Shape optimization under convexity constraint Jimmy LAMBOLEY - PowerPoint PPT Presentation

Shape optimization under convexity constraint Jimmy LAMBOLEY Université Paris-Dauphine ANR GAOS Work with D. Bucur, I. Fragalà, E. Harrell, A. Henrot, M. Pierre, A. Novruzi 03/04/2012, PICOF J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 1 / 20

Examples Isoperimetric problems : min P (Ω) . Ω ⊂ R d , | Ω | = V 0 Spectral problems : min λ 1 (Ω) . Ω ⊂ R d , | Ω | = V 0 J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 2 / 20

Newton’s problem, of the body of minimal resitance Let D = D ( 0 , 1 ) in R 2 . �� 1 � 1 + |∇ f | 2 , f : D → [ 0 , M ] , f concave min D Numerical computations : T. Lachand-Robert, E. Oudet, 2004 : M = 3 / 2 M = 1 J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 3 / 20

Mahler conjecture Conjecture : Is the cube Q d := [ − 1 , 1 ] d solution of � � M ( K ) := | K || K ◦ | , K convex of R d , − K = K min ? K ◦ := � � ξ ∈ R d , � ξ, x � ≤ 1 , ∀ x ∈ K . J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 4 / 20

Pólya-Szegö conjecture The electrostatic capacity of a bounded set Ω ⊂ R 3 is defined by R 3 \ Ω  ∆ u Ω = 0 in  �  u Ω = 1 on ∂ Ω |∇ u Ω | 2 Cap (Ω) := where | x |→ + ∞ u Ω lim = 0 R 3 \ Ω   Is the disk D ⊂ R 3 solution of : min Cap ( K ) ? K convex of R 3 , P ( K )= P 0 J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 5 / 20

Pólya-Szegö conjecture The electrostatic capacity of a bounded set Ω ⊂ R 3 is defined by R 3 \ Ω  ∆ u Ω = 0 in  �  u Ω = 1 on ∂ Ω |∇ u Ω | 2 Cap (Ω) := where | x |→ + ∞ u Ω lim = 0 R 3 \ Ω   Is the disk D ⊂ R 3 solution of : min Cap ( K ) ? K convex of R 3 , P ( K )= P 0 J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 5 / 20

Examples Reverse Isoperimetric problems : max P (Ω) . Ω ⊂ D , | Ω | = V 0 Reverse Spectral problems : λ 1 (Ω) . max Ω ⊂ D , | Ω | = V 0 J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 6 / 20

We want to analyze problem such as K convex of R d J ( K ) , min where J is a shape functional which satisfies a concavity property. Existence is usually easy, Geometric informations on the minimizers ? Find the minimizer. Difficulty : If K is convex, the neighbors of K are mostly not convex. How can we write and use optimality conditions ? J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 7 / 20

We want to analyze problem such as K convex of R d J ( K ) , min where J is a shape functional which satisfies a concavity property. Existence is usually easy, Geometric informations on the minimizers ? Find the minimizer. Difficulty : If K is convex, the neighbors of K are mostly not convex. How can we write and use optimality conditions ? J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 7 / 20

2-dimensional case Outline 2-dimensional case 1 A calculus of variations formulation Polygons as optimal shapes Higher dimensional case 2 J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 8 / 20

2-dimensional case A calculus of variations formulation Outline 2-dimensional case 1 A calculus of variations formulation Polygons as optimal shapes Higher dimensional case 2 J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 9 / 20

2-dimensional case A calculus of variations formulation Linear Parametrization of the convexity To a periodic function u : T → R ∗ + , we associate � � K u = ( r , θ ) ; 0 ≤ r < 1 / u ( θ ) . 1 u ( θ ) θ K u • O Parametrization of a starshaped set. Then K u convex ⇔ u ′′ + u ≥ 0 . J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 10 / 20

2-dimensional case A calculus of variations formulation Linear Parametrization of the convexity To a periodic function u : T → R ∗ + , we associate � � K u = ( r , θ ) ; 0 ≤ r < 1 / u ( θ ) . 1 u ( θ ) θ K u • O Parametrization of a starshaped set. Then K u convex ⇔ u ′′ + u ≥ 0 . J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 10 / 20

2-dimensional case A calculus of variations formulation Linear Parametrization of the convexity Therefore we get a one-to-one correspondance ∼ { v > 0 ∈ H 1 ( T ) such that v ′′ + v ≥ 0 } { 2 d convex sets } − → K u �− → u J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 11 / 20

2-dimensional case A calculus of variations formulation New setting of the problem � � min J ( K ) min j ( u ) := J ( K u ) K ∈F ad u ∈ Sad K convex u ′′ + u ≥ 0 where S ad is a functional space taking into account the other constraints. Examples : S ad = { u : T → R / u 2 ≤ u ≤ u 1 } K 1 K K 2 × � � 1 � S ad = u : T → R / | K u | = 2 u 2 ( θ ) d θ = V 0 T J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 12 / 20

2-dimensional case A calculus of variations formulation New setting of the problem � � min J ( K ) min j ( u ) := J ( K u ) K ∈F ad u ∈ Sad K convex u ′′ + u ≥ 0 where S ad is a functional space taking into account the other constraints. Examples : S ad = { u : T → R / u 2 ≤ u ≤ u 1 } K 1 K K 2 × � � 1 � S ad = u : T → R / | K u | = 2 u 2 ( θ ) d θ = V 0 T J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 12 / 20

2-dimensional case Polygons as optimal shapes Outline 2-dimensional case 1 A calculus of variations formulation Polygons as optimal shapes Higher dimensional case 2 J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 13 / 20

2-dimensional case Polygons as optimal shapes Case of geometric functionals � G ( θ, u ( θ ) , u ′ ( θ )) d θ u ′′ + u ≥ 0 j ( u ) := min T Theorem (L., Novruzi, 2008) If G is strictly concave in the third variable, then solutions are polygons. Application to Reverse isoperimetry : � � min µ | K | − P ( K ) , K convex , D 1 ⊂ K ⊂ D 2 Application to Mahler in R 2 (with E. Harrell, A. Henrot) : [ − 1 , 1 ] 2 . J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 14 / 20

2-dimensional case Polygons as optimal shapes Case of geometric functionals � G ( θ, u ( θ ) , u ′ ( θ )) d θ u ′′ + u ≥ 0 j ( u ) := min T Theorem (L., Novruzi, 2008) If G is strictly concave in the third variable, then solutions are polygons. Application to Reverse isoperimetry : � � min µ | K | − P ( K ) , K convex , D 1 ⊂ K ⊂ D 2 Application to Mahler in R 2 (with E. Harrell, A. Henrot) : [ − 1 , 1 ] 2 . J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 14 / 20

2-dimensional case Polygons as optimal shapes Case of non geometric functionnals u ′′ + u ≥ 0 j ( u ) := J ( K u ) min Theorem (L., Novruzi, Pierre, 2011) We assume j smooth and j ′′ ( u )( v , v ) ≤ α � v � 2 H 1 − a ( T ) − β | v | 2 H 1 ( T ) , for some β > 0 and 0 < a ≤ 1 . Then solutions are polygons. Application to � � λ 1 ( K ) − P ( K ) , K convex ⊂ D , | K | = V 0 min J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 15 / 20

2-dimensional case Polygons as optimal shapes Reverse Faber-Krahn � � max λ 1 ( K ) , K convex ⊂ D , | K | = V 0 We look at j ( u ) := − λ 1 ( K u ) + µ | K u | Lemma (L., Novruzi, Pierre, 2011) If K u is convex and v supported where ∂ K u is smooth, then − d 2 du 2 λ 1 ( K u ) · ( v , v ) ≤ C � v � 2 L 2 ( T ) − β | v | 2 2 ( T ) . 1 H Conclusion : any solution is nowhere “smooth and strictly convex”. J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 16 / 20

2-dimensional case Polygons as optimal shapes Reverse Faber-Krahn � � max λ 1 ( K ) , K convex ⊂ D , | K | = V 0 We look at j ( u ) := − λ 1 ( K u ) + µ | K u | Lemma (L., Novruzi, Pierre, 2011) If K u is convex and v supported where ∂ K u is smooth, then − d 2 du 2 λ 1 ( K u ) · ( v , v ) ≤ C � v � 2 L 2 ( T ) − β | v | 2 2 ( T ) . 1 H Conclusion : any solution is nowhere “smooth and strictly convex”. J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 16 / 20

Higher dimensional case Outline 2-dimensional case 1 A calculus of variations formulation Polygons as optimal shapes Higher dimensional case 2 J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 17 / 20

Higher dimensional case About Pólya-Szegö conjecture � � J ( K ) := f ( | K | , λ 1 ( K ) , Cap ( K )) , K convex ⊂ R d , P ( K ) = P 0 min Theorem (Bucur, Fragalà, L. 2010) Assume J is positive, (1-)homogeneous and smooth, and K 0 is a solution. Then, if ∂ K 0 contains a relatively open set ω of class C 2 , then the Gauss curvature vanishes on ω . Pólya-Szegö conjecture : J ( K ) = Cap ( K ) . J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 18 / 20

Higher dimensional case About Mahler conjecture � � J ( K ) := | K || K ◦ | , K convex ⊂ R d , K = − K min , Theorem (Harrell, Henrot, L. 2011) Let K 0 be a minimizer. If ∂ K 0 contains a relatively open set ω of class C 2 , then the Gauss curvature vanishes on ω . Improvement using Monge-Ampere equation and Transport Theory (work in Progress with Carlier and Gangbo). J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 19 / 20