Results on existence of solution for an optimal design problem

E extracta mathematicae Vol. 18, N´um. 3, 263 – 271 (2003) Results on Existence of Solution for an Optimal Design Problem Carmen Calvo Jurado, Juan Casado D´ ıaz Departamento de Matem´aticas, Escuela Polit´ecnica Universidad de Extremadura, 10071 C´aceres, Spain Departamento de Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla, 41012 Sevilla, Spain e-mail: cc[email protected], jc[email protected] (Presented by W. Okrasi`nski) AMS Subject Class. (2000): 35K55, 49J20, 76M50 Received June 4, 2003 1. Introduction In this paper we study a control problem for elliptic nonlinear monotone problems with Dirichlet boundary conditions where the control variables are the coefficients of the equation and the open set where the partial differential problem is studied. More exactly, we consider a bounded open set Ω ⊂RNand a monotone operator Afrom H1(Ω) to H−1(Ω), mapping y∈H1(Ω) in A y = −div a(x, ∇y)∈H−1(Ω), where a: Ω×RN→RNis a Carath´eodory function which defines a monotone Leray-Lions operator of order 2. Our problem is to find an open set e Ω⊂Ω and Aon the conditions above, such that for f∈H−1(Ω), the solution yof (A y =fin e Ω y∈H1 0(e Ω) (1.1) minimize a functional J:H1 0(Ω) →R(the solution of (1.1) will be considered extended by zero outside e Ω and then, defined as an element of H1 0(Ω)). When e Ω is fixed (see [18], [19], [20]) or Ais fixed (see [3], [4]), the problem has been studied in several papers, usually for linear problems. It is well know that these problems has no solution in general. In the present paper, we show the existence of solution when the controls are searched in a large set. 263 264 c. calvo jurado, j. casado d´ ıaz Our results can be generalized to systems of Mequations and operators of order p∈(1,+∞) (see [5]). Here, by simplicity, we study the scalar case with p= 2. From the point of view of the applications, the results exposed in the present paper are related with the selection of optimal shape material (take into account that the coefficients of the equation depend on the choice of the materials). 2. Notation and preliminaries Let Ω be a bounded open subset of RN. For a measure µ, we denote by L2 µ(Ω), the space of the functions which are µ-measurable and have its power two µ-integrable. If µis the Lebesgue measure, we write Lp(Ω,RM). We denote by H1 0(Ω) the closure of the C∞functions with compact support for the norm kukH1 0(Ω) =kukL2(Ω) +k∇ukL2(Ω)N. The dual space of H1 0(Ω), it is denoted by H−1(Ω). For every subset B⊂Ω, and p∈(1,+∞), we denote by C(B, Ω) the capacity of B(in Ω), which is defined as the infimum of ZΩ|∇y|2dx over the set of the functions y∈H1 0(Ω) such that y≥1 a.e. in a neighbourhood of B. We say that a property P(x) holds C-quasi everywhere (abbreviated as q.e.) in a set E, if there exists N⊂Ewith C(N, Ω) = 0 such that P(x) holds for all x∈E\N. A function y: Ω →Ris said to be quasi continous, if for every ε > 0 there exists N⊂Ω, with C(N, Ω) < ε, such that the restriction of yto Ω\Nis continuous. It is well know that every y∈H1 0(Ω) has a quasi continuous representative (see [15], [16], [24]). We always identify ywith its quasi continuous representative. A subset A⊂Ω is said to be quasi open in Ω, if for every ε > 0 there exists an open subset U⊂Ω, with C(U, Ω) < ε, such that A∪Nis open. We denote by M2 0(Ω) the class of all Borel measures which vanish on the sets of capacity zero and satisfy µ(B) = inf{µ(A) : Aquasi open, B⊆A⊆Ω} for every Borel set B⊆Ω. existence of solution 265 Definition 2.1. For α, γ > 0 , we denote by A(α, γ) the set of Carath´eodory functions a: Ω ×RN→RNsuch that (i) a(x, 0) = 0 for 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} for all ξ1, ξ2∈RN, a.e. x∈Ω. Remark 2.2. If abelongs to A(α, γ), then asatisfies (iii) |a(x, ξ1)−a(x, ξ2)| ≤ 1 γ|ξ1−ξ2|for all ξ1, ξ2∈R, a.e. x∈Ω. Reciprocally, if a function asatisfies (a(x, ξ1)−a(x, ξ2))(ξ1−ξ2)≥α|ξ1−ξ2|2 for all ξ1, ξ2∈R, a.e. x∈Ω, and there exists β > 0 such that |a(x, ξ1)−a(x, ξ2)| ≤ β|ξ1−ξ2| for all ξ1, ξ2∈R, a.e. x∈Ω, then asatisfies (ii) with γ=α β2. Definition 2.3. We denote by U(α, γ) the set of pairs (µ, F ) such that µ∈ M2 0(Ω) and F: Ω ×R→Rsatisfies (a) F(·, s) is µ-measurable for very 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} for all s1, s2∈R,µ-a.e. x∈Ω. Remark 2.4. Hypothesis (c) is equivalent to: 1 γ(s1−s2)≥F(x, s1)−F(x, s2)≥α(s1−s2) for all s1, s2∈R,s1≥s2,µ-a.e. x∈Ω. We consider a functional J:H1 0(Ω) →Rwhich is sequentially weakly lower semicontinuous, i.e.: yn* y ⇒lim inf n→∞ J(yn)≥J(y).(2.2) 266 c. calvo jurado, j. casado d´ ıaz 3. Existence of solution for the optimal design problem For f∈H−1(Ω), e Ω⊂Ω and a∈ A , we consider the partial differential problem (−div a(x, ∇y) = fin e Ω y∈H1 0(e Ω).(3.3) Our purpose is to find e Ω and a∈ A which solve the minimum problem ½min J(y) a∈ A,e Ω⊂Ω.(3.4) In order to show the existence of solution for (3.4), we can try to use the direct method of calculus of variations. For that, we consider Ωn⊂Ω opens, and an∈ A such that the sequence ynof solutions of ½−div an(x, ∇yn) = fin Ωn yn∈H1 0(Ωn)(3.5) is minimizing, i.e.: lim inf n→∞ J(yn) = I where I= inf{J(y) : a∈ A,e Ω⊂Ω, y satisfies (3.3)}. Taking ynas test function in (3.5), we deduce ZΩ an(x, ∇yn)∇yndx=ZΩ fyndx, which by (ii) implies kynkH1 0(Ω) ≤kfkH−1(Ω) √α, where we have identified ynwith its extension by zero to Ω\Ωn. So, there exists a subsequence (still denoted by yn) which converges weakly to a function yin H1 0(Ω). By the lower semicontinuity (2.2) of J, we have J(y)≤I. If there exists e Ω⊂Ω and a∈ A such that ysatisfies (3.3), then J(y) = Iand the problem is solved. Therefore, we need to find the equation satisfied by the function yand to know if it is of the same type that (3.3). Thus, we need to study the homogenization problem ½−div an(x, ∇yn) = fin D0(Ωn) yn∈H1 0(Ωn),(3.6) existence of solution 267 where an∈ A and Ωnis a sequence of arbitrary open sets contained in a given bounded open set Ω ⊂RN. This is a question which is well known when Ωn or anis fixed. When Ωnis fixed it has been proved (see for example [21], for the linear problem and [22], [23] for the nonlinear one) that there exists a function a∈ A such that (for a sequence) the solutions ynof (3.6) with Ωn=e Ω fixed, converge weakly in H1 0(Ω) to the solution yof (−div a(x, ∇y) = fin e Ω y∈H1 0(e Ω), where adoes not depend of f. In particular, this implies that the initial problem (3.4), has a solution if we assume that e Ω is not variable (control coefficients problem). However, when anis fixed, it is not true in general that there exists a subsequence of Ωn, still denoted by Ωn, and open set e Ω⊂Ω such that the solutions of (3.6) with an=afixed, converge weakly in H1 0(Ω) to the solution yof (−div a(x, ∇y) = fin D0(e Ω) y∈H1 0(e Ω). For example, if N= 3, and Ωn= Ω\Sk∈ Z NB(k n,1 n3), it has been proved in [8], that the sequence of solutions ynof (3.6) with a(x, ξ) = ξ, for all ξ∈R, a.e. x∈Ω, (laplacian operator) converges weakly in H1 0(Ω) to the unique solution yof ½−∆y+4π 3y=fin Ω y∈H1 0(Ω).(3.7) As a consequence of this result, let us now prove Theorem 3.1. The problem (3.4) has no solution in general. Proof. Let Ω ⊂RNbe a bounded open set. We respectively denote by y0 the solution of (3.7) with f= 1 and by ¯ythe solution of ½−∆¯y= 1 in Ω ¯y∈H1 0(Ω).(3.8) We consider J:H1 0(Ω) →Ras J(y) = RΩ|y−y0|2dy, for all y∈H1 0(Ω) and α= 1 −ε,γ=1 1+ε, with εsmall enough such that µε2+ 4ε 1−ε¶2ZΩ|∇¯y|2dy < ZΩ|∇(¯y−y0)|2dy. (3.9) 268 c. calvo jurado, j. casado d´ ıaz Remark that yn* y in H1 0(Ω) implies J(yn)→J(y) in R. It is clear for the result of Cianorescu-Murat mentioned above, that in this case I= 0. So, if there exists (a, e Ω) solution of (3.4), then (−div a(x, ∇y0) = 1 in e Ω y0∈H1 0(e Ω). Now, y0∈H1 0(e Ω), implies that y0= 0 q.e. in Ω\e Ω, but the strong maximum principle implies that y0>0 in Ω. So, Ω\e Ω has capacity zero, but then H1 0(Ω) is equal to H1 0(e Ω), and y0is also the solution of the problem ½−div a(x, ∇y0) = 1 in Ω y0∈H1 0(Ω).(3.10) On the other hand, by (ii), for every ξ∈RNand a.e. x∈Ω, we have |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 test function in the difference of (3.8) and (3.10), we deduce ZΩ [a(x, ∇y0)−∇¯y]∇(y0−¯y) dy= 0, and then, using (ii) and (iii), we obtain (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 . From (3.9) and (3.12) we deduce the absurd. Our interest in the following is to show that the control problem has a solution if we search for the control variables in a more large set. For this purpose, following G. Dal Maso and U. Mosco (see [11]), we remark that defining for e Ω⊂Ω open, the measure µ∈ M2 0(Ω) as µ(B) = (0 if cap((Ω\e Ω) ∩B) = 0 +∞if cap((Ω\e Ω) ∩B)>0, existence of solution 269 and taking F, such that the pair (µ, F ) belongs to U(α, γ) (it always exists) the problem (3.3) is equivalent to the variational problem          y∈H1 0(Ω) ∩L2 µ(Ω) ZΩ a(x, ∇y)∇vdx+ZΩ F(x, y)vdµ=hf, vi ∀v∈H1 0(Ω) ∩L2 µ(Ω) (3.13) Then, at the place of the original control problem, we can consider the following one min{J(y) : a∈ A(α, γ),(F, µ)∈ U(α, γ)},(3.14) where for A(α, γ), (F, µ)∈ U(α, γ), yis the unique solution of (3.13). The advantage of the new formulation is clear from the following theorem. Theorem 3.2. For every sequences anin A(α, γ)and (Fn, µn)in U(α, γ), there exists a subsequence, still denoted by n, such that for every f∈H−1(Ω), the solution ynof          yn∈H1 0(Ω) ∩ ∩L2 µn(Ω) ZΩ an(x, ∇yn)∇vdx+ZΩ Fn(x, yn)vdµn=hf, vi ∀v∈H1 0(Ω) ∩ ∩L2 µn(Ω) (3.15) converges weakly in H1 0(Ω) to the solution yof (3.13). Theorem 3.2 has been proved by the authors in [5], in fact it is true for operators of order p∈(1,+∞) and for systems. In particular, it gives the form of the limit problem of (3.6) for arbitrary Ωnand an. When µnis zero for every n, the result can be found in [22] and [23]. For the case anconstant, the theorem has been shown in [7], although it is not proved that the pair (F, µ) which appears in the limit problem is in U(α, γ) (see also [6], [8], [9], [10], [11], [12], . . . ). When anand Fnare linear, the result appears in [14]. For the double homogenization problem, with monotone operators, a previous result has been proved in [17], but in this work µnare not general, they correspond to a sequence of open sets Ωn, such that the measure µin the limit is the Lebesgue measure. Using Theorem 3.2, we can now apply the direct method of the calculus of variations as above to prove 270 c. calvo jurado, j. casado d´ ıaz Theorem 3.3. The problem (3.14) admits at least a solution a∈ A(α, γ), (F, µ)∈ U(α, γ). The question which remains is to know if the problem (3.15) is a relaxation of (3.4), i.e., if for every a∈ A(α, γ) and (F, µ)∈ U(α, γ), there exists an∈ A and Ωn⊂Ω open, such that the solutions ynof (3.6) converge weakly in H1 0(Ω) to the solution yof (3.13). This is true if we ask for the elements of A and Uto be linear in its second variable (see [14]). Acknowledgements This paper has been partially supported by the project PB98-1162 of the D.G.E.S.I.C. of Spain. 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