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=
H
(iii) We only require
u(
0
)
u
0
in
. In particular, for well-posedness of the equa-
L
2
,
not
that
u
0
∈
V
tion, it is only required that
u
0
∈
. This is important, e.g. for
binary contracts, where the payoff
u
0
is discontinuous (and, therefore, does not
belong to
in general).
(iv) The bilinear form
a(
V
)
is, in general,
not
symmetric due to the presence of a
drift term in the operator
·
,
·
A
.
We have the following general result for the existence of weak solutions of the
abstract parabolic problem (
3.7
). A proof is given in Appendix B, Theorem B.2.2.
See also [64, 65, 115].
Theorem 3.2.2
Assume that the bilinear form a(
·
,
·
) in
(
3.7
)
is continuous
,
i
.
e
.
there
is C
1
>
0
such that
|
a(u,v)
|≤
C
1
u
V
v
V
,
∀
u, v
∈
V
,
(3.8)
and satisfies the “Gårding inequality”
,
i
.
e
.
there are C
2
>
0,
C
3
≥
0
such that
2
2
H
a(u, u)
≥
C
2
u
V
−
C
3
u
,
∀
u
∈
V
.
(3.9)
L
2
(J
H
1
(J
;
V
∗
)
.
Moreover
,
∈
;
V
∩
Then
,
p
rob
lem
(
3.7
)
admits a unique solution u
)
C
0
(J
u
∈
)
,
and there holds the a priori estimate
u
C
0
(J
;
H
)
+
u
L
2
(J
;
V
)
+
u
H
1
(J
;
V
∗
)
≤
C
u
0
H
+
f
L
2
(J
;
V
∗
)
.
;
H
(3.10)
We give a simple example for the weak formulation (
3.7
).
V
∗
Example 3.2.3
Consider the heat equation as in (2.4). Then, the spaces
H
,
V
,
in
H
0
(G)
,
V
∗
=
H
−
1
(G)
, and the bilinear form
a(
L
2
(G)
,
(
3.6
)are
H
=
V
=
·
,
·
)
is
given by
u
(x)v
(x)
d
x,
H
0
(G).
a(u,v)
=
u,v
∈
G
The bilinear form is continuous on
V
, since by the Hölder inequality, for all
u, v
∈
H
0
(G)
u
v
|
u
L
2
(G)
v
L
2
(G)
≤
|
a(u,v)
|≤
G
|
d
x
≤
u
H
1
(G)
v
H
1
(G)
.
Furthermore, we have for all
u
∈
H
0
(G)
, by the Poincaré inequality (
3.4
)
1
2
1
2
u
(x)
2
d
x
u
2
u
2
L
2
(G)
a(u, u)
=
G
|
|
=
L
2
(G)
+
1
2
C
1
2
2
u
2
L
2
(G)
≥
u
L
2
(G)
+
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