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Optimal location of Dirichlet

regions for elliptic PDEs

Giuseppe Buttazzo

Dipartimento di Matematica

Universita di Pisa

buttazzo@dm.unipi.it

http://cvgmt.sns.it

Transport, optimization, equilibrium in economicsPIMS, July 1420, 2008

We want to study shape optimization prob-

lems of the form

min{F (, u) : A

}where F is a suitable shape functional and Ais a class of admissible choices. The function

u is the solution of an elliptic problem

Lu = f in u = 0 on

or more generally of a variational problem

min{G(u) : u = 0 on

}.

1

The cases we consider are when

G(u) =

(|Du|pp

f(x)u)

dx

corresponding to the p-Laplace equation{div

(|Du|p2Du

)= f in

u = 0 on

and the similar problem for p = + with

G(u) =

({|Du|1} f(x)u

)dx

2

which corresponds to the Monge-Kantorovich

equation

div(Du) = f in \u = 0 on u Lip1|Du| = 1 on spt() = 0.

We limit the presentation to the cases

p = + and p = 2

occurring in mass transportation theory and

in the equilibrium of elastic structures.

3

The case of mass transportation problems

We consider a given compact set Rd

(urban region) and a probability measure f

on (population distribution). We want to

find in an admissible class and to transport

f on in an optimal way.

It is known that the problem is governed by

the Monge-Kantorovich functional

G(u) =

({|Du|1} f(x)u

)dx

4

which provides the shape cost

F () =

dist(x,) df(x).

Note that in this case the shape cost doesnot depend on the state variable u.

Concerning the class of admissible controlswe consider the following cases:

A ={ : # n

}called location prob-

lem;

A ={ : connected, H1() L

}called irrigation problem.

5

Asymptotic analysis of sequences Fnthe -convergence protocol

1. order of vanishing n of minFn;

2. rescaling: Gn = 1n Fn;

3. identification of G = -limit of Gn;

4. computation of the minimizers of G.

6

The location problem

We call optimal location problem the mini-mization problem

Ln = min{F () : , # n

}.

It has been extensively studied, see for in-stance

Suzuki, Asami, Okabe: Math. Program. 1991Suzuki, Drezner: Location Science 1996Buttazzo, Oudet, Stepanov: Birkhauser 2002Bouchitte, Jimenez, Rajesh: CRAS 2002Morgan, Bolton: Amer. Math. Monthly 2002. . . . . . . . .

7

Optimal locations of 5 and 6 points in a disk for f = 1

8

We recall here the main known facts.

Ln n1/d as n +; n1/dFn Cd

1/df(x) dx as n +, inthe sense of -convergence, where the limitfunctional is defined on probability measures;

opt = Kdfd/(1+d) hence the optimal con-figurations n are asymptotically distributedin as fd/(1+d) and not as f (for instanceas f2/3 in dimension two).

in dimension two the optimal configurationapproaches the one given by the centers ofregular exagons.

9

In dimension one we have C1 = 1/4.

In dimension two we have

C2 =E|x| dx =

3 log3 + 4

6

233/4 0.377

where E is the regular hexagon of unit area

centered at the origin.

If d 3 the value of Cd is not known.

If d 3 the optimal asymptotical configu-ration of the points is not known.

The numerical computation of optimal con-figurations is very heavy.

10

If the choice of location points is made ran-domly, surprisingly the loss in average with

respect to the optimum is not big and a sim-

ilar estimate holds, i.e. there exists a con-

stant Rd such that

E(F (N

) RdN1/d

1/dd

(

fd/(1+d))(1+d)/d

while

F (optN ) CdN1/d1/dd

(

fd/(1+d))(1+d)/d

11

We have Rd = (1 + 1/d) so thatC1 = 0.5 while R1 = 1C2 ' 0.669 while R2 ' 0.886

d1+d Cd (1 + 1/d) = Rd for d 3

20 40 60 80 100

0.8

0.85

0.9

0.95

Untitled-1 1

Plot of d1+d and of (1 + 1/d) in terms of d12

The irrigation problem

Taking again the cost functional

F () :=

dist(x,) f(x) dx.

we consider the minimization problem

min{F () : connected, H1() `

}Connected onedimensional subsets of are called networks.

Theorem For every ` > 0 there exists an op-timal network ` for the optimization prob-lem above.

13

Some necessary conditions of optimality on

` have been derived:

Buttazzo-Oudet-Stepanov 2002,Buttazzo-Stepanov 2003,Santambrogio-Tilli 2005Mosconi-Tilli 2005.........

For instance the following facts have been

proved:

14

no closed loops;

at most triple point junctions;

120 at triple junctions;

no triple junctions for small `;

asymptotic behavior of ` as ` +(Mosconi-Tilli JCA 2005);

regularity of ` is an open problem.15

Optimal sets of length 0.5 and 1 in a unit square with f = 1.

16

Optimal sets of length 1.5 and 2.5 in a unit square with f = 1.

17

Optimal sets of length 3 and 4 in a unit square with f = 1.

18

Optimal sets of length 1 and 2 in the unit ball of R3.

19

Optimal sets of length 3 and 4 in the unit ball of R3.

20

Analogously to what done for the location

problem (with points) we can study the asymp-

totics as ` + for the irrigation problem.This has been made by S.Mosconi and P.Tilli

who proved the following facts.

L` `1/(1d) as ` +;

`1/(d1)F` Cd

1/(1d)f(x) dx as ` +, in the sense of -convergence, wherethe limit functional is defined on probability

measures;

21

opt = Kdf(d1)/d hence the optimal con-figurations n are asymptotically distributed

in as f(d1)/d and not as f (for instanceas f1/2 in dimension two).

in dimension two the optimal configurationapproaches the one given by many parallel

segments (at the same distance) connected

by one segment.

22

Asymptotic optimal irrigation network in dimension two.

23

The case when irrigation networks are nota priori assumed connected is much moreinvolved and requires a different setting upfor the optimization problem, considering thetransportation costs for distances of the form

d(x, y) = inf{A

(H1( \)

)+ B

(

)}being the infimum on paths joining x to y.On the subject we refer to the monograph:

G. BUTTAZZO, A. PRATELLI, S. SOLI-MINI, E. STEPANOV: Optimal urban net-works via mass transportation. Springer Lec-ture Notes Math. (to appear).

24

The case of elastic compliance

The goal is to study the configurations thatprovide the minimal compliance of a struc-ture. We want to find the optimal regionwhere to clamp a structure in order to ob-tain the highest rigidity.

The class of admissible choices may be, asin the case of mass transportation, a set ofpoints or a one-dimensional connected set.

Think for instance to the problem of locat-ing in an optimal way (for the elastic com-pliance) the six legs of a table, as below.

25

An admissible configuration for the six legs.

Another admissible configuration.

26

The precise definition of the cost functional

can be given by introducing the elastic com-

pliance

C() =

f(x)u(x) dx

where is the entire elastic membrane,

the region (we are looking for) where the

membrane is fixed to zero, f is the exterior

load, and u is the vertical displacement that

solves the PDE{u = f in \u = 0 in

27

The optimization problem is then

min{C() : admissible

}where again the set of admissible configura-

tions is given by any array of a fixed number

n of balls with total volume V prescribed.

As before, the goal is to study the optimal

configurations and to make an asymptotic

analysis of the density of optimal locations.

28

Theorem. For every V > 0 there exists aconvex function gV such that the sequence offunctional (Fn)n above -converges, for theweak* topology on P(), to the functional

F () =

f2(x) gV (a) dx

where a denotes the absolutely continuouspart of .

The Euler-Lagrange equation of the limitfunctional F is very simple: is absolutelycontinuous and for a suitable constant c

gV () =c

f2(x).

29

Open problems

Exagonal tiling for f = 1?

Non-circular regions , where also the ori-entation should appear in the limit.

Computation of the limit function gV .

Quasistatic evolution, when the points areadded one by one, without modifying the

ones that are already located.

30

Optimal location of 24 small discs for the compliance,

with f = 1 and Dirichlet conditions at the boundary.

31

Optimal location of many small discs for the compliance,

with f = 1 and periodic conditions at the boundary.

32

Optimal compliance networks

We consider the problem of finding the bestlocation of a Dirichlet region for a two-dimensional membrane subjected to a givenvertical force f . The admissible belong tothe class of all closed connected subsets of with H1() L.

The existen