Shape optimization under convexity constraint

Transcription

1 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

2 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

3 Newton s problem, of the body of minimal resitance Let D = D(0, 1) in R 2. { 1 min, f : D [0, M], f concave 1 + f 2 D Numerical computations : T. Lachand-Robert, E. Oudet, 2004 : M =3/2 M =1 J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 3 / 20

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

5 Pólya-Szegö conjecture The electrostatic capacity of a bounded set Ω R 3 is defined by Cap(Ω) := R 3 \Ω u Ω 2 u Ω = 0 in R 3 \ Ω where u Ω = 1 on Ω lim u Ω = 0 x + 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

6 Pólya-Szegö conjecture The electrostatic capacity of a bounded set Ω R 3 is defined by Cap(Ω) := R 3 \Ω u Ω 2 u Ω = 0 in R 3 \ Ω where u Ω = 1 on Ω lim u Ω = 0 x + 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

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

8 We want to analyze problem such as min J(K ), K convex of R d 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

9 We want to analyze problem such as min J(K ), K convex of R d 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

10 2-dimensional case Outline 1 2-dimensional case A calculus of variations formulation Polygons as optimal shapes 2 Higher dimensional case J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 8 / 20

11 2-dimensional case A calculus of variations formulation Outline 1 2-dimensional case A calculus of variations formulation Polygons as optimal shapes 2 Higher dimensional case J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 9 / 20

12 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) K u O θ Parametrization of a starshaped set. Then K u convex u + u 0. J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 10 / 20

13 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) K u O θ Parametrization of a starshaped set. Then K u convex u + u 0. J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 10 / 20

14 2-dimensional case A calculus of variations formulation Linear Parametrization of the convexity Therefore we get a one-to-one correspondance {2d convex sets {v > 0 H 1 (T) such that v + v 0 K u u J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 11 / 20

15 2-dimensional case A calculus of variations formulation New setting of the problem min J(K ) K F ad K convex { min j(u) := J(K u ) u S ad 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 2 K S ad = { u : T R / K u = 1 T 2u 2 (θ) dθ = V 0 J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 12 / 20

16 2-dimensional case A calculus of variations formulation New setting of the problem min J(K ) K F ad K convex { min j(u) := J(K u ) u S ad 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 2 K S ad = { u : T R / K u = 1 T 2u 2 (θ) dθ = V 0 J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 12 / 20

17 2-dimensional case Polygons as optimal shapes Outline 1 2-dimensional case A calculus of variations formulation Polygons as optimal shapes 2 Higher dimensional case J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 13 / 20

18 2-dimensional case Polygons as optimal shapes Case of geometric functionals min j(u) := G(θ, u(θ), u (θ))dθ u +u 0 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

19 2-dimensional case Polygons as optimal shapes Case of geometric functionals min j(u) := G(θ, u(θ), u (θ))dθ u +u 0 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

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

21 2-dimensional case Polygons as optimal shapes Reverse Faber-Krahn We look at max {λ 1 (K ), K convex D, K = V 0 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 H 1 2 (T). Conclusion : any solution is nowhere smooth and strictly convex. J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 16 / 20

22 2-dimensional case Polygons as optimal shapes Reverse Faber-Krahn We look at max {λ 1 (K ), K convex D, K = V 0 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 H 1 2 (T). Conclusion : any solution is nowhere smooth and strictly convex. J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 16 / 20

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

24 Higher dimensional case About Pólya-Szegö conjecture min {J(K ) := f ( K, λ 1 (K ), Cap(K )), K convex R d, P (K ) = P 0 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

25 Higher dimensional case About Mahler conjecture { min J(K ) := K K, K convex R d, K = K, 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

26 Higher dimensional case Open questions max{λ 1 (K ) / K convex D, K = V 0 in R 2? Nowhere strictly convex in higher dimension? Polyhedral solutions in higher dimension? J. Lamboley (Université Paris-Dauphine) Optimal convex shapes 20 / 20