Polarization tensors of planar domains as functions of the admittivity contrast

TL;DR

This paper provides an analytic representation of polarization tensors for planar inhomogeneities, linking their spectral properties to shape and contrast, and proves bounds relevant for inverse problems in electrical imaging.

Contribution

It introduces an explicit spectral-based formula for polarization tensors in 2D and characterizes when inhomogeneities are elliptical based on the tensor's rational function form.

Findings

Analytic representation of polarization tensors in terms of spectral properties.

Characterization of elliptical inhomogeneities via rational function form of the tensor.

Proof of Hashin-Shtrikman bounds for polarization tensors.

Abstract

(Electric) polarization tensors describe part of the leading order term of asymptotic voltage perturbations caused by low volume fraction inhomogeneities of the electrical properties of a medium. They depend on the geometry of the support of the inhomogeneities and on their admittivity contrast. Corresponding asymptotic formulas are of particular interest in the design of reconstruction algorithms for determining the locations and the material properties of inhomogeneities inside a body from measurements of current flows and associated voltage potentials on the body's surface. In this work we consider the two-dimensional case only and provide an analytic representation of the polarization tensor in terms of spectral properties of the double layer integral operator associated with the support of simply connected conductivity inhomogeneities. Furthermore, we establish that an (infinitesimal) simply connected inhomogeneity has the shape of an ellipse, if and only if the polarization tensor is a rational function of the admittivity contrast with at most two poles whose residues satisfy a certain algebraic constraint. We also use the analytic representation to provide a proof of the so-called Hashin-Shtrikman bounds for polarization tensors; a similar approach has been taken previously by Golden and Papanicolaou and Kohn and Milton in the context of anisotropic composite materials.