5 Sobolev Imbedding Theorems

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8 Οκτ 2013 (πριν από 3 χρόνια και 10 μήνες)

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5 Sobolev Imbedding Theorems
Target of the following investigations is to show imbeddings of
W
m,p
(Ω) (or
W
m,p
c
(Ω))
into Banach spaces of the types:
(1)
W
j,q
(Ω)
,j

m
(in particular
L
q
(Ω)),
(2)
W
j,q
(Ω
k
)
,j

m
and Ω
k
= Ω

H
k
,H
k
k
-dimensional plane,
(3)
C
j
b
(Ω) (see
§
1),
j

m,
(4)
C
j
(Ω

) (see
§
1),
j

m,
and others.
The imbedding properties depend on properties of Ω.
Definition 5.1


R
n
is said to satisfy the
cone condition
if there exists a finite cone
C
,such that each
x


is the vertex of a finite cone
C
x
contained in

congruent to
C
.
This means that
C
x
is obtained from
C
by rigid motion.
Remark:
A finite cone
C
has the form
C
=
{
x

R
n
:
x
= 0
or
0
<
￿
x
￿
2

￿,

(
x,v
)
<
κ
2
}
,
where
￿ >
0
,v

R
n
,v
￿
= 0
and
0
< κ

π.κ
is called aperture angle,
v
axis direction,
￿
height.
Lemma 5.2
Let


R
n
satisfy the cone condition.Let
κ
and
￿
be the aperture angle
and height specified by the cone condition.Let
m

N
.Then there exists a constant
K
(depending on
n,m
and
κ,￿
) such that for
u

C

(Ω)
,x


and every
r
with
0
< r

￿
we have
|
u
(
x
)
| ≤
K


￿
|
α
| ≤
m

1
r
|
α
|−
n
￿
C
x,r
|
D
α
u
(
y
)
|
dy
+
￿
|
α
|
=
m
￿
C
x,r
|
D
α
u
(
y
)
| ￿
x

y
￿
m

n
dy


,
where
C
x,r
=
{
y

C
x
:
￿
x

y
￿ ≤
r
}
,C
x
is a cone congruent to
C
having vertex at
x
.
Proof
.For
x


,y

C
x,r
define the function
f
(
t
) =
u
(
tx
+(1

t
)
y
)
,t

[0
,
1]
.
23
Taylor’s formula yields
f
(1) =
m

1
￿
j
=0
1
j
!
f
(
j
)
(0) +
1
(
m

1)!
￿
1
0
(1

t
)
m

1
f
(
m
)
(
t
)
dt.
Furthermore,
f
(
j
)
(
t
) =
￿
|
α
|
=
j
j
!
α
!
D
α
u
(
tx
+(1

t
)
y
) (
x

y
)
α
,
where
α
!=
α
1
!
∙ ∙ ∙
α
n
!and (
x

y
)
α
= (
x
1

y
1
)
α
1
∙ ∙ ∙
(
x
n

y
n
)
α
n
.
Thus we obtain
|
u
(
x
)
|
=
|
f
(1)
| ≤
￿
|
α
| ≤
m

1
1
α
!
|
D
α
u
(
y
)
| ￿
x

y
￿
|
α
|
+
￿
|
α
|
=
m
m
α
!
￿
x

y
￿
m
￿
1
0
(1

t
)
m

1
|
D
α
u
(
tx
+(1

t
)
y
)
|
dt.
If
C
has volume
c￿
n
,then
C
x,r
has volume
cr
n
.Integrating along
y
over
C
x,r
leads to
cr
n
|
u
(
x
)
| ≤
￿
|
α
| ≤
m

1
r
|
α
|
α
!
￿
C
x,r
|
D
α
u
(
y
)
|
dy
+
￿
|
α
|
=
m
m
α
!
￿
C
x,r
￿
x

y
￿
m
￿
1
0
(1

t
)
m

1
|
D
α
u
(
tx
+(1

t
)
y
)
|
dt dy.
We consider now the second double integral.First we can change the order of integration,
and then we substitute
z
=
tx
+ (1

t
)
y.
So we have
z

x
= (1

t
)(
y

x
) and
dz
= (1

t
)
n
dy,
which leads to
￿
C
x,r
￿
x

y
￿
m
￿
1
0
(1

t
)
m

1
|
D
α
u
(
tx
+(1

t
)
y
)
|
dt dy
=
￿
1
0
(1

t
)

n
(1

t
)

1
￿
C
x,
(1

t
)
r
￿
z

x
￿
m
|
D
α
u
(
z
)
|
dz dt.
Changing the order of integration the above integral is equal to
￿
C
x,r
￿
x

z
￿
m
|
D
α
u
(
z
)
|
￿
1

(
￿
z

x
￿
/r
)
0
(1

t
)

n

1
dt dz

r
n
n
￿
C
x,r
￿
x

z
￿
m

n
|
D
α
u
(
z
)
|
dz.
Now we have
|
u
(
x
)
| ≤
1
cr
n
￿
|
α
| ≤
m

1
r
|
α
|
α
!
￿
C
x,r
|
D
α
u
(
y
)
|
dy
+
1
cr
n
￿
|
α
|
=
m
mr
n
α
!
n
￿
C
x,r
￿
x

z
￿
m

n
|
D
α
u
(
z
)
|
dz
24
and hence (with an appropriate
K
)
|
u
(
x
)
| ≤
K


￿
|
α
| ≤
m

1
r
|
α
|−
n
￿
C
x,r
|
D
α
u
(
y
)
|
dy
+
￿
|
α
|
=
m
￿
C
x,r
|
D
α
u
(
y
)
| ￿
x

y
￿
m

n
dy


.
￿
We will use an interpolation inequality,which is of interest by itself.
Theorem 5.3
Let
1

p < q < r
≤ ∞
,so that
1
q
=
θ
p
+
1

θ
r
(

)
for some
θ

]0
,
1[
.
If
u

L
p
(Ω)

L
r
(Ω)
,
then
u

L
q
(Ω)
and
￿
u
￿
q
≤ ￿
u
￿
θ
p
￿
u
￿
1

θ
r
.
Proof
.Let
s
=
p
θ
q
.
Then by (

) we have
s
=
p
θq
=
1
q
￿
1
1
q

1

θ
r
￿

1
q
q
= 1
.
It is straightforward to check that (for
r <

)
q
1

θ
r
=
p

θq
p
=
s

1
s
,
hence
s
￿
=
s
s

1
=
r
(1

θ
)
q
By H¨older’s inequality we get (since
1
s
+
1
s
￿
= 1)
￿
u
￿
q
q
=
￿

|
u
(
x
)
|
θ
q
|
u
(
x
)
|
(1

θ
)
q
dx

￿
￿

|
u
(
x
)
|
θqs
dx
￿
1
/s
￿
￿

|
u
(
x
)
|
(1

θ
)
qs
￿
dx
￿
1
/s
￿
=
￿
u
￿
θ
q
p
￿
u
￿
(1

θ
)
q
r
,
and the result is shown if
r <

.
The proof is similar if
r
=

.
￿
Now we can formulate a first Sobolev imbedding result.
Theorem 5.4 (Sobolev)
Let


R
n
satisfy the cone condition,and for
1

k

n
let

k
= Ω

H
k
,H
k
a
k
-
dimensional plane (if
k
=
n
,then
Ω = Ω
k
)
.
Let
j

0
,m

1
be integers,and
1

p <

.
Then the following results hold:
25
If
mp > n
or
(
m
=
n
and
p
= 1
),then
W
j
+
m,p
(Ω)
￿

C
j
b
(Ω)
,
and
W
j
+
m,p
(Ω)
￿

W
j,q
(Ω
k
)
for
p

q
≤ ∞
.
In particular,
W
m,p
(Ω)
￿

L
q
(Ω)
for
p

q
≤ ∞
.
Proof
.We begin by observing that it is sufficient to prove the imbeddings for the
special case
j
= 0
.
The general case follows by applying this special case to the derivative
D
α
u
of
u
for
|
α
| ≤
j.
For example,if the imbedding
W
m,p
(Ω)
￿

L
q
(Ω) is shown,then
for any
u

W
j
+
m,p
(Ω) we have
D
α
u

W
m,p
(Ω) for
|
α
| ≤
j
,and hence
D
α
u

L
q
(Ω)
.
Therefore
u

W
j,q
(Ω) and
￿
u
￿
j,q
=


￿
|
α
|≤
j
￿
D
α
u
￿
q
0
,q


1
/q

K
1


￿
|
α
| ≤
j
￿
D
α
u
￿
p
m,p


1
/p

K
2
￿
u
￿
j
+
m,p
.
So we can assume
j
= 0
.
We deal at first with the imbedding into
C
j
b
(Ω)
.
Let
u

W
m,p
(Ω)

C

(Ω) (not necessarily bounded).
We have to show that
u
(
x
)

K
￿
u
￿
m,p
for all
x


.
(

)
In case of
m
=
n
and
p
= 1 this follows directly from Lemma 5.2.Now let
p >
1 and
mp > n.
Applying H¨older’s inequality to the two integrals of Lemma 5.2 we have with
r
=
￿
￿
C
x,￿
|
D
α
u
(
y
)
|
dy

vol(
C
x,￿
)
1
/q
￿
D
α
u
￿
p,C
x,￿
and
￿
C
x,￿
￿
D
α
u
￿
p,C
x,￿
￿
￿
C
x,￿
￿
x

y
￿
(
m

n
)
q
dy
￿
1
/q
.
Since vol(
C
x,￿
) =
c￿
n
and
￿
|
α
|−
n
(
￿
n
c
)
1
/q
=
c
1
/q
￿
|
α
|−
n/p
,
we obtain from Lemma 5.2
|
u
(
x
)
| ≤
K


￿
|
α
| ≤
m

1
c
1
/q
￿
|
α
|−
n
p
￿
D
α
u
￿
p,C
x,￿
+
￿
|
α
|
=
m
￿
D
α
u
￿
p,C
x,￿
￿
￿
C
x,￿
￿
x

y
￿
(
m

n
)
q
dy
￿
1
q


Since by
mp > n
it follows (
m

n
)
q
= (
m

n
)
p
p

1
>
n
p

1

np
p

1
>

n
and hence
￿
C
x,￿
￿
x

y
￿
(
m

n
)
q
dy
is finite,we obtain
|
u
(
x
)
| ≤
K
￿
|
α
| ≤
m
￿
D
α
u
￿
p,C
x,￿
(
∗∗
)
,
26
(where
K
has to be changed only depending on
n,m,p,￿
).
Since
C
x,￿

Ω it follows
|
u
(
x
)
| ≤
K
￿
u
￿
m,p
,the inequality (

)
.
By Theorem 4.3 each
u

W
m,p
(Ω) is the limit of a Cauchy sequence of continuous functions.By (

) this
Cauchy sequence converges uniformly to a continuous function.Hence
u
is equal almost
everywhere to a continuous function on Ω.Finally this implies (once again applying (

)),
that
u

C
b
(Ω)
.
The remaining case
p
= 1
,m > n
follows similarly,where we have to
note that
C
x,￿
is a bounded cone.
Now we study the imbedding into
W
j,q
(Ω
k
)
.
We can assume
j
= 0.Denote Ω
k,￿
=
{
x

R
n
:
d
(
x,

k
)
< ￿
}
.
Let
u

W
m,p
(Ω),and extend
u
and
D
α
u
to be zero outside Ω.
Obviously
C
x,￿

B
￿
(
x
)
.
Denote by
d
k
x
the
k
-dimensional Lebesgue measure.Applying
(
∗∗
) we obtain
￿

k
|
u
(
x
)
|
p
d
k
x

K
￿
|
α
| ≤
m
￿

k
￿
B
￿
(
x
)
|
D
α
u
(
y
)
|
p
dy d
k
x
=
K
￿
|
α
| ≤
m
￿

k,￿
￿
￿
H

B
￿
(
y
)
d
k
x
￿
|
D
α
u
(
y
)
|
p
dy

K
1
￿
u
￿
p
m,p,

.
Hence we have the imbedding
W
m,p
(Ω)
￿

L
p
(Ω
k
)
.
Inequality (

) yields the imbedding
W
m,p
(Ω)
￿

L

(Ω
k
)
.
By Theorem 5.3 it follows that
W
m,p
(Ω) is imbedded into
L
q
(Ω
k
)
.
￿
The case that
mp

n
is more involved.We have to derive some further auxiliary results.
Let
χ
r
:=
χ
B
r
(0)
,
where
B
r
(0) =
{
x

R
n
:
￿
x
￿
< r
}
,
and
ω
m
(
x
) =
￿
x
￿
m

n
.
Observe
that if
m

n
and 0
< r

1 then
χ
r
(
x
)

χ
r
ω
m
(
x
)

ω
m
(
x
)
.
Lemma 5.5
Let
1

p <

,
1

k

n
and
n

mp < k.
There exists a constant
K
,such that for every
r >
0
,k
-dimensional plane
H

R
n
and
v

L
p
(
R
n
)
we have
(
χ
r
ω
m
)
∗ |
v
| ∈
L
p
(
H
)
and
￿
χ
r
ω
m
∗ |
v
| ￿
p,H

K r
m

(
n

k
)
/p
￿
v
￿
p,
R
n
.
Proof
.If
p >
1,then H¨older’s inequality gives
χ
r
ω
m
∗ |
v
|
(
x
) =
￿
B
r
(
x
)
|
v
|
(
y
)
￿
x

y
￿

s
￿
x

y
￿
s
+
m

n
dy

￿
￿
B
r
(
x
)
|
v
(
y
)
|
p
￿
x

y
￿

sp
dy
￿
1
p
￿
￿
B
r
(
x
)
￿
x

y
￿
(
s
+
m

n
)
q
dy
￿
1
q

K r
s
+
m

n
p
￿
￿
B
r
(
x
)
|
v
(
y
)
|
p
￿
x

y
￿

sp
dy
￿
1
p
,
27
provided
s
+
m

n
p
>
0
.
(Note that
k
p
+
m

n
p
>
0
.
) The same estimate holds for
p
= 1,
provided
s
+
m

n >
0.Integrating the above estimate over
H
with the
k
-dimensional
Lebesgue measure
d
k
x
yields
￿
χ
r
ω
m
∗ |
v
| ￿
p
p,H
=
￿
H
|
χ
r
ω
m
∗ |
v
|
(
x
)
|
p
d
k
x

K r
(
s
+
m
)
p

n
￿
H
￿
B
r
(
x
)
|
v
(
y
)
|
p
￿
x

y
￿

sp
dy d
k
x

K r
(
s
+
m
)
p

n
r
k

sp
￿
v
￿
p
p,
R
n
=
K r
mp

(
n

k
)
￿
v
￿
p
p,
R
n
provided
k > sp
.
Since
n

mp < k
there exists
s
satisfying
n
p

m < s <
k
p
.
Hence the two additional
assumptions can be satisfied,and the inequality is shown.
￿
We have to use an improvement of Lemma 5.5.The proof is omitted.(One has to apply
an interpolation theorem of Marcinkiewicz.)
Lemma 5.6
Let
1
< p <

,mp < n,n

mp < k

n
and
p

=
kp
n

mp
.
There exists
a constant
K
such that for every
k
-dimensional plane
H
in
R
n
and every
v

L
p
(
R
n
)
we
have
ω
m
∗ |
v
| ∈
L
p

(
H
)
and
￿
χ
1
∗ |
v
| ￿
p

,H
≤ ￿
χ
1
ω
m
∗ |
v
| ￿
p

,H
≤ ￿
ω
m
∗ |
v
| ￿
p

,H

K
￿
v
￿
p,
R
n
.
By
m

mp < n,
the first and second inequalities are obvious.The proof of the third
inequality is not trivial,and is based on Lemma 5.5.
Theorem 5.7 (Sobolev)
Let


R
n
satisfy the cone condition.For
1

k

n
let

k
= Ω

H
k
,H
k
a
k
-
dimensional plane.Let
j

0
,m

1
be integers and
1

p <

.
Then the following
holds:
If
mp < n
and either
n

mp < k

n
or
(
p
= 1
and
n

m

k

n
)
,
then
W
j
+
m,p
(Ω)
￿

W
j,q
(Ω
k
)
for
p

q

p

=
kp
n

mp
.
In particular,
W
m,p
(Ω)
￿

L
q
(Ω)
for
p

q

p

=
np
n

mp
.
Proof
.Here we deal only with the case
p >
1
.
Let
u

C

(Ω) and extend
u
and all its
derivatives to be zero on Ω
c
.
Applying Lemma 5.2 with
r
=
￿
and replacing
C
x,￿
by the
larger set
B
1
(
x
) (we can assume that
￿

1) we obtain
|
u
(
x
)
| ≤
K


￿
|
α
|≤
m

1
χ
1
∗ |
D
α
u
|
(
x
) +
￿
|
α
|
=
m
χ
1
ω
m
∗ |
D
α
u
|
(
x
)


28
Now,if
1
q
=
θ
p
+
1

θ
p

for some
θ

[0
,
1]
,
Theorem 5.3 yields
￿
u
￿
q,

k
≤ ￿
u
￿
θ
p,H
k
￿
u
￿
1

θ
p

,H
k
.
Lemma 5.5 (and the inequalities before Lemma 5.5) implies
￿
u
￿
θ
p,H
k

K


￿
|
α
|≤
m
￿
D
α
u
￿
p,
R
n


θ
and Lemma 5.6 implies
￿
u
￿
1

θ
p

,H
k

˜
K


￿
|
α
|≤
m
￿
D
α
u
￿
p,
R
n


1

θ
.
Hence we have
￿
u
￿
q,

k

K
￿
u
￿
m,p,

(

)
for
u

W
m,p
(Ω)

C

(Ω)
.
As in the proof of Theorem 5.4 it is sufficient to have shown
inequality (

) for
u

W
m,p
(Ω)

C

(Ω)
.
￿
Remarks:
(1) The case
p
= 1 in Theorem 5.7 requires a technique which is called imbedding by
averaging.
(2) If
mp
=
n
one can prove that
W
j
+
m,p
(Ω)
￿

W
j,q
(Ω
k
)
holds for
p

q <

.
(3) The statements of Theorem 5.4 and Theorem 5.7 are valid for
arbitrary
domains
Ω if the
W
-space undergoing the imbedding is replaced with the corresponding
W
c
-space.This follows from the fact that Theorem 5.4 and Theorem 5.7 hold for
Ω =
R
n
,and we have the result of Lemma 4.5.
The Sobolev imbedding theorems tell us that
W
m,p
c
(
R
n
)
￿

L
q
(
R
n
) for certain values
q
depending on
m,p
and
n
.For such
q
there is a constant
K
such that for all
ϕ

C

c
(
R
n
)
￿
ϕ
￿
q

K
￿
ϕ
￿
m,p
.
Sobolev’s inequalities extend this inequality for certain seminorms
|
ϕ
|
m,p
in place of
￿
ϕ
￿
m,p
,where
ϕ

C

c
(
R
n
)
.
For 1

p <

and for integers
j,
0

j

m
we define
|
u
|
j,p
:=


￿
|
α
|
=
j
￿

|
D
α
u
(
x
)
|
p
dx


1
p
=


￿
|
α
|
=
j
￿
D
α
u
￿
p
0
,p


1
p
,u

W
m,p
(Ω)
.
29
Obviously
|
u
|
0
,p
=
￿
u
￿
0
,p
=
￿
u
￿
p
is a norm on
L
p
(Ω)
.
Moreover,
￿
u
￿
m,p
=
￿
m
￿
j
=0
|
u
|
p
j,p
￿
1
p
| ∙ |
j,p
is a seminorm,
j

1
.
(Consider,for example,
u
=
c,c
￿
= 0
.
)
For
j
=
m
and certain domains
| |
m,p
is a norm on
W
m,p
c
(Ω)
.
We say that Ω

R
n
has
finite width
,if Ω lies between two parallel planes of dimension
n

1.
Theorem 5.8 (Poincar´e’s Inequality)
If


R
n
has finite width,then there exists a constant
K
=
K
(
p
)
such that
￿
ϕ
￿
0
,p
=
￿
ϕ
￿
p

K
|
ϕ
|
1
,p
=
K
￿
n
￿
i
=1
￿

|
D
i
ϕ
(
x
)
|
p
dx
￿
1
p
for all
ϕ

C

c
(Ω)
.
Proof
.We can assume that Ω lies between the planes
x
n
= 0 and
x
n
=
c >
0
.
Let
x
= (˜
x,x
n
),where ˜
x
= (
x
1
,...,x
n

1
)

R
n

1
.
For
ϕ

C

c
(Ω) we get
ϕ
(
x
) =
￿
x
n
0
d
dt
ϕ

x,t
)
dt
=
￿
x
n
0
D
n
ϕ

x,t
)
dt
By H¨older’s inequality we have
￿
￿
￿
￿
￿
x
n
0
D
n
ϕ

x,t
)
dt
￿
￿
￿
￿
p

￿
x
n
0
|
D
n
ϕ

x,t
)
|
p
dt

￿
χ
[0
,x
n
]
￿
p
q

x
p

1
n
￿
x
n
0
|
D
n
ϕ

x,t
)
|
p
dt.
Hence
￿
ϕ
￿
p
p
=
￿
R
n

1
￿
c
0
|
ϕ
(
x
)
|
p
dx
n
d
˜
x

￿
R
n

1
￿
c
0
x
p

1
n
dx
n
￿
c
0
|
D
n
ϕ

x,t
)
|
p
dt dx
n
d
˜
x

c
p
p
|
ϕ
|
p
1
,p
.
Thus,choose
K
=
c
p

p
.
￿
Corollary 5.9
Suppose that

has finite width.Then
|∙|
m,p
is a norm on
W
m,p
c
(Ω)
,
which
is equivalent to
￿ ∙ ￿
m,p
.
Proof
.Let
ϕ

C

c
(Ω)
.
By Theorem 5.8 we get
|
ϕ
|
p
1
,p
≤ ￿
ϕ
￿
p
1
,p
=
￿
ϕ
￿
p
p
+
|
ϕ
|
p
1
,p

(
K
p
+1)
|
ϕ
|
p
1
,p
.
30
Successive iterations of this inequality to derivatives
D
α
ϕ,
|
α
| ≤
m

1
,
leads to
|
ϕ
|
p
m,p
≤ ￿
ϕ
￿
p
m,p

˜
K
|
ϕ
|
p
m,p
.
By completion,this holds for any
u

W
m,p
c
(Ω).
￿
Recall we had the inequality
￿
ϕ
￿
q

K
￿
ϕ
￿
m,p
for
ϕ

C

c
(
R
n
) in Theorem 5.4 and
Theorem 5.7.What about
￿
ϕ
￿
q

K
|
ϕ
|
m,p
?
Of course,now Ω =
R
n
.
Suppose that
￿
ϕ
￿
q
q

K
q
|
ϕ
|
q
m,p
for all
ϕ

C

c
(
R
n
)
.
Then this has to hold also for
ϕ
t
(
x
) =
ϕ
(
tx
)
,
0
< t <

.
Furthermore we have
￿
ϕ
t
￿
q
=
t

n
q
￿
ϕ
￿
q
and
￿
D
α
ϕ
t
￿
p
=
t
m

n
p
￿
D
α
ϕ
￿
p
if
|
α
|
=
m.
Hence by the assumption
￿
ϕ
￿
q
q
=
t
n
￿
ϕ
t
￿
q
q

t
n
K
q
|
ϕ
t
|
q
m,p
=
t
n
K
q


￿
|
α
|
=
m
￿
D
α
ϕ
t
￿
p
p


q
p
=
K
q
t
n


(
t
m

n
p
)
p
￿
|
α
|
=
m
￿
D
α
ϕ
￿
p
p


q
p
=
K
q
t
n
+
mq

nq
p


￿
|
α
|
=
m
￿
D
α
ϕ
￿
p
p


q
p
=
K
q
t
n
+
mq

nq
p
|
ϕ
|
q
m,p
This has to hold for each
t >
0,unless the exponent of
t
is zero,that is,unless
q
=
p

=
np
n

mp
.
Thus the assumption is only possible provided
mp < n
and
q
=
p

=
np
n

mp
.
We have just shown the necessity of the following equivalence condition.
Theorem 5.10 (Sobolev inequality)
If
mp < n
,there exists a constant
K
such that
￿
ϕ
￿
q

K
|
ϕ
|
m,p
for all
ϕ

C

c
(
R
n
)
,
if and onluy if
q
=
p

=
np
n

mp
.
31