Attainability by simply connected domains of optimal bounds for the polarization tensor

Habib Ammari (1) , Yves Capdeboscq (2) , Hyeonbae Kang (3) , Eunjoo Kim (4) , Mikyoung Lim (5)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
3 Department of Mathematics [Incheon]
4 EWHA - EWHA Womans University
5 Department of Mathematical Sciences, KAIST

The notion of polarization tensor is employed for the derivation of the leading-order boundary perturbations in the steady-state voltage potentials that are due to the presence of conductivity inclusions of small diameter. Recently, Capdeboscq and Vogelius obtained optimal bounds of Hashin-Shtrikman type for the trace of the polarization tensor, showing that every pair satisfying these optimal bounds arises as the eigenvalues of a polarization tensor associated with a coated ellipse. In this paper, we give numerical evidence of the fact that the set of possible polarization tensor eigenvalue pairs can also be obtained using simply connected domains. Our numerical computations are based on a boundary integral method.

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Habib Ammari, Yves Capdeboscq, Hyeonbae Kang, Eunjoo Kim, Mikyoung Lim. Attainability by simply connected domains of optimal bounds for the polarization tensor. European Journal of Applied Mathematics, 2006, 17 (2), pp.201--219. ⟨10.1017/S0956792506006541⟩. ⟨hal-02083815⟩