Hyperbolic–Parabolic Coupling and the Occurrence of Resonance in Partially Dissipative Systems - Fluid-Structure Interaction and Biomedical Applications - page 232

We shall next show that ( 3.80 )-( 3.81 ) can be put in the abstract form ( 3.69 ), for
an appropriate choice of spaces and operators involved, and that the assumptions of
Theorem 3.5 are satisfied. As a consequence, resonance cannot occur, whenever the
periodic load has a finite (but, in principle, arbitrarily large) number of modes. To
achieve our goal, we begin to choose H 2 WD ŒL 2 ./ 3 ,and

H 1 WD H./.asinthe

1

previous subsection, we set A 1 WD

L

, with L the linearized elasticity operator

introduced in Sect. 3.1 ,and

1
0 curl curl

A 2 WD

with

D .A 2 / Dfh 2 H 2 ./ \ H./ W curlh n D 0 at @ :

We h ave R .A 2 / H./. In fact, for an arbitrary ' 2 C 1 ./ and h 2 D .A 2 / we
show, by integration by parts,
Z Z

curl curlh r' D

@ r' curlh n D 0;

whichby[ 18 , Lemma III.2.1] proves the assertion. Actually, A 2 is a homeomor-
phism of D .A 2 / (endowed with the H 2 ./-norm) onto H. This property is a
particular case of [ 20 , Theorem 3.2.3]. Moreover, A 2 is strictly accretive. In fact,
for all h 2 D .A 2 /,
Z Z Z Z

2 2

curl curlh h D n curlh h C

jcurlhj

D

jcurlhj

0:

@

However, choosing the equality sign in the last step would imply, in view of the
assumption on , h Dr for some 2 H 1 ./, which, since h 2 H./furnishes
D const and h 0, thus proving the desired property. We next define

Bh WD curlh H 0

with

D .B/ Dfh 2 H./ W curlh H 0 2 ŒL 2 ./ 3

g

and (clearly) R .B/ ŒL 2 ./ 3 . Finally, we set

B v D curl.v H 0 /

with

D .B / Dfv 2 ŒL 2 ./ 3

W curl.v H 0 / 2 H./g: