C. R. Acad. Sci. Pa is, Se . I 342 (2006) 353–358
h p:// ance.else ie .com/di ec /CRASS1/
Nume ical Analysis/Calculus o Va ia ions
A densi y esul o he a ia ion o a ma e ial
wi h espec o small inclusions
Juan Casado-Díaz, Julio Couce-Cal o, José Domingo Ma ín-Gómez
Depa amen o de Ecuaciones Di e enciales y Análisis Numé ico, Uni e sidad de Se illa, c/Ta ia s/n, 41012 Se illa, Spain
Recei ed 18 Ma ch 2005; accep ed a e e ision 7 Decembe 2005
A ailable online 18 Janua y 2006
P esen ed by Oli ie Pi onneau
Abs ac
We conside he amily o ma e ials ob ained, ia homogeniza ion, by eplacing a small po ion, o size ε, o a ixed ma e ial by
o he ma e ials. In a p e ious pape we ha e ob ained a subse o he se o ‘de i a i es’ o his amily wi h espec o εin ε=0.
In he p esen No e we p o e ha his se is, in ac , dense. This esul can be applied, o example, o ob ain op imali y condi ions
o composi e ma e ials. To ci e his a icle: J. Casado-Díaz e al., C. R. Acad. Sci. Pa is, Se . I 342 (2006).
2005 Académie des sciences. Published by Else ie SAS. All igh s ese ed.
Résumé
Un ésul a su la a ia ion d’un ma é iau en onc ion de pe i es inclusions. On considè e une amille de ma é iaux ob enus
pa homogénéisa ion consis an à emplace une pe i e pa ie de ma é iau, de aille ε, pa d’au es ma é iaux. Dans un a icle
an é ieu on a ca ac é isé un sous-ensemble de l’ensemble des «dé i ées», pa appo à εde ce e amille, pou ε=0. Dans ce e
No e on démon e que ce sous-ensemble es en ai dense. Le ésul a peu ê e appliqué, pa exemple, à l’ob en ion des condi ions
d’op imali é pou des ma é iaux composi es. Pou ci e ce a icle:J. Casado-Díaz e al., C. R. Acad. Sci. Pa is, Se . I 342 (2006).
2005 Académie des sciences. Published by Else ie SAS. All igh s ese ed.
Ve sion ançaise ab égée
Il es bien connu que les ca ac é is iques d’un ma é iau (pa exemple en conduc i i é élec ique ou he mique) son
no malemen associées à une ma ice A(ma ice de di usion). Quand on considè e des ma é iaux composi es, il es
in é essan de dé e mine les a ia ions de ce e ma ice quand on change des pe i es pa ies du ma é iau pa d’au es
ma é iaux di é en s. Ce p oblème se pose, pa exemple, dans l’é ude des condi ions d’op imali é pou A. Dans ce
cas on considè e habi uellemen des pe i es a ia ions de la ma ice A( oi [1–3,8–10]). Un au e exemple es la
cons uc ion des di ec ions de descen e qui pe me en d’amélio e Aen in oduisan d’au es ma é iaux. Puisque
le mélange des ma é iaux es bien ca ac é isé pa la héo ie de l’homogénéisa ion ( oi pa exemple [1,3,5,7,8,10])
e puisque la H-con e gence es un p océdé local, la ques ion à pose es la sui an e : Pou A,B1,...,B
nma ices
dé inies posi i es, θ1,...,θ
n⩾0, a ec n
i=1θi=1, il s’agi d’ob eni l’ensemble D(A;θ1,...,θ
n;B1,...,B
n)de
ou es les ma ices Dqui son une limi e de la o me
E-mail add esses: [email p o ec ed] (J. Casado-Díaz), [email p o ec ed] (J. Couce-Cal o), [email p o ec ed] (J.D. Ma ín-Gómez).
1631-073X/$ – see on ma e 2005 Académie des sciences. Published by Else ie SAS. All igh s ese ed.
doi:10.1016/j.c ma.2005.12.021
354 J. Casado-Díaz e al. / C. R. Acad. Sci. Pa is, Se . I 342 (2006) 353–358
D=lim
ε→0
Aε−A
ε,(1)
où, pou chaque ε∈(0,1),Aεes ob enue pa homogénéisa ion, en mélangean les ma é iaux co espondan s aux
ma ices A,B1,...,B
na ec des p opo ions espec i es égales à 1 −ε,εθ1,...,εθ
n.
Dans un a icle an é ieu , [2], on a mon é (pou n=1, mais le cas géné al es analogue) qu’un sous-ensemble de
D(A;θ1,...,θ
n;B1,...,B
n)es donné pa l’ensemble H(A;θ1,...,θ
n;B1,...,B
n)de ou es les ma ices H, elles
qu’il exis e ω1,...,ω
n⊂RN, mesu ables, disjoin s, a ec |ωi|=θi,i=1,...,n, e el que
Hξ =
n

i=1
(Bi−A)θiξ+
ωi
∇wξdz,∀ξ∈RN,
où wξes la solu ion (unique à une cons an e addi i e p ès) de
−di AχRN n
i=1ωi+
n

i=1
Biχωi∇wξ=
n

i=1
di (Bi−A)χωiξin DRN,∇wξ∈L2RNN.
L’ensemble H(A;θ1,...,θ
n;B1,...,B
n)a é é u ilisé dans [2] pou ob eni des condi ions d’op imali é su A.De
plus, on a aussi ob enu quelques élémen s explici es de H(A;θ1,...,θ
n;B1,...,B
n).
L’objec i de ce e No e es de mon e le ésul a sui an :
Théo ème 0.1. L’ensemble H(A;θ1,...,θ
n;B1,...,B
n)es dense dans D(A;θ1,...,θ
n;B1,...,B
n).
La démons a ion u ilise un ésul a bien connu dû à Dal Maso e Kohn [1,4,5] é ablissan que les ma ices ob-
enues pa homogénéisa ion pé iodique son denses dans l’ensemble ob enu pa homogénéisa ion quelconque. Ceci
donne une app oxima ion du quo ien qui appa aî dans (1), mais on ema que que ce e app oxima ion n’es pas dans
D(A;θ1,...,θ
n;B1,...,B
n)e donc, pa exemple, elle ne peu pas ê e u ilisée comme di ec ion admissible pou un
p oblème d’op imali é.
1. In oduc ion and main esul
I is well known ha he cha ac e is ics o a ma e ial ( o example, in hea o elec ic conduc i i y p oblems)
a e associa ed o a ma ix A( he di usion ma ix). When we deal wi h composi e ma e ials, i is in e es ing o
know he a ia ion o his ma ix when we eplace a small po ion o he ma e ial by o he ma e ials. This ques-
ion a ises, o example, in he s udy o op imali y condi ions o A, whe e o cons uc admissible di ec ions, we
conside small pe u ba ions o A(see e.g. [1–3,8–10]). We also men ion he cons uc ion o descen di ec ions which
pe mi o imp o e a ma e ial Aby in oducing o he ma e ials. Since he mix u e o ma e ials can be cha ac e ized
ia homogeniza ion (see e.g. [1,3,5,7,8,10]) and his is a local p ocess, he ma hema ical ques ion is he ollowing:
Gi en A,B1,...,B
nde ini e posi i e ma ices in RN,θ1,...,θ
n⩾0, wi h n
i=1θi=1, ou p oblem is o ob ain he
se D(A;θ1,...,θ
n;B1,...,B
n)o all he possible ma ices Dwhich can be ob ained as a limi o he o m
D=lim
ε→0
Aε−A
ε,(2)
whe e, o e e y ε∈(0,1),Aεis ob ained ia homogeniza ion, by mixing he ma e ials co esponding o he ma ices
A,B1,...,B
nwi h espec i e p opo ions 1 −ε,εθ1,...,εθ
n.
In a p e ious a icle, [2], we ha e p o ed ( o n=1, bu he gene al case is analogous) ha a subse o
D(A;θ1,...,θ
n;B1,...,B
n)is gi en by he se H(A;θ1,...,θ
n;B1,...,B
n)o all he ma ices H, such ha he e
exis ω1,...,ω
n⊂RNmeasu able, disjoin , wi h |ωi|=θi,i=1,...,n, which sa is y:
Hξ =
n

i=1
(Bi−A)θiξ+
ωi
∇wξdz,∀ξ∈RN,
whe e wξis he solu ion (i is unique up o an addi i e cons an ) o
−di AχRN n
i=1ωi+
n

i=1
Biχωi∇wξ=
n

i=1
di (Bi−A)χωiξin DRN,∇wξ∈L2RNN.(3)
J. Casado-Díaz e al. / C. R. Acad. Sci. Pa is, Se . I 342 (2006) 353–358 355
The se H(A;θ1,...,θ
n;B1,...,B
n)has been used in [2] o ob ain op imali y condi ions o A. Some explici ele-
men s o H(A;θ1,...,θ
n;B1,...,B
n)ha e also been ob ained in he men ioned pape .
We obse e ha , using a dila a ion, H(A;θ1,...,θ
n;B1,...,B
n)can also be de ined as he se o ma ices H, such
ha he e exis λ>0, and ω1,...,ω
n⊂RNmeasu able, disjoin , wi h |ωi|=λθi,i=1,...,n, which sa is y:
Hξ =
n

i=1
(Bi−A)θiξ+1
λ
ωi
∇wξdz,(4)
whe e wξis he solu ion o (3).
The goal o he p esen No e is o p o e he ollowing esul :
Theo em 1.1. The se H(A;θ1,...,θ
n;B1,...,B
n)is dense in D(A;θ1,...,θ
n;B1,...,B
n).
The p oo uses a well known esul o Dal Maso and Kohn [1,4,5], which p o es ha he ma ices ob ained
ia pe iodic homogeniza ion a e dense in hose which can be ob ained by gene al homogeniza ion. This p o-
ides an app oxima ion o he quo ien which appea s in (2), bu we ema k ha his app oxima ion is no in
D(A;θ1,...,θ
n;B1,...,B
n)and hus, o example, i canno be used as an admissible di ec ion o op imal p ob-
lems.
2. P oo o Theo em 1.1
In he whole o he p oo , we deno e by Ca gene ic cons an , which can change om a line o ano he one. The
p oo is di ided in h ee s eps.
S ep1. Le Dbein D(A;θ1,...,θ
n;B1,...,B
n), and conside a sequence Aεo ma icesob ained ia homogeniza-
ion, by mixing he ma e ials co esponding o A,B1,...,B
nwi h espec i e p opo ions 1−ε,εθ1,...,εθ
n, and such
ha (2) is sa is ied. Using he esul o Dal Maso and Kohn men ionedabo e, we can assume ha o e e y ε>0, he e
exis ωε
i⊂Y=(−1
2,1
2)N, measu able, wi h |ωε
i|=εθi,i=1,...,n, such ha de ining ωε
i=k∈ZN(ωε
i+k) ( he ex-
ension by Y-pe iodici y o ωε
i⊂Y o RN)Mε=AχRN n
i=1ωε
i+n
i=1Biχωε
i,
Mε=AχRN n
i=1ωε
i+n
i=1Biχωε
i
( ema k ha Mε=
Mεin Y), we ha e
Aεξ=Y
Mε∇ ε
ξ+ξdy, ∀ξ∈RN,(5)
whe e ε
ξis he solu ion o (i is unique up o a cons an ):
−di 
Mε∇ ε
ξ=
n

i=1
di (Bi−A)χωε
iξin DRN,
ε
ξ∈H1
(Y ). (6)
He e H1
(Y ) deno es he space o unc ions o H1
loc(RN), which a e Y-pe iodic. A simple calcula ion shows
Dεξ=Aε−A
εξ=
n

i=1
(Bi−A)θiξ+1
ε
ωε
i
∇ ε
ξdy.(7)
Now, we de ine wε
ξas he solu ion o
−di Mε∇wε
ξ=
n

i=1
di (Bi−A)χωε
iξin DRN,∇wε
ξ∈L2RNN.(8)
Then, om (4), he ma ix Hε, de ined by:
Hεξ=
n

i=1
(Bi−A)θiξ+1
ε
ωε
i
∇wε
ξdy,(9)
356 J. Casado-Díaz e al. / C. R. Acad. Sci. Pa is, Se . I 342 (2006) 353–358
belongs o H(A;θ1,...,θ
n;B1,...,B
n). In o de o p o e Theo em 1.1, i is enough o show ha (H ε−Dε)ξ ends
o ze o o e e y ξ∈RN, when ε ends o ze o.
S ep 2. Fo e e y ρ>0, we deno e Yρ={y∈Y:dis (y, ∂Y ) > ρ}.Le us assume ha he e exis s δ>0 such
ha n
i=1ωε
i⊂Yδ. Taking in o accoun ha wε
ξsa is ies −di A∇wε
ξ=di gεin D(RN), wi h gε=n
i=1(Bi−A)
(∇wε
ξ+ξ)χωε
i, and ha ∇wε
ξbelongs o L2(RN)N, we deduce ha (a ep esen a i e o wε
ξis) wε
ξ=∇KA∗gε, wi h
KA he undamen al solu ion o −di A∇. Since wε
ξsa is ies ( ake wε
ξas es unc ion in (8))
∇wε
ξL2(RN)N⩽C√ε, ∇KA(z) =µL Lz
|Lz|N,
wi h µ∈R,L∈MNnon-singula , and he suppo o gεis con ained in n
i=1ωε
i, we deduce:
wε
ξ(z)⩽Cε
δN−1,∇wε
ξ(z)⩽Cε
δN,∀zwi h dis z,
n

i=1
ωε
i>δ
2.(10)
We ake ψ∈C∞
0(RN)a cu -o unc ion such ha
ψ=1inYδ/2,supp(ψ) ⊂
Y, |∇ψ|⩽C
δin RN.(11)
F om (10) and (11), we deduce:
∇wε
ξψ−wε
ξL2(Y )N⩽Cε
δN−1/2.(12)
On he o he hand, o e e y ϕ∈H1(Y ),weha e:
Y
Mε∇ψwε
ξ∇ϕdy=Y
Mε∇wε
ξ∇ψϕ−Y
ϕdzdy−Y
Mε∇wε
ξ∇ψϕ−Y
ϕdzdy
+Y
Mε∇ψ∇ϕw
ε
ξdy.
Using (8), and ψ=1inn
i=1ωε
i, he i s e m on he igh -hand side is
n

i=1
(A −Bi)ξ 
ωε
i
∇ϕdy.
The second and hi d e ms on he igh -hand side can be es ima ed by using (10), (11) and Poinca é–Wi inge ’s
inequali y. We hen ge :
Y
Mε∇ψwε
ξ∇ϕdy−
n

i=1
(A −Bi)ξ 
ωε
i
∇ϕdy⩽Cε
δN+1/2∇ϕL2(Y )N,∀ϕ∈H1(Y ),
and hen, om (6), we ge :
Y
Mε∇ψwε
ξ− ε
ξ∇ϕdy⩽Cε
δN+1/2∇ϕL2(Y )N,∀ϕ∈H1
(Y ).
Taking he e ϕ=ψwε
ξ− ε
ξ, we deduce ∇(ψwε
ξ− ε
ξ)L2(Y )N⩽Cεδ−N−1/2,which joining o (12) shows
lim
ε→0
1
εY∇wε
ξ− ε
ξ2dy=0,∀ξ∈RN.(13)
F om (7) and (9) we hen conclude ha (H ε−Dε)ξ ends o ze o, o e e y ξ∈RN.
S ep 3. Le us now conside he gene al case.
J. Casado-Díaz e al. / C. R. Acad. Sci. Pa is, Se . I 342 (2006) 353–358 357
We ix δ∈(0,1). Since in (5) and (6), ωε
ican be eplaced by ωε
i+a, o e e y a∈RN, hen, choosing aapp op i-
a ely, we can assume ha he e exis s C>0 which does no depend on εno on δ, such ha

n

i=1ωε
i∩(Y Yδ)⩽Cδε. (14)
We ake
ωε,δ
i=ωε
i∩Yδ,M
ε,δ =AχRN n
i=1ωε,δ
i+
n

i=1
Biχωε,δ
i
.
As abo e, we also deno e by ωε,δ
i, he ex ension by Y-pe iodici y o ωε,δ
i⊂Y o RNand by 
Mε,δ, he ex ension by
Y-pe iodici y o Mε,δχY o RN. We de ine wε,δ
ξ, ε,δ
ξas he solu ions o
−di Mε,δ∇wε,δ
ξ=
n

i=1
di (Bi−A)χωε,δ
i
ξin DRN,∇wε,δ
ξ∈L2RNN,(15)
−di 
Mε,δ∇ ε,δ
ξ=
n

i=1
di (Bi−A)χωε,δ
i
ξin DRN,
ε,δ
ξ∈H1
(Y ). (16)
F om S ep 2 (see (13)), we know
lim
ε→0
1
εY∇wε,δ
ξ− ε,δ
ξ2dy=0,∀ξ∈RN.(17)
Le us now es ima e wε
ξ−wε,δ
ξ. Taking i as es unc ion in he di e ence o (8) and (15) we ge :

RN
Mε∇wε
ξ−wε,δ
ξ∇wε
ξ−wε,δ
ξdx=
n

i=1
ωε
i ωε,δ
i
(A −Bi)∇wε
ξ+ξ∇wε
ξ−wε,δ
ξdx,
which gi es

RN∇wε
ξ−wε,δ
ξ2dx⩽C
n
i=1(ωε
i∩(Y Yδ)) 1+∇wε
ξ2dx. (18)
F om Meye s’ heo em [6] applied o (8), we know ha he e exis s p>2 such ha
∇wε
ξLp(Y )N⩽C
n

i=1
di (Bi−A)χωε,δ
i
ξW−1,p(2Y) +∇wε
ξL2(2Y)
N,
and as wε
ξsa is ies ∇wε
ξL2(RN)N⩽C√ε, we deduce ∇wε
ξLp(Y )N⩽Cp
√ε. Thus, es ima e (18) shows
∇wε
ξ−wε,δ
ξL2(RN)N⩽Cδ(p−2)/(2p)ε1/2.
A simila easoning p o es
∇ ε
ξ− ε,δ
ξL2(Y )N⩽Cδ(p−2)/(2p)ε1/2.
These es ima es and (17), hen p o e
limsup
ε→0∇ ε
ξ− ε,δ
ξL2(Y )N⩽Cδ(p−2)/(2p)ε1/2.
So, om (7) and (9), we deduce ha
limsup
ε→0Hε−Dεξ⩽Cδ(p−2)/(2p),∀δ∈(0,1)
and hen (H ε−Dε)ξ ends o ze o, o e e y ξ∈RN.

358 J. Casado-Díaz e al. / C. R. Acad. Sci. Pa is, Se . I 342 (2006) 353–358
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