H.T. Banks
Department of Mathematics
North Carolina State University
Raleigh NC 27695
D. Cioranescu
Laboratoire d'Analyse Numerique
Universite Pierre et Marie Curie, Paris
D.A. Rebnord
Department of Mathematics
Syracuse University
Syracuse NY 13224
In the past several years wehave pursued eorts related to the development
ofaccurate mo delsforthedynamicsof exiblestructures madeofcomp osite
materials. Mostof these eorts havefo cused on structures with very simple
geometrysuchasb eamswithtipmasses,andnot onthemorecomplex
struc-tures typically used inengineering applications. One exampleof these more
complexstructures isgrids (essentiallythin plates with holes). A
character-istic feature of grids is that they are comp osed of memb ers that are quite
thin and have a highly p erio dic structure. Unfortunately these
characteris-tics,coupledwiththe dicultiesrelatedto the unknown materialprop erties
such as stiness and internal damping,makeapplicationof traditional
com-putationalmetho dsto ndapproximatesolutionsto thevibrationalproblem
challenging.
In thispap er, ratherthanviewing p erio dicityandsparseness asobstacles
to b eovercome,weexploitthemtoouradvantage. Weconsidera variational
problem on a domain that has large, p erio dically distributed holes. Using
homogenization techniques we show that the solution to this problem is in
some top ology `close' to the solution of a similar problem that holds on a
muchsimplerdomain. In Section2 we study the b ehavior of the solution of
the variational problemas the holes increaseinnumb er,butdecreasein size
insuchawaythatthetotalamountofmaterialremainsconstant. Theresult
willb ean equation thatisingeneralmorecomplex,but with adomainthat
is simplyconnected ratherthan p erforated. In Section 3 we study the limit
ofthesolutionas theamountofmaterialgo esto zero. Thissecondlimitwill,
in most cases, retrievemuchof the simplicitythat was lost in the rst limit
withoutsacricingthesimplicityofthedomain. FinallyinSection4weshow
that these results can b e applied to the case of a vibrating Love-Kirchho
plate with Kelvin-Voigt damping.
We rely heavily on earlier results of [Du], [CS] for the static, undamp ed
Love-Kirchho equation. Our eorts here result in a mo dication of those
2
S 0
3
3
3
0 2 0
!
!
2f g
2
f 2
g
-6
-? 6
-
-6
6
? 2
1 2
1 2
1 2
1 2
1
2
1 2
1 2
1 2
2 2 2
2
2.1 Problem Statement and Notation
x
n
x
b
@
@x
l l
l l
Y Y l l
y
Y
x x
l
l
T
Y
Y
l l
y y
=
y
x y
Y ;l ;l
Y T
Y Y
H L V H H ;
x
Let b e a b ounded,op en subset of IR . Let denote that part of that
is covered by material (see the left half of Figure 1). A natural subset of
is enclosed in the dashed b ox. We shall call this subset a \cell" of .
As depicted in the gure the size of each cell is by , where is some
\small"(smallincomparisontotheotherlengthsintheproblem)parameter.
Each cell contains a hole of dimension ( ) ( ). Our goal is to
study the b ehavior of the solution of a variational problem rst as 0
and then as 0. In order to avoid diculties with shifting domains we
intro duce the xed (relative to ) cell where has length by . By
applying the transformation = we can go froma cellin to the xed
cell . We assume that the holes of do not meet the b oundary; this
assumption restricts b oth the geometry of and the p ermissiblevalues of
(e.g., 2 )
2
=
=
=[0 ] [0 ]
=
Figure1: Structure of , , , and .
Dene = ( ) and = ( ) = : ( ) =
=0 on =0 . Noneofwhatfollowsdep endsonthisparticularchoiceof
1 Z
Z
1 2
n 2
0 0
2 2
2
2 2
2
2
1
=1
1 2
8 2
2 2
j j 2
M
M
j j
2
D
j j
j j
!
tt
;
t
ijk h
k h i j
ijk h
ijk h
ij k h ij ij ij ij ji
E
E
E
E
n j
n
j j
k
j
k k
j
n
<u ; > u ; <f; > V
u u V u v H
'; a x
@ '
@x @x @
@x@x dx
a Y L
a A ; A> ; ; :
f f L ;T L
E
E E E g
L E g E g
g
E
g y dy
g L g
C
;:::;
D @ =@x k
j n
D D D :::D
u
u u V V
are valid for any b oundary conditions in which and the normalderivative
of vanish on an op en intervalof the outer b oundary. Using this notation
we considerthe following problem
+ ( )= (1)
(0)= (0)= (2)
where isa p ositiveconstant and
( )= ( )
and where in writing the ab ove equations we have adopted the summation
convention(i.e.,wesumonrep eatedindices). Weassumethateachofthe
co-ecients is -p erio dicandanelementof (IR )andthatthefollowing
ellipticitycondition holds
0 for all =( ) with = (3)
If the forcing function = is in (0 ; ( )) then there is a unique
weak solution to (1). (See e.g.,[LM],or [W], p.434{442 ).
We adopt the following notation. For any measurable set , 1 denotes
the characteristic function of and denotes the measure of . If
( ), wewill denotethe mean value of on by ( ):
( )= 1
( )
For any function ( ) we will denote by ~ the extension by zero to
all of . We will denote many dierentconstants which are indep endent of
by the one symb ol . The set of test functions on (i.e., the set of all
b ounded,innitely dierentiablefunctions on with compactsupp ort) will
b e denoted by (). The symb ol = ( ), 0 will denote a
multi-index,with =6 . Similarly, = for scalar and for
1 , and so
=
denotesadierentialop erator oforder . Lastly,to simplifynotation while
we study the limitof as 0,wewilltemp orarily suppress the index
2 2 3
3
3
3
3
Z
X X
1 1
[
3
[
3
3
3
j j j j
3 Lemma 2.1
Lemma 2.2
2
2
2
2
2
2
2
2
2 2
=2
( )
=2
( )
2
x
Y
Y
Y
Y
Y x
Y Y
Y Y
L Y
L Y 2
2 !
!
j j
M
2 ! !M
!M
j j
j j
M
2 !
!
2
2L
j j j j
2
g L Y Y
g x g ;x
g
Y
g y dy g L
g L Y g g L
Y x
Y
Y
L :
g L g g L u
u V H u
u
Y
Q H Y ;H Y
D Q' C D ' ;
' H Y C '
Suppose that is -periodic and is extended
periodi-cally to all of . If we dene , then as ,
in weakly.
Moreover,if then as , in weak-*.
Proof:
Assume has theuniform coneproperty, then there exists an
extension operator such that
for any , where is a constant independent of .
Proof:
Wewillhave o ccasion to use the following lemmafrequently.
( )
IR ( )= 0
1
( ) = ( ) ()
( ) 0 ( ) ()
SeeChapter 5, Lemma4.1 of [SP].
Henceifwelet1 denote theextensionbyp erio dicity ofthe characteristic
function of to all of IR , then 1 ( )=1 and so by the previous
lemma
1 (1 )= in () weakly
Thequantity (1 )willapp ear frequentlyinwhatfollows, andtherefore
we will simply denote it by . It follows from the ab ove lemma that if
(), then1 weakly in (). In orderto study the limit
as 0,weneed to extenditfromthe p erforateddomain tothe domain
. Since ( ) we cannot simply extend by zero into the
holes and preserve the smo othness of . A more sophisticated extension is
required, the existence of whichisguaranteed by the following two lemmas.
( ( ) ( ))
( )
1 1
X X
X X
X X
n n
2 2
2 2
2 2
Lemma 2.4
1 1
j j j j
1
j j j j
1
j j j j
1 1
2L
2
j j j j
2
2L
j j j j 2
1 2
f 1 g 1
2
j j j j
2L
f g 2 j j
! !
2 2
=2
(0 ; ())
=2
(0 ; ())
2
2 2
=2
()
=2
()
2
2
=2
()
=2
()
2
2 2
2
2
t
t
L ;T L
L ;T L
L
L
t
t
L
L
b
b
b
V
n
P L ;T H ;L ;T H
P ' P ' ;T
D P ' C D ' ;
' L ;T H C '
Q H H
D Q C D H
C
P
P ' t;x Q ' t; x ' L ;T H :
Q
@
@t
P ' t;x @
@t
Q' t; x Q ' t; x P ' t;x :
t ;T
D Q ' t C D ' t :
;T
Q
P H
P L ;T H ;L ;T H
V C C
L
H
an extension operator such that
in , and
for any , where is a constant independent of and
.
Proof:
Suppose , , independent of ; then there
exists a subsequence , such that in weakly as .
Moreover,the previous lemma ensures that is actually in . ( (0 ; ( )) (0 ; ()))
=( ) (0 )
(0 ; ( ))
Lemma2 of [Du] yieldsthe existence of an extension
op-erator ( ( ), ())with the prop erty
for all ( )
(4)
where is a constant indep endent of and . We dene the
op erator by
( )=[ ( )]( ) for (0 ; ( )) (5)
Then fromthe construction of in[Du] it follows that
[ ( )]= ( ) ( )=[ ( )]( )= ( )( )
Nowfrom(4) we havefor each (0 ),
( ) ( )
So taking the essential supremumover(0 ) of the rightside of
the inequalityab ove,and thenof the lefthandsideweobtainthe
desiredresult.
Weremarkthatthe construction ofthe extensionop erator and hence
thatof issuchthattheb oundary conditionsof ( )arepreserved;i.e.,
( (0 ; ( )) (0 ; ())).
~
() 0
2.3 A Priori Estimates
2
2 2
!
2
L
!
j j
2 !
2 !
j j j j !
2 j j j j
b
b
b
b
b
b
b
V
H
V
H 0 0
0
0
2
0
0
2
2
0 0
0
0
2
0
0
2 2
2 2
0
0
0
0
2 2
0
0
0
0
0 0 2
0
0 0
2
0 2
0
0 0
2
0 0
2 2
0 0 0 0
f;u ;v
u u H V
v v L H f L
;
u ;v f
u ;v ;f
f f L u
u
H H
P
Q H ;H
t u Q u
v v H L
; f
u Q u v v
< < u v L u u u
L u H v v v L
u ; v
> u u
f L ;T L u v u v
;
considerations, we discuss briey how the approximation results develop ed
heremayb eused inpractice. Of particularinterestishowonetreats forcing
functions and initial data ( in (1)-(2)). In the typical application,
one willhaveagivenxedstructure corresp ondingto adomain forxed
values of and . For this structure initial data = ( )= ,
= ( ) = as well as a forcing function ( ) will
b e available. One wishes to use a homogenized mo del or limiting mo del of
(1)-(2) as 0 as an approximation to the xed grid with domain .
This mo del will b e dened on (this is the whole p oint of our eorts) and
one must somehow construct data and forcing function (each with
domain ) to b e used in this approximation system. One can use the data
forthe actualstructure to construct data forthe approximation
system.
For the forcing function it is reasonable to use the \zero outside "
extension,i.e. = (). Fortheinitialdata the extensionisnot
reasonable (nor willitprovideneededsmo othness for ). However,Lemma
2.3 allows one to extend functions in ( ) to (), thus maintaining
smo othness and exterior b oundary conditions. The extension op erators
guaranteed by this lemmareduce to in ( ( ) ()) on elements
thatareconstantfunctionsin . Onecanthendeneinitialdata ^
and ^ ~ in () and (), resp ectively, that are natural to use in
the limitequation on . In the sequence of approximate systems(1)-(2) as
0, one might then use the forcing function and the restrictions (to
) ofthe initialdata i.e., , ~ foreach , with
, . Note that since ^ , ^ () we have ~ =1 ^ ^
in () weakly with ^ () and ~ =1 ^ ^ weakly in ().
Also observethat are uniformlyb ounded as , 0.
Welet 0b exedandconsidersolutions = of(1)-(2)corresp onding
to (0 ; ()) andinitialdata , suchthat and
are uniformlyb ounded (in ). Weremind the reader that inour notation
b elow we will only temp orarily suppress the dep endence on the xed value
! 0
P u
Theorem 2.1
1 1
1
1
1 1
1
1
1
1
Z
!
Z Z Z Z Z Z
2.4 Limit of as
2L
!
!
!
j j j j j j j j j j
! !
!
0 n n
n n n n
n n
n n
n n
n n
2
2
2
2
0 2
0 2
0
2
2
2
2
2
2 2
0
2
0
0
b
n
b
t
t
t
t
H
V
V
H
T
H
t
t b
t
n
b
t
t
ij
ijk h
k h
ijk h
k h
T
ij
i j
T T
tt There exists an extension operator
and a function such that for some sequence we have
in weak-*
in weak-*
Proof:
P
P L ;T V ;L ;T H
u u
P u u L ;T H ;
P u P u u L ;T L :
u t u t C u v f t dt
t ;T C
u u
L ;T V L ;T H
P
P u P u L ;T H
L ;T L P u
P u P u u L ;T H
P u u L ;T L
u
P u u
a x
@ u
@x @x a
x
@ u
@x @x :
@
@x@x
vdxdt f vdxdt P u vdxdt:
( (0 ; ) (0 ; ()))
= 0
(0 ; ())
( ) = (0 ; ())
Theargumentsunderlyingthe fundamentalexistence and
uniquenessresults(see[W],p. 439-442andalso[BIW])for(1)-(2)
can b e used to showthat solutions also satisfy
( ) + ( ) + + ( )
foralmost every in (0 ),wherethe constant isindep endent
of . From the uniform b oundedness assumption on the initial
data, we thus conclude that and are uniformly b ounded
in (0 ; ) and (0 ; ), resp ectively. We can thus use
the extension op erator constructed in Lemma 2.3 to
con-clude that and are b ounded in (0 ; ()) and
(0 ; ()),resp ectively. Hencetherearesubsequences ,
,suchthatas 0wehave in (0 ; ())
weak-*, and in (0 ; ()) weak-*.
Nowthat we have established the existence of a that is the subsequential
limitof ,we need to determinethe equation that satises. Dene
= ( ) =
Wethenextendthe weakformulationofthe plateequation(1) toallof by
writing
~
= 1 1
2 1
1
3
n
n
n
Z Z Z Z Z Z
Z
1
3 1
3 1
3
0
3
3
3
3
2 2D j j
j j
j j
2 !
!
!
0
2
8 2
0P
P
2
!
!
j j
! P
()
(0 ; ())
2
2
0
2
0 0
2
2 2
2
1
2
2
2
1
2
2
2
2 b
V
ijk h
ijk h L
ij
L ;T L
ij
n
ij ij
n
T
ij
i j
T T
tt
ij
lm
Y ijk h
lm
i j k h
lm lm lm
lm l m
lm
lm
lm
lm
l m
lm
lm
n
lm
lm
lm
lm
H v ;T v v T v T u t
C t a
L a C
C
L ;T L
L ;T L
@
@x@x
vdxdt f vdxdt u vdxdt:
t v v
T
u w y H Y
a y
@ w y
@y@y @
@y @y
dy H Y
W y w y y
y yy
W x QW
x
x Y
Q
w x W x xx
W w H
w w H
D w D L
for all (), (0 ) with (0)= ( )= ( )=0. Since ( )
can b eb oundedby aconstant indep endentof and ,and the are in
() with indep endentof , we have
~
whichimpliesthe existenceof a ~
(0 ; ()) andasequence 0
such that
~ ~
in (0 ; ()) weak *. (7)
Henceas 0 we haveusing Lemma2.1 and Theorem2.1,
~
= (8)
Wecan obtain the claimedinitial conditions by p erformingtwo integrations
by parts with resp ect to and applying the conditions on and at 0and
given ab ove. The remaining task then, is to somehow relate the limit ~
to . We intro ducethe function ( ) ( )which isthe solution of
( )
( )
=0 ( ) (9)
( ) ( ) ( ) Y-p erio dic (10)
where ( )= . Set
( )= ( ) for
and extend p erio dically to all of IR , where is the extension op erator of
Lemma2.2. Thendene ( ) ( )+ . It followsfrom (9)that
, andhence ,canb e b ounded,indep endentof ,in () so thatfor
somesequence 0,
in () weakly, (11)
and for any , =2,
3
1 1
1
1 n n
n n
Z
e
Z
Z Theorem 2.2
!
!
j j
2
! 2
!
2L
!
!
!
8 2 2
2
2
2
2
2 2
0 0
0 0
2
0 2
1 0
2
2 2
2
2
2
0 0
2 2
k h k hlm
k h
ijk h
lm
i j
ijk h
lm x
i j
k h
k h
k h
k h
Y ijk h
lm
i j
lmk h
b
b
b
n
b
t
t
tt
b
t
ijk h
k h i j
a x
@ w x
@x@x
a
x
@ w
@y@y
@
@x @x
dx :
Y
a y
@ w y
@y@y
dy L
u f L ;T L
u ;v V H
u u L u H
u v L
P L ;T H ;L ;T H
P u u L ;T H
P u u L ;T L
u
< u t ; > u t ; < f; > ;T ; H
u u = u v = :
'; q
@ '
@x @x @
@x@x dx Weintro duce = by
= ( )
( )
=
( )
By constructionits extension ~ to , satises
~ =0 (13)
Furthermore, using Lemma2.1 as 0
1
( )
( )
in () weakly. (14)
Weare now ready to prove
( ) ( ) (0 ; ())
~ () ()
~ ()
( (0 ; ( )) (0 ; ()))
0
(0 ; ())
(0 ; ())
( ) + ( ( ) )= (0 ) () (15)
(0)= (0) =
( )=
Let bethesolutionof 1 - 2 correspondingto
and initial data uniformly bounded in , satisfying
in weakly, with
in weakly.
Thenthereexistsanextensionoperator
such that for some sequence ,
in weak-*
in weak-*
where is the solution to the homogenized equation:
on
3 3 3 3 j j 0 0P 8 2 P 2 D 2 D 2 D 0 0 0 0 0 0 0 ! Z " # Z
Z Z Z Z
Z Z Z Z
Z Z Z Z
Z Z Z Z
Z Z Z Z
Z Z Z Z
Z Z Z Z
2 2 2 2 1 2 0 2 0 2 0 0 0 2 0
0 0
2
0
2
0
0 0
2
0
0
where the functions are the -periodic solutions of
periodic in
and where .
Proof: ijk h
Y
ijk h lmk h
ij l m ij Y lmk h ij ij
l m k h
ij i j lm T ij lm i j T ij i j T lm T tt lm T ij i j lm T ij i lm j T ij j lm i T ij lm i j T ij i j T k h k h T k h h k T k h k h T lm T tt lm q Y
a y a y
@ y
@y@y dy
y Y
a y
@ y y
@y@y
@
@y @y
dy H Y
Y
y yy
P u
v ;T T
w
@ w
@x@x
vdxdt
@ P u
@x@x
vdxdt
fw vdxdt P u w vdxdt:
@
@x@x
w vdxdt
@
@x @w
@x
vdxdt
@
@x @w
@x
vdxdt
@ w
@x@x
vdxdt
@
@x@x
P u vdxdt
@
@x
@ P u
@x
vdxdt
@
@x
@ P u
@x
vdxdt
@ P u
@x @x
vdxdt
fw vdxdt P u w vdxdt:
= 1 ( ) ( ) ( ) (16) ( ) ( ) ( ( ) ( ))
=0 ( ) (17)
( )=
In (13) cho ose = , where () and then
multiply by (0 ) and integrate from 0 to . Now in
(6) cho ose = , where () and subtract the two
equationsto obtain
~
( )
~
( )
= 1 1
If weapply the pro duct rulefor the dierentiationswe have
~ + ~ + ~ + ~ ~ ~ ( ) ~ ( ) ~ ( )
= 1 1
Examining the ab ove integrals one by one, we discover that all
1 0 0 0 0 0 0 0
3 3 3
3 3 3 3 3 3 3 3 3 3 3 3
Z Z Z Z
Z Z Z Z
Z Z Z Z
Z Z Z Z
Z Z Z Z
Z Z Z Z
Z Z Z Z
Z Z Z Z
Z Z Z Z
Z Z Z Z
Z Z Z Z
0 2 0 2 0 2 0 2 2 0 2 0 2 2 0 2 0
0 0
2
0 0
0 0
0 2 0 2 0 2 0 2
0 0
0 2 0 2 T ij lm i j T k h k h T ij lm i j T ijk h k h lm i j T k h k h T ijk h lm i j k h T ij i j lm T ij i lm j T ij j lm i T lmk h k h T lmk h k h T lmk h h k T lm T tt lm T ij lm i j T ij lm i j T lmk h k h T lmk h k h T lm T tt lm T ij lm i j T lmk h k h ~ and ~ ( )
Now since ~denotes the extension from by 0 to all of we
have ~ = and ~ ( ) =
But hereis wherethe adjointsystemcontributesto our analysis,
yielding that the two terms are in fact equal. So now using the
convergences(7), (11), and (14) we have
~ + ~ + ~ =
Wecan rewrite this as
~ ( ) ~ ( ) + =
Then using (8) and (13) we have
~
=
@ w
@x@x
vdxdt
@ P u
@x @x
vdxdt:
@ w
@x@x
vdxdt a
@ u
@x @x
@ w
@x@x
vdxdt
@ P u
@x @x
vdxdt a
@ w
@x@x
@ u
@x @x
vdxdt:
@
@x@x
w vdxdt
@
@x @w
@x
vdxdt
@
@x @w
@x
vdxdt
@
@x @x
uvdxdt
@
@x @u
@x
vdxdt
@
@x @u
@x
vdxdt
fw vdxdt u w vdxdt:
@ w
@x@x
vdxdt
@ w
@x@x
vdxdt
@ u
@x @x
vdxdt
@ u
@x @x
vdxdt
fw vdxdt u w vdxdt:
@ w
@x@x
vdxdt
@ u
@x @x
3
3 Z
!
Z
Z
Z
2
2
2
2 2
2 2
2 3
3
3
3
3
3
3
3
3
3
3
P 0
j j
0
0
j j
0P
j j
0P 0P
j j
0P 0P
0P 8 2
0P
j j
0P 0P
@ u
@x @x :
y y w y
q q
q
q
Y
a a
@ y
@y@y
dy
Y
a
@ y
@y@y
dy
Y
a
@ y
@y@y
@
@y @y dy:
;
; a
@
@y@y @
@y @y dy:
q
Y
; :
Y
; H Y ; :
; :
q
Y
; :
a
q
lm
lmk h
k h
lm lm
lm lmk h
lmk h lmk h
ijk h
ijk h
Y
ijk h lmk h ij
l m
Y
lmk h
ij ij
l m
Y
lmno
ij ij
l m
k h
n o
Y
Y
Y
lmno
l m n o
ijk h Y
ij ij k h
ij
Y
ij ij
k h
Y
ij ij k h
ijk h Y
ij ij k h k h
ijk h
Y
ijk h
~
=
Nowlet ( )= ( ) ( );itiseasytoverifythat =
wherethe are as givenin equation(16). Nowmaking
thissubstitutioninto(8)wehavethehomogenizedequation(15).
Finally we showthat the satisfy an ellipticity condition. Recallthat
= 1
( )
=
1 ( ( ) )
=
1 ( ( ) ) ( )
Wedene the sesquilinear form ( )by
( )=
Therefore
= 1
( ) (18)
Nowobserve that can b echaracterizedas the -p erio dic solution of
( )=0 ( ) Y-p erio dic
In particular if we cho ose = we have
( )=0 (19)
So nowcombining(18) and (19)
= 1
( )
Since the satisfyan ellipticitycondition itiseasy to showthat isa
co ercivesesquilinear form,which inturn makes iteasy to demonstratethat
3
P u
Z
Z
Z 2
8
8 2
2 2
f 2 g
( )
2 2
1 2
0 0
2
1
2
2
2
2 2
2
2
2 2.5 Limit of for the Case with Damping
ijk h
ijk h
ijk h
ijk h
ijk h ij k h ij ij ij ij ji
Y ijk h
k h i j
tt
t
t
ijk h
k h i j
ijk h
k h i j
a
c ;y
c ;y Y
c ;y
c ;y C ; C > ; :
; c ;y
@
@y @y @
@y@y dy
<u ; > u ; u ; <f; > V
u u V u v H L
; a x
@
@x @x @
@x@x dx;
; b x
@
@x @x @
@x@x dx;
V H ;
@
x
x
erage of the plus a corrector term. Furthermore note that the
homog-enization process takes us from having to solve our problem on a perforated
domain to solving a problem on a continuous domain.
We now extend our previous results to include a damping term. In
prov-ing this extension we will use Laplace transforms and the following idea of
homogenization with parameter takenfrom [BLP].
Supp osewehavetheco ecient ( ),where isaparameterb
elong-ing to sometop ological set3. Assumethat the ( )are -p erio dic for
any 3 andthat the ( )satisfy the followingellipticity condition
( ) 0 =( )with = (20)
Then the sesquilinearform
( )= ( )
ishomogenizedbythe formulasderivedinthe previoussection,with b eing
carried along as a parameter. This pro cess is called \ homogenization with
parameter".
We nowconsider the problem
+ ( )+ ( )= (21)
(0) = = ( ) (22)
where
( )= ( )
( )= ( )
and
n n
n n 1
1 1
1
1
1
1
0
b
Z
b b
b b
b
Z
b Theorem 2.3
2
2
2L
2 !
!
!
8 2
8 2 2
2
0 0
2
2
2
2
2
0
2
1 2
2
2 2
ijk h
b
b
n
b
t
t
st
ijk h
ijk h
k h i j Y
b L
u u f L ;T L
u v
P
P L ;T V ;L ;T H
u L ;T H
P u u L ;T H
P u u L ;T L :
V
u
u s e u t dt
<s u s ; > u s ; s u s ; <f s ; > V
s
<s u s ; > s u s ; <f s ; > V
s u s ; a x sb x
@ u s
@x @x @
@x@x dx: Let bethesolutionto 21 - 22 with
and initial data , as in Theorem 2.1. Then there exists an extension
operator
and such that for some sequence
in weak-*
and
in weak-*
Proof:
term that corresp onds to damping. The -p erio dic damping co ecients
(IR ) are assumed to satisfy an ellipticity condition like that of
(3). Just as in the previous case we can nd an extension of the solution
satisfying a priori b ounds.
= ( ) ( ) (0 ; ())
( (0 ; )) (0 ; ()))
(0 ; ()) 0
(0 ; ())
(0 ; ())
Since the ellipticity assumption yields that is co
er-cive, we may use an estimate (given in [BIW]) exactly like that
in the pro of of Theorem 2.1. The arguments are then the same
as thosefor the undamp edcase.
We nowtakethe Laplace transform of which isgivenby
( )= ( )
Taking the Laplacetransform of (21) we have
( ) + ( ( ) )+ (^( ) )= ^
( )
(23)
whichholds for all with p ositivereal part. We can rewritethis as
( ) + ( )( ( ) )= ^
( ) (24)
where
( )( ( ) )= ( ( )+ ( ))
3
3
3 3
3
1
1 n n
n n
n n
n n
8 2
j j
0
P
8 2
P
!
!
!
!
!
2 2
2 2
2
2 2
2
1
2
2
2
2
2
b b
b
Z
b
b
b
b
Z
!
Z
d
b
d
c
b
b b
b
ijk h
k h i j
ijk h
ijk h
Y
ijk h ijk h lmk h lmk h
ij
l m
ij
Y
lmk h lmk h
ij ij
l m k h
ij
i j
b
t
t
n
b
t
t
< s v s ; > s v s ; < f s ; > H
s v s ; q s
@ v s
@x @x @
@x@x dy
q s
q s
Y
a y sb y a y sb y
@ s;y
@y@y
dy
Y
a y sb y
@ s;y y
@y@y
@
@y @y
dy Y H Y
Y
y yy
v s s
P u u L ;T H
P u u L ;T L
s
P u s u s H
P u s u s L
u s
s u s v s s
s
u v
solution to
( ) + ( )( ( ) )= ^
( ) () (25)
where
( )( ( ) )= ( )
( )
and where the ( ) are dened by
( )= 1
( )+ ( ) ( ( )+ ( ))
( )
and the functions are the -p erio dic solutions of
( ( )+ ( ))
( ( )+ ( ))
=0 in ( )
p erio dicin
with ( ) = . The arguments in Chapter 6, Section 4 of [SP] can b e
used without essential mo dication to show existence and uniqueness of a
solution ^( )of (25) for with p ositivereal part. Since byTheorem 2.3
in (0 ; ()) weak-*
and
in (0 ; ()) weak-*
as 0, we have,taking Laplacetransforms, that for real p ositive
( ) ( ) in () weakly
and
( ) ( ) in () weakly.
Butbyhomogenizationwithparameter ( )alsosatises(25)forrealp ositive
. By the uniquenessof solutions of (25), ( )= ( ) forreal p ositive and
by analytic continuation for any with p ositive real part. Finally by the
uniqueness of the inverse Laplace transform we have = . We have thus
3
3
n n
n n
Z Z
b
Z
!
Z
1 1
1
1
0
3 3
3
Theorem 2.4 2
! 2
!
2L
!
!
!
1 8 2
1 0
L
j j
0
0P
8 2
P
2 2
0 0
0 0
2
0 2
1 0
2
2
2
2
2
0 0
0
2 2
1
2
2 2
2
1
2
b
b
n
b
t
t
tt
b
t
ijk h
k h i j
ijk h ijk h
ijk h
Y
ijk h ijk h lmk h lmk h
ij
l m
ij
Y
lmk h lmk h
ij ij
l m k h
ij
i j
( ) ( )
(0 ; ())
~ () ()
~ ()
( (0 ; ) (0 ; ()))
0
(0 ; ())
(0 ; ())
( ) + ( )( () )= (0 ) ()
(0)= _(0) =
( )( () )= ( )
( )
( )= [^ ( )]
( ) 1
( )+ ( ) ( ( )+ ( ))
( )
( )
( ( )+ ( ))
( ( ) ( ))
=0 ( )
( )=
u f
L ;T L u v
u u L u H
u v L
P L ;T V ;L ;T H
P u u L ;T H
P u u L ;T L
u
< u t ; > t u ; < f; > ;T H
u u = u v =;
t u ; q t
@ u ;x
@x @x @
@x@x
ddx:
q t q s
q s
Y
a y sb y a y sb y
@ s;y
@y@y
dy
s;y Y
a y sb y
@ s;y y
@y@y
@
@y @y
dy Y H Y
Y
y yy
Let be the solution of 21 - 22 , corresponding to
and initial data , as in Theorem 2.2 satisfying
in weakly, with
in weakly.
Thenthereexistsanextensionoperator
such that for some sequence ,
in weak-*
in weak-*
where is the solution to the homogenized equation:
on
corresponding to the homogenized hysteresis sesquilinear form
Thecoecients are the inverse Laplacetransformsof
the functions
and the functions are the -periodic solutions of
in
periodic in
2
2
2
3 3
3
3
3
3
Z Z
X X
X
Z
! 3
j j j j
3
j j
3 1 2
2 2 2
=2
2
( )
=2
( ) 1 2
=2
( )
1 2 1 2
2 !
j j j j 0 0
!
j j j j j j
j j j j
j j
0
ecients on
3.1 Introduction
3.2 A Priori Estimates and the Limiti ng Equation for
the Case without Damping
a b
l l Y Y
u
q q u u
y
a y
@ y
@y@y
@ y
@y @y
dy a y
@ y
@y @y
a
A D C D Y :
D C Y C ;
C
q
q
Y
a y a y
@ y
@y@y
dy
ijk h
ijk h
ijk h
ijk h ij
ij
ij
Y
lmk h
ij
l m ij
k h
Y ijk h
ij
k h
ijk h
ij
L Y
ij
L Y
=
ij
L Y
= =
ij
ijk h
ijk h
Y
ijk h lmk h
ij
l m
Inthis sectionweconsiderthelimitb ehaviorof thehomogenizedco ecients
as the thickness of the memb ers 0. For the sake of simplicity we will
assumethatthetheoriginalco ecients and areconstantsandthat
= =1. Hence =1, = (2 )and = (2 ). Since wewant
to study thedep endenceof (the solution of the homogenizedequation)on
thethickness ofthematerialas 0wenowreintro ducethe sup erscript
(e.g. , , etc.).
We againprove somea priori estimates to show the existence of convergent
subsequences. Web egin by cho osing = ( ) inequation (17) yielding .
( )
( ) ( )
= ( )
( )
Thenfromtheellipticityofthe andanapplicationofHolder'sinequality
(26)
Thus
(27)
where, as explained earlier, we use to denote a generic constant indep
en-dent of throughout. We can then use this b ound on the to b ound the
. Recallthat
= 1
( ) ( )
( )
3
q
3 2
n
ijk h 2
4
X
3
5
3.3 Calculation of the .
1 1 1 2
=2
( )
1
2
2
4
0 0
1 2
1 2
1 2
0 0 3
j j
0 3
3
3 1
3
3 3
3
3 3
3 3
3
3
3
3
3
3
0 j j j j
!
!
!
!
2
!
Y 0 0
Y
fj j j j g Y
fj j j j g Y
fj j j j g Y
ijk h
=
ij
L Y
n n
n
ijk h ijk h
ijk h
b
b
tt ijk h
i j k h
ijk h
ij
ij
q C Y D C :
q q :
q
u u L ;T H
u H
u q
@ u
xx x x
f
u u = u v = :
Y Y
Y a
Y
Y
= ; = Y
H y = ; y =
V y = ; y =
K y = ; y = :
(2 )+
Thus thereis a sequence such that as 0,
(29)
Examining the homogenized equation (15) we see that if the give rise
to aco ercivesesquilinearform, then as 0
in (0 ; ()) weak-*
where () is the weak solution to
2 + =2 in
(0) = 2 and _ (0)= 2
Wewishto investigatethelimitofequation(28)as 0. But(28)dep ends
on integralsover so it willb e necessary to map to some xedregion.
Moreovernote that the integrands over contain only constants and
the -p erio dic functions . Thus we may take integrals over any ane
transformation of and still preservethe values of the integralsand our a
priori b ounds on the . In particular we willnd itconvenientto consider
the translated cell =( 1 2 1 2) + depicted in the gure b elow. We
willnditusefulto decomp ose into horizontal,verticalandcentral parts
by
= 1 2 2 the horizontalpart of
= 2 1 2 the vertical part of
-6
-6
6
8
8
1
2
1
2
1
2 1
2
2 2
0
0
3
0
3 3
3
0 0 3 0 0
0 Y
Y Y
Y Y
Y
jY j 0 0
!
Y
Z Z
Z
Z 2
4
! ! ! !
3
5 2
1
2
1
1 1 1
2
1
2
1
2
1 2
1 2 1 2
1 2 1 2
1 2 1 2
2
2
1 2
2
1 2 2
2
2 1 2
2
2
2 2
ijk h
ijk h
ijk h
H lmk h
ij
l m
V lmk h
ij
l m
K
lmk h ij
l m
H
V
K
H
ij
ij
ij
ij
y
y Y
y
Y Y
q
q a a
@
@y@y
dy a
@
@y@y dy
a
@
@y@y dy :
K H V
y ;y H
y ;y = y ;y :
V K
y =;y y ;y
y =;y = y ;y :
@
@y
@
@y @y
@
@y @y
@
@y
dy C:
Figure2: Structure of , , , and .
Nowusing the denition of the in(28) andthe ab ovedecomp osition of
wehave
=
+ (30)
As notedab ove,b ecause the domainsinthe ab oveintegralsdep endon
and since we are interested in the limit of these integrals as 0 we will
nd itconvenientto map , ,and onto the xeddomain . For any
function ( ) denedon wedene
( )= ( )
Similarlyfor and resp ectively
( )= ( )
and
( )= ( )
Usingthe apriori b ound (27) wehave that
! Y
! Y
! Y
! Y
! Y
! Y
! Y
! Y
! Y
! Y
Z 2
6
4 0
@
1
A
0
@
1
A
0
@
1
A
0
@
1
A 3
7
5
Y
0 0 0
0
0
0
0 0
0
0
0
0 0
0
0 0
0
00 00
0
00
0
00 00 ij
;H
ij
;H
ij
;H
ij
;H
ij
;H
ij
ij
;H ij
ij
;H ij
ij
;H ij
ij
;V ij
ij
;V ij
ij
;V ij
ij
;V ij
= ij
;K ij
= ij
;K
ij
1 1 2 2
2
2
1 2
1 2
1 2 2
1 2
2 1 2
2 2
2
2 2
2
2
1
11
2
1 2
2 1
21
2
1 2
1 2
12
2
2 2
2
2
22
2
1 1
2 2
2 2
2
1
11
2
1 2
1 2
12
2
1 2
1 2
21
2
2
2
2
22
2
1 1
2 2
3 2 2
2
1
11
2
3 2 2
1 2
12
2 @
@z
@
@z @z
@
@z @z
@
@z
dz C
@
@z
h L
@
@z @z
h L
@
@z @z
h L
@
@z
h L
V z y = z y
@
@z
v L
@
@z @z
v L
@
@z @z
v L
@
@z
v L
K z y = z y =
@
@z
k L
@
@z @z
k L
b ound can b e rewritten as
+ + +
thus
in ( ) weakly (31)
in ( ) weakly (32)
in ( ) weakly (33)
in ( )weakly. (34)
Similarlyfor we have = , = and
in ( ) weakly (35)
in ( ) weakly (36)
in ( ) weakly (37)
in ( ) weakly (38)
and lastly for ( = , = ) wehave
in ( ) weakly (39)
3 3
3
! Y
! Y
!
j j
0 0
Y
Y
!
Z Z Z
Z Z Z
Z
Z Z
Z Z
Z Z
Z Z
Z
"
Z
# 0
00 00
0
00
Y Y Y
Y Y Y
3
0
3
Y Y
Y Y
0
Y
0
Y
3
! 0
Y
! 0
!
0
0
0 3 2
2
1 2
21
2
3 2 2
2
2
22
2
11 12 21
12 22 22
1
2
1 2
11
11
22
22
11 22
1
1
22 2
2
2
2
1
22 2
2
2
0 1
22 2
2
2
0 1
22 2
2
2
0 22
1 2
1 2 1
1 2
2 2
2
2 2
22 2
2
2 =
;K ij
= ij
;K ij
ij
ij ij ij
ij ij ij
ijk h
K lmk h
ij
l m
=
ijk h
ijk h k h ij
k h ij
ij ij
lm
ij
l m
ij
ij
H ij
ij
=
=
=
=
ij
@
@z @z
k L
@
@z
k L
h dy h dy h dy :
v dy v dy v dy :
q
a
@
@y@y
dy C :
q a a v dy a h dy:
v dy h dy:
y
a
@
@y@y @
@y
dy a
@
@y dy:
H V K
V
K
a
@
@y
dy a
@
@y dy
a dy
@
@y dy
a
@
@y :
in ( ) weakly (41)
in ( ) weakly. (42)
Nowobserve thatb ecause ofthe p erio dicityof wehave
= = =0
Similarly,
= = =0
We are now ready to b egin the evaluation of the using (30). We rst
note that (27) impliesthat the last termof (30) go es to 0as 0 since
So nowapplying the convergencies(31)-(38) to (29) wehave
=2 (43)
Our remainingtask then is to evaluate the integral terms
and
To that end let b e a smo oth function, p erio dic in and indep endent
of . Cho ose = in (17) then
= (44)
Asb eforewedecomp osetheintegralson intointegralson , ,and .
For the right side observe that the contribution from the term is 0 since
is p erio dicand that as 0 the integralson go to zero. Therefore
lim = lim
= lim
3 Y 0 0 0 0 0 ! j j Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z 0 Y 0 0 0 0 Y 0 0 Y 0 0 0 Y
0 0 0
Y 0 0 0 Y 0 0 0 0 0 Y 0 0 0 0 0 0 0 Y 0 0 0 0 0 0 0 Y 0 0 0 0 0 0 Y 0 00 0 00 00 0 Y 0 00 00 0 00 00 0 Y 0 00 00 0 00 00 0 Y 0 00 0 00 00 0 ! 0 0 Y 0 0 lm ij l m H lm ij l m V lm ij l m K lm ij l m ij ;H H ij ;H H ij ;H H ij ;H H ij ;V V ij ;V V ij ;V V ij ;V V ij ;K K ij ;K K ij ;K K ij ;K K
V H K
H K lmk h ij l m = ij ;H H H ij
Wenowdeal with the leftsidetermsusingthe samedecomp ositionof as
b efore = + = + + + + + + +
Since is indep endentof , = and = . Furthermore
( )= ( ) (0) (46)
The integrals over converge to 0 by the estimate
Then observethat
= 1 22 2 2 2 2 1 22 2 2 2 2 1 22 2 2 2 2 1 22 2 2 2 2 1 1122 2 2 1 2 2 2 2 1 1222 1 2 1 2 2 2 2 2 1 2122 1 2 2 1 2 2 2 2 1 2222 2 2 2 2 2 2 2 2 1 2222 2 2 2 2 2 2 2 1 2122 1 2 2 1 2 2 2 2 1 1222 1 2 1 2 2 2 2 2 1 1122 2 2 2 1 2 2 2 2 1 1122 2 2 2 1 2 2 2 2 2 1 1222 2 2 1 2 2 2 2 2 2 1 2122 2 2 2 1 2 2 2 2 2 1 2222 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 0 2 2 2 1 2 1 2 1 1122 2 2 1 2 2 2 2 1 1122 2 2 1 2 2 2 a @ @y@y @ @y dy a @
@y@y @
@y
dy a
@
@y@y @
@y dy
a
@
@y@y @ @y dy a @ @z @ @z
dz a
@
@z @z @ @z dz a @
@z @z
@
@z
dz a
@ @z @ @z dz a @ @z @ @z
dz a
@ @z @z @ @z dz a @
@z @z
@
@z
dz a
@ @z @ @z dz a @ @z @ @z
dz a
@ @z @z @ @z dz a @
@z @z
@
@z
dz a
@ @z @ @z dz y @ @z z @ @y y @ @y : K a @
@y@y
dy C :
a @ @z @ @z
dz a
@
@y @
3
0
0
0 Lemma 3.1
Z Z
Z
Z
Z Z Z
Z
Z
Z
Z
0
0 0
0
Y
0 0
0
Y
0 0
0
Y Y
0
0 0
Y
Y 0
0 0
Y
0
0
=
=
=
= ij
ij
ij
;H H
ij
;H H
lm
ij
l m
lm
ij
lm
ij
lm
ij
lm
ij
ij
=
=
=
=
1
1122 2
2 2
2
2 1 2
1 2 2
2
1
1 2
1
1222 1
2
1 2 2
2
2
2
1
2122 1
2
2 1 2
2
2
2
1
22 2
2
2
2
22
2
2
2
2222 2
2
2
22
22
2
2
2
2
2
2
2222
22
22
2 1
2 1
2
1 2
1 2
1
2 1
2
1 2
1 2
Let be a function in , periodic and let be a real
constant. If
holds for any smooth function periodic on and such that
then
and
Proof:
a
@
@y
@
@y
dy dy
a
@
@z @z
@
@z dz
a
@
@z @z
@
@z
dz :
a
@
@y@y @
@y
dy a v z
@
@z
dz a
@
@y
h z dz:
a v z
@
@z dz
@
@y
a h z dz a :
w L ; a
a w x x dx
x dx ;
a w constant:
=
= 0
by the p erio dicity ofthe . By similar argumentswe can showthat
=0
and
=0
Thenusingtheweakconvergencies(34),(35)-(38)andthestrongconvergence
(46) we obtain
= ( ) + (0) ( )
Atlong last recalling(44), (45) and the equation ab ove we have
( ) + (0) ( ) =0 (47)
Wenow use the following lemmafrom [CS].
( )
7(0)+ ( )7( ) =0
7 [ ]
7( ) =0
=0 =
0
0
0 Z
Z
Z
Z
X
X Y
0
0
0 0 0
Y
Y
3
3
3
3.4 Extension to the Case with Damping
4 Consequences for the Love-Kirchho Plate
2222
22
22
2
2
2
2
22 1 2
1 2
1 2 1
22
22
2222
2
11
11
1111
2
=1
2
=1
a a h z dz a
@
@y
w z a v z ;z dz
h z dz a
a :
y
v z dz a
a :
q a
a a
a :
a
a sb
q s a sb
a sb a sb
a sb
a b
q s
h
ij
ij
lm =
= ij
lm
ij
ij
ij
ij
ijk h
ijk h
l
ijll llk h
llll
ijk h
ijk h ijk h
ijk h
ijk h ijk h
l
ijll ijll llk h llk h
llll llll
ijk h ijk h
ijk h
= ( ) 7=
and
( )= ( )
it follows that
( ) =
Nowif we were to rep eat allthe ab ove calculations with a smo oth function
that was indep endent of we would seethat
( ) =
Inserting these values into (43) we have
=2
Theextensionofthe ab overesultstoincludethecaseof dampingis
straight-forward. Weagain use Laplacetransformsand homogenization with
param-eter and instead of doing the ab ove calculations for we do them for
+ to obtain the homogenized co ecients
( )=2( + )
( + )( + )
+
(48)
Underthe usual assumptionson the and itis nothard to seethat
the ( ) satisfy an ellipticitycondition.
The ab ove results can b e sp ecialized to the case of a Love-Kirchho plate
( ) ( )
( ) ( )
0
0
0
0
0 0
0 0
ijk h
ijk h
ijk h
D
D
D
ijk h
D
D
D
D
1111 2222
2
1122 2211
2
1212 2121 1221 2112
1111 2222
2
1122 2211
2
1212 2121 1221 2112
2
2
2 1
2
1
2 12
1 2
2 2
2
2
1
2 2
2
1
2
2
2
2 3
2
1
3
2
2
2
2 2
2
2
2
2
1
2 3
2
2
3
2
1
12
2
1 2
3
1 2
a b
a a
EI
a a
EI
a a a a
EI
a
b b
c I
b b
c I
b b b b
c I
b EI c I
h @ u
@t
@ M
@x
@ M
@x @x
@ M
@x
f t >
M
EI
@ u
@x
@ u
@x
c I
@ u
@x @t
@ u
@x @t
M
EI
@ u
@x
@ u
@x
c I
@ u
@x @t
@ u
@x @t
M
EI
@ u
@x @x
c I
@ u
@x @x @t and we set the co ecients , as follows
= =
1
= =
1
= = = =
2(1+ )
and all other zero, and
= =
1
= =
1
= = = =
2(1+ )
and all other zero where is Poisson's ratio and and are the
usual stiness and Kelvin-Voigtdamping. Thenweobtain the weakform of
the Love-Kirchho plate equation with Kelvin-Voigt damping.To clarifythe
exp osition of the implications of the results of sections2 and 3 we now put
the resulting equationin strongform.
+ +2 + = 0
where
=
1
+ +
1
+
=
1
+ +
1
+
=
1+
+
! !
b
t
D
D
D
1 3
3
3 3 3
3 3
3
3 3
3
3 3
3
3 3
3
3 3 3
2
2
1
1
1
12
2
2
2
2
2
12
1
1
2
2
2
2 1
2
1
2 12
1 2
2 2
2
2
0 0
1
2
2
1
3
2
1
2
2
2
2
3
2
2
12
2
1 2
3
1 2
2
1 2 12 u
@u
@x
x :
M ;
@M
@x
@M
@x
x
M ;
@M
@x
@M
@x
x
P u
L ;T H u
h @ u
@t
@ M
@x
@ M
@x @x
@ M
@x
f t>
u u = ; u v =
M
EI@ u
@x
c I @ u
@x @t
M
EI@ u
@x
c I @ u
@x @t
M
EI
@ u
@x @x
c I
@ u
@x @x @t :
u
x M M M
= =0for =0
The conditions on the rest of theb oundary are natural ones andcorresp ond
to zero momentand zero shear. That is,
=0 +2 =0 (49)
on edgesparallel to the axis, and
=0 +2 =0 (50)
on edges parallel to the axis. The results of sections 2 and 3 then imply
that there exists an extension of the solution to this problem that
converges in (0 ; ()) as 0 and then 0 to , the unique
solution to
+ +2 + = 0
(0)= 2 (0)= 2
where
=
2
+
2
=
2
+
2
=
1+
+
1+
Of course must also satisfy the clamp ed b oundary condition on the edge
= 0 and the moments , , must satisfy the zero moment, zero
shear sp ecied in(49) and (49).
Finally we remark that we have made initial numerical investigations
into the question of the validity and utility of this approximate mo del by
comparingexp erimentallyobservedmo dalprop ertiesofametalplatecutinto
theshap eofarectangulargridwiththemo dalprop ertiesofthecorresp onding
Homogenizationtechniques andEstimationofMaterialParametersin
Distributed Structures Computation and Control II
Well-posedness andapproximation
for damped second order systems with unbounded input operators
Asymptotic Analysis for Periodic Structures
Asymptotic Analysis of
Elastic Wireworks
Exact Internal Controllability in
Per-forated Domains
ComportementMacroscopiqueD'unePlaque Perforee
Pe-riodiquement
LinearDierentialEquations inBanachSpace
NonhomogeneousBoundaryValue
Prob-lems, Vol. I
Non-Homogeneous Media and Vibration
The-ory
PartialDierential Equations
[B] Banks, H.T., D. Cioranescu, A. Das, R. Miller, and D.A. Rebnord,
, in (K. Bowers&
J. Lund, eds), Birkhauser,(1991), pp. 13-30 .
[BIW] Banks,H.T.,K.Ito,andY.Wang,
, to
app ear.
[BLP] Bensoussan, A. et al., ,
North-Holland, Amsterdam(1978).
[CS] Cioranescu, D. and Saint Jean Paulin, J.,
,preprint.
[CD] Cioranescu,D. andDonato, P.,
,preprint.
[Du] Duvaut,G.,
,inSingularPerturbationsandBoundaryLayerTheory.,
Ed.Brauneret.al.,LectureNotesinMathematicsNo.594,
Springer-Verlag,(1978), pp. 131-145 .
[K] Krein,S.G., ,American
MathematicalSo ciety,Providence,R.I.(1971).
[LM] Lions,J.L.,andMagenes,E.,
, Springer-Verlag, NewYork(1972).
[SP] Sanchez-Palencia, E.,
, Lecture Notes in Physics No. 127, Springer-Verlag, New
York,(1980).
[W] Wloka,J., ,CambridgeUniversityPress
