤ m and let ϕ ∈ C_{c}^{∞}
(U )
. Then
∫
(D (uψ))(ϕ) ≡ − uψϕ dx
xi ∫U ,xi
( )
= − U u (ψϕ),xi − ϕ ψ,xi dx
∫
= (Dxiu)(ψϕ)+ uψ,xiϕdx
∫ U
= (ψDxiu+ uψ,xi)ϕdx
U
Therefore, D_{xi}
(uψ)
= ψD_{xi}u + uψ_{,xi}∈ L^{p}
(U )
. In other words, the product rule holds.
Now considering the terms in the last expression, you can do the same argument with
each of these as long as they all have derivatives in L^{p}
(U )
. Therefore, continuing this
process the lemma is proved.
Theorem 37.0.11Let U be an open set and suppose there exists a locally finitecovering^{2}of Uwhich is of the form
{Ui}
_{i=1}^{∞}such that each U_{i}is a bounded open set whichsatisfies the conditions of Definition 37.0.9. Thus there exist vectors, a_{i}such that for allt ∈
(0,1)
,
U-∩ U + ta ⊆ U.
i i
Then C^{∞}
(-)
U
is dense in X^{m,p}
(U)
and so W^{m,p}
(U)
= X^{m,p}
(U)
.
Proof:Let
{ψ }
i
_{i=1}^{∞} be a partition of unity subordinate to the given open cover
with ψ_{i}∈ C_{c}^{∞}
(U )
i
and let u ∈ X^{m,p}
(U )
. Thus
∞∑
u = ψku.
k=1
Consider U_{k} for some k. Let a_{k} be the special vector associated with U_{k} such
that
-- ---
tak + U ∩Uk ⊆ U (37.0.7)
(37.0.7)
for all t ∈
(0,1)
and consider only t small enough that
spt(ψk)− tak ⊆ Uk (37.0.8)
(37.0.8)
Pick l
(t)
> 1∕t which is also large enough that
( ) -----------(------)------
tak + U-∩ Uk + B 0,-1-- ⊆ U,spt(ψk)+ B 0,--1-- − tak ⊆ Uk. (37.0.9)
l(t) l(tk)