Contemporary Mathematics
Volume 408, 2006
Generalized polarization tensors, inverse conductivity
problems, and dilute composite materials: a review
Habib Ammari and Hyeonbae Kang
We provide a survey of some recent developments in inverse isotropic
and anisotropic conductivity problems to detect diametrically small electric
inclusions and review calculations of effective properties of dilute composite
materials. The central concept in these developments is the generalized polar-
ization tensors and our main approach is based on layer potential techniques.
1. Introduction
2. Layer potentials and transmission problems
3. Generalized polarization tensors
4. Reconstruction of small inclusions
5. Effective properties of composites
6. Near-boundary conductivity inclusions
The present survey paper is concerned with recent developments in electrical
impedance imaging of small conductivity inclusions and the theory of dilute com-
posite materials. The unifying thread is the use of the generalized polarization
tensors (GPT's) which depend only on the geometry and the conductivity of the
Electrical impedance imaging uses measurements of boundary voltage poten-
tials and associated boundary currents to infer information about the internal
conductivity profile of an object. Complete information about all voltages and
currents is known to uniquely characterize an isotropic conductivity distribution
[100, 129, 117, 36]. In its most general form electrical impedance imaging is
2000 Mathematics Subject Classification. Primary 35R30; Secondary 35B30.
Key words and phroses. Inverse isotropic and anisotropic conductivity problems, polarization
tensors, direct reconstruction algorithms, effective electrical properties.
@2006 American Mathematical Society
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