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=
We will also need spaces with boundary conditions where we impose
u
0on
∂G
.
. Then,
W
1
,p
0
is the closure of
C
0
in the
W
1
,p
-
Definition 3.1.5
Let 1
≤
p<
∞
norm,
W
1
,p
0
C
0
(G)
·
W
1
,p
(G)
.
(G)
=
The space
W
1
,p
0
W
1
,p
(G)
is a closed linear subspace. In particular,
⊂
(G)
W
1
,
2
0
H
0
(G)
:=
(G)
is again a Hilbert space with the norm
·
H
1
(G)
.Wehave
the important
Poincaré inequality
.
Theorem 3.1.6
(Poincaré inequality)
Assume that G
⊂ R
bounded
.
Then
,
|
|
(i)
There exists a constant C(
G
,p)>
0
such that
W
1
,p
0
u
L
p
(G)
,
u
L
p
(G)
≤
C
∀
u
∈
(G).
(3.4)
(ii)
Define
u
0
.
W
1
,p
W
1
,p
(G)
(G)
:=
∈
:
u
d
x
=
(3.5)
∗
G
W
1
,p
∈
Then
,(
3.4
)
holds also for all u
(G)
,
with different C
.
∗
Proof
W
1
,p
0
(i) Let
u
∈
(
G
)
,
G
=
(x
1
,x
2
)
be arbitrary,
bu
t fixed. By Theorem
3.1.4
, there
C
0
(G)
such that
u
exists
u
∈
=
u
for a.e.
x
∈
G
and such that
u(x
1
)
=
0. There-
fore, using Hölder's inequality,
u
(ξ )
d
ξ
≤
|
x
1
q
u
L
p
(G)
,
|
|
=
|
|
=
x
1
|
u(x)
u(x)
−
u(x
1
)
x
−
x
1
(
G
|
p
q
1
p
.
1
1
where
q
+
p
=
1. Hence, the result follows with
C
=
x
−
x
1
|
d
x)
(ii) Let
u
∈
W
1
,p
u
∈
C
0
(G)
suc
h
that
u
=
u
for a.e.
x
∈
G
(G)
. Then, there exists
and such that
G
u
d
x
=
∗
0. Therefore, there is
x
∗
∈
G
such that
u(x
∗
)
=
0. We
may repeat therefore the proof of (i) with
u(x
1
)
replaced by
u(x
∗
)
. Taking the
supremum over all possible values of
x
∗
gives the result.
In Chap. 2, we have already introduced the Bochner spaces
L
p
(J
;
H
)
which
such that the
L
p
(J
consist of functions
u
)
-norm is finite. For the theory
of parabolic PDEs, it will prove essential to consider maps
u
:
J
→
H
;
H
:
J
→
H
which are also
differentiable (in time). We call
u
the
weak derivative
of
u
if
u
(t)ϕ(t)
d
t
u(t)ϕ
(t)
d
t,
C
0
(J ).
=−
∀
ϕ
∈
J
J
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