Results on existence of solution for an optimal design problem

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. Re e ences [1] Bocca do, L., Mu a , F., Almos e e ywhe e con e gence o he g adi- en s o solu ions o ellip ic and pa abolic equa ions, Nonlinea Anal. Theo . Ma h. Appl. 19 (6) (1992), 581 – 597. [2] B aides, A., Malusa, A., App oxima ion o elaxed Di ichle p oblems, in “Calculus o Va ia ions, Homogeniza ion and Con inuum Mechanics”, Ma - seille, 1993, 83 – 97. (Se . Ad . Ma h. Appl. Sci. 18, Wo ld Sci., Ri e Edge, 1994.) [3] Bu azzo, G., Dal Maso, G., Shape op imiza ion o Di ichle p oblems. Relaxed SIS and op imaly condi ions, Appl. Ma h. Op im. 23 (1991), 17 – 49. [4] Bu azzo, G., Dal Maso, G., Ga oni, A., Malusa, A., On he elaxed o mula ion o some shape op imiza ion p oblems, Ad . Ma h. Sci. Appl. 7(1) (1997), 1 – 24. [5] Cal o-Ju ado, C., Casado-D´ iaz, J., The limi o Di ichle sys ems o a iable mono one ope a o s in gene al pe o a ed domains, J. Ma h. Pu es Appl. 81 (2002), 471 – 493. [6] Casado-D´ iaz, J., Homogenisa ion o Di ichle p oblems o mono one op- e a o s in a ying domains, P oc. Roy. Soc. Edinbu gh Sec . A 127 (1997), 457 – 478. [7] Casado-D´ iaz, J., Ga oni, A., Asymp o ic beha iou o nonlinea ellip ic sys ems on a ying domains, SIAM J. Anal. Ma h. 31 (3) (2000), 249 – 276. [8] Ciona escu, D., Mu a , F., Un e me ´e ange enu d’ailleu s, in “Non- linea Pa ial Di e en ial Equa ions and hei Applica ions”, Coll`ege de F ance Semina , Vols II and III, 98 – 138 and 154 – 78 (Ed. by H. B ´ezis and J.L. Lions, Resea ch No es in Ma h. 60 and 70, Pi man, London, 1982.) [9] Dal Maso, G., De anceschi, A., Limi s o nonlinea Di ichle p oblems in a ying domains, Manusc ip a Ma h. 61 (1988), 251 – 278. [10] Dal Maso, G., Ga oni, A., New esul s on he assymp o ic beha iou o Di ichle p oblems in pe o a ed domains, Ma h. Models Me h. Appl. Sci. 3(1994), 373 – 407. exis ence o solu ion 271 [11] Dal Maso, G., Mosco, U., Wiene -c i e ion and Γ-con e gence, Appl. Ma h. Op im. 15 (1987), 15 – 63. [12] Dal Maso, G., Mu a , F., Asymp o ic beha iou and co ec o s o Di- ichle p oblems in pe o a ed domains wi h homogeneous mono one ope a - o s, Ann. Sc. No m. Sup. Pisa 7(4) (1997), 765 – 803. [13] Dal Maso, G., Mu a , F., Almos e e ywhe e con e gence o g adien s o solu ions o nonlinea ellip ic sys ems, Nonlinea Anal. Se ie A 31 (3/4) (1998), 405-412. [14] Dal Maso, G., Mu a , F., Asymp o ic beha iou and co ec o s o linea Di ichle p oblems wi h simul aneously a ying ope a o s and domains ( o appea ). [15] E ans, L.C., Ga iepy, R.F., “Measu e Theo y and Fine P ope ies o Func ions”, CRC P ess, Boca Ra on, 1992. [16] Fede e , H., Zieme , W.P., The Lebesgue se o a unc ion whose dis i- bu ion de i a ies a e p- h powe sumable, Indiana Uni . Ma h. J. 22 (1972), 139 – 158. [17] Ko ale sky, A., An e ec o double homogeniza ion o Di ichle p oblems in a iable domains o gene al s uc u e, C.R. Acad. Sci. Pa is Se . I 328 (1999), 1151 – 1156. [18] Lu ie, K.A., “Applied Op imal Con ol Theo y o Dis ibu ed Sys ems”, Plenum P ess, New Yo k, 1993. [19] Mu a , F., Th´eo `emes de non-exis ence pou des p obl`emes de con ˆole dans le coe icien s, C.R. Acad. Sci. Pa is Se . A 274 (1972), 395 – 398. [20] Mu a , F., Ta a , L., On he con ol o coe icien es in pa ial di e - en ial equa ions, in “Topics in he Ma hema ical Modelling o Composi e Ma e ials” (Ed. by A. Che kaek, R. Kohn, P og ess in Nonlinea Di e en ial Equa ions and hei Appl., Bi kh¨ause , Bos on, 1997), 139 – 173. [21] Mu a , F., Ta a , L., H-con e gence, in “Topics in he Ma hema ical Modelling o Composi e Ma e ials”, Ed. by A. Che kae , R. Kohn, P og ess in Nonlinea Di e en ial Equa ions and hei Applica ions (Bi k¨ause , Bo- s on, 1997) 21 – 43. [22] Panko , A., “G-Con e gence and Homogeniza ion o Nonlinea Pa ial Di - e en ial Ope a o s”, Ma h. and i s Appl. 422, Kluwe Academic Publishe s, London, 1997. [23] del Vecchio, T., On he homogeniza ion in a class o pseudomono one ope a o s in di e gence o m, Boll. Unione Ma . I al. B (7) 5(2) (1991), 369 – 388. [24] Zieme , W.P., “Weakly Di e en iable Func ions”, Sp inge -Ve lag, Be lin, 1989.