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

Abstract

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. This paper is concerned with the notion of polarization tensor (PT) associated with a bounded Lipschitz domain and an isotropic constant conductivity. The notion of PT appeared in problems of potential theory related to certain questions arising in hydrodynamics, in electrostatics, and in low-frequency scattering [14, 21, 22]. The PT is a key mathematical concept in efficiently imaging small conductivity inclusions from boundary measurements and also in calculating effective electrical properties of composite materials consisting of inclusions of one material of known shape embedded homogeneously into a continuous matrix of another having electrical properties different from its own. It is now known that the leading-order term in the boundary perturbations due to the presence of an inclusion inside a conductor as well as in the asymptotic expansion of the effective conductivity of a dilute composite material in terms of the volume fraction of the inclusions can be expressed by means of the PT of the inclusions shape [1, 4, 6, 8, 11, 13, 15, 16, 20]. The asymptotic expansion of the effective conductivity is motivated by the practically important inverse problem of determining the volume fraction of a suspension of complicated shaped particles from boundary measurements of voltage potentials. Therefore, in