A NNALI DELLA
S CUOLA N ORMALE S UPERIORE DI P ISA Classe di Scienze
P. A. M ARKOWICH F. P OUPAUD
The Maxwell equation in a periodic medium : homogenization of the energy density
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4
esérie, tome 23, n
o2 (1996), p. 301-324
<http://www.numdam.org/item?id=ASNSP_1996_4_23_2_301_0>
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Homogenization of the Energy Density
P. A. MARKOWICH - F. POUPAUD
1. - Introduction
We consider the evolution of the
electro-magnetic
fieldquantities
in aperiodic
medium with a smallperiod.
Therefore let a > 0 be a small parameter such that the latticespacing
of the medium is in0(a),
and denoteby
E’ theelectric field and
by
Ha themagnetic
field.Assuming
that the medium haszero
conductivity,
the Maxwellequations
for the fields then read:We
impose
the initial conditionThe stand for the
permittivity and, respectively,
perme-ability
of themedium; s = s(x),
JL =/1(x)
> 0 are assumed to beperiodic
ona lattice with
O(l)-spacing, uniformly
bounded away from 0 and inR).
We also assume that the initial data
E’, Hj
are inII~)3
andsatisfy
thecompatibility
conditionThe
homogenization
limit a -~ 0 of the fieldquantities E",
Ha is well-known(see,
e.g.[BLP]).
Inparticular
there exist functions E and H such that(maybe
after selection of a
subsequence):
Supported by the "Human Capital and Mobility" - Project ERBCHRXCT 930413 entitled "Non-
linear Spatio-Temporal Structures in Semiconductors, Fluids and Oscillator Ensembles", funded
by E.C.
Pervenuto alla Redazione il 15 dicembre 1994 e in forma definitiva il 19 settembre 1995.
E and H are weak solutions of a
homogenized
version of the Maxwellequations
with initial data
EI , HI
such that -For the
precise
form of thehomogenized permittivity
andpermeability
functionswe refer to
[BLP].
An
important quantity
is theelectro-magnetic
energydensity:
whose
(i.e.
the totalelectro-magnetic energy)
ispreserved by
theMaxwell flow:
The
goal
of this paper is toanalyze
thehomogenization
limit a - 0 of theenergy
density
n".Obviously
thetopology
in which the limit(1.4)
of the fieldquantities
takesplace
is not strongenough
to carry out the limit of n"directly.
Thus,
an alternativeapproach
has to be taken.We remark that
compensated
compactness methods can be used to pass to the limit a - 0 in certain nonlinearexpression,
e.g. the limit oft) ~2 - t) ~2
can be carried outdirectly [BLP]).
However,
these methods are notapplicable
to the energydensity
n".In this paper we
proceed
inanalogy
to thehomogenization
limit for theSchrodinger equation
in acrystal presented
in[MMP].
We constructsubspaces
invariant under the Maxwell
operator by
the well-known Bloch’decomposition [RS],
set up the so calledband-Wigner
transforms of theprojections
of the fieldsonto these
subspaces
as introduced in[MMP]
and pass to the limit a - 0 in the evolutionequation
for theWigner-functions obtaining
a denumerable set of kineticequations. Finally,
we showusing
anargument
of[G]
that thehomogenization
limit of the energydensity
is obtained as sum over all bands of theposition
densities of the limits of theband-Wigner
functions. In this waywe
exploit
a somewhat hidden kinetic structure of the Maxwellequations.
We remark that the
homogenization
limits of theband-Wigner
functionsare the so-called semi-classical measures introduced in
[G]. Also,
there are obviousanalogies
to the construction of thefull-space Wigner
transform andtheir
limiting Wigner
measuresgiven
in[LP].
Another
approach
based on H-measures has beenproposed
in[FM].
Inthis work the authors
give
the measure limit of the energydensity
for the waveequation
when the coefficients do notdepend
on the small parameter.Our main result is the
following.
We constructinfinitely
manynon-negative
measures
wl,l (x, k), x
E ER ,1
EZ,
each of themcorresponding
to aneigenvalue wi(k)
of anelliptic problem
indexedby
k. For initial dataoscillating
at the scale a we obtain that:
The bounded domain B is the Brillouin zone defined in the next section.
2. - Bloch
decomposition
of the Maxwellequations
in aperiodic
mediumLet a(l), a(2), a(3) be a basis in
II~3.
Then we define the latticeand the dual lattice
where the dual basis vector
a , a , a
are determinedby
theequations
The basic
period
cell of the lattice L is denotedby
and the Brillouin zone B is the
Wigner-Seitz
cell of the dual lattice:Note
that
_(2~ ) 3
holds(I. I
denotes thevolume).
For the
following let s = sex), f1
= be the(real-valued)
dielectricand, respectively, permeability
functions onJae3,
with theproperties:
We set
s(xla), f1(xla).
For a
domain S2 c JR3
and a function a ER)
we considerthe Hilbert space
H(Q, a, div 0) :=
=0 {
which weequip
with theL 2-scalar product
onJR3
withweight
a. The local versionHloc (Q,
a, div0)
is defined in the obvious way.Now,
let a E(0, ao)
for some fixed ao > 0. We introduce the Maxwell.
operator
with domain
where
Note that L" is obtained from
L
1by
therescaling
of theposition
variablex -
xla.
The Maxwell
equations ( 1.1 ), (1.3)
then can be written asAssuming
that the(real-valued)
initial datasatisfy
there exists a
unique
solutionsince L" maps into
As usual we start the
Bloch-decomposition
with the introduction of spaces ofquasi-periodic
functions onR3x
B. We define:with the norms
Also,
foraL-periodic
functions a E L 00( JR3; R)
we define the spacewhich we
equip
with the scalarproduct
where " " denotes
complex conjugation.
The
following proposition
isstraightforward:
PROPOSITION 2.1. The space
L#,",
div0)
are isometric toL 2 (IE~3 ) , H(IR3, curl) and, respectively, H (1I~3 ,
a, div0).
Theisometry
I" isgiven by:
The
proof
followsdirectly
from Parseval’sidentity.
Thus,
theproblem
offinding ( E" , H" )
EC(Rt; (L2(R’))’ )2,
whichsolves
(2.8)
under theassumption (2.9),
isequivalent
tosolving:
in where we set
The
assumption (2.9)
then readsand
Ecx,
H" are recovered from:Obviously,
we havefor
e, h
EFor fixed k E B we introduce a space of
k-quasi-periodic
functions ofWe denote
by I’ (k)
theoperator (2.17)
with domainNote that
l" (k)
is adensely
defined unboundedoperator
on the Hilbert spacewhich we
equip
with the scalarproduct
i.e.
An
application
of Green’s-theorem for thecurl-operator
on theLipschitz-domain
aC
using
thek-quasi-periodicity
shows that1’(k)
isself-adjoint
onHU,a(k).
We now consider the
eigenvalue problem
for Since andhave the same
eigenvalues,
it suffices toanalyze
for
w (k)
E R and(e, h)
E Theeigenfunctions
ofl" (k)
are obtainedfrom the
eigenfunctions
ofby applying
therescaling
x -x /a (and
viceversa).
The
eigenvalue problem (2.20)
reads:At first we prove:
LEMMA 2.1. For k E B assume that
w (k)
= 0 holds. Then k = 0.PROOF. We obtain from
(2.21)(a),(b).
Because of(2.21)(c)
we can writewhere Ul, U2 are
L-periodic,
i.e.From
(2.22)
we concludeand Ej (a)
follows for all cr E L * with scalar functions aj if, We obtain
for k #
0. Note that no condition onuj (01’
is obtained if k =0,
whichimplies
- ,. , , - ,-,
Multiplying
theequation
div = div = 0by
q;l and ~02, respec-tively,
andintegration
over Cimplies
~01 >w2 n0 for k E B, 1:1 Now let k E 0. Then since(o (k) =,4
0 we can eliminate husing
and the
eigenvalue problem
to be solved readssubject
to thek-quasi-periodicity
conditionWe now consider the unbounded operator
on
equipped
with thescalar-product
A (k)
is defined on its form domaini.e.
LEMMA 2. 2. Let k E B. Then
A (k)
isself-adjoint
and bounded below(by 0)
on
Hi,V (k).
Its resolvant ( is Hilbert-Schmidtuniformly
in k E Bfor
every ~, > 0.PROOF. The
self-adjointness
as boundedness from below(with
bound0)
follow
immediately
from anapplication
of Green’s theorem for thecurl-operator.
Consider now the resolvant
equation (A(k) ~- ~,)e
=f
forf
E andÀ > 0 written in weak form
’
for w E
D(A(k)).
The
Lax-Milgram
lemmaimmediately implies
the existence of aunique
solution with
Since
Hloc (JR3, curl)
n iscompactly
embedded inHJ,V (k)
for allwe conclude that
(A(k)
+X)-1
is compact. For k E B denoteby
the sequence of
eigenvalues
ofA (k),
here listedaccording
to their finite multi-plicities.
The min-maxprinciple
foreigenvalues [RS] implies 8m(h)
>:Ym (k),
whereym (k)
are theeigenvalues
of the operatorThus,
we have toanalyze
theeigenvalue problem
We set z =
..j8u
and obtain theeigenvalue problem
where
C (k)
is theoperator
on
Hloc (IR3, ~, div 0)
nL a, (k) equipped
with the usualproduct.
A
simple
calculationusing
the condition div(#z)
= 0 shows thatfor ~ = ~(~)
sufficiently large
.holds for all z in the form-domain of
C (k), Thus, again, by using
the min-max
principle
we conclude thatym (k) > ( 1 /~)~Bm (k) - ~,
wheref3m (k)
are theeigenvalues
of - 0 onL~,,(k).
Fourieranalysis gives (after re-indexing)
with three-dimensional
eigenspaces.
Therefore we have
which is
uniformly
bounded in k E B . DApplying
Lemma 4.1 of[G]
we conclude thatthe
functions8m -
have
uniformly Lipschitz
continuousL*-periodic
from B to for every M E N.Also,
the methods of[W] developed
for theanalysis
of the Hamilton operator with aperiodic
electricpotential,
can beadapted
to theanalysis
ofA(k). They
show that for every M E N there exists a closed set
Fm
C B ofLebesgue
measurezero such that the
L*-periodic
extension of8m
isanalytic R’ k -
Moreover in
R) x B
and such that for all k EB - they
form acomplete
orthonormed system in the spaceH (C,
8, div0) (equipped
with thescalar
product ( ~ , ~ ) ~, £ ) .
Since the
positive
andnegative square-roots
of theeigenvalues 8m (k) #
0of
A (k)
areeigenvalues
of11 (k) (and
viceversa)
we conclude that 0 theoperator l (k)
has a sequence ofeigenvalues
listed
according
to their(finite) multiplicities.
Theregularity properties
ofare as follows:
The
L*-periodic
extension of úJm ==cvm (k)
areuniformly Lipschitz-continu-
ous on
R3
if8m (0) #
0and, respectively,
on every closed subset of L * if8m(0)
= 0. It isanalytic
inIRk - 8m (0) #
0and, respectively,
in L * - +
a)
if8m (0)
= 0.Obviously,
is inCO,I/2(B)
even if8m(0)
= 0.The
eigenfunction
ofll (k) corresponding
to theeigenvalue
0 isgiven by
It is an easy exercise to show that
f hm (~, k) 1,,,N
is acomplete
orthonormed set in the spaceH (C,
/1, div0) equipped
with the scalarproduct (., .)c,JL
ifThe
measurability
ofhm
is a directconsequence of the
continuity
of cvm and of themeasurability
of em.These facts
imply
that{(em (~, k), ±h,,, (-, k))
is acomplete
orthonormedsystem
inH (C,
6~, div0)
xH (C,
it div0) equipped
with the scalarproduct (., .)c
(see (2.19)(b))
for k E B - F -(0).
The
null-space
of can beeasily computed.
From Lemma 2.1 weconclude
(e, h)
E Null(11 (0))
if andonly
ifwhere aI, a2 E
C3
arearbitrary
and u 1, u2 areL-periodic
solutions ofThus dim
( Null (11 (0)))
= 6.The
eigenfunctions
ofl" (k)
are obtainedby
therescaling x - x j a
andby
normalization with respect to the scalarproduct (-, )exc. They
aregiven by
The
following decomposition
theorem is a direct consequence of thespectral analysis
of the operatorla (k)
and ofProposition
2.1(see,
e.g.[RS]
for aproof):
THEOREM 2.1. For i ~ set
Then the
following
statements hold:(i)
The mapsare isometries:
with inverses
Let and denote
We now define the
(Floquet) subspaces
offor M E N. A
simple
calculationusing
Theorem 2.1 show thatSm , S’.
areinvariant under the action of
L",
thatSm and ’
areorthogonal
with respectto the scalar
product (., -)R3 (given by (2.19)(b)
with aCreplaced by II~3)
forand that
Also the
following
result is obtainedby
astraight-forward
calculation:LEMMA 2.3. Let mEN and denote
by wm (y)
theFourier-coefficient of wm (k) :
. Then:
More
explicit
calculation can be carried out in the case of ahomogeneous
medium:
LEMMA 2.4. Assume that s
and It
arepositive
constants, Then theeigenvalues of
theproblem (2.21 )
are:where c :
(e~c,c) 1~2
denotes thelight velocity.
Themultiplicity of each eigenvalue
is 2a.e. in B.
PROOF. For 0 the
problem (2.21)
reduced toExpanding e
in the Fourier seriesleads to
Moreover with
y’
if andonly
ifwhich is the
equation
of aplane
inR3
and therefore a closed set ofLebesgue
measure 0. D
This shows
that,
ingeneral, eigenvalues
are notsimple.
Since the sub- sequentanalysis
we need anon-degeneracy hypothesis (which
has topermit multiple eigenvalues),
we assume:(A2) For
all theeigenvalue wm(k)
has a constantmultiplicity z(m)
a.e. inB.
To
simplify
the notation we set Z*:= Z 2013 {0}
and(D-m(k) := -wm(k)
form E N
Now let be a sequence of
integers
in Z* with theproperty
thatm-i = -ml and
We define:
Since the Fourier-coefficients of the
eigenvalues
ml, ... , m =ml+1 - 1 are the same we obtain the
following
extension of lemma 2.3:LEMMA 2.5. Let Then
Obviously,
we set forThus, given
an initial datumwe compute its
projection
..
and re-write the .Maxwell
equation
as:
for all I E Z*. The solution
(E",
is reconstructed from3. -
Wigner-Functions
We now define the 1-th band
Wigner-function:
for x E E B, t E R and I E Z*. For the basic
properties
ofw’
we referto
[MMP].
Inparticular
we remark thatwhich we shall call the l-th band energy
density.
A
simple
calculationusing
theL/2-periodicity
of 8 and itgives:
LEMMA 3. l. The
function w’ satisfies
the initial valueproblem
Also,
we set up theWigner
function:Note that the energy
density
satisfiesLater on we shall relate the limits of to the limits of The initial
Wigner
function is defined in the obvious way:We denote the
weighted
norm onL2(R 3)2 by
and
impose
thefollowing
conditions on the initial datawhere K is
independent
ofThe first term on the left-hand side of
(A3) (i)
is the initial energy wheren 1
stands for the initial energydensity
Obviously,
the energy and are conservedby
the motiongenerated by
the Maxwellequations (2.8):
Now take a function with for and
Multiplication
of( 1.1 ) by .
of(1.2) by
summation and
integration by
partsgives
Thus, (A3)(i), (ii) imply
We conclude
that,
if theassumption (A3)
isimposed
on the initial dataE’, Hj’,
then it holds true for the solution
( E" (t ) ,
for all t E R.We
point
out that theassumption (A3)
isequivalent
to the condition ofa-oscillating
and compact atinfinity
data of[G].
A
simple
calculationusing
Theorem 2.1gives
where
E", H"
are definedaccording
to(2.31).
Thus,
we have-
(from
now on we denoteby K
notnecessarily equal
constantindependent
ofThe
assumption (A3)
is sufficient to guarantee the existence of a subse- quence of a - 0(which, by
abuse ofnotation,
we denoteby
the samesymbol)
and of
non-negative
measures wj,l, wj,wi(t), w(t)
such that for all l E Z*Here we denote the
separable
Banach spaceThe arguments used to show
(3.11)
aregiven
in[MMP].
Inparticular,
weremark that the
non-negativity
of w, and w is a consequence of the Husimi-regularization [MM, LP, MMP].
Multiplying (1.1) by (1.2) by
where ES(R3), adding
theequations
andintegrating by
partsgives
We recall that every bounded set of is
precompact,
therefore we obtain(up
to asubsequence)
the uniform convergence in t:Similarly
we obtainby using (3.2):
It was also shown in
[MMP]
that thelimiting Wigner-measures
w,satisfy
the
transport equations
(the subscript "per"
refers toL*-periodicity
ink).
We remark that the
point k
= 0 and the closed setFlmll
I of measure 0 areexempt because the necessary
C)j-regularity
of úJml forpassing
to the limita - 0 in
(3.2)(a)
cannot beguaranteed
there. Thisproblem
will be remediedlater on.
We prove:
PROOF. Let a =
a (t)
be in T’ =t)
and (D’ =t)
beuniformly
bounded sequence inL°°(Rt; (L~(R~))~). Following
an idea ofGerard
[G]
we introduce the(x, t)-Wigner
transformWe set also
such that
holds. We have
(see [MMP], [LP]), maybe
after extraction of asubsequence:
where the
separable Banach-space
C is definedby
The
corresponding
Husimi-functions readwith
The x, t and r convolutions are defined in the usual way and
for
L*-periodic
functionsThe Husimi-functions are
non-negative
and converge
(after
selection of asubsequence)
toWo
andwo respectively [MMP], [LP].
Now let
E1, Ht satisfy (2.36)(a)-(c).
Then a somewhat tedious calculationgives
where
Wï[Ef],
stand for the firstand, respectively,
second term onthe
right-hand
side of(3.15)
when wa isreplaced by Ei
and O"by
Wecan
immediately
pass to the limit a - 0 in the aboveequations
obtainwhere we denote
by
theweak
limitsof subsequences
ofand, respectively,
. Since 1 we havewhere we write
Wl
for the weak limit(of
asubsequence)
ofIt is easy to check that the
assumption (A2)
is sufficient to pass to the limit in(3.17):
p - _
I
and conclude that
Thus,
theassumption (A3) implies
that the measuresWl, Wj
aremutually
sin-gular
if I E Z*.now let
Ei , Ht
begiven by (2.32)(d). Then,
due to thequadratic
natureof the Husimi transform we
obtain(see [LP])
where
Here are the Husimi transforms of and
Taking
andL * -periodic
in kgives
The methods
[LP]
show thatsince are
mutually singular.
Also we havewhere W
is the weak limit of the(x, t)-Wigner
transformsH" ] .
We estimateusing (3.23), (3.17)
and(3.10)
The same bounds hold for the other terms on the
right-hand
side of(3.24).
From the
proof
of Lemma 2.2 we obtainwhere I denotes the finite set in Z* such that
81(0)
= 0 for I E I.The dominated convergence theorem
applied
to(3.24)
and(3.17) gives
Note that
(3.14) implies
for the energydensity
To characterize the
limiting
energydensity n completely
we have to determinethe
Wigner
measures wl in the setsFill
n{0}.
Therefore we make anotherassumption
on the initial data:(A4) R3
x(F
U{OJ)
is a null-set of thelimiting
initialWigner
measure wj.We prove:
LEMMA 3. 3. Let
(A 1 )-(A4)
hold and I E Z*. Then wl is thesolution
of (3.13) given by
PROOF. The
energy-conservation properties
holdwhere . Since
(after selecting
asubsequence)
we conclude from
(3.9)
and from(3.25):
Now we denote
by
z, =k, t)
the solution ofin i.e.
Let
Q8
be open in B, ofLebesgue
measure less orequal
8 andThen
Passing
to the limit 8 -~ 0gives
since, by (3.14)
and(A4),
x(F
U{OJ)
is a null-set ofand, by (3.27),
_
Also, we
have onand,
thuson x B.
Then, (3.26)(b)
and(3.28) give
and the assertion of the lemma follows. F-1
THEOREM 3.1. Let the
assumption (A 1 )-(A4)
hold.Then, maybe after
selectionof a subsequence,
the energy densities nasatisfy:
where the
band-Wigner
measures 1 aregiven by
Here wl,l is the B* - w* limit
(of a subsequence) of the
initial bandWigner function w’,, given
in(3.2)(c).
In the case of
homogeneous media,
the above theorem and Lemma 2.4imply:
.°
COROLLARY 3.1. Let E and /1 be
positive
constants and c =1/ fi
thelight ve-
locity. Then for
any initial data sequencesatisfying (A3), (A4),
the energy densitiesna
verify
where the band
Wigner
measures 1 I aregiven by
The measure
)
are the B* - w * limitof the
initial bandWigner
functions given
in(3.2)(c)
withbeing the projection corresponding
to theeigenvalue c y +k | (respectively,
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[BLP] A. BENSOUSSAN - J.-L. LIONS - G. PAPANICOLAOU, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam-New York-Oxford, 1978.
[FM] G.A. FRANCFORT - F. MURAT, Oscillations and energy densities in the wave equation,
Comm. Partial Differential Equations 17 (1992), 1785-1865.
[G] P. GERARD, Mesures Semi-Classiques et Ondes de Bloch, Sem. Ecole Polytechnique 16 (1991), 1-19.
[LP] P.L. LIONS - T. PAUL, Sur le Measures de Wigner, Rev. Mat. Iberoamericana 9 (1993), 553-618.
[MM] P.A.MARKOWICH - N.J. MAUSER, The classical limit of a self-consistent quantum- Vlasov equation in 3-D, Math. Methods Appl. Sci. 3 (1993), 109-124.
[MMP] P.A. MARKOWICH - N.J. MAUSER - F. POUPAUD, A Wigner function approach to
semiclassical limits: electrons in a periodic potential, J. Math. Phys. 35 (1994), 1066- 1094.
[RS] M. REED - B. SIMON, Methods of Modem Mathematical Physics IV, Academic Press, 4th ed., New York-San Francisco-London, 1987.
[W] C.H. WILCO, Theory of Bloch waves, J. Analyse Math. 33 (1978), 146-167.
Fachbereich Mathematik Technische Universitat Berlin
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