On the optimal shape of a rigid body supported by an elastic membrane

Nonlinear Analysis 39 (2000) 47 – 63

www.elsevier.nl/locate/na

On the optimal shape of a rigid body supported by an elastic membrane Giuseppe Buttazzoa , Alfred Wagner b; ∗ a Dipartimento b Institut

di Matematica, Universita di Pisa, Via Buonarroti, 2, 56127 Pisa, Italy fur Mathematik, Universitat Koln, Weyertal 86-90, 50931 Koln, Germany Received 6 December 1996; accepted 11 December 1997

1. Introduction Given an elastic membrane which supports a rigid body, under the action of the gravity force eld the elastic deformation of the membrane is such that the sum of its elastic energy and of the potential energy of the body attains its minimum. This minimum value depends, of course, on the shape of the rigid body, as well as on its position over the membrane. We study the problem of nding the rigid body which makes the total energy minimal. We denote by the reference con guration of the membrane and we assume that its vertical displacement u is zero on the boundary @ ; the elastic energy associated to u is then given by the Dirichlet integral Z 1 |∇u(x)|2 dx (u ∈ H01 ( )): 2

The presence of a rigid body over the membrane contributes to the total energy by the potential energy term mg · b where m is the mass of the body, g is the gravity acceleration, and b is the body barycenter. After the description of the rigid body by analytical parameters, we transform the search of the optimal body into a variational minimization problem which is attacked by the direct methods of the calculus of variations. Under geometrical extra assumptions (for instance if we assume that the body is a ball) the optimization problem for the position of the body over the membrane was already considered by Bemelmans and Chipot [1] and by Elliott and Friedman [2].

∗

Corresponding author. E-mail: [email protected]

0362-546X/99/$ - see front matter ? 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 1 6 3 - 1

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G. Buttazzo, A. Wagner / Nonlinear Analysis 39 (2000) 47 – 63

In the present paper, if no additional constraints on the rigid body are imposed, we show that the problem is trivial in the sense that below a gravity threshold the best body is the empty one, while above the threshold the optimal body does not exist, being the in mum of the energy equal to −∞. Therefore, we consider the optimization problem in the following restricted classes of rigid bodies: • the class of all bodies whose height is constrained to be less than or equal to a given M ; • the class of all bodies whose volume is prescribed. We show that in these cases an optimal solution exists.

2. The mathematical model Let be a bounded open subset of Rn (n = 2 in the physical case) which describes an elastic membrane in its reference con guration; the vertical displacement will be described by a function u(x) which is taken in the Sobolev space H01 ( ) and the energy due to the elastic deformation will be given by the Dirichlet integral Eelastic =

1 2

Z

|∇u(x)|2 dx;

where we have normalised the elasticity coecient of the membrane to one. Let ! ⊂ be a measurable set which is xed for the moment. It describes the vertical projection of the rigid body on . The body B is described as a graph: B := {(x; y) ∈ Rn+1 : x ∈ !; ’(x) ≤ y ≤ c}; where ’ : ! → R is a measurable function and c ∈ R is some unknown constant. The volume of the body is given by Z V=

!

(c − ’(x)) dx

and its barycenter b = (b1 ; : : : ; bn+1 ) has the vertical component 1 bn+1 = − 2V

Z !

(c2 − ’2 (x)) dx:

Therefore, if g = (0; : : : ; 0; −g) denotes the (constant) gravity acceleration,  the (constant) density of the body, m its mass, and k = g, then the potential energy of the body is Ebody = mg · b = kVbn+1 =

k 2

Z !

(c2 − ’2 (x)) dx:

G. Buttazzo, A. Wagner / Nonlinear Analysis 39 (2000) 47 – 63

49

Fig. 1. A membrane supporting a rigid body.

We are interested in nding the shape of the body which minimizes the total energy. Thus, we have to consider the minimization problem for the functional Z Z (2.1) E = |∇u(x)|2 dx + k (c2 − ’2 (x)) dx;

!

H01 ( ),

’ varies among all measurable functions on !, and c varies where u varies in in R, with the constraints u ≤ 0, ’ ≤ c. In addition, we require that the membrane must be below the body, that is u ≤ ’ on ! in the sense of obstacles (see Fig. 1). This means, that u ≤ ’ on ! q.e. in capacity, that is cap{x ∈ !: u(x)¿’(x)} = 0; where cap denotes the Newton capacity. In the following, inequalities and equalities between functions, sup and inf , will be always intended up to sets of capacity zero. We want to point out that, without imposing further constraints on the body, the problem is trivial. It is easy to see that in this case the optimal choice for c is c = 0 and for ’ is ’ = u, so that the minimum problem reduces to  Z Z |∇u(x)|2 dx − k u2 (x) dx: u ∈ H01 ( ); u(x) ≤ 0 : (2.2) min

!

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G. Buttazzo, A. Wagner / Nonlinear Analysis 39 (2000) 47 – 63

Setting

Z

k = min



|∇u(x)|2 dx: u ∈ H01 ( );

Z

 |u(x)|2 dx = 1 ;

we then have the following classi cation: • if k¡k the only solution of Eq. (2.2) is u = 0; in other words, the optimal body is the empty body, corresponding to ’ = c = 0; • if k¿k the in mum of Eq. (2.2) is −∞; in other words, the optimal body does not exist; • if k = k the function u = 0 (corresponding to the empty body ’ = c = 0) is still a solution of Eq. (2.2), together with all negative solutions of the elliptic equation −
u(x) − k! (x)u(x) = 0 in ; u ∈ H01 ( ); where ! denotes the characteristic function of !. 3. The case of constrained height In this section we consider the shape optimization problem associated to energy (2.1) for bodies whose height is constrained by a positive constant M . This means, that in Eq. (2.1) we add the constraint c − ’ ≤ M on !, i.e. c ≤ M + inf ’:

(3.1)

!

Clearly, if a portion of the body is above zero level, this would give a positive contribution to the potential energy, so that removing it would be more performant for the minimization problem. In other words, we may limit ourselves to consider only functions ’ ≤ 0 and constraints c ≤ 0, so that Eq. (3.1) becomes −  (3.2) c ≤ − M + inf ’ : !

By expressions (2.1) and (3.2) we obtain, that for xed u and ’, the best choice for c is c = −(M + inf ! ’)− . This reduces the minimum problem to ! ) (Z  Z − 2 2 2 1 M + inf ’ − ’ (x) dx: u ∈ H0 ( ) |∇u(x)| dx + k (3.3) min

!

!

with the constraints

−  u(x) ≤ ’(x) ≤ − M + inf ’ !

on !:

(3.4)

For a xed function ’, if we take u ∨ inf ! ’ instead of u, we obtain a lower value for the energy; therefore Eq. (3.4) becomes  − on !: (3.5) inf ’ ≤ u(x) ≤ ’(x) ≤ − M + inf ’ !

!

G. Buttazzo, A. Wagner / Nonlinear Analysis 39 (2000) 47 – 63

51

By Eq. (3.5) we have inf ! ’ = inf ! u; moreover it is easy to see, that the choice ’ = u is compatible with constraints (3.5) and gives the minimum of the energy (3.3) for u xed. Then the minimum problem (3.3) becomes ! ) (Z  Z − 2 M + inf u − u2 (x) dx: u ∈ H01 ( ) |∇u(x)|2 dx + k (3.6) min !

!

with the constraint −  u(x) ≤ − M + inf u !

on !:

(3.7)

It must be noticed that in Eq. (3.6) we may substitute inf ! u by inf u; indeed if u ∈ H01 ( ) with u ≤ 0, the function v = u ∨ inf u !

satis es inf ! v = inf v and provides a lower energy. The problem of nding the best rigid body supported by a membrane, with height less than or equal to M and projection ! then reduces to the minimum problem     − 2 Z   Z  M + inf u − u2 (x) dx: u ∈ H01 ( ) |∇u(x)|2 dx + k (3.8) min  

! with constraint −  u ≤ − M + inf u

on !:

(3.9)

We shall prove, that problem (3.8) with the constraint (3.9) admits a solution by applying the direct methods of the calculus of variations. The coercivity follows easily by noticing that if M + inf u ≥ 0 we have  − 2 M + inf u − u2 (x) = − u2 (x) ≥ −M 2 :

On the other hand, if M + inf u ¡ 0 we have by Eq. (3.7)  − 2  2  2 M + inf u − u(x)2 ≥ M + inf u − inf u



= M 2 + 2M inf u

≥ −M 2 + 2Mu(x);

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G. Buttazzo, A. Wagner / Nonlinear Analysis 39 (2000) 47 – 63

where the last inequality follows since inf u ≥ u − M . Therefore,   − 2 Z Z   M + inf u − u2 (x) |∇u(x)|2 dx + k



! Z Z |∇u(x)|2 dx + k (2Mu(x) − M 2 ) dx ≥

!

and the coercivity of the right-hand side follows immediately by Poincare inequality. The lower semicontinuity with respect to the weak convergence follows directly from the following lemma. Lemma 3.1. If uh → u weakly in H01 ( ); then   inf u≥ lim sup inf uh :



h

Proof. Passing to a subsequence we may assume, that the lim sup is a limit. By contradiction assume, that there exists an ¿0 such that    + inf u¡ lim inf uh ;

h



that is  + inf u≤ inf uh



for h large is enough. This would imply uh (x) ≥  + inf u

on

for h is large enough, so that, as h → ∞ u(x) ≥  + inf u

on

which is a contradiction. Summarizing, we have obtained the following result. Theorem 3.2. Among all membranes u ∈ H01 ( ) satisfying u ≤ 0 on and u ≤ −(M + in f u)− on a given ! there exists one which minimizes the total energy     − 2 Z   Z  M + inf u − u2 (x) dx : |∇u(x)|2 dx + k min  

! The optimal body Bopt is then described by the function ’ = u and the constant c = −(M + inf u)− ; that is (  − ) : Bopt = (x; y) ∈ Rn+1 : x ∈ !; u(x) ≤ y ≤ − M + inf u

G. Buttazzo, A. Wagner / Nonlinear Analysis 39 (2000) 47 – 63

53

A particular case of a solution is shown in Fig. 2 below. We consider now the case in which also ! may vary among all measurable subsets of a given subset E of . Therefore, we deal with the minimization problem    − 2 Z  Z  M + inf u − u2 (x) dx: |∇u(x)|2 dx + k min 

!   (3.10) u ∈ H01 ( ); ! ⊂ E ;  with the constraint u ≤ − (M + inf u)− on !. Clearly, the best choice for !, when u is xed, is    − 2   (3.11) ! = x ∈ E : M + inf u ≤ u2 (x) ;  

Fig. 2. The case M = 2, k = 3:5, ! = [−0:5; 0:5], = (−1:1).

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G. Buttazzo, A. Wagner / Nonlinear Analysis 39 (2000) 47 – 63

so that problem (3.10) can be written as   −  − 2 Z   Z |∇u(x)|2 dx − k  M + inf u − u2 (x) dx: u ∈ H01 ( ) : min  



E (3.12) Again we prove the existence of a solution of Eq. (3.12) by using the direct method of the calculus of variations. The lower semicontinuity with respect to the weak convergence in H01 ( ) follows easily from Lemma 3.1. In order to prove the coercivity we distinguish three cases. (i) If M + inf u ≥ 0, then −  − 2  2   M + inf u − u2 (x) = u2 (x) ≤ inf u ≤ M 2 ;



(ii) if M + inf u¡0 and M + inf u ≤ u(x), then −  − 2   M + inf u − u2 (x) = 0;

(iii) if M + inf u¡0 and M + inf u¿u(x), then −  − 2  2   M + inf u − u2 (x) = u2 (x) − M + inf u



 2  2 ≤ inf u − M + inf u



2

≤ M − 2Mu(x): Then we have Z

−  − 2  |∇u(x)|2 dx − k  M + inf u − u2 (x) dx



E Z Z |∇u(x)|2 dx − k (M 2 − 2Mu(x)) dx ≥ Z



E

and the coercivity of the right-hand side follows by Poincare inequality. Thus, we have proved. Theorem 3.3. Among all membranes u ∈ H01 ( ) satisfying u ≤ 0 on there exists one which minimizes the total potential energy −  − 2 Z Z  |∇u(x)|2 dx − k  M + inf u − u2 (x) dx:



E

G. Buttazzo, A. Wagner / Nonlinear Analysis 39 (2000) 47 – 63

55

The optimal body Bopt is then described by the function ’ = u and the constant c = −(M + inf u)− ; that is (  − ) n+1 : Bopt = (x; y) ∈ R : x ∈ E; u(x) ≤ y ≤ − M + inf u

4. The case of prescribed volume In this section we consider the optimization problem associated to energy (2.1) for bodies whose volume is prescribed, i.e. Z (c − ’(x)) dx = V (4.1) !

or equivalently c=

V + ’; |!|

where ’ =

1 |!|

Z !

’(x) dx:

We assume ! is xed and ’(x) ≤ c on !. Note that the sign of c is not a priori known. Our problem then reduces to nd a minimizer for the energy 2 Z  Z V + ’ − ’2 (x) dx E(u; ’) = |∇u|2 dx + k |!|

! under the condition u ≤ ’ ≤ V=|!| + ’ on ! and u ≤ 0 on . We introduce the notation 2 Z  V + ’ − ’2 (x) dx: J (’) := k |!| ! for the potential energy of the body. Lemma 4.1. Let (u; ’) be an admissible pair. Then there exists a set !0 ⊂ ! such that the function  u: on !;0 ’ˆ = c: on !\!0 is still admissible and the pair (u; ’) ˆ gives a lower energy. Proof. After having xed u and ’; the problem reduces to the maximization of Z 2 (x) dx !

under the constraints Z (x) dx = |!|’: u ≤ ≤ c; !

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G. Buttazzo, A. Wagner / Nonlinear Analysis 39 (2000) 47 – 63

The conclusion then follows from the well-known fact that in this case the optimal coincides either with u or with c. The functional J now takes the form 2 Z  u V  − u2 dx; + J () = k  |!| ! where  denotes the characteristic function of !0 . We want to describe the position of !0 in terms of u. In other words, since  describes the characteristic function of !0 we like to nd further necessary conditions on the minimizer which allow us to replace  by some expression in u. We de ne the following rearrangement: {x ∈ !:  = 1}∗ := {x ∈ !: u(x) ≤ }; where  is implicitely given by Z Z V 1 1 V u(x) dx = 0 + 0 u(x) dx: + |{u ≤ }| |{u ≤ }| {u≤} |! | |! | !0 We denote by ∗ (x) the characteristic function of (!0 )∗ = {u ≤ }. Writing J in the form J () =

k |!0 |

Z !

2 u dx + V

Z −k

!

u2 dx;

(4.2)

we will prove Lemma 4.2. J decreases under rearrangement. Proof. The volume constraint gives Z Z (x)u(x) dx − ∗ (x)u(x) dx = c(|!0 | − |(!0 )∗ |): !

(4.3)

!

We may suppose that !0 and (!0 )∗ are disjoint because otherwise we reduce the computations on the symmetric dierence !0 4(!0 )∗ . Since u ≥  on !0 and u ≤  on (!0 )∗ Eq. (4.3) gives (|!0 | − |(!0 )∗ |) = c(|!0 | − |(!0 )∗ |); R R which implies |!0 | ≥ |(!0 )∗ | and ! (x)u(x) dx ≥ ! ∗ (x)u(x) dx: Comparing the rst term in Eq. (4.2) with its rearrangement gives k |!0 |

Z !

2 u dx + V

k = k|! |c ≥ k|(! ) |c = |(!0 )∗ | 0

2

0 ∗

2

Z !

∗

 u dx + V

2 :

G. Buttazzo, A. Wagner / Nonlinear Analysis 39 (2000) 47 – 63

57

Concerning the second term in Eq. (4.2) we obtain Z Z 2 (x)u (x) dx ≥ −k (x)u(x) dx −k ! ! Z Z ∗ (x)u2 (x) dx ≥ −k ∗ (x)u(x) dx ≥ −k !

!

which concludes the proof. As a consequence, we nd an expression for  only depending on u and : (x) =

[u − ]− ; |u − |

!0 = {x ∈ !: u(x) ≤ }:

As a nal result we write J as a functional on the parameter  for a given u: Z 2 Z [u − ]− [u − ]− 2 k u dx + V − k u dx; J () = |{u ≤ }| ! |u − | ! |u − | so that we may reformulate our problem as the minimization problem in (u; ) for the energy Z 2 Z 1 [u − ]− u dx + V E(u; ) = |∇u|2 dx + k |{u ≤ }|

! |u − | Z [u − ]− 2 u dx −k ! |u − | among all u ∈ H01 ( ), u ≤ 0 and  ≤ 0 which satisfy the constraint Z 1 V + u dx on !: u(x) ≤ |{u ≤ }| |{u ≤ }| {u≤} To prove the existence of a minimizer we have to prove coercivity and lower semicontinuity of the functional E. Lemma 4.3. We have E(u; ) → ∞

as kukH01 ( ) + || → ∞:

Proof. Taking into account that !0 = {u ≤ } we have !2 Z Z Z 1 2 u dx + V − k u2 dx E(u; ) = |∇u| dx + k 0 |! |

{u≤} {u≤} !2 Z Z 1 2 0 (u − ) dx + |! | + V = |∇u| dx + k 0 |! |

{u≤} Z Z −k (u − )2 dx − 2k (u − ) dx − k2 |!0 | {u≤}

{u≤}

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G. Buttazzo, A. Wagner / Nonlinear Analysis 39 (2000) 47 – 63

!2 Z Z k = |∇u| dx + 0 (u − ) dx − k (u − )2 dx |! |

{u≤} {u≤} Z 2 V kV u dx: + 0 + 2k 0 |! | |! | {u≤} Z

2

By using the interpolation inequality Z {u≤}

(u − )2 dx ≤ c1

!2

Z

Z

c2 |∇u|2 dx +  {u≤}

{u≤}

(u − ) dx

;

where c1 and c2 only depend on n (see e.g. [3], p. 45, Eq. (2.10)), we obtain !2  Z  Z c2 k 2 −k (u − ) dx E(u; ) ≥ (1 − kc1 ) |∇u| dx + |!0 | 

{u≤} Z V 2k V + 0 + 2k 0 u dx: |! | |! | {u≤} Choosing  = 1=4kc1 and c3 = 4c1 c2 we have 3 E(u; ) ≥ 4

Z



2

|∇u| dx +

V kV 2 + 0 + 2k 0 |! | |! |

k − k 2 c3 |!0 |



Z {u≤}

!2

Z {u≤}

(u − ) dx

u dx:

below the level  we may use the volume constraint RSince the body lls the membrane 0 u dx = −V + |! |, where  ≥  denotes the top of the rearranged body. In {u≤} particular, this implies Z u dx ≥ −V + |!0 | {u≤}

and

!2

Z −

{u≤}

u −  dx

≥ −V 2 :

We then have 3 E(u; ) ≥ 4

Z

k |∇u|2 dx + 0 |! |

−k 2 c3 V 2 +

!2

Z {u≤}

V kV 2 + 2k 0 0 |! | |! |

(u − ) dx

Z

{u≤}

u dx:

G. Buttazzo, A. Wagner / Nonlinear Analysis 39 (2000) 47 – 63

59

We estimate the last integral as 2k

V |!0 |

Z {u≤}

u dx ≥ −2kVc4 |!0 |(2−n)=2n

Z

|∇u|2 dx Z

1 4

≥ −4c42 V 2 k 2 |!0 |(2−n)=n −



1=2

|∇u|2 dx;

where c4 = c4 (n) is the constant of the imbedding H01 ( ) ,→ L2n=(n−2) ( ). This gives the estimate Z kV 2 1 |∇u|2 dx − k 2 c3 V 2 + 0 − 4c42 V 2 k 2 |!0 |(2−n)=n : E(u; ) ≥ 2

|! | Thus, we have E(u; ) → ∞

as k u kH01 ( ) +|| → ∞:

since |!0 | → 0 as || → ∞. Since functions in H01 ( ) are not necessarily bounded, the existence proof does not exclude the possibility of an in nitely deep membrane as a minimizer. However, we will show that this cannot happen. Roughly speaking, we demonstrate that there exists a level  such that if the membrane u is below that level on a set !0 the total energy is lower if we replace u by  on !0 . It is easy to verify that the constraints are uneected by this modi cation. Thus, for a given (u; ) we compare the energy E(u; ) with E(u ; ), where ( u if u(x) ≥ ; u :=  if u(x) ≤ : Inserting this de nition we obtain for E(u ; ) the expression !2 Z k 0 |∇u| dx + 0 u dx + |! | + V E(u ; ) = |! |

\!0 !0 \!0 Z −k u2 dx − k  2 |!0 |: Z

2

!0 \!0

Thus, for the dierence E(u; ) − E(u ; ) we have Z E(u; ) − E(u ; ) =

!0

|∇u|2 dx +

k − 0 |! |

Z !0 \!0

k |!0 |

Z !0

2 u dx + V !2

u dx+|!0 |

+V

Z −k

!0

Z +k

!0 \!0

u2 dx

u2 dx+k  2 |!0 |

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G. Buttazzo, A. Wagner / Nonlinear Analysis 39 (2000) 47 – 63

Z

k = |∇u| dx + 0 |! | 0 ! 2

Z !2

u dx + V

!0 \!0

|!0 |2 −k 2 0 |! |

u dx + V

!0

Z

k − 0 |! |

2

+ k

2

|!0 |

|!0 | − 2k 0 |! |

Z +k

2

!0 \!0

!

Z !0 \!0

Z

u dx − k

!0

u dx + V

u2 dx:

Since |!0 | → 0 as  → −∞ we note that the term ! Z |!0 |2 |!0 | u dx + V − k 2 0 −2k  0 |! | |! | !0 \!0 tends to zero to a higher order with respect to  than the term k  2 |!0 |. Concerning the term k |!0 |

Z

2

!0

u dx + V

!0 \!0

we write it as k |!0 |

Z 2V +

!0

which is of the order of Z

Z !0

|u| dx ≤

!0

u dx

!0 \!0

R

u dx + V

;

!Z

Z u dx +

!2

Z

k − 0 |! |

!0

u dx;

u dx as  → −∞. This last term can be controlled as

!0

|u − | dx + |||!0 | 2n=(n−2)

!0

≤ c|!0 |(n+2)=2n ≤ c|!0 |(n+2)=n

!(n−2)=2n

Z

≤ |!0 |(n+2)=2n

|u − |

+ |||!0 |

!1=2

Z

1 + 4

dx

2

!0

|∇u| dx

Z

!0

+ |||!0 |

|∇u|2 dx + |||!0 |:

As an intermediate estimate we obtain Z Z 3 |∇u|2 dx − k |u|2 dx; E(u; ) − E(u ; ) ≥ 4 !0 !0 where we used that the term k  2 |!0 | dominates the remaining terms as  →−∞.

G. Buttazzo, A. Wagner / Nonlinear Analysis 39 (2000) 47 – 63

61

Thus, we are done if we can prove, that (for suciently negative ) Z Z |∇u|2 dx − k u2 dx¿0: !0

!0

We proceed as before, estimating Z

Z

!2

Z

c u dx ≤  |∇u| dx +  !0 !0 2

2

!0

u −  dx

−

2

|!0 |2

+ 2

Thus, Z

c |∇u| dx − k E(u; ) − E(u ; ) ≥ (1 − k)  0 ! ! Z −2k 

!0

2

!0

u dx :

!2

Z !0

!

Z

u −  dx

u −  dx :

R Using now the fact that  ≥ u on !0 and  ≥ 1=!0 !0 u dx gives the estimate  Z Z Z u −  dx ≥ −2kc2 |!0 |2=n |∇u|2 dx |∇u|2 dx: −2k  !0



!0

Similarly, we obtain ! Z Z c2 c u −  dx ≥ −k |!0 |(n+2)=2n |∇u|2 dx: k   0 0 ! ! Choosing now  = 2=k gives the nal estimate   Z 2 1 2 0 2=n 2 2c 0 (n+2)=2n − 2kc |! | |! | |∇u| dx − k E(u; ) − E(u ; ) ≥ 2 2 

Z |∇u|2 dx: !0

Since |!0 | tends to zero as  → −∞ the right-hand side of the inequality becomes positive for suciently large values of . We summarize our results. Theorem 4.4. Among all membranes u=H01 ( ) satisfying u ≤ 0 on and among all parameters  ≤ 0 there exists a unique minimizer (u; ) which minimizes the total energy Z 2 Z [u − ]− 1 u dx + V E(u; ) = |∇u|2 dx + k |{u ≤ }|

! |u − | Z [u − ]− 2 u dx: −k ! |u − |

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G. Buttazzo, A. Wagner / Nonlinear Analysis 39 (2000) 47 – 63

Fig. 3. The case V = 0:5; k = 9; ! = [−0:5; 0:5]; = (−1:1).

Moreover, the function u is bounded, and the optimal body is given by Bopt = {(x; y) ∈ Rn+1 : x ∈ !; u + (1 − )c ≤ y ≤ c}; where [u − ]− ; = |u − |

1 c= |{u ≤ }|

Z V+

{u≤}

! u(x) dx :

A particular case of a solution with volume constraint is shown in Fig. 3 below.

Acknowledgements This work has been supported by the VIGONI scienti c cooperation program between the Universities of Koln and Pisa, by the MAP (“Matematica A Pisa”) scienti c program, and by the EEC scienti c program “Phase Transition Problems and Singular Perturbations”, contract CHRX-CT94-0608. The authors wish to thank the Departments of Mathematics of the Universities of Koln and Pisa for the warm hospitality during their visits.

G. Buttazzo, A. Wagner / Nonlinear Analysis 39 (2000) 47 – 63

63

References [1] J. Bemelmans, M. Chipot, On a variational problem for an elastic membrane supporting a heavy ball, Calc. Var. Partial Dierential Equation 3 (4) (1995) 447– 473. [2] C.M. Elliott, A. Friedman, The contact set of a rigid body partially supported by a membrane, Nonlinear Anal. Theory Meth. Appl. 10 (1986) 251–276. [3] O.A. Ladyzhenskaya, N.N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. [4] W.P. Ziemer, Weakly Dierentiable Functions, Springer, Berlin, 1989.