Civil Engineering Reference
In-Depth Information
Problems
1.10
Let
be a bounded domain. With the help of Friedrichs' inequality, show
that the constant function
u
=
1 is not contained in
H
0
()
, and thus
H
0
()
is a
proper subspace of
H
1
()
.
n
s
1.11
Let
⊂ R
be a sphere with center at the origin. Show that
u(x)
=
x
possesses a weak derivative in
L
2
()
if 2
s>
2
−
n
or if
s
=
0 (the trivial case).
1.12 A variant of Friedrichs' inequality.
Let
be a domain which satisfies the
hypothesis of Theorem 1.9. Then there is a constant
c
=
c()
such that
v
0
≤
c
|
v
|+|
v
|
1
for all
v
∈
H
1
()
(
1
.
11
)
1
µ()
with
v
=
v(x)dx.
Hint: This variant of Friedrichs' inequality can be established using the technique
from the proof of the inequality 1.5 only under restrictive conditions on the domain.
Use the compactness of
H
1
()
→
L
2
()
in the same way as in the proof of
Lemma 6.2 below.
n
be bounded, and suppose that for the bijective continuously
differentiable mapping
F
:
1
→
2
,
1.13
Let
1
,
2
⊂ R
(DF (x))
−
1
DF (x)
and
are bounded
for
x
∈
. Verify that
v
∈
H
1
(
2
)
implies
v
◦
F
∈
H
1
(
1
)
.
1.14
Exhibit a function in
C
[0
,
1] which is not contained in
H
1
[0
,
1]. - To
illustrate that
H
0
()
H
0
()
, exhibit a sequence in
C
0
(
0
,
1
)
which converges
to the constant function
v
=
=
1inthe
L
2
[0
,
1] sense.
1.15
Let
p
denote the space of infinite sequences
(x
1
,x
2
,...)
satisfying the
condition
k
|
x
k
|
p
<
∞
. It is a Banach space with the norm
k
|
x
k
|
p
1
/p
x
p
:
=
x
p
:
=
,
1
≤
p<
∞
.
Since
·
2
≤·
1
, the imbedding
1
→
2
is continuous. Is it also compact?
1.16
Consider
(a) the Fourier series
+∞
c
k
e
ikx
on [0
,
2
π
],
(b) the Fourier series
+∞
k
=−∞
c
k
e
ikx
+
iy
on [0
,
2
π
]
2
.
Express the condition
u
∈
H
m
in terms of the coefficients. In particular, show the
equivalence of the assertions
u
∈
L
2
and
c
∈
2
.
Show that in case (b),
u
xx
+
u
yy
∈
L
2
k,
=−∞
implies
u
xy
∈
L
2
.
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