Optimal compliance problem

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Optimal compliance problem

Antonin Chambolle , Jimmy Lamboley , Antoine Lemenant , Eugene Stepanov

Universit´e Paris Dauphine,CEREMADE

Octobre 2015, ENS Rennes

Jimmy Lamboley (Paris-Dauphine) Octobre 2015, ENS Rennes 1 / 14

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Statement of the problem

We consider the optimization problem min

n C (Σ), Σ ∈ A (Ω), H ¹ (Σ) ≤ L o

where

Ω is a bounded set in R ² and A (Ω) :=

n

Σ ⊂ Ω, Σ compact and connected o

C (Σ) = Z

Ω

fu _Σ where

( − ∆u _Σ = f in Ω \ Σ

u _Σ = 0 on ∂Ω ∪ Σ

H ¹ (Σ) = length of Σ.

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Statement of the problem

We consider the optimization problem min

n C (Σ), Σ ∈ A (Ω), H ¹ (Σ) ≤ L o

where

Ω is a bounded set in R ² and A (Ω) :=

n

Σ ⊂ Ω, Σ compact and connected o

C (Σ) = Z

Ω

fu _Σ where

( − ∆u _Σ = f in Ω \ Σ u _Σ = 0 on ∂Ω ∪ Σ H ¹ (Σ) = length of Σ.

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Statement of the problem

We consider the optimization problem min

n C (Σ), Σ ∈ A (Ω), H ¹ (Σ) ≤ L o

where

Ω is a bounded set in R ² and A (Ω) :=

n

Σ ⊂ Ω, Σ compact and connected o

C (Σ) = Z

Ω

fu _Σ where

( − ∆u _Σ = f in Ω \ Σ u _Σ = 0 on ∂Ω ∪ Σ

H ¹ (Σ) = length of Σ.

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Statement of the problem

We consider the optimization problem min

n C (Σ), Σ ∈ A (Ω), H ¹ (Σ) ≤ L o

where

Ω is a bounded set in R ² and A (Ω) :=

n

Σ ⊂ Ω, Σ compact and connected o

C (Σ) = Z

Ω

fu _Σ where

( − ∆u _Σ = f in Ω \ Σ u _Σ = 0 on ∂Ω ∪ Σ H ¹ (Σ) = length of Σ.

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Questions

Existence : Blaschke + Sver´ ak + Golab

Behavior when L → ∞ :

G. Buttazzo, F. Santambrogio, Asymptotical compliance optimization for connected networks, Netw. Heterog. Media, 2007

Geometrical description of the solution : Saturation of the constraint ?

Are the optimal sets regular ?

Are there loops in the optimal set ?

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Questions

Existence : Blaschke + Sver´ ak + Golab Behavior when L → ∞ :

G. Buttazzo, F. Santambrogio, Asymptotical compliance optimization for connected networks, Netw. Heterog. Media, 2007

Geometrical description of the solution : Saturation of the constraint ?

Are the optimal sets regular ? Are there loops in the optimal set ?

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Questions

Existence : Blaschke + Sver´ ak + Golab Behavior when L → ∞ :

G. Buttazzo, F. Santambrogio, Asymptotical compliance optimization for connected networks, Netw. Heterog. Media, 2007

Geometrical description of the solution : Saturation of the constraint ?

Are the optimal sets regular ?

Are there loops in the optimal set ?

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Our result

About a penalized version

min n

C (Σ) + λ H ¹ (Σ), Σ ∈ A (Ω) o

Theorem

Assume f ∈ L

^p

(Ω) with p > 2, and Σ

_opt

is optimal. Then Σ

_opt

contains no closed curves,

Σ

_opt

consists in a finite number of C ^1,α -curves, possibly intersecting at “triple points” where the curves form 120

^◦

angles.

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Our result

About a penalized version

min n

C (Σ) + λ H ¹ (Σ), Σ ∈ A (Ω) o

Theorem

Assume f ∈ L

^p

(Ω) with p > 2, and Σ

_opt

is optimal. Then Σ

_opt

contains no closed curves,

Σ

_opt

consists in a finite number of C ^1,α -curves, possibly intersecting

at “triple points” where the curves form 120

^◦

angles.

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Outline

1

Monotonicity formula, No loop

3

Regularity : main ingredients

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Average distance problem

Irrigation problem

min Z

Ω

dist(x, Σ)f (x)dx + λ H ¹ (Σ), Σ ∈ A (Ω)

.

G. Buttazzo, E. Stepanov, Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem, Ann. Sc. Norm. Super. Pisa Cl. Sci., 2003, F. Santambrogio, P. Tilli. Blow-up of optimal sets in the irrigation problem, J. Geom. Anal., 2005

D. Slepcev, Counterexample to regularity in average-distance problem, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 2014

No loop,

Classification of blow-ups,

No full-regularity in general (corners),

Γ-limit of the p-compliance problem when p → ∞ .

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Average distance problem

Irrigation problem

min Z

Ω

dist(x, Σ)f (x)dx + λ H ¹ (Σ), Σ ∈ A (Ω)

.

G. Buttazzo, E. Stepanov, Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem, Ann. Sc. Norm. Super. Pisa Cl. Sci., 2003, F. Santambrogio, P. Tilli. Blow-up of optimal sets in the irrigation problem, J.

Geom. Anal., 2005

D. Slepcev, Counterexample to regularity in average-distance problem, Ann. Inst.

H. Poincar´ e Anal. Non Lin´ eaire, 2014

No loop,

Classification of blow-ups,

No full-regularity in general (corners),

Γ-limit of the p-compliance problem when p → ∞ .

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Average distance problem

Irrigation problem

min Z

Ω

dist(x, Σ)f (x)dx + λ H ¹ (Σ), Σ ∈ A (Ω)

.

G. Buttazzo, E. Stepanov, Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem, Ann. Sc. Norm. Super. Pisa Cl. Sci., 2003, F. Santambrogio, P. Tilli. Blow-up of optimal sets in the irrigation problem, J.

Geom. Anal., 2005

D. Slepcev, Counterexample to regularity in average-distance problem, Ann. Inst.

H. Poincar´ e Anal. Non Lin´ eaire, 2014

No loop,

Classification of blow-ups,

No full-regularity in general (corners),

Γ-limit of the p-compliance problem when p → ∞ .

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Mumford-Shah problem

Segmentation

min 1

2 Z

Ω |∇ u | ² dx + Z

Ω

(u − g ) ² dx + H ¹ (Σ),

Σ ⊂ Ω compact, u ∈ H ¹ (Ω \ Σ)

Conjecture (open) : Σ is a finite union of C ¹ -curves Classification of connected blow-up limits

A. Bonnet, On the regularity of edges in image segmentation, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 1996

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Mumford-Shah problem

Segmentation

min 1

2 Z

Ω |∇ u | ² dx + Z

Ω

(u − g ) ² dx + H ¹ (Σ),

Σ ⊂ Ω compact, u ∈ H ¹ (Ω \ Σ)

Conjecture (open) : Σ is a finite union of C ¹ -curves Classification of connected blow-up limits

A. Bonnet, On the regularity of edges in image segmentation, Ann. Inst. H.

Poincar´ e Anal. Non Lin´ eaire, 1996

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Monotonicity formula, No loop

Outline

1

Monotonicity formula, No loop

3

Regularity : main ingredients

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Monotonicity formula

Estimate of the energy

Let 0 ≤ r ₀ < r ₁ and x ₀ such that

∀ r ∈ [r 0 , r 1 ], Σ ∩ ∂B

r

(x 0 ) 6 = ∅ . and γ ∈ (0, 2π] such that

γ ≥ sup

H ¹ (S )

r : r ∈ (r ₀ , r ₁ ) and S connected ⊂ ∂B

_r

(x ₀ ) \ Σ

, Lemma

The function

r ∈ [r ₀ , r ₁ ] 7→ 1 r

^2π^γ

Z

Br

(x

0

) |∇ u _Σ | ² dx + Cr

2 p0

! ,

is nondecreasing for some C = C ( | Ω | , p, k f k

p

, γ).

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Monotonicity formula

Estimate of the energy

Let 0 ≤ r ₀ < r ₁ and x ₀ such that

∀ r ∈ [r 0 , r 1 ], Σ ∩ ∂B

r

(x 0 ) 6 = ∅ .

and γ ∈ (0, 2π] such that γ ≥ sup

H ¹ (S )

r : r ∈ (r ₀ , r ₁ ) and S connected ⊂ ∂B

_r

(x ₀ ) \ Σ

, Lemma

The function

r ∈ [r ₀ , r ₁ ] 7→ 1 r

^2π^γ

Z

Br

(x

0

) |∇ u _Σ | ² dx + Cr

2 p0

! ,

is nondecreasing for some C = C ( | Ω | , p, k f k

p

, γ).

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Monotonicity formula

Estimate of the energy

Let 0 ≤ r ₀ < r ₁ and x ₀ such that

∀ r ∈ [r 0 , r 1 ], Σ ∩ ∂B

r

(x 0 ) 6 = ∅ . and γ ∈ (0, 2π] such that

γ ≥ sup

H ¹ (S )

r : r ∈ (r ₀ , r ₁ ) and S connected ⊂ ∂B

_r

(x ₀ ) \ Σ

,

Lemma

The function

r ∈ [r ₀ , r ₁ ] 7→ 1 r

^2π^γ

Z

Br

(x

0

) |∇ u _Σ | ² dx + Cr

2 p0

! ,

is nondecreasing for some C = C ( | Ω | , p, k f k

p

, γ).

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Monotonicity formula

Estimate of the energy

Let 0 ≤ r ₀ < r ₁ and x ₀ such that

∀ r ∈ [r 0 , r 1 ], Σ ∩ ∂B

r

(x 0 ) 6 = ∅ . and γ ∈ (0, 2π] such that

γ ≥ sup

H ¹ (S )

r : r ∈ (r ₀ , r ₁ ) and S connected ⊂ ∂B

_r

(x ₀ ) \ Σ

, Lemma

The function

r ∈ [r ₀ , r ₁ ] 7→ 1 r

^2π^γ

Z

Br

(x

0

) |∇ u _Σ | ² dx + Cr

2 p0

! ,

is nondecreasing for some C = C ( | Ω | , p, k f k

p

, γ).

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Variations of the compliance

Proposition

For every Σ

⁰

∈ A (Ω) satisfying Σ∆Σ

⁰

⊂ B

r

(x 0 ) we have

|C (Σ

⁰

) − C (Σ) | ≤ C

r

^2π^γ

+ r

2 p0

where C is depending on Ω, f , γ , p, r ₁ .

No loop’s proof : γ = π + ε λr ≤ λ H ¹ (Σ

opt

) − H ¹ (Σ

^r_opt

)

≤ C (Σ

^r_opt

) − C (Σ

opt

) ≤ C

r ^2−ε + r

2 p0

Contradiction

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Variations of the compliance

Proposition

For every Σ

⁰

∈ A (Ω) satisfying Σ∆Σ

⁰

⊂ B

r

(x 0 ) we have

|C (Σ

⁰

) − C (Σ) | ≤ C

r

^2π^γ

+ r

2 p0

where C is depending on Ω, f , γ , p, r ₁ .

No loop’s proof : γ = π + ε

λr ≤ λ H ¹ (Σ

opt

) − H ¹ (Σ

^r_opt

)

≤ C (Σ

^r_opt

) − C (Σ

opt

) ≤ C

r ^2−ε + r

2 p0

Contradiction

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Variations of the compliance

Proposition

For every Σ

⁰

∈ A (Ω) satisfying Σ∆Σ

⁰

⊂ B

r

(x 0 ) we have

|C (Σ

⁰

) − C (Σ) | ≤ C

r

^2π^γ

+ r

2 p0

where C is depending on Ω, f , γ , p, r ₁ .

No loop’s proof : γ = π + ε λr ≤ λ H ¹ (Σ

opt

) − H ¹ (Σ

^r_opt

)

≤ C (Σ

^r_opt

) − C (Σ

opt

) ≤ C

r ^2−ε + r

2 p0

Contradiction

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Variations of the compliance

Proposition

For every Σ

⁰

∈ A (Ω) satisfying Σ∆Σ

⁰

⊂ B

r

(x 0 ) we have

|C (Σ

⁰

) − C (Σ) | ≤ C

r

^2π^γ

+ r

2 p0

where C is depending on Ω, f , γ , p, r ₁ .

No loop’s proof : γ = π + ε λr ≤ λ H ¹ (Σ

opt

) − H ¹ (Σ

^r_opt

)

≤ C (Σ

^r_opt

) − C (Σ

opt

) ≤ C

r ^2−ε + r

2 p0

Contradiction

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Regularity : main ingredients

Outline

1

Monotonicity formula, No loop

3

Regularity : main ingredients

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Flatness implies regularity

Flatness : β

Σ

(x, r) := inf 1

r d

H

(Σ ∩ B

r

(x), P), lines P

ω

Σ

(x, r ) = max ( 1

r Z

Br(x)

|∇ u

Σ⁰

|

²

dx, Σ

⁰

∈ A (Ω), Σ

⁰

∆Σ ⊂ B

r

(x) )

. β

Σ

and ω

Σ

small enough implies

× Σ P

0

x

0

B

r1

(x

0

)

×

× x

1

x

2

Σ

^′

Figure : Construction of the competitor Σ

⁰

.

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Flatness implies regularity

Flatness : β

Σ

(x, r) := inf 1

r d

H

(Σ ∩ B

r

(x), P), lines P

ω

Σ

(x, r ) = max ( 1

r Z

Br(x)

|∇ u

Σ⁰

|

²

dx, Σ

⁰

∈ A (Ω), Σ

⁰

∆Σ ⊂ B

r

(x) )

.

β

Σ

and ω

Σ

small enough implies

× Σ P

0

x

0

B

r1

(x

0

)

×

× x

1

x

2

Σ

^′

Figure : Construction of the competitor Σ

⁰

.

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Flatness implies regularity

Flatness : β

Σ

(x, r) := inf 1

r d

H

(Σ ∩ B

r

(x), P), lines P

ω

Σ

(x, r ) = max ( 1

r Z

Br(x)

|∇ u

Σ⁰

|

²

dx, Σ

⁰

∈ A (Ω), Σ

⁰

∆Σ ⊂ B

r

(x) )

. β

Σ

and ω

Σ

small enough implies

× Σ P

0

x

0

B

r1

(x

0

)

×

× x

1

x

2

Σ

^′

Figure : Construction of the competitor Σ

⁰

.

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Classification of blow-ups

Around x

0

= 0 : Σ

n

:= 1

r

n

Σ, Ω

n

= 1 r

n

Ω,

u

n

(x ) := r

⁻

1

n 2

u(r

n

x) ∈ H

₀¹

(Ω

n

\ Σ

n

),

f

n

(x) = r

_n^3/2

f (r

n

x).

Theorem

Any blow-up limit (u

₀

, Σ

₀

) is one of the following list :

1

Σ

₀

is a line and u

₀

is a constant on each side.

2

Σ

0

is a propeller (three half lines meeting by 3 by 120 degree angles) and u

0

is a constant on each side.

3

Σ

0

is a half-line and u

0

is the “Dirichlet-craktip” function p

r /2π cos(θ/2)

in polar coordinates.

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Classification of blow-ups

Around x

0

= 0 : Σ

n

:= 1

r

n

Σ, Ω

n

= 1 r

n

Ω,

u

n

(x ) := r

⁻

1

n 2

u(r

n

x) ∈ H

₀¹

(Ω

n

\ Σ

n

),

f

n

(x) = r

_n^3/2

f (r

n

x).

Theorem

Any blow-up limit (u

₀

, Σ

₀

) is one of the following list :

1

Σ

₀

is a line and u

₀

is a constant on each side.

2

Σ

0

is a propeller (three half lines meeting by 3 by 120 degree angles) and u

0

is a constant on each side.

3

Σ

0

is a half-line and u

0

is the “Dirichlet-craktip” function p

r /2π cos(θ/2) in polar coordinates.

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Classification of blow-ups

Around x

0

= 0 : Σ

n

:= 1

r

n

Σ, Ω

n

= 1 r

n

Ω,

u

n

(x ) := r

⁻

1

n 2

u(r

n

x) ∈ H

₀¹

(Ω

n

\ Σ

n

),

f

n

(x) = r

_n^3/2

f (r

n

x).

Theorem

Any blow-up limit (u

₀

, Σ

₀

) is one of the following list :

1

Σ

₀

is a line and u

₀

is a constant on each side.

2

Σ

0

is a propeller (three half lines meeting by 3 by 120 degree angles) and u

0

is a constant on each side.

3

Σ

0

is a half-line and u

0

is the “Dirichlet-craktip” function p

r /2π cos(θ/2)

in polar coordinates.

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Classification of blow-ups

Around x

0

= 0 : Σ

n

:= 1

r

n

Σ, Ω

n

= 1 r

n

Ω,

u

n

(x ) := r

⁻

1

n 2

u(r

n

x) ∈ H

₀¹

(Ω

n

\ Σ

n

),

f

n

(x) = r

_n^3/2

f (r

n

x).

Theorem

Any blow-up limit (u

₀

, Σ

₀

) is one of the following list :

1

Σ

₀

is a line and u

₀

is a constant on each side.

2

Σ

0

is a propeller (three half lines meeting by 3 by 120 degree angles) and u

0

is a constant on each side.

3

Σ

0

is a half-line and u

0

is the “Dirichlet-craktip” function p

r /2π cos(θ/2) in polar coordinates.

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Classification of blow-ups

Around x

0

= 0 : Σ

n

:= 1

r

n

Σ, Ω

n

= 1 r

n

Ω,

u

n

(x ) := r

⁻

1

n 2

u(r

n

x) ∈ H

₀¹

(Ω

n

\ Σ

n

),

f

n

(x) = r

_n^3/2

f (r

n

x).

Theorem

Any blow-up limit (u

₀

, Σ

₀

) is one of the following list :

1

Σ

₀

is a line and u

₀

is a constant on each side.

2

Σ

0

is a propeller (three half lines meeting by 3 by 120 degree angles) and u

0

is a constant on each side.

3

Σ

0

is a half-line and u

0

is the “Dirichlet-craktip” function p

r /2π cos(θ/2)

in polar coordinates.

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Perspectives

Constrained problem

max n

λ ₁ (Ω \ Σ), Σ ∈ A (Ω), H ¹ (Σ) ≤ L o .

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Optimal compliance problem