A NNALES DE L ’I. H. P., SECTION A
G. F. D ELL ’A NTONIO
R. F IGARI E. O RLANDI
An approach through orthogonal projections to the study of inhomogeneous or random media with linear response
Annales de l’I. H. P., section A, tome 44, n
o1 (1986), p. 1-28
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1
An approach through orthogonal projections
to the study of inhomogeneous
orrandom media
with linear response
G. F. DELL’ANTONIO
R. FIGARI
E. ORLANDI
Centre de Physique theorique, CNRS Luminy,
On leave from Dip. di Matematica, Univ. di Roma
Dip. di Fisica, Univ. di Napoli and Bibos, University of Bielefeld
Dip. di Matematica, Univ. di Roma, CNR, GNFM
Ann. Henri Poincaré,
Vol. 44, n° 1, 1986, Physique theorique
ABSTRACT. 2014 We present, for the
study
of effective constants in anhomogeneous
or randommedium,
a formalism based onorthogonal projections
which stresses the mathematicalequivalence
of differentphy-
sical
problems.
,
We
develop
also anapproximation
scheme andapply
it to a case ofphysical
interest.RESUME. - On
introduit,
dans l’étude des constants effectives pour des milieuxinhomogenes
oualeatoires,
unprocede soulignant l’équivalence mathematique
deplusieurs problemes physiques.
On considere aussiun schema
d’approximation
et on1’applique
a unprobleme
d’interetphysique.
Annales de l’Institut Poincaré - Physique theorique - Vol. 44, 0246-0211 86/01/ 1 /28/$ 4,80 (Ç) Gauthier-Villars
2 G. F. DELL’ANTONIO, R. FIGARI AND E. ORLANDI
1 INTRODUCTION AND SUMMARY
Consider a dielectric medium in a
region
D cR3.
Denoteby E(x)
thelocal dielectric tensor. Choose a
boundary
condition r on ~D and let E be the electric field in D. Whenonly partial
information on E isavailable,
or when e is a
rapidly varying function,
one is interested inaveraged
quan- - tities. One of them is the effective dielectric constantE*,
definedroughly speaking
as theproportionality
ratio between the averageof |~E| and of 1 E I.
This situation
prevails
also in a number of otherphysical problems,
e. g.in the
study
ofdissipative properties (conductivity,
thermalconductivity,
viscous
flows)
ordynamical properties (magnetic susceptibility, elasticity, velocity
ofsound).
Infact,
from thepoint
of view consideredhere,
all theseproblems
can be considered asmathematically equivalent.
In each of them one considers a field S in
D,
with values in a vector spaceK; physically,
the field S may describe the electricfield,
the tempe-rature
gradient,
the fluidvelocity,
the elastic strain tensor, etc. The field S is in some sense agradient (more generally,
a closedform).
It also satisfiesan
equation
of the formwhere 6 is a field of maps from K to
itself, describing
the localproperties
of the medium. If 6 does not
depend
onS,
and is bounded andstrictly positive, ( 1.1 )
isequivalent
to anelliptic
system and has therefore aunique
solution under very mild conditions on D and r
[1 ].
Denoteby
F theaverage of the
quantity
F over D. Let K be a unit vector inR 3..
If r is homo-geneous, as we shall
always
assume, isproportional
to . Theeffective constant 7* is then defined
by K . 6S
= o-*K. S. Almost allpublished
results on
approximations
or bounds for r* refer to the linear case. A notableexception
is[2 ].
In this paper, in which we consider
only
the linear case, we solve(1.1) by
a construction whichemphasizes
the mathematicalequivalence
ofthe
physical problems
described above. We workentirely
in a Hilbert-space
setting,
andexploit
the fact that rot and div are(Hodge) adjoints [6a]
[6b].
In the case in which takesonly
a finite number ofvalues 6i
indomains
Di,
andaDi
aresufficiently regular,
theapproach
we followis
equivalent but
not identical to the use ofboundary-layer potentials.
The paper is
organized
as follows :In section 2 we
develop
the formalism when S is the electric field(denoted by E)
and ~7 the dielectric tensor(denoted bye).
We construct E asuniformly
convergent series. When the
system
is made of Nhomogeneous
components with dielectric tensors Ek, we determine theanalyticity
domain of E as a.
Annales de l’Institut Henri Poincaré - Physique theorique
3
INHOMOGENEOUS OR RANDOM MEDIA WITH LINEAR RESPONSE
function of the
Ek’s
.We deduce then thecorresponding
resultsfor
E*.We also touch
briefly
on the variationalapproach.
In section 3 we indicate how other
physical problems
can be solvedby
the sameformalism;
we outline theprocedure
for elastostatics and viscous flow.We also outline the
analysis
for astationary
random medium] [Sb ].
In section 4 we
develop
anapproximation scheme,
based on the intro-duction of
approximate
local fields 6’and/or
a differentboundary
condi-tion r’. In way of
example,
wecompute
to order 21 the(uniformly
conver-gent) expansion
ofs* in powersof p
for the case whenhomogeneous spherical
inclusions of radius p are
placed
at the centers of a unit cubic lattice in ahomogeneous
host medium.In an
Appendix,
we sketch the relation of ourapproach
topotential theory.
In a
separate
paper we use theapproach developed
here to prove existence of solutions of(1.1)
in case of non-linear response, under suitable assump- tions on the non-linear terms. We shall also extend to the non-linearcase the
approximation
scheme of Section 4.Our interest in this
subject
wasoriginated
some time agothrough
dis-cussions with G.
Papanicolaou,
which aregratefully acknowledged.
We haveespecially profitted
from severalenlightening
discussions with D.Bergman.
Part of the work was done while one of the authors
(GFDA)
wastaking
part in the
project
n.2,
Mathematics andPhysics,
atZIF, University
ofBielefeld.
The other authors
enjoyed
at various times thehospitality
and support of the Courant Institute(E. 0.)
and of the Ruhr-Universitat Bochum(R. F.).
We express here our thanks to Prof. L.
Streit,
G.Papanicolaou
and S. Albe- verio for the warmhospitality.
2.
ELECTROSTATICS,
EXPLICIT DETERMINATION OF THE ELECTRIC FIELDAND OF THE EFFECTIVE DIELECTRIC CONSTANT Let D be a bounded domain in
R3, E(x)
the local dielectric tensor field.We
study
the static electric fieldE(x)
for suitableboundary
conditionson aD. Maxwell’s
equations
areor, in abstract form
where * is the
Hodge-duality.
Vol. 44, n° 1-1986.
4 G. F. DELL’ANTONIO, R. FIGARI AND E. ORLANDI
On e we assume that one can find
positive
numbers a,j8
such thatWe shall use the notation
where lei
denotes the norm of the matrix E.The system
(2.2)
isstrictly elliptic,
and has aunique
solutionby
theLax-Milgram
lemma.We shall
give
anequivalent
version of(2.2), exploiting
the fact that the kernel of d * is contained in theorthogonal complement
of the range of d.The
projection operator
we use has anexplicit form,
when aD issmooth,
as an
integral
operator; in this case, theapproach
we follow is related topotential theory.
We shallgive
in theAppendix
an outline of this relation.We denote
by L2(D))3
theHilbert-space
of vector-valued functions on D, with scalarproduct
where D | is
theLebesgue
measure of D.Let
H 1 (D)
be the space of those distributions whose(generalized)
gra-dient is in
(L2(D))3.
LetHo(D)
be the closure inH1(D)
of the function of class C1 withsupport strictly
contained inD,
and let V be ahomogeneous boundary-value
space, i. e. a closed proper linearsubspace
ofsuch that V ;2
Ho(D).
Let
Ytv
be the smallest closedsubspace
of(L2(D))3
which contains which have thefollowing property :
for everyxED,
onecan find a
neighbourhood ~
and afunction ~
E V suchthat ~ ==
in ~.Let A be the
orthogonal projection
of(L2(D))3
onto Inanalogy
withhydrodynamics,
we call A aHodge projection.
We shall not indicateexpli- citely
itsdependence
on V.Let
Eo
EEo $ Ytv.
We shall say that E is an irrotational field in D
satisfying
theboundary
conditions
(V, Eo)
if E -Eo
EYtv [6 ]. By
definition thereforeDefine
so that One has then
In many
practical cases,
e. g.parallel-plate
condenser orperiodic
Annales de l’Institut Henri Poincare - Physique théorique
INHOMOGENEOUS OR RANDOM MEDIA WITH LINEAR RESPONSE 5
boundary conditions, Eo
is a constant vector, andAEo
= 0. This rela-tion however does not hold in
general,
even ifEo
is a constant vector.For
practical
purposes, it is convenient to know theexplicit
action of Awhen ç
E(C1)3.
If aD is such that a Green function G exists for the homo- geneousboundary
condition(V, 0),
one hasHere and in the
following
summation overrepeated
indices is understood.To find the action of A on a
generic
elementof (L 2(D))3
one can use(2. 8)
on
approximating
sequences.,
For
periodic boundary
conditions over aregular lattice,
the action of Ais easily
describedusing
dual lattice variables. Forexample,
for a cubiclattice of side one, one has
where,
for each f =1,2, 3,
is the Fourier transformSince
V 4>
EHv
for all Vif 03C6
ECõ, integration by
part shows thatif ~ ~ Jfy
then ---.!.
~,
=~t
Therefore,
for allboundary
conditions(V, 1>0)
weinterpret
div s. E = 0 asNotice that
(2.10) provides
also information on the behaviour of eE at theboundary.
The system
(2.7), (2.10)
isequivalent
to Maxwell’sequations
withboundary
conditions(V, Eo).
Inparticular,
any solution E of(2. 7), (2.10)
of class
C1
is also a strong solution of Maxwell’sequation.
Let ~ > 0 be any real number.
Ùsing (2. 6’)
and A2 = A one verifies that(2. 7), (2.10)
areequivalent
toLet ~n > E + , and define the
positive
operatorQ by
(we
do notdistinguish notationally
between the function and the operator ofmultiplication by
Since is
invertible,
one obtains from(2.11),
with 5 =~-10
theequi-
valent form
Vol. 44, n° 1-1986.
6 G. F. DELL’ANTONIO, R. FIGARI AND E. ORLANDI
which can be written
Similarly,
for 0 ~o E _ , letOne has then
Remark
that ~Q~0~ 1, where ~ ~
denotes the operator norm. Therefore(2.14) provides
auniformly convergent
series which determines E.Consider now the
important special
case whenE(x)
takesonly
a finitenumber of values Ei, i = 1 ...
N,
and let xi be the characteristic function of the domainDi
in whichE(x) _
~i. ThenTo
study
the response to external fields whichdepend slowly
ontime,
it is often convenient to consider Maxwell’s
equations
when the tensors ~iare allowed to be
complex-valued.
We consideronly
the case in which. the
complex extensions,
still denotedby
Ei, can bediagonalized.
Denoteby
a =1, 2,
3 theeigenvalues of ~i,
andlet s = {
E For afixed geometry
(i.
e. choiceof { D~}),
letE(s)
be the(complex-valued)
solu-tion of Maxwell’s
equation
whens(;c) =
Let E> cC3N
be definedby
One has then
PROPOSITION 2.1. - For all E E
8, E(E)
exists and isunique.
MoreoverE(s)
isanalytic
in e(as
an(L 2(D))3-valued function),
and for each E eone can find Eo E C such
that,
The series is
uniformly
convergent in aneighbourhood
ofE~ 1 ~
and isdefined in
(2.12).
-Proof
- The first statement, and alsoanalyticity,
follows from thel’Institut Henri Poincaré - Physique theorique
7 INHOMOGENEOUS OR RANDOM MEDIA WITH LINEAR RESPONSE
coercivity
of the operator2014
in eachregion
for a suitablechoice of
03C6a.
For the
representation through
a convergentseries,
notice that(2.13), (2.15)
hold also when E takescomplex values, provided
one can find Eo E C such thatFor E let ~o = C > 0. Then
and
by
construction > 0.One can then find a constant
Co, depending only
onmax
suchthat
(2.17)
holds in a fullneighbourhood
of if C > - REMARK 2.2. - One haswhere
P, Q
= 1 ... 3 N stand forpair
of indices(i, oc)
andIt is easy to
verify
that e is the(inductive)
limit ofanalytic polyhedra,
in the sense
of Weyl [7 ].
This allows to write
integral representations
for E(and
for8*)
wherethe
integration
is on a measuresupported by
50 anddepending only
on the domains
Di, D,
while theintegrand
is a suitable function of the REMARK 2 . 3. - Ifmultiplication by
8 is a bounded ope- rator on(Hk)3. If k ~ 3,
the same is trueif ~ij
E Hk.If ~0
is chosensufficiently large (2.12)
defines then a boundedpositive operator
on(Hk)3,
with normsmaller than one. The
Hodge projection
A can be defined on(Hk)3
in muchthe same way as on
(L2)3.
Therefore under theassumptions
indicatedhere, (2.14), (2.15)
andProposition
2.1 hold also in(Hk)3.
IIWe now turn to the
description
of the effective dielectric constant8*,
which we
define,
forboundary
conditions(V, Eo), by
where
B = B . ~ ~ B ~ ~ -1.
Vo).44,n"l-!986.
8 G. F. DELL’ANTONIO, R. FIGARI AND E. ORLANDI
From
(2. 7), (2.6)
one hasand therefore G* satisfies
In
(2.21)
the role of E* as effectiveparameter
in linear responsetheory
is
clearly
exhibited.Let
In most cases of interest
B
is constant sothat ~B
is the average of( B, E(x) B) = SaM.
From(2.14), (2.15)
one deriveswhere /~ is the
spectral
measure of in the stateQ~B (i.
e. the~
moment of ~ is
«
and/~o
is thespectral
measureof Q’~’0AQ’~’0 in Q’~’0 B.
From
(2.19)
andProposition
1 one hasPROPOSITION 2.4. 2014 With the notations of
Proposition 2.1, e*(s)
isanalytic
in0;
for each ~ e 0 one can find So e C such that. and the series is
uniformly convergent
in aneighbourhood
of E. .In
(2.24)
we have used the notationRemark
that,
when B is constant, one haswhere pi
is the relative volume of the domainDi.
Wepoint
out that themeasures in
(2.23’,") depend
both on thegeometry
and on the(2.23)
and(2.23’)
do notgive
arepresentation
in which theanalyticity
of E* can be used
directly.
l’Institut Henri Poincaré - Physique theorique
INHOMOGENEOUS OR RANDOM MEDIA WITH LINEAR RESPONSE 9
The
only exception
is the case N =2, E~
=ciI,
In this case thedependence
on 81, 82 isonly through
their totalweight, and, choosing
80 = max(~i, 82),
one can writewhere
is thespectral
measure ofx2Ax2 in
the state In all other cases, this factorization does not occur in(2.23’/’).
We
point
out, for later use, that one has theexplicit
formulaeWe conclude this
chapter
with a brief comment on the variational for- mulation.It is evident that
(2.13)
is theEuler-Lagrange equation
associatedto the
quadratic
functionalSince
I,
isstrictly
convex so thatUsing (2.28), (2.19), (2.20)
one concludesWe remark that does not coincide with the
quadratic
functionals which intervene in the direct or in the dual variationalprinciples
asso-ciated to
(2.1).
Thereforeinequality (2.30)
is different from the ones consi- dered e. g. in[3 ].
’
Similarly,
for 0 E _ , one considers thequadratic
functionalon
(L2(D))3
definedby
Since
Q’~’0AQ~’0
>0,
P isstrictly
concave, so thatfrom which one concludes
Vol. 44, n° 1-1986.
10 G. F. DELL’ANTONIO, R. FIGARI AND E. ORLANDI
3.
ELASTOSTATICS,
VISCOUSFLOW,
RANDOM MEDIA
In this section we shall outline the steps
by
which theequations
forelastostatics and viscous flows can be
placed
in thegeneral
frame describedin §
2. We shall alsoindicate, using
electrostatics as anexample,
how thescheme
of §
2 extends to random media.a)
Elastostatics(see
e. g.[8 ]).
In an
orthogonal
local coordinate system we denoteby
u the elasticdisplacement
vector, andby v
the strain tensor, with componentsLet L, with components denote the stress tensor. Then Hooke’s law reads
where C is the elastic stiffness tensor.
We shall
only
treat the case in which there are nobody
forces. In this case,T satisfies
where D is the
region occupied by
the elastic medium.We consider the
problem
describedby equations (3 .1 ), (3 . 2), (3 . 3)
ina bounded
simply
connected domainD,
withprescribed boundary
condi-tions on the strain tensor
It is known
(cf. [6 ],
ch.7) that,
if ~D is aLyapunov surface, given v(x)
one can solve
(3.1)
for u,uniquely
modulorigid
translations. We write therefore(3.1) together
with theboundary
condition(3.4)
aswhere V is the closure in
(L2(D))6
of the vector fields of the formwith
~i
ECo(D).
Let now A be the
orthogonal projection
onto 1/. Then(3.1), (3 . 3) read
Annales de l’Institut Henri Poincaré - Physique theorique
INHOMOGENEOUS OR RANDOM MEDIA WITH LINEAR RESPONSE 11
as one
easily
verifiesthrough integration by
parts. Forcomputational
purpose, it is worth
remarking
that A in(3.6), (3. 7)
is the closure of theoperator A
defined on(C1(D))3 by
Here
Gkm
is the Green function of the elastostaticproblem (double-layer potential operator),
i. e. the solutionof 0394Gkm
+ withboundary
conditions defined
by
1/.Typical
choices forv°
are(isotropic strain),
orFor the choices
(3.9), (3.9’),
one has 0.We remark that
(3 . 6), (3 . 7)
are identical in form to(2. 7), (2.10),
so thatthe formalism of
§ 2 applies.
To define the effective elastostatic structure, notice firstthat,
as in Section2,
where
v2)
=For
simplicity,
we assume thatv°
is a constantsymmetric
tensor, and that theboundary
conditions are such that 0.Then
(2.14) implies
for everysymmetric tensor 11
From
(3 .10), (3 .11 )
one derivesThe effective elastic stiffness C* associated to the
given boundary
condi-tions is then defined
by
-Of
particular
interest is the case when themicroscopic
elastic structureis
represented by
two Lame scalarfunctions,
i. e.for every
symmetric
tensor vB ’.
From
(3.13)
and(3.14)
one deriveseasily
that there are constants/)*, ,u*
such that
This defines the effective Lame coefficients.
Vol. 44, n° 1-1986.
12 G. F. DELL’ANTONIO, R. FIGARI AND E. ORLANDI
We refer to
[3c] ] [4c ~ (and
the referencesgiven there)
for a detailedstudy
of bounds for ~* and
/1*,
which are obtained in[~c] ] by exploiting
thestructure as
analytic
functions of~,,
/1, and in[~c] ] by
a combination ofperturbation theory
and variationaltechniques.
Here we limit ourselves to a
simple
remark.Consider,
forexample,
a
two-component
system in which eachcomponent Da
a =1, 2,
is charac- terizedby
the Lame coefficientsComparing
with Section2,
we see that theproblem
ismathematically equivalent
to electrostatics for atwo-phase
system in which eachphase
is
dielectrically anisotropic.
Define
IIo
on(L2(R3))3 by
Let
Ap
be theHodge projection
forperiodic boundary
conditions on acubic
lattice,
andA (0)
theHodge projection
in(LZ(R3))3.
ThenTaking v0 =
Ior v0
a tracelesssymmetric
tensor, one verifiesthen,
forthe cubic lattice or if one
neglects
the effect of theboundary, that ( ~, 7 )
is
represented
as in(2 . 23)’,
the measuredepending
ononly through
its total mass.
On the other
hand,
in view of(3.15), (3.16),
for every choiceof v0
theexpression ~ v°, ~ ~
is a linear combination of ~,* and/1*.
We conclude that for N = 2 and
neglecting boundary effects,
thereare
two linear
combinations of ~* and/1*
which admit arepresentation
of the form
(2.23)’
such that thedependence
on thegeometry
isseparated
from the
dependence
on Bounds on these two linear combinationscan therefore be obtained
by
thetechniques
of[4a, b (see [4c ]).
b) Viscous flow (see e. g. [9 ]).
We consider the Stokes flow in a
region
D filled with anincompressible
fluid of
viscosity
tensor Denoteby
u thevelocity field,
andby p(x)
the pressure field.
As in
elastostatics,
we introduce thesymmetric
tensor V definedby
and remark
that,
if aD issufficiently regular, v
defines u modulorigid translations,
so that v can be considered as the relevantphysical
variable.Define 6
through
and let
Annales de l’Institut Henri Poincaré - Physique ’ theorique ’
INHOMOGENEOUS OR RANDOM MEDIA WITH LINEAR RESPONSE 13
Then 7T is a field of
symmetric
tensors and Stokesequation
readsTo write
(3.18)(-3.20)
in the form(2.7), (2.10),
denoteby V
the closurein
(L2(D))6
of the vector fields of the formwith 03C6; E 1 D and 0..
with
03C6i
E and 0 - = 0.Take ’
boundary
condition of the formwhere
u°
is adivergenceless
vector field. Letv°
be definedby
so that
v(x)
= ;c E ~D. Assume that 1/.PROPOSITION 3.1. - Let aD be of class
C1. Equations (3.18)-(3.20)
.with the
boundary
condition(3.21)
areequivalent
toProof
From the definition of A it follows that(3.23)
isequivalent
to the fact that v can be written
uniquely
as in(3 .18)
where u satisfies(3 . 21).
Indeed if aD is of class
C 1,
u is definedby v
in(3.18)
modulorigid
trans-lations,
and(3.21)
determines ucompletely.
Let’now ,
be any field ofsymmetric
tensors. ThenA’
= 0 isequivalent
toIntegrating by
parts this leads towhich
implies
that there exists a scalar valued fieldp(x), uniquely
definedmodulo an additive constant, such that
Vol. 44, n° 1-1986.
14 G. F. DELL’ANTONIO, R. FIGARI AND E. ORLANDI
We will not pursue here further the
study
of the viscous flow. Weonly
remark that the flow
through
porous media can beregarded
as the limitof a viscous flow when the
viscosity
tensor has the formand lim Cn = + 00.
In the limit the flow
occupies only
theregion
D -D1,
and M = 0 at~D1.
If for all n ~ Z+
one can
compute
the effectiveviscosity 1]*
for the viscous flow as lim it willdepend linearly
on1]0.
Therefore one can define a constant K
(Darcy’s constant) through 1]* = K -11]0.
The formalism described in section 2 can then be used
directly
tostudy r~n
in the limit c~, -~ oo, and therefore to
give
estimates on K.c)
Random media.We denote
by
Q theunderlying probability
space, which carries a measureP and a
norm-preserving
continuousergodic representation ~ -~ Ta
ofthe translation group
IR 3.
Q can beregarded
as the set ofpossible configu-
rations of the system.
The dielectric
properties
of astationary
random medium are then des- cribedby
astationary
random dielectric tensors(jc).
As in
§ 1,
on E we assume that one can find 0 a b oo such thatWe have used the notation
We denote
again by
e+, s- thelargest (resp. smallest)
real number such thatSince
E(x, . )
isstationary,
one has thenMaxwell’s
equations
becomeThe derivatives must be understood o in a weak sense, as usual. We look for solutions of
(3 . 29) 1, 2
whichare stationary random
fields. Theequi-
Annales de l’Institut Henri Physique theorique
INHOMOGENEOUS OR RANDOM MEDIA WITH LINEAR RESPONSE 15
valent of a
boundary
condition is now theassignment
of the average value ofE(x)
with respect to P. This average isindependent
of x in viewof
(3 . 27)
and will be denotedby
B.Let lk, k
=1, 2, 3,
be theanti-self-adjoint
operators defined ondP) by T’a
= expThe
lj’s
have a dense common coreS, by
Stone’s theorem. On S define Aby
_where l2
= .k
It is
straightforward
toverify
that A is closable and boundedby
I onS,
and that its extension todP))3,
which we denoteby A,
is aprojection
operator.
If the
periodic
case isgiven
aprobabilistic description,
the operator K here isprecisely
theHodge projection.
We therefore call A theproba-
bilistic
Hodge projection.
Notice that[A, Ta =
0Va
ER3,
and moreoverTa
leaves S invariant.
- -
We use the notation
One has
(see
alsob]).
LEMMA 3 . 2. -
Equations (3 . 29) 1, 2. supplemented by
the conditionare
equivalent
towhere all
equalities
are understood in thedP)
sense. -Proof
DefineH1(Q) through
where the closure is taken in the
(Hilbert)
normWe denote
by
H -1 the dualVol. 44, n° 1-1986.
16 G. F. DELL’ANTONIO, R. FIGARI AND E. ORLANDI
One verifies that the
following
limit exists inand moreover
(3.29)2
isequivalent
towhich is
equivalent
to(3 . 32) 1, 2 .
In the same way one proves that for all
x E R3,
and that
(3 . 29) 1
isequivalent
tofor some constant vector C. -
Since A is
self-adjoint, taking
scalarproducts
with a constantfunction,
one derives from
(3 . 3 5)
In view of Lemma
3 . 2,
the formalism of section 2applies
withoutchange
to the case of
stationary
media.For
computational
purpose, it is convenient to have anexplicit
formof
expressions
suchas ( B, p E Z+,
in terms of the corre-lation functions of the random field e. we write
The correlation functions are defined
by
and
g(x)
be astationary
random field such thatThen and
Therefore,
if a > 0It follows
that,
ifE(h .g(x)) E L 1(R 3),
thenwhere
Ga( . )
is theintegral
Kernel of(
- d +a) - ~ .
Annales de l’Institut Henri Poincare - Physique theorique
17 INHOMOGENEOUS OR RANDOM MEDIA WITH LINEAR RESPONSE
In the same way, is a
stationary
vector-valued randomfield,
such that and andif E(fgk(x))E L2(R3)
nH1(R3),
one has
Let A 0 be the
Hodge projection
associated to theLaplacian
inR3.
Since A is bounded on and A 0 is bounded on
L2(R3),
the iden-tity (3 . 40)
can be extended to a = 0 to obtainFrom
(3.41)
itfollows,
where W is the Fourier transform of W.
In the
special
case in which = one haswhere
W2
is the Fourier transform of To thefollowing order,
one hasThe first term has been evaluated in
(3 . 42).
We
give
the second termonly
in the case inwhich ~ij
=Let
W3(~, r~)
=~(8(0)s(~)8(~)),
and let W3(/?~’)
be its Fourier transform.Assume that the function
W3
can be written aswhere x
R3).
Vol. 44, n° 1-1986.
18 G. F. DELL’ANTONIO, R. FIGARI AND E. ORLANDI
Then one has
All terms of the form
E(B,
can be obtained in this way.4. RESOLVENT
EXPANSIONS,
THE CLAUSIUS-MOSSOTTI APPROXIMATION AND AN APPLICATION
In this section we want to elaborate further on the formalism of sec-
tion 2 and in
particular
on thepossibility
ofdeveloping approximation
schemes and of
testing
their convergence. We shall consider hereonly approximations
for G*. A naturalprocedure
is to substitute in(2.27)1,2
for
QEOAQEO (resp. Q’~0AQ’~0)
anotherpositive
operator Z with which theexpression
can be calculated orapproximated,
and then to estimate theerror made.
In view of the resolvent structure of
(2 . 27) 1, 2
we call thisprocedure
a resolvent
expansion.
One hasOur task is to control the second term on the r. h. s. of
(4.1), (4.2).
A
possible
choice for Z is whereAo
may be theHodge projection corresponding
to differentboundary
conditions on aD. The
following
easy lemma indicates the results to beexpected.
LEMMA 4.1. - Let Z be defined as in
(4.3)
and assumeThen the iteration of
(4.2) gives
auniformly
convergent series. IIProof
From(4.4)
and theassumptions
of thelemma,
it follows thatAnnales de l’Institut Henri Poincaré - Physique ’ theorique ’
INHOMOGENEOUS OR RANDOM MEDIA WITH LINEAR RESPONSE 19
The formal series obtained
by iterating (4.2)
iswhere K =
(1
+ +The conclusion of the lemma is then a consequence of
(4 . 6).
- Remark. From theproof given,
it is clear that a resultcorresponding
to lemma 4.1 holds for an
expansion
ofif the weaker
assumption
holdsfor all
!/
in the smallestsubspace
of(L2(D))3
whichcontains B
and is left invariantby Q~AQ~ ancfby
We shall use this fact in theexample
illustrated below. A second case of interest is the
following.
LetD~,
i >2,
have diameter10
andsatisfy Dj)
>Lo
for i~ j and
alsod(Di’ aD)
>Lo,
where
Lo
» Let E~ =cj
i = 1 ...N,
and E 1 Ek, /c > 2. We do not assumethat Ei ~ E j
for i ~j.
This
description corresponds
to ahomogeneous
host medium in which Nhomogeneous
inclusions aresparsely placed.
Let
{ D~,
i > 2}
be a collection of domains in D such that are smooth andDi
cD,
iff i= j, d(Di,
>Lo/2 d(Di, aD)
>Lo/2,
and letD1 =
J>2
In
words, Di,
i >-2,
arewidely separated
and for eachi, Di
is interior toD ~
andwidely separated
from itsboundary.
Let
A1,
i = 2 ...N,
be theHodge
operator inD,
with Dirichletboundary
conditions on and
A 1
be theHodge
operator with Dirichletboundary
condition on and with the
boundary
conditions on ~D which entered the definition of A. Consider the naturaldecomposition
and let
Take Eo = that
VoL 44, n° 1-1986.
20 G. F. DELL’ANTONIO, R. FIGARI AND E. ORLANDI
and notice that Since
one has
where ,ui
is thespectral
measureof ~iAi~i
in the statexiB.
With these nota-tions,
one hasLEMMA 4. 2. - For
Lo sufficiently large,
the seriesis
uniformly convergent
and converges to zero whenLo ~
oo . -Proof
2014 Weregard
anoperator
T on(L2(D))3
as a matrix ofoperators
from
(L 2(Dd)3
to(L2(D~ ))3.
Since(I
+QAoQ)’’ I,
it suffices to provethat one can find A > 0 such
that,
forLo
>A,
theoperator
whose matrixis has norm smaller than one.
Remark that
~iA0~j
=0,
i ~j.
It suffices therefore to prove that for any
integer
N one can find A > 0 suchthat,
forLo
> ABy
alimiting argument,
it iseasily
established thatwhere G is the Green function associated to A and
Gi
the Green function inDi
with Dirichletboundary
conditions. The conclusions of lemma 4.2 follow then from standard results on thedecay properties
of Green func- tions forregular
domains. -Annales de l’Institut Henri Poincaré - Physique theorique
INHOMOGENEOUS OR RANDOM MEDIA WITH LINEAR RESPONSE 21
Remark. 2014 We remark that the first term in the resolvent
expansion,
i. e.the second term in
(4.9),
isusually
called the Clausius-Mossottiapproxi- mation [3a].
In the
remaining
part of thissection,
we work out the details of theapproxi-
mation
procedure
for thefollowing
case[77]a, b,
c. A cubicperiodic
array of balls of dielectrictensor ~2.I
and radius p isplaced
in ahomogeneous
host medium with dielectric tensor
E1.I.
Periodicboundary
conditionsare assumed. We seek an
expansion
ofs* in powers of p. Onehas, according
to
§ 1,
andassuming
G1 > EZwhere/?2
=4/3~~,
X is the characteristic function of the ball at theorigin,
and we have chosen the vector B to be
parallel
to one of theprincipal
axis of the
lattice,
which we denoteby k.
We denoteby
W the unit cube centered at theorigin,
andby b
the ball ofradius p
at theorigin.
We compute the action ofAX
onspecial
vectors.Let f E (L 2(W)3,
= 0(weakly). By
a limit process one verifies that ,~) == (Axf)(x)
has thefollowing properties,
which characterize ituniquely
curl-and
divergence
free except at most onab ;
b)
on ab the normal componentof (
has adiscontinuity equal to -f.
n,where
_n(x)
is the outwardnormal ;
c) (has
mean value zero in Wand isperiodic
on aW.1
Formally, ~(x)
is the electric field due tocharges - 4 f . n
distributedon the surface of the balls at each lattice site. 4n -
Let
Ao
be the restriction to(L2(W))3
of theHodge projection
on(L2(R3))3.
(Aoxf)(x)
is then1/4~c
times the electric field due to acharge 2014/.~
on ab.Define
through
.Recall that any vector-valued
function g
which has supportin b,
is ofclass
C 1,
isdivergence
free and is continuous up toab,
can beuniquely decomposed
aswhere gd
is the «dipole »
partof g,
relative to thegiven
axisk.
If I is divergence-free,
we defindwhere
~.(~/)’
is the field in W due to the array of «image » sphares (with
Vol. 44, n° 1-1986.
22 G. F. DELL’ANTONIO, R. FIGARI AND E. ORLANDI
the exclusion of the one at the
origin)
eachcharged
with a surfacecharge - f . n..
This field is
given by
auniformly convergent
series. In thisnotation, by
definition the field in W due to allspheres,
eachcharged
withsurface
charge - f .
n.Moreover,
sincef
isdivergence-free
The action
of Aox
on the(divergence-free)
vectors~Ylmrl
can becomputed
to be
In
particular
Using
invariance underpermutation,
one proves alsoTaking
into account invariance under rotationof 03C0/2 around k,
one verifiesthat
The coefficients are obtained
comparing (4.16)
with thegradient
of the
potential given by
theimage charges
One obtains
where
~n,
are theangular
coordinates of thepoint n
EZ3.
The action of on more
general divergence-free
vectorsrequires only
a
slight generalization.
Anexpression
similar to(4.16)
must becompared
with the
potential
of animage
array ofmultipoles
at the latticepoints.
Annales de l’Institut Henri Poincaré - Physique theorique