BRUCE K. DRIVER
†
19.
Weak and Strong Derivatives
For this section, let
Ω
be an open subset of
R
d
, p,q,r
∈
[1
,
∞
]
,L
p
(
Ω
)=
L
p
(
Ω
,
B
Ω
,m
)
and
L
p
loc
(
Ω
)=
L
p
loc
(
Ω
,
B
Ω
,m
)
,
where
m
is Lebesgue measure on
B
R
d
and
B
Ω
is the Borel
σ
—a
lgebraon
Ω
.
If
Ω
=
R
d
,
we will simply write
L
p
and
L
p
loc
for
L
p
(
R
d
)
and
L
p
loc
(
R
d
)
respectively. Also let
h
f,g
i
:=
Z
Ω
fgdm
for any pair of measurable functions
f,g
:
Ω
→
C
such that
fg
∈
L
1
(
Ω
)
.
For
example, by Hölder’s inequality, if
h
f,g
i
is de
f
ned for
f
∈
L
p
(
Ω
)
and
g
∈
L
q
(
Ω
)
when
q
=
p
p
−
1
.
De
f
nition 19.1.
A sequence
{
u
n
}
∞
n
=1
⊂
L
p
loc
(
Ω
)
is said to converge to
u
∈
L
p
loc
(
Ω
)
if
lim
n
→∞
k
u
−
u
n
k
L
q
(
K
)
=0
for all compact subsets
K
⊂
Ω
.
The following simple but useful remark will be used (typically without further
comment) in the sequel.
Remark
19.2
.
Suppose
r, p, q
∈
[1
,
∞
]
are such that
r
−
1
=
p
−
1
+
q
−
1
and
f
t
→
f
in
L
p
(
Ω
)
and
g
t
→
g
in
L
q
(
Ω
)
as
t
→
0
,
then
f
t
g
t
→
fg
in
L
r
(
Ω
)
.
Indeed,
k
f
t
g
t
−
fg
k
r
=
k
(
f
t
−
f
)
g
t
+
f
(
g
t
−
g
)
k
r
≤
k
f
t
−
f
k
p
k
g
t
k
q
+
k
f
k
p
k
g
t
−
g
k
q
→
0
as
t
→
0
19.1.
Basic De
f
nitions and Properties.
De
f
nition 19.3
(Weak Di
f
erentiability)
.
Let
v
∈
R
d
and
u
∈
L
p
(
Ω
)(
u
∈
L
p
loc
(
Ω
))
then
∂
v
u
is said to
exist weakly
in
L
p
(
Ω
)(
L
p
loc
(
Ω
))
if there exists a function
g
∈
L
p
(
Ω
)(
g
∈
L
p
loc
(
Ω
))
such that
(19.1)
h
u, ∂
v
φ
i
=
−
h
g,φ
i
for all
φ
∈
C
∞
c
(
Ω
)
.
The function
g
if it exists will be denoted by
∂
(
w
)
v
u.
Similarly if
α
∈
N
d
0
and
∂
α
is
as in Notation 11.10, we say
∂
α
u
exists weakly
in
L
p
(
Ω
)(
L
p
loc
(
Ω
))
i
f
there exists
g
∈
L
p
(
Ω
)(
L
p
loc
(
Ω
))
such that
h
u, ∂
α
φ
i
=(
−
1)

α

h
g,φ
i
for all
φ
∈
C
∞
c
(
Ω
)
.
More generally if
p
(
ξ
)=
P

α

≤
N
a
α
ξ
α
is a polynomial in
ξ
∈
R
n
,
then
p
(
∂
)
u
exists
weakly
in
L
p
(
Ω
)(
L
p
loc
(
Ω
))
i
f
there exists
g
∈
L
p
(
Ω
)(
L
p
loc
(
Ω
))
such that
(19.2)
h
u, p
(
−
∂
)
φ
i
=
h
g,φ
i
for all
φ
∈
C
∞
c
(
Ω
)
and we denote
g
by
w
−
p
(
∂
)
u.
By Corollary 11.28, there is at most one
g
∈
L
1
loc
(
Ω
)
such that Eq. (19.2) holds,
so
w
−
p
(
∂
)
u
is well de
f
ned.
Lemma 19.4.
Let
p
(
ξ
)
be a polynomial on
R
d
,k
=deg(
p
)
∈
N
,
and
u
∈
L
1
loc
(
Ω
)
such that
p
(
∂
)
u
exists weakly in
L
1
loc
(
Ω
)
.
Then
(1)
supp
m
(w
−
p
(
∂
)
u
)
⊂
supp
m
(
u
)
,
where
supp
m
(
u
)
is the essential support of
u
relative to Lebesgue measure, see De
f
nition 11.14.
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.
 Three '10
 Smith
 Algebra, Derivative, The Land, Trigraph, Ω, rd, BRUCE K. DRIVER

Click to edit the document details