Multiphase Shape Optimization Problems · 2014. 6. 22. · 1;u 2 2H1(B 1) be two non-negative...

LAMA Universite de Savoie

Multiphase Shape OptimizationProblems

Dorin Bucurjoint work with Bozhidar Velichkov

Toulouse, June 17, 2014

Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 1

Generic problem

ming((F1(Ω1), . . . ,Fh(Ωh)

)+c∣∣ h⋃i=1

Ωi

∣∣ : Ωi ⊂ D, Ωi ∩Ωj = ∅,

where c ≥ 0.


Example (perimeter and measure, c = 0)

Image from http://en.wikipedia.org/wiki/Honeycomb

minPer(Ω1)+...+Per(Ωh) : Ωi∩Ωj = ∅,

h⋃i=1

Ωi = D, |Ωi | =|D|h

,

Hales 1999


Examples

Kelvin structure Weaire and Phelan structureImages from http://en.wikipedia.org/wiki/Weaire-Phelan structure


Examples

−∆u = λu in Ω

u = 0 ∂Ω

0 < λ1 ≤ λ2 ≤ ... ≤ λk ≤→ +∞

Variational definition :

λ1(Ω) = minu∈H1

0 (Ω),u 6=0

∫Ω |∇u|

2dx∫Ω |u|2dx

λk(Ω) = minSk∈H1

0 (Ω)maxu∈Sk

∫Ω |∇u|

2dx∫Ω |u|2dx


Examples

Examples

min n∑

i=1

λki (Ωi ) + c |Ωi | : Ωi ⊂ D, Ωi quasi-open, Ωi ∩Ωj = ∅.

min n∑

i=1

E (Ωi , fi )+c |Ωi | : Ωi ⊂ D, Ωi quasi-open, Ωi ∩Ωj = ∅.

where

E (Ω, f ) = min1

2

∫Ω|∇u|2dx −

∫Ωfudx : u ∈ H1

0 (Ω)


Questions

I existence of a solution : partition if c = 0, not a partition ifc > 0 (different regimes)

I properties of Ωi coming from optimality : regularity,asymptotic behavior, ...

I numerical computations


One phase, c > 0 large

I k = 1 Faber-Krahn 1921-1923, ball

I k = 2 Faber-Krahn inequality 1921-1923, two equal balls

I k = 3 conjecture : ball in 2D

I k = 4 conjecture : two non equal balls in 2D

I k = 13 it is not a ball or union of balls, Wolf-Keller 1992


Oudet 2004, Antunes-Freitas 2012 : λ5 to λ15


Existence of a solution

Buttazzo Dal Maso 1992 (single phase)B. Buttazzo, Henrot 1998For every c ≥ 0, problem

min n∑

i=1

g(λk1(Ω1), ..., λkh(Ωh))+c∣∣ h⋃i=1

Ωi

∣∣ : Ωi ⊂ D, Ωi∩Ωj = ∅.

has a solution, provided g is l.s.c. and increasing in each variable.

Examples :

I g(x1, x2) = x1 + x2

I g(x1) = x1

I not admissible g(x1, x2) = x1 − x2


Idea of the proof

Weak-γ convergence : let wΩ be the torsion function−∆wΩ = 1 in Ω

wΩ = 0 ∂Ω

If Ω1n ∩ Ω2

n = ∅, such that wΩin

H1

wi , then

|w1 > 0 ∩ w2 > 0| = 0

andλk(wi > 0) ≤ lim inf λk(Ωi

n).


What about the regularity of each cell ?

Ramos, Tavares, Terracini 2014

If c = 0, there exists eigenfunctions uki which are Lipschitz, thesets Ωi are open, and the nodal lines are C 1,α, with the exceptionof a set of small dimension.

B., Mazzoleni, Pratelli, Velichkov 2013

(One phase) If n = 1 and c > 0 then for every solution Ω of

minλk(Ω) + c |Ω|

there exists a Lipschitz eigenfunction.



DefinitionThe set Ω is a shape subsolution for λk if ∃ c > 0 such that

∀Ω ⊆ Ω λk(Ω) + c |Ω| ≤ λk(Ω) + c |Ω|. (1)

If g is bi-Lipschitz in

min n∑

i=1

g(λk1(Ω1), ..., λkh(Ωh))+c∣∣ h⋃i=1

Ωi

∣∣ : Ωi ⊂ D, Ωi∩Ωj = ∅.

every cell Ωi is a shape subsolution for λki .



The torsion energy of Ω is

E (Ω) = minu∈H1

0 (Ω)

1

2

∫|∇u|2dx −

∫udx .

Theorem (B. 2011)

If Ω is a subsolution for the torsion energy, then Ω has finiteperimeter and satisfies some inner density condition.


Alt-Caffarelli type argument :

I Only inner perturbations allowed !

I If supB2r (x)

u ≤ c0r then u ≡ 0 on Br

=⇒ inner density, boundedness and control of the diameter.

I Control on

∫0<w<ε

|∇u|dx =⇒ finite perimeter.

Roughly speaking |∇wΩ| ≥ α > 0 near the boundary of Ω.


TheoremIf Ω is a sub solution for λk with constant c , there exists Λ suchthat every solution of Ω is a subsolution of the torsion energy withconstant Λ.

=⇒ finite perimeter, inner density and control of the diameter ofevery mini minimizer

Idea of the proof, if Ω ⊆ Ω are γ-close :

λk(Ω)− λk(Ω) ≤ CΩ(E (Ω)− E (Ω)).


Consequences

I if Ω1,Ω2 are two disjoint sub solutions, there exists D1 openset such that Ω1 ⊆ D1 and Ω2 ∩ D1 = ∅.

I one phase points (between the cells and the void), two phasepoints (between cells)

Ω i

Ui

Ω

Ω i

ε

yx

Ω

ik

j

U

Uij

k

D x


Junction of sub solutionsAlt, Caffarelli, Friedman 1984 Let u1, u2 ∈ H1(B1) be twonon-negative functions such that −∆ui ≤ 0, for i = 1, 2, andu1u2 = 0. Then

2∏i=1

(1

r2

∫Br

|∇ui |2

|x |d−2dx

)is non decreasing in r .Caffarelli, Jerison, Kenig 2002Let u1, u2 ∈ H1(B1) be two non-negative functions such that−∆ui ≤ 1, for i = 1, 2, and uiuj = 0. Then there is a dimensionalconstant Cd such that for each r ∈ (0, 1) we have

2∏i=1

(1

r2

∫Br

|∇ui |2

|x |d−2dx

)≤ Cd

(1 +

2∑i=1

∫B1

|∇ui |2

|x |d−2dx

)2

.


Junction of sub solutions

Consequence :

For multiphase problems with positive states (energy, firsteigenfunctions) the states are Lipschitz continuous.


Absence of triple points

Conti, Terracini, Verzini 2005In R2, let u1, u2, u3 ∈ H1(B1) be three non-negative subharmonicfunctions such that uiuj = 0. Then the function

r 7→3∏

i=1

(1

r3

∫Br

|∇ui |2 dx)

(2)

is nondecreasing on [0, 1].


Absence of triple points

Theorem B. Velichkov [Three-phase]Let ui ∈ H1(B1), i = 1, 2, 3, be three non-negative Sobolevfunctions such that −∆ui ≤ 1, for each i = 1, 2, 3, and uiuj = 0,for each i 6= j . Then

3∏i=1

(1

r2+ε

∫Br

|∇ui |2

|x |d−2dx

)≤ Cd

(1 +

3∑i=1

∫B1

|∇ui |2

|x |d−2dx

)3

. (3)


Numerical results (Bogosel, Velichkov 2013)

3 cells


Optimal partitions : c = 0

Ω

Ω

Ω

Ω Ω

ΩΩ

Ω

1

2

3

4

5

6

7

8

minN∑i=1

λ1(Ωi ) : Ωi ∩ Ωj = ∅, Ωi ⊆ D

Conjecture Caffarelli-Lin (2007) :

minN∑i=1

λ1(Ωi ) ' N2λ1(H), (4)

where H is the regular hexagon of surface equal to 1 in R2.


Numerical resultsB., Bourdin, Oudet 2009

16 cells, non periodical


Numerical results

16 cells, periodical


Numerical results

384 cells


Numerical results

512 cells


Thank you for your attention !


Multiphase Shape Optimization Problems · 2014. 6. 22. · 1;u 2 2H1(B 1) be two non-negative...

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