Multiphase Shape Optimization Problems Dorin Bucur joint work with - PowerPoint PPT Presentation

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 problem h � � � � : Ω i ⊂ D , Ω i ∩ Ω j = ∅ � � � � min g ( F 1 (Ω 1 ) , . . . , F h (Ω h ) + c Ω i , � i =1 where c ≥ 0. Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 2

Example (perimeter and measure, c = 0) Image from http://en.wikipedia.org/wiki/Honeycomb h Ω i = D , | Ω i | = | D | � � � min Per (Ω 1 )+ ... + Per (Ω h ) : Ω i ∩ Ω j = ∅ , , h i =1 Hales 1999 Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 3

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

Examples � − ∆ u = λ u in Ω u = 0 ∂ Ω 0 < λ 1 ≤ λ 2 ≤ ... ≤ λ k ≤→ + ∞ Variational definition : Ω |∇ u | 2 dx � λ 1 (Ω) = min � Ω | u | 2 dx u ∈ H 1 0 (Ω) , u � =0 Ω |∇ u | 2 dx � λ k (Ω) = min 0 (Ω) max � Ω | u | 2 dx S k ∈ H 1 u ∈ S k Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 5

Examples Examples n � � � min λ k i (Ω i ) + c | Ω i | : Ω i ⊂ D , Ω i quasi-open, Ω i ∩ Ω j = ∅ . i =1 n � � � min E (Ω i , f i )+ c | Ω i | : Ω i ⊂ D , Ω i quasi-open, Ω i ∩ Ω j = ∅ . i =1 where E (Ω , f ) = min { 1 � � |∇ u | 2 dx − fudx : u ∈ H 1 0 (Ω) } 2 Ω Ω Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 6

Questions ◮ existence of a solution : partition if c = 0, not a partition if c > 0 (different regimes) ◮ properties of Ω i coming from optimality : regularity, asymptotic behavior, ... ◮ numerical computations Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 7

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 Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 8

Oudet 2004, Antunes-Freitas 2012 : λ 5 to λ 15 Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 9

Existence of a solution Buttazzo Dal Maso 1992 (single phase) B. Buttazzo, Henrot 1998 For every c ≥ 0, problem n h � � � � � : Ω i ⊂ D , Ω i ∩ Ω j = ∅ � � min g ( λ k 1 (Ω 1 ) , ..., λ k h (Ω h ))+ c Ω i . � i =1 i =1 has a solution, provided g is l.s.c. and increasing in each variable. Examples : ◮ g ( x 1 , x 2 ) = x 1 + x 2 ◮ g ( x 1 ) = x 1 ◮ not admissible g ( x 1 , x 2 ) = x 1 − x 2 Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 10

Idea of the proof Weak- γ convergence : let w Ω be the torsion function � − ∆ w Ω = 1 in Ω w Ω = 0 ∂ Ω H 1 If Ω 1 n ∩ Ω 2 n = ∅ , such that w Ω i ⇀ w i , then n |{ w 1 > 0 } ∩ { w 2 > 0 }| = 0 and λ k ( { w i > 0 } ) ≤ lim inf λ k (Ω i n ) . Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 11

What about the regularity of each cell ? Ramos, Tavares, Terracini 2014 If c = 0, there exists eigenfunctions u k i which are Lipschitz, the sets Ω i are open, and the nodal lines are C 1 ,α , with the exception of 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 λ k (Ω) + c | Ω | there exists a Lipschitz eigenfunction. Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 12

What about the regularity of each cell ? Definition The set Ω is a shape subsolution for λ k if ∃ c > 0 such that ∀ ˜ λ k (Ω) + c | Ω | ≤ λ k (˜ Ω) + c | ˜ Ω ⊆ Ω Ω | . (1) If g is bi-Lipschitz in n h � � � � � : Ω i ⊂ D , Ω i ∩ Ω j = ∅ � � min g ( λ k 1 (Ω 1 ) , ..., λ k h (Ω h ))+ c Ω i . � i =1 i =1 every cell Ω i is a shape subsolution for λ k i . Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 13

What about the regularity of each cell ? The torsion energy of Ω is 1 � � |∇ u | 2 dx − E (Ω) = min udx . 2 u ∈ H 1 0 (Ω) Theorem (B. 2011) If Ω is a subsolution for the torsion energy, then Ω has finite perimeter and satisfies some inner density condition. Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 14

Alt-Caffarelli type argument : ◮ Only inner perturbations allowed ! ◮ If sup u ≤ c 0 r then u ≡ 0 on B r B 2 r ( x ) ⇒ inner density, boundedness and control of the diameter. = � ◮ Control on |∇ u | dx = ⇒ finite perimeter. 0 < w <ε Roughly speaking |∇ w Ω | ≥ α > 0 near the boundary of Ω. Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 15

Theorem If Ω is a sub solution for λ k 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 : λ k (˜ Ω) − λ k (Ω) ≤ C Ω ( E (˜ Ω) − E (Ω)) . Dorin Bucur joint work with Bozhidar Velichkov: Multiphase Shape Optimization Problems, 16