Civil Engineering Reference
In-Depth Information
|·|
m
is a norm on H
0
() which is equivalent
to
·
m
.If is contained in a cube with side length s, then
1.7 Theorem.
If is bounded, then
+
s)
m
for all v
∈
H
0
().
|
v
|
m
≤
v
m
≤
(
1
|
v
|
m
(
1
.
7
)
Possible Singularities of
H
1
functions
It is well known that
L
2
()
also contains unbounded functions. Whether such
functions also belong to higher order Sobolev spaces depends on the dimension
of the domain. We illustrate this with the most important space
H
1
()
.
[
a, b
] is a real interval, then
H
1
[
a, b
]
1.8 Remark.
If
=
⊂
C
[
a, b
], i.e., each
element in
H
1
[
a, b
] has a representer which lies in
C
[
a, b
].
Proof.
Let
v
∈
C
∞
[
a, b
] or more generally in
C
1
[
a, b
]. Then for
|
x
−
y
|≤
δ
, the
Cauchy-Schwarz inequality gives
Dv(t)dt
≤
1
2
dt
[
Dv(t)
]
2
dt
y
y
y
1
/
2
1
/
2
≤
√
δ
v
1
.
|
v(x)
−
v(y)
|=
·
x
x
x
Thus, every Cauchy sequence in
H
1
[
a, b
]
∩
C
∞
[
a, b
] is equicontinuous and
bounded. The theorem of Arzela-Ascoli implies that the limiting function is con-
tinuous.
The analogous assertion already fails for a two-dimensional domain
. The
function
log log
2
u(x, y)
=
r
,
(
1
.
8
)
where
r
2
=
x
2
+
y
2
, is an unbounded
H
1
function on the unit disk
D
:
={
(x, y)
∈
2
;
x
2
+
y
2
<
1
. The fact that
u
lies in
H
1
(D)
follows from
1
/
2
R
}
dr
r
log
2
r
<
∞
.
0
For an
n
-dimensional domain with
n
≥
3,
u(x)
=
r
−
α
,α< n
−
2
)/
2
,
(
1
.
9
)
is an
H
1
function with a singularity at the origin. Clearly, the singularity in (1.9)
becomes stronger with increasing
n
.
The fact that functions in
H
2
2
are continuous will be
established in §3 in connection with an imbedding and a trace theorem.
over a domain in
R
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