An extremal eigenvalue problem for a two-phase conductor 1 Carlos - PowerPoint PPT Presentation

Plan The Problem Existence? Existence! An extremal eigenvalue problem for a two-phase conductor 1 Carlos Conca , Rajesh Mahadevan , Leon Sanz Sixi` emes Journ´ ees Franco-Chiliennes d’Optimisation 19-21 mai 2008 Universit´ e du Sud Toulon-Var Complexe Ag´ elonde - La Londe les Maures rmahadevan@udec.cl 1 This work was realised with the support of CMM and FONDECYT N o 1070675 Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Plan Problem Statement. Existence-Difficulties. Symmetry and Existence. Improvements. Numerical Experiments. Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Plan Problem Statement. Existence-Difficulties. Symmetry and Existence. Improvements. Numerical Experiments. Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Plan Problem Statement. Existence-Difficulties. Symmetry and Existence. Improvements. Numerical Experiments. Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Plan Problem Statement. Existence-Difficulties. Symmetry and Existence. Improvements. Numerical Experiments. Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Plan Problem Statement. Existence-Difficulties. Symmetry and Existence. Improvements. Numerical Experiments. Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Problem Statement Eigenvalue Problem for Conductors Ω ⊂ R n - design region. 0 < α < β - conductivity coefficients. ω ⊂ Ω - region occupied by β . � Ω ( αχ Ω \ ω + βχ ω ) |∇ u | 2 dx λ 1 ( ω ) := � min . Ω | u | 2 dx u ∈ H 1 0 (Ω) Optimization Problem. m -constant, 0 < m < | Ω | . � � λ 1 ( ω ) : ω ⊂ Ω , ω measurable , | ω | = m inf . Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Problem Statement Eigenvalue Problem for Conductors Ω ⊂ R n - design region. 0 < α < β - conductivity coefficients. ω ⊂ Ω - region occupied by β . � Ω ( αχ Ω \ ω + βχ ω ) |∇ u | 2 dx λ 1 ( ω ) := � min . Ω | u | 2 dx u ∈ H 1 0 (Ω) Optimization Problem. m -constant, 0 < m < | Ω | . � � λ 1 ( ω ) : ω ⊂ Ω , ω measurable , | ω | = m inf . Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Problem Statement Eigenvalue Problem for Conductors Ω ⊂ R n - design region. 0 < α < β - conductivity coefficients. ω ⊂ Ω - region occupied by β . � Ω ( αχ Ω \ ω + βχ ω ) |∇ u | 2 dx λ 1 ( ω ) := � min . Ω | u | 2 dx u ∈ H 1 0 (Ω) Optimization Problem. m -constant, 0 < m < | Ω | . � � λ 1 ( ω ) : ω ⊂ Ω , ω measurable , | ω | = m inf . Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Problem Statement Eigenvalue Problem for Conductors Ω ⊂ R n - design region. 0 < α < β - conductivity coefficients. ω ⊂ Ω - region occupied by β . � Ω ( αχ Ω \ ω + βχ ω ) |∇ u | 2 dx λ 1 ( ω ) := � min . Ω | u | 2 dx u ∈ H 1 0 (Ω) Optimization Problem. m -constant, 0 < m < | Ω | . � � λ 1 ( ω ) : ω ⊂ Ω , ω measurable , | ω | = m inf . Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Problem Statement Eigenvalue Problem for Conductors Ω ⊂ R n - design region. 0 < α < β - conductivity coefficients. ω ⊂ Ω - region occupied by β . � Ω ( αχ Ω \ ω + βχ ω ) |∇ u | 2 dx λ 1 ( ω ) := � min . Ω | u | 2 dx u ∈ H 1 0 (Ω) Optimization Problem. m -constant, 0 < m < | Ω | . � � λ 1 ( ω ) : ω ⊂ Ω , ω measurable , | ω | = m inf . Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Problem Statement Eigenvalue Problem for Conductors Ω ⊂ R n - design region. 0 < α < β - conductivity coefficients. ω ⊂ Ω - region occupied by β . � Ω ( αχ Ω \ ω + βχ ω ) |∇ u | 2 dx λ 1 ( ω ) := � min . Ω | u | 2 dx u ∈ H 1 0 (Ω) Optimization Problem. m -constant, 0 < m < | Ω | . � � λ 1 ( ω ) : ω ⊂ Ω , ω measurable , | ω | = m inf . Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Problem Statement Eigenvalue Problem for Conductors Ω ⊂ R n - design region. 0 < α < β - conductivity coefficients. ω ⊂ Ω - region occupied by β . � Ω ( αχ Ω \ ω + βχ ω ) |∇ u | 2 dx λ 1 ( ω ) := � min . Ω | u | 2 dx u ∈ H 1 0 (Ω) Optimization Problem. m -constant, 0 < m < | Ω | . � � λ 1 ( ω ) : ω ⊂ Ω , ω measurable , | ω | = m inf . Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Questions of Interest Does there exist a minimizer for the problem? How does it look like? - To obtain characterizations of minimizers. Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Questions of Interest Does there exist a minimizer for the problem? How does it look like? - To obtain characterizations of minimizers. Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Existence General Formulation inf { F ( ω ) : ω ∈ A} . A - admissible shapes. Weierstrass-Tonnelli Existence Theorem If we can give a topology on A for which F is lower-semicontinuous and, 1 the level sets of F in A are compact 2 then the existence of a minimizer to the problem follows. Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Existence General Formulation inf { F ( ω ) : ω ∈ A} . A - admissible shapes. Weierstrass-Tonnelli Existence Theorem If we can give a topology on A for which F is lower-semicontinuous and, 1 the level sets of F in A are compact 2 then the existence of a minimizer to the problem follows. Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Existence-Difficulties Finding a topology which serves. Haussdorff convergence of sets H → ω if d H ( ω n , ω ) → 0 , ω n where � � d H ( ω n , ω ) = max sup d ( x , ω ) , sup x ∈ ω d ( x , ω n ) , x ∈ ω n ω �→ λ 1 ( ω ) is continuous but, { ω : ω ⊂ Ω , ω measurable , | ω | = m } is not compact. Supplementary constraints Perimeter constraint, convex inclusions, number of connected components, capacity conditions etc...make the constraint set compact for the above topology cf. Bucur and Buttazzo, Henrot and Pierre. Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Existence-Difficulties Finding a topology which serves. Haussdorff convergence of sets H → ω if d H ( ω n , ω ) → 0 , ω n where � � d H ( ω n , ω ) = max sup d ( x , ω ) , sup x ∈ ω d ( x , ω n ) , x ∈ ω n ω �→ λ 1 ( ω ) is continuous but, { ω : ω ⊂ Ω , ω measurable , | ω | = m } is not compact. Supplementary constraints Perimeter constraint, convex inclusions, number of connected components, capacity conditions etc...make the constraint set compact for the above topology cf. Bucur and Buttazzo, Henrot and Pierre. Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor

Plan The Problem Existence? Existence! Existence-Difficulties Finding a topology which serves. Haussdorff convergence of sets H → ω if d H ( ω n , ω ) → 0 , ω n where � � d H ( ω n , ω ) = max sup d ( x , ω ) , sup x ∈ ω d ( x , ω n ) , x ∈ ω n ω �→ λ 1 ( ω ) is continuous but, { ω : ω ⊂ Ω , ω measurable , | ω | = m } is not compact. Supplementary constraints Perimeter constraint, convex inclusions, number of connected components, capacity conditions etc...make the constraint set compact for the above topology cf. Bucur and Buttazzo, Henrot and Pierre. Carlos Conca , Rajesh Mahadevan , Leon Sanz An extremal eigenvalue problem for a two-phase conductor