Thin cylindrical conductivity inclusions in a 3-dimensional domain: - PowerPoint PPT Presentation

Thin cylindrical conductivity inclusions in a 3-dimensional domain: polarization tensor and unique determination from boundary data Elisa Francini in collaboration with Elena Beretta, Yves Capdeboscq and Fr´ ed´ eric de Gournay July 20-24, 2009 E. Francini (Universit` a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 1 / 1

Low volume fraction inclusions Ω is a bounded smooth domain, ω ε ω ǫ ⊂ Ω is an inclusion | ω ǫ | → 0 as ǫ → 0 γ 0 and γ 1 are smooth functions defined in Ω. Ω γ ǫ = γ 0 + ( γ 1 − γ 0 ) 1 ω ǫ is the conductivity of the body Given a current g on ∂ Ω, study how the electric potential generated in Ω by this current depends on the presence of the inclusion. Recover information on the inclusion from boundary measurements of the potential. E. Francini (Universit` a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 2 / 1

Low volume fraction inclusions Ω is a bounded smooth domain, ω ε ω ǫ ⊂ Ω is an inclusion | ω ǫ | → 0 as ǫ → 0 γ 0 and γ 1 are smooth functions defined in Ω. Ω γ ǫ = γ 0 + ( γ 1 − γ 0 ) 1 ω ǫ is the conductivity of the body Given a current g on ∂ Ω, study how the electric potential generated in Ω by this current depends on the presence of the inclusion. Recover information on the inclusion from boundary measurements of the potential. E. Francini (Universit` a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 2 / 1

Low volume fraction inclusions Ω is a bounded smooth domain, ω ε ω ǫ ⊂ Ω is an inclusion | ω ǫ | → 0 as ǫ → 0 γ 0 and γ 1 are smooth functions defined in Ω. Ω γ ǫ = γ 0 + ( γ 1 − γ 0 ) 1 ω ǫ is the conductivity of the body Given a current g on ∂ Ω, study how the electric potential generated in Ω by this current depends on the presence of the inclusion. Recover information on the inclusion from boundary measurements of the potential. E. Francini (Universit` a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 2 / 1

Low volume fraction inclusions Ω is a bounded smooth domain, ω ε ω ǫ ⊂ Ω is an inclusion | ω ǫ | → 0 as ǫ → 0 γ 0 and γ 1 are smooth functions defined in Ω. Ω γ ǫ = γ 0 + ( γ 1 − γ 0 ) 1 ω ǫ is the conductivity of the body Given a current g on ∂ Ω, study how the electric potential generated in Ω by this current depends on the presence of the inclusion. Recover information on the inclusion from boundary measurements of the potential. E. Francini (Universit` a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 2 / 1

Low volume fraction inclusions Ω is a bounded smooth domain, ω ε ω ǫ ⊂ Ω is an inclusion | ω ǫ | → 0 as ǫ → 0 γ 0 and γ 1 are smooth functions defined in Ω. Ω γ ǫ = γ 0 + ( γ 1 − γ 0 ) 1 ω ǫ is the conductivity of the body Given a current g on ∂ Ω, study how the electric potential generated in Ω by this current depends on the presence of the inclusion. Recover information on the inclusion from boundary measurements of the potential. E. Francini (Universit` a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 2 / 1

Low volume fraction inclusions Ω is a bounded smooth domain, ω ε ω ǫ ⊂ Ω is an inclusion | ω ǫ | → 0 as ǫ → 0 γ 0 and γ 1 are smooth functions defined in Ω. Ω γ ǫ = γ 0 + ( γ 1 − γ 0 ) 1 ω ǫ is the conductivity of the body Given a current g on ∂ Ω, study how the electric potential generated in Ω by this current depends on the presence of the inclusion. Recover information on the inclusion from boundary measurements of the potential. E. Francini (Universit` a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 2 / 1

Essential references A. Friedman and M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence , Arch. Rat. Mech. Anal. , 1989. H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements , Lecture Notes in Math. 1846, Springer-Verlag, Berlin, 2004. Y. Capdeboscq and M. S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction , M2AN Math. Model. Numer. Anal. , 2003. E. Francini (Universit` a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 3 / 1

Essential references A. Friedman and M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence , Arch. Rat. Mech. Anal. , 1989. H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements , Lecture Notes in Math. 1846, Springer-Verlag, Berlin, 2004. Y. Capdeboscq and M. S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction , M2AN Math. Model. Numer. Anal. , 2003. E. Francini (Universit` a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 3 / 1

Essential references A. Friedman and M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence , Arch. Rat. Mech. Anal. , 1989. H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements , Lecture Notes in Math. 1846, Springer-Verlag, Berlin, 2004. Y. Capdeboscq and M. S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction , M2AN Math. Model. Numer. Anal. , 2003. E. Francini (Universit` a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 3 / 1

Essential references A. Friedman and M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence , Arch. Rat. Mech. Anal. , 1989. H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements , Lecture Notes in Math. 1846, Springer-Verlag, Berlin, 2004. Y. Capdeboscq and M. S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction , M2AN Math. Model. Numer. Anal. , 2003. E. Francini (Universit` a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 3 / 1

Compare two potentials � A current g ∈ H − 1 / 2 ( ∂ Ω) such that ∂ Ω g d σ = 0 induces two different potentials:   div ( γ 0 ∇ u 0 ) = 0 in Ω , div ( γ ǫ ∇ u ǫ ) = 0 in Ω ,         ∂ u 0 ∂ u ǫ γ 0 = g on ∂ Ω , γ ǫ = g on ∂ Ω , � ∂ n � ∂ n         u 0 d σ = 0 , u ǫ d σ = 0 . ∂ Ω ∂ Ω ω ε Ω Ω Can we evaluate u ǫ − u 0 on ∂ Ω? E. Francini (Universit` a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 4 / 1

Compare two potentials � A current g ∈ H − 1 / 2 ( ∂ Ω) such that ∂ Ω g d σ = 0 induces two different potentials:   div ( γ 0 ∇ u 0 ) = 0 in Ω , div ( γ ǫ ∇ u ǫ ) = 0 in Ω ,         ∂ u 0 ∂ u ǫ γ 0 = g on ∂ Ω , γ ǫ = g on ∂ Ω , � ∂ n � ∂ n         u 0 d σ = 0 , u ǫ d σ = 0 . ∂ Ω ∂ Ω ω ε Ω Ω Can we evaluate u ǫ − u 0 on ∂ Ω? E. Francini (Universit` a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 4 / 1

Assumptions and notation Assumption on the behavior of the inclusion : d ( ω ǫ , ∂ Ω) ≥ d 0 > 0 for every ǫ and | ω ǫ | − 1 1 ω ǫ ( · ) converges in the sense of measure to µ when ǫ → 0 . Assumption on the conductivities c 0 < γ i ( x ) < c − 1 γ i smooth , 0 , for x ∈ Ω , i = 0 , 1 . Let N denote the Neumann function of the unperturbed domain: given y ∈ Ω, let N ( · , y ) be the solution to  div x ( γ 0 ( x ) ∇ x N ( x , y )) = δ y ( x ) for x ∈ Ω ,    γ 0 ( x ) ∂ N 1  ( x , y ) = | ∂ Ω | for x ∈ ∂ Ω , ∂ n x �     N ( x , y ) d σ x = 0 . ∂ Ω E. Francini (Universit` a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 5 / 1

Assumptions and notation Assumption on the behavior of the inclusion : d ( ω ǫ , ∂ Ω) ≥ d 0 > 0 for every ǫ and | ω ǫ | − 1 1 ω ǫ ( · ) converges in the sense of measure to µ when ǫ → 0 . Assumption on the conductivities c 0 < γ i ( x ) < c − 1 γ i smooth , 0 , for x ∈ Ω , i = 0 , 1 . Let N denote the Neumann function of the unperturbed domain: given y ∈ Ω, let N ( · , y ) be the solution to  div x ( γ 0 ( x ) ∇ x N ( x , y )) = δ y ( x ) for x ∈ Ω ,    γ 0 ( x ) ∂ N 1  ( x , y ) = | ∂ Ω | for x ∈ ∂ Ω , ∂ n x �     N ( x , y ) d σ x = 0 . ∂ Ω E. Francini (Universit` a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 5 / 1