NUMERICAL APPROXIMATION OF A ONE-DIMENSIONAL
ELLIPTIC OPTIMAL DESIGN PROBLEM*
J. CASADO-DÍAZ†, C. CASTRO‡, M. LUNA-LAYNEZ†,AND E. ZUAZUA§
Abs ac . We add ess he nume ical app oxima ion by ini e-elemen me hods o an op imal design
p oblem o a wo phase ma e ial in one space dimension. This p oblem, in he con inuous se ing, due o high
equency oscilla ions, o en does no ha e a classical solu ion, and a elaxed o mula ion is needed o ensu e
exis ence. On he con a y, he disc e e e sions ob ained by nume ical app oxima ion ha e a solu ion. In his
a icle we p o e he con e gence o he disc e iza ions and ob ain con e gence a es. We also show a as e
con e gence when he elaxed e sion o he con inuous p oblem is aken in o accoun when building he dis-
c e iza ion s a egy. In pa icula i is wo h emphasizing ha , e en when he o iginal p oblem has a classical
solu ion so ha elaxa ion is no necessa y, nume ical algo i hms con e ge as e when implemen ed on he
elaxed e sion.
Key wo ds. con ol in he coe icien s, composi e op imal design, elaxa ion, nume ical app oxima ion,
ini e elemen s
AMS subjec classi ica ions. 49M25, 49J20
DOI. 10.1137/10081928X
1. In oduc ion. This pape is de o ed o he ini e-elemen nume ical analysis o
a p oblem o op imal mix u e o wo ( he mal o elec ical) ma e ials in o de o mini-
mize a gi en unc ional in one space dimension.
Le Ωbe a bounded open se o RN,N≥1(al hough ou analysis is limi ed o he
case N¼1, he p oblem makes sense in any space dimension), and conside he ollow-
ing op imiza ion p oblem:
Find ω0∈Usuch ha
Jðω0Þ¼min
ω∈U
JðωÞ:
ð1:1Þ
He e ω, he con ol, is a measu able subse o Ω,JðωÞ, he cos unc ional, is o he o m
JðωÞ¼Zω
F1ðx; u; ∇uÞdxþZΩ ω
F2ðx; u; ∇uÞdx;ð1:2Þ
whe e F1;F2∶Ω×R×RN→Ra e gi en unc ions, and u, he s a e, is he solu ion o
*Recei ed by he edi o s Decembe 23, 2010; accep ed o publica ion (in e ised o m) May 26, 2011;
published elec onically Sep embe 15, 2011.
h p://www.siam.o g/jou nals/mms/9-3/81928.h ml
†Dp o. de Ecuaciones Di e enciales y Análisis Numé ico, Facul ad de Ma emá icas, Uni e sidad de Se illa,
C. Ta ía s/n, 41012 Se illa, Spain ([email p o ec ed], [email p o ec ed]). The wo k o he i s and hi d au ho s
was pa ially suppo ed by p ojec MTM2008-00306 o he MICINN (Spain) and he esea ch g oup
FQM-309 o he CICE (Andalusia).
‡Dp o. de Ma emá icas e In o má ica, ETSI caminos, canales y pue os, Uni e sidad Poli écnica de Mad id,
Ciudad Uni e si a ia, 28040 Mad id, Spain ([email p o ec ed]). The wo k o he second au ho was
pa ially suppo ed by g an MTM2008-03541 o he MICINN (Spain).
§Basque Cen e o Applied Ma hema ics, Bizkaia Technology Pa k, Building 500. E-48160 De io, Basque
Coun y, Spain ([email p o ec ed]); IKERBASQUE, Basque Founda ion o Science, E-48011 Bilbao,
Basque Coun y, Spain. The wo k o he las au ho was pa ially suppo ed by ERC ad anced
g an FP7-246775 NUMERIWAVES, g an PI2010-04 o he Basque Go e nmen , ESF Resea ch
Ne wo king P og amme OPTPDE, and g an MTM2008-03541 o he MICINN (Spain).
1181
MULTISCALE MODEL.SIMUL.
Vol. 9, No. 3, pp. 1181–1216
© 2011 Socie y o Indus ial and Applied Ma hema ics
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
−di ððαχωþβð1−χωÞÞ∇uÞ¼ in Ω;
u¼0on∂Ω
ð1:3Þ
o some gi en sou ce e m ∶Ω→R. The posi i e cons an s α,β ep esen he wo
ma e ials, de e mining he coe icien s o he co esponding di usion ma ices. Some
es ic ions can and mus be imposed o he con ol ωdepending on he p oblem.
Fo example, an in e es ing case is when he ma e ial αis mo e e icien han he
ma e ial βbu i is also mo e expensi e. Then, i is usual o conside a es ic ion o
he o m jωj≤κ, limi ing he use o he ma e ial α. We include his es ic ion in
he admissible se o con ols U,
U¼ ω⊂Ω∶ωmeasu able;jωj≤κg:ð1:4Þ
The exis ence o an op imal se ω ul illing hese cons ain s, o which he unc ion
usolu ion o (1.3) minimizes J, does no hold in gene al (see [15], [16]). In hese cases, i
is na u al o look o minimizing sequences, i.e., sequences ωlg∞
l¼1⊂Usuch ha
lim
l→∞
JðωlÞ¼in
ω∈U
JðωÞ
since hey p o ide nea op imal designs. A usual p ocedu e o ind such sequences is o
in oduce a elaxed e sion o he p oblem o which a minimize exis s. Then, a sui able
app oxima ion o he minimize s p o ides minimizing sequences o he o iginal
p oblem.
Fo a sequen ially con inuous unc ional J, in he weak opology o he Sobole
space H1ðΩÞ(see [1], [13], [20]), his elaxa ion can be ob ained by eplacing in
(1.3) he unc ion χωwi h a measu able unc ion θ aking i s alues in he closed in e al
½0;1and he unc ion ðαχωþβð1−χωÞÞ wi h a ma ix unc ion Ain he se KðθÞo
ma ices cons uc ed by homogeniza ion (see, e.g., [17], [19], [21]) mixing he ma e ials α
and βwi h espec i e p opo ions θand 1−θ. Rema k ha he se KðθÞis known in he
case desc ibed abo e, co esponding o he mix u e o wo iso opic ma e ials (see [14],
[22]), bu no in o he in e es ing cases such as he mix u e o mo e han wo ma e ials,
aniso opic ma e ials, e c. Hence o h we deno e by ^
U he se o elaxed con ols ðθ;AÞ.
No e ha unc ionals o he o m (1.2) a e no sequen ially con inuous in he weak
opology o H1ðΩÞ, in gene al. In hose cases, o ob ain he elaxed e sion (see [6]) we
mus eplace he se o con ols χωand coe icien s ðαχωþβð1−χωÞÞ wi h he pai s
ðθ;AÞ∈^
Uas abo e, and he unc ional Jwi h ano he one o he o m
^
Jðθ;AÞ¼ZΩ
Hðx; u; ∇u; A∇u; θÞdx;ð1:5Þ
whe e uis solu ion o he homogenized p oblem
8
<
:
−di A∇u¼ in Ω;
u¼0on∂Ω:
ð1:6Þ
An explici exp ession o he unc ion His only known in some pa icula cases
(see he e e ences [2], [6], [7], [8], [11], [12], [18], [23]). I sa is ies
Hðx; u; ∇u; A∇u; θÞ¼F1ðx; u; ∇uÞχωþF2ðx; u; ∇uÞχΩ ω;i θ¼χω;
1182 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
and A¼ðαχωþβð1−χωÞÞIand, so, he elaxed unc ional is in ac an ex ension o he
o iginal one o he la ge se o elaxed con ols. The elaxed con ol p oblem eads
Find ðθ0;A
0Þ∈^
Usuch ha
^
Jðθ0;A
0Þ¼ min
ðθ;AÞ∈^
U
^
Jðθ;AÞ:
ð1:7Þ
In p ac ical applica ions, in o de o sol e nume ically he abo e con ol p oblem
(1.1), i is necessa y o in oduce a disc e iza ion o bo h he con ol se and he unc-
ional. In he p esen con ex , we ha e a leas wo app oaches o his nume ical ap-
p oxima ion issue. One based on he disc e iza ion o he o iginal p oblem and one
elying on he disc e iza ion o he elaxed e sion. Recen ly, in [8] and [9], bo h disc e-
iza ion p ocedu es ha e been shown o con e ge. In hese a icles, some pa ially
elaxed e sions ha e also been s udied in which he class o con ols unde conside a ion
is enla ged bu no o he ex en o exhaus ing he class o he elaxed e sion o he
p oblem; we e e o [12] o a ela ed esul . We also e e o [26] o he nume ical s udy
o he elaxed o mula ion o a pa icula case o p oblem (1.1).
In his pape we compa e and ge con e gence a es o he sequences o disc e e
minimize s ob ained wi h bo h app oxima ion me hods. These issues a e add essed
in he simples one-dimensional se ing, whe e he pa ial di e en ial equa ion (1.3)
is educed o an o dina y di e en ial equa ion, he se KðθÞis well known o be educed
o he ha monic mean o αand βwi h espec i e p opo ions θand 1−θ, and he unc-
ion His explici ly known. No e ha in his case we can w i e ^
Jðθ;AÞ¼ ^
JðθÞin (1.5),
since Ais comple ely de e mined by θ, and ^
Uis jus he se o measu able unc ions
θ∶Ω→½0;1wi h in eg al less o equal han κ.
To make ou esul s p ecise, we i s conside he disc e iza ion o he se o con ols
bu no o he s a e equa ion (1.3). In he con ex o ini e-elemen app oxima ion me h-
ods, we can conside a decomposi ion o Ωin elemen s wi h maximum size and subse s
ωcons i u ed by unions o a subse o such elemen s. I we deno e by U he se o such
subse s, he disc e e p oblem eads
Find ω
0∈U such ha
Jðω
0Þ¼min
ω∈U JðωÞ:
ð1:8Þ
The disc e e space o con ols ob ained in his way U is compac in he s ong opology
o L1ðΩÞ, and he co esponding s a e unc ions a e compac in H1ðΩÞ. The e o e, he
disc e ized p oblem has a solu ion wi hou he need o a elaxed e sion.
In his way we ob ain a sequence o disc e e minimize s ω
0g ha a e likely o
cons i u e a minimizing sequence o Jin U,as →0. We show ha his is he case,
and we gi e con e gence a es o
Jðω
0Þ−in
ω∈U
JðωÞas →0:ð1:9Þ
On he o he hand, ins ead o disc e izing he o iginal con ol p oblem, we can dis-
c e ize he elaxed e sion. A e in oducing a decomposi ion o Ωin elemen s, wi h
maximal size , we can conside he se ^
U o unc ions θ∈^
Uwhich a e cons an on
each elemen . The disc e e elaxed p oblem eads
Find ^
θ
0∈^
U such ha
^
Jð^
θ
0Þ¼min
θ∈^
U
^
JðθÞ.
ð1:10Þ
APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1183
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
As abo e, we show ha
^
Jð^
θ
0Þ−in
ω∈^
U∇
JðuÞ→0as →0;ð1:11Þ
and we gi e con e gence a es.
Once a disc e e elaxed minimize is known ^
θ
0we can cons uc a sequence
ωk; g∞
k¼1⊂Usuch ha
lim
k→∞
Jðωk; Þ¼ ^
Jð^
θ
0Þ:
This p o ides a minimizing sequence o he o iginal p oblem. As we show, he sequence
ωk; g∞
k¼1can be cons uc ed explici ly om ^
θ
0wi h almos no compu a ional cos .
Ou esul s show ha i is be e o disc e ize he elaxed p oblem, in he sense ha
we ge a as e con e gence a e, as →0, o (1.11) han he one ob ained o (1.9).
This is ue e en in he case whe e he o iginal p oblem has a solu ion, and so he e-
laxa ion is unnecessa y om a heo e ical poin o iew. Despi e his, he elaxed e sion
o he o iginal minimiza ion p oblem can always be o mula ed, and ou esul s show
ha i is indeed be e o app oxima e he op imal design p oblem nume ically in hese
cases as well.
F om a compu a ional poin o iew, besides disc e izing he se o con ols, we mus
also disc e ize he s a e equa ion (1.3) o (1.6). This equi es a second decomposi ion o
Ωcons i u ed by elemen s o maximum size h. A na u al assump ion is o conside his
new decomposi ion as a e inemen o he one used o he con ol se , o ice e sa.
In he con ex o he o iginal un elaxed con ol p oblem, deno ing by uh he
P1- ini e-elemen app oxima ion o he solu ion o (1.3), and de ining Jhas
JhðωÞ¼Zω
F1ðx; uh;∇uhÞdxþZΩ ω
F2ðx; uh;∇uhÞdx;
he ull disc e e con ol p oblem eads
Find ω ;h
0∈U such ha
Jhðω ;h
0Þ¼min
ω∈U JhðωÞ:
ð1:12Þ
Analogously, we can de ine a ull disc e iza ion o he elaxed p oblem by
conside ing
^
JhðθÞ¼ZΩ
Hðx; uh;∇uh;A∇uh;θÞdx;ð1:13Þ
whe e uhis he P1- ini e-elemen app oxima ion o (1.6). The ully disc e e elaxed
p oblem in his case is
Find ^
θ ;h
0∈^
U such ha
^
Jhð^
θ ;h
0Þ¼min
θ∈^
U
^
JhðθÞ:
ð1:14Þ
We ocus on he con e gence a es o he sequences ω ;h
0g ;h and ^
θ ;h
0g ;h ob ained
wi h he wo app oaches abo e, espec i ely. Mo e p ecisely, we compa e he sequences
1184 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
Jðω ;h
0Þ−in
ω∈U
JðωÞand ^
Jð^
θ ;h
0Þ−in
ω∈U
JðωÞ;
as ; h →0.
The ollowing esul s a e p o en:
•Disc e izing he elaxed o mula ion, we show ha , sol ing he s a e equa ion
wi h he P1- ini e-elemen me hod in a mesh o size hand aking he con ol
θ o be piecewise cons an on elemen s o a coa se mesh o size ffiffiffi
h
p, he e o
is o o de h.
This cons i u es a big id o mul iscale s a egy, implemen ed on he elaxed
e sion, in he sense ha he disc e iza ion o he PDE and ha o he con ol
a e pe o med on wo di e en g ids. The PDE is disc e ized in he ine g id o
size h, while he con ol is disc e ized in he coa se one o size ffiffiffi
h
p.
•Disc e izing he o iginal un elaxed p oblem, sol ing he s a e equa ion wi h a
P1- ini e-elemen me hod in a mesh o size h, and aking he con ol χωpiece-
wise cons an in he elemen s o such mesh, we show ha he e o is o o de
h1−εwi h εa bi a ily small i he unc ions Fiin (1.2) do no depend on he
a iable uand ε¼1∕2o he wise.
A big id s a egy consis ing in disc e izing he PDE in he coa se g id (ins ead o
he ine one) can p oduce lack o con e gence o bo h he un elaxed and elaxed p o-
blems. In pa icula , he minimize s o he disc e e p oblem will possibly gi e a non-
minimizing sequence o he con inuous con ol p oblem, as ; h →0.
We also gi e an explici example in which he unc ional is independen o u, show-
ing ou es ima es a e nea ly sha p. To be mo e p ecise, ou example shows he op imali y
o he es ima es in he case in which he elaxed e sion o he p oblem is disc e ized,
while an o de ho con e gence is ob ained when he o iginal p oblem is disc e ized, hus
showing ha ou es ima es a e nea ly op imal.
The e o e he app oach based on he disc e iza ion o he elaxed o mula ion p o-
ides a be e app oxima ion and a as e con e gence a e wi h a lowe compu a ional
cos . The compu a ional cos and he complexi y o his app oach is lowe since he
con ols a e disc e ized in a mesh o o de ffiffiffi
h
pins ead o h. Fu he mo e, he minimize s
o he co esponding disc e e op imiza ion p oblems a e easie o ind nume ically.
Indeed, hanks o he con exi y o he elaxed con ol se , g adien -like algo i hms
can be implemen ed. This is in con as o he un elaxed p oblem, whe e he con ol
se is no con ex and we canno compu e a ia ions. Ins ead, much less e icien me hods
such as Mon e Ca lo o gene ic algo i hms should be used.
On he con a y, he ad an ages o disc e izing he o iginal p oblem di ec ly a e
ha , on one hand, one does no need o know he elaxed o mula ion and, on he o he
hand, i p o ides a physical con ol (i.e., a cha ac e is ic unc ion) ins ead o a elaxed
one. Howe e , his la e d awback can be o e come by app oxima ing he elaxed con-
ols by physical ones, wi h almos no compu a ional cos .
This pape p o ides a comple e analysis o he a e o con e gence o he ini e-
elemen app oxima ion o he op imal design p oblem unde conside a ion. Whe he
his classical enginee ing p ac ice leads o con e gen algo i hms is unknown in many
o he op imal design p oblems, excep in some o he pa icula examples as i occu s
when dealing wi h he op imal shape design o he domain o Di ichle Laplacian in
wo space dimensions (see [10]). No e, howe e , ha , in he la e , he e is no esul abou
he con e gence a e.
Al hough he p esen a icle is de o ed o he s udy o he 1−dop imal design
p oblem, some ema ks abou he N-dimensional case a e gi en in he las sec ion o
he pape . As abo e hese ema ks a e de o ed o he case o di usion coe icien s ha
APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1185
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.

a e uni o mly ellip ic and bounded; he case whe e we conside ma ix di usions such
ha hei smalle and/o la ge eigen alues can app oxima e o ze o o in ini y, espec-
i ely, is mo e in ol ed. Indeed, e en he de ini ion o solu ion o he s a e equa ion is
no clea in his case, whe e in pa icula La en ie ’s phenomenon can occu ; i.e.,
smoo h unc ions canno be dense in he space o unc ions wi h bounded ene gy
(see, e.g., [25] and he e e ences he ein). In his sense, we ema k ha in o de o p o e
he con e gence o he ini e-elemen me hod, i is necessa y o ha e he densi y o he
Lipschi z unc ions in he space whe e we a e looking o he solu ion o he s a e equa-
ion. A ecip oca e o his esul has been ob ained in [5] o a calculus o a ia ions
p oblem wi hou es ic ions.
As we ha e al eady ema ked, con ol p oblem (1.1) does no ha e a solu ion in
gene al. To ha e a well-posed p oblem, such as we do in he p esen pape , an app oach
consis s o ob aining a elaxa ion o (1.1) by using homogeniza ion echniques. Howe e ,
he e exis o he app oaches, o ins ance, he il e ing echnique. Loosely speaking, he
idea o he il e ing echnique consis s o eplacing he se o con ols in (1.1) wi h a
smoo he class, de ined by mean o a con olu ion ope a o . Mo e p ecisely, in (1.3)
he cha ac e is ic unc ions χω, wi h ω∈U, a e eplaced by he smoo h unc ions
ρRθ, wi h θ∈L∞ðΩ;½0;1Þ sa is ying he olume es ic ion, whe e ρRis he ypical
molli ie unc ion ρRðxÞ¼ρðx∕RÞ∕RNwi h ρa ixed C∞nonnega i e unc ion wi h
suppo in he ball o cen e 0 and adius 1, and in eg al equals 1. Thus, we ob ain a new
p oblem ( il e ed p oblem) wi h a compac se o con ols in C∞ð¯
ΩÞ, which gua an ees
he exis ence o a solu ion when he cos unc ional Jis sequen ially lowe con inuous in
he weak opology o H1ðΩÞ. The il e ed p oblems a e hen smoo h app oxima ions o
(1.1) when R>0is small, a leas o mally. Gi en R ixed, he ini e-elemen app ox-
ima ion o he il e ed p oblem has been s udied in [4], in he amewo k o a con ol
p oblem in he coe icien s in elas ici y—namely he compliance p oblem. In [4], he
con e gence o he ini e-elemen app oxima ion, as he mesh size ends o ze o, is
p o ed bu wi hou explici a es.
Some de ini ions and no a ions:
•Fo a numbe ∈R, we deno e by ½  he in ege pa o .
•Fo a (Lebesgue) measu able subse Eo ð0;1Þ, wi h posi i e measu e, and a
unc ion win L1ð0;1Þ, we deno e he mean alue o win Eby
⨍E
wdx¼1
jEjZE
wdx:
•The se o unc ions o bounded a ia ion in ð0;1Þis deno ed by BVð0;1Þ.I ψis
in BVð0;1Þand Iis a subin e al o ½0;1, hen VIðψÞ ep esen s he o al a -
ia ion o ψin I.
•Th oughou he pape , αand βa e wo posi i e cons an s.
•Fo p∈½0;1, we deno e by MðpÞ∈R he ha monic mean o αand βwi h
p opo ions pand 1−p, espec i ely, gi en by
MðpÞ¼p
αþ1−p
β−1
¼αβ
ð1−pÞαþpβ:
No e ha Mð1Þ¼α,Mð0Þ¼β, and
α≤MðpÞ≤β∀p∈½0;1:ð1:15Þ
1186 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
Fo e e y θ∈L∞ð0;1; ½0;1Þ we de ine Mθ∈L∞ðΩÞby
MθðxÞ¼MðθðxÞÞ o a:e:x∈ð0;1Þ:
•Fo a ma ix A∈RN×N, we deno e by EigðAÞ he se o i s eigen alues.
•Le Φbe a unc ion de ined in he in e al ð0;δÞ o some δ>0. The equali y
Φ¼oðhÞ(Landau symbol) means
lim
h→0
ΦðhÞ
h¼0:
•We deno e by Ca gene ic posi i e cons an ha can change om line o line.
2. Disc e iza ion and e o es ima es.
2.1. The main esul s. In his sec ion we s a e he main esul s o he pape . They
a e e e ed o he nume ical analysis o a con ol p oblem o he 1−dellip ic s a e
equa ion in Ω¼ð0;1Þbelow, he con ol being he space-dependen coe icien
(−d
dx ðαχωþβð1−χωÞÞdu
dx¼ inð0;1Þ;
uð0Þ¼uð1Þ¼0;
ð2:1Þ
whe e αand βa e wo ixed posi i e cons an s and a gi en unc ion in (a leas )
L1ð0;1Þ.
De ining, o a ixed cons an κ>0, he se o admissible con ols as (1.4), ou aim is
o choose ω∈Usuch ha he unique solu ion uω∈H1
0ð0;1Þo p oblem (2.1) minimizes
he unc ional J∶UR
!de ined as he 1−d e sion o (1.2); i.e.,
JðωÞ¼Zω
F1x; uω;duω
dx dxþZð0;1Þ ω
F2x; uω;duω
dx dx∀ω∈U:ð2:2Þ
He e F1;F2∶ð0;1Þ×R×R→Rsa is y
Fi∈W1;∞ðð0;1Þ×ð−R; RÞ×ð−R; RÞÞ ∀i∈ 1;2g∀R>0:ð2:3Þ
As we said in he in oduc ion, αand β ep esen wo ma e ials ha we wan o mix
in o de o minimize J. The cons an κis he maximum quan i y o ma e ial α ha can
be used in he mix u e. No e ha aking κ≥1would be equi alen o no imposing any
es ic ion in he se o admissible se s ω.
Rema k 1. In (2.1), we conside homogeneous Di ichle condi ions o ix ideas, bu
ou esul s also hold o nonhomogeneous Di ichle condi ions o o he bounda y
condi ions, such as Fou ie o Neumann ones. We can also conside he unc ions Fi
sa is ying weake assump ions han (2.3), bu hen he e o es ima es we ind o
he nume ical app oxima ions de ined below a e wo se.
I is well known ha he o iginal minimiza ion p oblem (1.1) does no ha e a solu-
ion in gene al (see [15], [16]). The e o e, i is necessa y o in oduce a elaxa ion. How-
e e , as we ha e men ioned in he in oduc ion, o nume ical pu poses i is o en
con enien o wo k in he elaxed e sion o he p oblem e en when he o iginal o mu-
la ion has a minimize . The elaxed e sion hus plays a key ole in he nume ical
analysis we de elop in his a icle.
The ollowing esul p o ides a cha ac e iza ion o he elaxa ion.
APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1187
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
THEOREM 2.1. A elaxa ion o p oblem (1.1) is gi en by
Find θ0∈^
Usuch ha
^
Jðθ0Þ¼min
θ∈^
U
^
JðθÞ;
ð2:4Þ
whe e
^
U¼θ∈L∞ð0;1; ½0;1Þ∶Z1
0
θdx≤κ;ð2:5Þ
and ^
J∶^
U→Ris de ined by
^
JðθÞ¼Z1
0θF1x; uθ;Mθ
α
duθ
dx þð1−θÞF2x; uθ;Mθ
β
duθ
dx dxð2:6Þ
o e e y θ∈^
Uwi h u¼uθ he solu ion o
(−d
dx Mθdu
dx¼ in ð0;1Þ;
uð0Þ¼uð1Þ¼0:
ð2:7Þ
Rema k 2. Theo em 2.1 also holds ue o e e y ∈H−1ð0;1Þand mo e gene al
nonlinea i ies F1,F2. Indeed, i is enough o assume ha F1,F2a e wo Ca a héodo y
unc ions (measu able wi h espec o xand con inuous wi h espec o ðs; ξÞ) such ha
o e e y R>0, he unc ions φ1;R,φ2;R de ined as
φi;RðxÞ¼ sup
jsjþjξj≤RjFiðx; s; ξÞj o a:e:x∈ð0;1Þ∀i∈ 1;2g
belong o L1ð0;1Þ.
Rema k 3. Fo e e y ω⊂ð0;1Þmeasu able, we ha e
JðωÞ¼ ^
JðχωÞ:
The e o e, ^
Jis in ac an ex ension o he unc ional χω↦JðωÞde ined on he space
L∞ð0;1; 0;1gÞ o he elaxed con ol se L∞ð0;1; ½0;1Þ.
Rema k 4. Theo em 2.1 is a gene aliza ion o P oposi ion 4.1 and Theo em 4.3 in
[6], whe e he mul idimensional case is also conside ed.
In he p esen pape , we a e in e es ed mainly in he nume ical analysis o p oblem
(1.1). Fo his pu pose, hanks o Theo em 2.1, wo choices a e possible: o disc e ize
di ec ly p oblem (1.1) o o disc e ize he elaxed p oblem (2.4). Ou goal is o compa e
hese wo possibili ies.
To his aim, gi en >0, we ake a pa i ion P ¼ ykgm
k¼0o ½0;1, wi h m ∈N,
such ha
¼max
1≤k≤m ðyk−yk−1Þ:ð2:8Þ
Then, we de ine ^
U and U as he subse s o ^
Ugi en by
^
U ¼θ∈^
U∶θ¼X
m
k¼1
kχðyk−1;ykÞa:e:inð0;1Þwi h k∈½0;1;1≤k≤m ;ð2:9Þ
1188 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
U ¼ ω⊂ð0;1Þ∶χω∈^
U g:ð2:10Þ
Associa ed o hese subse s we can conside he wo disc e iza ions o he con ol p o-
blem gi en by (1.10) and (1.8).
No e ha p oblem (1.8) is a disc e iza ion o he o iginal minimiza ion p oblem
(1.1), while (1.10) is a disc e iza ion o he elaxed p oblem (2.4).
The ollowing heo ems p o ide es ima es on he di e ence be ween hese p oblems
and (2.4). Some e sions o Theo em 2.2 can also be ob ained in he N-dimensional case;
see sec ion 8.
THEOREM 2.2. Assuming ∈L1ð0;1Þ, p oblem (1.10) has a solu ion o e e y >0,
and we ha e
0≤min
θ∈^
U
^
JðθÞ−min
θ∈^
U
^
JðθÞ¼oð Þ:ð2:11Þ
Mo eo e , i ∈L∞ð0;1Þand p oblem (2.4) has a solu ion θ0in BVð0;1Þ, hen
0≤min
θ∈^
U
^
JðθÞ−min
θ∈^
U
^
JðθÞ≤C 2:ð2:12Þ
THEOREM 2.3. Assuming ∈L1ð0;1Þ, p oblem (1.8) has a solu ion o e e y >0,
and we ha e
0≤min
ω∈U JðωÞ−in
ω∈U
JðωÞ≤C 1
2:ð2:13Þ
Mo eo e , i o some in ege l≥1, we ha e ha belongs o he space Wl;1ð0;1Þand
F1ðx; s; ξÞ,F2ðx; s; ξÞa e independen o sand belong o Cl;1
locð½0;1×RÞ; hen we ha e
0≤min
ω∈U JðωÞ−in
ω∈U
JðωÞ≤C lþ1
lþ2:ð2:14Þ
2.2. Op imali y. We now gi e an example showing ha he p e ious esul s a e
nea ly op imal.
Example 1. We conside p oblem (1.1) wi h α<β, ¼1,κ¼2∕3, and Jgi en by
JðωÞ¼−αZω
duω
dx 
2
dx−βZð0;1Þ ω
duω
dx 
2
dx:ð2:15Þ
Fo e e y n∈N, we de ine Pnas he pa i ion o ½0;1gi en by
Pn¼ k10−n∶0≤k≤10ng:
We de ine
^
Un¼θ∈^
U∶θ¼X
10n
k¼1
kχððk−1Þ10−n;k10−nÞwi h k∈½0;1∀k∈ 1; :::;10ng;ð2:16Þ
Un¼ ω∈U∶χω∈^
Ung:ð2:17Þ
We will p o e in sec ion 6 he ollowing esul .
APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1189
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
Rema k 7. In Example 3 we a e disc e izing he s a e equa ion (2.1) o (2.7) using a
pa i ion o ½0;1o size hbigge han he size ¼h∕2employed in disc e izing he se o
con ols. S a emen (2.38) shows ha in his case he minimum o he disc e ized
p oblem does no end o he in imum o (1.1). Thus, his ype o disc e iza ion is
no con e gen in gene al.
3. P oo o he elaxa ion esul . This sec ion is de o ed o p o ing
Theo em 2.1, which cha ac e izes he elaxa ion o p oblem (1.1). To do i , we use
he ollowing lemma.
LEMMA 3.1. The unc ional ^
J∶^
U⊂L∞ð0;1Þ→Ris sequen ially con inuous o he
-weak opology o L∞ð0;1Þ.
P oo . Gi en a sequence θn∈^
Uwhich con e ges weakly-in L∞ð0;1Þ o a unc ion
θ∈^
U, we ha e o see ha ^
JðθnÞcon e ges o ^
JðθÞ. Fo a such sequence θn, we obse e
ha he co esponding solu ion uθno (2.7) is gi en by
uθnðxÞ¼−Zx
0
Fð Þ−cn
Mθn
d ¼−Zx
0ðFð Þ−cnÞαð1−θnð ÞÞþβθnð Þ
αβ d
wi h Fa p imi i e o in ð0;1Þand
cn¼Z1
0
d
Mθnð Þ−1Z1
0
Fð Þ
Mθnð Þd :
The e o e, i is immedia e o show ha
kuθnkW1;∞ð0;1Þ≤C; uθn→uθin C0ð½0;1Þ;M
θn
dun
dx −Mθ
duθ
dx →0inC0ð½0;1Þ
wi h uθ he unique solu ion o (2.7). Then, by (2.3) we ob ain
lim
n→∞
^
JðθnÞ
¼limn→∞Z1
0θnF1x; uθn;Mθn
α
duθn
dx þð1−θnÞF2x; uθn;Mθn
β
duθn
dx dx
¼Z1
0θF1x; uθ;Mθ
α
duθ
dx þð1−θÞF2x; uθ;Mθ
β
duθ
dx dx¼^
JðθÞ:▯
P oo o Theo em 2.1. Taking in o accoun ha he space o con ols ^
Ugi en by
(2.5) is sequen ially compac in he -weak opology o L∞ð0;1Þ, om Lemma 3.1 we
deduce ha p oblem (2.4) has a leas a solu ion. On he o he hand, by Rema k 3 i is
clea ha
in
ω∈U
JðωÞ¼ in
χω∈^
U
^
JðχωÞ≥min
θ∈^
U
^
JðθÞ:
The e o e, in o de o check ha p oblem (2.4) is a elaxa ion o (1.1), i is enough o
p o e ha o e e y θ∈^
U, he e exis s a sequence ωnin Usuch ha
1196 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.

χωn⇀
θin L∞ð0;1Þ;ð3:1Þ
JðωnÞ→^
JðθÞ:ð3:2Þ
The exis ence o his sequence ωnis well known ( o example, i is a consequence o
Lemma 5.1 below), while by he con inui y p ope y o ^
Jp o ed in s ep 1, (3.2) is
a consequence o (3.1). So, he p oo o Theo em 2.1 is comple e. ▯
4. P oo o he con e gence es ima es o he disc e ized elaxed con ol
p oblem. In his sec ion we p o e Theo em 2.2 e e ed o he con e gence o he dis-
c e iza ion o p oblem (2.4) gi en by (1.10). No e ha we a e disc e izing he con ols
bu no he s a e equa ion. We also gi e he p oo o P oposi ion 2.5, which pe mi s us o
ob ain a physical con ol om a elaxed one.
Along his sec ion, we conside a pa i ion P ¼ ykgm
k¼0, wi h m ∈N, sa is ying
(2.8). The space ^
U is de ined by (2.9).
In o de o show Theo em 2.2, we will use he ope a o Π de ined by he ollowing.
DEFINITION 4.1. We de ine he p ojec ion ope a o Π ∶L1ð0;1Þ→^
U by
Π ψ¼X
m
k¼1⨍yk
yk−1
ψdsχðyk−1;ykÞ∀ψ∈L1ð0;1Þ:ð4:1Þ
The ollowing lemma es ima es he di e ence Π θ−θwhen ends o ze o.
LEMMA 4.2. Le θbe in L∞ð0;1; ½0;1Þ. Then, o e e y φ∈W1;1ð0;1Þ, i holds ha
Z1
0ðθ−Π θÞφdx¼oð Þ;ð4:2Þ
Z1
0Zx
0ðθð Þ−Π θð ÞÞφð Þd 
dx¼oð Þ:ð4:3Þ
Mo eo e , i θis in BVð0;1Þ, and φis in W1;∞ð0;1Þ, we ha e he ollowing imp o emen
o he p e ious es ima es:
Z1
0ðθ−Π θÞφdx
≤C



dφ
dx


L∞ð0;1Þ
2;ð4:4Þ
Z1
0Zx
0ðθð Þ−Π θð ÞÞφð Þd 
dx≤CkφkW1;∞ð0;1Þ 2:ð4:5Þ
P oo . We ake φ∈W1;1ð0;1Þ; o a gi en x∈½0;1, we conside yjde ined by
yj¼sup yk∶yk≤x; 0≤k≤m g:
Then, using he inequali y

φð Þ−⨍yk
yk−1
φds
≤



dφ
d 


L1ðyk−1;ykÞ
∀ ∈½yk−1;y
k
we ha e
APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1197
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
Zx
0ðθ−Π θÞφd 
¼X
j
k¼1Zyk
yk−1θ−⨍yk
yk−1
θdsφd þZx
yjθ−⨍yk
yk−1
θdsφd
¼X
j
k¼1Zyk
yk−1ðθ−⨍yk
yk−1
θdsÞðφ−⨍yk
yk−1
φdsÞd
þZx
yjθ−⨍yk
yk−1
θdsφd
≤X
j
k¼1



dφ
dx


L1ðyk−1;ykÞkθ−Π θkL1ðyk−1;ykÞþkφkL∞ð0;1Þkθ−Π θkL1ðyj;xÞ:ð4:6Þ
In eg a ing his inequali y in ð0;1Þ,wege
Z1
0Zx
0ðθð Þ−Π θð ÞÞφð Þd 
dx
≤X
m
k¼1



dφ
dx


L1ðyk−1;ykÞkθ−Π θkL1ðyk−1;ykÞ
þkφkL∞ð0;1ÞX
m −1
j¼0Zyjþ1
yjkθ−Π θkL1ðyj;xÞdx
≤X
m
k¼1



dφ
dx


L1ðyk−1;ykÞkθ−Π θkL1ðyk−1;ykÞþkφkL∞ð0;1Þkθ−Π θkL1ð0;1Þ :ð4:7Þ
I φbelongs o W1;∞ð0;1Þand θbelongs o BVð0;1Þ, using in (4.7)




dφ
dx


L1ðyk−1;ykÞ
≤



dφ
dx


L∞ð0;1Þ
; kθ−Π θkL1ð0;1Þ≤Vð0;1ÞðθÞ ;ð4:8Þ
we deduce (4.5).
Inequali y (4.4) is a consequence o (4.6) wi h x¼1¼yjand (4.8).
In o de o show (4.2) and (4.3) we now ake a sequence φnin W1;∞ð0;1Þwhich
con e ges o φin W1;1ð0;1Þand a sequence θnin BVð0;1Þ, wi h 0≤θn≤1in ð0;1Þ,
which con e ges o θin L1ð0;1Þ. Then, we es ima e he igh -hand side o (4.7) as ollows:
X
m
k¼1



dφ
dx


L1ðyk−1;ykÞkθ−Π θkL1ðyk−1;ykÞþkφkL∞ð0;1Þkθ−Π θkL1ð0;1Þ
≤2



dðφ−φnÞ
dx 


L1ð0;1Þ
þkφkL∞ð0;1Þkθ−θn−Π ðθ−θnÞkL1ð0;1Þ
þX
m
k¼1



dφn
dx 


L1ðyk−1;ykÞkθ−Π θkL1ðyk−1;ykÞþkφkL∞ð0;1Þkθn−Π θnkL1ð0;1Þ
≤2



dðφ−φnÞ
dx 


L1ð0;1Þ
þkφkL∞ð0;1Þkθ−θn−Π ðθ−θnÞkL1ð0;1Þ
þ



dφn
dx 


L∞ð0;1Þ
Vð0;1ÞðθÞþkφkL∞ð0;1ÞVð0;1ÞðθnÞ 2:
1198 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
Di iding his inequali y by and passing o he limi i s when ends o ze o and hen
when n ends o in ini y, we deduce (4.3). The p oo o (4.2) can be ob ained easoning in a
simila way wi h (4.6). ▯
Fo θ∈L∞ð0;1; ½0;1Þ, he ollowing lemma es ima es he di e ence be ween he
solu ion o (2.7) and he solu ion o he analogous p oblem when θis eplaced by Π θ.
LEMMA 4.3. Assume ∈L1ð0;1Þ. Fo θ∈L∞ð0;1; ½0;1Þ, we conside θ ¼Π θ.
Then, he solu ions uθand uθ o (2.7) o θand θ , espec i ely, sa is y
kuθ−uθ kL1ð0;1Þ≤oð Þ;ð4:9Þ




Mθ
duθ
dx −Mθ
duθ
dx 


L∞ð0;1Þ
≤oð Þ:ð4:10Þ
I is in L∞ð0;1Þand θis in BVð0;1Þ, hen in (4.9) and (4.10) we can ake
oð Þ¼CVð0;1ÞðθÞ 2:
P oo . The unc ions uθand uθ a e gi en by
uθðxÞ¼−Zx
0
g
Mθ
dsþcZx
0
1
Mθ
ds o a:e:x∈ð0;1Þ;ð4:11Þ
uθ ðxÞ¼−Zx
0
g
Mθ
dsþc Zx
0
1
Mθ
ds o a:e:x∈ð0;1Þð4:12Þ
wi h ga p imi i e o and c; c ∈Rde ined by
c¼Z1
0
1
Mθ
dx−1Z1
0
g
Mθ
dx; c ¼Z1
0
1
Mθ
dx−1Z1
0
g
Mθ
dx:ð4:13Þ
Using hese exp essions and aking in o accoun ha
min α;βg≤Mθ;Mθ ≤max α;βg;
we easily deduce
kuθ−uθ kL1ð0;1Þ≤CZ1
0ðθ−θ ÞgdxþZ1
0ðθ−θ Þdx
:
þZ1
0Zx
0ðθð Þ−θ ð ÞÞgð Þd 
dxþZ1
0Zx
0ðθð Þ−θ ð ÞÞd 
dx
and




Mθ
duθ
dx −Mθ
duθ
dx 


L∞ð0;1Þ
≤CZ1
0ðθ−θ ÞgdxþZ1
0ðθ−θ Þdx:
Lemma 4.3 is hen a simple consequence o Lemma 4.2. ▯
We a e now in posi ion o p o e he ollowing.
P oo o Theo em 2.2. The exis ence o solu ion o p oblem (1.10) is a simple con-
sequence o he compac ness o (2.9) in L1ð0;1Þ.
On he o he hand, using ha F1and F2a e locally Lipschi z, and ha he unc-
ions uθ,uθ de ined as in Lemma 4.2 a e bounded in W1;∞ð0;1Þindependen ly o ,
we ha e
APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1199
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
j^
JðθÞ−^
Jðθ Þj
≤Z1
0
F1x; uθ;Mθ
α
duθ
dx ðθ−θ ÞdxþZ1
0
F2x; uθ;Mθ
α
duθ
dx ðθ−θ Þdx
þCZ1
0juθ−uθ jþMθ
duθ
dx −Mθ
duθ
dx dx:
Thanks o Lemma 4.3, we hen deduce (2.11) and (2.12). ▯
To inish his sec ion, we now gi e he p oo o P oposi ion 2.5.
P oo o P oposi ion 2.5. Reasoning as in he p oo o Theo em 2.2, we ha e ha
he esul is an immedia e consequence o he ollowing lemma, which is simila o
Lemma 4.2. ▯
LEMMA 4.4. Assume θand ωas in he s a emen o P oposi ion 2.5; hen o e e y
φ∈W1;∞ð0;1Þ, i holds ha
Z1
0ðθ−χωÞφdx
≤



dφ
dx


L∞ð0;1Þ
2;ð4:14Þ
Z1
0Zx
0ðθð Þ−χωð ÞÞφð Þd 
dx≤kφkW1;∞ð0;1Þ 2:ð4:15Þ
P oo . Since in each in e al ½yk−1þði−1Þsk;y
kþisk, wi h 1≤k≤m ,
1≤i≤jk, he unc ions θand χωha e he same in eg al, we can eason as in he p oo
o (4.6) o deduce ha o e e y x∈½0;1, we ha e
Zx
0ðθ−χωÞφd 
≤



dφ
dx


L∞ð0;1Þkθ−χωkL1ð0;1Þ 2þkφkL∞ð0;1Þkθ−χωkL1ðIÞ ;ð4:16Þ
whe e Iis an in e al o he o m ½yk−1þði−1Þsk;y
kþiskcon aining x. Taking x¼1
we ge (4.14). On he o he hand, since θand χωbelong o L∞ð0;1; ½0;1Þ, inequali y
(4.16) implies
Zx
0ðθ−χωÞφd 
≤ 2kφkW1;∞ð0;1Þ
o e e y x∈½0;1. This inequali y immedia ely p o es (4.15). ▯
5. P oo o he con e gence es ima es o he disc e ized un elaxed
con ol p oblem. Le us now p o e Theo em 2.3. As o Theo em 2.2, we will need
some p elimina y lemmas.
LEMMA 5.1. We conside θ∈L∞ð0;1Þand l∈N; hen, he e exis s ω⊂ð0;1Þmea-
su able such ha
Z1
0
jθð Þd ¼Zω
jd ∀j∈ 0; :::;lg:ð5:1Þ
Mo eo e ωcan be chosen in he ollowing way:
1200 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
I l¼2n, wi h n∈N,
ω¼ð0;b
0Þ[[
m
i¼1ðai;b
iÞ;
whe e m≤nand 0≤b0<a1<b1<···<am<bm≤1.
I l¼2nþ1, wi h n∈N,
ω¼[
m
i¼1ðai;b
iÞ;
whe e m≤nþ1and 0≤a1<b1<···<am<bm≤1.
P oo . Le us p o e he esul in he case l¼2nþ1, he o he one being simila .
We de ine D⊂L1ð0;1Þas
D¼ϕ¼X
m
i¼1
χðai;biÞwi h m≤nþ1;0≤a1<b1<···<am<bm≤1
and Ψ∶D→Rby
ΨðϕÞ¼X
2nþ1
j¼0Z1
0
jðθð Þ−ϕð ÞÞd 2
∀ϕ∈D:
Since Dis compac in L1ð0;1Þand Ψis con inuous, we know ha Ψa ains i s minimum
in some unc ion
ϕ¼X
m
i¼1
χðai;biÞ∈D:
Then, we de ine he polynomial Pas
PðλÞ¼X
2nþ1
j¼0Z1
0
jðθð Þ−ϕð ÞÞd λj.
We ix k, wi h 1≤k≤m. Fo ε∈R, wi h jεjsmall (εmus also be posi i e i k¼1,
a1¼0), he unc ion
ϕε¼χ∪i≠kðai;biÞþχðakþε;bkÞ
belongs o D. Taking in o accoun ha
ΨðϕεÞ¼X
2nþ1
j¼0Z1
0
jðθð Þ−ϕð ÞÞd þZakþε
ak
jd 2
;
and ha ϕis a minimum poin o Ψ, he de i a i e o ΨðϕεÞwi h espec o εyields
PðakÞ¼0i ak≠0;Pða1Þ≥0i a1¼0:
Analogously, we can p o e
APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1201
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.

PðbkÞ¼0i bk≠1;PðbmÞ≥0i bm¼1:
I Phas 2nþ2ze os, hen i is he ze o polynomial and we ob ain he conclusion o
he lemma. So, we assume in he ollowing ha Phas a mos 2nþ1ze os. By he abo e
p o ed we deduce ha
m¼nþ1;a
1¼0;and∕o bnþ1¼1;
o
m<nþ1:
Le us p o e ha in all hese cases Psa is ies
PðλÞ≥0in [
m
i¼1ðai;b
iÞ;PðλÞ≤0inð0;1Þ [
m
i¼1ðai;b
iÞ.ð5:2Þ
(i) Case m¼nþ1,a1¼0,bnþ1¼1. Since we a e supposing ha he numbe o
ze os o Pis s ic ly less han 2nþ2and P anishes in he 2npoin s akwi h
k¼2; :::;nþ1,bkwi h k¼1; :::;n, we ha e ha Phas 2no 2nþ1ze os
in ½0;1. I he numbe o ze os is 2nþ1, hen using ha Pð0Þ;Pð1Þ≥0,we
deduce ha he o he ze o o Pis in 0 o 1 and ha Psa is ies (5.2). I he
numbe o ze os is 2n, hen we ha e Pð0Þ;Pð1Þ>0and (5.2) is sa is ied.
(ii) Case m¼nþ1,a1¼0,bnþ1<1. In his case we ha e ha he 2nþ1ze os
o Pa e gi en by he poin s akwi h k¼2; :::;nþ1,bkwi h
k¼1; :::;nþ1. Since Pð0Þ≥0, we deduce (5.2).
(iii) Case m¼nþ1,a1>0,bnþ1¼1. I is simila o he case (ii).
(i ) Case m<nþ1. In his case, we ake a poin c∈ðai;b
iÞ o some
i∈ 1; :::mg. Then o ε>0, small enough, he unc ion
ϕε¼ϕ−χðc−ε;cþεÞ
belongs o D. Using ha
ΨðϕεÞ¼X
2nþ1
j¼0Z1
0
jðθð Þ−ϕð ÞÞd þZcþε
c−ε
jd 2
;
and de i ing wi h espec o ε, we deduce ha
PðcÞ≥0∀c∈[
m
i¼1ðai;b
iÞ:
Analogously, i c∈ð0;1Þ Sm
i¼1½ai;b
i, aking
ϕε¼ϕþχðc−ε;cþεÞ;
we deduce ha
1202 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
PðcÞ≤0∀c∈ð0;1Þ [
m
i¼1½ai;b
i:
Thus, (5.2) is also p o en in his case.
To inish, le us p o e ha (5.2) implies he conclusion o he lemma. Fo his pu -
pose, we jus w i e
X
2nþ1
j¼0Z1
0
jðθð Þ−ϕð ÞÞd 2
¼Z1
0X
2nþ1
j¼0Z1
0
jðθð Þ−ϕð ÞÞd sjðθðsÞ−ϕðsÞÞds
¼Z1
0
PðsÞðθðsÞ−ϕðsÞÞds.ð5:3Þ
I s∈Sm
i¼1ðai;b
iÞ(i.e., ϕðsÞ¼1), hen by (5.2), PðsÞ≥0and since θðsÞ≤1, we ha e
PðsÞθðsÞ≤PðsÞϕðsÞ:
I s∈=Sm
i¼1ðai;b
iÞ(i.e., ϕðsÞ¼0), hen by (5.2), PðsÞ≤0and since θðsÞ≥0, we also
ha e
PðsÞθðsÞ≤PðsÞϕðsÞ:
The e o e he las in eg al in (5.3) is nonposi i e, which p o es
X
2nþ1
j¼0Z1
0
jðθð Þ−ϕð ÞÞd 2
¼0:
This p o es Lemma 5.1. ▯
As a consequence, we deduce he ollowing.
LEMMA 5.2. Le a,bbe in Rwi h a<band le ykgm
k¼0be a pa i ion o ½a; bo size
δ¼max
1≤k≤mðyk−yk−1Þ:
Le also θbe in L∞ða; b;½0;1Þ. Then o e e y l∈N, he e exis s I⊂ 1; :::;mgsuch
ha
~
ω¼[
k∈Iðyk−1;y
kÞð5:4Þ
sa is ies
j~
ωj≤Zb
a
θdx;ð5:5Þ
Zb
aðθ−χ~
ωÞφdx
≤Cðb−aÞlþ1kDlþ1φkL1ða;bÞþCδkφkL∞ða;bÞ∀φ∈Wlþ1;1ð0;1Þ;
ð5:6Þ
whe e Cis a posi i e cons an ha depends on l, bu i is independen o θ,δ,a, and b.
APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1203
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
P oo . I is enough o show he case a¼0,b¼1. The gene al one ollows using a
ansla ion and a dila a ion which ans o ms ða; bÞin ð0;1Þ.
Fo a gi en l∈N, by Lemma 5.1 we know he e exis s ω⊂ð0;1Þsa is ying (5.1) and
such ha he numbe o discon inui y poin s o χωin ½0;1is a mos lþ1. We hen
de ine
I¼ k∈ 1; :::;mg∶ðyk−1;y
kÞ⊂ωg
and ~
ωby (5.4). By he de ini ion o ~
ω, we ha e ~
ω⊂ω, and hen using (5.1) when j¼0,
we ob ain (5.5). Mo eo e , using ha χωhas a mos lþ1discon inui y poin s in ½0;1,
we ha e
jω ~
ωj≤ðlþ1Þδ:ð5:7Þ
We now ix φ∈Wlþ1;1ð0;1Þ. Taking a polynomial po deg ee lsuch ha
Z1
0jφ−pjdx≤CkDlþ1φkL1ð0;1Þ
wi h Cindependen o φ( ake, o example, he Taylo polynomial o deg ee lo φ∈
Wlþ1;1ð0;1Þ⊂Clð½0;1Þ in some poin o ½0;1), we ge
Z1
0ðθ−χ~
ωÞφdx
≤Z1
0ðθ−χωÞðφ−pÞdxþZ1
0ðχω−χ~
ωÞφdx
≤CkDlþ1φkL1ð0;1Þþðlþ1ÞδkφkL∞ð0;1Þ:ð5:8Þ
This p o es (5.6) o a¼0,b¼1.▯
LEMMA 5.3. Fo >0small we ake a pa i ion P ¼ ykgm
k¼0wi h m ∈Nsuch ha
(2.8) is sa is ied. We de ine ^
Uby (2.5) and U by (2.10).
(a) Fo e e y θ∈^
U, he e exis s ω∈U such ha
Zx
0ðθ−χωÞφds
≤C 1
2kφkW1;1ð0;1Þ∀x∈½0;1;∀φ∈W1;1ð0;1Þ;ð5:9Þ
whe e Cis a posi i e cons an independen o θand .
(b) Fo e e y θ∈^
Uand e e y l∈N, he e exis s ω∈U such ha
Z1
0ðθ−χωÞφds
≤C lþ1
lþ2kφkWlþ1;1ð0;1Þ∀φ∈Wlþ1;1ð0;1Þ;ð5:10Þ
whe e Cis a posi i e cons an ha depends on l, bu i is independen o θand .
P oo . We ake l∈N,γ∈ð2 ; 1Þ, and a subpa i ion Pγ¼ zigmγ
i¼0⊂P o P
which sa is ies
γ− ≤zi−zi−1≤γ∀i∈ 1; :::;m
γ−1g; ≤zmγ−zmγ−1≤γ:
This implies in pa icula
mγ≤1
γ− þ1≤3
γ:ð5:11Þ
1204 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
Using ha o e e y i∈ 1; :::;m
γ−1g he poin s ykwi h zi−1≤yk≤zia e a pa i ion
o ½zi−1;z
iwi h mesh , we can apply Lemma 5.2 in each in e al ½zi−1;z
i o cons uc a
se ω∈Usuch ha o e e y i∈ 1; :::;m
γ−1g, we ha e
Zzi
zi−1ðθ−χωÞφdx
≤Cðγlþ1kDlþ1φkL1ðzi−1;ziÞþkφkL∞ðzi−1;ziÞ Þð5:12Þ
o e e y φ∈Wlþ1;1ð0;1Þ.
Fo x∈½0;1, we ake he la ge jsuch ha zj≤x; hen, hanks o (5.12) and
(5.11), we ha e
Zx
0ðθ−χωÞφds¼Zzj
0ðθ−χωÞφdsþZx
zjðθ−χωÞφds
≤Cγlþ1kDlþ1φkL1ð0;1ÞþkφkL∞ð0;1Þ3
γþðx−zjÞ:ð5:13Þ
Fo l¼0, he abo e inequali y and x−zj<γp o e
Zx
0ðθ−χωÞφds
≤CγkD1φkL1ð0;1ÞþCkφkL∞ð0;1Þ
γþγ:
Minimizing in γ his quan i y, we deduce (5.9).
On he o he hand, o x¼1¼zjinequali y (5.13) gi es
Z1
0ðθ−χωÞφds
≤Cγlþ1kDlþ1φkL1ð0;1ÞþCkφkL∞ð0;1Þ
γ;
which minimizing in γp o es (5.10). ▯
Using Lemma 5.3 and easoning simila ly o Lemma 4.3, we easily deduce he
ollowing.
LEMMA 5.4. Le θbe in ^
Uand ∈L1ð0;1Þ. Then, o e e y >0, he e exis s ω∈U
such ha , de ining uθ,u as he solu ions o (2.7) o θand χω, espec i ely, we ha e he
ollowing:
(a)
kuθ−u kL1ð0;1Þ≤Cð1þk kL1ð0;1ÞÞ 1
2:ð5:14Þ
(b) I belongs o Wl;1ð0;1Þ, hen




Mθ
duθ
dx −Mχω
du
dx 


L∞ð0;1Þ
≤Cð1þk kWl;1ð0;1ÞÞ lþ1
lþ2:ð5:15Þ
LEMMA 5.5. Le ∈L1ð0;1Þand θbe in ^
U; hen o e e y >0, he e exis s ω∈U
such ha
j^
JðθÞ−JðωÞj ≤C 1
2ð1þk kL1ð0;1ÞÞ:ð5:16Þ
I o some l∈Nwe ha e ha belongs o Wl;1ð0;1Þ,F1ðx; s; ξÞ,F2ðx; s; ξÞa e
independen o sand belong o Cl;1
locð½0;1×RÞ; hen
APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1205
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
This pe mi s us, o example, o subs i u e in he de ini ion o he elaxed con ol se ^
U
he se KðpÞby he (mo e simple) se o symme ic ma ices whose eigen alues a e com-
p essed be ween λðpÞand ΛðpÞ.
Rema k 9. De ining
E¼ ðξ;η;pÞ∈RN×RN×½0;1∶ðη−λðpÞξÞ·ðη−ΛðpÞξÞ≤0g;
he unc ion H ha appea s in (8.5) is a Ca a héodo y unc ion wi h domain Ω×R×E.
An explici exp ession o Hin he whole o i s domain is no known in gene al.
In he pa icula case whe e F1ðx; s; ξÞ,F2ðx; s; ξÞa e a ine unc ions in he a iable
ξ, we ha e
Hðx; s; ξ;η;pÞ¼pF1ðx; s; ξÞþð1−pÞF2ðx; s; ξÞ∀ðs; ξ;η;pÞ∈R×Ea:e:x∈Ω;
while o nonlinea unc ions Fiin he a iable ξ, an exp ession o His only known in
some pa icula cases (which essen ially a e conce ned wi h he nonlinea unc ion jξj2);
see [3], [6], [8], [11], and [18].
Howe e , an explici ep esen a ion is always known in he bounda y o i s domain
ðx; s; ξ;η;pÞ∶∈Ω×R×R×½0;1∶ðη−λðpÞξÞ·ðη−ΛðpÞξÞ¼0g;
whe e Hðx; s; ξ;η;pÞis gi en by
8
>
>
<
>
>
:
F1ðx; s; ξÞi p¼1;
F2ðx; s; ξÞi p¼0;
pF1x; s; βξ−η
pðβ−αÞþð1−pÞF2x; s; η−αξ
ð1−pÞðβ−αÞi p≠0;1:
ð8:7Þ
Obse e ha he las line can be aken as he gene al exp ession o H, aking he alues
o p¼0and p¼1by con inui y.
Analogously as we did in he one-dimensional case, in o de o nume ically sol e
p oblem (8.5), o >0we decompose Ωas
Ω¼[
m
i¼1
Ki;K
idisjoin ;measu able;diamðKiÞ< ; i ∈ 1; :::;m
g:ð8:8Þ
Then, we disc e ize p oblem (8.5) as
min ZΩ
Hðx; u; ∇u; M∇u; θÞdx
−di M∇u¼ in Ω;u¼0on∂Ω;
ðθ;MÞ∈^
U;ðθ;MÞcons an in Ki;1≤i≤m ;RΩθdx≤κ:
ð8:9Þ
As we said in Rema k 9 in he case whe e he unc ions Fiðx; s; ξÞa e nonlinea in he
a iable ξ, one o he main di icul ies o sol e p oblem (8.9) is ha His no known. To
sol e his di icul y we can eplace Hwi h ano he unc ion. The ollowing esul is
p o ed in [8] in he pa icula case F1ðx; s; ξÞ¼F2ðx; s; ξÞ¼FðξÞ. The gene al case
ollows simila ly.
1212 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.

THEOREM 8.1. We conside a unc ion ^
H∶Ω×R×E→R∪ þ∞gsuch ha
^
Hð:; s; ξ;η;pÞis measu able in Ω∀ðs; ξ;η;pÞ∈R×E;ð8:10Þ
^
Hðx; :; :; :; :Þis lowe semicon inuous in R×E o a:e: x ∈Ω;ð8:11Þ
^
Hðx; s; ξ;αξ;1Þ¼F1ðx; s; ξÞ;^
Hðx; s; ξ;βξ;0Þ¼F2ðx; s; ξÞ;ð8:12Þ
^
Hðx; s; ξ;η;pÞ≥Hðx; s; ξ;η;pÞ∀ðs; ξ;η;pÞ∈R×E a:e: x ∈Ω:ð8:13Þ
Fo e e y >0, we decompose Ωby (8.9). Then, he p oblem
min ZΩ
^
Hðx; u; ∇u; M∇u; θÞdx
−di M∇u¼ in Ω;u¼0on∂Ω;
ðθ;MÞ∈^
U;ðθ;MÞcons an in Ki;1≤i≤m ;RΩθdx≤κ
ð8:14Þ
has a solu ion (no unique in gene al) ðθ ;M Þ. Taking u as he solu ion o
−di M ∇u ¼ inΩ;u
¼0on ∂Ω;
we ha e
∃lim
→0ZΩ
^
Hðx; u ;∇u ;M ∇u ;θ Þdx¼I
wi h I he minimum alue o p oblem de ined by (8.5). The sequence ðθ ;M ;u
Þis
bounded in L∞ðΩÞ×L∞ðΩ;RN×NÞ×H1
0ðΩÞ. E e y unc ion ðθ;M;uÞ∈L∞ðΩÞ×
L∞ðΩ;RN×NÞ×H1
0ðΩÞsuch ha he e exis s a subsequence o , s ill deno ed by ,
sa is ying
θ ⇀
θin L∞ðΩÞ;M
⇀
MinL
∞ðΩ;RN×NÞ;u
⇀uinH
1
0ðΩÞ
is such ha he unc ion ðθ;σ;uÞwi h σ¼M∇uis a solu ion o (8.6).
Rema k 10. A i s choice o unc ion ^
His o ake
^
Hðx; s; ξ;η;pÞ¼8
<
:
F1ðx; s; ξÞi p¼1;η¼αξ;
F2ðx; s; ξÞi p¼0;η¼βξ;
þ∞o he wise:
In his case, aking in o accoun ha ^
Hðx; u; ∇u; M∇u; θÞ<þ∞a.e. in Ωimplies ha θ
is a cha ac e is ic unc ion we ge ha p oblem (8.14) can be w i en as
min Zω
F1ðx; u; ∇uÞdxþZΩ ω
F2ðx; u; ∇uÞdx
−di ðαχωþβχΩ ωÞ∇u¼ in Ω;u¼0on∂Ω;
∃I⊂ 1; :::;m
gsuch ha ω¼S
i∈I
Ki;jωj≤κ:
The e o e, wi h his choice o unc ion ^
H, Theo em 8.1 gi es he con e gence o he nu-
me ical me hod consis ing in disc e izing di ec ly he o iginal (un elaxed) p oblem (8.1).
APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1213
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
Thanks o (8.7), ano he possibili y o ^
His o ake ^
H¼Hin ∂DðHÞ, and ^
H¼þ∞,
o he wise. Fo his choice o unc ion ^
H, aking in o accoun ha o p≠0;1a ma ix
M∈KðpÞsa is ies
ðMξ−λðpÞξÞ·ðMξ−ΛðpÞξÞ¼ξ o some ξ≠0
⇔Mis a lamina ion o αI; βIwi h p opo ions pand 1 −p
⇔EigðMÞ¼ðλðpÞ;ΛðpÞ; :::;ΛðpÞÞ:
We can w i e p oblem (8.14) as
min ZΩpF1x; u; β∇u−M∇u
pðβ−αÞþð1−pÞF2x; u; M∇u−α∇u
ð1−pÞðβ−αÞdx
8
<
:
−di M∇u¼ in Ω;u¼0on∂Ω;
θ∈L∞ðΩ;½0;1Þ;Msymme ic;EigðMÞ¼ðλðθÞ;ΛðθÞ; :::;ΛðθÞÞa:e:in Ω;
θ;Mcons an s in Ki;i¼1; :::;m
;RΩθdx≤κ:
In his case, p oblem (8.14) consis s in disc e izing a pa ial elaxa ion o p oblem (8.1)
consis ing in conside ing no only he o iginal con ols bu also he ones ob ained by a
simple lamina ion.
Clea ly, when His known, ano he possibili y is o ake di ec ly ^
H¼H. In his case
we a e disc e izing he elaxed con ol p oblem (8.9).
Rema k 11. Al hough Theo em 8.1 gi es he con e gence o he disc e ized p oblem
(8.14), i does no p o ide any e o es ima e. In pa icula , i does no show which
choice o he unc ions ^
Hmen ioned in Rema k 10 is be e .
As we saw in he p oo o he es ima es o he one-dimensional p oblem, in o de o
ob ain an es ima e o he con e gence a e o he nume ical me hod, one idea is o con-
s uc om a elaxed con ol ðθ;MÞano he con ol ðθ ;M Þin he se o disc e ized
con ols such ha he solu ions o he s a e equa ions ela i e o ðθ;MÞand
ðθ ;M Þa e close. In he case whe e ^
H¼H(which can only be used i His known),
one idea is o ake ðθ ;M Þas he mean alue o ðθ;MÞin each elemen o he iangula-
ion. Deno ing by uand u he solu ions o
−di M∇u¼ in Ω;
u¼0on∂Ω;−di M ∇u ¼ in Ω;
u¼0on∂Ω
wi h in H−1ðΩÞand aking in o accoun ha
−di M ∇ðu−u Þ¼−di ðM −MÞ∇uin Ω;
we deduce ha
ZΩj∇ðu−u Þj2dx≤CZΩjðM −MÞ∇uj2dx;
which pe mi s o es ima e he di e ence o u−u depending on he smoo hness p ope -
ies o Mand uand hen o es ima e he e o o he disc e ized me hod.
When His no known and he e o e we need o disc e ize di ec ly he o iginal
p oblem o o conside some pa ial elaxa ion, he choice o ðθ ;M Þis no clea .
1214 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
Rema k 12. In Theo em 8.1, we ha e disc e ized he se o con ols, bu he s a e
equa ion is di ec ly sol ed. I will be in e es ing o s udy he con e gence when we also
disc e ize his equa ion, and in pa icula o s udy wha he ela ion is ha we mus use
be ween he iangula ion chosen o he con ols and he one chosen o he esolu ion o
he s a e equa ion. A esul in his sense can be ound in [8], showing ha in some cases
he me hod con e ges using he same iangula ion o disc e ize he con ols and he
s a e equa ion.
Acknowledgmen . The au ho s a e g a e ul o he Basque Cen e o Applied
Ma hema ics o i s hospi ali y and suppo in se e al isi s.
REFERENCES
[1] G. ALLAIRE,Shape Op imiza ion by he Homogeniza ion Me hod, Appl. Ma h. Sci. 146, Sp inge -Ve lag,
New Yo k, 2002.
[2] G. ALLAIRE AND S. GUTIÉRREZ,Op imal design in small ampli ude homogeniza ion, ESAIM Ma h. Model.
Nume . Anal., 41 (2007), pp. 543–574.
[3] J. C. BELLIDO AND P. PEDREGAL,Explici quasicon exi ica ion o some cos unc ionals depending on
de i a i es o he s a e in op imal designing, Disc e e Con in. Dyn. Sys ., 8 (2002), pp. 967–982.
[4] B. BOURDIN,Fil e s in opology op imiza ion, In e na . J. Nume . Me hods Eng g., 50 (2001),
pp. 2143–2158.
[5] C. CARSTENSEN AND C. ORTHNER,Analysis o a class o penal y me hods o compu ing singula minimize s,
Compu . Me hods Appl. Ma h., 10 (2010), pp. 137–163.
[6] J. CASADO-DÍAZ,J.COUCE-CALVO,AND J. D. MARTÍN-GÓMEZ,Relaxa ion o a con ol p oblem in he coe i-
cien s wi h a unc ional o quad a ic g ow h in he g adien , SIAM J. Con ol Op im., 47 (2008),
pp. 1428–1459.
[7] J. CASADO-DÍAZ,J.COUCE-CALVO,AND J. D. MARTÍN-GÓMEZ,Relaxa ion o op imal design p oblems wi h
nonlinea unc ionals in he g adien , o appea .
[8] J. CASADO-DÍAZ,J.COUCE-CALVO,M.LUNA-LAYNEZ,AND J. D. MARTÍN-GÓMEZ,Op imal design
p oblems o a non-linea cos in he g adien : Nume ical esul s, Appl. Anal., 87 (2008),
pp. 1461–1487.
[9] J. CASADO-DÍAZ,J.COUCE-CALVO,M.LUNA-LAYNEZ,AND J. D. MARTÍN-GÓMEZ,Disc e iza ion o coe icien
con ol p oblems wi h a nonlinea cos in he g adien , in In eg al Me hods in Science and
Enginee ing, Compu a ional Me hods 2, C. Cons anda and M. E. Pé ez, eds., Bi khäuse , Bos on,
2010, pp. 55–63.
[10] D. CHENAIS AND E. ZUAZUA,Fini e elemen app oxima ion o 2D ellip ic op imal design, J. Ma h. Pu es
Appl. (9), 85 (2006), pp. 225–249.
[11] Y. GRABOVSKY,Op imal design o wo-phase conduc ing composi es wi h weakly discon inuous objec i e
unc ionals, Ad . in Appl. Ma h., 27 (2001), pp. 683–704.
[12] R. LIPTON AND A. P. VELO,Op imal design o g adien ields wi h applica ions o elec os a ics,in
Nonlinea Pa ial Di e en ial Equa ions and Thei Applica ions, Collège de F ance Semina
XIV, D. Cio anescu and J. L. Lions, eds., S ud. Ma h. Appl. 31, No h-Holland, Ams e dam,
2002, pp. 509–532.
[13] K. A. LURIE,Applied Op imal Con ol Theo y o Dis ibu ed Sys ems, Plenum P ess, New Yo k,
1993.
[14] K. A. LURIE AND A. V. CHERKAEV,Exac es ima es o he conduc i i y o a bina y mix u e o iso opic
ma e ials, P oc. Roy. Soc. Edinbu gh Sec . A, 104 (1986), pp. 21–38.
[15] F. MURAT,Un con e-example pou le p oblème du con ôle dans les coe icien s, C. R. Ma h. Acad. Sci.
Pa is, 273 (1971), pp. 708–711.
[16] F. MURAT,Théo èmes de non exis ence pou des p oblèmes de con ôle dans les coe icien s, C. R. Ma h.
Acad. Sci. Pa is, 274 (1972), pp. 395–398.
[17] F. MURAT AND L. TARTAR,H-con e gence, in Topics in he Ma hema ical Modelling o Composi e
Ma e ials, P og . Nonlinea Di e en ial Equa ions Appl. 31, L. Che kae and R. V. Kohn, eds.,
Bi khäuse , Bos on, 1998, pp. 21–43.
[18] P. PEDREGAL,Op imal design in wo-dimensional conduc i i y o a gene al cos depending on he ield,
A ch. Ra ion. Mech. Anal., 182 (2006), pp. 367–385.
APPROXIMATION OF AN OPTIMAL DESIGN PROBLEM 1215
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
[19] S. SPAGNOLO,Sulla con e genza di soluzioni di equazioni pa aboliche ed elli iche, Ann. Sc. No m. Supe .
Pisa Cl. Sci. (5), 22 (1968), pp. 571–597.
[20] L. TARTAR,P oblèmes de con ôle des coe icien s dan des équa ions aux dé i ées pa ielles, in Con ol
Theo y, Nume ical Me hods and Compu e Sys ems Modelling, Lec u es No es in Econ. and Ma h.
Sys ems 10, Sp inge -Ve lag, Be lin, 1975. English ansla ion: F. MURAT AND L. TARTAR,On he
con ol o coe icien s in pa ial di e en ial equa ions, in Topics in he Ma hema ical Modelling
o Composi e Ma e ials, P og . Nonlinea Di e en ial Equa ions Appl. 31, L. Che kae
and R. V. Kohn, eds., Bi khäuse , Bos on, 1998, pp. 1–8.
[21] L. TARTAR,Cou s Pecco , Collège de F ance, 1977, pa ly w i en in [17].
[22] L. TARTAR,Es ima ions ines de coe icien s homogénéisés, in Ennio de Gio gi Colloquium (Pa is 1983),
Res. No es Ma h. 125, P. K ee, ed., Pi man, London, 1985, pp. 168–187.
[23] L. TARTAR,Rema ks on op imal design p oblems, in Calculus o Va ia ions, Homogeniza ion and
Con inuum Mechanics, Se . Ad . Ma h. Appl. Sci. 18, G. Bu azzo, G. Bouchi e, and P. Suque ,
eds., Wo ld Scien i ic, Hackensack, NJ, 1994, pp. 279–296.
[24] L. TARTAR,An in oduc ion o he homogeniza ion me hod in op imal design, in Op imal Shape Design,
Lec u e No es in Ma h. 1740, A. Cellina and A. O nelas, eds., Sp inge -Ve lag, Be lin, 2000,
pp. 47–156.
[25] V. V. ZHIKOV,Weigh ed Sobole spaces, Sb. Ma h., 189 (1998), pp. 1139–1170.
[26] J. GOODMAN,R.V.KOHN,AND L. REYNA,Nume ical s udy o a elaxed a ia ional p oblem om op imal
design, Compu . Me hods Appl. Mech. Eng g., 57 (1986), pp. 107–127.
1216 CASADO-DÍAZ, CASTRO, LUNA-LAYNEZ, ZUAZUA
Copy igh ©by SIAM. Unau ho ized ep oduc ion o his a icle is p ohibi ed.
Rep oduced wi h pe mission o he copy igh owne . Fu he ep oduc ion p ohibi ed wi hou
pe mission.