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

Ω⊂ Rd, |Ω|=V0

P(Ω). Spectral problems : min

Ω⊂ Rd, |Ω|=V0

λ1(Ω).

• 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 R2. min

• D

1 1 + |∇f|2 , f : D → [0, M], f concave

• 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 Qd := [−1, 1]d solution of min

• M(K) := |K||K ◦|, K convex of Rd, −K = K
• ?

K ◦ :=

• ξ ∈ Rd, 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 ⊂ R3 is defined by Cap(Ω) :=

• R3\Ω

|∇uΩ|2 where      ∆uΩ = in R3 \ Ω uΩ = 1

• n

∂Ω lim

|x|→+∞uΩ

= Is the disk D ⊂ R3 solution of : min

K convex of R3, P(K)=P0

Cap(K) ?

• J. Lamboley (Université Paris-Dauphine)

Optimal convex shapes 5 / 20

Pólya-Szegö conjecture

The electrostatic capacity of a bounded set Ω ⊂ R3 is defined by Cap(Ω) :=

• R3\Ω

|∇uΩ|2 where      ∆uΩ = in R3 \ Ω uΩ = 1

• n

∂Ω lim

|x|→+∞uΩ

= Is the disk D ⊂ R3 solution of : min

K convex of R3, P(K)=P0

Cap(K) ?

• J. Lamboley (Université Paris-Dauphine)

Optimal convex shapes 5 / 20

Examples

Reverse Isoperimetric problems : max

Ω⊂D, |Ω|=V0

P(Ω). Reverse Spectral problems : max

Ω⊂D, |Ω|=V0

λ1(Ω).

• J. Lamboley (Université Paris-Dauphine)

Optimal convex shapes 6 / 20

We want to analyze problem such as min

K convex of Rd J(K),

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 min

K convex of Rd J(K),

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

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

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

2-dimensional case A calculus of variations formulation

Linear Parametrization of the convexity

To a periodic function u : T → R∗

+, we associate

Ku =

• (r, θ) ; 0 ≤ r < 1/u(θ)
• .
• O

Ku θ

1 u(θ)

Parametrization of a starshaped set. Then Ku 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

Ku =

• (r, θ) ; 0 ≤ r < 1/u(θ)
• .
• O

Ku θ

1 u(θ)

Parametrization of a starshaped set. Then Ku 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 {2d convex sets}

∼

− → {v > 0 ∈ H1(T) such that v′′ + v ≥ 0} Ku − → 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

K∈Fad

K convex

J(K) min

u∈Sad

u′′+u≥0

• j(u) := J(Ku)
• where Sad is a functional space taking into account the other

constraints. Examples : Sad = {u : T → R / u2 ≤ u ≤ u1} × K2 K1 K Sad =

• u : T → R / |Ku| =
• T

1 2u2(θ)dθ = V0

• J. Lamboley (Université Paris-Dauphine)

Optimal convex shapes 12 / 20

2-dimensional case A calculus of variations formulation

New setting of the problem

min

K∈Fad

K convex

J(K) min

u∈Sad

u′′+u≥0

• j(u) := J(Ku)
• where Sad is a functional space taking into account the other

constraints. Examples : Sad = {u : T → R / u2 ≤ u ≤ u1} × K2 K1 K Sad =

• u : T → R / |Ku| =
• T

1 2u2(θ)dθ = V0

• J. Lamboley (Université Paris-Dauphine)

Optimal convex shapes 12 / 20

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

2-dimensional case Polygons as optimal shapes

Case of geometric functionals

min

u′′+u≥0 j(u) :=

• T

G(θ, u(θ), u′(θ))dθ

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, D1 ⊂ K ⊂ D2
• Application to Mahler in R2 (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

min

u′′+u≥0 j(u) :=

• T

G(θ, u(θ), u′(θ))dθ

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, D1 ⊂ K ⊂ D2
• Application to Mahler in R2 (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

min

u′′+u≥0 j(u) := J(Ku)

Theorem (L., Novruzi, Pierre, 2011)

We assume j smooth and j′′(u)(v, v) ≤ αv2

H1−a(T) − β|v|2 H1(T), for some β > 0 and 0 < a ≤ 1.

Then solutions are polygons. Application to min

• λ1(K) − P(K), K convex ⊂ D, |K| = V0
• J. Lamboley (Université Paris-Dauphine)

Optimal convex shapes 15 / 20

2-dimensional case Polygons as optimal shapes

Reverse Faber-Krahn

max

• λ1(K), K convex ⊂ D, |K| = V0
• We look at

j(u) := −λ1(Ku) + µ|Ku|

Lemma (L., Novruzi, Pierre, 2011)

If Ku is convex and v supported where ∂Ku is smooth, then − d2 du2 λ1(Ku) · (v, v) ≤ Cv2

L2(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

2-dimensional case Polygons as optimal shapes

Reverse Faber-Krahn

max

• λ1(K), K convex ⊂ D, |K| = V0
• We look at

j(u) := −λ1(Ku) + µ|Ku|

Lemma (L., Novruzi, Pierre, 2011)

If Ku is convex and v supported where ∂Ku is smooth, then − d2 du2 λ1(Ku) · (v, v) ≤ Cv2

L2(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

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

Higher dimensional case

About Pólya-Szegö conjecture

min

• J(K) := f(|K|, λ1(K), Cap(K)), K convex ⊂ Rd, P (K) = P0
• Theorem (Bucur, Fragalà, L. 2010)

Assume J is positive, (1-)homogeneous and smooth, and K0 is a solution. Then, if ∂K0 contains a relatively open set ω of class C2, 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

min

• J(K) := |K||K ◦|, K convex ⊂ Rd, K = −K
• ,

Theorem (Harrell, Henrot, L. 2011)

Let K0 be a minimizer. If ∂K0 contains a relatively open set ω of class C2, 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

Higher dimensional case

Open questions

max{λ1(K) / K convex ⊂ D, |K| = V0} in R2 ? Nowhere strictly convex in higher dimension ? Polyhedral solutions in higher dimension ?

• J. Lamboley (Université Paris-Dauphine)

Optimal convex shapes 20 / 20