E
ex ac a ma hema icae Vol. 18, N´um. 3, 263 – 271 (2003)
Resul s on Exis ence o Solu ion o an
Op imal Design P oblem
Ca men Cal o Ju ado, Juan Casado D´
ıaz
Depa amen o de Ma em´a icas, Escuela Poli ´ecnica
Uni e sidad de Ex emadu a, 10071 C´ace es, Spain
Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico
Uni e sidad de Se illa, 41012 Se illa, Spain
e-mail: cc[email p o ec ed], jc[email p o ec ed]
(P esen ed by W. Ok asi`nski)
AMS Subjec Class. (2000): 35K55, 49J20, 76M50 Recei ed June 4, 2003
1. In oduc ion
In his pape we s udy a con ol p oblem o ellip ic nonlinea mono one
p oblems wi h Di ichle bounda y condi ions whe e he con ol a iables a e
he coe icien s o he equa ion and he open se whe e he pa ial di e en ial
p oblem is s udied.
Mo e exac ly, we conside a bounded open se Ω ⊂RNand a mono-
one ope a o A om H1(Ω) o H−1(Ω), mapping y∈H1(Ω) in A y =
−di a(x, ∇y)∈H−1(Ω), whe e a: Ω×RN→RNis a Ca a h´eodo y unc ion
which de ines a mono one Le ay-Lions ope a o o o de 2.
Ou p oblem is o ind an open se e
Ω⊂Ω and Aon he condi ions abo e,
such ha o ∈H−1(Ω), he solu ion yo
(A y = in e
Ω
y∈H1
0(e
Ω) (1.1)
minimize a unc ional J:H1
0(Ω) →R( he solu ion o (1.1) will be conside ed
ex ended by ze o ou side e
Ω and hen, de ined as an elemen o H1
0(Ω)).
When e
Ω is ixed (see [18], [19], [20]) o Ais ixed (see [3], [4]), he p oblem
has been s udied in se e al pape s, usually o linea p oblems. I is well know
ha hese p oblems has no solu ion in gene al. In he p esen pape , we show
he exis ence o solu ion when he con ols a e sea ched in a la ge se .
263
264 c. cal o ju ado, j. casado d´
ıaz
Ou esul s can be gene alized o sys ems o Mequa ions and ope a o s
o o de p∈(1,+∞) (see [5]). He e, by simplici y, we s udy he scala case
wi h p= 2.
F om he poin o iew o he applica ions, he esul s exposed in he
p esen pape a e ela ed wi h he selec ion o op imal shape ma e ial ( ake
in o accoun ha he coe icien s o he equa ion depend on he choice o he
ma e ials).
2. No a ion and p elimina ies
Le Ω be a bounded open subse o RN. Fo a measu e µ, we deno e by
L2
µ(Ω), he space o he unc ions which a e µ-measu able and ha e i s powe
wo µ-in eg able. I µis he Lebesgue measu e, we w i e Lp(Ω,RM).
We deno e by H1
0(Ω) he closu e o he C∞ unc ions wi h compac suppo
o he no m kukH1
0(Ω) =kukL2(Ω) +k∇ukL2(Ω)N. The dual space o H1
0(Ω),
i is deno ed by H−1(Ω).
Fo e e y subse B⊂Ω, and p∈(1,+∞), we deno e by C(B, Ω) he
capaci y o B(in Ω), which is de ined as he in imum o
ZΩ|∇y|2dx
o e he se o he unc ions y∈H1
0(Ω) such ha y≥1 a.e. in a neighbou hood
o B.
We say ha a p ope y P(x) holds C-quasi e e ywhe e (abb e ia ed as
q.e.) in a se E, i he e exis s N⊂Ewi h C(N, Ω) = 0 such ha P(x) holds
o all x∈E N.
A unc ion y: Ω →Ris said o be quasi con inous, i o e e y ε > 0 he e
exis s N⊂Ω, wi h C(N, Ω) < ε, such ha he es ic ion o y o Ω Nis
con inuous. I is well know ha e e y y∈H1
0(Ω) has a quasi con inuous ep-
esen a i e (see [15], [16], [24]). We always iden i y ywi h i s quasi con inuous
ep esen a i e.
A subse A⊂Ω is said o be quasi open in Ω, i o e e y ε > 0 he e
exis s an open subse U⊂Ω, wi h C(U, Ω) < ε, such ha A∪Nis open.
We deno e by M2
0(Ω) he class o all Bo el measu es which anish on he
se s o capaci y ze o and sa is y
µ(B) = in {µ(A) : Aquasi open, B⊆A⊆Ω}
o e e y Bo el se B⊆Ω.
exis ence o solu ion 265
De ini ion 2.1. Fo α, γ > 0 , we deno e by A(α, γ) he se o Ca a h´eo-
do y unc ions a: Ω ×RN→RNsuch ha
(i) a(x, 0) = 0 o a.e. x∈Ω;
(ii) (a(x, ξ1)−a(x, ξ2))(ξ1−ξ2)≥max{α|ξ1−ξ2|2, γ|a(x, ξ1)−a(x, ξ2)|2}
o all ξ1, ξ2∈RN, a.e. x∈Ω.
Rema k 2.2. I abelongs o A(α, γ), hen asa is ies
(iii) |a(x, ξ1)−a(x, ξ2)| ≤ 1
γ|ξ1−ξ2| o all ξ1, ξ2∈R, a.e. x∈Ω.
Recip ocally, i a unc ion asa is ies
(a(x, ξ1)−a(x, ξ2))(ξ1−ξ2)≥α|ξ1−ξ2|2
o all ξ1, ξ2∈R, a.e. x∈Ω, and he e exis s β > 0 such ha
|a(x, ξ1)−a(x, ξ2)| ≤ β|ξ1−ξ2|
o all ξ1, ξ2∈R, a.e. x∈Ω, hen asa is ies (ii) wi h γ=α
β2.
De ini ion 2.3. We deno e by U(α, γ) he se o pai s (µ, F ) such ha
µ∈ M2
0(Ω) and F: Ω ×R→Rsa is ies
(a) F(·, s) is µ-measu able o e y s∈R;
(b) F(x, 0) = 0, µ-a.e. x∈Ω;
(c) (F(x, s1)−F(x, s2))(s1−s2)≥max{α|s1−s2|2, γ|F(x, s1)−F(x, s2)|2}
o all s1, s2∈R,µ-a.e. x∈Ω.
Rema k 2.4. Hypo hesis (c) is equi alen o:
1
γ(s1−s2)≥F(x, s1)−F(x, s2)≥α(s1−s2)
o all s1, s2∈R,s1≥s2,µ-a.e. x∈Ω.
We conside a unc ional J:H1
0(Ω) →Rwhich is sequen ially weakly
lowe semicon inuous, i.e.:
yn* y ⇒lim in
n→∞ J(yn)≥J(y).(2.2)
266 c. cal o ju ado, j. casado d´
ıaz
3. Exis ence o solu ion o he op imal design p oblem
Fo ∈H−1(Ω), e
Ω⊂Ω and a∈ A , we conside he pa ial di e en ial
p oblem (−di a(x, ∇y) = in e
Ω
y∈H1
0(e
Ω).(3.3)
Ou pu pose is o ind e
Ω and a∈ A which sol e he minimum p oblem
½min J(y)
a∈ A,e
Ω⊂Ω.(3.4)
In o de o show he exis ence o solu ion o (3.4), we can y o use he di ec
me hod o calculus o a ia ions. Fo ha , we conside Ωn⊂Ω opens, and
an∈ A such ha he sequence yno solu ions o
½−di an(x, ∇yn) = in Ωn
yn∈H1
0(Ωn)(3.5)
is minimizing, i.e.:
lim in
n→∞ J(yn) = I
whe e
I= in {J(y) : a∈ A,e
Ω⊂Ω, y sa is ies (3.3)}.
Taking ynas es unc ion in (3.5), we deduce
ZΩ
an(x, ∇yn)∇yndx=ZΩ
yndx,
which by (ii) implies
kynkH1
0(Ω) ≤k kH−1(Ω)
√α,
whe e we ha e iden i ied ynwi h i s ex ension by ze o o Ω Ωn. So, he e exis s
a subsequence (s ill deno ed by yn) which con e ges weakly o a unc ion yin
H1
0(Ω). By he lowe semicon inui y (2.2) o J, we ha e J(y)≤I. I he e
exis s e
Ω⊂Ω and a∈ A such ha ysa is ies (3.3), hen J(y) = Iand he
p oblem is sol ed.
The e o e, we need o ind he equa ion sa is ied by he unc ion yand
o know i i is o he same ype ha (3.3). Thus, we need o s udy he
homogeniza ion p oblem
½−di an(x, ∇yn) = in D0(Ωn)
yn∈H1
0(Ωn),(3.6)
exis ence o solu ion 267
whe e an∈ A and Ωnis a sequence o a bi a y open se s con ained in a gi en
bounded open se Ω ⊂RN. This is a ques ion which is well known when Ωn
o anis ixed.
When Ωnis ixed i has been p o ed (see o example [21], o he linea
p oblem and [22], [23] o he nonlinea one) ha he e exis s a unc ion a∈ A
such ha ( o a sequence) he solu ions yno (3.6) wi h Ωn=e
Ω ixed, con e ge
weakly in H1
0(Ω) o he solu ion yo
(−di a(x, ∇y) = in e
Ω
y∈H1
0(e
Ω),
whe e adoes no depend o . In pa icula , his implies ha he ini ial
p oblem (3.4), has a solu ion i we assume ha e
Ω is no a iable (con ol
coe icien s p oblem).
Howe e , when anis ixed, i is no ue in gene al ha he e exis s a
subsequence o Ωn, s ill deno ed by Ωn, and open se e
Ω⊂Ω such ha he
solu ions o (3.6) wi h an=a ixed, con e ge weakly in H1
0(Ω) o he solu ion
yo (−di a(x, ∇y) = in D0(e
Ω)
y∈H1
0(e
Ω).
Fo example, i N= 3, and Ωn= Ω Sk∈
Z
NB(k
n,1
n3), i has been p o ed in
[8], ha he sequence o solu ions yno (3.6) wi h a(x, ξ) = ξ, o all ξ∈R,
a.e. x∈Ω, (laplacian ope a o ) con e ges weakly in H1
0(Ω) o he unique
solu ion yo ½−∆y+4π
3y= in Ω
y∈H1
0(Ω).(3.7)
As a consequence o his esul , le us now p o e
Theo em 3.1. The p oblem (3.4) has no solu ion in gene al.
P oo . Le Ω ⊂RNbe a bounded open se . We espec i ely deno e by y0
he solu ion o (3.7) wi h = 1 and by ¯y he solu ion o
½−∆¯y= 1 in Ω
¯y∈H1
0(Ω).(3.8)
We conside J:H1
0(Ω) →Ras J(y) = RΩ|y−y0|2dy, o all y∈H1
0(Ω) and
α= 1 −ε,γ=1
1+ε, wi h εsmall enough such ha
µε2+ 4ε
1−ε¶2ZΩ|∇¯y|2dy < ZΩ|∇(¯y−y0)|2dy. (3.9)

268 c. cal o ju ado, j. casado d´
ıaz
Rema k ha yn* y in H1
0(Ω) implies J(yn)→J(y) in R.
I is clea o he esul o Ciano escu-Mu a men ioned abo e, ha in his
case I= 0. So, i he e exis s (a, e
Ω) solu ion o (3.4), hen
(−di a(x, ∇y0) = 1 in e
Ω
y0∈H1
0(e
Ω).
Now, y0∈H1
0(e
Ω), implies ha y0= 0 q.e. in Ω e
Ω, bu he s ong maximum
p inciple implies ha y0>0 in Ω. So, Ω e
Ω has capaci y ze o, bu hen H1
0(Ω)
is equal o H1
0(e
Ω), and y0is also he solu ion o he p oblem
½−di a(x, ∇y0) = 1 in Ω
y0∈H1
0(Ω).(3.10)
On he o he hand, by (ii), o e e y ξ∈RNand a.e. x∈Ω, we ha e
|ξ−a(x, ξ)|=|ξ|2+|a(x, ξ)|2−2a(x, ξ)ξ
≤ |ξ|2+ (1 + ε)2|ξ|2−2(1 −ε)|ξ|2= (4ε+ε2)|ξ|2.(3.11)
Taking y0−¯yas es unc ion in he di e ence o (3.8) and (3.10), we deduce
ZΩ
[a(x, ∇y0)−∇¯y]∇(y0−¯y) dy= 0,
and hen, using (ii) and (iii), we ob ain
(1 −ε)ZΩ|∇(y0−¯y)|2dy≤ZΩ
[a(x, ∇y0)−a(x, ∇¯y)]∇(y0−¯y) dy
≤ZΩ
[∇¯y−a(x, ∇¯y)]∇(y0−¯y) dy(3.12)
≤(4ε+ε2)µZΩ|∇¯y|2dy¶1
2µZΩ|∇(y0−¯y)|2dy¶1
2
.
F om (3.9) and (3.12) we deduce he absu d.
Ou in e es in he ollowing is o show ha he con ol p oblem has a
solu ion i we sea ch o he con ol a iables in a mo e la ge se . Fo his
pu pose, ollowing G. Dal Maso and U. Mosco (see [11]), we ema k ha
de ining o e
Ω⊂Ω open, he measu e µ∈ M2
0(Ω) as
µ(B) = (0 i cap((Ω e
Ω) ∩B) = 0
+∞i cap((Ω e
Ω) ∩B)>0,
exis ence o solu ion 269
and aking F, such ha he pai (µ, F ) belongs o U(α, γ) (i always exis s)
he p oblem (3.3) is equi alen o he a ia ional p oblem









y∈H1
0(Ω) ∩L2
µ(Ω)
ZΩ
a(x, ∇y)∇ dx+ZΩ
F(x, y) dµ=h , i
∀ ∈H1
0(Ω) ∩L2
µ(Ω)
(3.13)
Then, a he place o he o iginal con ol p oblem, we can conside he ollow-
ing one
min{J(y) : a∈ A(α, γ),(F, µ)∈ U(α, γ)},(3.14)
whe e o A(α, γ), (F, µ)∈ U(α, γ), yis he unique solu ion o (3.13). The
ad an age o he new o mula ion is clea om he ollowing heo em.
Theo em 3.2. Fo e e y sequences anin A(α, γ)and (Fn, µn)in U(α, γ),
he e exis s a subsequence, s ill deno ed by n, such ha o e e y ∈H−1(Ω),
he solu ion yno









yn∈H1
0(Ω) ∩ ∩L2
µn(Ω)
ZΩ
an(x, ∇yn)∇ dx+ZΩ
Fn(x, yn) dµn=h , i
∀ ∈H1
0(Ω) ∩ ∩L2
µn(Ω)
(3.15)
con e ges weakly in H1
0(Ω) o he solu ion yo (3.13).
Theo em 3.2 has been p o ed by he au ho s in [5], in ac i is ue o
ope a o s o o de p∈(1,+∞) and o sys ems. In pa icula , i gi es he
o m o he limi p oblem o (3.6) o a bi a y Ωnand an. When µnis ze o
o e e y n, he esul can be ound in [22] and [23]. Fo he case ancons an ,
he heo em has been shown in [7], al hough i is no p o ed ha he pai
(F, µ) which appea s in he limi p oblem is in U(α, γ) (see also [6], [8], [9],
[10], [11], [12], . . . ). When anand Fna e linea , he esul appea s in [14].
Fo he double homogeniza ion p oblem, wi h mono one ope a o s, a p e-
ious esul has been p o ed in [17], bu in his wo k µna e no gene al, hey
co espond o a sequence o open se s Ωn, such ha he measu e µin he limi
is he Lebesgue measu e.
Using Theo em 3.2, we can now apply he di ec me hod o he calculus
o a ia ions as abo e o p o e
270 c. cal o ju ado, j. casado d´
ıaz
Theo em 3.3. The p oblem (3.14) admi s a leas a solu ion a∈ A(α, γ),
(F, µ)∈ U(α, γ).
The ques ion which emains is o know i he p oblem (3.15) is a elaxa ion
o (3.4), i.e., i o e e y a∈ A(α, γ) and (F, µ)∈ U(α, γ), he e exis s an∈ A
and Ωn⊂Ω open, such ha he solu ions yno (3.6) con e ge weakly in
H1
0(Ω) o he solu ion yo (3.13). This is ue i we ask o he elemen s o A
and U o be linea in i s second a iable (see [14]).
Acknowledgemen s
This pape has been pa ially suppo ed by he p ojec PB98-1162 o
he D.G.E.S.I.C. o Spain.
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