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Optimal Dirichlet regions for
elliptic PDEs
Giuseppe Buttazzo
Dipartimento di Matematica
Universita di Pisa
http://cvgmt.sns.it
Sixiemes Journees Franco-Chiliennes d’OptimisationToulon, 19-21 mai 2008
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We want to study shape optimization prob-
lems of the form
minF (Σ, uΣ) : Σ ∈ A
where F is a suitable shape functional and Ais a class of admissible choices. The function
uΣ is the solution of an elliptic problem
Lu = f in Ω u = 0 on Σ
or more generally of a variational problem
minG(u) : u = 0 on Σ
.
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The cases we consider are when
G(u) =∫Ω
(|Du|p
p− f(x)u
)dx
corresponding to the p-Laplace equation−div
(|Du|p−2Du
)= f in Ω
u = 0 on Σ
and the similar problem for p = +∞ with
G(u) =∫Ω
(χ|Du|≤1 − f(x)u
)dx
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which corresponds to the Monge-Kantorovich
equation
−div(µDu) = f in Ω \Σu = 0 on Σu ∈ Lip1|Du| = 1 on sptµµ(Σ) = 0.
We limit the presentation to the cases
p = +∞ and p = 2
occurring in mass transportation theory and
in the equilibrium of elastic structures.
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The case of mass transportation problems
We consider a given compact set Ω ⊂ Rd
(urban region) and a probability measure f
on Ω (population distribution). We want to
find Σ in an admissible class and to transport
f on Σ in an optimal way.
It is known that the problem is governed by
the Monge-Kantorovich functional
G(u) =∫Ω
(χ|Du|≤1 − f(x)u
)dx
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which provides the shape cost
F (Σ) =∫Ω
dist(x,Σ) df(x).
Note that in this case the shape cost doesnot depend on the state variable uΣ.
Concerning the class of admissible controlswe consider the following cases:
• A =Σ : #Σ ≤ n
called location prob-
lem;
• A =Σ : Σ connected, H1(Σ) ≤ L
called irrigation problem.
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Asymptotic analysis of sequences Fn
the Γ-convergence protocol
1. order of vanishing ωn of minFn;
2. rescaling: Gn = ω−1n Fn;
3. identification of G = Γ-limit of Gn;
4. computation of the minimizers of G.
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The location problem
We call optimal location problem the mini-mization problem
Ln = minF (Σ) : Σ ⊂ Ω, #Σ ≤ 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. . . . . . . . .
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Optimal locations of 5 and 6 points in a disk for f = 1
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We recall here the main known facts.
• Ln ≈ n−1/d as n → +∞;
• n1/dFn → Cd
∫Ω
µ−1/df(x) dx as n → +∞, in
the sense of Γ-convergence, where the limitfunctional is defined on probability measures;
• µopt = Kdfd/(1+d) hence the optimal con-
figurations Σn are asymptotically distributedin Ω 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.
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• 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.
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• 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 (ΣN
)≈ RdN
−1/dω−1/dd
( ∫Ω
fd/(1+d))(1+d)/d
while
F (ΣoptN ) ≈ CdN
−1/dω−1/dd
( ∫Ω
fd/(1+d))(1+d)/d
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We have Rd = Γ(1 + 1/d) so thatC1 = 0.5 while R1 = 1C2 ' 0.669 while R2 ' 0.886
d1+d ≤ Cd ≤ Γ(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 Γ(1 + 1/d) in terms of d
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The irrigation problem
Taking again the cost functional
F (Σ) :=∫Ω
dist(x,Σ) f(x) dx.
we consider the minimization problem
minF (Σ) : Σ connected, H1(Σ) ≤ `
Connected onedimensional subsets Σ of Ωare called networks.
Theorem For every ` > 0 there exists an op-timal network Σ` for the optimization prob-lem above.
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Some necessary conditions of optimality on
Σ` have been derived:
Buttazzo-Oudet-Stepanov 2002,Buttazzo-Stepanov 2003,Santambrogio-Tilli 2005Mosconi-Tilli 2005.........
For instance the following facts have been
proved:
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• no closed loops;
• at most triple point junctions;
• 120 at triple junctions;
• no triple junctions for small `;
• asymptotic behavior of Σ` as ` → +∞(Mosconi-Tilli JCA 2005);
• regularity of Σ` is an open problem.
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Optimal sets of length 0.5 and 1 in a unit square with f = 1.
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Optimal sets of length 1.5 and 2.5 in a unit square with f = 1.
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Optimal sets of length 3 and 4 in a unit square with f = 1.
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Optimal sets of length 1 and 2 in the unit ball of R3.
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Optimal sets of length 3 and 4 in the unit ball of R3.
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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/(1−d) as ` → +∞;
• `1/(d−1)F` → Cd
∫Ω
µ1/(1−d)f(x) dx as ` →+∞, in the sense of Γ-convergence, where
the limit functional is defined on probability
measures;
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• µopt = Kdf(d−1)/d hence the optimal con-
figurations Σn are asymptotically distributed
in Ω as f(d−1)/d and not as f (for instance
as f1/2 in dimension two).
• in dimension two the optimal configuration
approaches the one given by many parallel
segments (at the same distance) connected
by one segment.
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Asymptotic optimal irrigation network in dimension two.
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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
dΣ(x, y) = infA
(H1(θ \Σ)
)+ B
(θ ∩Σ
)being the infimum on paths θ 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).
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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.
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An admissible configuration for the six legs.
Another admissible configuration.
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The precise definition of the cost functional
can be given by introducing the elastic com-
pliance
C(Σ) =∫Ω
f(x)uΣ(x) dx
where Ω is the entire elastic membrane, Σ
the region (we are looking for) where the
membrane is fixed to zero, f is the exterior
load, and uΣ is the vertical displacement that
solves the PDE−∆u = f in Ω \Σu = 0 in Σ ∪ ∂Ω
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The optimization problem is then
minC(Σ) : Σ 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.
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Theorem. For every V > 0 there exists aconvex function gV such that the sequence offunctional (Fn)n above Γ-converges, for theweak* topology on P(Ω), to the functional
F (µ) =∫Ω
f2(x) gV (µa) dx
where µa denotes the absolutely continuouspart of µ.
The Euler-Lagrange equation of the limitfunctional F is very simple: µ is absolutelycontinuous and for a suitable constant c
g′V (µ) =c
f2(x).
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Open problems
• Exagonal tiling for f = 1?
• Non-circular regions Σ, where also the ori-
entation should appear in the limit.
• Computation of the limit function gV .
• Quasistatic evolution, when the points are
added one by one, without modifying the
ones that are already located.
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Optimal location of 24 small discs for the compliance,
with f = 1 and Dirichlet conditions at the boundary.
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Optimal location of many small discs for the compliance,
with f = 1 and periodic conditions at the boundary.
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Optimal compliance networks
We consider the problem of finding the bestlocation of a Dirichlet region Σ for a two-dimensional membrane Ω subjected to a givenvertical force f . The admissible Σ belong tothe class of all closed connected subsets ofΩ with H1(Σ) ≤ L.
The existence of an optimal configurationΣL for the optimization problem describedabove is well known; for instance it can beseen as a consequence of the Sverak com-pactness result.
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An admissible compliance network.
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As in the previous situations we are inter-
ested in the asymptotic behaviour of ΣL as
L → +∞; more precisely our goal is to obtain
the limit distribution (density of lenght per
unit area) of ΣL as a limit probability mea-
sure that minimize the Γ-limit functional of
the suitably rescaled compliances.
To do this it is convenient to associate to
every Σ the probability measure
µΣ =H1xΣ
H1(Σ)
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and to define the rescaled compliance func-
tional FL : P(Ω) → [0,+∞]
FL(µ) =
L2∫Ω
fuΣ dx if µ = µΣ, H1(Σ) ≤ L
+∞ otherwise
where uΣ is the solution of the state equa-
tion with Dirichlet condition on Σ. The scal-
ing factor L2 is the right one in order to avoid
the functionals to degenerate to the trivial
limit functional which vanishes everywhere.
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Theorem. The family of functionals (FL)above Γ-converges, as L → +∞ with respectto the weak* topology on P(Ω), to the func-tional
F (µ) = C∫Ω
f2
µ2a
dx
where C is a constant.
In particular, the optimal compliance net-works ΣL are such that µΣL
converge weakly*to the minimizer of the limit functional, givenby
µ = cf2/3 dx.
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Computing the constant C is a delicate issue.If Y = (0,1)2, taking f = 1, it comes fromthe formula
C = inf
lim infL→+∞
L2∫Y
uΣLdx : ΣL admissible
.
A grid is less performant than a comb structure,
that we conjecture to be the optimal one.
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Open problems
• Optimal periodic network for f = 1? Thiswould give the value of the constant C.
• Numerical computation of the optimal net-works ΣL.
• Quasistatic evolution, when the length in-creases with the time and ΣL also increaseswith respect to the inclusion (irreversibility).
• Same analysis with −∆p, and limit be-haviour as p → +∞, to see if the geometricproblem of average distance can be recov-ered.
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![Page 41: Optimal Dirichlet regions for elliptic PDEschampion.univ-tln.fr/JFCO-Exposes/Buttazzo-slides.pdf · We want to study shape optimization prob-lems of the form min n F(Σ,uΣ) : Σ](https://reader030.vdocuments.net/reader030/viewer/2022020319/5c7070ff09d3f20d2f8baeb4/html5/thumbnails/41.jpg)
References
G. Bouchitte, C. Jimenez, M. Rajesh CRAS
2002.
G. Buttazzo, F. Santambrogio, N. Varchon
COCV 2006, http://cvgmt.sns.it
F. Morgan, R. Bolton Amer. Math. Monthly
2002.
S. Mosconi, P. Tilli J. Conv. Anal. 2005,
http://cvgmt.sns.it
G. Buttazzo, F. Santambrogio NHM 2007,
http://cvgmt.sns.it.
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