Polygons as optimal shapes with convexity constraint Jimmy Lamboley - - PowerPoint PPT Presentation

Polygons as optimal shapes with convexity constraint

Jimmy Lamboley Arian Novruzi ´ Ecole Normale Sup´ erieure de Cachan University of Ottawa Antenne de Bretagne, France Ontario, Canada Conference on Applied Inverse Problems

Non-smooth Optimization in Inverse Problems

Vienna, Austria, July 20-24

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem

We consider the problem min

• j(u) :=
• ❚ G(θ, u, u′)dθ,

u ∈ W 1,∞(❚), u′′ + u ≥ 0, a ≤ u ≤ b} , (1) where G(θ, u, p) : ❚ × (0, ∞) × ❘ → ❘, ❚ = [0, 2π). When (1) has for solution a polygon? We are interested also in min {j(u), u ∈ W 1,∞(❚), u′′ + u ≥ 0, m(u) := 1

2

• ❚

dθ u2 = m0 > 0

• .

(2)

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem

We consider the problem min

• j(u) :=
• ❚ G(θ, u, u′)dθ,

u ∈ W 1,∞(❚), u′′ + u ≥ 0, a ≤ u ≤ b} , (1) where G(θ, u, p) : ❚ × (0, ∞) × ❘ → ❘, ❚ = [0, 2π). When (1) has for solution a polygon? We are interested also in min {j(u), u ∈ W 1,∞(❚), u′′ + u ≥ 0, m(u) := 1

2

• ❚

dθ u2 = m0 > 0

• .

(2)

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem

We consider the problem min

• j(u) :=
• ❚ G(θ, u, u′)dθ,

u ∈ W 1,∞(❚), u′′ + u ≥ 0, a ≤ u ≤ b} , (1) where G(θ, u, p) : ❚ × (0, ∞) × ❘ → ❘, ❚ = [0, 2π). When (1) has for solution a polygon? We are interested also in min {j(u), u ∈ W 1,∞(❚), u′′ + u ≥ 0, m(u) := 1

2

• ❚

dθ u2 = m0 > 0

• .

(2)

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem

We consider the problem min

• j(u) :=
• ❚ G(θ, u, u′)dθ,

u ∈ W 1,∞(❚), u′′ + u ≥ 0, a ≤ u ≤ b} , (1) where G(θ, u, p) : ❚ × (0, ∞) × ❘ → ❘, ❚ = [0, 2π). When (1) has for solution a polygon? We are interested also in min {j(u), u ∈ W 1,∞(❚), u′′ + u ≥ 0, m(u) := 1

2

• ❚

dθ u2 = m0 > 0

• .

(2)

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Motivation

• T. Lachand-Robert and M.A. Peletier

Newton’s problem: Find U0 solution of: min{E(U), U ∈ Fad}, E(U) =

• Ω

dx 1 + |∇U|2 Fad = {U : Ω → [0, M] : with graph in conv. env. of ∂Ω × {0} ∪ {U = M} × {M}} Ω ⊂ ❘2 a disk, M > 0 Newton found a radially symmetric minimizer In 1996, Brock, Ferone, and Kawohl: the minimizer is not radially symmetric min{G(θ, u, p) := h1(u) − p2h2(u), u ∈ Fad}, with Fad = {u regular enough, u′′ + u ≥ 0, 0 < a ≤ u ≤ b} Here u characterizes N := {U = M} The minimizer u0: supp(u′′

0 + u0) is finite

The set N0 := {U0 = M} is a regular polygon centered in Ω

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Motivation

• T. Lachand-Robert and M.A. Peletier

Newton’s problem: Find U0 solution of: min{E(U), U ∈ Fad}, E(U) =

• Ω

dx 1 + |∇U|2 Fad = {U : Ω → [0, M] : with graph in conv. env. of ∂Ω × {0} ∪ {U = M} × {M}} Ω ⊂ ❘2 a disk, M > 0 Newton found a radially symmetric minimizer In 1996, Brock, Ferone, and Kawohl: the minimizer is not radially symmetric min{G(θ, u, p) := h1(u) − p2h2(u), u ∈ Fad}, with Fad = {u regular enough, u′′ + u ≥ 0, 0 < a ≤ u ≤ b} Here u characterizes N := {U = M} The minimizer u0: supp(u′′

0 + u0) is finite

The set N0 := {U0 = M} is a regular polygon centered in Ω

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Motivation

• T. Lachand-Robert and M.A. Peletier

Newton’s problem: Find U0 solution of: min{E(U), U ∈ Fad}, E(U) =

• Ω

dx 1 + |∇U|2 Fad = {U : Ω → [0, M] : with graph in conv. env. of ∂Ω × {0} ∪ {U = M} × {M}} Ω ⊂ ❘2 a disk, M > 0 Newton found a radially symmetric minimizer In 1996, Brock, Ferone, and Kawohl: the minimizer is not radially symmetric min{G(θ, u, p) := h1(u) − p2h2(u), u ∈ Fad}, with Fad = {u regular enough, u′′ + u ≥ 0, 0 < a ≤ u ≤ b} Here u characterizes N := {U = M} The minimizer u0: supp(u′′

0 + u0) is finite

The set N0 := {U0 = M} is a regular polygon centered in Ω

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Motivation

• T. Lachand-Robert and M.A. Peletier

Newton’s problem: Find U0 solution of: min{E(U), U ∈ Fad}, E(U) =

• Ω

dx 1 + |∇U|2 Fad = {U : Ω → [0, M] : with graph in conv. env. of ∂Ω × {0} ∪ {U = M} × {M}} Ω ⊂ ❘2 a disk, M > 0 Newton found a radially symmetric minimizer In 1996, Brock, Ferone, and Kawohl: the minimizer is not radially symmetric min{G(θ, u, p) := h1(u) − p2h2(u), u ∈ Fad}, with Fad = {u regular enough, u′′ + u ≥ 0, 0 < a ≤ u ≤ b} Here u characterizes N := {U = M} The minimizer u0: supp(u′′

0 + u0) is finite

The set N0 := {U0 = M} is a regular polygon centered in Ω

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Motivation

• T. Lachand-Robert and M.A. Peletier

Newton’s problem: Find U0 solution of: min{E(U), U ∈ Fad}, E(U) =

• Ω

dx 1 + |∇U|2 Fad = {U : Ω → [0, M] : with graph in conv. env. of ∂Ω × {0} ∪ {U = M} × {M}} Ω ⊂ ❘2 a disk, M > 0 Newton found a radially symmetric minimizer In 1996, Brock, Ferone, and Kawohl: the minimizer is not radially symmetric min{G(θ, u, p) := h1(u) − p2h2(u), u ∈ Fad}, with Fad = {u regular enough, u′′ + u ≥ 0, 0 < a ≤ u ≤ b} Here u characterizes N := {U = M} The minimizer u0: supp(u′′

0 + u0) is finite

The set N0 := {U0 = M} is a regular polygon centered in Ω

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Motivation

• T. Lachand-Robert and M.A. Peletier

Newton’s problem: Find U0 solution of: min{E(U), U ∈ Fad}, E(U) =

• Ω

dx 1 + |∇U|2 Fad = {U : Ω → [0, M] : with graph in conv. env. of ∂Ω × {0} ∪ {U = M} × {M}} Ω ⊂ ❘2 a disk, M > 0 Newton found a radially symmetric minimizer In 1996, Brock, Ferone, and Kawohl: the minimizer is not radially symmetric min{G(θ, u, p) := h1(u) − p2h2(u), u ∈ Fad}, with Fad = {u regular enough, u′′ + u ≥ 0, 0 < a ≤ u ≤ b} Here u characterizes N := {U = M} The minimizer u0: supp(u′′

0 + u0) is finite

The set N0 := {U0 = M} is a regular polygon centered in Ω

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Motivation

• T. Lachand-Robert and M.A. Peletier

Newton’s problem: Find U0 solution of: min{E(U), U ∈ Fad}, E(U) =

• Ω

dx 1 + |∇U|2 Fad = {U : Ω → [0, M] : with graph in conv. env. of ∂Ω × {0} ∪ {U = M} × {M}} Ω ⊂ ❘2 a disk, M > 0 Newton found a radially symmetric minimizer In 1996, Brock, Ferone, and Kawohl: the minimizer is not radially symmetric min{G(θ, u, p) := h1(u) − p2h2(u), u ∈ Fad}, with Fad = {u regular enough, u′′ + u ≥ 0, 0 < a ≤ u ≤ b} Here u characterizes N := {U = M} The minimizer u0: supp(u′′

0 + u0) is finite

The set N0 := {U0 = M} is a regular polygon centered in Ω

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Motivation

• T. Lachand-Robert and M.A. Peletier

Newton’s problem: Find U0 solution of: min{E(U), U ∈ Fad}, E(U) =

• Ω

dx 1 + |∇U|2 Fad = {U : Ω → [0, M] : with graph in conv. env. of ∂Ω × {0} ∪ {U = M} × {M}} Ω ⊂ ❘2 a disk, M > 0 Newton found a radially symmetric minimizer In 1996, Brock, Ferone, and Kawohl: the minimizer is not radially symmetric min{G(θ, u, p) := h1(u) − p2h2(u), u ∈ Fad}, with Fad = {u regular enough, u′′ + u ≥ 0, 0 < a ≤ u ≤ b} Here u characterizes N := {U = M} The minimizer u0: supp(u′′

0 + u0) is finite

The set N0 := {U0 = M} is a regular polygon centered in Ω

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Motivation

• T. Lachand-Robert and M.A. Peletier

Newton’s problem: Find U0 solution of: min{E(U), U ∈ Fad}, E(U) =

• Ω

dx 1 + |∇U|2 Fad = {U : Ω → [0, M] : with graph in conv. env. of ∂Ω × {0} ∪ {U = M} × {M}} Ω ⊂ ❘2 a disk, M > 0 Newton found a radially symmetric minimizer In 1996, Brock, Ferone, and Kawohl: the minimizer is not radially symmetric min{G(θ, u, p) := h1(u) − p2h2(u), u ∈ Fad}, with Fad = {u regular enough, u′′ + u ≥ 0, 0 < a ≤ u ≤ b} Here u characterizes N := {U = M} The minimizer u0: supp(u′′

0 + u0) is finite

The set N0 := {U0 = M} is a regular polygon centered in Ω

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Motivation

• T. Lachand-Robert and M.A. Peletier

Newton’s problem: Find U0 solution of: min{E(U), U ∈ Fad}, E(U) =

• Ω

dx 1 + |∇U|2 Fad = {U : Ω → [0, M] : with graph in conv. env. of ∂Ω × {0} ∪ {U = M} × {M}} Ω ⊂ ❘2 a disk, M > 0 Newton found a radially symmetric minimizer In 1996, Brock, Ferone, and Kawohl: the minimizer is not radially symmetric min{G(θ, u, p) := h1(u) − p2h2(u), u ∈ Fad}, with Fad = {u regular enough, u′′ + u ≥ 0, 0 < a ≤ u ≤ b} Here u characterizes N := {U = M} The minimizer u0: supp(u′′

0 + u0) is finite

The set N0 := {U0 = M} is a regular polygon centered in Ω

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Motivation

Figure: Minimizer U0 as a function of M

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Motivation

Figure: Minimizer U0 as a function of M

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Motivation

• M. Crouzeix

motivated by abstract operator theory: G(θ, u, p) = h(p/u), u = 1

r

and Fad = {u regular enough, u′′ + u ≥ 0, 0 < a ≤ u ≤ b} Optimal shapes are polygons Furthermore:

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Motivation

• M. Crouzeix

motivated by abstract operator theory: G(θ, u, p) = h(p/u), u = 1

r

and Fad = {u regular enough, u′′ + u ≥ 0, 0 < a ≤ u ≤ b} Optimal shapes are polygons Furthermore:

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Motivation

• M. Crouzeix

motivated by abstract operator theory: G(θ, u, p) = h(p/u), u = 1

r

and Fad = {u regular enough, u′′ + u ≥ 0, 0 < a ≤ u ≤ b} Optimal shapes are polygons Furthermore:

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Motivation

• M. Crouzeix

motivated by abstract operator theory: G(θ, u, p) = h(p/u), u = 1

r

and Fad = {u regular enough, u′′ + u ≥ 0, 0 < a ≤ u ≤ b} Optimal shapes are polygons Furthermore:

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Motivation

• M. Crouzeix

motivated by abstract operator theory: G(θ, u, p) = h(p/u), u = 1

r

and Fad = {u regular enough, u′′ + u ≥ 0, 0 < a ≤ u ≤ b} Optimal shapes are polygons Furthermore:

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem: shape problem with convexity constraint

Our initial problem is: min{J(Ω), Ω convex, Ω ∈ Sad}, where Sad is a set of 2-dimensional admissible shapes, J : Sad → ❘ is a shape functional Parametrization of Ω:

W 1,∞(❚) := W 1,∞

loc (❘) ∩ {2π-periodic}

Ωu :=

• (r, θ) ∈ [0, 2π] ×
• 0,

1 u(θ)

• ,

0 < u ∈ W 1,∞(❚)

• .

The curvature: κ(Ωu) =

u′′+u (1+u′2)3/2 .

If u ∈ W 1,∞(❚), we say that u′′ + u ≥ 0 if ∀ v ∈ W 1,∞(❚) with v ≥ 0,

• ❚
• uv − u′v′

dθ ≥ 0.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem: shape problem with convexity constraint

Our initial problem is: min{J(Ω), Ω convex, Ω ∈ Sad}, where Sad is a set of 2-dimensional admissible shapes, J : Sad → ❘ is a shape functional Parametrization of Ω:

W 1,∞(❚) := W 1,∞

loc (❘) ∩ {2π-periodic}

Ωu :=

• (r, θ) ∈ [0, 2π] ×
• 0,

1 u(θ)

• ,

0 < u ∈ W 1,∞(❚)

• .

The curvature: κ(Ωu) =

u′′+u (1+u′2)3/2 .

If u ∈ W 1,∞(❚), we say that u′′ + u ≥ 0 if ∀ v ∈ W 1,∞(❚) with v ≥ 0,

• ❚
• uv − u′v′

dθ ≥ 0.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem: shape problem with convexity constraint

Our initial problem is: min{J(Ω), Ω convex, Ω ∈ Sad}, where Sad is a set of 2-dimensional admissible shapes, J : Sad → ❘ is a shape functional Parametrization of Ω:

W 1,∞(❚) := W 1,∞

loc (❘) ∩ {2π-periodic}

Ωu :=

• (r, θ) ∈ [0, 2π] ×
• 0,

1 u(θ)

• ,

0 < u ∈ W 1,∞(❚)

• .

The curvature: κ(Ωu) =

u′′+u (1+u′2)3/2 .

If u ∈ W 1,∞(❚), we say that u′′ + u ≥ 0 if ∀ v ∈ W 1,∞(❚) with v ≥ 0,

• ❚
• uv − u′v′

dθ ≥ 0.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem: shape problem with convexity constraint

Our initial problem is: min{J(Ω), Ω convex, Ω ∈ Sad}, where Sad is a set of 2-dimensional admissible shapes, J : Sad → ❘ is a shape functional Parametrization of Ω:

W 1,∞(❚) := W 1,∞

loc (❘) ∩ {2π-periodic}

Ωu :=

• (r, θ) ∈ [0, 2π] ×
• 0,

1 u(θ)

• ,

0 < u ∈ W 1,∞(❚)

• .

The curvature: κ(Ωu) =

u′′+u (1+u′2)3/2 .

If u ∈ W 1,∞(❚), we say that u′′ + u ≥ 0 if ∀ v ∈ W 1,∞(❚) with v ≥ 0,

• ❚
• uv − u′v′

dθ ≥ 0.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem: shape problem with convexity constraint

Our initial problem is: min{J(Ω), Ω convex, Ω ∈ Sad}, where Sad is a set of 2-dimensional admissible shapes, J : Sad → ❘ is a shape functional Parametrization of Ω:

W 1,∞(❚) := W 1,∞

loc (❘) ∩ {2π-periodic}

Ωu :=

• (r, θ) ∈ [0, 2π] ×
• 0,

1 u(θ)

• ,

0 < u ∈ W 1,∞(❚)

• .

The curvature: κ(Ωu) =

u′′+u (1+u′2)3/2 .

If u ∈ W 1,∞(❚), we say that u′′ + u ≥ 0 if ∀ v ∈ W 1,∞(❚) with v ≥ 0,

• ❚
• uv − u′v′

dθ ≥ 0.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem: shape problem with convexity constraint

Our initial problem is: min{J(Ω), Ω convex, Ω ∈ Sad}, where Sad is a set of 2-dimensional admissible shapes, J : Sad → ❘ is a shape functional Parametrization of Ω:

W 1,∞(❚) := W 1,∞

loc (❘) ∩ {2π-periodic}

Ωu :=

• (r, θ) ∈ [0, 2π] ×
• 0,

1 u(θ)

• ,

0 < u ∈ W 1,∞(❚)

• .

The curvature: κ(Ωu) =

u′′+u (1+u′2)3/2 .

If u ∈ W 1,∞(❚), we say that u′′ + u ≥ 0 if ∀ v ∈ W 1,∞(❚) with v ≥ 0,

• ❚
• uv − u′v′

dθ ≥ 0.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem: shape problem with convexity constraint

Our initial problem is: min{J(Ω), Ω convex, Ω ∈ Sad}, where Sad is a set of 2-dimensional admissible shapes, J : Sad → ❘ is a shape functional Parametrization of Ω:

W 1,∞(❚) := W 1,∞

loc (❘) ∩ {2π-periodic}

Ωu :=

• (r, θ) ∈ [0, 2π] ×
• 0,

1 u(θ)

• ,

0 < u ∈ W 1,∞(❚)

• .

The curvature: κ(Ωu) =

u′′+u (1+u′2)3/2 .

If u ∈ W 1,∞(❚), we say that u′′ + u ≥ 0 if ∀ v ∈ W 1,∞(❚) with v ≥ 0,

• ❚
• uv − u′v′

dθ ≥ 0.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem: reformulation

For u ∈ W 1,∞(❚):

Ωu is convex ⇐ ⇒ u′′ + u ≥ 0 in M(❚) straight lines on ∂Ωu: {u′′ + u = 0} corners on the boundary ∂Ωu: u′′ + u = αnδn

We consider: find u0 ∈ Fad: j(u0) = min{j(u) :=

• ❚ G(θ, u, u′)dθ,

u ∈ W 1,∞(❚), u′′ + u ≥ 0, u ∈ Fad}, where Fad is a set of convenient admissible functions. Choices of Fad: i) Fad :=

• u : ∂Ωu ⊂ A(a, b) :=
• (r, θ) :

1 b ≤ r ≤ 1 a

ii) Fad := {u : u ∈ C(m0)}, C(m0) :=

• u :

1 2

• ❚

dθ u2 = m0

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem: reformulation

For u ∈ W 1,∞(❚):

Ωu is convex ⇐ ⇒ u′′ + u ≥ 0 in M(❚) straight lines on ∂Ωu: {u′′ + u = 0} corners on the boundary ∂Ωu: u′′ + u = αnδn

We consider: find u0 ∈ Fad: j(u0) = min{j(u) :=

• ❚ G(θ, u, u′)dθ,

u ∈ W 1,∞(❚), u′′ + u ≥ 0, u ∈ Fad}, where Fad is a set of convenient admissible functions. Choices of Fad: i) Fad :=

• u : ∂Ωu ⊂ A(a, b) :=
• (r, θ) :

1 b ≤ r ≤ 1 a

ii) Fad := {u : u ∈ C(m0)}, C(m0) :=

• u :

1 2

• ❚

dθ u2 = m0

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem: reformulation

For u ∈ W 1,∞(❚):

Ωu is convex ⇐ ⇒ u′′ + u ≥ 0 in M(❚) straight lines on ∂Ωu: {u′′ + u = 0} corners on the boundary ∂Ωu: u′′ + u = αnδn

We consider: find u0 ∈ Fad: j(u0) = min{j(u) :=

• ❚ G(θ, u, u′)dθ,

u ∈ W 1,∞(❚), u′′ + u ≥ 0, u ∈ Fad}, where Fad is a set of convenient admissible functions. Choices of Fad: i) Fad :=

• u : ∂Ωu ⊂ A(a, b) :=
• (r, θ) :

1 b ≤ r ≤ 1 a

ii) Fad := {u : u ∈ C(m0)}, C(m0) :=

• u :

1 2

• ❚

dθ u2 = m0

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem: reformulation

For u ∈ W 1,∞(❚):

Ωu is convex ⇐ ⇒ u′′ + u ≥ 0 in M(❚) straight lines on ∂Ωu: {u′′ + u = 0} corners on the boundary ∂Ωu: u′′ + u = αnδn

We consider: find u0 ∈ Fad: j(u0) = min{j(u) :=

• ❚ G(θ, u, u′)dθ,

u ∈ W 1,∞(❚), u′′ + u ≥ 0, u ∈ Fad}, where Fad is a set of convenient admissible functions. Choices of Fad: i) Fad :=

• u : ∂Ωu ⊂ A(a, b) :=
• (r, θ) :

1 b ≤ r ≤ 1 a

ii) Fad := {u : u ∈ C(m0)}, C(m0) :=

• u :

1 2

• ❚

dθ u2 = m0

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem: reformulation

For u ∈ W 1,∞(❚):

Ωu is convex ⇐ ⇒ u′′ + u ≥ 0 in M(❚) straight lines on ∂Ωu: {u′′ + u = 0} corners on the boundary ∂Ωu: u′′ + u = αnδn

We consider: find u0 ∈ Fad: j(u0) = min{j(u) :=

• ❚ G(θ, u, u′)dθ,

u ∈ W 1,∞(❚), u′′ + u ≥ 0, u ∈ Fad}, where Fad is a set of convenient admissible functions. Choices of Fad: i) Fad :=

• u : ∂Ωu ⊂ A(a, b) :=
• (r, θ) :

1 b ≤ r ≤ 1 a

ii) Fad := {u : u ∈ C(m0)}, C(m0) :=

• u :

1 2

• ❚

dθ u2 = m0

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem: reformulation

For u ∈ W 1,∞(❚):

Ωu is convex ⇐ ⇒ u′′ + u ≥ 0 in M(❚) straight lines on ∂Ωu: {u′′ + u = 0} corners on the boundary ∂Ωu: u′′ + u = αnδn

We consider: find u0 ∈ Fad: j(u0) = min{j(u) :=

• ❚ G(θ, u, u′)dθ,

u ∈ W 1,∞(❚), u′′ + u ≥ 0, u ∈ Fad}, where Fad is a set of convenient admissible functions. Choices of Fad: i) Fad :=

• u : ∂Ωu ⊂ A(a, b) :=
• (r, θ) :

1 b ≤ r ≤ 1 a

ii) Fad := {u : u ∈ C(m0)}, C(m0) :=

• u :

1 2

• ❚

dθ u2 = m0

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem: reformulation

For u ∈ W 1,∞(❚):

Ωu is convex ⇐ ⇒ u′′ + u ≥ 0 in M(❚) straight lines on ∂Ωu: {u′′ + u = 0} corners on the boundary ∂Ωu: u′′ + u = αnδn

We consider: find u0 ∈ Fad: j(u0) = min{j(u) :=

• ❚ G(θ, u, u′)dθ,

u ∈ W 1,∞(❚), u′′ + u ≥ 0, u ∈ Fad}, where Fad is a set of convenient admissible functions. Choices of Fad: i) Fad :=

• u : ∂Ωu ⊂ A(a, b) :=
• (r, θ) :

1 b ≤ r ≤ 1 a

ii) Fad := {u : u ∈ C(m0)}, C(m0) :=

• u :

1 2

• ❚

dθ u2 = m0

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem: reformulation

For u ∈ W 1,∞(❚):

Ωu is convex ⇐ ⇒ u′′ + u ≥ 0 in M(❚) straight lines on ∂Ωu: {u′′ + u = 0} corners on the boundary ∂Ωu: u′′ + u = αnδn

We consider: find u0 ∈ Fad: j(u0) = min{j(u) :=

• ❚ G(θ, u, u′)dθ,

u ∈ W 1,∞(❚), u′′ + u ≥ 0, u ∈ Fad}, where Fad is a set of convenient admissible functions. Choices of Fad: i) Fad :=

• u : ∂Ωu ⊂ A(a, b) :=
• (r, θ) :

1 b ≤ r ≤ 1 a

ii) Fad := {u : u ∈ C(m0)}, C(m0) :=

• u :

1 2

• ❚

dθ u2 = m0

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

The problem: reformulation

For u ∈ W 1,∞(❚):

Ωu is convex ⇐ ⇒ u′′ + u ≥ 0 in M(❚) straight lines on ∂Ωu: {u′′ + u = 0} corners on the boundary ∂Ωu: u′′ + u = αnδn

We consider: find u0 ∈ Fad: j(u0) = min{j(u) :=

• ❚ G(θ, u, u′)dθ,

u ∈ W 1,∞(❚), u′′ + u ≥ 0, u ∈ Fad}, where Fad is a set of convenient admissible functions. Choices of Fad: i) Fad :=

• u : ∂Ωu ⊂ A(a, b) :=
• (r, θ) :

1 b ≤ r ≤ 1 a

ii) Fad := {u : u ∈ C(m0)}, C(m0) :=

• u :

1 2

• ❚

dθ u2 = m0

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: existence

The problem (1) (with inclusion in A(a, b)) has a solution if, for example,

j(u) is continuous in H1(❚), because {u ∈ W 1,∞(❚), u′′ + u ≥ 0, a ≤ u ≤ b} is strongly compact in H1(❚)

The problem (2) (with area constraint)

the question is more case specific. For example: maximization of the perimeter: non-existence However, existence may be proved for many further functionals

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: existence

The problem (1) (with inclusion in A(a, b)) has a solution if, for example,

j(u) is continuous in H1(❚), because {u ∈ W 1,∞(❚), u′′ + u ≥ 0, a ≤ u ≤ b} is strongly compact in H1(❚)

The problem (2) (with area constraint)

the question is more case specific. For example: maximization of the perimeter: non-existence However, existence may be proved for many further functionals

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: existence

The problem (1) (with inclusion in A(a, b)) has a solution if, for example,

j(u) is continuous in H1(❚), because {u ∈ W 1,∞(❚), u′′ + u ≥ 0, a ≤ u ≤ b} is strongly compact in H1(❚)

The problem (2) (with area constraint)

the question is more case specific. For example: maximization of the perimeter: non-existence However, existence may be proved for many further functionals

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: existence

The problem (1) (with inclusion in A(a, b)) has a solution if, for example,

j(u) is continuous in H1(❚), because {u ∈ W 1,∞(❚), u′′ + u ≥ 0, a ≤ u ≤ b} is strongly compact in H1(❚)

The problem (2) (with area constraint)

the question is more case specific. For example: maximization of the perimeter: non-existence However, existence may be proved for many further functionals

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: existence

The problem (1) (with inclusion in A(a, b)) has a solution if, for example,

j(u) is continuous in H1(❚), because {u ∈ W 1,∞(❚), u′′ + u ≥ 0, a ≤ u ≤ b} is strongly compact in H1(❚)

The problem (2) (with area constraint)

the question is more case specific. For example: maximization of the perimeter: non-existence However, existence may be proved for many further functionals

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: existence

The problem (1) (with inclusion in A(a, b)) has a solution if, for example,

j(u) is continuous in H1(❚), because {u ∈ W 1,∞(❚), u′′ + u ≥ 0, a ≤ u ≤ b} is strongly compact in H1(❚)

The problem (2) (with area constraint)

the question is more case specific. For example: maximization of the perimeter: non-existence However, existence may be proved for many further functionals

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: existence

The problem (1) (with inclusion in A(a, b)) has a solution if, for example,

j(u) is continuous in H1(❚), because {u ∈ W 1,∞(❚), u′′ + u ≥ 0, a ≤ u ≤ b} is strongly compact in H1(❚)

The problem (2) (with area constraint)

the question is more case specific. For example: maximization of the perimeter: non-existence However, existence may be proved for many further functionals

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: existence

The problem (1) (with inclusion in A(a, b)) has a solution if, for example,

j(u) is continuous in H1(❚), because {u ∈ W 1,∞(❚), u′′ + u ≥ 0, a ≤ u ≤ b} is strongly compact in H1(❚)

The problem (2) (with area constraint)

the question is more case specific. For example: maximization of the perimeter: non-existence However, existence may be proved for many further functionals

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: existence

The problem (1) (with inclusion in A(a, b)) has a solution if, for example,

j(u) is continuous in H1(❚), because {u ∈ W 1,∞(❚), u′′ + u ≥ 0, a ≤ u ≤ b} is strongly compact in H1(❚)

The problem (2) (with area constraint)

the question is more case specific. For example: maximization of the perimeter: non-existence However, existence may be proved for many further functionals

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity

Theorem 0.1

1 Let G = G(θ, u, p) ∈ C2(❚ × ❘ × ❘). Set j(u) =

• ❚ G(θ, u, u′).

2 Let u0 be a solution of (1) or (2) and assume that G satisfies:

Gpp(θ, u0, u′

0) < 0,

∀θ ∈ ❚. (3)

3 If u0 is a solution of (1), then:

Su0 ∩ I is finite for any I = (γ1, γ2) ⊂ {a < u0(θ) < b}, and in particular, Ωu0 is locally polygonal inside the annulus A(a, b).

4 If u0 > 0 is a solution of (2), then Su0 ∩ ❚ is finite, and so

Ωu0 is a polygon.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity

Theorem 0.1

1 Let G = G(θ, u, p) ∈ C2(❚ × ❘ × ❘). Set j(u) =

• ❚ G(θ, u, u′).

2 Let u0 be a solution of (1) or (2) and assume that G satisfies:

Gpp(θ, u0, u′

0) < 0,

∀θ ∈ ❚. (3)

3 If u0 is a solution of (1), then:

Su0 ∩ I is finite for any I = (γ1, γ2) ⊂ {a < u0(θ) < b}, and in particular, Ωu0 is locally polygonal inside the annulus A(a, b).

4 If u0 > 0 is a solution of (2), then Su0 ∩ ❚ is finite, and so

Ωu0 is a polygon.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity

Theorem 0.1

1 Let G = G(θ, u, p) ∈ C2(❚ × ❘ × ❘). Set j(u) =

• ❚ G(θ, u, u′).

2 Let u0 be a solution of (1) or (2) and assume that G satisfies:

Gpp(θ, u0, u′

0) < 0,

∀θ ∈ ❚. (3)

3 If u0 is a solution of (1), then:

Su0 ∩ I is finite for any I = (γ1, γ2) ⊂ {a < u0(θ) < b}, and in particular, Ωu0 is locally polygonal inside the annulus A(a, b).

4 If u0 > 0 is a solution of (2), then Su0 ∩ ❚ is finite, and so

Ωu0 is a polygon.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity

Theorem 0.1

1 Let G = G(θ, u, p) ∈ C2(❚ × ❘ × ❘). Set j(u) =

• ❚ G(θ, u, u′).

2 Let u0 be a solution of (1) or (2) and assume that G satisfies:

Gpp(θ, u0, u′

0) < 0,

∀θ ∈ ❚. (3)

3 If u0 is a solution of (1), then:

Su0 ∩ I is finite for any I = (γ1, γ2) ⊂ {a < u0(θ) < b}, and in particular, Ωu0 is locally polygonal inside the annulus A(a, b).

4 If u0 > 0 is a solution of (2), then Su0 ∩ ❚ is finite, and so

Ωu0 is a polygon.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity

Theorem 0.1

1 Let G = G(θ, u, p) ∈ C2(❚ × ❘ × ❘). Set j(u) =

• ❚ G(θ, u, u′).

2 Let u0 be a solution of (1) or (2) and assume that G satisfies:

Gpp(θ, u0, u′

0) < 0,

∀θ ∈ ❚. (3)

3 If u0 is a solution of (1), then:

Su0 ∩ I is finite for any I = (γ1, γ2) ⊂ {a < u0(θ) < b}, and in particular, Ωu0 is locally polygonal inside the annulus A(a, b).

4 If u0 > 0 is a solution of (2), then Su0 ∩ ❚ is finite, and so

Ωu0 is a polygon.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity

Theorem 0.1

1 Let G = G(θ, u, p) ∈ C2(❚ × ❘ × ❘). Set j(u) =

• ❚ G(θ, u, u′).

2 Let u0 be a solution of (1) or (2) and assume that G satisfies:

Gpp(θ, u0, u′

0) < 0,

∀θ ∈ ❚. (3)

3 If u0 is a solution of (1), then:

Su0 ∩ I is finite for any I = (γ1, γ2) ⊂ {a < u0(θ) < b}, and in particular, Ωu0 is locally polygonal inside the annulus A(a, b).

4 If u0 > 0 is a solution of (2), then Su0 ∩ ❚ is finite, and so

Ωu0 is a polygon.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity

Theorem 0.1

1 Let G = G(θ, u, p) ∈ C2(❚ × ❘ × ❘). Set j(u) =

• ❚ G(θ, u, u′).

2 Let u0 be a solution of (1) or (2) and assume that G satisfies:

Gpp(θ, u0, u′

0) < 0,

∀θ ∈ ❚. (3)

3 If u0 is a solution of (1), then:

Su0 ∩ I is finite for any I = (γ1, γ2) ⊂ {a < u0(θ) < b}, and in particular, Ωu0 is locally polygonal inside the annulus A(a, b).

4 If u0 > 0 is a solution of (2), then Su0 ∩ ❚ is finite, and so

Ωu0 is a polygon.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity (cont’d)

Theorem 0.2 (inclusion in annulus A(a, b))

1 Let j(u) =

• ❚ G(u, u′), u0 be a solution of (1). Assume that

2 (i) G = G(u, p) is a C2 function and

Gpp < 0 on {(u0(θ), u′

0(θ)), θ ∈ ❚},

3 (ii) The function p → G(a, p) is even and one of the

followings holds

(ii.1): Gu(a, 0) < 0, or (ii.2): Gu(a, 0) = 0 and Gu(u0, u′

0)u0 + Gp(u0, u′ 0)u′ 0 ≤ 0,

4 (iii) The function p → G(·, p) is even and Gu ≥ 0 near (b, 0). 5 Then Su0 is finite, i.e. Ωu0 is a polygon.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity (cont’d)

Theorem 0.2 (inclusion in annulus A(a, b))

1 Let j(u) =

• ❚ G(u, u′), u0 be a solution of (1). Assume that

2 (i) G = G(u, p) is a C2 function and

Gpp < 0 on {(u0(θ), u′

0(θ)), θ ∈ ❚},

3 (ii) The function p → G(a, p) is even and one of the

followings holds

(ii.1): Gu(a, 0) < 0, or (ii.2): Gu(a, 0) = 0 and Gu(u0, u′

0)u0 + Gp(u0, u′ 0)u′ 0 ≤ 0,

4 (iii) The function p → G(·, p) is even and Gu ≥ 0 near (b, 0). 5 Then Su0 is finite, i.e. Ωu0 is a polygon.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity (cont’d)

Theorem 0.2 (inclusion in annulus A(a, b))

1 Let j(u) =

• ❚ G(u, u′), u0 be a solution of (1). Assume that

2 (i) G = G(u, p) is a C2 function and

Gpp < 0 on {(u0(θ), u′

0(θ)), θ ∈ ❚},

3 (ii) The function p → G(a, p) is even and one of the

followings holds

(ii.1): Gu(a, 0) < 0, or (ii.2): Gu(a, 0) = 0 and Gu(u0, u′

0)u0 + Gp(u0, u′ 0)u′ 0 ≤ 0,

4 (iii) The function p → G(·, p) is even and Gu ≥ 0 near (b, 0). 5 Then Su0 is finite, i.e. Ωu0 is a polygon.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity (cont’d)

Theorem 0.2 (inclusion in annulus A(a, b))

1 Let j(u) =

• ❚ G(u, u′), u0 be a solution of (1). Assume that

2 (i) G = G(u, p) is a C2 function and

Gpp < 0 on {(u0(θ), u′

0(θ)), θ ∈ ❚},

3 (ii) The function p → G(a, p) is even and one of the

followings holds

(ii.1): Gu(a, 0) < 0, or (ii.2): Gu(a, 0) = 0 and Gu(u0, u′

0)u0 + Gp(u0, u′ 0)u′ 0 ≤ 0,

4 (iii) The function p → G(·, p) is even and Gu ≥ 0 near (b, 0). 5 Then Su0 is finite, i.e. Ωu0 is a polygon.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity (cont’d)

Theorem 0.2 (inclusion in annulus A(a, b))

1 Let j(u) =

• ❚ G(u, u′), u0 be a solution of (1). Assume that

2 (i) G = G(u, p) is a C2 function and

Gpp < 0 on {(u0(θ), u′

0(θ)), θ ∈ ❚},

3 (ii) The function p → G(a, p) is even and one of the

followings holds

(ii.1): Gu(a, 0) < 0, or (ii.2): Gu(a, 0) = 0 and Gu(u0, u′

0)u0 + Gp(u0, u′ 0)u′ 0 ≤ 0,

4 (iii) The function p → G(·, p) is even and Gu ≥ 0 near (b, 0). 5 Then Su0 is finite, i.e. Ωu0 is a polygon.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity (cont’d)

Theorem 0.2 (inclusion in annulus A(a, b))

1 Let j(u) =

• ❚ G(u, u′), u0 be a solution of (1). Assume that

2 (i) G = G(u, p) is a C2 function and

Gpp < 0 on {(u0(θ), u′

0(θ)), θ ∈ ❚},

3 (ii) The function p → G(a, p) is even and one of the

followings holds

(ii.1): Gu(a, 0) < 0, or (ii.2): Gu(a, 0) = 0 and Gu(u0, u′

0)u0 + Gp(u0, u′ 0)u′ 0 ≤ 0,

4 (iii) The function p → G(·, p) is even and Gu ≥ 0 near (b, 0). 5 Then Su0 is finite, i.e. Ωu0 is a polygon.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity (cont’d)

Theorem 0.2 (inclusion in annulus A(a, b))

1 Let j(u) =

• ❚ G(u, u′), u0 be a solution of (1). Assume that

2 (i) G = G(u, p) is a C2 function and

Gpp < 0 on {(u0(θ), u′

0(θ)), θ ∈ ❚},

3 (ii) The function p → G(a, p) is even and one of the

followings holds

(ii.1): Gu(a, 0) < 0, or (ii.2): Gu(a, 0) = 0 and Gu(u0, u′

0)u0 + Gp(u0, u′ 0)u′ 0 ≤ 0,

4 (iii) The function p → G(·, p) is even and Gu ≥ 0 near (b, 0). 5 Then Su0 is finite, i.e. Ωu0 is a polygon.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity (cont’d)

Theorem 0.2 (inclusion in annulus A(a, b))

1 Let j(u) =

• ❚ G(u, u′), u0 be a solution of (1). Assume that

2 (i) G = G(u, p) is a C2 function and

Gpp < 0 on {(u0(θ), u′

0(θ)), θ ∈ ❚},

3 (ii) The function p → G(a, p) is even and one of the

followings holds

(ii.1): Gu(a, 0) < 0, or (ii.2): Gu(a, 0) = 0 and Gu(u0, u′

0)u0 + Gp(u0, u′ 0)u′ 0 ≤ 0,

4 (iii) The function p → G(·, p) is even and Gu ≥ 0 near (b, 0). 5 Then Su0 is finite, i.e. Ωu0 is a polygon.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity (cont’d)

Example Consider J(Ωu) = λm(Ωu) − P(Ωu) = λ 2

• ❚

1 u2 −

• ❚
• u2 + |u′|2

u2 , with m(Ωu) the area, P(Ωu) the perimeter, and λ ∈ [0, +∞] The minimization of J within convex sets and ∂Ω ⊂ A(a, b) is in general a non trivial problem When λ = 0, the solution is the disk of radius 1

a

When λ = +∞, the solution is the disk of radius 1

b

λ ∈ (0, ∞): any solution is locally polygonal inside A(a, b) Gpp = − 1 (u2 + p2)3/2 , Gu(u, 0) = u − λ u3 From Theorem 0.2, if λ ∈ (a, b) then any solution is a polygon

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity (cont’d)

Example Consider J(Ωu) = λm(Ωu) − P(Ωu) = λ 2

• ❚

1 u2 −

• ❚
• u2 + |u′|2

u2 , with m(Ωu) the area, P(Ωu) the perimeter, and λ ∈ [0, +∞] The minimization of J within convex sets and ∂Ω ⊂ A(a, b) is in general a non trivial problem When λ = 0, the solution is the disk of radius 1

a

When λ = +∞, the solution is the disk of radius 1

b

λ ∈ (0, ∞): any solution is locally polygonal inside A(a, b) Gpp = − 1 (u2 + p2)3/2 , Gu(u, 0) = u − λ u3 From Theorem 0.2, if λ ∈ (a, b) then any solution is a polygon

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity (cont’d)

Example Consider J(Ωu) = λm(Ωu) − P(Ωu) = λ 2

• ❚

1 u2 −

• ❚
• u2 + |u′|2

u2 , with m(Ωu) the area, P(Ωu) the perimeter, and λ ∈ [0, +∞] The minimization of J within convex sets and ∂Ω ⊂ A(a, b) is in general a non trivial problem When λ = 0, the solution is the disk of radius 1

a

When λ = +∞, the solution is the disk of radius 1

b

λ ∈ (0, ∞): any solution is locally polygonal inside A(a, b) Gpp = − 1 (u2 + p2)3/2 , Gu(u, 0) = u − λ u3 From Theorem 0.2, if λ ∈ (a, b) then any solution is a polygon

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity (cont’d)

Example Consider J(Ωu) = λm(Ωu) − P(Ωu) = λ 2

• ❚

1 u2 −

• ❚
• u2 + |u′|2

u2 , with m(Ωu) the area, P(Ωu) the perimeter, and λ ∈ [0, +∞] The minimization of J within convex sets and ∂Ω ⊂ A(a, b) is in general a non trivial problem When λ = 0, the solution is the disk of radius 1

a

When λ = +∞, the solution is the disk of radius 1

b

λ ∈ (0, ∞): any solution is locally polygonal inside A(a, b) Gpp = − 1 (u2 + p2)3/2 , Gu(u, 0) = u − λ u3 From Theorem 0.2, if λ ∈ (a, b) then any solution is a polygon

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity (cont’d)

Example Consider J(Ωu) = λm(Ωu) − P(Ωu) = λ 2

• ❚

1 u2 −

• ❚
• u2 + |u′|2

u2 , with m(Ωu) the area, P(Ωu) the perimeter, and λ ∈ [0, +∞] The minimization of J within convex sets and ∂Ω ⊂ A(a, b) is in general a non trivial problem When λ = 0, the solution is the disk of radius 1

a

When λ = +∞, the solution is the disk of radius 1

b

λ ∈ (0, ∞): any solution is locally polygonal inside A(a, b) Gpp = − 1 (u2 + p2)3/2 , Gu(u, 0) = u − λ u3 From Theorem 0.2, if λ ∈ (a, b) then any solution is a polygon

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity (cont’d)

Example Consider J(Ωu) = λm(Ωu) − P(Ωu) = λ 2

• ❚

1 u2 −

• ❚
• u2 + |u′|2

u2 , with m(Ωu) the area, P(Ωu) the perimeter, and λ ∈ [0, +∞] The minimization of J within convex sets and ∂Ω ⊂ A(a, b) is in general a non trivial problem When λ = 0, the solution is the disk of radius 1

a

When λ = +∞, the solution is the disk of radius 1

b

λ ∈ (0, ∞): any solution is locally polygonal inside A(a, b) Gpp = − 1 (u2 + p2)3/2 , Gu(u, 0) = u − λ u3 From Theorem 0.2, if λ ∈ (a, b) then any solution is a polygon

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity (cont’d)

Example Consider J(Ωu) = λm(Ωu) − P(Ωu) = λ 2

• ❚

1 u2 −

• ❚
• u2 + |u′|2

u2 , with m(Ωu) the area, P(Ωu) the perimeter, and λ ∈ [0, +∞] The minimization of J within convex sets and ∂Ω ⊂ A(a, b) is in general a non trivial problem When λ = 0, the solution is the disk of radius 1

a

When λ = +∞, the solution is the disk of radius 1

b

λ ∈ (0, ∞): any solution is locally polygonal inside A(a, b) Gpp = − 1 (u2 + p2)3/2 , Gu(u, 0) = u − λ u3 From Theorem 0.2, if λ ∈ (a, b) then any solution is a polygon

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Main results: regularity (cont’d)

Example Consider J(Ωu) = λm(Ωu) − P(Ωu) = λ 2

• ❚

1 u2 −

• ❚
• u2 + |u′|2

u2 , with m(Ωu) the area, P(Ωu) the perimeter, and λ ∈ [0, +∞] The minimization of J within convex sets and ∂Ω ⊂ A(a, b) is in general a non trivial problem When λ = 0, the solution is the disk of radius 1

a

When λ = +∞, the solution is the disk of radius 1

b

λ ∈ (0, ∞): any solution is locally polygonal inside A(a, b) Gpp = − 1 (u2 + p2)3/2 , Gu(u, 0) = u − λ u3 From Theorem 0.2, if λ ∈ (a, b) then any solution is a polygon

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions

Theorem 0.3 (inclusion in A(a, b))

1 If u0 is solves (1) and j ∈ C1(H1(❚); ❘), then there exist

0 ≤ ζ0 ∈ H1(❚), µa, µb ∈ M+(❚) such that {ζ0 = 0}⊂Su0, Supp(µa)⊂{u0 = a}, Supp(µb)⊂{u0 = b}.

2 v ∈ H1(❚):

j′(u0)v = ζ0 + ζ′′

0 , vU′×U +

• ❚

vdµa −

• ❚

vdµb.

3 Moreover, ∀ v ∈ H1(❚) such that ∃λ ∈ ❘ with

       v′′ + v ≥ λ(u′′

0 + u0)

v ≥ λ(u0 − a), v ≤ λ(u0 − b), ζ0 + ζ′′

0 , vU′×U

+

• ❚ vd(µa − µb) = 0,

⇒ j′′(u0)(v, v) ≥ 0.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions

Theorem 0.3 (inclusion in A(a, b))

1 If u0 is solves (1) and j ∈ C1(H1(❚); ❘), then there exist

0 ≤ ζ0 ∈ H1(❚), µa, µb ∈ M+(❚) such that {ζ0 = 0}⊂Su0, Supp(µa)⊂{u0 = a}, Supp(µb)⊂{u0 = b}.

2 v ∈ H1(❚):

j′(u0)v = ζ0 + ζ′′

0 , vU′×U +

• ❚

vdµa −

• ❚

vdµb.

3 Moreover, ∀ v ∈ H1(❚) such that ∃λ ∈ ❘ with

       v′′ + v ≥ λ(u′′

0 + u0)

v ≥ λ(u0 − a), v ≤ λ(u0 − b), ζ0 + ζ′′

0 , vU′×U

+

• ❚ vd(µa − µb) = 0,

⇒ j′′(u0)(v, v) ≥ 0.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions

Theorem 0.3 (inclusion in A(a, b))

1 If u0 is solves (1) and j ∈ C1(H1(❚); ❘), then there exist

0 ≤ ζ0 ∈ H1(❚), µa, µb ∈ M+(❚) such that {ζ0 = 0}⊂Su0, Supp(µa)⊂{u0 = a}, Supp(µb)⊂{u0 = b}.

2 v ∈ H1(❚):

j′(u0)v = ζ0 + ζ′′

0 , vU′×U +

• ❚

vdµa −

• ❚

vdµb.

3 Moreover, ∀ v ∈ H1(❚) such that ∃λ ∈ ❘ with

       v′′ + v ≥ λ(u′′

0 + u0)

v ≥ λ(u0 − a), v ≤ λ(u0 − b), ζ0 + ζ′′

0 , vU′×U

+

• ❚ vd(µa − µb) = 0,

⇒ j′′(u0)(v, v) ≥ 0.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions

Theorem 0.3 (inclusion in A(a, b))

1 If u0 is solves (1) and j ∈ C1(H1(❚); ❘), then there exist

0 ≤ ζ0 ∈ H1(❚), µa, µb ∈ M+(❚) such that {ζ0 = 0}⊂Su0, Supp(µa)⊂{u0 = a}, Supp(µb)⊂{u0 = b}.

2 v ∈ H1(❚):

j′(u0)v = ζ0 + ζ′′

0 , vU′×U +

• ❚

vdµa −

• ❚

vdµb.

3 Moreover, ∀ v ∈ H1(❚) such that ∃λ ∈ ❘ with

       v′′ + v ≥ λ(u′′

0 + u0)

v ≥ λ(u0 − a), v ≤ λ(u0 − b), ζ0 + ζ′′

0 , vU′×U

+

• ❚ vd(µa − µb) = 0,

⇒ j′′(u0)(v, v) ≥ 0.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions

Theorem 0.3 (inclusion in A(a, b))

1 If u0 is solves (1) and j ∈ C1(H1(❚); ❘), then there exist

0 ≤ ζ0 ∈ H1(❚), µa, µb ∈ M+(❚) such that {ζ0 = 0}⊂Su0, Supp(µa)⊂{u0 = a}, Supp(µb)⊂{u0 = b}.

2 v ∈ H1(❚):

j′(u0)v = ζ0 + ζ′′

0 , vU′×U +

• ❚

vdµa −

• ❚

vdµb.

3 Moreover, ∀ v ∈ H1(❚) such that ∃λ ∈ ❘ with

       v′′ + v ≥ λ(u′′

0 + u0)

v ≥ λ(u0 − a), v ≤ λ(u0 − b), ζ0 + ζ′′

0 , vU′×U

+

• ❚ vd(µa − µb) = 0,

⇒ j′′(u0)(v, v) ≥ 0.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions

Theorem 0.3 (inclusion in A(a, b))

1 If u0 is solves (1) and j ∈ C1(H1(❚); ❘), then there exist

0 ≤ ζ0 ∈ H1(❚), µa, µb ∈ M+(❚) such that {ζ0 = 0}⊂Su0, Supp(µa)⊂{u0 = a}, Supp(µb)⊂{u0 = b}.

2 v ∈ H1(❚):

j′(u0)v = ζ0 + ζ′′

0 , vU′×U +

• ❚

vdµa −

• ❚

vdµb.

3 Moreover, ∀ v ∈ H1(❚) such that ∃λ ∈ ❘ with

       v′′ + v ≥ λ(u′′

0 + u0)

v ≥ λ(u0 − a), v ≤ λ(u0 − b), ζ0 + ζ′′

0 , vU′×U

+

• ❚ vd(µa − µb) = 0,

⇒ j′′(u0)(v, v) ≥ 0.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions

Theorem 0.3 (inclusion in A(a, b))

1 If u0 is solves (1) and j ∈ C1(H1(❚); ❘), then there exist

0 ≤ ζ0 ∈ H1(❚), µa, µb ∈ M+(❚) such that {ζ0 = 0}⊂Su0, Supp(µa)⊂{u0 = a}, Supp(µb)⊂{u0 = b}.

2 v ∈ H1(❚):

j′(u0)v = ζ0 + ζ′′

0 , vU′×U +

• ❚

vdµa −

• ❚

vdµb.

3 Moreover, ∀ v ∈ H1(❚) such that ∃λ ∈ ❘ with

       v′′ + v ≥ λ(u′′

0 + u0)

v ≥ λ(u0 − a), v ≤ λ(u0 − b), ζ0 + ζ′′

0 , vU′×U

+

• ❚ vd(µa − µb) = 0,

⇒ j′′(u0)(v, v) ≥ 0.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks)

Ioffe A. D - Tihomirov V. M., Maurer H. - Zowe J. Abstract setting: U, Y real Banach, ∅ = K ⊂ Y closed convex cone, f : U → ❘, g : U → Y , min{f (u), u ∈ U, g(u) ∈ K}. Proposition 0.4 Let u0 ∈ U solve the min. problem. Assume f , g are C2 at u0 and g′(u0)(U) = Y . Then,

(i) ∃ l ∈ Y ′

+ : f ′(u0) = l ◦ g ′(u0) and l(g(u0)) = 0, where

Y ′

+ = {l ∈ Y ′; ∀ k ∈ K, l(k) ≥ 0 }

(ii) Set L(u) := f (u) − l(g(u)). Then F ′′(u0)(v, v) ≥ 0, ∀v ∈ Tu0, Tu0 =

• v ∈ U; f ′(u0)(v) = 0,

g ′(u0)(v) ∈ Kg(u0) = {K + λg(u0); λ ∈ ❘}

• .
• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks)

Ioffe A. D - Tihomirov V. M., Maurer H. - Zowe J. Abstract setting: U, Y real Banach, ∅ = K ⊂ Y closed convex cone, f : U → ❘, g : U → Y , min{f (u), u ∈ U, g(u) ∈ K}. Proposition 0.4 Let u0 ∈ U solve the min. problem. Assume f , g are C2 at u0 and g′(u0)(U) = Y . Then,

(i) ∃ l ∈ Y ′

+ : f ′(u0) = l ◦ g ′(u0) and l(g(u0)) = 0, where

Y ′

+ = {l ∈ Y ′; ∀ k ∈ K, l(k) ≥ 0 }

(ii) Set L(u) := f (u) − l(g(u)). Then F ′′(u0)(v, v) ≥ 0, ∀v ∈ Tu0, Tu0 =

• v ∈ U; f ′(u0)(v) = 0,

g ′(u0)(v) ∈ Kg(u0) = {K + λg(u0); λ ∈ ❘}

• .
• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks)

Ioffe A. D - Tihomirov V. M., Maurer H. - Zowe J. Abstract setting: U, Y real Banach, ∅ = K ⊂ Y closed convex cone, f : U → ❘, g : U → Y , min{f (u), u ∈ U, g(u) ∈ K}. Proposition 0.4 Let u0 ∈ U solve the min. problem. Assume f , g are C2 at u0 and g′(u0)(U) = Y . Then,

(i) ∃ l ∈ Y ′

+ : f ′(u0) = l ◦ g ′(u0) and l(g(u0)) = 0, where

Y ′

+ = {l ∈ Y ′; ∀ k ∈ K, l(k) ≥ 0 }

(ii) Set L(u) := f (u) − l(g(u)). Then F ′′(u0)(v, v) ≥ 0, ∀v ∈ Tu0, Tu0 =

• v ∈ U; f ′(u0)(v) = 0,

g ′(u0)(v) ∈ Kg(u0) = {K + λg(u0); λ ∈ ❘}

• .
• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks)

Ioffe A. D - Tihomirov V. M., Maurer H. - Zowe J. Abstract setting: U, Y real Banach, ∅ = K ⊂ Y closed convex cone, f : U → ❘, g : U → Y , min{f (u), u ∈ U, g(u) ∈ K}. Proposition 0.4 Let u0 ∈ U solve the min. problem. Assume f , g are C2 at u0 and g′(u0)(U) = Y . Then,

(i) ∃ l ∈ Y ′

+ : f ′(u0) = l ◦ g ′(u0) and l(g(u0)) = 0, where

Y ′

+ = {l ∈ Y ′; ∀ k ∈ K, l(k) ≥ 0 }

(ii) Set L(u) := f (u) − l(g(u)). Then F ′′(u0)(v, v) ≥ 0, ∀v ∈ Tu0, Tu0 =

• v ∈ U; f ′(u0)(v) = 0,

g ′(u0)(v) ∈ Kg(u0) = {K + λg(u0); λ ∈ ❘}

• .
• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks)

Ioffe A. D - Tihomirov V. M., Maurer H. - Zowe J. Abstract setting: U, Y real Banach, ∅ = K ⊂ Y closed convex cone, f : U → ❘, g : U → Y , min{f (u), u ∈ U, g(u) ∈ K}. Proposition 0.4 Let u0 ∈ U solve the min. problem. Assume f , g are C2 at u0 and g′(u0)(U) = Y . Then,

(i) ∃ l ∈ Y ′

+ : f ′(u0) = l ◦ g ′(u0) and l(g(u0)) = 0, where

Y ′

+ = {l ∈ Y ′; ∀ k ∈ K, l(k) ≥ 0 }

(ii) Set L(u) := f (u) − l(g(u)). Then F ′′(u0)(v, v) ≥ 0, ∀v ∈ Tu0, Tu0 =

• v ∈ U; f ′(u0)(v) = 0,

g ′(u0)(v) ∈ Kg(u0) = {K + λg(u0); λ ∈ ❘}

• .
• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks)

Ioffe A. D - Tihomirov V. M., Maurer H. - Zowe J. Abstract setting: U, Y real Banach, ∅ = K ⊂ Y closed convex cone, f : U → ❘, g : U → Y , min{f (u), u ∈ U, g(u) ∈ K}. Proposition 0.4 Let u0 ∈ U solve the min. problem. Assume f , g are C2 at u0 and g′(u0)(U) = Y . Then,

(i) ∃ l ∈ Y ′

+ : f ′(u0) = l ◦ g ′(u0) and l(g(u0)) = 0, where

Y ′

+ = {l ∈ Y ′; ∀ k ∈ K, l(k) ≥ 0 }

(ii) Set L(u) := f (u) − l(g(u)). Then F ′′(u0)(v, v) ≥ 0, ∀v ∈ Tu0, Tu0 =

• v ∈ U; f ′(u0)(v) = 0,

g ′(u0)(v) ∈ Kg(u0) = {K + λg(u0); λ ∈ ❘}

• .
• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks)

U = H1(❚), Y = H−1(❚) × H1(❚) × H1(❚), K = Y+, g(u) = (u′′ + u, u − a, b − u) Easy to find

• ❚

ζ0d(u′′

0 + u0) = 0,

• ❚

ζ0d(v′′ + v) ≥ 0, for all v′′ + v ≥ 0 We find that ζ0 satisfies furthermore ζ0 ≥ 0, ζ0(u′′

0 + u0) = 0

The second order optimality condition is necessary Example: Consider G(u, p) = −1

2

p

u

2. = −

• {u′′

0 +u0>0}

u′ u0 v u0 ′ =

• {u′′

0 +u0>0}

u′ u0 ′ v u0 u0 = AeBθ

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks)

U = H1(❚), Y = H−1(❚) × H1(❚) × H1(❚), K = Y+, g(u) = (u′′ + u, u − a, b − u) Easy to find

• ❚

ζ0d(u′′

0 + u0) = 0,

• ❚

ζ0d(v′′ + v) ≥ 0, for all v′′ + v ≥ 0 We find that ζ0 satisfies furthermore ζ0 ≥ 0, ζ0(u′′

0 + u0) = 0

The second order optimality condition is necessary Example: Consider G(u, p) = −1

2

p

u

2. = −

• {u′′

0 +u0>0}

u′ u0 v u0 ′ =

• {u′′

0 +u0>0}

u′ u0 ′ v u0 u0 = AeBθ

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks)

U = H1(❚), Y = H−1(❚) × H1(❚) × H1(❚), K = Y+, g(u) = (u′′ + u, u − a, b − u) Easy to find

• ❚

ζ0d(u′′

0 + u0) = 0,

• ❚

ζ0d(v′′ + v) ≥ 0, for all v′′ + v ≥ 0 We find that ζ0 satisfies furthermore ζ0 ≥ 0, ζ0(u′′

0 + u0) = 0

The second order optimality condition is necessary Example: Consider G(u, p) = −1

2

p

u

2. = −

• {u′′

0 +u0>0}

u′ u0 v u0 ′ =

• {u′′

0 +u0>0}

u′ u0 ′ v u0 u0 = AeBθ

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks)

U = H1(❚), Y = H−1(❚) × H1(❚) × H1(❚), K = Y+, g(u) = (u′′ + u, u − a, b − u) Easy to find

• ❚

ζ0d(u′′

0 + u0) = 0,

• ❚

ζ0d(v′′ + v) ≥ 0, for all v′′ + v ≥ 0 We find that ζ0 satisfies furthermore ζ0 ≥ 0, ζ0(u′′

0 + u0) = 0

The second order optimality condition is necessary Example: Consider G(u, p) = −1

2

p

u

2. = −

• {u′′

0 +u0>0}

u′ u0 v u0 ′ =

• {u′′

0 +u0>0}

u′ u0 ′ v u0 u0 = AeBθ

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks)

U = H1(❚), Y = H−1(❚) × H1(❚) × H1(❚), K = Y+, g(u) = (u′′ + u, u − a, b − u) Easy to find

• ❚

ζ0d(u′′

0 + u0) = 0,

• ❚

ζ0d(v′′ + v) ≥ 0, for all v′′ + v ≥ 0 We find that ζ0 satisfies furthermore ζ0 ≥ 0, ζ0(u′′

0 + u0) = 0

The second order optimality condition is necessary Example: Consider G(u, p) = −1

2

p

u

2. = −

• {u′′

0 +u0>0}

u′ u0 v u0 ′ =

• {u′′

0 +u0>0}

u′ u0 ′ v u0 u0 = AeBθ

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks)

U = H1(❚), Y = H−1(❚) × H1(❚) × H1(❚), K = Y+, g(u) = (u′′ + u, u − a, b − u) Easy to find

• ❚

ζ0d(u′′

0 + u0) = 0,

• ❚

ζ0d(v′′ + v) ≥ 0, for all v′′ + v ≥ 0 We find that ζ0 satisfies furthermore ζ0 ≥ 0, ζ0(u′′

0 + u0) = 0

The second order optimality condition is necessary Example: Consider G(u, p) = −1

2

p

u

2. = −

• {u′′

0 +u0>0}

u′ u0 v u0 ′ =

• {u′′

0 +u0>0}

u′ u0 ′ v u0 u0 = AeBθ

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks)

U = H1(❚), Y = H−1(❚) × H1(❚) × H1(❚), K = Y+, g(u) = (u′′ + u, u − a, b − u) Easy to find

• ❚

ζ0d(u′′

0 + u0) = 0,

• ❚

ζ0d(v′′ + v) ≥ 0, for all v′′ + v ≥ 0 We find that ζ0 satisfies furthermore ζ0 ≥ 0, ζ0(u′′

0 + u0) = 0

The second order optimality condition is necessary Example: Consider G(u, p) = −1

2

p

u

2. = −

• {u′′

0 +u0>0}

u′ u0 v u0 ′ =

• {u′′

0 +u0>0}

u′ u0 ′ v u0 u0 = AeBθ

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A(a, b))

By contradiction Case a < u0(θ0 = 0) < b (Fig. 1)

∃ǫn → 0, Su0 ∩ (0, ǫn) = ∅ To find vn: (v ′′

n + vn) ≥ λ(u′′ 0 + u0) 1 a 1 b

ǫn Ωu0 ǫi

n

θ0 A(a, b)

• Fig. 1

We consider vn,i ∈ H1(❚) solving v ′′

n,i + vn,i

= χ(ǫi

n,ǫi+1 n

)(u′′ 0 + u0),

(0, ǫn), vn,i = 0, (0, ǫn)c. Look for vn =

i=1,3 λn,ivn,i such that

v ′

n(0+) = v ′ n(ǫ− n ) = 0, which implies (v ′′ n + vn) ≥ λ(u′′ 0 + u0)

• ❚ vn(ζ0 + ζ′′

0 ) =

• ❚ vndµa =
• ❚ vndµb = 0, and

≤ j′′(u0)(vn, vn) =

• ❚

Guuv 2

n + 2Gupvnv ′ n + Gppv ′ n 2

≤

• ❚

(o(1) − Kpp)|v ′

n|2 < 0

(!) (vnL2(❚) ≤ √ǫnv ′

nL2(❚))

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A(a, b))

By contradiction Case a < u0(θ0 = 0) < b (Fig. 1)

∃ǫn → 0, Su0 ∩ (0, ǫn) = ∅ To find vn: (v ′′

n + vn) ≥ λ(u′′ 0 + u0) 1 a 1 b

ǫn Ωu0 ǫi

n

θ0 A(a, b)

• Fig. 1

We consider vn,i ∈ H1(❚) solving v ′′

n,i + vn,i

= χ(ǫi

n,ǫi+1 n

)(u′′ 0 + u0),

(0, ǫn), vn,i = 0, (0, ǫn)c. Look for vn =

i=1,3 λn,ivn,i such that

v ′

n(0+) = v ′ n(ǫ− n ) = 0, which implies (v ′′ n + vn) ≥ λ(u′′ 0 + u0)

• ❚ vn(ζ0 + ζ′′

0 ) =

• ❚ vndµa =
• ❚ vndµb = 0, and

≤ j′′(u0)(vn, vn) =

• ❚

Guuv 2

n + 2Gupvnv ′ n + Gppv ′ n 2

≤

• ❚

(o(1) − Kpp)|v ′

n|2 < 0

(!) (vnL2(❚) ≤ √ǫnv ′

nL2(❚))

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A(a, b))

By contradiction Case a < u0(θ0 = 0) < b (Fig. 1)

∃ǫn → 0, Su0 ∩ (0, ǫn) = ∅ To find vn: (v ′′

n + vn) ≥ λ(u′′ 0 + u0) 1 a 1 b

ǫn Ωu0 ǫi

n

θ0 A(a, b)

• Fig. 1

We consider vn,i ∈ H1(❚) solving v ′′

n,i + vn,i

= χ(ǫi

n,ǫi+1 n

)(u′′ 0 + u0),

(0, ǫn), vn,i = 0, (0, ǫn)c. Look for vn =

i=1,3 λn,ivn,i such that

v ′

n(0+) = v ′ n(ǫ− n ) = 0, which implies (v ′′ n + vn) ≥ λ(u′′ 0 + u0)

• ❚ vn(ζ0 + ζ′′

0 ) =

• ❚ vndµa =
• ❚ vndµb = 0, and

≤ j′′(u0)(vn, vn) =

• ❚

Guuv 2

n + 2Gupvnv ′ n + Gppv ′ n 2

≤

• ❚

(o(1) − Kpp)|v ′

n|2 < 0

(!) (vnL2(❚) ≤ √ǫnv ′

nL2(❚))

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A(a, b))

By contradiction Case a < u0(θ0 = 0) < b (Fig. 1)

∃ǫn → 0, Su0 ∩ (0, ǫn) = ∅ To find vn: (v ′′

n + vn) ≥ λ(u′′ 0 + u0) 1 a 1 b

ǫn Ωu0 ǫi

n

θ0 A(a, b)

• Fig. 1

We consider vn,i ∈ H1(❚) solving v ′′

n,i + vn,i

= χ(ǫi

n,ǫi+1 n

)(u′′ 0 + u0),

(0, ǫn), vn,i = 0, (0, ǫn)c. Look for vn =

i=1,3 λn,ivn,i such that

v ′

n(0+) = v ′ n(ǫ− n ) = 0, which implies (v ′′ n + vn) ≥ λ(u′′ 0 + u0)

• ❚ vn(ζ0 + ζ′′

0 ) =

• ❚ vndµa =
• ❚ vndµb = 0, and

≤ j′′(u0)(vn, vn) =

• ❚

Guuv 2

n + 2Gupvnv ′ n + Gppv ′ n 2

≤

• ❚

(o(1) − Kpp)|v ′

n|2 < 0

(!) (vnL2(❚) ≤ √ǫnv ′

nL2(❚))

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A(a, b))

By contradiction Case a < u0(θ0 = 0) < b (Fig. 1)

∃ǫn → 0, Su0 ∩ (0, ǫn) = ∅ To find vn: (v ′′

n + vn) ≥ λ(u′′ 0 + u0) 1 a 1 b

ǫn Ωu0 ǫi

n

θ0 A(a, b)

• Fig. 1

We consider vn,i ∈ H1(❚) solving v ′′

n,i + vn,i

= χ(ǫi

n,ǫi+1 n

)(u′′ 0 + u0),

(0, ǫn), vn,i = 0, (0, ǫn)c. Look for vn =

i=1,3 λn,ivn,i such that

v ′

n(0+) = v ′ n(ǫ− n ) = 0, which implies (v ′′ n + vn) ≥ λ(u′′ 0 + u0)

• ❚ vn(ζ0 + ζ′′

0 ) =

• ❚ vndµa =
• ❚ vndµb = 0, and

≤ j′′(u0)(vn, vn) =

• ❚

Guuv 2

n + 2Gupvnv ′ n + Gppv ′ n 2

≤

• ❚

(o(1) − Kpp)|v ′

n|2 < 0

(!) (vnL2(❚) ≤ √ǫnv ′

nL2(❚))

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A(a, b))

By contradiction Case a < u0(θ0 = 0) < b (Fig. 1)

∃ǫn → 0, Su0 ∩ (0, ǫn) = ∅ To find vn: (v ′′

n + vn) ≥ λ(u′′ 0 + u0) 1 a 1 b

ǫn Ωu0 ǫi

n

θ0 A(a, b)

• Fig. 1

We consider vn,i ∈ H1(❚) solving v ′′

n,i + vn,i

= χ(ǫi

n,ǫi+1 n

)(u′′ 0 + u0),

(0, ǫn), vn,i = 0, (0, ǫn)c. Look for vn =

i=1,3 λn,ivn,i such that

v ′

n(0+) = v ′ n(ǫ− n ) = 0, which implies (v ′′ n + vn) ≥ λ(u′′ 0 + u0)

• ❚ vn(ζ0 + ζ′′

0 ) =

• ❚ vndµa =
• ❚ vndµb = 0, and

≤ j′′(u0)(vn, vn) =

• ❚

Guuv 2

n + 2Gupvnv ′ n + Gppv ′ n 2

≤

• ❚

(o(1) − Kpp)|v ′

n|2 < 0

(!) (vnL2(❚) ≤ √ǫnv ′

nL2(❚))

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A(a, b))

By contradiction Case a < u0(θ0 = 0) < b (Fig. 1)

∃ǫn → 0, Su0 ∩ (0, ǫn) = ∅ To find vn: (v ′′

n + vn) ≥ λ(u′′ 0 + u0) 1 a 1 b

ǫn Ωu0 ǫi

n

θ0 A(a, b)

• Fig. 1

We consider vn,i ∈ H1(❚) solving v ′′

n,i + vn,i

= χ(ǫi

n,ǫi+1 n

)(u′′ 0 + u0),

(0, ǫn), vn,i = 0, (0, ǫn)c. Look for vn =

i=1,3 λn,ivn,i such that

v ′

n(0+) = v ′ n(ǫ− n ) = 0, which implies (v ′′ n + vn) ≥ λ(u′′ 0 + u0)

• ❚ vn(ζ0 + ζ′′

0 ) =

• ❚ vndµa =
• ❚ vndµb = 0, and

≤ j′′(u0)(vn, vn) =

• ❚

Guuv 2

n + 2Gupvnv ′ n + Gppv ′ n 2

≤

• ❚

(o(1) − Kpp)|v ′

n|2 < 0

(!) (vnL2(❚) ≤ √ǫnv ′

nL2(❚))

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A(a, b))

By contradiction Case a < u0(θ0 = 0) < b (Fig. 1)

∃ǫn → 0, Su0 ∩ (0, ǫn) = ∅ To find vn: (v ′′

n + vn) ≥ λ(u′′ 0 + u0) 1 a 1 b

ǫn Ωu0 ǫi

n

θ0 A(a, b)

• Fig. 1

We consider vn,i ∈ H1(❚) solving v ′′

n,i + vn,i

= χ(ǫi

n,ǫi+1 n

)(u′′ 0 + u0),

(0, ǫn), vn,i = 0, (0, ǫn)c. Look for vn =

i=1,3 λn,ivn,i such that

v ′

n(0+) = v ′ n(ǫ− n ) = 0, which implies (v ′′ n + vn) ≥ λ(u′′ 0 + u0)

• ❚ vn(ζ0 + ζ′′

0 ) =

• ❚ vndµa =
• ❚ vndµb = 0, and

≤ j′′(u0)(vn, vn) =

• ❚

Guuv 2

n + 2Gupvnv ′ n + Gppv ′ n 2

≤

• ❚

(o(1) − Kpp)|v ′

n|2 < 0

(!) (vnL2(❚) ≤ √ǫnv ′

nL2(❚))

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A(a, b))

By contradiction Case a < u0(θ0 = 0) < b (Fig. 1)

∃ǫn → 0, Su0 ∩ (0, ǫn) = ∅ To find vn: (v ′′

n + vn) ≥ λ(u′′ 0 + u0) 1 a 1 b

ǫn Ωu0 ǫi

n

θ0 A(a, b)

• Fig. 1

We consider vn,i ∈ H1(❚) solving v ′′

n,i + vn,i

= χ(ǫi

n,ǫi+1 n

)(u′′ 0 + u0),

(0, ǫn), vn,i = 0, (0, ǫn)c. Look for vn =

i=1,3 λn,ivn,i such that

v ′

n(0+) = v ′ n(ǫ− n ) = 0, which implies (v ′′ n + vn) ≥ λ(u′′ 0 + u0)

• ❚ vn(ζ0 + ζ′′

0 ) =

• ❚ vndµa =
• ❚ vndµb = 0, and

≤ j′′(u0)(vn, vn) =

• ❚

Guuv 2

n + 2Gupvnv ′ n + Gppv ′ n 2

≤

• ❚

(o(1) − Kpp)|v ′

n|2 < 0

(!) (vnL2(❚) ≤ √ǫnv ′

nL2(❚))

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A(a, b))

By contradiction Case a < u0(θ0 = 0) < b (Fig. 1)

∃ǫn → 0, Su0 ∩ (0, ǫn) = ∅ To find vn: (v ′′

n + vn) ≥ λ(u′′ 0 + u0) 1 a 1 b

ǫn Ωu0 ǫi

n

θ0 A(a, b)

• Fig. 1

We consider vn,i ∈ H1(❚) solving v ′′

n,i + vn,i

= χ(ǫi

n,ǫi+1 n

)(u′′ 0 + u0),

(0, ǫn), vn,i = 0, (0, ǫn)c. Look for vn =

i=1,3 λn,ivn,i such that

v ′

n(0+) = v ′ n(ǫ− n ) = 0, which implies (v ′′ n + vn) ≥ λ(u′′ 0 + u0)

• ❚ vn(ζ0 + ζ′′

0 ) =

• ❚ vndµa =
• ❚ vndµb = 0, and

≤ j′′(u0)(vn, vn) =

• ❚

Guuv 2

n + 2Gupvnv ′ n + Gppv ′ n 2

≤

• ❚

(o(1) − Kpp)|v ′

n|2 < 0

(!) (vnL2(❚) ≤ √ǫnv ′

nL2(❚))

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A(a, b))

Case u0(θ0 = 0) = a (Fig. 2)

Near θ0: u′′

0 + u0 = n∈◆∗ αnδθn near θ0 .

Consider vn ∈ H1

0(θn+1, θn−1):

v ′′

n + vn = δθn in (θn+1, θn−1). 1 a 1 b

θn θ0

• Fig. 2

We can choose λ ≪ 0 (depending on n) such that       

• ❚ vd(ζ0 + ζ′′

0 + µa − µb)

= 0, v ′′

n + vn

≥ λ(u′′

0 + u0)

vn ≥ λ(u0 − a), vn ≤ λ(u0 − b). Proceeding as in (a) above, we find a contradiction! Case u0(θ0 = 0) = b Proceed as in u0(θ0) = a.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A(a, b))

Case u0(θ0 = 0) = a (Fig. 2)

Near θ0: u′′

0 + u0 = n∈◆∗ αnδθn near θ0 .

Consider vn ∈ H1

0(θn+1, θn−1):

v ′′

n + vn = δθn in (θn+1, θn−1). 1 a 1 b

θn θ0

• Fig. 2

We can choose λ ≪ 0 (depending on n) such that       

• ❚ vd(ζ0 + ζ′′

0 + µa − µb)

= 0, v ′′

n + vn

≥ λ(u′′

0 + u0)

vn ≥ λ(u0 − a), vn ≤ λ(u0 − b). Proceeding as in (a) above, we find a contradiction! Case u0(θ0 = 0) = b Proceed as in u0(θ0) = a.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A(a, b))

Case u0(θ0 = 0) = a (Fig. 2)

Near θ0: u′′

0 + u0 = n∈◆∗ αnδθn near θ0 .

Consider vn ∈ H1

0(θn+1, θn−1):

v ′′

n + vn = δθn in (θn+1, θn−1). 1 a 1 b

θn θ0

• Fig. 2

We can choose λ ≪ 0 (depending on n) such that       

• ❚ vd(ζ0 + ζ′′

0 + µa − µb)

= 0, v ′′

n + vn

≥ λ(u′′

0 + u0)

vn ≥ λ(u0 − a), vn ≤ λ(u0 − b). Proceeding as in (a) above, we find a contradiction! Case u0(θ0 = 0) = b Proceed as in u0(θ0) = a.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A(a, b))

Case u0(θ0 = 0) = a (Fig. 2)

Near θ0: u′′

0 + u0 = n∈◆∗ αnδθn near θ0 .

Consider vn ∈ H1

0(θn+1, θn−1):

v ′′

n + vn = δθn in (θn+1, θn−1). 1 a 1 b

θn θ0

• Fig. 2

We can choose λ ≪ 0 (depending on n) such that       

• ❚ vd(ζ0 + ζ′′

0 + µa − µb)

= 0, v ′′

n + vn

≥ λ(u′′

0 + u0)

vn ≥ λ(u0 − a), vn ≤ λ(u0 − b). Proceeding as in (a) above, we find a contradiction! Case u0(θ0 = 0) = b Proceed as in u0(θ0) = a.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A(a, b))

Case u0(θ0 = 0) = a (Fig. 2)

Near θ0: u′′

0 + u0 = n∈◆∗ αnδθn near θ0 .

Consider vn ∈ H1

0(θn+1, θn−1):

v ′′

n + vn = δθn in (θn+1, θn−1). 1 a 1 b

θn θ0

• Fig. 2

We can choose λ ≪ 0 (depending on n) such that       

• ❚ vd(ζ0 + ζ′′

0 + µa − µb)

= 0, v ′′

n + vn

≥ λ(u′′

0 + u0)

vn ≥ λ(u0 − a), vn ≤ λ(u0 − b). Proceeding as in (a) above, we find a contradiction! Case u0(θ0 = 0) = b Proceed as in u0(θ0) = a.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A(a, b))

Case u0(θ0 = 0) = a (Fig. 2)

Near θ0: u′′

0 + u0 = n∈◆∗ αnδθn near θ0 .

Consider vn ∈ H1

0(θn+1, θn−1):

v ′′

n + vn = δθn in (θn+1, θn−1). 1 a 1 b

θn θ0

• Fig. 2

We can choose λ ≪ 0 (depending on n) such that       

• ❚ vd(ζ0 + ζ′′

0 + µa − µb)

= 0, v ′′

n + vn

≥ λ(u′′

0 + u0)

vn ≥ λ(u0 − a), vn ≤ λ(u0 − b). Proceeding as in (a) above, we find a contradiction! Case u0(θ0 = 0) = b Proceed as in u0(θ0) = a.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A(a, b))

Case u0(θ0 = 0) = a (Fig. 2)

Near θ0: u′′

0 + u0 = n∈◆∗ αnδθn near θ0 .

Consider vn ∈ H1

0(θn+1, θn−1):

v ′′

n + vn = δθn in (θn+1, θn−1). 1 a 1 b

θn θ0

• Fig. 2

We can choose λ ≪ 0 (depending on n) such that       

• ❚ vd(ζ0 + ζ′′

0 + µa − µb)

= 0, v ′′

n + vn

≥ λ(u′′

0 + u0)

vn ≥ λ(u0 − a), vn ≤ λ(u0 − b). Proceeding as in (a) above, we find a contradiction! Case u0(θ0 = 0) = b Proceed as in u0(θ0) = a.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A(a, b))

Case u0(θ0 = 0) = a (Fig. 2)

Near θ0: u′′

0 + u0 = n∈◆∗ αnδθn near θ0 .

Consider vn ∈ H1

0(θn+1, θn−1):

v ′′

n + vn = δθn in (θn+1, θn−1). 1 a 1 b

θn θ0

• Fig. 2

We can choose λ ≪ 0 (depending on n) such that       

• ❚ vd(ζ0 + ζ′′

0 + µa − µb)

= 0, v ′′

n + vn

≥ λ(u′′

0 + u0)

vn ≥ λ(u0 − a), vn ≤ λ(u0 − b). Proceeding as in (a) above, we find a contradiction! Case u0(θ0 = 0) = b Proceed as in u0(θ0) = a.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.2 (inclusion in A(a, b))

No accumulation points in {a < u0 < b}:

follows from Theorem 0.1

No accumulation points on {u0 = a}:

θ1

n

θ1

n−1

θ2

n−1

θ0 ǫn

1 a

ωn−1 ωn

• Fig. 3

θn θ0 ǫn

1 a

Fa Fa ωi−1 ωi

• Fig. 4
• Fig. 3: v ′′

n + vn = u′′ 0 + u0 = Ni i=1 αiδθi

n in ωn, vn = 0 in ωc

n.

• Fig. 4: v ′′

n + vn = −(u′′ 0 + u0) in (0, ǫn), vn = 0 in (0, ǫn)c.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.2 (inclusion in A(a, b))

No accumulation points in {a < u0 < b}:

follows from Theorem 0.1

No accumulation points on {u0 = a}:

θ1

n

θ1

n−1

θ2

n−1

θ0 ǫn

1 a

ωn−1 ωn

• Fig. 3

θn θ0 ǫn

1 a

Fa Fa ωi−1 ωi

• Fig. 4
• Fig. 3: v ′′

n + vn = u′′ 0 + u0 = Ni i=1 αiδθi

n in ωn, vn = 0 in ωc

n.

• Fig. 4: v ′′

n + vn = −(u′′ 0 + u0) in (0, ǫn), vn = 0 in (0, ǫn)c.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.2 (inclusion in A(a, b))

No accumulation points in {a < u0 < b}:

follows from Theorem 0.1

No accumulation points on {u0 = a}:

θ1

n

θ1

n−1

θ2

n−1

θ0 ǫn

1 a

ωn−1 ωn

• Fig. 3

θn θ0 ǫn

1 a

Fa Fa ωi−1 ωi

• Fig. 4
• Fig. 3: v ′′

n + vn = u′′ 0 + u0 = Ni i=1 αiδθi

n in ωn, vn = 0 in ωc

n.

• Fig. 4: v ′′

n + vn = −(u′′ 0 + u0) in (0, ǫn), vn = 0 in (0, ǫn)c.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.2 (inclusion in A(a, b))

No accumulation points in {a < u0 < b}:

follows from Theorem 0.1

No accumulation points on {u0 = a}:

θ1

n

θ1

n−1

θ2

n−1

θ0 ǫn

1 a

ωn−1 ωn

• Fig. 3

θn θ0 ǫn

1 a

Fa Fa ωi−1 ωi

• Fig. 4
• Fig. 3: v ′′

n + vn = u′′ 0 + u0 = Ni i=1 αiδθi

n in ωn, vn = 0 in ωc

n.

• Fig. 4: v ′′

n + vn = −(u′′ 0 + u0) in (0, ǫn), vn = 0 in (0, ǫn)c.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.2 (inclusion in A(a, b))

No accumulation points in {a < u0 < b}:

follows from Theorem 0.1

No accumulation points on {u0 = a}:

θ1

n

θ1

n−1

θ2

n−1

θ0 ǫn

1 a

ωn−1 ωn

• Fig. 3

θn θ0 ǫn

1 a

Fa Fa ωi−1 ωi

• Fig. 4
• Fig. 3: v ′′

n + vn = u′′ 0 + u0 = Ni i=1 αiδθi

n in ωn, vn = 0 in ωc

n.

• Fig. 4: v ′′

n + vn = −(u′′ 0 + u0) in (0, ǫn), vn = 0 in (0, ǫn)c.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.2 (inclusion in A(a, b))

No accumulation points in {a < u0 < b}:

follows from Theorem 0.1

No accumulation points on {u0 = a}:

θ1

n

θ1

n−1

θ2

n−1

θ0 ǫn

1 a

ωn−1 ωn

• Fig. 3

θn θ0 ǫn

1 a

Fa Fa ωi−1 ωi

• Fig. 4
• Fig. 3: v ′′

n + vn = u′′ 0 + u0 = Ni i=1 αiδθi

n in ωn, vn = 0 in ωc

n.

• Fig. 4: v ′′

n + vn = −(u′′ 0 + u0) in (0, ǫn), vn = 0 in (0, ǫn)c.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.2 (inclusion in A(a, b))

No accumulation points in {a < u0 < b}:

follows from Theorem 0.1

No accumulation points on {u0 = a}:

θ1

n

θ1

n−1

θ2

n−1

θ0 ǫn

1 a

ωn−1 ωn

• Fig. 3

θn θ0 ǫn

1 a

Fa Fa ωi−1 ωi

• Fig. 4
• Fig. 3: v ′′

n + vn = u′′ 0 + u0 = Ni i=1 αiδθi

n in ωn, vn = 0 in ωc

n.

• Fig. 4: v ′′

n + vn = −(u′′ 0 + u0) in (0, ǫn), vn = 0 in (0, ǫn)c.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.2 (inclusion in A(a, b))

No accumulation points on {u0 = b}:

σn0

1 b

Fb ǫn θi

n

τn 2ǫn ωi

n

• Fig. 5

   un = u0, (0, 2ǫn)c, un = b cos θ, (0, ǫn), un = b cos(θ − 2ǫn), (ǫn, 2ǫn),

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.2 (inclusion in A(a, b))

No accumulation points on {u0 = b}:

σn0

1 b

Fb ǫn θi

n

τn 2ǫn ωi

n

• Fig. 5

   un = u0, (0, 2ǫn)c, un = b cos θ, (0, ǫn), un = b cos(θ − 2ǫn), (ǫn, 2ǫn),

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.2 (inclusion in A(a, b))

No accumulation points on {u0 = b}:

σn0

1 b

Fb ǫn θi

n

τn 2ǫn ωi

n

• Fig. 5

   un = u0, (0, 2ǫn)c, un = b cos θ, (0, ǫn), un = b cos(θ − 2ǫn), (ǫn, 2ǫn),

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.2 (inclusion in A(a, b))

No accumulation points on {u0 = b}:

σn0

1 b

Fb ǫn θi

n

τn 2ǫn ωi

n

• Fig. 5

   un = u0, (0, 2ǫn)c, un = b cos θ, (0, ǫn), un = b cos(θ − 2ǫn), (ǫn, 2ǫn),

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions

Theorem 0.5 (inclusion in A(a, b))

1

Let j(u) =

• ❚ G(u, u′), u0 be a solution of (1). Assume that

2

(i) G is a C2 function and Gpp < 0 on {(u0(θ), u′

0(θ)), θ ∈ ❚},

3

(ii) The function p → G(a, p) is even and one of the followings holds

(ii.1): Gu(a, 0) < 0, or (ii.2): Gu(a, 0) = 0 and Gu(u0, u′

0)u0 + Gp(u0, u′ 0)u′ 0 ≤ 0, 4

(iii) The function p → G(·, p) is even and Gu ≥ 0 near (b, 0).

5

Then Su0 is finite, i.e. Ωu0 is a polygon.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions

Theorem 0.5 (inclusion in A(a, b))

1

Let j(u) =

• ❚ G(u, u′), u0 be a solution of (1). Assume that

2

(i) G is a C2 function and Gpp < 0 on {(u0(θ), u′

0(θ)), θ ∈ ❚},

3

(ii) The function p → G(a, p) is even and one of the followings holds

(ii.1): Gu(a, 0) < 0, or (ii.2): Gu(a, 0) = 0 and Gu(u0, u′

0)u0 + Gp(u0, u′ 0)u′ 0 ≤ 0, 4

(iii) The function p → G(·, p) is even and Gu ≥ 0 near (b, 0).

5

Then Su0 is finite, i.e. Ωu0 is a polygon.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Concavity condition Gpp < 0 Set c = a+b

2

and consider G(u, p) = 1

2

• (u − c)2 + p2

.

G satisfies (ii.1) because Gu(a, 0) = a − c < 0, G satisfies (iii) because Gu(b, 0) = b − c > 0.

G does not satisfy (i) because Gpp = 1. It is obvious that the corresponding solution of (1) is not a polygon, but rather (the circle) {u0 = c}.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Concavity condition Gpp < 0 Set c = a+b

2

and consider G(u, p) = 1

2

• (u − c)2 + p2

.

G satisfies (ii.1) because Gu(a, 0) = a − c < 0, G satisfies (iii) because Gu(b, 0) = b − c > 0.

G does not satisfy (i) because Gpp = 1. It is obvious that the corresponding solution of (1) is not a polygon, but rather (the circle) {u0 = c}.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Concavity condition Gpp < 0 Set c = a+b

2

and consider G(u, p) = 1

2

• (u − c)2 + p2

.

G satisfies (ii.1) because Gu(a, 0) = a − c < 0, G satisfies (iii) because Gu(b, 0) = b − c > 0.

G does not satisfy (i) because Gpp = 1. It is obvious that the corresponding solution of (1) is not a polygon, but rather (the circle) {u0 = c}.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Concavity condition Gpp < 0 Set c = a+b

2

and consider G(u, p) = 1

2

• (u − c)2 + p2

.

G satisfies (ii.1) because Gu(a, 0) = a − c < 0, G satisfies (iii) because Gu(b, 0) = b − c > 0.

G does not satisfy (i) because Gpp = 1. It is obvious that the corresponding solution of (1) is not a polygon, but rather (the circle) {u0 = c}.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Concavity condition Gpp < 0 Set c = a+b

2

and consider G(u, p) = 1

2

• (u − c)2 + p2

.

G satisfies (ii.1) because Gu(a, 0) = a − c < 0, G satisfies (iii) because Gu(b, 0) = b − c > 0.

G does not satisfy (i) because Gpp = 1. It is obvious that the corresponding solution of (1) is not a polygon, but rather (the circle) {u0 = c}.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Concavity condition Gpp < 0 Set c = a+b

2

and consider G(u, p) = 1

2

• (u − c)2 + p2

.

G satisfies (ii.1) because Gu(a, 0) = a − c < 0, G satisfies (iii) because Gu(b, 0) = b − c > 0.

G does not satisfy (i) because Gpp = 1. It is obvious that the corresponding solution of (1) is not a polygon, but rather (the circle) {u0 = c}.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Concavity condition Gpp < 0 Set c = a+b

2

and consider G(u, p) = 1

2

• (u − c)2 + p2

.

G satisfies (ii.1) because Gu(a, 0) = a − c < 0, G satisfies (iii) because Gu(b, 0) = b − c > 0.

G does not satisfy (i) because Gpp = 1. It is obvious that the corresponding solution of (1) is not a polygon, but rather (the circle) {u0 = c}.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Condition at u = a: Gu(a, 0) < 0 Consider the function G(u, p) = 1

2(u2 − p2).

G(u, p) satisfies (i) because Gpp(u, p) = −2 G(u, p) satisfies (iii) because Gu(b, 0) = b G does not satisfy (ii.1), neither (ii.2).

The solution of (1) for this G(u, p) is (the circle) u0 = a. Indeed, for admissible u we have j(u) = 1 2

• ❚

(u+u′′)u ≥ a 2

• ❚

(u+u′′) = a 2

• ❚

u ≥ πa2 = j(u0), which proves that u0 ≡ a is the minimizer of j(u). Another example: Take G(u, p) = −(u2+p2)1/2

u2

, so j(u) = −P(Ωu). Then u0 = a.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Condition at u = a: Gu(a, 0) < 0 Consider the function G(u, p) = 1

2(u2 − p2).

G(u, p) satisfies (i) because Gpp(u, p) = −2 G(u, p) satisfies (iii) because Gu(b, 0) = b G does not satisfy (ii.1), neither (ii.2).

The solution of (1) for this G(u, p) is (the circle) u0 = a. Indeed, for admissible u we have j(u) = 1 2

• ❚

(u+u′′)u ≥ a 2

• ❚

(u+u′′) = a 2

• ❚

u ≥ πa2 = j(u0), which proves that u0 ≡ a is the minimizer of j(u). Another example: Take G(u, p) = −(u2+p2)1/2

u2

, so j(u) = −P(Ωu). Then u0 = a.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Condition at u = a: Gu(a, 0) < 0 Consider the function G(u, p) = 1

2(u2 − p2).

G(u, p) satisfies (i) because Gpp(u, p) = −2 G(u, p) satisfies (iii) because Gu(b, 0) = b G does not satisfy (ii.1), neither (ii.2).

The solution of (1) for this G(u, p) is (the circle) u0 = a. Indeed, for admissible u we have j(u) = 1 2

• ❚

(u+u′′)u ≥ a 2

• ❚

(u+u′′) = a 2

• ❚

u ≥ πa2 = j(u0), which proves that u0 ≡ a is the minimizer of j(u). Another example: Take G(u, p) = −(u2+p2)1/2

u2

, so j(u) = −P(Ωu). Then u0 = a.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Condition at u = a: Gu(a, 0) < 0 Consider the function G(u, p) = 1

2(u2 − p2).

G(u, p) satisfies (i) because Gpp(u, p) = −2 G(u, p) satisfies (iii) because Gu(b, 0) = b G does not satisfy (ii.1), neither (ii.2).

The solution of (1) for this G(u, p) is (the circle) u0 = a. Indeed, for admissible u we have j(u) = 1 2

• ❚

(u+u′′)u ≥ a 2

• ❚

(u+u′′) = a 2

• ❚

u ≥ πa2 = j(u0), which proves that u0 ≡ a is the minimizer of j(u). Another example: Take G(u, p) = −(u2+p2)1/2

u2

, so j(u) = −P(Ωu). Then u0 = a.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Condition at u = a: Gu(a, 0) < 0 Consider the function G(u, p) = 1

2(u2 − p2).

G(u, p) satisfies (i) because Gpp(u, p) = −2 G(u, p) satisfies (iii) because Gu(b, 0) = b G does not satisfy (ii.1), neither (ii.2).

The solution of (1) for this G(u, p) is (the circle) u0 = a. Indeed, for admissible u we have j(u) = 1 2

• ❚

(u+u′′)u ≥ a 2

• ❚

(u+u′′) = a 2

• ❚

u ≥ πa2 = j(u0), which proves that u0 ≡ a is the minimizer of j(u). Another example: Take G(u, p) = −(u2+p2)1/2

u2

, so j(u) = −P(Ωu). Then u0 = a.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Condition at u = a: Gu(a, 0) < 0 Consider the function G(u, p) = 1

2(u2 − p2).

G(u, p) satisfies (i) because Gpp(u, p) = −2 G(u, p) satisfies (iii) because Gu(b, 0) = b G does not satisfy (ii.1), neither (ii.2).

The solution of (1) for this G(u, p) is (the circle) u0 = a. Indeed, for admissible u we have j(u) = 1 2

• ❚

(u+u′′)u ≥ a 2

• ❚

(u+u′′) = a 2

• ❚

u ≥ πa2 = j(u0), which proves that u0 ≡ a is the minimizer of j(u). Another example: Take G(u, p) = −(u2+p2)1/2

u2

, so j(u) = −P(Ωu). Then u0 = a.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Condition at u = a: Gu(a, 0) < 0 Consider the function G(u, p) = 1

2(u2 − p2).

G(u, p) satisfies (i) because Gpp(u, p) = −2 G(u, p) satisfies (iii) because Gu(b, 0) = b G does not satisfy (ii.1), neither (ii.2).

The solution of (1) for this G(u, p) is (the circle) u0 = a. Indeed, for admissible u we have j(u) = 1 2

• ❚

(u+u′′)u ≥ a 2

• ❚

(u+u′′) = a 2

• ❚

u ≥ πa2 = j(u0), which proves that u0 ≡ a is the minimizer of j(u). Another example: Take G(u, p) = −(u2+p2)1/2

u2

, so j(u) = −P(Ωu). Then u0 = a.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Condition at u = a: Gu(a, 0) < 0 Consider the function G(u, p) = 1

2(u2 − p2).

G(u, p) satisfies (i) because Gpp(u, p) = −2 G(u, p) satisfies (iii) because Gu(b, 0) = b G does not satisfy (ii.1), neither (ii.2).

The solution of (1) for this G(u, p) is (the circle) u0 = a. Indeed, for admissible u we have j(u) = 1 2

• ❚

(u+u′′)u ≥ a 2

• ❚

(u+u′′) = a 2

• ❚

u ≥ πa2 = j(u0), which proves that u0 ≡ a is the minimizer of j(u). Another example: Take G(u, p) = −(u2+p2)1/2

u2

, so j(u) = −P(Ωu). Then u0 = a.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Condition at u = a: Gu(a, 0) < 0 Consider the function G(u, p) = 1

2(u2 − p2).

G(u, p) satisfies (i) because Gpp(u, p) = −2 G(u, p) satisfies (iii) because Gu(b, 0) = b G does not satisfy (ii.1), neither (ii.2).

The solution of (1) for this G(u, p) is (the circle) u0 = a. Indeed, for admissible u we have j(u) = 1 2

• ❚

(u+u′′)u ≥ a 2

• ❚

(u+u′′) = a 2

• ❚

u ≥ πa2 = j(u0), which proves that u0 ≡ a is the minimizer of j(u). Another example: Take G(u, p) = −(u2+p2)1/2

u2

, so j(u) = −P(Ωu). Then u0 = a.

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Condition at u = b: Gu(b, 0) ≥ 0 Consider G(u, p) = −1

2(u2 + p2).

G(u, p) satisfies (i), (ii.1) since Gpp = −2 and Gu(a, 0) = −a. G(u, p) does not satisfy (iii). A solution of the corresponding minimization problem is u0 ≡ b. In fact, any u0 representing a convex polygon with edges tangent to the circle {u0 = b} is a solution! Indeed: j(u) = lim

n→∞ j(un) = −1

2 lim

n→∞

• ❚

(u2

n + |u′ n|)2 ≥ −πb2

= j(u0).

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Condition at u = b: Gu(b, 0) ≥ 0 Consider G(u, p) = −1

2(u2 + p2).

G(u, p) satisfies (i), (ii.1) since Gpp = −2 and Gu(a, 0) = −a. G(u, p) does not satisfy (iii). A solution of the corresponding minimization problem is u0 ≡ b. In fact, any u0 representing a convex polygon with edges tangent to the circle {u0 = b} is a solution! Indeed: j(u) = lim

n→∞ j(un) = −1

2 lim

n→∞

• ❚

(u2

n + |u′ n|)2 ≥ −πb2

= j(u0).

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Condition at u = b: Gu(b, 0) ≥ 0 Consider G(u, p) = −1

2(u2 + p2).

G(u, p) satisfies (i), (ii.1) since Gpp = −2 and Gu(a, 0) = −a. G(u, p) does not satisfy (iii). A solution of the corresponding minimization problem is u0 ≡ b. In fact, any u0 representing a convex polygon with edges tangent to the circle {u0 = b} is a solution! Indeed: j(u) = lim

n→∞ j(un) = −1

2 lim

n→∞

• ❚

(u2

n + |u′ n|)2 ≥ −πb2

= j(u0).

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Condition at u = b: Gu(b, 0) ≥ 0 Consider G(u, p) = −1

2(u2 + p2).

G(u, p) satisfies (i), (ii.1) since Gpp = −2 and Gu(a, 0) = −a. G(u, p) does not satisfy (iii). A solution of the corresponding minimization problem is u0 ≡ b. In fact, any u0 representing a convex polygon with edges tangent to the circle {u0 = b} is a solution! Indeed: j(u) = lim

n→∞ j(un) = −1

2 lim

n→∞

• ❚

(u2

n + |u′ n|)2 ≥ −πb2

= j(u0).

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Condition at u = b: Gu(b, 0) ≥ 0 Consider G(u, p) = −1

2(u2 + p2).

G(u, p) satisfies (i), (ii.1) since Gpp = −2 and Gu(a, 0) = −a. G(u, p) does not satisfy (iii). A solution of the corresponding minimization problem is u0 ≡ b. In fact, any u0 representing a convex polygon with edges tangent to the circle {u0 = b} is a solution! Indeed: j(u) = lim

n→∞ j(un) = −1

2 lim

n→∞

• ❚

(u2

n + |u′ n|)2 ≥ −πb2

= j(u0).

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Condition at u = b: Gu(b, 0) ≥ 0 Consider G(u, p) = −1

2(u2 + p2).

G(u, p) satisfies (i), (ii.1) since Gpp = −2 and Gu(a, 0) = −a. G(u, p) does not satisfy (iii). A solution of the corresponding minimization problem is u0 ≡ b. In fact, any u0 representing a convex polygon with edges tangent to the circle {u0 = b} is a solution! Indeed: j(u) = lim

n→∞ j(un) = −1

2 lim

n→∞

• ❚

(u2

n + |u′ n|)2 ≥ −πb2

= j(u0).

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: sharpness of conditions - counterexamples

Condition at u = b: Gu(b, 0) ≥ 0 Consider G(u, p) = −1

2(u2 + p2).

G(u, p) satisfies (i), (ii.1) since Gpp = −2 and Gu(a, 0) = −a. G(u, p) does not satisfy (iii). A solution of the corresponding minimization problem is u0 ≡ b. In fact, any u0 representing a convex polygon with edges tangent to the circle {u0 = b} is a solution! Indeed: j(u) = lim

n→∞ j(un) = −1

2 lim

n→∞

• ❚

(u2

n + |u′ n|)2 ≥ −πb2

= j(u0).

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: generality of approach

j(u) =

• ❚ G(θ, u, u′)dθ is a geometric shape functionnal.

The method works even for non-geometric shape functionnals. Namely, Theorem 0.1 is valid for

j : u ∈ W 1,∞(❚) → j(u) ∈ ❘, of class C2 and satisfying juu(u0)(v, v) ≤ −αv2

H1(❚) + βv2 L2(❚),

for some α > 0, β ∈ ❘, because vL∞(0,ǫ) ≤ √ǫv ′L2(0,ǫ), v ∈ H1

0(0, ǫ),

is used

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: generality of approach

j(u) =

• ❚ G(θ, u, u′)dθ is a geometric shape functionnal.

The method works even for non-geometric shape functionnals. Namely, Theorem 0.1 is valid for

j : u ∈ W 1,∞(❚) → j(u) ∈ ❘, of class C2 and satisfying juu(u0)(v, v) ≤ −αv2

H1(❚) + βv2 L2(❚),

for some α > 0, β ∈ ❘, because vL∞(0,ǫ) ≤ √ǫv ′L2(0,ǫ), v ∈ H1

0(0, ǫ),

is used

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: generality of approach

j(u) =

• ❚ G(θ, u, u′)dθ is a geometric shape functionnal.

The method works even for non-geometric shape functionnals. Namely, Theorem 0.1 is valid for

j : u ∈ W 1,∞(❚) → j(u) ∈ ❘, of class C2 and satisfying juu(u0)(v, v) ≤ −αv2

H1(❚) + βv2 L2(❚),

for some α > 0, β ∈ ❘, because vL∞(0,ǫ) ≤ √ǫv ′L2(0,ǫ), v ∈ H1

0(0, ǫ),

is used

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: generality of approach

j(u) =

• ❚ G(θ, u, u′)dθ is a geometric shape functionnal.

The method works even for non-geometric shape functionnals. Namely, Theorem 0.1 is valid for

j : u ∈ W 1,∞(❚) → j(u) ∈ ❘, of class C2 and satisfying juu(u0)(v, v) ≤ −αv2

H1(❚) + βv2 L2(❚),

for some α > 0, β ∈ ❘, because vL∞(0,ǫ) ≤ √ǫv ′L2(0,ǫ), v ∈ H1

0(0, ǫ),

is used

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: generality of approach

j(u) =

• ❚ G(θ, u, u′)dθ is a geometric shape functionnal.

The method works even for non-geometric shape functionnals. Namely, Theorem 0.1 is valid for

j : u ∈ W 1,∞(❚) → j(u) ∈ ❘, of class C2 and satisfying juu(u0)(v, v) ≤ −αv2

H1(❚) + βv2 L2(❚),

for some α > 0, β ∈ ❘, because vL∞(0,ǫ) ≤ √ǫv ′L2(0,ǫ), v ∈ H1

0(0, ǫ),

is used

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: generality of approach

j(u) =

• ❚ G(θ, u, u′)dθ is a geometric shape functionnal.

The method works even for non-geometric shape functionnals. Namely, Theorem 0.1 is valid for

j : u ∈ W 1,∞(❚) → j(u) ∈ ❘, of class C2 and satisfying juu(u0)(v, v) ≤ −αv2

H1(❚) + βv2 L2(❚),

for some α > 0, β ∈ ❘, because vL∞(0,ǫ) ≤ √ǫv ′L2(0,ǫ), v ∈ H1

0(0, ǫ),

is used

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: generality of approach

j(u) =

• ❚ G(θ, u, u′)dθ is a geometric shape functionnal.

The method works even for non-geometric shape functionnals. Namely, Theorem 0.1 is valid for

j : u ∈ W 1,∞(❚) → j(u) ∈ ❘, of class C2 and satisfying juu(u0)(v, v) ≤ −αv2

H1(❚) + βv2 L2(❚),

for some α > 0, β ∈ ❘, because vL∞(0,ǫ) ≤ √ǫv ′L2(0,ǫ), v ∈ H1

0(0, ǫ),

is used

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: generality of approach

Example Consider L2(❚) ∋ p → ψ := Ψ(p) ∈ H1(❚) given by −ψ′′ + ψ = 1 + 1 2p2 on ❚ and j(u) = −1

2

• ❚(ψ′)2 + ψ2 for p = u′. Then

ψ ≥ 1, and juu(u0)(v, v) = −

• ❚ Ψ′

ppψ′ + Ψppψ −

• ❚(Ψ′

p)2 + Ψ2 p

≤ −

• ❚ Ψ′

ppψ′ + Ψppψ

=

• ❚ ψ|v ′|2

≤ −v2

H1 + v2 L2.

So, if u0 is a minimizer of j(u) then Ωu0 is a polygon inside A(a, b).

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: generality of approach

Example Consider L2(❚) ∋ p → ψ := Ψ(p) ∈ H1(❚) given by −ψ′′ + ψ = 1 + 1 2p2 on ❚ and j(u) = −1

2

• ❚(ψ′)2 + ψ2 for p = u′. Then

ψ ≥ 1, and juu(u0)(v, v) = −

• ❚ Ψ′

ppψ′ + Ψppψ −

• ❚(Ψ′

p)2 + Ψ2 p

≤ −

• ❚ Ψ′

ppψ′ + Ψppψ

=

• ❚ ψ|v ′|2

≤ −v2

H1 + v2 L2.

So, if u0 is a minimizer of j(u) then Ωu0 is a polygon inside A(a, b).

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: generality of approach

Example Consider L2(❚) ∋ p → ψ := Ψ(p) ∈ H1(❚) given by −ψ′′ + ψ = 1 + 1 2p2 on ❚ and j(u) = −1

2

• ❚(ψ′)2 + ψ2 for p = u′. Then

ψ ≥ 1, and juu(u0)(v, v) = −

• ❚ Ψ′

ppψ′ + Ψppψ −

• ❚(Ψ′

p)2 + Ψ2 p

≤ −

• ❚ Ψ′

ppψ′ + Ψppψ

=

• ❚ ψ|v ′|2

≤ −v2

H1 + v2 L2.

So, if u0 is a minimizer of j(u) then Ωu0 is a polygon inside A(a, b).

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: generality of approach

Example Consider L2(❚) ∋ p → ψ := Ψ(p) ∈ H1(❚) given by −ψ′′ + ψ = 1 + 1 2p2 on ❚ and j(u) = −1

2

• ❚(ψ′)2 + ψ2 for p = u′. Then

ψ ≥ 1, and juu(u0)(v, v) = −

• ❚ Ψ′

ppψ′ + Ψppψ −

• ❚(Ψ′

p)2 + Ψ2 p

≤ −

• ❚ Ψ′

ppψ′ + Ψppψ

=

• ❚ ψ|v ′|2

≤ −v2

H1 + v2 L2.

So, if u0 is a minimizer of j(u) then Ωu0 is a polygon inside A(a, b).

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: generality of approach

Example Consider L2(❚) ∋ p → ψ := Ψ(p) ∈ H1(❚) given by −ψ′′ + ψ = 1 + 1 2p2 on ❚ and j(u) = −1

2

• ❚(ψ′)2 + ψ2 for p = u′. Then

ψ ≥ 1, and juu(u0)(v, v) = −

• ❚ Ψ′

ppψ′ + Ψppψ −

• ❚(Ψ′

p)2 + Ψ2 p

≤ −

• ❚ Ψ′

ppψ′ + Ψppψ

=

• ❚ ψ|v ′|2

≤ −v2

H1 + v2 L2.

So, if u0 is a minimizer of j(u) then Ωu0 is a polygon inside A(a, b).

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Remarks: generality of approach

Example Consider L2(❚) ∋ p → ψ := Ψ(p) ∈ H1(❚) given by −ψ′′ + ψ = 1 + 1 2p2 on ❚ and j(u) = −1

2

• ❚(ψ′)2 + ψ2 for p = u′. Then

ψ ≥ 1, and juu(u0)(v, v) = −

• ❚ Ψ′

ppψ′ + Ψppψ −

• ❚(Ψ′

p)2 + Ψ2 p

≤ −

• ❚ Ψ′

ppψ′ + Ψppψ

=

• ❚ ψ|v ′|2

≤ −v2

H1 + v2 L2.

So, if u0 is a minimizer of j(u) then Ωu0 is a polygon inside A(a, b).

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit

Thank you!

• J. Lamboley, A. Novruzi

Polygons as solutions to shape optimization problems with convexit