Polarization tensors of planar domains as functions of the admittivity contrast

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.

Keywords:

• Polarization tensor
• Fredholm eigenvalues
• shape reconstruction
• Hashin–Shtrikman bounds

AMS Subject Classifications:

• 35R30
• 65N21

Acknowledgements

We like to thank Grame W. Milton (University of Utah) for pointing out to us relevant literature from the theory of composite materials.

Notes

No potential conflict of interest was reported by the authors.

1 Throughout denotes the space of bounded linear operators between normed spaces X and Y, and .

2 In terms of the measurable functional calculus, if , where denotes the indicator function of , and if .

3 Here, odd means that for a.e. ; in particular, c can be chosen to satisfy .