4
HABIB AMMARI AND HYEONBAE KANG
with this, the work of Sanchez-Palencia [125] and its extension to the Navier-Stokes
equation by Levy and Sanchez-Palencia [105].
We review a general unified layer potential technique for rigorously deriving
very accurate asymptotic expansions of electrical effective properties of dilute me-
dia for non-spherical Lipschitz isotropic and anisotropic conductivity inclusions.
The approach is valid for high contrast mixtures and inclusions with Lipschitz
boundaries. We shall emphasize the fact that it gives us any higher-order term in
the asymptotic expansion of the effective conductivity. Our results have important
implications for imaging composites. They show that the volume fractions and the
GPT's form the only information that can be reconstructed in a stable way from
boundary measurements. The volume fraction is the simplest but most important
piece of microstructural information. The GPT's involve microstructural informa-
tion beyond that contained in the volume fractions (material contrast, inclusion
shape and orientation).
The paper is organized as follows. In Section 2, we introduce the main tools
for studying the isotropic and anisotropic conductivity problems and collect some
notation and preliminary results regarding layer potentials. In Section
3,
we intro-
duce the GPT's associated with a bounded domain and an isotropic or anisotropic
conductivity and provide a survey of their main properties. In Section
4,
we present
non-iterative reconstruction algorithms based on asymptotic formulae of the bound-
ary perturbations due to the presence of the conductivity inclusions. The methods
presented in this section detect the locations and the GPT's from boundary mea-
surements. It is the detected first-order polarization tensor which yields an infor-
mation on the size and orientation of the inclusion. However, the information from
the first-order polarization tensor is a mixture of the conductivity and the volume.
Section
5
is devoted to the determination of the effective electrical conductivity of
a two-phase composite material using boundary layer potentials.
Finally, we discuss the case where the conductivity inclusion is at a distance
comparable to its diameter apart from the boundary of the background conductor.
We provide in Section
6
some essential insight for understanding the interaction
between the inclusion and the boundary of the background medium.
The paper is intended to be reasonably self-contained. No familiarity with
layer potential techniques is required. The results discussed in this survey paper
complete the material presented in our book
[18].
2. Layer potentials and transmission problems
Our aim in this section is to collect together the various concepts, basic defi-
nitions and key theorems on layer potentials, with which the reader might not be
familiar. We then provide an important decomposition formula due to Kang and
Seo
[87]
of the steady-state voltage potential into a harmonic part and a refraction
part.
Let us begin our study with the following boundary value problem:
(2.l) {
~:~=
o
in
n,
av
an=
g.
Here 0 is a bounded domain representing a conducting body, the function
g
repre-
sents the applied boundary current; it belongs to
£
2
(80) and has mean value zero,
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