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

Motivation T. Lachand-Robert and M.A. Peletier Newton’s problem: Find U 0 solution of: � dx min { E ( U ) , U ∈ F ad } , E ( U ) = 1 + |∇ U | 2 Ω F ad = { 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 ) := h 1 ( u ) − p 2 h 2 ( u ) , u ∈ F ad } , with F ad = { u regular enough , u ′′ + u ≥ 0 , 0 < a ≤ u ≤ b } Here u characterizes N := { U = M } The minimizer u 0 : supp ( u ′′ 0 + u 0 ) is finite The set N 0 := { U 0 = 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 U 0 solution of: � dx min { E ( U ) , U ∈ F ad } , E ( U ) = 1 + |∇ U | 2 Ω F ad = { 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 ) := h 1 ( u ) − p 2 h 2 ( u ) , u ∈ F ad } , with F ad = { u regular enough , u ′′ + u ≥ 0 , 0 < a ≤ u ≤ b } Here u characterizes N := { U = M } The minimizer u 0 : supp ( u ′′ 0 + u 0 ) is finite The set N 0 := { U 0 = M } is a regular polygon centered in Ω J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Motivation Figure: Minimizer U 0 as a function of M J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Motivation Figure: Minimizer U 0 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: u = 1 G ( θ, u , p ) = h ( p / u ), and r F ad = { 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: u = 1 G ( θ, u , p ) = h ( p / u ), and r F ad = { 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: u = 1 G ( θ, u , p ) = h ( p / u ), and r F ad = { 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: u = 1 G ( θ, u , p ) = h ( p / u ), and r F ad = { 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: u = 1 G ( θ, u , p ) = h ( p / u ), and r F ad = { 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 ∈ S ad } , where S ad is a set of 2-dimensional admissible shapes , J : S ad → ❘ is a shape functional Parametrization of Ω: W 1 , ∞ ( ❚ ) := W 1 , ∞ loc ( ❘ ) ∩ { 2 π -periodic } � � � 1 Ω u := ( r , θ ) ∈ [0 , 2 π ] × 0 , , u ( θ ) 0 < u ∈ W 1 , ∞ ( ❚ ) � . u ′′ + u The curvature: κ (Ω 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 ′ � � d θ ≥ 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 ∈ S ad } , where S ad is a set of 2-dimensional admissible shapes , J : S ad → ❘ is a shape functional Parametrization of Ω: W 1 , ∞ ( ❚ ) := W 1 , ∞ loc ( ❘ ) ∩ { 2 π -periodic } � � � 1 Ω u := ( r , θ ) ∈ [0 , 2 π ] × 0 , , u ( θ ) 0 < u ∈ W 1 , ∞ ( ❚ ) � . u ′′ + u The curvature: κ (Ω 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 ′ � � d θ ≥ 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 ∈ S ad } , where S ad is a set of 2-dimensional admissible shapes , J : S ad → ❘ is a shape functional Parametrization of Ω: W 1 , ∞ ( ❚ ) := W 1 , ∞ loc ( ❘ ) ∩ { 2 π -periodic } � � � 1 Ω u := ( r , θ ) ∈ [0 , 2 π ] × 0 , , u ( θ ) 0 < u ∈ W 1 , ∞ ( ❚ ) � . u ′′ + u The curvature: κ (Ω 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 ′ � � d θ ≥ 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 ∈ S ad } , where S ad is a set of 2-dimensional admissible shapes , J : S ad → ❘ is a shape functional Parametrization of Ω: W 1 , ∞ ( ❚ ) := W 1 , ∞ loc ( ❘ ) ∩ { 2 π -periodic } � � � 1 Ω u := ( r , θ ) ∈ [0 , 2 π ] × 0 , , u ( θ ) 0 < u ∈ W 1 , ∞ ( ❚ ) � . u ′′ + u The curvature: κ (Ω 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 ′ � � d θ ≥ 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 ∈ S ad } , where S ad is a set of 2-dimensional admissible shapes , J : S ad → ❘ is a shape functional Parametrization of Ω: W 1 , ∞ ( ❚ ) := W 1 , ∞ loc ( ❘ ) ∩ { 2 π -periodic } � � � 1 Ω u := ( r , θ ) ∈ [0 , 2 π ] × 0 , , u ( θ ) 0 < u ∈ W 1 , ∞ ( ❚ ) � . u ′′ + u The curvature: κ (Ω 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 ′ � � d θ ≥ 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 ∈ S ad } , where S ad is a set of 2-dimensional admissible shapes , J : S ad → ❘ is a shape functional Parametrization of Ω: W 1 , ∞ ( ❚ ) := W 1 , ∞ loc ( ❘ ) ∩ { 2 π -periodic } � � � 1 Ω u := ( r , θ ) ∈ [0 , 2 π ] × 0 , , u ( θ ) 0 < u ∈ W 1 , ∞ ( ❚ ) � . u ′′ + u The curvature: κ (Ω 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 ′ � � d θ ≥ 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 ∈ S ad } , where S ad is a set of 2-dimensional admissible shapes , J : S ad → ❘ is a shape functional Parametrization of Ω: W 1 , ∞ ( ❚ ) := W 1 , ∞ loc ( ❘ ) ∩ { 2 π -periodic } � � � 1 Ω u := ( r , θ ) ∈ [0 , 2 π ] × 0 , , u ( θ ) 0 < u ∈ W 1 , ∞ ( ❚ ) � . u ′′ + u The curvature: κ (Ω 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 ′ � � d θ ≥ 0 . ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

The problem: reformulation For u ∈ W 1 , ∞ ( ❚ ): ⇒ u ′′ + u ≥ 0 in M ( ❚ ) Ω u is convex ⇐ { u ′′ + u = 0 } straight lines on ∂ Ω u : u ′′ + u = � α n δ n corners on the boundary ∂ Ω u : We consider: find u 0 ∈ F ad : � ❚ G ( θ, u , u ′ ) d θ, u ∈ W 1 , ∞ ( ❚ ) , j ( u 0 ) = min { j ( u ) := u ′′ + u ≥ 0 , u ∈ F ad } , where F ad is a set of convenient admissible functions. Choices of F ad : � F ad := u : ∂ Ω u ⊂ A ( a , b ) := i) 1 b ≤ r ≤ 1 � � � ( r , θ ) : a 1 d θ � � � ii) F ad := { u : u ∈ C ( m 0 ) } , C ( m 0 ) := u : u 2 = m 0 2 ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

The problem: reformulation For u ∈ W 1 , ∞ ( ❚ ): ⇒ u ′′ + u ≥ 0 in M ( ❚ ) Ω u is convex ⇐ { u ′′ + u = 0 } straight lines on ∂ Ω u : u ′′ + u = � α n δ n corners on the boundary ∂ Ω u : We consider: find u 0 ∈ F ad : � ❚ G ( θ, u , u ′ ) d θ, u ∈ W 1 , ∞ ( ❚ ) , j ( u 0 ) = min { j ( u ) := u ′′ + u ≥ 0 , u ∈ F ad } , where F ad is a set of convenient admissible functions. Choices of F ad : � F ad := u : ∂ Ω u ⊂ A ( a , b ) := i) 1 b ≤ r ≤ 1 � � � ( r , θ ) : a 1 d θ � � � ii) F ad := { u : u ∈ C ( m 0 ) } , C ( m 0 ) := u : u 2 = m 0 2 ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

The problem: reformulation For u ∈ W 1 , ∞ ( ❚ ): ⇒ u ′′ + u ≥ 0 in M ( ❚ ) Ω u is convex ⇐ { u ′′ + u = 0 } straight lines on ∂ Ω u : u ′′ + u = � α n δ n corners on the boundary ∂ Ω u : We consider: find u 0 ∈ F ad : � ❚ G ( θ, u , u ′ ) d θ, u ∈ W 1 , ∞ ( ❚ ) , j ( u 0 ) = min { j ( u ) := u ′′ + u ≥ 0 , u ∈ F ad } , where F ad is a set of convenient admissible functions. Choices of F ad : � F ad := u : ∂ Ω u ⊂ A ( a , b ) := i) 1 b ≤ r ≤ 1 � � � ( r , θ ) : a 1 d θ � � � ii) F ad := { u : u ∈ C ( m 0 ) } , C ( m 0 ) := u : u 2 = m 0 2 ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

The problem: reformulation For u ∈ W 1 , ∞ ( ❚ ): ⇒ u ′′ + u ≥ 0 in M ( ❚ ) Ω u is convex ⇐ { u ′′ + u = 0 } straight lines on ∂ Ω u : u ′′ + u = � α n δ n corners on the boundary ∂ Ω u : We consider: find u 0 ∈ F ad : � ❚ G ( θ, u , u ′ ) d θ, u ∈ W 1 , ∞ ( ❚ ) , j ( u 0 ) = min { j ( u ) := u ′′ + u ≥ 0 , u ∈ F ad } , where F ad is a set of convenient admissible functions. Choices of F ad : � F ad := u : ∂ Ω u ⊂ A ( a , b ) := i) 1 b ≤ r ≤ 1 � � � ( r , θ ) : a 1 d θ � � � ii) F ad := { u : u ∈ C ( m 0 ) } , C ( m 0 ) := u : u 2 = m 0 2 ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

The problem: reformulation For u ∈ W 1 , ∞ ( ❚ ): ⇒ u ′′ + u ≥ 0 in M ( ❚ ) Ω u is convex ⇐ { u ′′ + u = 0 } straight lines on ∂ Ω u : u ′′ + u = � α n δ n corners on the boundary ∂ Ω u : We consider: find u 0 ∈ F ad : � ❚ G ( θ, u , u ′ ) d θ, u ∈ W 1 , ∞ ( ❚ ) , j ( u 0 ) = min { j ( u ) := u ′′ + u ≥ 0 , u ∈ F ad } , where F ad is a set of convenient admissible functions. Choices of F ad : � F ad := u : ∂ Ω u ⊂ A ( a , b ) := i) 1 b ≤ r ≤ 1 � � � ( r , θ ) : a 1 d θ � � � ii) F ad := { u : u ∈ C ( m 0 ) } , C ( m 0 ) := u : u 2 = m 0 2 ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

The problem: reformulation For u ∈ W 1 , ∞ ( ❚ ): ⇒ u ′′ + u ≥ 0 in M ( ❚ ) Ω u is convex ⇐ { u ′′ + u = 0 } straight lines on ∂ Ω u : u ′′ + u = � α n δ n corners on the boundary ∂ Ω u : We consider: find u 0 ∈ F ad : � ❚ G ( θ, u , u ′ ) d θ, u ∈ W 1 , ∞ ( ❚ ) , j ( u 0 ) = min { j ( u ) := u ′′ + u ≥ 0 , u ∈ F ad } , where F ad is a set of convenient admissible functions. Choices of F ad : � F ad := u : ∂ Ω u ⊂ A ( a , b ) := i) 1 b ≤ r ≤ 1 � � � ( r , θ ) : a 1 d θ � � � ii) F ad := { u : u ∈ C ( m 0 ) } , C ( m 0 ) := u : u 2 = m 0 2 ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

The problem: reformulation For u ∈ W 1 , ∞ ( ❚ ): ⇒ u ′′ + u ≥ 0 in M ( ❚ ) Ω u is convex ⇐ { u ′′ + u = 0 } straight lines on ∂ Ω u : u ′′ + u = � α n δ n corners on the boundary ∂ Ω u : We consider: find u 0 ∈ F ad : � ❚ G ( θ, u , u ′ ) d θ, u ∈ W 1 , ∞ ( ❚ ) , j ( u 0 ) = min { j ( u ) := u ′′ + u ≥ 0 , u ∈ F ad } , where F ad is a set of convenient admissible functions. Choices of F ad : � F ad := u : ∂ Ω u ⊂ A ( a , b ) := i) 1 b ≤ r ≤ 1 � � � ( r , θ ) : a 1 d θ � � � ii) F ad := { u : u ∈ C ( m 0 ) } , C ( m 0 ) := u : u 2 = m 0 2 ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

The problem: reformulation For u ∈ W 1 , ∞ ( ❚ ): ⇒ u ′′ + u ≥ 0 in M ( ❚ ) Ω u is convex ⇐ { u ′′ + u = 0 } straight lines on ∂ Ω u : u ′′ + u = � α n δ n corners on the boundary ∂ Ω u : We consider: find u 0 ∈ F ad : � ❚ G ( θ, u , u ′ ) d θ, u ∈ W 1 , ∞ ( ❚ ) , j ( u 0 ) = min { j ( u ) := u ′′ + u ≥ 0 , u ∈ F ad } , where F ad is a set of convenient admissible functions. Choices of F ad : � F ad := u : ∂ Ω u ⊂ A ( a , b ) := i) 1 b ≤ r ≤ 1 � � � ( r , θ ) : a 1 d θ � � � ii) F ad := { u : u ∈ C ( m 0 ) } , C ( m 0 ) := u : u 2 = m 0 2 ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

The problem: reformulation For u ∈ W 1 , ∞ ( ❚ ): ⇒ u ′′ + u ≥ 0 in M ( ❚ ) Ω u is convex ⇐ { u ′′ + u = 0 } straight lines on ∂ Ω u : u ′′ + u = � α n δ n corners on the boundary ∂ Ω u : We consider: find u 0 ∈ F ad : � ❚ G ( θ, u , u ′ ) d θ, u ∈ W 1 , ∞ ( ❚ ) , j ( u 0 ) = min { j ( u ) := u ′′ + u ≥ 0 , u ∈ F ad } , where F ad is a set of convenient admissible functions. Choices of F ad : � F ad := u : ∂ Ω u ⊂ A ( a , b ) := i) 1 b ≤ r ≤ 1 � � � ( r , θ ) : a 1 d θ � � � ii) F ad := { u : u ∈ C ( m 0 ) } , C ( m 0 ) := u : u 2 = m 0 2 ❚ 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 H 1 ( ❚ ), because { u ∈ W 1 , ∞ ( ❚ ) , u ′′ + u ≥ 0 , a ≤ u ≤ b } is strongly compact in H 1 ( ❚ ) 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 H 1 ( ❚ ), because { u ∈ W 1 , ∞ ( ❚ ) , u ′′ + u ≥ 0 , a ≤ u ≤ b } is strongly compact in H 1 ( ❚ ) 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 H 1 ( ❚ ), because { u ∈ W 1 , ∞ ( ❚ ) , u ′′ + u ≥ 0 , a ≤ u ≤ b } is strongly compact in H 1 ( ❚ ) 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 H 1 ( ❚ ), because { u ∈ W 1 , ∞ ( ❚ ) , u ′′ + u ≥ 0 , a ≤ u ≤ b } is strongly compact in H 1 ( ❚ ) 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 H 1 ( ❚ ), because { u ∈ W 1 , ∞ ( ❚ ) , u ′′ + u ≥ 0 , a ≤ u ≤ b } is strongly compact in H 1 ( ❚ ) 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 H 1 ( ❚ ), because { u ∈ W 1 , ∞ ( ❚ ) , u ′′ + u ≥ 0 , a ≤ u ≤ b } is strongly compact in H 1 ( ❚ ) 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 H 1 ( ❚ ), because { u ∈ W 1 , ∞ ( ❚ ) , u ′′ + u ≥ 0 , a ≤ u ≤ b } is strongly compact in H 1 ( ❚ ) 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 H 1 ( ❚ ), because { u ∈ W 1 , ∞ ( ❚ ) , u ′′ + u ≥ 0 , a ≤ u ≤ b } is strongly compact in H 1 ( ❚ ) 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 H 1 ( ❚ ), because { u ∈ W 1 , ∞ ( ❚ ) , u ′′ + u ≥ 0 , a ≤ u ≤ b } is strongly compact in H 1 ( ❚ ) 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 ) ∈ C 2 ( ❚ × ❘ × ❘ ) . Set j ( u ) = ❚ G ( θ, u , u ′ ) . � 2 Let u 0 be a solution of (1) or (2) and assume that G satisfies: G pp ( θ, u 0 , u ′ 0 ) < 0 , ∀ θ ∈ ❚ . (3) 3 If u 0 is a solution of (1) , then: S u 0 ∩ I is finite for any I = ( γ 1 , γ 2 ) ⊂ { a < u 0 ( θ ) < b } , and in particular, Ω u 0 is locally polygonal inside the annulus A ( a , b ) . 4 If u 0 > 0 is a solution of (2) , then S u 0 ∩ ❚ is finite, and so Ω u 0 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 ) ∈ C 2 ( ❚ × ❘ × ❘ ) . Set j ( u ) = ❚ G ( θ, u , u ′ ) . � 2 Let u 0 be a solution of (1) or (2) and assume that G satisfies: G pp ( θ, u 0 , u ′ 0 ) < 0 , ∀ θ ∈ ❚ . (3) 3 If u 0 is a solution of (1) , then: S u 0 ∩ I is finite for any I = ( γ 1 , γ 2 ) ⊂ { a < u 0 ( θ ) < b } , and in particular, Ω u 0 is locally polygonal inside the annulus A ( a , b ) . 4 If u 0 > 0 is a solution of (2) , then S u 0 ∩ ❚ is finite, and so Ω u 0 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 ) ∈ C 2 ( ❚ × ❘ × ❘ ) . Set j ( u ) = ❚ G ( θ, u , u ′ ) . � 2 Let u 0 be a solution of (1) or (2) and assume that G satisfies: G pp ( θ, u 0 , u ′ 0 ) < 0 , ∀ θ ∈ ❚ . (3) 3 If u 0 is a solution of (1) , then: S u 0 ∩ I is finite for any I = ( γ 1 , γ 2 ) ⊂ { a < u 0 ( θ ) < b } , and in particular, Ω u 0 is locally polygonal inside the annulus A ( a , b ) . 4 If u 0 > 0 is a solution of (2) , then S u 0 ∩ ❚ is finite, and so Ω u 0 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 ) ∈ C 2 ( ❚ × ❘ × ❘ ) . Set j ( u ) = ❚ G ( θ, u , u ′ ) . � 2 Let u 0 be a solution of (1) or (2) and assume that G satisfies: G pp ( θ, u 0 , u ′ 0 ) < 0 , ∀ θ ∈ ❚ . (3) 3 If u 0 is a solution of (1) , then: S u 0 ∩ I is finite for any I = ( γ 1 , γ 2 ) ⊂ { a < u 0 ( θ ) < b } , and in particular, Ω u 0 is locally polygonal inside the annulus A ( a , b ) . 4 If u 0 > 0 is a solution of (2) , then S u 0 ∩ ❚ is finite, and so Ω u 0 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 ) ∈ C 2 ( ❚ × ❘ × ❘ ) . Set j ( u ) = ❚ G ( θ, u , u ′ ) . � 2 Let u 0 be a solution of (1) or (2) and assume that G satisfies: G pp ( θ, u 0 , u ′ 0 ) < 0 , ∀ θ ∈ ❚ . (3) 3 If u 0 is a solution of (1) , then: S u 0 ∩ I is finite for any I = ( γ 1 , γ 2 ) ⊂ { a < u 0 ( θ ) < b } , and in particular, Ω u 0 is locally polygonal inside the annulus A ( a , b ) . 4 If u 0 > 0 is a solution of (2) , then S u 0 ∩ ❚ is finite, and so Ω u 0 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 ) ∈ C 2 ( ❚ × ❘ × ❘ ) . Set j ( u ) = ❚ G ( θ, u , u ′ ) . � 2 Let u 0 be a solution of (1) or (2) and assume that G satisfies: G pp ( θ, u 0 , u ′ 0 ) < 0 , ∀ θ ∈ ❚ . (3) 3 If u 0 is a solution of (1) , then: S u 0 ∩ I is finite for any I = ( γ 1 , γ 2 ) ⊂ { a < u 0 ( θ ) < b } , and in particular, Ω u 0 is locally polygonal inside the annulus A ( a , b ) . 4 If u 0 > 0 is a solution of (2) , then S u 0 ∩ ❚ is finite, and so Ω u 0 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 ) ∈ C 2 ( ❚ × ❘ × ❘ ) . Set j ( u ) = ❚ G ( θ, u , u ′ ) . � 2 Let u 0 be a solution of (1) or (2) and assume that G satisfies: G pp ( θ, u 0 , u ′ 0 ) < 0 , ∀ θ ∈ ❚ . (3) 3 If u 0 is a solution of (1) , then: S u 0 ∩ I is finite for any I = ( γ 1 , γ 2 ) ⊂ { a < u 0 ( θ ) < b } , and in particular, Ω u 0 is locally polygonal inside the annulus A ( a , b ) . 4 If u 0 > 0 is a solution of (2) , then S u 0 ∩ ❚ is finite, and so Ω u 0 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 ′ ) , u 0 be a solution of (1). Assume that � 2 (i) G = G ( u , p ) is a C 2 function and G pp < 0 on { ( u 0 ( θ ) , u ′ 0 ( θ )) , θ ∈ ❚ } , 3 (ii) The function p �→ G ( a , p ) is even and one of the followings holds (ii.1): G u ( a , 0) < 0 , or (ii.2): G u ( a , 0) = 0 and G u ( u 0 , u ′ 0 ) u 0 + G p ( u 0 , u ′ 0 ) u ′ 0 ≤ 0 , 4 (iii) The function p �→ G ( · , p ) is even and G u ≥ 0 near ( b , 0) . 5 Then S u 0 is finite, i.e. Ω u 0 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 ′ ) , u 0 be a solution of (1). Assume that � 2 (i) G = G ( u , p ) is a C 2 function and G pp < 0 on { ( u 0 ( θ ) , u ′ 0 ( θ )) , θ ∈ ❚ } , 3 (ii) The function p �→ G ( a , p ) is even and one of the followings holds (ii.1): G u ( a , 0) < 0 , or (ii.2): G u ( a , 0) = 0 and G u ( u 0 , u ′ 0 ) u 0 + G p ( u 0 , u ′ 0 ) u ′ 0 ≤ 0 , 4 (iii) The function p �→ G ( · , p ) is even and G u ≥ 0 near ( b , 0) . 5 Then S u 0 is finite, i.e. Ω u 0 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 ′ ) , u 0 be a solution of (1). Assume that � 2 (i) G = G ( u , p ) is a C 2 function and G pp < 0 on { ( u 0 ( θ ) , u ′ 0 ( θ )) , θ ∈ ❚ } , 3 (ii) The function p �→ G ( a , p ) is even and one of the followings holds (ii.1): G u ( a , 0) < 0 , or (ii.2): G u ( a , 0) = 0 and G u ( u 0 , u ′ 0 ) u 0 + G p ( u 0 , u ′ 0 ) u ′ 0 ≤ 0 , 4 (iii) The function p �→ G ( · , p ) is even and G u ≥ 0 near ( b , 0) . 5 Then S u 0 is finite, i.e. Ω u 0 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 ′ ) , u 0 be a solution of (1). Assume that � 2 (i) G = G ( u , p ) is a C 2 function and G pp < 0 on { ( u 0 ( θ ) , u ′ 0 ( θ )) , θ ∈ ❚ } , 3 (ii) The function p �→ G ( a , p ) is even and one of the followings holds (ii.1): G u ( a , 0) < 0 , or (ii.2): G u ( a , 0) = 0 and G u ( u 0 , u ′ 0 ) u 0 + G p ( u 0 , u ′ 0 ) u ′ 0 ≤ 0 , 4 (iii) The function p �→ G ( · , p ) is even and G u ≥ 0 near ( b , 0) . 5 Then S u 0 is finite, i.e. Ω u 0 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 ′ ) , u 0 be a solution of (1). Assume that � 2 (i) G = G ( u , p ) is a C 2 function and G pp < 0 on { ( u 0 ( θ ) , u ′ 0 ( θ )) , θ ∈ ❚ } , 3 (ii) The function p �→ G ( a , p ) is even and one of the followings holds (ii.1): G u ( a , 0) < 0 , or (ii.2): G u ( a , 0) = 0 and G u ( u 0 , u ′ 0 ) u 0 + G p ( u 0 , u ′ 0 ) u ′ 0 ≤ 0 , 4 (iii) The function p �→ G ( · , p ) is even and G u ≥ 0 near ( b , 0) . 5 Then S u 0 is finite, i.e. Ω u 0 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 ′ ) , u 0 be a solution of (1). Assume that � 2 (i) G = G ( u , p ) is a C 2 function and G pp < 0 on { ( u 0 ( θ ) , u ′ 0 ( θ )) , θ ∈ ❚ } , 3 (ii) The function p �→ G ( a , p ) is even and one of the followings holds (ii.1): G u ( a , 0) < 0 , or (ii.2): G u ( a , 0) = 0 and G u ( u 0 , u ′ 0 ) u 0 + G p ( u 0 , u ′ 0 ) u ′ 0 ≤ 0 , 4 (iii) The function p �→ G ( · , p ) is even and G u ≥ 0 near ( b , 0) . 5 Then S u 0 is finite, i.e. Ω u 0 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 ′ ) , u 0 be a solution of (1). Assume that � 2 (i) G = G ( u , p ) is a C 2 function and G pp < 0 on { ( u 0 ( θ ) , u ′ 0 ( θ )) , θ ∈ ❚ } , 3 (ii) The function p �→ G ( a , p ) is even and one of the followings holds (ii.1): G u ( a , 0) < 0 , or (ii.2): G u ( a , 0) = 0 and G u ( u 0 , u ′ 0 ) u 0 + G p ( u 0 , u ′ 0 ) u ′ 0 ≤ 0 , 4 (iii) The function p �→ G ( · , p ) is even and G u ≥ 0 near ( b , 0) . 5 Then S u 0 is finite, i.e. Ω u 0 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 ′ ) , u 0 be a solution of (1). Assume that � 2 (i) G = G ( u , p ) is a C 2 function and G pp < 0 on { ( u 0 ( θ ) , u ′ 0 ( θ )) , θ ∈ ❚ } , 3 (ii) The function p �→ G ( a , p ) is even and one of the followings holds (ii.1): G u ( a , 0) < 0 , or (ii.2): G u ( a , 0) = 0 and G u ( u 0 , u ′ 0 ) u 0 + G p ( u 0 , u ′ 0 ) u ′ 0 ≤ 0 , 4 (iii) The function p �→ G ( · , p ) is even and G u ≥ 0 near ( b , 0) . 5 Then S u 0 is finite, i.e. Ω u 0 is a polygon. J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Main results: regularity (cont’d) Example Consider u 2 + | u ′ | 2 � J (Ω u ) = λ m (Ω u ) − P (Ω u ) = λ � 1 � u 2 − , u 2 2 ❚ ❚ with m (Ω u ) the area, P (Ω u ) 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 ) 1 G u ( u , 0) = u − λ G pp = − ( u 2 + p 2 ) 3 / 2 , u 3 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 u 2 + | u ′ | 2 � J (Ω u ) = λ m (Ω u ) − P (Ω u ) = λ � 1 � u 2 − , u 2 2 ❚ ❚ with m (Ω u ) the area, P (Ω u ) 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 ) 1 G u ( u , 0) = u − λ G pp = − ( u 2 + p 2 ) 3 / 2 , u 3 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 u 2 + | u ′ | 2 � J (Ω u ) = λ m (Ω u ) − P (Ω u ) = λ � 1 � u 2 − , u 2 2 ❚ ❚ with m (Ω u ) the area, P (Ω u ) 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 ) 1 G u ( u , 0) = u − λ G pp = − ( u 2 + p 2 ) 3 / 2 , u 3 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 u 2 + | u ′ | 2 � J (Ω u ) = λ m (Ω u ) − P (Ω u ) = λ � 1 � u 2 − , u 2 2 ❚ ❚ with m (Ω u ) the area, P (Ω u ) 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 ) 1 G u ( u , 0) = u − λ G pp = − ( u 2 + p 2 ) 3 / 2 , u 3 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 u 2 + | u ′ | 2 � J (Ω u ) = λ m (Ω u ) − P (Ω u ) = λ � 1 � u 2 − , u 2 2 ❚ ❚ with m (Ω u ) the area, P (Ω u ) 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 ) 1 G u ( u , 0) = u − λ G pp = − ( u 2 + p 2 ) 3 / 2 , u 3 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 u 2 + | u ′ | 2 � J (Ω u ) = λ m (Ω u ) − P (Ω u ) = λ � 1 � u 2 − , u 2 2 ❚ ❚ with m (Ω u ) the area, P (Ω u ) 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 ) 1 G u ( u , 0) = u − λ G pp = − ( u 2 + p 2 ) 3 / 2 , u 3 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 u 2 + | u ′ | 2 � J (Ω u ) = λ m (Ω u ) − P (Ω u ) = λ � 1 � u 2 − , u 2 2 ❚ ❚ with m (Ω u ) the area, P (Ω u ) 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 ) 1 G u ( u , 0) = u − λ G pp = − ( u 2 + p 2 ) 3 / 2 , u 3 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 u 2 + | u ′ | 2 � J (Ω u ) = λ m (Ω u ) − P (Ω u ) = λ � 1 � u 2 − , u 2 2 ❚ ❚ with m (Ω u ) the area, P (Ω u ) 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 ) 1 G u ( u , 0) = u − λ G pp = − ( u 2 + p 2 ) 3 / 2 , u 3 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 u 0 is solves (1) and j ∈ C 1 ( H 1 ( ❚ ); ❘ ), then there exist 0 ≤ ζ 0 ∈ H 1 ( ❚ ) , µ a , µ b ∈ M + ( ❚ ) such that { ζ 0 = 0 }⊂ S u 0 , Supp ( µ a ) ⊂{ u 0 = a } , Supp ( µ b ) ⊂{ u 0 = b } . 2 v ∈ H 1 ( ❚ ) : � � j ′ ( u 0 ) v = � ζ 0 + ζ ′′ 0 , v � U ′ × U + vd µ a − vd µ b . ❚ ❚ 3 Moreover, ∀ v ∈ H 1 ( ❚ ) such that ∃ λ ∈ ❘ with v ′′ + v λ ( u ′′  ≥ 0 + u 0 )   v ≥ λ ( u 0 − a ) ,  j ′′ ( u 0 )( v , v ) ≥ 0 . ⇒ v ≤ λ ( u 0 − b ) ,   � ζ 0 + ζ ′′ � 0 , v � U ′ × U + ❚ vd ( µ a − µ b ) = 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 u 0 is solves (1) and j ∈ C 1 ( H 1 ( ❚ ); ❘ ), then there exist 0 ≤ ζ 0 ∈ H 1 ( ❚ ) , µ a , µ b ∈ M + ( ❚ ) such that { ζ 0 = 0 }⊂ S u 0 , Supp ( µ a ) ⊂{ u 0 = a } , Supp ( µ b ) ⊂{ u 0 = b } . 2 v ∈ H 1 ( ❚ ) : � � j ′ ( u 0 ) v = � ζ 0 + ζ ′′ 0 , v � U ′ × U + vd µ a − vd µ b . ❚ ❚ 3 Moreover, ∀ v ∈ H 1 ( ❚ ) such that ∃ λ ∈ ❘ with v ′′ + v λ ( u ′′  ≥ 0 + u 0 )   v ≥ λ ( u 0 − a ) ,  j ′′ ( u 0 )( v , v ) ≥ 0 . ⇒ v ≤ λ ( u 0 − b ) ,   � ζ 0 + ζ ′′ � 0 , v � U ′ × U + ❚ vd ( µ a − µ b ) = 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 u 0 is solves (1) and j ∈ C 1 ( H 1 ( ❚ ); ❘ ), then there exist 0 ≤ ζ 0 ∈ H 1 ( ❚ ) , µ a , µ b ∈ M + ( ❚ ) such that { ζ 0 = 0 }⊂ S u 0 , Supp ( µ a ) ⊂{ u 0 = a } , Supp ( µ b ) ⊂{ u 0 = b } . 2 v ∈ H 1 ( ❚ ) : � � j ′ ( u 0 ) v = � ζ 0 + ζ ′′ 0 , v � U ′ × U + vd µ a − vd µ b . ❚ ❚ 3 Moreover, ∀ v ∈ H 1 ( ❚ ) such that ∃ λ ∈ ❘ with v ′′ + v λ ( u ′′  ≥ 0 + u 0 )   v ≥ λ ( u 0 − a ) ,  j ′′ ( u 0 )( v , v ) ≥ 0 . ⇒ v ≤ λ ( u 0 − b ) ,   � ζ 0 + ζ ′′ � 0 , v � U ′ × U + ❚ vd ( µ a − µ b ) = 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 u 0 is solves (1) and j ∈ C 1 ( H 1 ( ❚ ); ❘ ), then there exist 0 ≤ ζ 0 ∈ H 1 ( ❚ ) , µ a , µ b ∈ M + ( ❚ ) such that { ζ 0 = 0 }⊂ S u 0 , Supp ( µ a ) ⊂{ u 0 = a } , Supp ( µ b ) ⊂{ u 0 = b } . 2 v ∈ H 1 ( ❚ ) : � � j ′ ( u 0 ) v = � ζ 0 + ζ ′′ 0 , v � U ′ × U + vd µ a − vd µ b . ❚ ❚ 3 Moreover, ∀ v ∈ H 1 ( ❚ ) such that ∃ λ ∈ ❘ with v ′′ + v λ ( u ′′  ≥ 0 + u 0 )   v ≥ λ ( u 0 − a ) ,  j ′′ ( u 0 )( v , v ) ≥ 0 . ⇒ v ≤ λ ( u 0 − b ) ,   � ζ 0 + ζ ′′ � 0 , v � U ′ × U + ❚ vd ( µ a − µ b ) = 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 u 0 is solves (1) and j ∈ C 1 ( H 1 ( ❚ ); ❘ ), then there exist 0 ≤ ζ 0 ∈ H 1 ( ❚ ) , µ a , µ b ∈ M + ( ❚ ) such that { ζ 0 = 0 }⊂ S u 0 , Supp ( µ a ) ⊂{ u 0 = a } , Supp ( µ b ) ⊂{ u 0 = b } . 2 v ∈ H 1 ( ❚ ) : � � j ′ ( u 0 ) v = � ζ 0 + ζ ′′ 0 , v � U ′ × U + vd µ a − vd µ b . ❚ ❚ 3 Moreover, ∀ v ∈ H 1 ( ❚ ) such that ∃ λ ∈ ❘ with v ′′ + v λ ( u ′′  ≥ 0 + u 0 )   v ≥ λ ( u 0 − a ) ,  j ′′ ( u 0 )( v , v ) ≥ 0 . ⇒ v ≤ λ ( u 0 − b ) ,   � ζ 0 + ζ ′′ � 0 , v � U ′ × U + ❚ vd ( µ a − µ b ) = 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 u 0 is solves (1) and j ∈ C 1 ( H 1 ( ❚ ); ❘ ), then there exist 0 ≤ ζ 0 ∈ H 1 ( ❚ ) , µ a , µ b ∈ M + ( ❚ ) such that { ζ 0 = 0 }⊂ S u 0 , Supp ( µ a ) ⊂{ u 0 = a } , Supp ( µ b ) ⊂{ u 0 = b } . 2 v ∈ H 1 ( ❚ ) : � � j ′ ( u 0 ) v = � ζ 0 + ζ ′′ 0 , v � U ′ × U + vd µ a − vd µ b . ❚ ❚ 3 Moreover, ∀ v ∈ H 1 ( ❚ ) such that ∃ λ ∈ ❘ with v ′′ + v λ ( u ′′  ≥ 0 + u 0 )   v ≥ λ ( u 0 − a ) ,  j ′′ ( u 0 )( v , v ) ≥ 0 . ⇒ v ≤ λ ( u 0 − b ) ,   � ζ 0 + ζ ′′ � 0 , v � U ′ × U + ❚ vd ( µ a − µ b ) = 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 u 0 is solves (1) and j ∈ C 1 ( H 1 ( ❚ ); ❘ ), then there exist 0 ≤ ζ 0 ∈ H 1 ( ❚ ) , µ a , µ b ∈ M + ( ❚ ) such that { ζ 0 = 0 }⊂ S u 0 , Supp ( µ a ) ⊂{ u 0 = a } , Supp ( µ b ) ⊂{ u 0 = b } . 2 v ∈ H 1 ( ❚ ) : � � j ′ ( u 0 ) v = � ζ 0 + ζ ′′ 0 , v � U ′ × U + vd µ a − vd µ b . ❚ ❚ 3 Moreover, ∀ v ∈ H 1 ( ❚ ) such that ∃ λ ∈ ❘ with v ′′ + v λ ( u ′′  ≥ 0 + u 0 )   v ≥ λ ( u 0 − a ) ,  j ′′ ( u 0 )( v , v ) ≥ 0 . ⇒ v ≤ λ ( u 0 − b ) ,   � ζ 0 + ζ ′′ � 0 , v � U ′ × U + ❚ vd ( µ a − µ b ) = 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 u 0 ∈ U solve the min. problem. Assume f , g are C 2 at u 0 and g ′ ( u 0 )( U ) = Y . Then, (i) ∃ l ∈ Y ′ + : f ′ ( u 0 ) = l ◦ g ′ ( u 0 ) and l ( g ( u 0 )) = 0 , where Y ′ + = { l ∈ Y ′ ; ∀ k ∈ K , l ( k ) ≥ 0 } (ii) Set L ( u ) := f ( u ) − l ( g ( u )) . Then F ′′ ( u 0 )( v , v ) ≥ 0 , ∀ v ∈ T u 0 , � v ∈ U ; f ′ ( u 0 )( v ) = 0 , T u 0 = g ′ ( u 0 )( v ) ∈ K g ( u 0 ) = { K + λ g ( u 0 ); 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 u 0 ∈ U solve the min. problem. Assume f , g are C 2 at u 0 and g ′ ( u 0 )( U ) = Y . Then, (i) ∃ l ∈ Y ′ + : f ′ ( u 0 ) = l ◦ g ′ ( u 0 ) and l ( g ( u 0 )) = 0 , where Y ′ + = { l ∈ Y ′ ; ∀ k ∈ K , l ( k ) ≥ 0 } (ii) Set L ( u ) := f ( u ) − l ( g ( u )) . Then F ′′ ( u 0 )( v , v ) ≥ 0 , ∀ v ∈ T u 0 , � v ∈ U ; f ′ ( u 0 )( v ) = 0 , T u 0 = g ′ ( u 0 )( v ) ∈ K g ( u 0 ) = { K + λ g ( u 0 ); 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 u 0 ∈ U solve the min. problem. Assume f , g are C 2 at u 0 and g ′ ( u 0 )( U ) = Y . Then, (i) ∃ l ∈ Y ′ + : f ′ ( u 0 ) = l ◦ g ′ ( u 0 ) and l ( g ( u 0 )) = 0 , where Y ′ + = { l ∈ Y ′ ; ∀ k ∈ K , l ( k ) ≥ 0 } (ii) Set L ( u ) := f ( u ) − l ( g ( u )) . Then F ′′ ( u 0 )( v , v ) ≥ 0 , ∀ v ∈ T u 0 , � v ∈ U ; f ′ ( u 0 )( v ) = 0 , T u 0 = g ′ ( u 0 )( v ) ∈ K g ( u 0 ) = { K + λ g ( u 0 ); 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 u 0 ∈ U solve the min. problem. Assume f , g are C 2 at u 0 and g ′ ( u 0 )( U ) = Y . Then, (i) ∃ l ∈ Y ′ + : f ′ ( u 0 ) = l ◦ g ′ ( u 0 ) and l ( g ( u 0 )) = 0 , where Y ′ + = { l ∈ Y ′ ; ∀ k ∈ K , l ( k ) ≥ 0 } (ii) Set L ( u ) := f ( u ) − l ( g ( u )) . Then F ′′ ( u 0 )( v , v ) ≥ 0 , ∀ v ∈ T u 0 , � v ∈ U ; f ′ ( u 0 )( v ) = 0 , T u 0 = g ′ ( u 0 )( v ) ∈ K g ( u 0 ) = { K + λ g ( u 0 ); 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 u 0 ∈ U solve the min. problem. Assume f , g are C 2 at u 0 and g ′ ( u 0 )( U ) = Y . Then, (i) ∃ l ∈ Y ′ + : f ′ ( u 0 ) = l ◦ g ′ ( u 0 ) and l ( g ( u 0 )) = 0 , where Y ′ + = { l ∈ Y ′ ; ∀ k ∈ K , l ( k ) ≥ 0 } (ii) Set L ( u ) := f ( u ) − l ( g ( u )) . Then F ′′ ( u 0 )( v , v ) ≥ 0 , ∀ v ∈ T u 0 , � v ∈ U ; f ′ ( u 0 )( v ) = 0 , T u 0 = g ′ ( u 0 )( v ) ∈ K g ( u 0 ) = { K + λ g ( u 0 ); 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 u 0 ∈ U solve the min. problem. Assume f , g are C 2 at u 0 and g ′ ( u 0 )( U ) = Y . Then, (i) ∃ l ∈ Y ′ + : f ′ ( u 0 ) = l ◦ g ′ ( u 0 ) and l ( g ( u 0 )) = 0 , where Y ′ + = { l ∈ Y ′ ; ∀ k ∈ K , l ( k ) ≥ 0 } (ii) Set L ( u ) := f ( u ) − l ( g ( u )) . Then F ′′ ( u 0 )( v , v ) ≥ 0 , ∀ v ∈ T u 0 , � v ∈ U ; f ′ ( u 0 )( v ) = 0 , T u 0 = g ′ ( u 0 )( v ) ∈ K g ( u 0 ) = { K + λ g ( u 0 ); λ ∈ ❘ } � . J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks) U = H 1 ( ❚ ), Y = H − 1 ( ❚ ) × H 1 ( ❚ ) × H 1 ( ❚ ), K = Y + , g ( u ) = ( u ′′ + u , u − a , b − u ) Easy to find � � ζ 0 d ( v ′′ + v ) ≥ 0 , for all v ′′ + v ≥ 0 ζ 0 d ( u ′′ 0 + u 0 ) = 0 , ❚ ❚ We find that ζ 0 satisfies furthermore ζ 0 ( u ′′ ζ 0 ≥ 0 , 0 + u 0 ) = 0 The second order optimality condition is necessary � p � 2 . Example: Consider G ( u , p ) = − 1 2 u � v � ′ v � ′ u ′ � u ′ � � 0 0 0 = − = u 0 u 0 u 0 u 0 { u ′′ { u ′′ 0 + u 0 > 0 } 0 + u 0 > 0 } Ae B θ u 0 = J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks) U = H 1 ( ❚ ), Y = H − 1 ( ❚ ) × H 1 ( ❚ ) × H 1 ( ❚ ), K = Y + , g ( u ) = ( u ′′ + u , u − a , b − u ) Easy to find � � ζ 0 d ( v ′′ + v ) ≥ 0 , for all v ′′ + v ≥ 0 ζ 0 d ( u ′′ 0 + u 0 ) = 0 , ❚ ❚ We find that ζ 0 satisfies furthermore ζ 0 ( u ′′ ζ 0 ≥ 0 , 0 + u 0 ) = 0 The second order optimality condition is necessary � p � 2 . Example: Consider G ( u , p ) = − 1 2 u � v � ′ v � ′ u ′ � u ′ � � 0 0 0 = − = u 0 u 0 u 0 u 0 { u ′′ { u ′′ 0 + u 0 > 0 } 0 + u 0 > 0 } Ae B θ u 0 = J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks) U = H 1 ( ❚ ), Y = H − 1 ( ❚ ) × H 1 ( ❚ ) × H 1 ( ❚ ), K = Y + , g ( u ) = ( u ′′ + u , u − a , b − u ) Easy to find � � ζ 0 d ( v ′′ + v ) ≥ 0 , for all v ′′ + v ≥ 0 ζ 0 d ( u ′′ 0 + u 0 ) = 0 , ❚ ❚ We find that ζ 0 satisfies furthermore ζ 0 ( u ′′ ζ 0 ≥ 0 , 0 + u 0 ) = 0 The second order optimality condition is necessary � p � 2 . Example: Consider G ( u , p ) = − 1 2 u � v � ′ v � ′ u ′ � u ′ � � 0 0 0 = − = u 0 u 0 u 0 u 0 { u ′′ { u ′′ 0 + u 0 > 0 } 0 + u 0 > 0 } Ae B θ u 0 = J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks) U = H 1 ( ❚ ), Y = H − 1 ( ❚ ) × H 1 ( ❚ ) × H 1 ( ❚ ), K = Y + , g ( u ) = ( u ′′ + u , u − a , b − u ) Easy to find � � ζ 0 d ( v ′′ + v ) ≥ 0 , for all v ′′ + v ≥ 0 ζ 0 d ( u ′′ 0 + u 0 ) = 0 , ❚ ❚ We find that ζ 0 satisfies furthermore ζ 0 ( u ′′ ζ 0 ≥ 0 , 0 + u 0 ) = 0 The second order optimality condition is necessary � p � 2 . Example: Consider G ( u , p ) = − 1 2 u � v � ′ v � ′ u ′ � u ′ � � 0 0 0 = − = u 0 u 0 u 0 u 0 { u ′′ { u ′′ 0 + u 0 > 0 } 0 + u 0 > 0 } Ae B θ u 0 = J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks) U = H 1 ( ❚ ), Y = H − 1 ( ❚ ) × H 1 ( ❚ ) × H 1 ( ❚ ), K = Y + , g ( u ) = ( u ′′ + u , u − a , b − u ) Easy to find � � ζ 0 d ( v ′′ + v ) ≥ 0 , for all v ′′ + v ≥ 0 ζ 0 d ( u ′′ 0 + u 0 ) = 0 , ❚ ❚ We find that ζ 0 satisfies furthermore ζ 0 ( u ′′ ζ 0 ≥ 0 , 0 + u 0 ) = 0 The second order optimality condition is necessary � p � 2 . Example: Consider G ( u , p ) = − 1 2 u � v � ′ v � ′ u ′ � u ′ � � 0 0 0 = − = u 0 u 0 u 0 u 0 { u ′′ { u ′′ 0 + u 0 > 0 } 0 + u 0 > 0 } Ae B θ u 0 = J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks) U = H 1 ( ❚ ), Y = H − 1 ( ❚ ) × H 1 ( ❚ ) × H 1 ( ❚ ), K = Y + , g ( u ) = ( u ′′ + u , u − a , b − u ) Easy to find � � ζ 0 d ( v ′′ + v ) ≥ 0 , for all v ′′ + v ≥ 0 ζ 0 d ( u ′′ 0 + u 0 ) = 0 , ❚ ❚ We find that ζ 0 satisfies furthermore ζ 0 ( u ′′ ζ 0 ≥ 0 , 0 + u 0 ) = 0 The second order optimality condition is necessary � p � 2 . Example: Consider G ( u , p ) = − 1 2 u � v � ′ v � ′ u ′ � u ′ � � 0 0 0 = − = u 0 u 0 u 0 u 0 { u ′′ { u ′′ 0 + u 0 > 0 } 0 + u 0 > 0 } Ae B θ u 0 = J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Proofs: optimality conditions (remarks) U = H 1 ( ❚ ), Y = H − 1 ( ❚ ) × H 1 ( ❚ ) × H 1 ( ❚ ), K = Y + , g ( u ) = ( u ′′ + u , u − a , b − u ) Easy to find � � ζ 0 d ( v ′′ + v ) ≥ 0 , for all v ′′ + v ≥ 0 ζ 0 d ( u ′′ 0 + u 0 ) = 0 , ❚ ❚ We find that ζ 0 satisfies furthermore ζ 0 ( u ′′ ζ 0 ≥ 0 , 0 + u 0 ) = 0 The second order optimality condition is necessary � p � 2 . Example: Consider G ( u , p ) = − 1 2 u � v � ′ v � ′ u ′ � u ′ � � 0 0 0 = − = u 0 u 0 u 0 u 0 { u ′′ { u ′′ 0 + u 0 > 0 } 0 + u 0 > 0 } Ae B θ u 0 = J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A ( a , b )) 1 a By contradiction ǫ n Case a < u 0 ( θ 0 = 0) < b (Fig. 1) ǫ i n θ 0 ∃ ǫ n → 0, S u 0 ∩ (0 , ǫ n ) � = ∅ 1 b Ω u 0 A ( a , b ) To find v n : ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) We consider v n , i ∈ H 1 ( ❚ ) solving Fig. 1 v ′′ n , i + v n , i = χ ( ǫ i ) ( u ′′ 0 + u 0 ) , (0 , ǫ n ) , n ,ǫ i +1 n (0 , ǫ n ) c . v n , i = 0 , Look for v n = � i =1 , 3 λ n , i v n , i such that n (0 + ) = v ′ v ′ n ( ǫ − n ) = 0, which implies ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) � � � ❚ v n ( ζ 0 + ζ ′′ 0 ) = ❚ v n d µ a = ❚ v n d µ b = 0, and � 2 G uu v 2 j ′′ ( u 0 )( v n , v n ) = n + 2 G up v n v ′ n + G pp v ′ 0 ≤ n ❚ ( � v n � L 2 ( ❚ ) ≤ √ ǫ n � v ′ � n | 2 < 0 ≤ ( o (1) − K pp ) | v ′ (!) n � L 2 ( ❚ ) ) ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A ( a , b )) 1 a By contradiction ǫ n Case a < u 0 ( θ 0 = 0) < b (Fig. 1) ǫ i n θ 0 ∃ ǫ n → 0, S u 0 ∩ (0 , ǫ n ) � = ∅ 1 b Ω u 0 A ( a , b ) To find v n : ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) We consider v n , i ∈ H 1 ( ❚ ) solving Fig. 1 v ′′ n , i + v n , i = χ ( ǫ i ) ( u ′′ 0 + u 0 ) , (0 , ǫ n ) , n ,ǫ i +1 n (0 , ǫ n ) c . v n , i = 0 , Look for v n = � i =1 , 3 λ n , i v n , i such that n (0 + ) = v ′ v ′ n ( ǫ − n ) = 0, which implies ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) � � � ❚ v n ( ζ 0 + ζ ′′ 0 ) = ❚ v n d µ a = ❚ v n d µ b = 0, and � 2 G uu v 2 j ′′ ( u 0 )( v n , v n ) = n + 2 G up v n v ′ n + G pp v ′ 0 ≤ n ❚ ( � v n � L 2 ( ❚ ) ≤ √ ǫ n � v ′ � n | 2 < 0 ≤ ( o (1) − K pp ) | v ′ (!) n � L 2 ( ❚ ) ) ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A ( a , b )) 1 a By contradiction ǫ n Case a < u 0 ( θ 0 = 0) < b (Fig. 1) ǫ i n θ 0 ∃ ǫ n → 0, S u 0 ∩ (0 , ǫ n ) � = ∅ 1 b Ω u 0 A ( a , b ) To find v n : ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) We consider v n , i ∈ H 1 ( ❚ ) solving Fig. 1 v ′′ n , i + v n , i = χ ( ǫ i ) ( u ′′ 0 + u 0 ) , (0 , ǫ n ) , n ,ǫ i +1 n (0 , ǫ n ) c . v n , i = 0 , Look for v n = � i =1 , 3 λ n , i v n , i such that n (0 + ) = v ′ v ′ n ( ǫ − n ) = 0, which implies ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) � � � ❚ v n ( ζ 0 + ζ ′′ 0 ) = ❚ v n d µ a = ❚ v n d µ b = 0, and � 2 G uu v 2 j ′′ ( u 0 )( v n , v n ) = n + 2 G up v n v ′ n + G pp v ′ 0 ≤ n ❚ ( � v n � L 2 ( ❚ ) ≤ √ ǫ n � v ′ � n | 2 < 0 ≤ ( o (1) − K pp ) | v ′ (!) n � L 2 ( ❚ ) ) ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A ( a , b )) 1 a By contradiction ǫ n Case a < u 0 ( θ 0 = 0) < b (Fig. 1) ǫ i n θ 0 ∃ ǫ n → 0, S u 0 ∩ (0 , ǫ n ) � = ∅ 1 b Ω u 0 A ( a , b ) To find v n : ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) We consider v n , i ∈ H 1 ( ❚ ) solving Fig. 1 v ′′ n , i + v n , i = χ ( ǫ i ) ( u ′′ 0 + u 0 ) , (0 , ǫ n ) , n ,ǫ i +1 n (0 , ǫ n ) c . v n , i = 0 , Look for v n = � i =1 , 3 λ n , i v n , i such that n (0 + ) = v ′ v ′ n ( ǫ − n ) = 0, which implies ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) � � � ❚ v n ( ζ 0 + ζ ′′ 0 ) = ❚ v n d µ a = ❚ v n d µ b = 0, and � 2 G uu v 2 j ′′ ( u 0 )( v n , v n ) = n + 2 G up v n v ′ n + G pp v ′ 0 ≤ n ❚ ( � v n � L 2 ( ❚ ) ≤ √ ǫ n � v ′ � n | 2 < 0 ≤ ( o (1) − K pp ) | v ′ (!) n � L 2 ( ❚ ) ) ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A ( a , b )) 1 a By contradiction ǫ n Case a < u 0 ( θ 0 = 0) < b (Fig. 1) ǫ i n θ 0 ∃ ǫ n → 0, S u 0 ∩ (0 , ǫ n ) � = ∅ 1 b Ω u 0 A ( a , b ) To find v n : ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) We consider v n , i ∈ H 1 ( ❚ ) solving Fig. 1 v ′′ n , i + v n , i = χ ( ǫ i ) ( u ′′ 0 + u 0 ) , (0 , ǫ n ) , n ,ǫ i +1 n (0 , ǫ n ) c . v n , i = 0 , Look for v n = � i =1 , 3 λ n , i v n , i such that n (0 + ) = v ′ v ′ n ( ǫ − n ) = 0, which implies ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) � � � ❚ v n ( ζ 0 + ζ ′′ 0 ) = ❚ v n d µ a = ❚ v n d µ b = 0, and � 2 G uu v 2 j ′′ ( u 0 )( v n , v n ) = n + 2 G up v n v ′ n + G pp v ′ 0 ≤ n ❚ ( � v n � L 2 ( ❚ ) ≤ √ ǫ n � v ′ � n | 2 < 0 ≤ ( o (1) − K pp ) | v ′ (!) n � L 2 ( ❚ ) ) ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A ( a , b )) 1 a By contradiction ǫ n Case a < u 0 ( θ 0 = 0) < b (Fig. 1) ǫ i n θ 0 ∃ ǫ n → 0, S u 0 ∩ (0 , ǫ n ) � = ∅ 1 b Ω u 0 A ( a , b ) To find v n : ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) We consider v n , i ∈ H 1 ( ❚ ) solving Fig. 1 v ′′ n , i + v n , i = χ ( ǫ i ) ( u ′′ 0 + u 0 ) , (0 , ǫ n ) , n ,ǫ i +1 n (0 , ǫ n ) c . v n , i = 0 , Look for v n = � i =1 , 3 λ n , i v n , i such that n (0 + ) = v ′ v ′ n ( ǫ − n ) = 0, which implies ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) � � � ❚ v n ( ζ 0 + ζ ′′ 0 ) = ❚ v n d µ a = ❚ v n d µ b = 0, and � 2 G uu v 2 j ′′ ( u 0 )( v n , v n ) = n + 2 G up v n v ′ n + G pp v ′ 0 ≤ n ❚ ( � v n � L 2 ( ❚ ) ≤ √ ǫ n � v ′ � n | 2 < 0 ≤ ( o (1) − K pp ) | v ′ (!) n � L 2 ( ❚ ) ) ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A ( a , b )) 1 a By contradiction ǫ n Case a < u 0 ( θ 0 = 0) < b (Fig. 1) ǫ i n θ 0 ∃ ǫ n → 0, S u 0 ∩ (0 , ǫ n ) � = ∅ 1 b Ω u 0 A ( a , b ) To find v n : ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) We consider v n , i ∈ H 1 ( ❚ ) solving Fig. 1 v ′′ n , i + v n , i = χ ( ǫ i ) ( u ′′ 0 + u 0 ) , (0 , ǫ n ) , n ,ǫ i +1 n (0 , ǫ n ) c . v n , i = 0 , Look for v n = � i =1 , 3 λ n , i v n , i such that n (0 + ) = v ′ v ′ n ( ǫ − n ) = 0, which implies ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) � � � ❚ v n ( ζ 0 + ζ ′′ 0 ) = ❚ v n d µ a = ❚ v n d µ b = 0, and � 2 G uu v 2 j ′′ ( u 0 )( v n , v n ) = n + 2 G up v n v ′ n + G pp v ′ 0 ≤ n ❚ ( � v n � L 2 ( ❚ ) ≤ √ ǫ n � v ′ � n | 2 < 0 ≤ ( o (1) − K pp ) | v ′ (!) n � L 2 ( ❚ ) ) ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A ( a , b )) 1 a By contradiction ǫ n Case a < u 0 ( θ 0 = 0) < b (Fig. 1) ǫ i n θ 0 ∃ ǫ n → 0, S u 0 ∩ (0 , ǫ n ) � = ∅ 1 b Ω u 0 A ( a , b ) To find v n : ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) We consider v n , i ∈ H 1 ( ❚ ) solving Fig. 1 v ′′ n , i + v n , i = χ ( ǫ i ) ( u ′′ 0 + u 0 ) , (0 , ǫ n ) , n ,ǫ i +1 n (0 , ǫ n ) c . v n , i = 0 , Look for v n = � i =1 , 3 λ n , i v n , i such that n (0 + ) = v ′ v ′ n ( ǫ − n ) = 0, which implies ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) � � � ❚ v n ( ζ 0 + ζ ′′ 0 ) = ❚ v n d µ a = ❚ v n d µ b = 0, and � 2 G uu v 2 j ′′ ( u 0 )( v n , v n ) = n + 2 G up v n v ′ n + G pp v ′ 0 ≤ n ❚ ( � v n � L 2 ( ❚ ) ≤ √ ǫ n � v ′ � n | 2 < 0 ≤ ( o (1) − K pp ) | v ′ (!) n � L 2 ( ❚ ) ) ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A ( a , b )) 1 a By contradiction ǫ n Case a < u 0 ( θ 0 = 0) < b (Fig. 1) ǫ i n θ 0 ∃ ǫ n → 0, S u 0 ∩ (0 , ǫ n ) � = ∅ 1 b Ω u 0 A ( a , b ) To find v n : ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) We consider v n , i ∈ H 1 ( ❚ ) solving Fig. 1 v ′′ n , i + v n , i = χ ( ǫ i ) ( u ′′ 0 + u 0 ) , (0 , ǫ n ) , n ,ǫ i +1 n (0 , ǫ n ) c . v n , i = 0 , Look for v n = � i =1 , 3 λ n , i v n , i such that n (0 + ) = v ′ v ′ n ( ǫ − n ) = 0, which implies ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) � � � ❚ v n ( ζ 0 + ζ ′′ 0 ) = ❚ v n d µ a = ❚ v n d µ b = 0, and � 2 G uu v 2 j ′′ ( u 0 )( v n , v n ) = n + 2 G up v n v ′ n + G pp v ′ 0 ≤ n ❚ ( � v n � L 2 ( ❚ ) ≤ √ ǫ n � v ′ � n | 2 < 0 ≤ ( o (1) − K pp ) | v ′ (!) n � L 2 ( ❚ ) ) ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit

Proofs: Theorem 0.1 (inclusion in A ( a , b )) 1 a By contradiction ǫ n Case a < u 0 ( θ 0 = 0) < b (Fig. 1) ǫ i n θ 0 ∃ ǫ n → 0, S u 0 ∩ (0 , ǫ n ) � = ∅ 1 b Ω u 0 A ( a , b ) To find v n : ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) We consider v n , i ∈ H 1 ( ❚ ) solving Fig. 1 v ′′ n , i + v n , i = χ ( ǫ i ) ( u ′′ 0 + u 0 ) , (0 , ǫ n ) , n ,ǫ i +1 n (0 , ǫ n ) c . v n , i = 0 , Look for v n = � i =1 , 3 λ n , i v n , i such that n (0 + ) = v ′ v ′ n ( ǫ − n ) = 0, which implies ( v ′′ n + v n ) ≥ λ ( u ′′ 0 + u 0 ) � � � ❚ v n ( ζ 0 + ζ ′′ 0 ) = ❚ v n d µ a = ❚ v n d µ b = 0, and � 2 G uu v 2 j ′′ ( u 0 )( v n , v n ) = n + 2 G up v n v ′ n + G pp v ′ 0 ≤ n ❚ ( � v n � L 2 ( ❚ ) ≤ √ ǫ n � v ′ � n | 2 < 0 ≤ ( o (1) − K pp ) | v ′ (!) n � L 2 ( ❚ ) ) ❚ J. Lamboley, A. Novruzi Polygons as solutions to shape optimization problems with convexit