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LAMA Universit e de Savoie Multiphase Shape Optimization Problems Dorin Bucur joint work with Bozhidar Velichkov Toulouse, June 17, 2014 Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 1 Generic

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LAMA Universit´ e de Savoie

Multiphase Shape Optimization Problems

Dorin Bucur joint work with Bozhidar Velichkov Toulouse, June 17, 2014

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

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Generic problem

min

  • g
  • (F1(Ω1), . . . , Fh(Ωh)
  • +c
  • h
  • i=1

Ωi

  • : Ωi ⊂ D, Ωi ∩Ωj = ∅
  • ,

where c ≥ 0.

Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 2

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Example (perimeter and measure, c = 0)

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

  • Per(Ω1)+...+Per(Ωh) : Ωi∩Ωj = ∅,

h

  • i=1

Ωi = D, |Ωi| = |D| h

  • ,

Hales 1999

Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 3

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Examples

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

Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 4

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Examples

−∆u = λu in Ω u = 0 ∂Ω 0 < λ1 ≤ λ2 ≤ ... ≤ λk ≤→ +∞ Variational definition : λ1(Ω) = min

u∈H1

0(Ω),u=0

  • Ω |∇u|2dx
  • Ω |u|2dx

λk(Ω) = min

Sk∈H1

0(Ω) max

u∈Sk

  • Ω |∇u|2dx
  • Ω |u|2dx

Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 5

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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 ) = min{1 2

  • Ω

|∇u|2dx −

  • Ω

fudx : u ∈ H1

0(Ω)}

Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 6

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Questions

◮ existence of a solution : partition if c = 0, not a partition if

c > 0 (different regimes)

◮ properties of Ωi coming from optimality : regularity,

asymptotic behavior, ...

◮ numerical computations

Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 7

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One phase, c > 0 large

◮ k = 1 Faber-Krahn 1921-1923, ball ◮ k = 2 Faber-Krahn inequality 1921-1923, two equal balls ◮ k = 3 conjecture : ball in 2D ◮ k = 4 conjecture : two non equal balls in 2D ◮ k = 13 it is not a ball or union of balls, Wolf-Keller 1992

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Oudet 2004, Antunes-Freitas 2012 : λ5 to λ15

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Existence of a solution

Buttazzo Dal Maso 1992 (single phase)

  • B. Buttazzo, Henrot 1998

For 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 :

◮ g(x1, x2) = x1 + x2 ◮ g(x1) = x1 ◮ not admissible g(x1, x2) = x1 − x2

Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 10

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Idea of the proof

Weak-γ convergence : let wΩ be the torsion function −∆wΩ = 1 in Ω wΩ = 0 ∂Ω If Ω1

n ∩ Ω2 n = ∅, such that wΩi

n

H1

⇀ wi, then |{w1 > 0} ∩ {w2 > 0}| = 0 and λk({wi > 0}) ≤ lim inf λk(Ωi

n).

Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 11

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What about the regularity of each cell ?

Ramos, Tavares, Terracini 2014 If c = 0, there exists eigenfunctions uki which are Lipschitz, the sets Ωi are open, and the nodal lines are C 1,α, with the exception

  • f a set of small dimension.

B., Mazzoleni, Pratelli, Velichkov 2013 (One phase) If n = 1 and c > 0 then for every solution Ω of min λk(Ω) + c|Ω| there exists a Lipschitz eigenfunction.

Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 12

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What about the regularity of each cell ?

Definition

The 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.

Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 13

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What about the regularity of each cell ?

The torsion energy of Ω is E(Ω) = min

u∈H1

0(Ω)

1 2

  • |∇u|2dx −
  • udx.

Theorem (B. 2011)

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

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Alt-Caffarelli type argument :

◮ Only inner perturbations allowed ! ◮ If sup B2r(x)

u ≤ c0r then u ≡ 0 on Br = ⇒ inner density, boundedness and control of the diameter.

◮ Control on

  • 0<w<ε

|∇u|dx = ⇒ finite perimeter. Roughly speaking |∇wΩ| ≥ α > 0 near the boundary of Ω.

Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 15

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Theorem

If Ω is a sub solution for λk with constant c, there exists Λ such that every solution of Ω is a subsolution of the torsion energy with constant Λ. = ⇒ finite perimeter, inner density and control of the diameter of every mini minimizer Idea of the proof, if ˜ Ω ⊆ Ω are γ-close : λk(˜ Ω) − λk(Ω) ≤ CΩ(E(˜ Ω) − E(Ω)).

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Consequences

◮ if Ω1, Ω2 are two disjoint sub solutions, there exists D1 open

set such that Ω1 ⊆ D1 and Ω2 ∩ D1 = ∅.

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

points (between cells)

  • Ωi

Ui Ω Ωi

ε

yx Ω

ik j

U U

ij k

D x

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Junction of sub solutions

Alt, Caffarelli, Friedman 1984 Let u1, u2 ∈ H1(B1) be two non-negative functions such that −∆ui ≤ 0, for i = 1, 2, and u1u2 = 0. Then

2

  • i=1

1 r2

  • Br

|∇ui|2 |x|d−2 dx

  • is non decreasing in r.

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

2

  • i=1

1 r2

  • Br

|∇ui|2 |x|d−2 dx

  • ≤ Cd
  • 1 +

2

  • i=1
  • B1

|∇ui|2 |x|d−2 dx 2 .

Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 18

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Junction of sub solutions

Consequence : For multiphase problems with positive states (energy, first eigenfunctions) the states are Lipschitz continuous.

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Absence of triple points

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

3

  • i=1

1 r3

  • Br

|∇ui|2 dx

  • (2)

is nondecreasing on [0, 1].

Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 20

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Absence of triple points

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

3

  • i=1

1 r2+ε

  • Br

|∇ui|2 |x|d−2 dx

  • ≤ Cd
  • 1 +

3

  • i=1
  • B1

|∇ui|2 |x|d−2 dx 3 . (3)

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Numerical results (Bogosel, Velichkov 2013)

3 cells

Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 22

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Numerical results (Bogosel, Velichkov 2013)

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Numerical results (Bogosel, Velichkov 2013)

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Optimal partitions : c = 0

Ω Ω Ω Ω Ω Ω Ω Ω

1 2 3 4 5 6 7 8

min{

N

  • i=1

λ1(Ωi) : Ωi ∩ Ωj = ∅, Ωi ⊆ D} Conjecture Caffarelli-Lin (2007) : min

N

  • i=1

λ1(Ωi) ≃ N2λ1(H), (4) where H is the regular hexagon of surface equal to 1 in R2.

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Numerical results

B., Bourdin, Oudet 2009 16 cells, non periodical

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Numerical results

16 cells, periodical

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Numerical results

384 cells

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Numerical results

512 cells

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Thank you for your attention !

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