Civil Engineering Reference
In-Depth Information
2
α
v
n
−
v
m
≤
a(v
n
−
v
m
,v
n
−
v
m
)
=
2
a(v
n
,v
n
)
+
2
a(v
m
,v
m
)
−
a(v
n
+
v
m
,v
n
+
v
m
)
8
J(
v
m
+
v
n
2
=
4
J(v
n
)
+
4
J(v
m
)
−
)
≤
4
J(v
n
)
+
4
J(v
m
)
−
8
c
1
,
1
since
V
is convex and thus
2
(v
n
+
v
m
)
∈
V
.Now
J(v
n
), J (v
m
)
→
c
1
implies
v
n
−
v
m
→
0 for
n, m
→∞
. Thus,
(v
n
)
is a Cauchy sequence in
H
, and
u
=
lim
n
→∞
v
n
exists. Since
V
is closed, we also have
u
∈
V
. The continuity of
J
implies
J (u)
=
inf
v
∈
V
J(v)
.
We now show that the solution is unique. Suppose
u
1
and
u
2
are both solutions.
Clearly,
u
1
,u
2
,u
1
,u
2
,...
is a minimizing sequence. As we saw above, every
minimizing sequence is a Cauchy sequence. This is only possible if
u
1
=
u
2
.
lim
n
→∞
J(v
n
)
=
2.6 Remarks.
(1) The above proof makes use of the following
parallelogram law
:
the sum of the squares of the lengths of the diagonals in any parallelogram is equal
to the sum of the squares of the lengths of the sides.
(2) In the special case
V
=
H
, Theorem 2.5 implies that given
∈
H
, there
exists an element
u
∈
H
with
a(u, v)
=
, v
for all
v
∈
H.
=
(u, v)
, where
(u, v)
is the
defining scalar product on
H
, then we obtain the
Riesz representation theorem
:
given
∈
H
, there exists an element
u
∈
H
with
(3) If we further specialize to the case
a(u, v)
:
for all
v
∈
H.
This defines a mapping
H
→
H,
→
u
which is called the
canonical imbedding
of H
in H
.
(4) The Characterization Theorem 2.2 can be generalized to convex sets as
follows. The function
u
is the minimal solution in a convex set
V
if and only if
the so-called
variational inequality
(u, v)
=
, v
a(u, v
−
u)
≥
, v
−
u
for all
v
∈
V
(
2
.
15
)
holds. We leave the proof to the reader.
If the underlying space has finite dimension, i.e., is the Euclidean space
N
,
R
then instead of (2.13) we only need to require that
for all
v
∈
H,v
=
a(v, v) >
0
0
.
(
2
.
16
)
Then the compactness of the unit ball implies (2.13) for some
α>
0. The fact
that (2.16) does not suffice in the infinite-dimensional case can already be seen in
the example (2.12). To make this point even clearer, we consider another simple
example.
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