Biomedical Engineering Reference
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and, therefore, we have
u
Dr' C curlv
' WD
X
k
2Z
2
e
{k
x
; v WD
X
k
2Z
u
k
k
jk
j
u
k
k
jk
j
(3.34)
e
{k
x
:
2
3
3
Theorem 3.2.
Assume
f
and
Q
are time-periodic functions of period
T
with
f;Q2 L
2
.0;T I H
#
.
C
//
, and let
f Dr‰ C curlF
(3.35)
be the Helmholtz-Weyl decomposition of
f
. Then resonance may occur for solutions
to system
(
3.12
)
satisfying space-periodic boundary conditions if and only if
curlF 6 0
. On the other hand, if
curlF 0
and, in addition,
//; Q 2 W
2;2
.0;T I H
#
.
‰ 2 L
2
.0;T I H
#
.
// \ W
3;2
.0;T I H
#
.
C
C
C
//;
the system
(
3.12
)
has at least one periodic solution,
.
u
;/
of period
T
, arbitrary
T>0
, with
u
2 W
2;2
.0;T I H
#
.
// \ W
1;2
.0;T I H
#
.
// \ L
2
.0;T I H
#
.
C
C
C
//
2 W
2;2
.0;T I H
#
.
// \ W
1;2
.0;T I H
#
.
C
C
//:
¤ jk
j
This solution is also unique in its own class if and only if
!
2
2
=n
2
, for all
3
.k;n/2
Z
Z
.
Proof.
By the Helmholtz-Weyl decomposition, we have
u
Dr' C curlv;
which once replaced into (
3.12
), with the help of (
3.34
), delivers
r
'
tt
. C 2/'
1
‰
C curl
v
tt
v F
D 0
t
D
2
'
t
C Q:
(3.36)
By the uniqueness of the decomposition, (
3.36
) is equivalent to the following set of
three equations
v
tt
v D F
'
tt
. C 2/' D
1
C ‰
t
D
2
'
t
C Q:
(3.37)
Further on we shall show that, under the given assumptions on ‰ and Q,(
3.37
)
2;3
has one and only one T -periodic solution .';/ in the appropriate class. As a
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