1
Lecture 12
Definition 1
Let
α
be a multiindex. The function
D
α
f
(which is actually an equivalence
class of functions) is called the
α
th weak derivative
of
f
if for any test function
φ
∈
C
∞
0
(Ω)
D
α
f, φ
≡
Ω
D
α
f
(
x
)
φ
(
x
)
dx
= (

1)

α

f, D
α
φ
An equivalent norm on
H
s
is defined by

u
2
H
s
=

α
≤
s
D
α
u
2
L
2
Examples:
1) Consider
u
(
x
) on Ω = (0
,
2) definde by
u
(
x
) =
x
2
0
< x
≤
1
2
x
2

2
x
+ 1
1
< x <
2
Note
u
∈
C
1
(Ω) and
u
(
x
) =
2
x
0
< x
≤
1
4
x

2
1
< x <
2
u
does not exist in the classical sense at 1. The weak derivative of
u
is
u
(
x
) =
2
0
< x
≤
1
4
1
< x <
2
u, u , u
∈
L
2
(0
,
2). However, the distributional derivative of
u
is 2
δ
(
x

1)
∈
L
2
(0
,
2).
Thus,
u
∈
H
2
(0
,
2)
u
2
H
2
=
2
0
(
u
2
+ (
u
)
2
+ (
u
)
2
)
dx
= 71
.
37
u
∈
H
1
(0
,
2)
u
2
H
1
=
2
0
((
u
)
2
+ (
u
)
2
)
dx
= 39
u
∈
H
0
(0
,
2)
u
2
H
0
=
2
0
(
u
)
2
dx
= 20
2) Consider
u
defined on Ω

(

1
,
1)
×
(

1
,
1) defined by
u
(
x, y
) =
x
x >
0
0
x
≤
0
Let
φ
∈
C
∞
0
(Ω). Then
Ω
∂u
∂y
φ
(
x, y
)
dxdy
=

Ω
u
∂φ
∂y
dxdy
=

1
0
1

1
x
∂φ
∂y
dydx
=

1
0
x
(
φ
(
x,
1)

φ
(
x,

1))
dx
=
0
since
φ
∈
C
∞
0
(Ω)
1
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Thus,
∂u
∂y
= 0.
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 Fall '08
 Staff
 Derivative, UK, dx, Ω

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