existence and approximation in optimal shape design problems

University of Jyvaskyla Department of Mathematics Laboratory of Scienti c Computing

Report 6/1998

EXISTENCE AND APPROXIMATION IN OPTIMAL SHAPE DESIGN PROBLEMS Pekka Neittaanmaki Dan Tiba

UNIVERSITY OF JYVA SKYLA JYVA SKYLA 1998

EXISTENCE AND APPROXIMATION IN OPTIMAL SHAPE DESIGN PROBLEMS Pekka Neittaanmaki1 and Dan Tiba2 1

Department of Mathematics University of Jyvaskyla P.O. Box 35, FIN{40351 Jyvaskyla Finland 2

Institute of Mathematics Romanian Academy P.O. Box 1{764, RO{70700 Bucharest Romania Abstract. We consider a system given by a second order elliptic equation with jumps in the coecients. This models a body made of two dierent materials and we study the question of the material distribution that minimizes a certain cost functional. We introduce a local compactness condition for a class of characteristic functions to obtain the existence of the optimum. We also indicate a new approximation procedure via a distributed control approach for the original shape optimization problem. By letting some of the coecients tend to 0 or to 1 we obtain existence results for optimal shape design problems governed by Neumann or Dirichlet boundary value problems. Keywords: optimal design, monotone operators, existence AMS Classi cation (1991): 49D37, 49M29, 65K10

1. Introduction Fixed domain methods are very useful in variable domain problems as free boundary problems, optimal shape design or material distribution problems. In shape optimization problems, we quote the mapping method, Murat and Simon [13], Pironneau [14], the penalization method, Kawarada [10], the topology optimization method, Bendse [2], Bendse and Rodriguez [3], the controllability approach, Tiba [18], Tiba and Neittaanmaki [19]. Some advantages over the classical boundary variation technique are: the use of a xed nite element mesh in all the iterations (which saves much of the computation time), the possibility not to choose in advance the topological properties of the desired domain (for instance: simply connected or doubly connected, etc.). On the other hand, certain xed domain methods may be sti or dicult to handle in certain applications.

Here, we discuss a new approach which may be mainly compared with the mapping method or the topology optimization method since it reduces the optimal shape design problem to a control in the coecients problem. However, no global description of the boundary of the variable domain is needed and no scaling has to be performed. Our basic idea is simply to generate characteristic functions of measurable subsets  D (a xed domain) via the maximal monotone extension of the Heaviside mapping:  1; p  0; (1.1) H (p) = 0; p < 0; which is again denoted by H and has the form 8 p > 0; > < 1; [0; 1]; p = 0; (1.2) H (p) = > : 0; p < 0: For any measurable subset  D, there are measurable functions p 2 L1 (D) such that the characteristic function of , denoted by  , is given via (1.1) by  = H (p ). One may directly take p = ?Dn , but, in the sequel, some conditions will be imposed on p that will specify the class of admissible choices for . For instance, certain weak compactness assumptions on families of characteristic functions (and, equivalently, of sets ) may be easily formulated in the language of the functions p . For H given by (1.2), there exists a well-known regularization technique via the Yosida approximation of maximal monotone operators: 8 p > "; > < 1; p p 2 [0; "]; (1.3) H" (p) = "; > : 0; p < 0: Relations (1.1){(1.3) provide a simple ecient way to generate and to approximate characteristic functions. They are as well very easy to implement in the setting of shape optimization problems governed by partial dierential equations. Moreover, since (1.2), (1.3) \ ll in" the jumps into the coecients of the governing equation (for instance, (1.5), (1.6), (2.2), (2.6), etc.), our approach may be considered as well as a relaxation procedure which allows the coecients to take any value in some interval (see Bendse [2], Raitums [15], Dal Maso [7]). In this way, it is known that a convex optimization problem is generated, but it may be dicult to interpret the solution of the convexi ed problem in terms of the original one. In our case, the approximating problem remains nonconvex, but we prove existence and strong convergence properties for the approximating minimizers. We shall demonstrate these facts on the following optimal design problem, denoted by (P): (1.4)

Minimize E  D

Z

E

jy ? zd j2 dx

2

subject to the transmission problem (1.5) (1.6) (1.7)

?a1
y1 + b1 y1 = f in ; ?a2
y2 + b2 y2 = f in D n ; @y1 @y2 a1 = a2 ; y1 = y2 in @ n (@ \ @D); @n @n

(1.8)

ai

@yi @n

= 0 in ?1 ;

yi

= 0 in ?2 ;

i

= 1; 2:

Above, E  D are xed domains, @D = ?1 [?2 is, for instance, Lipschitz, ?1 \?2 = ?, ai ; bi > 0, i = 1; 2, are some constants, f 2 L2 (D ), zd 2 L2 (E ) are given and y 2 H 1 (D ) is de ned by: (1.9)

y (x)

=



y1 (x)

y2 (x)

in ; in D n :

The equation (1.5){(1.8) models a body made of two materials occupying respectively the regions and D n and having dierent physical properties expressed by the dierent coecients associated to each region. The problem (1.4){(1.9) is, therefore, a material distribution problem (lay-out problem). A variant of this problem was previously studied by Pironneau [14] by dierent methods and a numerical approach via (1.3) was developped by Makinen, Neittaanmaki and Tiba [11]. The plan of the paper is as follows. In Section 2, we discuss the problem (1.4){ (1.9) and we prove existence and approximation properties. Section 3 explores the behaviour for a2 ! 0, b2 ! 0 or a2 ! 1, b2 ! 1 which enables us to reobtain existence results for optimal shape design problems governed by Dirichlet or Neumann boundary value problems. In this way, we develop a uni ed approach to the existence and approximation theory in shape optimization, independent of the boundary conditions on @ . It should be noted that our hypotheses allow, in certain cases,

to be merely measurable subsets of D and, in general, our compactness condition has a local character and is weaker than the Chenais [5] condition. In Section 3, in order to use extension or trace theorems, we impose Lipschitz regularity on , similar to Chenais [5], but we have no limitation on the space dimension or on the number of connected components of as in Sverak [16], [17]. Throughout this paper, we denote by j
j the Euclidean norm in RN , N  1, and by j
jX , (
;
)X X  the norm and the duality pairing between the Banach space X and its dual X .

2. Transmission conditions We de ne the space V  H 1 (D): (2.1)

V

= fw 2 H 1 (D) : w = 0 in ?2 g: 3

The weak formulation of (1.5){(1.9) is: (2.2)

Z

 D

[a1 + a2(1 ?  )]ry
rw + [b1  + b2(1 ?  )]y w

? fw



dx

= 0; 8w 2 V:

The bilinear form is bounded and coercive in V if the coecients are strictly positive. Then, for any f 2 L2( ), there is a unique weak solution y 2 V , which formally satis es (1.5){(1.9). In Pironneau [14], the characteristic functions  are directly interpreted as a control parameter, however, the constraint to take only the values 0 or 1 in D is dicult to handle. Our approach is to take  of the form H (p ), where p 2 Uad , an admissible set of controls. We x now Uad  H 1(D) by the following conditions:

jpjH 1+K (K )  MK ; 8K  D compact; jp(x)j + jrp(x)j   > 0 a.e. D; p(x)  0 a.e. E;

(2.3) (2.4) (2.5)

where MK ,  , K are positive constants such that Uad is nonvoid. There are no assumptions on the dimension of the Euclidean space containing D. We formulate the following (restricted) variant of the problem (P): (P1 )

Minimize p2U ad

Z E

jyp ? zd j2 dx

subject to (2.6)

Z

 D

[a1H (p) + a2(1 ? H (p))]ryprw + [b1H (p) + b2(1 ? H (p))]ypw

? fw



dx

= 0; 8w 2 V:

The new notations are obvious and H (
) is de ned by (1.1).

Theorem 2.1. The problem (P1) has at least one optimal pair [y; p] 2 H 1(D)  Uad . Proof. Since Uad = 6 ?, we may consider a minimizing sequence of pairs [yn; pn ]  Uad ,

for (P1). By (2.3) and the Rellich theorem, we may take a subsequence (again denoted by n) such that pn ! pK strongly in H 1 (K ), for all K S D compact. By taking an increasing sequence of compacts with 1 l=1 Kl = D , we may de ne p 2 Uad such that pn ! p and rpn ! rp pointwisely, a.e. in D . Denote by a = minfa1 ; a2 ; b1 ; b2 g > 0 and take w = yn (= ypn ) in (2.6). Since a1 H (pn ) + (1 ? H (pn ))a2  a and b1H (pn ) + (1 ? H (pn ))  a a.e. D, then fyn g is bounded in V  H 1 (D ) and we may assume that yn ! y weakly in H 1 (D ), on a subsequence. 4

As p 2 Uad, the set fx 2 D : p(x) = 0g has zero measure, by condition (2.4) and the regularity of p, Brezis [4, p. 195]. Obviously fH (pn )g is bounded in L1 (D) and, the demiclosedness of maximal monotone operators, gives H (pn ) ! H (p) weakly* in L1 (K ), for all K  D compact, with H (p) expressed via (1.2). Since the set fx 2 D : p(x) = 0g has zero measure, it follows that H (p) is a characteristic function in D, despite the fact that H (p) given by (1.2) may (conceptually) take arbitrary values in [0; 1]. Moreover, it follows that H (pn ) ! H (p) a.e. in D: We know that pn (x) ! p(x) 6= 0 a.e. in D. Take rst p(x) > 0, then pn (x) > 0 for n  nx and H (pn (x)) = H (p(x)) = 1, for n  nx . In the case p(x) < 0, it follows similarly that pn (x) < 0 for n  nx and H (pn (x)) = H (p(x)) = 0 for n  nx . These two situations are valid a.e. in D. Consequently, H (pn ) ! H (p) strongly in Ls(D), for any s  1, by the Lebesgue dominated convergence theorem. Since [yn; pn ] satisfy (2.6), we can now pass to the limit in (2.6). Together with p 2 Uad , this shows that the pair [y; p] is admissible for (P1 ). We also have Inf(P1 ) = nlim !1

Z

E

jyn ? zd j2 dx =

Z

E

jy ? zd j2 dx

which gives that the pair [y; p] is optimal for (P1). We redenote it by [y; p] and the proof is nished. Remarks. 1) The above argument remains valid for any cost functional weakly lower semicontinuous in H 1 (D). Other boundary conditions may be imposed on @D. 2) The conditions (2.3){(2.5) de ne a class of characteristic functions depending on  which is nonconvex, but \locally" compact with respect to a convenient topology.

We continue by de ning an approximation of (P1), denoted by (P" ): (P")

Minimize p2U ad

Z

E

jyp" ? zd j2 dx

subject to (2.7)

Z

 D

[a1H" (p) + a2 (1 ? H"(p))]ryp"rw + [b1H" (p) + b2 (1 ? H"(p))]yp"w

? fw



dx

= 0; 8w 2 V;

where H" is the Yosida approximation of H as in (1.3). As in the proof of Theorem 2.1, it yields that (P") has at least one optimal pair [y"; p"], p" 2 Uad , for any " > 0.

Theorem 2.2. For " ! 0, we have [y"; p"] ! [^y; p^] in the weak-strong topology of H 1 (D )  H 1 (K ), any K  D compact, on a subsequence, and [^ y ; p^] is an optimal pair for (P1).

Proof. By (2.3), there is p^ 2 Uad such that, for any K  D compact, we have p" ! p^ strongly in H 1(K ), on a subsequence. As 1  H"(p)  0 and a = min(a1 ; a2; b1; b2) >

5

0, we see that fy"g is bounded in H 1 (D), by xing w = y" in (2.7). We may assume that y" ! y^ weakly in H 1(D), on a subsequence. It is known from the theory of monotone operators that H" (p") ! H (^p) weakly* in L1 (K ) since it is bounded in L1 (K ) and p" ! p^ strongly in L2 (K ), for instance. Moreover, H (^ p) is de ned via (1.2), but it is a characteristic function since the set fx 2 D : p^(x) = 0g has zero measure by p^ 2 Uad . We can also prove the pointwise convergence for H" (p"), a.e. in D. If p^(x) > 0, then p"(x) > 21 p^(x) for " < "x , so p" (x) > " for " < "x and H" (p" (x)) = 1 = H (^ p(x)), by (1.3). If p^(x) < 0, then p" (x) < 0 for " < "x and " H" (p (x)) = 0 = H (^ p(x)). Consequently H" (p" ) ! H (^ p) strongly in Ls (D ); 8s  1; by the Lebesgue dominated convergence theorem. Then, we can pass to the limit in (2.7) and infer that [^y; p^] is admissible for the problem (P1). To show that it is optimal, we use the de nition of [y"; p"], i.e.: Z

(2.8)

E

jy

"

? zd j2 dx 

Z

E

jyp" ? zd j2 dx; 8p 2 Uad ;

where is associated to p via (2.7). For a xed p, a similar argument gives that yp" ! yp (given by (2.6)) weakly in 1 H (D ), on a subsequence. Passing to the limit in (2.8) yields the optimality of [^ y ; p^] in (P1) and the proof is nished. Corollary 2.3. On a subsequence, we have: (2.9) y " ! y^ strongly in H 1 (D ): Proof. yp"

(2.10)

j

ay

"

? y^j2 1 H

(D) 

Z

D

+ =

[a1H"(p" ) + (1 ? H" (p"))a2][r(y" ? y^)]2 dx

Z

D

Z



D

[b1H" (p") + (1 ? H"(p" ))b2](y" ? y^)2 dx

[a1H" (p") + (1 ? H" (p"))a2]ry"r(y" ? y^)

+ [b1H"(p" ) + b2 (1 ? H"(p" ))]y"(y" ? y^)

?

Z



D

=

dx

[a1H"(p" ) + a2 (1 ? H"(p" ))]ry^r(y" ? y^)

+ [b1H"(p" ) + b2 (1 ? H"(p" ))]^y(y" ? y^) Z





dx

?

f (y " y^) dx D Z  [a1H"(p" ) + a2 (1

?

? H"(p" ))]ry^r(y" ? y^) D + [b1H"(p" ) + b2 (1 ? H"(p" ))]^y(y" ? y^) dx = I 1 ? I2 : 6

By Theorem 2.2, clearly I1 ! 0 for " ! 0. The same is true for I2 since the coecients a1 H" (p" ) + a2 (1 ? H" (p" )) and b1 H" (p" ) + b2 (1 ? H" (p" )) are a.e. convergent in D to their limit. The only dicult term to pass to the limit in I2 is the rst one (2.11)

Z

D

[a1H" (p") + a2(1 ? H"(p"))]ry^r(y" ? y^) dx:

We notice that the factors f[a1H" (p") + a2(1 ? H" (p"))]r(y" ? y^)g are bounded in L2 (D )N by (1.3) and Theorem 2.2. Then, on a subsequence, we may take that [a1H" (p") + a2(1 ? H"(p" ))]r(y" ? y^) ! u weakly in L2(D)N . Due to the pointwise convergence of the coecients, the Egorov theorem shows that for any  > 0, there is D  D measurable, with meas(D n D ) <  , such that a1 H" (p" ) + a2 (1 ? H" (p" )) ! a1 H (^ p) + (1 ? H (^ p))a2 uniformly in D . Combining this with the weak convergence of r(y" ? y^) ! 0 in L2(D)N , we get that u(x) = 0 a.e. in D. Finally, the integral (2.11) has limit zero and (2.10) achieves the proof. Remark. 3) Numerical tests using this approach are reported in Makinen, Neittaanmaki and Tiba [11]. It is possible to perform a further smoothing of H" by means of a Friedrichs molli er and to compute the gradient of the cost functional with respect to p.

Let us now brie y comment on the signi cance of the assumptions (2.3), (2.4) in terms of the sets p = fx 2 D : p(x) > 0g. On the boundary of p (given by p(x) = 0 if p is continuous), we have rp(x) 6= 0. If p is Lipschitz and this maximal rank condition remains valid for any selection of the generalized gradient, the Clarke [6, p. 256] implicit function theorem gives that the domains p are Lipschitz, for any p 2 Uad . Conversely, for Lipschitz domains, in any local chart, we have for the boundary the representation xn = (x1; :::; xn?1),  Lipschitz, which ensures automatically that (2.4) is ful lled. Therefore, our hypotheses are slightly weaker than in the classical result of Chenais [5], Pironneau [14]. Moreover, our approach is certainly dierent and it is independent of the type of boundary conditions on the variable part of the boundary as it is explained in the next section. Remark. 4) The essential ingredient in the proofs is to pass to the limit in the transmission problem (1.5){(1.8) when the coecients are changing. This type of questions are also very important in homogenization theory, Raitums [15], Bendse [2], Dal Maso [7] and, we emphasize that, under our assumptions, oscillations of the unknown boundary @ are allowed, due to the local character of the compactness assumption (2.3).

3. Neumann and Dirichlet problems In this setting, we need a stronger variant of hypothesis (2.3), namely: (2.3)'

jpjH 1+ (D)  M; 8p 2 Uad; 7

for some positive , M . We shall also ask that all the connected components of the zero level sets of p (p 2 Uad ) are Lipschitz domains with some uniform Lipschitz constants. This follows by (2.3)', (2.4), if  is large enough (for instance), via the Sobolev embedding theorem and the implicit functions theorem. Based on the results of x2, we show rst that it is possible to prove the existence in optimal design problems governed by Neumann boundary value problems. In (2.2), we x f of the form: (3.1)

f (x) =  g (x) + (1

?  )g(x); a.e. in ;

with some given  > 0 and g 2 L2 (D). We also choose a2 = b2 =  and, in the sequel, we analyse what happens when  ! 0. Let us mention that such an approach has already been proposed by Pironneau [14, p. 134] in connection with the optimality conditions question. We denote by (P ) the problem (P1) with f and the coecients xed as above and by [y ; p] 2 H 1(D)  Uad some optimal pair of (P ), which exists according to a variant of the argument from Section 2. That is: (3.2) (3.3)

Z

(y ? zd )2 dx 

ZE



D

=

Z

Z E

(yp ? zd )2 dx;

[a1H (p) + (1 ? H (p))]ryp
rw + [b1H (p) + (1 ? H (p))]ypw

D



dx

[H (p)g + (1 ? H (p))g]w dx; 8w 2 V;

for any p 2 Uad and with yp being the solution of (3.3) associated to the given p and  > 0. By hypothesis (2.3)', we may assume that p ! p~ 2 Uad uniformly in D, on a subsequence. However, since  ! 0, the estimates on fyg are weaker than1 in x2. Namely, by xing as usual, w = y in (3.3) with p = p , we obtain that f 2 y g is bounded in H 1 (D). The same is true for yp . Let us denote by  = supp(p )+ and by = supp(~p)+ . Then, clearly (3.4)

Z

?



jry j2 + jy j2




dx

 ct:

Notice that by (2.5), E   , for all  > 0, and E  . We also have: (3.5) meas(  n ) + meas( n  ) =

Z

 nZ

=

dx +

D

Z

n 

dx

jH (p) ? H (~p)j dx ! 0 for  ! 0

since H (p ) ! H (~p) in Ls(D), for all s  1, by the arguments developped in x2. The rst two terms appearing in (3.5) can be written shortly as ( ; ) = meas(  4 ) 8

(symmetric dierence of sets) and give a metric on the family of measurable subsets of D, dierent from the Hausdor{Pompeiu metric for closed sets, Hewitt and Stromberg [9, p. 144]. It coincides with the Ekeland metric on L1 (D) (for the corresponding characteristic mappings): measfx 2 D : u(x) 6= v(x)g

dE (u; v ) =

and the set of characteristic functions in D is closed with respect to this metric. To pass to the limit in (3.2), we have to analyse both fyg and fypg. We take various terms of interest, one after another: Z

(3.6)

D

(3.7)

(1 Z D

Z

(3.8)

? H (p ))ry rw dx ! 0; (1

(1 D Z

(3.9)

D Z

(3.10)

? H (p ))y w dx ! 0;

? H (p))ryprw dx ! 0; ? H (p))ypw dx ! 0;

(1

D

(1

? H (p))gw dx ! 0;

for any xed p 2 Uad , w 2 V , since estimates of order ? 21 are valid in D. We continue with: (3.11) (3.12)

Z

D

r rw =

H (p ) y

Z

 n

w

Z

\ 

ry r  Cw  Cw

ry rw +

Z

 n

Z



Z

 n

ry rw

jry j

jry j2

 12

meas(  n ) 12 ! 0

due to (3.4) and (3.5) and for smooth w. To deal with the rst integral in the right-hand side of (3.11), it is known that an extension operator (for instance, the Calderon extension) from  \ to exists and we denote by y~ the extension of y j \ to , which may be dierent from y ! Moreover, the following estimate is valid

jy~jH 1 ( )  C jy jH 1 ( \ ) with C independent of  , Chenais [5], Adams [1, p. 81], Pironneau [14, p. 40]. We

(3.13) have:

(3.14)

Z

\ 

ry rw =

Z



ry~ rw ? 9

Z

n 

ry~ rw:

For smooth w, it yields: (3.15)

Z

n 

w

ry~ r  Cw

Z

n 

jry~ j  Cw jy~ jH 1 ( ) meas( n  ) 21

 CCw jyjH 1 ( \ ) meas( n  ) 21  CCw jyjH 1 ( ) meas( n  ) 21 ! 0;



! 0:

This argument applies to each connected component of and  and any xed  > 0, suciently small. Let K  , K1  D n be compacts obtained by subtracting from D a \thin" band around @ . Since p ! p~ uniformly, sign p
sign p~ > 0 on K [ K1 , for  small and we put into a one to one correspondence the components of ,  which contain the same component of K . Let us notice that  cannot have other components in D n (K [ K1) or in K1 by the uniform Lipschitz hypothesis and the de nition of K , K1 . In (3.15) we have used (3.13), (3.4) and (3.5). Consequently, (3.14), (3.15), (3.11) and (3.12) give Z

(3.16)

D

r

H (p ) y

!0 rw ?! 

Z



ry~rw

where y~ is the weak limit in H 1( ) of y~ , on a subsequence, and w 2 C 1 (D). In a similar manner, we infer that Z

(3.17)

!0 H (p )y w ?! 

D

Z



y~w:

By (3.6){(3.10) and (3.16), (3.17) we can pass to the limit in (3.3) with p = p and obtain Z

(3.18)



rr

a1 y~ w dx +

Z



b1 y~w dx

=

Z



gw dx

for any w 2 C 1 ( ) and, by density, for any w 2 H 1 ( ), Maz'ja [12, p. 14]. Denote now by p the set p = supp(p)+ , p 2 Uad, and by yp the weak limit in H 1( p ) of fypj p g. By xing w = yp in (3.3), it follows easily that f 21 yp g is bounded in H 1 (D) and fypj p g is bounded in H 1( p ). Using (3.8){(3.10), one can directly pass to the limit in (3.3) to infer: (3.19)

Z

p

r r

a1 yp w dx +

Z

p

b1 yp w dx

Moreover, passing to the limit in (3.2), yields (3.20)

Z

E

(~y ? zd )2 dx 

Z

E

=

Z

p

gw dx;

8w 2 H 1(D):

(yp ? zd )2 dx; 8p 2 Uad :

Relations (3.18), (3.19) show that y~, yp are weak solutions for the Neumann boundary value problem in , respectively in p . Together with (3.20), they prove the desired existence result: 10

Theorem 3.1. Under the above conditions, there is at least one solution for the optimal shape design problem (3.19), (3.20) in the class of open sets p , p 2 Uad. We continue by examining the case of the Dirichlet conditions on the boundary of the unknown set. We formulate the approximating optimal shape design problem (corresponding to  ! 1) via the transmission conditions setting, as discussed in Section 2: (3.21) Minimize p2U (3.22)

Z



D

=

Z

ad

Z E

(yp ? zd )2 dx;

[a1H (p) + (1 ? H (p))]ryprw + [b1H (p) + b2(1 ? H (p))]ypw f w dx;

D

8w 2 V = H01 (D);



dx

 > 0:

The notations are standard and the existence of at least one optimal pair, denoted by [y ; p] 2 H01(D)  Uad , follows from Section 2. By the de nition of Uad , we get !1 p^ 2 U strongly in H 1 (D), H (p ) ! H (^p) strongly in Ls (D), for all that p ?! ad  s  1, on a subsequence. Since the coecients are majorized from below by a strictly positive constant, the bilinear form in (3.22) is coercive uniformly with respect to  ! 1. Then, both sequences fy g and fyp g are bounded in H01 (D ) and, on a subsequence, we may assume that y ! y^ and yp ! yp weakly in H01 (D). Remark. 5) These estimates remain valid if b1 = 0, b2 = 0.

However, it is not possible to pass directly to the limit in (3.22) due to the terms containing  ! 1. By dividing (3.22) by  ! 1, we get (3.23) (3.24)

Z

(1 ? H (p ))ry rw +

D

Z

D

Z

(1 ? H (p))ryprw +

D Z

(1 ? H (p ))yw ! 0;

D

(1 ? H (p))ypw ! 0; 8w 2 H01 (D):

The above mentioned convergences allow to pass to the limit in (3.23), (3.24) and to obtain: (3.25) (3.26)

Z D

Z D

(1 ? H (^p))ry^rw +

(1 ? H (p))ryprw +

Z

Z

D

D

(1 ? H (^p))^yw = 0;

(1 ? H (p))ypw = 0; 8w 2 H01(D):

If we denote = supp(^p)+ , p = supp(p)+, relations (3.25), (3.26) show that = 0 in D n , yp = 0 in D n p , that is y^ 2 H01 ( ), yp 2 H01 ( p ) by the trace theorem and the assumed regularity hypotheses.

y^

11

Remark. 6) It is only for proving that from y^ 2 H01 (D), y^ = 0 a.e. D n it yields y^ 2 H01 ( ), that we need the strengthened variant (2.3)' of the assumption (2.3), in the case of homogeneous Dirichlet boundary value problems. Otherwise (2.3) and Uad  C (D ) would have been sucient. The diculty and the importance of this point is clearly described by Henrot [8].

When p = p , we choose in (3.22) w = H"(p )v with v 2 C01 (D). Since H" de ned by (1.3) is Lipschitzian and p 2 H 1 (D), it is known that w 2 H01(D), therefore this choice is possible. Moreover, we have: (1 ? H (p ))H"(p ) = 0 a.e.

(3.27)

D;

by (1.1), (1.3). Then, the following relation is obtained by a simple calculation: (3.28)

Z

a1 H (p )[H"0 (p ) p D Z

r
ry v + H"(p )ry
rv]

+

D

b1 H (p )H" (p )y v

=

Z

D

f H" (p )v;

8v 2 C01 (D):

For the moment, " > 0 is a xed parameter and we shall take  ! 1 in (3.28) since !1 H (^p) the terms containing  itself disappeared by (3.27). Obviously, H"(p ) ?! " strongly in Ls (D), for all s  1. If "   (see (2.4)), the usual argument, Brezis [4, p. 195], gives that the sets fx 2 D : p^(x) = "g, fx 2 D : p^(x) = 0g have zero measure. We also notice that, for any p 2 R: H"0 (p)

(3.29)

8 >
";

= > 1" : 0

0 < p < "; p < 0:

When p^(x) 2 (?1; 0) [ ("; +1), the pointwise convergence yields that p (x) 2 (?1; 0) [ ("; +1) for  > x and a.e. x 2 D with this property. Therefore H"0 (p (x)) = H"0 (^ p(x)) = 0, for  > x . Similarly, if p^(x) 2 (0; "), then p (x) 2 (0; ") for  > x and H"0 (px (x)) = H"0 (^p(x)) = 1" for  > x . These two properties are valid a.e. x 2 D, that is H"0 (p ) ! H"0 (^p) a.e. x 2 D. Recall that, in fact, fp g is bounded in H 1+ (D) and, by the Sobolev embedding theorem, frp g is bounded in some LtN (D)N , tN > 2, for any N , the dimension of D. Since fH (p )g, fH"0 (p )g, v are bounded in L1 (D), it follows that the rst product in (3.28) is bounded in LrN (D), rN > 1, and it is weakly convergent, on a subsequence, to some limit function q 2 LrN (D). Due to the strong convergence of p in H 1(D) and to the above mentioned pointwise convergences, the Egorov theorem allows to identify (3.30)

q

= a1H (^p)H"0 (^p)rp^
ry^v a.e. 12

D;

as in the proof of Corollary 2.3. Clearly, (3.28), (3.30) yield (3.31)

Z

a1 H (^ p)[H"0 (^ p) p^ y^v + H" (^ p) y^ v ] D Z

rr

rr

+

D

b1 H (^ p)H" (^ p)^ yv

=

Z D

f H" (^ p)v;

8v 2 C01 (D):

To take " ! 0 in (3.30), we assume again that Uad  C (D). This is, for instance, valid by (2.3)' and the Sobolev theorem if N  3 and  > 21 . It follows that (and

p , for all p 2 Uad) have nonvoid interiors and we may take v 2 C01 ( ), in (3.30). Since p^ is continuous, the set S" = fx 2 : 0 < p^(x) < "g is an open neighbourhood of @ in and S" \ supp v = ? for 0 < " < "v = x2min p^(x) since supp v  is supp v compact. Then, (3.29) shows that, for any v 2 C01 ( ) and " < "v , relation (3.31) becomes (3.31)'

Z



Z

rr

a1 H (^ p)H" (^ p) y^ v +



b1 H (^ p)H" (^ p)^ yv

=

Z

D

f H" (^ p)v:

In (3.31)', we use H"(^p) ! H (^p) strongly in Ls(D), for all s  1 and H 2(^p) = H (^p) = 1 a.e. in , to infer Z

(3.32)



rr

a1 y^ v +

Z



b1 y^v

=

Z



f v;

8v 2 H01 ( )

by the density of C01 ( ) in H01 ( ). Then, (3.32) and (3.25) give that y^ is the solution of the Dirichlet problem in the \limit" set . Finally, for any xed p 2 Uad, the above technique allows as well to take  ! 1 in (3.22) and to infer (3.33)

Z

p

r r

a 1 yp v +

Z

p

b1 yp v

=

Z

p

f v;

8v 2 H01 ( p )

(under the above condition that Uad  C (D)). Therefore, yp is the solution of the Dirichlet problem in p since yp 2 H01 ( p ) by (3.26). One can clearly pass to the limit in the corresponding inequality (3.2) (associated with (3.21)) and we have:

Theorem 3.2. Under the above hypotheses, there is at least one solution for the shape optimization problem (3.33), (3.21) in the class of all open sets p , p 2 Uad . Remarks. 7) We emphasize that throughout this paper conditions (2.3) or (2.3)' and (2.4) are mainly needed to ensure pointwise convergence a.e. in D and the fact that certain level sets (for mappings p 2 Uad ) have zero measure. From this point of view, our approach is very exible and one may nd other classes of \control" mappings, instead of Uad , having such properties and, consequently, generating other families of subsets of D and relaxing the regularity assumptions.

13

References 1. R. Adams, \Sobolev spaces," Academic Press, New York, 1975. 2. M. Bendse, \Optimization of structural topology, shape and material," Springer, New York, 1995. 3. M. Bendse and H. C. Rodriguez, Integrated topology and boundary shape optimization of 2-D solids, Comp. Meth. Appl. Mech. Engrg. 87 (1991), 15{34. 4. H. Brezis, \Analyse fonctionnelle. Theorie et applications," Masson, Paris, 1983. 5. D. Chenais, On the existence of a solution in a domain identi cation problem, J. Math. Anal. Appl. 52 (1975), 189{219. 6. F. H. Clarke, \Optimization and nonsmooth analysis," J. Wiley & Sons, New York, 1983. 7. G. Dal Maso, \An introduction to ?-convergence," Birkhauser, Boston, 1993. 8. A. Henrot, Continuity with respect to the domain for the Laplacian: a survey, Control and Cybernetics 23 (1994), No. 3, 427{443. 9. E. Hewitt and K. Stromberg, \Real and abstract analysis," Springer, Berlin, 1965. 10. H. Kawarada, \Numerical solution of a free boundary problem for an ideal uid," Lecture Notes in Phys. 81, Springer, Berlin, 1977. 11. R. Makinen, P. Neittaanmaki and D. Tiba, On a xed domain approach for a shape optimization problem, in \Computational and Applied Mathematics II: Dierential Equations," W. F. Ames and P. J. van der Houwen (eds.), North-Holland, Amsterdam, 1992, pp. 317{326. 12. V. G. Maz'ja, \Sobolev spaces," Springer, Berlin, 1985. 13. F. Murat and J. Simon, \Studies in optimal shape design," Lecture Notes in Comput. Sci. 41, Springer, Berlin, 1976. 14. O. Pironneau, \Optimal shape design for elliptic systems," Springer, Berlin, 1984. 15. U. Raitums, \Lecture notes on G-convergence, convexi cation and optimal control problems for elliptic equations," Lecture Notes 39, Univ. of Jyvaskyla, Dept. of Mathematics, Jyvaskyla, 1997. 16. V. Sverak, On optimal shape design, C. R. Acad. Sci. Paris Ser. I Math. 315 (1992), No. 5, 545{549. 17. V. Sverak, On optimal shape design, J. Math. Pures Appl. 72 (1993), No. 6, 537{551. 18. D. Tiba, Une approche par contr^olabilite frontiere dans les problemes de design optimal, C. R. Acad. Sci. Paris Ser. I Math. 310 (1990), No. 4, 175{177. 19. D. Tiba and P. Neittaanmaki, An embedding of domains approach in free boundary problems and optimal design, SIAM J. Control Optimiz. 33 (1995), No. 5, 1587{1602.

14