Spectral Optimization Problems

Giuseppe Buttazzo

Dipartimento di Matematica

Universita di Pisa

buttazzo@dm.unipi.it

http://cvgmt.sns.it

Workshop on “Calculus of Variations”Ancona, June 6–8, 2011

Spectral optimization

=

How to designyour favourite drum

1

We deal with the optimization problem

minF (Ω) : Ω ∈ A(D)

where D is a given bounded domain of Rd

and A(D) is a class of admissible choicesmade of subsets of D.

Typical examples are:

F (Ω) = Φ(λ(Ω)

)where λ(Ω) is the spec-

trum of the Dirichlet Laplacian in Ω andΦ : RN → [0,+∞] is a given function; forinstance

F (Ω) = λk(Ω).

2

F (Ω) =∫Dj(x, uΩ) dx where j is a given in-

tegrand and uΩ is the solution in H10(Ω) of

−∆u = f ; for instance

j(x, u) = −a(x)up.

In both cases a very natural choice for ad-

missible domains is

A(D) =

Ω ⊂ D, Ω quasi-open, |Ω| ≤ m

.

Some good sources for results and problems...

3

Dorin B

ucur and Giuseppe B

uttazzoV

ariational Methods in Shape O

petimization P

roblems

PNLDE65

Dorin BucurGiuseppe Buttazzo

Variational Methodsin Shape OptimizationProblems

Birkhäuser

Progress in Nonlinear Differential Equationsand Their Applications

BirkhäuserISBN 0-8176-4359-1www.birkhauser.com

The study of shape optimization problems encompasses a wide spectrum ofacademic research with numerous applications to the real world. In thiswork these problems are treated from both the classical and modernperspectives and target a broad audience of graduate students in pure andapplied mathematics, as well as engineers requiring a solid mathematicalbasis for the solution of practical problems.

Key topics and features:

• Presents foundational introduction to shape optimization theory

• Studies certain classical problems: the isoperimetric problem and the Newton problem involving the best aerodynamical shape, and optimization problems over classes of convex domains

• Treats optimal control problems under a general scheme, giving a topological framework, a survey of γ-convergence, and problems governed by ODE

• Examines shape optimization problems with Dirichlet and Neumann conditions on the free boundary, along with the existence of classical solutions

• Studies optimization problems for obstacles and eigenvalues of elliptic operators

• Poses several open problems for further research

• Substantial bibliography and index

Driven by good examples and illustrations and requiring only a standardknowledge in the calculus of variations, differential equations, andfunctional analysis, the book can serve as a text for a graduate course incomputational methods of optimal design and optimization, as well as anexcellent reference for applied mathematicians addressing functional shapeoptimization problems.

Dorin Bucur and Giuseppe ButtazzoVariational Methods in Shape Optimization Problems

4

2005. XII, 334 p. (Mathématiques et

Applications, Vol. 48) Softcover

59,72 €

$129.00

SFr. 99.50

£47.99

ISBN 978-3-540-26211-4

A. Henrot, Université Henri Poincaré, Vandoeuvre-les-Nancy, France; M. Pierre, Antenne de Bre-

tagne de I'ENS Cachan, Bruz, France

Variation et optimisation de formes

Une analyse géométrique

Ce livre est une initiation aux approches modernes de l’optimisation mathématique de

formes. On y développe la méthodologie ainsi que les outils d’analyse mathématique et

de géométrie nécessaires à l’étude des variations de domaines. On y trouve une étude

systématique des questions géométriques associées à l’opérateur de Laplace, de la capa-

cité classique, de la dérivation par rapport à une forme, ainsi qu’un FAQ sur les topolo-

gies usuelles sur les domaines et sur les propriétés géométriques des formes optimales

avec ce qui se passe quand elles n’existent pas, le tout avec une importante bibliogra-

phie.... more on http://springer.com/978-3-540-26211-4

Yes, please send me copies

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One issue we intend to develop is the studyof gradient flow evolutions for shape opti-mization problems, via minimizing movements.

The framework is a metric space (space ofshapes) and a functional F : X → [0,+∞];via the Euler scheme of time step ε andinitial condition u0 ∈ X we may constructa step function uε(t) = w([t/ε]) by settingw(0) = u0 and

w(n+ 1) ∈ argminF(v) +

d2(v, w(n)

)2ε

.

The gradient flow u(t) is then a limit of asequence uεn with εn → 0.

6

In spectral optimization we take a suitabledistance on the space of shapes and

F(Ω) = Φ(λ(Ω)

).

The goal is to study existence and propertiesof spectral flows Ω(t). At this stage the onlyavailable results are obtained for debondingproblems, where mushy regions may appearduring the evolution.

D.Bucur, G.Buttazzo: J. Convex Anal. 2008D.Bucur, G.Buttazzo, A.Lux: Arch. Ratio-nal Mech. Anal. 2008.

From now on we restrict to static shape op-timization problems.

7

In general one cannot expect the existence

of a solution; for instance, the problem

min ∫

D|u− c|2 dx : −∆u = 1 in H1

0(Ω)

has no solution if c is small, and one has

to deal with relaxed solutions that in this

case are capacitary mesures µ, i.e. Borel

countably additive set functions with values

in [0,+∞] and such that

µ(E) = 0 whenever cap(E) = 0.

8

On the other hand, adding some geomet-rical constraint to the admissible domains,gives extra compactness that provides theexistence of a solution in a very large num-ber of situation. For instance:• The class Aconvex of convex sets containedin D.• The class Aunif cone of domains satisfyinga uniform exterior cone property.• The class Aunif flat cone of domains satis-fying a uniform flat cone condition, i.e., asabove, but with the weaker requirement thatthe cone may be flat, that is of dimensiond− 1.

9

• The class Acap density of domains satisfying

a uniform capacity density condition.

• The class Aunif Wiener of domains satisfy-

ing a uniform Wiener condition.

Aconvex ⊂ Aunif cone ⊂ Aunif flat cone⊂ Acap density ⊂ Aunif Wiener

• Another interesting class, which is only

of topological type and is not contained in

any of the previous ones, is (for d = 2) the

class of domains for which the number of

connected components of D \Ω is uniformly

bounded (Sverak 1993).

10

A powerful result which applies in the sit-

uations when no geometric constraints are

imposed(Buttazzo-Dal Maso 1993):

Theorem Assume that

1. F is decreasing for the set inclusion;

2. F is l.s.c. for the γ-convergence.

Then the minimum problem

minF (Ω) : |Ω| ≤ m, Ω ⊂ D quasi open

admits at least a solution Ωopt.

11

Example 1. Take F (Ω) = Φ(λ(Ω)

)with Φ

l.s.c. and increasing in each of its variables;

then the assumptions are verified. For in-

stance we may take F (Ω) = λk(Ω).

For k = 1 (and large D) the best domain is

a ball of volume m (Faber-Krahn 1923);

For k = 2 (and large D) the best domain is

the union of two disjoint balls of volume m/2

(Polya-Szego 1955).

12

Here is a numerical computation of optimalshapes for k = 3− 10 (Oudet)

k best array of balls best shape

12/03/09 12:20Eigenvalues

Page 1 of 1http://www.lama.univ-savoie.fr/~oudet/index.php?page=Galerie/../Eigenvalues/eigenvalues&lang=fr

Minimisation des modes propres

d'une membrane par rapport au

domaine

Le tableau ci-dessous présente les formes obtenues numériquement par uneméthode mixte de type level set/ relaxation lors de la minimisation des premiersmodes propres (k=3 à 10) d'une membrane à bord fixe (condition au bord de typeDirichlet) d'aire fixée (pour plus de détails voir Numerical minimization ofeigenmodes of a membrane with respect to the domain ainsi que Minimizing thesecond eigenvalue of the Laplace operator with Dirichlet boundary conditions).On compare ces résultats aux valeurs obtenues par Wolf et Keller en serestreignant à des réunions de disques dans l'article :

Wolf, Sven Andreas(1-STF); Keller, Joseph B.(1-STF) Range of the first two eigenvalues of the Laplacian. (English. English summary) Proc. Roy. Soc. London Ser. A 447 (1994), no. 1930, 397--412. 35P15 (58G25)

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La conjecture de Kelvin

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RESUME DE THÈSE

13

Very little is known on regularity of optimaldomains; apart the case of λ1 the proof thatthey are actually open sets is still missing.

For spectral optimization problems that donot fulfill the decreasing property of theoremabove, the case of

min

Φ(λ1, λ2) : |Ω| ≤ m, Ω ⊂ D

is interesting. Indeed, if D is large enough, itis possible to prove (Bucur-Buttazzo-Figueiredo1999) that the existence of an optimal do-main always occur, i.e. for every (continu-ous) function Φ.

14

Here is the plot of the set E of points of R2

that are of the form(λ1(Ω), λ2(Ω)

)for some

Ω ⊂ D with |Ω| ≤ m (Keller-Wolf 1994).6.4 The first two eigenvalues 155

Figure 6.1. The set E for N = 2 and c = 1.

Theorem 6.4.1 The set E is closed in R2.

The proof of the theorem above is based on the following lemma.

Lemma 6.4.2 If the set E is convex on the vertical and horizontal directions, thenE is closed in R2.

Proof Consider (x, y) ! E . There exists a sequence of sets (An)n!N " Ac(B)

such that s(An) # (x, y). From the weak ! -compactness of the set Ac(B), for asubsequence still denoted by the same indices we can write An # A in the weak! -sense. Then A ! Ac(B) and since the eigenvalues of the Laplacian are weakly! -lower semicontinuous we get

"1(A) $ liminfn#%

"1(An) = x and "2(A) $ liminfn#%

"2(An) = y.

From the vertical convexity of E , the vertical segment joining s(A) with the halfline ts(B1) : t & 1 is contained in E . If y < "2(B1) we can find the point("1(A), y) on this segment and using now the horizontal convexity, the segmentjoining ("1(A), y) to ts(B2) : t & 1 is in E . But this segment contains the point(x, y) since "1(A) $ x .

If y & "2(B1), then the horizontal convexity gives directly (x, y) ! E .

Following Lemma 6.4.2 it suffices to prove the convexity of E on vertical andhorizontal directions. For this purpose, we split the proof in two steps:

15

Optimal partitions

We look for a partition Ω1, . . . ,ΩN of Ω whichminimizes a cost of the form

F (Ω1, . . . ,ΩN)

among all partitions of Ω (volume constraints|Ωi| = mi can be added). The cost F couldbe very general, for instance

F (Ω1, . . . ,ΩN) = Φ(λk1

(Ω1), . . . , λkN(ΩN)).

General existence theoremBucur-B.-Henrot Adv. Math. Sci. Appl. ’98

16

RelaxationB., Timofte Adv. Math. Sci. Appl. ’02

Regularity for λ1(Ω1) + · · ·+ λ1(ΩN)Caffarelli, Lin J. Sci. Comp. ’06

Other variants (manifolds, nonlinear, . . . )Conti, Terracini, Verzini JFA ’03, Calc.Var ’05Helffer, Hoffmann-Ostenhof Preprint

Conjecture: dim=2, optimal partitions forλ1(Ω1) + · · ·+ λ1(ΩN) approach as N → ∞a regular exagonal tiling.

17

Replacing the volume constraint |Ω| ≤ m by

a perimeter constraint Per(Ω) ≤ L provides

interesting variants to the spectral optimiza-

tion problem (Bucur-B.-Henrot ’09, Van den

Berg-Iversen ’09). Note that spectral func-

tional are in general not l.s.c. for the L1

convergence.

Theorem Assume that F is as above (de-

creasing and γ-l.s.c.). Then the minimum

problem

minF (Ω) : Per(Ω) ≤ L, Ω ⊂ D

.

admits at least a solution Ωopt.

18

For F (Ω) = λ2(Ω) and D = Rd we have:

• in the case d = 2 Ωopt is convex, of class

C∞, its boundary does not contain any seg-

ment, does not contain any arc of circle,

contains exactly two points where the cur-

vature vanishes.

• In the case d ≥ 3 Ωopt is not convex, is

connected, regularity of Ωopt is not known

(even not if it is an open set).

19

• A similar problem has been studied by Athana-

sopoulos, Caffarelli, Kenig, Salsa (CPAM 2001):

minE(Ω) + Per(Ω) : Ω ⊂ D

where E(Ω) is the Dirichlet energy

E(Ω) = min ∫

D|Dv|2 dx :

v = 0 on D \Ωv = g on ∂D

.

• numerical plot of Ωopt for d = 2 (courtesy

of E. Oudet).

20

21

We consider now Cheeger-like rescaled shape

optimization problems

minM(Ω)J(Ω) : Ω ⊂ D

where M(Ω) is a scaling factor and J(Ω) a

shape functional. The coercivity condition

limM(Ω)→0

M(Ω)J(Ω) = +∞

is assumed, which means that the scaling is

“above” the scaling invariance, as it happens

in the Cheeger problem where

M(Ω) = Per(Ω), J(Ω) = |Ω|−α, α > 1−1/d.

22

Theorem (B.-Wagner) If M and J are non-negative, fulfill the coercivity condition aboveand• J is γ-l.s.c. and decreasing for inclusion;• M is wγ-l.s.c..the minimum problem above has a solution.

Examples• |Ω|αλk(Ω), α < 2/d

• |Ω|α(C(Ω)

)−1, α < 1 + 2

dwhere C(Ω) is the compliance functional

∫Ω fuΩ dx

•(

Per(Ω))αλk(Ω), α < 2/(d− 1);

•(

Per(Ω))α(

C(Ω))−1

, α < 1 + 2d−1.

23

Note that Per(Ω) is not a wγ-l.s.c. and the

existence proof requires some extra devices.

Necessary conditions of optimality can be

found; for instance in the case of |Ω|α(C(Ω)

)−1

if Ω is an optimal domain in D and u is the

corresponding solution:

|∇u|2 = αC(Ω)|Ω| on ∂Ω ∩D (free boundary);

|∇u|2 ≥ αC(Ω)|Ω| on ∂Ω∩∂D (common bound.).

For instance if D has a corner the optimal

domain Ω cannot fill the entire D.

24

To finish, a challenging open question. Con-sider the problem

minC(Ω)λαk(Ω) : Ω ⊂ D

where α is above the scaling invariance 1 +d/2. Thanks to a result of Kohler-Jobin the

minimum of C(Ω)λ1+d/21 (Ω) is reached by a

ball, and so when α > 1 + d/2 the coercivitycondition

limM(Ω)→0

M(Ω)J(Ω) = +∞

is fulfilled, taking

M(Ω) = C(Ω) and J(Ω) = λαk(Ω).

25

However, the other conditions are not ful-

filled; in particular C(Ω) is not wγ-l.s.c. and,

even if strongly expected, the existence of an

optimal domain is still missing.

Once the existence of an optimal domain is

established, the subsequent step would be

the regularity of the free boundary and the

necessary conditions of optimality.

26