Shape optimization under convexity constraint

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Shape optimization under convexity constraint

Jimmy LAMBOLEY University Paris-Dauphine

with D. Bucur, G. Carlier, I. Fragal` a, W. Gangbo, E. Harrell, A. Henrot, A. Novruzi, M. Pierre

07/10/2013, Ottawa

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Examples I

Isoperimetric inequality :

min

Ω ⊂ R

^N

, | Ω | =V

0

P (Ω).

Faber-Krahn inequality :

min

Ω ⊂ R

^N

, | Ω | =V

0

λ 1 (Ω).

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Examples I

Isoperimetric inequality :

min

Ω ⊂ R

^N

, | Ω | =V

0

P (Ω).

Faber-Krahn inequality :

min

Ω ⊂ R

^N

, | Ω | =V

0

λ 1 (Ω).

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Main goals

We seek to “solve” :

Ω min ∈F

ad

J(Ω),

where J is a shape functional, F ad ⊂ { domains of R ^N } .

Goals :

Existence of a minimizer ?

Geometrical/analytical informations on minimizers.

“Find” the minimizer (explicitly/numerically).

What about the case F ad ⊂ { convex domains of R ^N } .

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Main goals

We seek to “solve” :

Ω min ∈F

ad

J(Ω),

where J is a shape functional, F ad ⊂ { domains of R ^N } . Goals :

Existence of a minimizer ?

Geometrical/analytical informations on minimizers.

“Find” the minimizer (explicitly/numerically).

What about the case F ad ⊂ { convex domains of R ^N } .

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Main goals

We seek to “solve” :

Ω min ∈F

ad

J(Ω),

where J is a shape functional, F ad ⊂ { domains of R ^N } . Goals :

Existence of a minimizer ?

Geometrical/analytical informations on minimizers.

“Find” the minimizer (explicitly/numerically).

What about the case F ad ⊂ { convex domains of R ^N } .

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Main goals

We seek to “solve” :

Ω min ∈F

ad

J(Ω),

where J is a shape functional, F ad ⊂ { domains of R ^N } . Goals :

Existence of a minimizer ?

Geometrical/analytical informations on minimizers.

“Find” the minimizer (explicitly/numerically).

What about the case F ad ⊂ { convex domains of R ^N } .

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Main goals

We seek to “solve” :

Ω min ∈F

ad

J(Ω),

where J is a shape functional, F ad ⊂ { domains of R ^N } . Goals :

Existence of a minimizer ?

Geometrical/analytical informations on minimizers.

“Find” the minimizer (explicitly/numerically).

What about the case F ad ⊂ { convex domains of R ^N } .

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Outline

1 Regularity for isoperimetric problems Dimension 2

Higher dimension

2 Case of concave functionals

Higher Dimension

Dimension 2

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Regularity for isoperimetric problems

Outline

1 Regularity for isoperimetric problems Dimension 2

Higher dimension

2 Case of concave functionals

Higher Dimension

Dimension 2

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Examples II

K convex, min K ⊂ D

h

P(K ) + F (K ) i

where F (K ) = f ( | K | , λ ₁ (K )) for some smooth function f : R ² → R . min

K convex, K ⊂ D,| K | =V

0

h

P (K ) + G (K ) i

where G(K ) = g (λ ₁ (K )).

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Without convexity constraint

Theorem

Let Ω be a quasi-minimizer of the perimeter, i.e.

P(Ω) ≤ P (Ω ⁰ ) + Cr ^N ⁻ ^1+2α , ∀ Ω ⁰ such that Ω∆Ω ⁰ b B r (x) for all x and r small.

Then it exists Σ sing ⊂ ∂Ω such that

∂Ω \ Σ sing is C ^1,α

dim _H Σ _sing ≤ N − 8

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With convexity constraint

Results

K convex min , K ⊂ D

h

P (K ) + F (K ) i

min

K convex ,K ⊂ D, | K | =V

0

h

P (K ) + G (K ) i

Theorem (L.-Novruzi-Pierre 2011)

N = 2. The solutions of the first order optimality condition are C ^1,1 .

Theorem (L. 2013)

N ≥ 2. The minimizers are C ^1,1 .

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Regularity for isoperimetric problems Dimension 2

Outline

1 Regularity for isoperimetric problems Dimension 2

Higher dimension

2 Case of concave functionals

Higher Dimension

Dimension 2

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Linear formulation of the convexity

To a periodic function u : T → R ^∗ + , we associate K u = { (r, θ) ; 0 ≤ r < 1/u(θ) } .

O •

K u θ

1 u(θ)

Parametrization of a starshaped set.

Then K _u convex ⇔ u ⁰⁰ + u ≥ 0.

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Linear formulation of the convexity

To a periodic function u : T → R ^∗ + , we associate K u = { (r, θ) ; 0 ≤ r < 1/u(θ) } .

O •

K u θ

1 u(θ)

Parametrization of a starshaped set.

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Linear formulation of the convexity

There is a one-to-one correspondance

{ Convex 2d sets } −→ { ^∼ v > 0 ∈ H ¹ (T) such that v ⁰⁰ + v ≥ 0 }

K _u 7−→ u

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Reformulation of the problem

min

K∈Fad

K convex

J(K ) min

u∈Sad

u

⁰⁰

+u ≥ 0

n

j(u) := J(K _u ) o

where S ad is a functional space taking the other constraints into account.

Examples :

S ad = { u : T → R / u 2 ≤ u ≤ u 1 }

K ₂ K 1

K

S ad = n

u : T → R / | K u | = R

T 1

2u

²

(θ) d θ = V 0

o

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Reformulation of the problem

min

K∈Fad

K convex

J(K ) min

u∈Sad

u

⁰⁰

+u ≥ 0

n

j(u) := J(K _u ) o

where S ad is a functional space taking the other constraints into account.

Examples :

S ad = { u : T → R / u 2 ≤ u ≤ u 1 }

K ₂ K 1

K

S ad = n

u : T → R / | K u | = R

T 1

2u

²

(θ) d θ = V 0

o

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Proof of the regularity

min

j (u) = P (K _u ) + F (K _u ), u ⁰⁰ + u ≥ 0 . Write the first optimality condition :

( j ⁰ (u)(v ) = h ζ ⁰⁰ + ζ, v i D

⁰

×D

ζ ≥ 0, ζ (u ⁰⁰ + u) = 0

Observe that

j (u) = Z

T

G (u, u ⁰ ) + g (u)

with j ⁰ (u )(v ) = h− ∂ _u

⁰

_u

⁰

G (u, u ⁰ )u ⁰⁰ + h, v i D

⁰

×D and h ∈ L ^∞ .

Compare the signs of the singular measures in the optimality

condition.

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Proof of the regularity

min

j (u) = P (K _u ) + F (K _u ), u ⁰⁰ + u ≥ 0 . Write the first optimality condition :

( j ⁰ (u)(v ) = h ζ ⁰⁰ + ζ, v i D

⁰

×D

ζ ≥ 0, ζ (u ⁰⁰ + u) = 0 Observe that

j (u) = Z

T

G (u, u ⁰ ) + g (u)

with j ⁰ (u )(v ) = h− ∂ _u

⁰

_u

⁰

G (u, u ⁰ )u ⁰⁰ + h, v i D

⁰

×D and h ∈ L ^∞ .

Compare the signs of the singular measures in the optimality

condition.

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Proof of the regularity

min

j (u) = P (K _u ) + F (K _u ), u ⁰⁰ + u ≥ 0 . Write the first optimality condition :

( j ⁰ (u)(v ) = h ζ ⁰⁰ + ζ, v i D

⁰

×D

ζ ≥ 0, ζ (u ⁰⁰ + u) = 0 Observe that

j (u) = Z

T

G (u, u ⁰ ) + g (u)

with j ⁰ (u )(v ) = h− ∂ _u

⁰

_u

⁰

G (u, u ⁰ )u ⁰⁰ + h, v i D

⁰

×D and h ∈ L ^∞ .

Compare the signs of the singular measures in the optimality

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Regularity for isoperimetric problems Higher dimension

Outline

1 Regularity for isoperimetric problems Dimension 2

Higher dimension

2 Case of concave functionals

Higher Dimension

Dimension 2

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Regularity in calculus of variations

min Z

Ω

L(x, u(x), ∇ u (x)), u ∈ H ₀ ¹ (Ω)

L convex in ∇ u.

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Model case with convexity constraint

min 1

2 Z

Ω |∇ u | ² + Z

Ω

fu, u ∈ H ₀ ¹ (Ω) ∩ C

o` u C =

u : Ω → R , u convex . Theorem (Caffarelli-Carlier-Lions 2013)

If f is in L ^p (p > N), and u is a minimizer, then u is C ^1,1 ⁻ ^N/p .

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Proof of the regularity

min { P (K ) + F (K ), K convex } . Write a minimizer K as a graph that solves locally

min Z

D

q

1 + |∇ u | ² + f (u), u convex

.

Adapt the Caffarelli-Carlier-Lions procedure to a convex Lagrangian : built K _ε = graph(u _ε ).

Prove the estimate

| F (K ) − F (K _ε ) | ≤ C | K \ K _ε | .

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Proof of the regularity

min { P (K ) + F (K ), K convex } . Write a minimizer K as a graph that solves locally

min Z

D

q

1 + |∇ u | ² + f (u), u convex

.

Adapt the Caffarelli-Carlier-Lions procedure to a convex Lagrangian : built K _ε = graph(u _ε ).

Prove the estimate

| F (K ) − F (K _ε ) | ≤ C | K \ K _ε | .

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Proof of the regularity

min { P (K ) + F (K ), K convex } . Write a minimizer K as a graph that solves locally

min Z

D

q

1 + |∇ u | ² + f (u), u convex

.

Adapt the Caffarelli-Carlier-Lions procedure to a convex Lagrangian : built K _ε = graph(u _ε ).

Prove the estimate

| F (K ) − F (K _ε ) | ≤ C | K \ K _ε | .

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Case of concave functionals

Outline

1 Regularity for isoperimetric problems Dimension 2

Higher dimension

2 Case of concave functionals

Higher Dimension

Dimension 2

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Example III

Newton’s problem of the body of minimal resistance

min Z

D

1 1 + |∇ f | ² / f : { x ∈ R ² , | x | ≤ 1 } → [0, M ], f concave

Numerical computations : Lachand-Robert, Oudet, 2004 :

M = 3/2 M = 1

M = 7/10 M = 4/10

Figure 6: Computed solutions of Newton’s problem of the body of minimal resistance.

used ! = 300.) In this particular problem, this yields an overestimated value of the functional. Indeed if A ⊂ Ω

!

× [0, M ] is convex, then ˜ A := A ∩ Q

1

belongs to A , and F ( ˜ A) ≤ F (A) since f ≥ 0 and vanishes on ∂ A ˜ \ ∂A, where the normal vectors belong to { e

3

}

^⊥

. Obviously for a minimization problem,

hal-00385109, version 1 - 18 May 2009

J. Lamboley (University Paris-Dauphine) Optimal convex shapes 19 / 29

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Example IV

Mahler conjecture

Conjecture : is the cube Q _N := [ − 1, 1] ^N solution of min n

M(K ) := | K || K ^◦ | , K convex of R ^N , − K = K o

?

K ^◦ := n

ξ ∈ R ^N , h ξ, x i ≤ 1, ∀ x ∈ K o

.

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Example V

Conjecture of P´ olya-Szeg¨ o

The electrostatic capacity of a set Ω ⊂ R ³ is defined by

Cap(Ω) :=

Z

R

³

\ Ω |∇ u _Ω | ² where



 

 

∆u Ω = 0 in R ³ \ Ω

u _Ω = 1 on ∂Ω

| x |→ lim + ∞ u _Ω = 0

Is the disk D ⊂ R ³ solution of : min

K convex of R

³

, P(K)=P

0

Cap(K ) ?

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Example V

Conjecture of P´ olya-Szeg¨ o

The electrostatic capacity of a set Ω ⊂ R ³ is defined by

Cap(Ω) :=

Z

R

³

\ Ω |∇ u _Ω | ² where



 

 

∆u Ω = 0 in R ³ \ Ω

u _Ω = 1 on ∂Ω

| x |→ lim + ∞ u _Ω = 0 Is the disk D ⊂ R ³ solution of :

min

K convex of R

³

, P (K)=P

0

Cap(K ) ?

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Example VI : Toy Problem

min

D(a)⊂K⊂D(b)

K convex

h

µ | K | − P (K ) i .

If µ = 0 : the solution is the big disk (maximizes the perimeter).

If “µ = + ∞ ” : the solution is the small disk (minimizes the area).

What happens if µ ∈ (0, + ∞ ) ?

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Example VI : Toy Problem

min

D(a)⊂K⊂D(b)

K convex

h

µ | K | − P (K ) i .

If µ = 0 : the solution is the big disk (maximizes the perimeter).

If “µ = + ∞ ” : the solution is the small disk (minimizes the area).

What happens if µ ∈ (0, + ∞ ) ?

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Results

K convex min

h − P(K ) + F (K ) i

min

K convex , | K | =V

0

h − P (K ) + G (K ) i where F (K ) = f ( | K | , λ 1 (K )), G(K ) = g (λ 1 (K )).

Theorem (L.-Novruzi-Pierre 2011) N = 2. Minimizers are polygons.

Theorem (Bucur, Fragala, L. 2010)

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Case of concave functionals Higher Dimension

Outline

1 Regularity for isoperimetric problems Dimension 2

Higher dimension

2 Case of concave functionals

Higher Dimension

Dimension 2

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min

K convex

h

J(K ) = − P (K ) + F (K ) i

Theorem (Bucur, Fragal` a, L. 2010)

We assume that ω ⊂ ∂K is C ² . Then

J ⁰⁰ (K )(v, v) < 0 if v : ∂K → R has a small support in ω.

Corollary

If K is a minimizer and ω ⊂ ∂K is C ² , then the Gauss curvature vanishes on ω.

Application :

P´ olya-Szeg¨ o’s conjecture : F (K ) = Cap(K ) ² .

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min

K convex

h

J(K ) = − P (K ) + F (K ) i

Theorem (Bucur, Fragal` a, L. 2010)

We assume that ω ⊂ ∂K is C ² . Then

J ⁰⁰ (K )(v, v) < 0 if v : ∂K → R has a small support in ω.

Corollary

If K is a minimizer and ω ⊂ ∂K is C ² , then the Gauss curvature vanishes on ω.

Shape optimization under convexity constraint