Biomedical Engineering Reference
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in linearized elasticity, where, in fact, the whole energy is conserved, and in linear
thermoelasticity with periodic boundary conditions, where only a part of the energy
is conserved. The next result confirms the above expectation, and makes it precise
from the rigorous mathematical viewpoint.
Theorem 3.7.
Let
X
be a Hilbert space, and let
X
0
be a subspaces that is left
invariant by
U.t/
. Suppose that
kU.t/xkDkxk
, for all
t 0
and all
x 2 X
0
.
Then, there exists at least one resonant period for
(
3.48
)
.
Proof.
Without loss, we can take X
0
D X. Assume, by contradiction that for any
T -periodic force f ,(
3.48
) admits a corresponding T -periodic strong solution. In
view of Lemma
3.2
, we may then expand both f and x in Fourier series:
f.t/D
X
k
2Z
f
k
e
{k!t
;x.t/D
X
k
2Z
x
k
e
{k!t
:
Thus, from (
3.48
) we deduce that the Fourier coefficients x
k
, f
k
satisfy the equations
1
{
f
k
;k2
Z
k!x
k
Lx
k
D
;
(3.78)
where L WD {A.
14
Since U.t/ is a semigroup of isometries, by the Cooper-Phillips
theorem [
14
, Theorem 2(a)], [
40
, Theorem 1.1.4], L is maximal symmetric. As a
result, at least one of the deficiency indices of L has to be zero and, therefore, the
spectrum of L must contain (at least) a non
-e
mpty subset,
O
, o
f
the real line [
27
,p.
271]. We then choose T in such a way that k
!
2
O
, and conclude
that (
3.78
)
does not have a solution for k D k, provided we pick f
k
appropriately.
In
fact, if k! is an eigenvalue, we take f
k
in the corresponding eige
n
space, while if
k! is in the continuous or residual spectrum of L,wetakef
k
62
R
.k!I L/.The
proof of the theorem is completed.
,forsomek 2
Z
t
3.5
Some Applications
Objective of this section is to provide several applications of the theory developed
in Sect.
3.4
to a number of problems involving hyperbolic-parabolic couplings,
including thermo- and magneto-elasticity and certain basic models of liquid-
structure interaction.
3.5.1
Three-Dimensional Linear Thermoelasticity (Revisited)
As we know from the results of Sect.
3.3
, while in the one-dimensional case one
eliminates the occurrence of resonance (at least for a class of sufficiently regular
14
Notice that, of course, x
k
2 D
.A/, because x.t/
2 D
.A/ for all t
0.
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