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.