Civil Engineering Reference
In-Depth Information
The inequality (6.10) below is sometimes called
Necas' inequality
; see Necas
[1965]. We will encounter Necas' inequality once more in the proof of Korn's
inequality in Ch. VI, §3.
n
be a bounded connected domain with Lipschitz contin-
6.4 Theorem.
Let
⊂ R
uous boundary.
(1) The image of the linear mapping
grad :
L
2
()
−→
H
−
1
()
n
(
6
.
9
)
is closed in H
−
1
()
n
.
(2) Let f
∈
H
−
1
()
n
.If
f, v
=
0
for all v
∈
V,
(
6
.
10
)
then there exists a unique q
∈
L
2
,
0
() with f
=
grad
q.
(3) There exists a constant c
=
c() such that
q
≤
c(
grad
q
−
1
,
+
q
−
1
,
)
for all q
∈
L
2
(),
(
6
.
11
)
0
,
q
0
,
≤
c
grad
q
−
1
,
for all q
∈
L
2
,
0
(). (
6
.
12
)
6.5 Remark.
The inf-sup condition (4.8) for the Stokes problem (6.5) follows
from Theorem 6.3 and Theorem 6.4, respectively.
Proof.
(1) Given
q
∈
L
2
,
0
, there exists
v
∈
H
0
()
n
that satisfies (6.8). Hence,
2
0
2
0
v
1
=
q
v
1
≥
q
(
div
v, q)
1
c
q
0
,
c
q
0
=
which establishes the Brezzi condition.
(2) For
q
∈
L
2
,
0
, it follows from (6.12) that
grad
q
−
1
≥
c
−
1
q
0
.
By the definition of negative norms, there exists
v
∈
H
0
()
n
with
v
1
=
1 and
1
2
v
1
1
2
c
q
0
.
(v,
grad
q)
0
,
≥
grad
q
−
1
≥
By (6.6),
b(
−
v, q)
v
1
=
(v,
grad
q)
0
,
≥
1
2
c
q
0
.
which establishes the Brezzi condition.
The properties above are also necessary for the stability of the Stokes problem;
see Problem 6.7.
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