New results about shape derivatives and convexity constraint

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New results about shape derivatives and convexity constraint

Jimmy LAMBOLEY University Paris-Dauphine

with A. Novruzi, M. Pierre

29/04/2016, Cortona

Geometric aspects of PDE’s and functional inequalities

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Main motivation

Convexity constraint in shape optimization

We are interested in problems of the form : min

n

J(Ω), Ω is convex , Ω ∈ F ad

o

where J is a shape functional.

Examples of constraints : F ad =

Ω, B (0, a) ⊂ Ω ⊂ B (0, b) , F ad =

| Ω | = V 0 , and Ω ⊂ B (0, b) .

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Main motivation

Convexity constraint in shape optimization

We are interested in problems of the form : min

n

J(Ω), Ω is convex , Ω ∈ F ad

o

where J is a shape functional.

Examples of constraints : F ad =

Ω, B (0, a) ⊂ Ω ⊂ B (0, b) , F ad =

| Ω | = V 0 , and Ω ⊂ B (0, b) .

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Outline

1 Convexity constraint Old, new, open problems Case of concave functionals Applications

2 Shape derivatives

Previous results (2-dim)

New results (N-dim)

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Convexity constraint

Outline

1 Convexity constraint Old, new, open problems Case of concave functionals Applications

2 Shape derivatives

Previous results (2-dim)

New results (N-dim)

(6)

Convexity constraint Old, new, open problems

Outline

1 Convexity constraint Old, new, open problems Case of concave functionals Applications

2 Shape derivatives

Previous results (2-dim)

New results (N-dim)

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Example I

Newton’s problem of the body of minimal resistance [1685]

min Z

D

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

D = { x ∈ R ² , | x | ≤ 1 }

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, this is not a predicament to overestimate the functional.

14 hal-00385109, version 1 - 18 May 2009

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Example I

Newton’s problem of the body of minimal resistance [1685]

min Z

D

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

D = { x ∈ R ² , | x | ≤ 1 }

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

hal-00385109, version 1 - 18 May 2009

J. Lamboley (University Paris-Dauphine) Optimal convex shapes 6 / 28

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Example II

Mahler’s conjecture [1939]

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

? where

K ^◦ := n

ξ ∈ R ^N , h ξ, x i ≤ 1, ∀ x ∈ K o .

Theorem (Mahler, 1939)

If N = 2, ok.

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Example II

Mahler’s conjecture [1939]

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

? where

K ^◦ := n

ξ ∈ R ^N , h ξ, x i ≤ 1, ∀ x ∈ K o .

Theorem (Mahler, 1939)

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Example III

P´ olya-Szeg¨ o’s conjecture [1951]

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

Cap(K ) :=

Z

R

³

\ K |∇ u K | ² where



 

 

∆u _K = 0 in R ³ \ K

u _K = 1 on ∂K

| x |→ lim + ∞ u K = 0

Is the disk D ⊂ R ³ solution of : min

Cap(K ), K convex of R ³ , P (K ) = P 0 ? or equivalently solution of

min

Cap(K ) ²

P (K ) , K convex

⇔ min

Cap(K ) ² − µP(K ), K convex

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Example III

P´ olya-Szeg¨ o’s conjecture [1951]

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

Cap(K ) :=

Z

R

³

\ K |∇ u K | ² where



 

 

∆u _K = 0 in R ³ \ K

u _K = 1 on ∂K

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

min

Cap(K ), K convex of R ³ , P (K ) = P ₀ ?

or equivalently solution of min

Cap(K ) ²

P (K ) , K convex

⇔ min

Cap(K ) ² − µP(K ), K convex

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Example III

P´ olya-Szeg¨ o’s conjecture [1951]

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

Cap(K ) :=

Z

R

³

\ K |∇ u K | ² where



 

 

∆u _K = 0 in R ³ \ K

u _K = 1 on ∂K

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

min

Cap(K ), K convex of R ³ , P (K ) = P ₀ ? or equivalently solution of

min

Cap(K ) ²

P (K ) , K convex

⇔ min

Cap(K ) ² − µP(K ), K convex

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Example IV

Variations on the charged liquid drop problem [Goldman, Novaga, Ruffini, 2016]

min { P (Ω) + Q ² I 0 (Ω), Ω convex ⊂ R ^N , | Ω | = V 0 } I 0 (Ω) = inf

µ ∈P (Ω)

Z

Ω × Ω

log 1

| x − y |

d µ(x)d µ(y) = 1 Lcap(Ω)

Existence ok

If N = 2, C ^1,1 -regularity of solutions Ball if Q is small enough

Convergence to a segment if Q → ∞ (after rescaling).

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Example IV

Variations on the charged liquid drop problem [Goldman, Novaga, Ruffini, 2016]

min { P (Ω) + Q ² I 0 (Ω), Ω convex ⊂ R ^N , | Ω | = V 0 } I 0 (Ω) = inf

µ ∈P (Ω)

Z

Ω × Ω

log 1

| x − y |

d µ(x)d µ(y) = 1 Lcap(Ω) Existence ok

If N = 2, C ^1,1 -regularity of solutions Ball if Q is small enough

Convergence to a segment if Q → ∞ (after rescaling).

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Example IV

Variations on the charged liquid drop problem [Goldman, Novaga, Ruffini, 2016]

min { P (Ω) + Q ² I 0 (Ω), Ω convex ⊂ R ^N , | Ω | = V 0 } I 0 (Ω) = inf

µ ∈P (Ω)

Z

Ω × Ω

log 1

| x − y |

d µ(x)d µ(y) = 1 Lcap(Ω) Existence ok

If N = 2, C ^1,1 -regularity of solutions

Ball if Q is small enough

Convergence to a segment if Q → ∞ (after rescaling).

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Example IV

Variations on the charged liquid drop problem [Goldman, Novaga, Ruffini, 2016]

min { P (Ω) + Q ² I 0 (Ω), Ω convex ⊂ R ^N , | Ω | = V 0 } I 0 (Ω) = inf

µ ∈P (Ω)

Z

Ω × Ω

log 1

| x − y |

d µ(x)d µ(y) = 1 Lcap(Ω) Existence ok

If N = 2, C ^1,1 -regularity of solutions Ball if Q is small enough

Convergence to a segment if Q → ∞ (after rescaling).

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Example V : Reverse isoperimetry

[Bianchini-Henrot 2012]

min n

µ | Ω | − P (Ω), Ω convex ⊂ R ^N , B(0, a) ⊂ Ω ⊂ B(0, b) o

If N = 2,

Big ball if µ small, Small ball if µ big,

Polygonal inside B(0, b) \ B (0, a) [L-Novruzi 2008]

Complete description available [Bianchini-Henrot] : ’true’ polygons

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Example V : Reverse isoperimetry

[Bianchini-Henrot 2012]

min n

µ | Ω | − P (Ω), Ω convex ⊂ R ^N , B(0, a) ⊂ Ω ⊂ B(0, b) o

If N = 2,

Big ball if µ small, Small ball if µ big,

Polygonal inside B(0, b) \ B (0, a) [L-Novruzi 2008]

Complete description available [Bianchini-Henrot] : ’true’ polygons

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Example V : Reverse isoperimetry

[Bianchini-Henrot 2012]

min n

µ | Ω | − P (Ω), Ω convex ⊂ R ^N , B(0, a) ⊂ Ω ⊂ B(0, b) o

If N = 2,

Big ball if µ small, Small ball if µ big,

Polygonal inside B(0, b) \ B (0, a) [L-Novruzi 2008]

Complete description available [Bianchini-Henrot] : ’true’ polygons

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Example V : Reverse isoperimetry

[Bianchini-Henrot 2012]

min n

µ | Ω | − P (Ω), Ω convex ⊂ R ^N , B(0, a) ⊂ Ω ⊂ B(0, b) o

If N = 2,

Big ball if µ small, Small ball if µ big,

Polygonal inside B(0, b) \ B (0, a) [L-Novruzi 2008]

Complete description available [Bianchini-Henrot] : ’true’ polygons

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Example VI

Variation on the Cheeger inequality [Parini, 2015]

min

λ ₁ (Ω)

h ₁ (Ω) ² , Ω convex ⊂ R ^N

If N = 2,

Existence ok

every connected component of ∂Ω ^∗ \ ∂C is made of two segments

understanding ∂Ω ^∗ ∩ ∂C ? Can it be strictly convex ?

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Example VI

Variation on the Cheeger inequality [Parini, 2015]

min

λ ₁ (Ω)

h ₁ (Ω) ² , Ω convex ⊂ R ^N

If N = 2,

Existence ok

every connected component of ∂Ω ^∗ \ ∂C is made of two segments

understanding ∂Ω ^∗ ∩ ∂C ? Can it be strictly convex ?

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Example VI

Variation on the Cheeger inequality [Parini, 2015]

min

λ ₁ (Ω)

h ₁ (Ω) ² , Ω convex ⊂ R ^N

If N = 2,

Existence ok

every connected component of ∂Ω ^∗ \ ∂C is made of two segments

understanding ∂Ω ^∗ ∩ ∂C ? Can it be strictly convex ?

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Example VI

Variation on the Cheeger inequality [Parini, 2015]

min

λ ₁ (Ω)

h ₁ (Ω) ² , Ω convex ⊂ R ^N

If N = 2,

Existence ok

every connected component of ∂Ω ^∗ \ ∂C is made of two segments

understanding ∂Ω ^∗ ∩ ∂C ? Can it be strictly convex ?

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Few formal remarks/results

Existence of a minimizer is usually easy.

’Non-existence’ may be difficult (Marini’s talk about τ and λ 1 , Fundamental Gap Theorem [Andrews-Clutterbuck 2011]).

To get geometrical/analytical informations on minimizers is usually difficult (optimality conditions ?)

If J = P + G where G is ‘smooth enough’, then we expect Ω ^∗ to be smooth (C ^1,1 ), [LNP 2011, GNR 2016, L. 201X ?]

Example IV : the charged drop like problem

If J is ’weakly concave’, then we expect Ω ^∗ to be ’weakly extremal’ Examples I,II,III,V, maybe VI

Use second order optimality condition !

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Few formal remarks/results

Existence of a minimizer is usually easy.

’Non-existence’ may be difficult

(Marini’s talk about τ and λ 1 , Fundamental Gap Theorem [Andrews-Clutterbuck 2011]).

To get geometrical/analytical informations on minimizers is usually difficult (optimality conditions ?)

If J = P + G where G is ‘smooth enough’, then we expect Ω ^∗ to be smooth (C ^1,1 ), [LNP 2011, GNR 2016, L. 201X ?]

Example IV : the charged drop like problem

If J is ’weakly concave’, then we expect Ω ^∗ to be ’weakly extremal’ Examples I,II,III,V, maybe VI

Use second order optimality condition !

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Few formal remarks/results

Existence of a minimizer is usually easy.

’Non-existence’ may be difficult (Marini’s talk about τ and λ 1 ,

Fundamental Gap Theorem [Andrews-Clutterbuck 2011]).

To get geometrical/analytical informations on minimizers is usually difficult (optimality conditions ?)

If J = P + G where G is ‘smooth enough’, then we expect Ω ^∗ to be smooth (C ^1,1 ), [LNP 2011, GNR 2016, L. 201X ?]

Example IV : the charged drop like problem

If J is ’weakly concave’, then we expect Ω ^∗ to be ’weakly extremal’ Examples I,II,III,V, maybe VI

Use second order optimality condition !

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Few formal remarks/results

Existence of a minimizer is usually easy.

’Non-existence’ may be difficult (Marini’s talk about τ and λ 1 , Fundamental Gap Theorem [Andrews-Clutterbuck 2011]).

To get geometrical/analytical informations on minimizers is usually difficult (optimality conditions ?)

If J = P + G where G is ‘smooth enough’, then we expect Ω ^∗ to be smooth (C ^1,1 ), [LNP 2011, GNR 2016, L. 201X ?]

Example IV : the charged drop like problem

If J is ’weakly concave’, then we expect Ω ^∗ to be ’weakly extremal’ Examples I,II,III,V, maybe VI

Use second order optimality condition !

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Few formal remarks/results

Existence of a minimizer is usually easy.

’Non-existence’ may be difficult (Marini’s talk about τ and λ 1 , Fundamental Gap Theorem [Andrews-Clutterbuck 2011]).

To get geometrical/analytical informations on minimizers is usually difficult (optimality conditions ?)

If J = P + G where G is ‘smooth enough’, then we expect Ω ^∗ to be smooth (C ^1,1 ), [LNP 2011, GNR 2016, L. 201X ?]

Example IV : the charged drop like problem

If J is ’weakly concave’, then we expect Ω ^∗ to be ’weakly extremal’ Examples I,II,III,V, maybe VI

Use second order optimality condition !

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Few formal remarks/results

Existence of a minimizer is usually easy.

’Non-existence’ may be difficult (Marini’s talk about τ and λ 1 , Fundamental Gap Theorem [Andrews-Clutterbuck 2011]).

To get geometrical/analytical informations on minimizers is usually difficult (optimality conditions ?)

If J = P + G where G is ‘smooth enough’, then we expect Ω ^∗ to be smooth (C ^1,1 ), [LNP 2011, GNR 2016, L. 201X ?]

Example IV : the charged drop like problem

If J is ’weakly concave’, then we expect Ω ^∗ to be ’weakly extremal’ Examples I,II,III,V, maybe VI

Use second order optimality condition !

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Few formal remarks/results

Existence of a minimizer is usually easy.

’Non-existence’ may be difficult (Marini’s talk about τ and λ 1 , Fundamental Gap Theorem [Andrews-Clutterbuck 2011]).

To get geometrical/analytical informations on minimizers is usually difficult (optimality conditions ?)

If J = P + G where G is ‘smooth enough’, then we expect Ω ^∗ to be smooth (C ^1,1 ), [LNP 2011, GNR 2016, L. 201X ?]

Example IV : the charged drop like problem

∗

Use second order optimality condition !

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Few formal remarks/results

Existence of a minimizer is usually easy.

’Non-existence’ may be difficult (Marini’s talk about τ and λ 1 , Fundamental Gap Theorem [Andrews-Clutterbuck 2011]).

To get geometrical/analytical informations on minimizers is usually difficult (optimality conditions ?)

If J = P + G where G is ‘smooth enough’, then we expect Ω ^∗ to be smooth (C ^1,1 ), [LNP 2011, GNR 2016, L. 201X ?]

Example IV : the charged drop like problem

If J is ’weakly concave’, then we expect Ω ^∗ to be ’weakly extremal’

Examples I,II,III,V, maybe VI

Use second order optimality condition !

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Convexity constraint Case of concave functionals

Outline

1 Convexity constraint Old, new, open problems Case of concave functionals Applications

2 Shape derivatives

Previous results (2-dim)

New results (N-dim)

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The gauge function

If N ≥ 2, and u : S ^N ⁻ ¹ → (0, ∞ ) is given, we define Ω u :=

(r, θ) ∈ [0, ∞ ) × S ^N ⁻ ¹ , r < 1 u(θ)

.

Then

Ω u is convex if and only if u is convex on S ^N ⁻ ¹ .

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The gauge function

If N ≥ 2, and u : S ^N ⁻ ¹ → (0, ∞ ) is given, we define Ω u :=

(r, θ) ∈ [0, ∞ ) × S ^N ⁻ ¹ , r < 1 u(θ)

. Then

Ω u is convex if and only if u is convex on S ^N ⁻ ¹ .

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Reformulation of the problem

min n

J(Ω), Ω convex , Ω ∈ F ad }

⇐⇒ min n

j (u ) := J(Ω u ), u convex , u ∈ S ad

o

For example,

S _ad = n

u : S ^N ⁻ ¹ → R , a ≤ 1/u ≤ b

o

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Reformulation of the problem

min n

J(Ω), Ω convex , Ω ∈ F ad }

⇐⇒ min n

j (u ) := J(Ω u ), u convex , u ∈ S ad

o

For example,

S _ad = n

u : S ^N ⁻ ¹ → R , a ≤ 1/u ≤ b

o

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Minimization of locally concave functionals

Let u 0 such that :

j (u ₀ ) = min n

j (u), u convex, u ∈ S _ad o

Theorem (L.,Novruzi,Pierre 2015) Assume, for any v ∈ W ^1,∞ ( S ^N ⁻ ¹ ),

j ⁰⁰ (u 0 )(v, v) ≤ − α | v | ² H

¹

( S

^N−1

) + β k v k ² H

^s

( S

^N−1

) . (H) for some α > 0, s ∈ [0, 1).Then the set

T _u

₀

= n

v/ ∃ ε > 0, ∀| t | < ε, u ₀ + tv is convex and ∈ S _ad o

,

is a linear vector space of finite dimension.

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Minimization of locally concave functionals

Let u 0 such that :

j (u ₀ ) = min n

j (u), u convex, u ∈ S _ad o

Theorem (L.,Novruzi,Pierre 2015) Assume, for any v ∈ W ^1,∞ ( S ^N ⁻ ¹ ),

j ⁰⁰ (u 0 )(v, v) ≤ − α | v | ² H

¹

( S

^N−1

) + β k v k ² H

^s

( S

^N−1

) . (H) for some α > 0, s ∈ [0, 1).

Then the set T _u

₀

= n

v/ ∃ ε > 0, ∀| t | < ε, u ₀ + tv is convex and ∈ S _ad o

,

is a linear vector space of finite dimension.

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Minimization of locally concave functionals

Let u 0 such that :

j (u ₀ ) = min n

j (u), u convex, u ∈ S _ad o

Theorem (L.,Novruzi,Pierre 2015) Assume, for any v ∈ W ^1,∞ ( S ^N ⁻ ¹ ),

j ⁰⁰ (u 0 )(v, v) ≤ − α | v | ² H

¹

( S

^N−1

) + β k v k ² H

^s

( S

^N−1

) . (H) for some α > 0, s ∈ [0, 1).Then the set

T _u

₀

= n

v/ ∃ ε > 0, ∀| t | < ε, u ₀ + tv is convex and ∈ S _ad o

,

is a linear vector space of finite dimension.

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Consequences

Let u 0 such that T u

0

is of finite dimension.

N = 2, u ₀ can be seen as a 2π-periodic function. Lemma (Lachand-Robert Peletier, L. Novruzi Pierre)

Then u ⁰⁰ ₀ + u 0 is a sum of Dirac masses, or equivalently Ω u

0

is polygonal, inside B(0, b) \ B(0, a).

N ≥ 3 Lemma (easy)

Then if ω ⊂ ∂Ω _u

₀

is C ² , then the Gauss curvature vanishes on ω

Problem : Ω u

0

is not necessarily a polyhedra.

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Consequences

Let u 0 such that T u

0

is of finite dimension.

N = 2, u ₀ can be seen as a 2π-periodic function.

Lemma (Lachand-Robert Peletier, L. Novruzi Pierre)

Then u ⁰⁰ ₀ + u 0 is a sum of Dirac masses, or equivalently Ω u

0

is polygonal, inside B(0, b) \ B(0, a).

N ≥ 3 Lemma (easy)

Then if ω ⊂ ∂Ω _u

₀

is C ² , then the Gauss curvature vanishes on ω

Problem : Ω u

0

is not necessarily a polyhedra.

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Consequences

Let u 0 such that T u

0

is of finite dimension.

N = 2, u ₀ can be seen as a 2π-periodic function.

Lemma (Lachand-Robert Peletier, L. Novruzi Pierre) Then u ⁰⁰ ₀ + u 0 is a sum of Dirac masses,

or equivalently Ω u

0

is polygonal, inside B(0, b) \ B(0, a).

N ≥ 3 Lemma (easy)

Then if ω ⊂ ∂Ω _u

₀

is C ² , then the Gauss curvature vanishes on ω

Problem : Ω u

0

is not necessarily a polyhedra.

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Consequences

Let u 0 such that T u

0

is of finite dimension.

N = 2, u ₀ can be seen as a 2π-periodic function.

Lemma (Lachand-Robert Peletier, L. Novruzi Pierre)

Then u ⁰⁰ ₀ + u 0 is a sum of Dirac masses, or equivalently Ω u

0

is polygonal, inside B(0, b) \ B(0, a).

N ≥ 3 Lemma (easy)

Then if ω ⊂ ∂Ω _u

₀

is C ² , then the Gauss curvature vanishes on ω

Problem : Ω u

0

is not necessarily a polyhedra.

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Consequences

Let u 0 such that T u

0

is of finite dimension.

N = 2, u ₀ can be seen as a 2π-periodic function.

Lemma (Lachand-Robert Peletier, L. Novruzi Pierre)

Then u ⁰⁰ ₀ + u 0 is a sum of Dirac masses, or equivalently Ω u

0

is polygonal, inside B(0, b) \ B(0, a).

N ≥ 3 Lemma (easy)

Problem : Ω u

0

is not necessarily a polyhedra.

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Consequences

Let u 0 such that T u

0

is of finite dimension.

N = 2, u ₀ can be seen as a 2π-periodic function.

Lemma (Lachand-Robert Peletier, L. Novruzi Pierre)

Then u ⁰⁰ ₀ + u 0 is a sum of Dirac masses, or equivalently Ω u

0

is polygonal, inside B(0, b) \ B(0, a).

N ≥ 3 Lemma (easy)

Then if ω ⊂ ∂Ω _u

₀

is C ² , then the Gauss curvature vanishes on ω

Problem : Ω is not necessarily a polyhedra.

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Convexity constraint Applications

Outline

1 Convexity constraint Old, new, open problems Case of concave functionals Applications

2 Shape derivatives

Previous results (2-dim)

New results (N-dim)

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Results

min n

J(Ω) := − P (Ω) + F (Ω), Ω convex o

where F (K ) = f ( | K | , λ 1 (K )).

Theorem (L.-Novruzi-Pierre 2011-2015) J satisfies (H).

Example V, reverse isoperimetry

min { J(Ω) := − P (Ω) + µ | Ω | ; Ω convex in R ^N } . Example V bis

min { J(Ω) := − P (Ω) + µλ 1 (Ω) ; Ω convex in R ^N , | Ω | = V 0 } .

Example VI (Enea’s problem) ?

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Results

min n

J(Ω) := − P (Ω) + F (Ω), Ω convex o

where F (K ) = f ( | K | , λ 1 (K )).

Theorem (L.-Novruzi-Pierre 2011-2015) J satisfies (H).

Example V, reverse isoperimetry

min { J(Ω) := − P (Ω) + µ | Ω | ; Ω convex in R ^N } . Example V bis

min { J(Ω) := − P (Ω) + µλ 1 (Ω) ; Ω convex in R ^N , | Ω | = V 0 } .

Example VI (Enea’s problem) ?

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Results

min n

J(Ω) := − P (Ω) + F (Ω), Ω convex o

where F (K ) = f ( | K | , λ 1 (K )).

Theorem (L.-Novruzi-Pierre 2011-2015) J satisfies (H).

Example V, reverse isoperimetry

min { J(Ω) := − P (Ω) + µ | Ω | ; Ω convex in R ^N } .

Example V bis

min { J(Ω) := − P (Ω) + µλ 1 (Ω) ; Ω convex in R ^N , | Ω | = V 0 } .

Example VI (Enea’s problem) ?

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Results

min n

J(Ω) := − P (Ω) + F (Ω), Ω convex o

where F (K ) = f ( | K | , λ 1 (K )).

Theorem (L.-Novruzi-Pierre 2011-2015) J satisfies (H).

Example V, reverse isoperimetry

min { J(Ω) := − P (Ω) + µ | Ω | ; Ω convex in R ^N } . Example V bis

Example VI (Enea’s problem) ?

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Results

min n

J(Ω) := − P (Ω) + F (Ω), Ω convex o

where F (K ) = f ( | K | , λ 1 (K )).

Theorem (L.-Novruzi-Pierre 2011-2015) J satisfies (H).

Example V, reverse isoperimetry

min { J(Ω) := − P (Ω) + µ | Ω | ; Ω convex in R ^N } . Example V bis

min { J(Ω) := − P (Ω) + µλ ₁ (Ω) ; Ω convex in R ^N , | Ω | = V ₀ } .

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Shape derivatives

Outline

1 Convexity constraint Old, new, open problems Case of concave functionals Applications

2 Shape derivatives

Previous results (2-dim)

New results (N-dim)

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Shape derivatives Previous results (2-dim)

Outline

1 Convexity constraint Old, new, open problems Case of concave functionals Applications

2 Shape derivatives

Previous results (2-dim)

New results (N-dim)

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2-dim result

The perimeter, the volume

p(u) = P (Ω u ) = Z

S

¹

√ u ² + u ⁰ ²

u ² d θ, a(u) = | Ω u | = 1 2

Z

S

¹

1 u ² d θ

For any u ₀ such that a ≤ 1/u ₀ ≤ b, p ⁰⁰ (u 0 )(v, v) ≥ α | v | ² _H

¹

− β k v k ² _L

²

| a ⁰⁰ (u ₀ )(v, v) | ≤ β k v k ² _L

²

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2-dim result

The perimeter, the volume

p(u) = P (Ω u ) = Z

S

¹

√ u ² + u ⁰ ²

u ² d θ, a(u) = | Ω u | = 1 2

Z

S

¹

1 u ² d θ For any u ₀ such that a ≤ 1/u ₀ ≤ b,

p ⁰⁰ (u 0 )(v , v) ≥ α | v | ² _H

¹

− β k v k ² _L

²

| a ⁰⁰ (u ₀ )(v, v) | ≤ β k v k ² _L

²

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2-dim result

PDE functional

`(u) := λ ₁ (Ω _u ) Theorem (L., Novruzi, Pierre, 2011)

Let Ω _u be convex in R ² . Then for any ε > 0, there exist β such that

| ` ⁰⁰ (u)(v, v) | ≤ β k v k ²

H

¹²^+ε

.

Applications :

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Shape derivatives New results (N-dim)

Outline

1 Convexity constraint Old, new, open problems Case of concave functionals Applications

2 Shape derivatives

Previous results (2-dim)

New results (N-dim)

(60)

N-dim result

The perimeter, the volume

p(u) = Z

S

^N−1

p u ² + |∇ τ u | ²

u ^N d θ, m(u) = 1 N

Z

S

^N−1

1 u ^N d θ,

For any u ₀ such that a ≤ 1/u ₀ ≤ b, p ⁰⁰ (u 0 )(v, v) ≥ α | v | ² _H

¹

− β k v k ² _L

²

| a ⁰⁰ (u 0 )(v, v) | ≤ β k v k ² _L

²

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N-dim result

The perimeter, the volume

p(u) = Z

S

^N−1

p u ² + |∇ τ u | ²

u ^N d θ, m(u) = 1 N

Z

S

^N−1

1 u ^N d θ, For any u ₀ such that a ≤ 1/u ₀ ≤ b,

p ⁰⁰ (u ₀ )(v , v) ≥ α | v | ² _H

¹

− β k v k ² _L

²

| a ⁰⁰ (u 0 )(v, v) | ≤ β k v k ² _L

²

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N-dim result

PDE functional, convex domains

`(u) := λ ₁ (Ω _u ) (for example)

Theorem (L., Novruzi, Pierre, 2015)

Let Ω _u be semi-convex in R ^N (exterior ball condition). Then there exist β such that

| ` ⁰⁰ (u)(v , v) | ≤ β k v k ²

H

¹²

.

Main tools for the proof :

New way to estimate shape derivative

W ^1,p regularity theory for elliptic PDE in semi-convex domains (valid

for any p ∈ (1, ∞ )).

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N-dim result

PDE functional, convex domains

`(u) := λ ₁ (Ω _u ) (for example)

Theorem (L., Novruzi, Pierre, 2015)

Let Ω _u be semi-convex in R ^N (exterior ball condition). Then there exist β such that

| ` ⁰⁰ (u)(v , v) | ≤ β k v k ²

H

¹²

. Main tools for the proof :

New way to estimate shape derivative

W ^1,p regularity theory for elliptic PDE in semi-convex domains (valid

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N-dim result

PDE functional, lipschitz domains

Theorem (L., Novruzi, Pierre, 2015)

Let Ω _u be Lipschitz in R ² . Then there exist β and ε > 0 such that

| ` ⁰⁰ (u)(v , v) | ≤ β k v k ² _H

^1−ε

.

Allows to deal with Exterior PDE problems

Estimate available in R ^N , but too weak to handle Example III

(P´ olya-Sz¨ ego)

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N-dim result

PDE functional, lipschitz domains

Theorem (L., Novruzi, Pierre, 2015)

Let Ω _u be Lipschitz in R ² . Then there exist β and ε > 0 such that

| ` ⁰⁰ (u)(v , v) | ≤ β k v k ² _H

^1−ε

. Allows to deal with Exterior PDE problems

Estimate available in R ^N , but too weak to handle Example III

(P´ olya-Sz¨ ego)

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N-dim result

PDE functional, lipschitz domains

Theorem (L., Novruzi, Pierre, 2015)

Let Ω _u be Lipschitz in R ² . Then there exist β and ε > 0 such that

| ` ⁰⁰ (u)(v , v) | ≤ β k v k ² _H

^1−ε

. Allows to deal with Exterior PDE problems

Estimate available in R ^N , but too weak to handle Example III

(P´ olya-Sz¨ ego)

(67)

Perspectives - More open problems

Reverse Faber-Krahn inequality

max { λ 1 (K ) / K convex ⊂ B(0, b), | K | = V 0 }

Polyhedra in dimension ≥ 3 ?

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Perspectives - More open problems

Reverse Faber-Krahn inequality

max { λ 1 (K ) / K convex ⊂ B(0, b), | K | = V 0 }

Polyhedra in dimension ≥ 3 ?

New results about shape derivatives and convexity constraint