±
±
±
LECTURE
NOTES
FOR
18.155,
FALL
2004
63
10.
Sobolev
embedding
The
properties
of
Sobolev
spaces
are
brieﬂy
discussed
above.
If
m
is
a
positive
integer
then
u
→
H
m
(
R
n
)
‘means’
that
u
has
up
to
m
derivatives
in
L
2
(
R
n
)
.
The
question
naturally
arises
as
to
the
sense
in
which
these
‘weak’
derivatives
correspond
to
oldfashioned
‘strong’
derivatives.
Of
course
when
m
is
not
an
integer
it
is
a
little
harder
to
imagine
what
these
‘fractional
derivatives’
are.
However
the
main
result
is:
Theorem
10.1
(Sobolev
embedding)
.
If
u
→
H
m
(
R
n
)
where
m > n/
2
then
u
→ C
0
0
(
R
n
)
,
i.e.,
H
m
(10.1)
(
R
n
)
C
0
0
(
R
n
)
,
m > n/
2
.
Proof.
By
deFnition,
u
→
H
m
(
R
n
)
means
v
→ S
(
R
n
)
and
⊂
π
≤
m
ˆ
u
(
π
)
→
L
2
(
R
n
).
Suppose
Frst
that
u
→ S
(
R
n
).
The
±ourier
inversion
formula
shows
that
ix
·
α
ˆ
²
(2
)
n

u
(
x
)

=
±
e
u
(
π
)
dπ
1
/
2
∀
³±
⊂
π
≤
2
m

ˆ(
π
)

2
dπ
·
´
µ
⊂
π
≤
−
2
m
dπ
¶
1
/
2
.
u
R
n
R
n
Now,
if
m
>
n/
2
then
the
second
integral
is
Fnite.
Since
the
Frst
integral
is
the
norm
on
H
m
(
R
n
)
we
see
that
(10.2)
sup

u
(
x
)

=
∈
u
∈
∀
(2
)
−
n
∈
u
∈
H
m
, m > n/
2
.
L
R
n
This
is
all
for
u
→
S
(
R
n
),
but
S
(
R
n
)
γ
±
H
m
(
R
n
)
is
dense.
The
estimate
(10.2)
shows
that
if
u
j
±
u
in
H
m
(
R
n
)
,
with
u
j
→
S
(
R
n
),
0
then
u
j
±
u
in
C
0
(
R
n
)
.
In
fact
u
=
u
in
S
(
R
n
)
since
u
j
±
u
in
0
L
2
(
R
n
)
and
u
j
±
u
in
C
0
(
R
n
)
both
imply
that
R
u
j
α
converges,
so
u
j
α
±
uα
=
u
α
²
α
→ S
(
R
n
)
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '13
 Melrose
 Derivative, Mac OS X, Sobolev inequality

Click to edit the document details