数理解析研究所講究録
第 1896 巻 2014 年 26-33

26Some problems related to polarization tensors
Hyeonbae Kang*

1Introduction

There are a few geometric or physical quantities associated with domains such as eigenvalues, capacities,
and moments. The generalized polarization tensors (GPT) are one of those quantities. They appear
naturally in the multi-polar asymptotic expansion of the perturbation of fields due to presence of inclusions(domains). The notion of GPTs has been applied to diverse fields of research, theory of composites, inverse
problems to recover diametrically small anomalies, bio-medical imaging, dictionary matching (patternrecognition), and electro sensing, to name a few.
The purpose of this article is to introduce the notion of GPTs and briefly review its recent applications,and then discuss two mathematical problems related to GPTs, spectral properties of the NeumannPoincar\’e operator and bounds on polarization tensor. This article is a summary of a talk delivered at
the workshop “Geometry of Solutions of Partial Differential Equations” at RIMS.

2Generalized polarization tensors

In this section we introduce the notion of GPTs and review some important properties. Details andproofs of the results described in this section can be found in [8].
$d=2,3$ , see Fig. 2.1. The
Consider a configuration where an inclusion embedded in the free space(or dielectric
conductivity
thecomponents,has
multipleconnected
singleorinclusion, which may have a. Let be the conductivity distribution of the
, and
has
constant)
and the medium
configuration so that$\mathbb{R}^{d},$$\mathbb{R}^{d}\backslash \overline{\Omega}$$\sigma_{c}$$\sigma_{m}$$\sigma_{c}\neq\sigma_{m}$$\sigma$$\sigma=\sigma_{m}\chi(\mathbb{R}^{d}\backslash \overline{\Omega})+\sigma_{c}\chi(\Omega)$.

We then consider the conductivity equation$[Matrix]$(2.1)

represents the background electrical field and $u-h$. So,
where is a given harmonic function infor some constantis the perturbation due to presence of the inclusion. In Fig. 2.1 $h(x)$ is given byvector , so the field is uniform.
The solution to (2.1) admits the dipolar asymptotic expansion at infinity when $h(x)=a\cdot x$ :$\mathbb{R}^{d}$$h$$-\nabla h$$a\cdot x$

$a$

$u$

$u(x)=a \cdot x-\frac{1}{\omega_{d}}\frac{\langle a,Mx\rangle}{|x|^{d}}+O(|x|^{-d})$

a

,

$s$

$|x|arrow\infty$

(2.2)

,

matrix $M$ appeared in the asymptotic
. The
where
is the surface area of the unit sphere in
expansion is called the Polarization Tensor ( $PT$ ) associated with (or more precisely ). More generally,
ltipolar expansion:
admits the$\mathbb{R}^{d}$$w_{d}$$d\cross d$$\Omega$$u$$\sigma$$m|\iota$$u(x)=h(x)+ \sum_{\alpha}\sum_{\beta}\frac{(-1)^{|\beta|}}{\alpha!\beta!}\partial^{\alpha}h(0)\partial^{\beta}\Gamma(x)m_{\alpha\beta}, |x|arrow\infty$

.

(2.3)

are caJled Generalized Polarization Tensors (GPT). We emphasize that $PT$ and GPTs
The quantities
are not dependent on the given function , they depend only on the inclusion and the conductivity ratio$m_{\alpha\beta}$$h$$\Omega$*Department of Mathematics, Inha University, Incheon 402-751, Korea (hbkang@inha.ac.kr)

27

Figure 2.1: An inclusion embedded in the free space with a uniform field. So, GPTs are the building blocks of asymptotic expansion of the perturbation of the potential(or field) due to presence of the inclusion, and the $PT$ is that of the leading order term. It is worth
mentioning that there is ambiguity in the asymptotic expansion (2.3): the point appearing in $h(O)$ . We
may use other points for expansion of . This ambiguity can be removed by using spherical harmonics
for the expansionl.
Explicit formula for $PT$ and GPTs can be found for shapes like disks, balls, ellipses, and ellipsoids.
For example, if is an ellipse, then$\sigma_{c}/\sigma_{m}$

$0$

$h$

$\frac{x}{a}\eta 2+*_{b}^{2}\leq 1$$\Omega$$M(k, \Omega)=(\sigma_{c}-\sigma_{m})|\Omega|[^{\frac{a+b}{\sigma_{m}a+\sigma_{c}b}}0 \frac{a+b0}{\sigma_{m}b+\sigma_{c}a}]$

.

(2.4)

where
. It is worth mentioning that GPTs for two disks were found by Lim [25].
GPTs carry important geometric information on the shape of the inclusion. For example, the full setof GPTs determines the shape of the inclusion uniquely as proved in [7]. More precisely, if$k=\sigma_{c}/\sigma_{m}$$\sum a_{\alpha}b_{\beta}m_{\alpha\beta}(k_{1}, \Omega_{1})=\sum a_{\alpha}b_{\beta}m_{\alpha\beta}(k_{2}, \Omega_{2})$for all harmonic coefficients2$a_{\alpha}$and$b_{\beta}$, then$k_{1}=k_{2}$

Here

and

$\Omega_{1}=\Omega_{2}.$is the conductivity ratio. We emphasize that this result holds for inclusions with multiple
components. It is worthwhile to compare this result, which may be called the inverse GPT problem, with
the inverse spectral problem. It was proved in [18] that the full set of eigenvalues does not determine theshape uniquely. It means that GPTs carry richer information than eigenvalues.
Since the full set of GPTs determines the shape, a natural question which follows would be what
about a finite number of GPTs. In this regard, it was found in [12] that there is canonical one-to-one
correspondence between the class of PTs and that of ellipses (ellipsoids). So one can recover the equivalent
ellipse (ellipsoid) of the inclusion using its $PT$ . The equivalent ellipse of a given domain is an ellipse whose$PT$ is the same as that of thedomain. It represents a kind of the averaged shape of the inclusion. The
left figure in Fig. 2.2 shows an equivalent ellipse (black curve) of the kite shaped domain (grey curve). It$k_{j}$

is difficult to see in an analytic way what kind of shape information the higher order GPTs contain. So, anumerical method based on optimization was used to reconstruct the shape using finite number of GPTs[11]. The middle figure in Fig. 2.2 shows the result of reconstruction using GPTs
. In
[6] a level set method is incorporated into the optimization algorithm to recover even the topology, theresult of which is shown in the right figure of Fig. 2.2. In relation to shape information of GPTs, it isshown in [21] that the coefficients of the Riemann mapping from outside the unit disk to outside of ansimply connected domain can be computed explicitly using GPTs.
$m_{\alpha\beta},$lThe GPTs defined using spherical harmonics are called contracted GPTs. See [20].
$2a_{\alpha}$

is called harmonic coefficients if$\sum_{\alpha}x^{\alpha}$

is aharmonic polynomial.

$|\alpha|+|\beta|\leq 6$

28$QO\copyright$$-1 -\tau -1$
Figure 2.2: Left: the equivalent ellipse (black) of an inclusion (grey). Middle and Right: reconstructionof a single and multiple inclusions using$m_{\alpha\beta},$$|\alpha|+|\beta|\leq 6.$

matrix, enjoys several important properties: it is symmetric and positive
The $PTM$ , which is). Generalizing these properties to GPTs, the following(negative definite if
definite if
properties have been proved in [7].
$d\cross d$.
.
.

$\sigma_{c}<\sigma_{m}$$\sigma_{c}>\sigma_{m}$Symmetry: If$\{a_{\alpha}\}$and$\{b_{\beta}\}$are such that$\sum a_{\alpha}x^{\alpha}$and$\sum b_{\beta}x^{\beta}$$\sum a_{\alpha}b_{\beta}m_{\alpha\beta}=\sum a_{\alpha}b_{\beta}m_{\beta\alpha}$

Positivity: If$\sigma_{c}>(<)\sigma_{m}$are harmonic polynomials, then

.

(2.5)

, then.

(2.6)$\sum a_{\alpha}a_{\beta}m_{\alpha\beta}>(<)O$

Bounds: If$f(x)= \sum_{\alpha\in I}a_{\alpha}x^{\alpha}$

is a harmonic polynomial, then$\int_{\Omega}|\nabla f|^{2}dx\leq\frac{\sigma_{c}+\sigma_{m}}{|\sigma_{c}-\sigma_{m}|}|\sum_{\alpha,\beta\in I}a_{\alpha}a_{\beta}m_{\alpha\beta}|\leq C\int_{\Omega}|\nabla f|^{2}dx$

.

(2.7)

The bounds (2.7) hold on the $PT$ . But there are tighter bounds on the $PT$ :$\frac{\sigma_{m}}{\sigma_{c}-\sigma_{m}}Tx(M)\leq|\Omega|(d-1+\frac{\sigma_{m}}{\sigma_{c}})$

,

(2.8)

and
$\frac{\sigma_{c}-\sigma_{m}}{\sigma_{m}}R(M^{-1})\leq\frac{d-1+\sigmaarrow\sigma_{m}}{|\Omega|}$

,

(2.9)

the volume of . The bounds are called the Hashin-Shtrikman bounds
where $R$ denotes the trace andafter names of the scientists who first found the optimal bounds on the effective conductivity of isotropic
two-phase composites [19]. These bounds have been obtained by Lipton [26], and later by CapdeboscqVogelius [14]. They are optimal in the sense that every matrix satisfying bounds is realized as the $PT$ of
an inclusion [2, 13].
It is interesting to observe that the $PT$ of the ellipse given by (2.4) satisfies the equality in (2.9) (itis true for ellipsoids as well). In [28] $P6lya$ and Szeg\"o raised the conjecture that the inclusion whosepolarization tensor has the minimal trace among inclusions of the same volume takes the shape of a disk
or a ball. In terms of the Hashin-Shtrikman bounds what they asked amounts to that if the equality in(2.9) holds and all eigenvalues are the same then the domain must be a disk or a ball. At the time theywrote the book the bounds (2.8) and (2.9) were not known. In the sense of isoperimetric inequalities
the correct conjecture is that if the equality holds in (2.9), then the domain must be an ellipse or an
ellipsoid, and the original conjecture P\’olya and Szeg\"o is a part of the extended version. This conjecture
was proved by Kang and Milton [23]. More interestingly this conjecture was related to the conjectureof Eshelby. Eshelby found that if the inclusion is elliptical or ellipsoidal, then the solution to (2.1) for$h(x)=a\cdot x$ is linear inside the inclusion (see Fig. 2.3), and then conjectured that ellipses and ellipsoids
are the only shape with this property. It was proved in [22, 23] that these two conjectures are equivalentand true.
$|\Omega|$$\Omega$29

Figure 2.3: Field is uniform inside the elliptic inclusions.

3

Applications of GPTs

The notion of polarization tensor (polarizability tensor) has been widely used in the theory of composites
to compute the effective properties of two phase composites, for which we refer to the comprehensive
book of Milton [27] (see also [8]). Other classical usage of the $PT$ includes the study of potential flow[28] and low frequency asymptotic of wave [15].
The notion of $PT$ (or GPT) has been attracting much attention lately due to its usage in the inverse
problem to detect diametrically small (point-like) inclusions. The related electrical impedance tomography problem is described as follows: Let $D$ be a diametrically small inclusion buried in a domain (atsome distance from) and let$\Omega$$\partial\Omega$$\sigma=\chi(\Omega\backslash D)+\sigma_{c}\chi(D)$For a given$g\in L_{0}^{2}(\partial\Omega)$(current) let$u$be the solution to the Neumann boundary value problem(3.1)$\{\begin{array}{l}\nabla\cdot(\sigma(x)\nabla u)=0 in \Omega,\sigma\frac{\partial u}{\partial\nu}|_{\partial\Omega}=g\int_{\partial\Omega}ud\sigma=0.\end{array}$

The problem is to detect the location and $/or$ some geometric information (and unknown ) of the
inclusion using a finite number of datameasured on
The connection of the $PT$ (and GPT) to the inverse problem lies in the asymptotic expansion of on, namely,$u(x)=u_{0}(x)-M(D)\nabla u_{0}(z)\cdot\nabla_{z}N(x, z)+O(\delta^{d+1}) , x\in\partial\Omega$ ,(3.2)$\sigma_{c}$$\partial\Omega.$$(u|_{\partial\Omega}, g)$$u$$\partial\Omega$where$u_{0}$

is the solution in absence of $D$ , i. e., ,$\{\begin{array}{l}\triangle u_{0}=0 in \Omega,\frac{\partial u_{0}}{\partial\nu}|_{\partial\Omega}=g\int_{\partial\Omega}u_{0}d\sigma=0,\end{array}$represents the location of $D,$ $N(x, z)$ is the Neumann function (the Green function for the Neumann
problem), and is the order of small diameter of $D$ . It is worth mentioning that $M(D)$ is of order
If there are more than one anomahes, say, respectively, then theat the locations
formula becomes$z$$\delta^{d}.$$\delta$$D_{1},$$\ldots,$$D_{N}$$z_{1},$$\ldots,$$z_{N}$.

$u(x)=u_{0}(x)- \sum_{j=1}^{N}M(D_{j})\nabla u_{0}(z_{j})\cdot\nabla_{z}N(x, z_{j})+O(\delta^{d+1}) , x\in\partial\Omega$(3.3)

This formula was first found by Riedman and Vogelius [17], and effectively used to detect anomalies[10, 12].

The asymptotic expansion method has been developed in various other contexts such as elasticity,heat conduction and waves, and applied to diverse problems arising from non-destructive evaluation,

30
multi-static and medical imaging. Also various reconstruction algorithms using (3.3) among which aresimple pole algorithm, MUSIC algorithm and topological derivative algorithm. Compressed sensing may
be applied for reconstruction. We refer interested readers to a survey paper [9] and a recent book [5] andreferences therein.
More recently the notion of GPT has been applied to the dictionary matching. Since GPTs obey
certain rules under rigid motions and scaling, an infinity number of invariants can be constructed using
GPTs, and these invariants can be used for dictionary matching. Another important application was
found in enhancement of near cloaking. It was shown that by coating a spherical inclusion with multilayer spherical structure one can make GPTs up to arbitrary order vanish. This GPT vanishing structureenhances the near cloaking effect dramatically. We also refer to [5] and references therein for thesedevelopment. For the last we mention another recent apphcation. GPTs was used for modelling sensing
mechanism of blind fishes living in mud [1], which is an important development in electro-sensing.

4

Neumann-Poincar\’e operator and GPTs

The Neumann-Poincar\’e ( $NP$ ) operator is a classical operator (as the name suggests) arising naturally
when solving the Dirchlet or Neumann problem using integral operators such as the single or double layer
be the fundamental solution to the Laplacian, i. e.,potentials. In order to introduce it, let$\Gamma(x)$(4.1)$\Gamma(x)=\{\begin{array}{ll}\frac{1}{2\pi}\ln|x|, d=2,\frac{1}{(2-d)\omega_{d}}|x|^{2-d} d\geq 3.\end{array}$

The single layer potential$S_{\partial\Omega}[\varphi]$of a function$\varphi\in L^{2}(\partial\Omega)$

is defined by$S_{\partial\Omega}[ \varphi](x);=\int_{\partial\Omega}\Gamma(x-y)\varphi(y)d\sigma(y) , x\in \mathbb{R}^{d}$

.

(4.2)

It enjoys the jump relation$\frac{\partial}{\partial\nu}\mathcal{S}_{\partial\Omega}[\varphi]|_{\pm}(x)=(\pm\frac{1}{2}I+\mathcal{K}_{\partial\Omega}^{*})[\varphi](x) , x\in\partial\Omega$where the operator$\mathcal{K}_{\partial\Omega}^{*}$

(4.3)

,

is defined by$\mathcal{K}_{\partial\Omega}^{*}[\varphi](x)=\frac{1}{\omega_{d}}\int_{\partial\Omega}\frac{\langle x-y,\nu_{x}\rangle}{|x-y|^{d}}\varphi(y)d\sigma(y)$

.

(4.4)

, respectively, andHere the subscripts and –denote the limits from outside and inside of to
Neumann-Poincar\’e
)
the
(or
is
calledoperator. The
denotes the outward unit normal towiththedomainoperator associated
Let $M=(m_{ij})$ be the polarization tensor of . It is connected to the $NP$ operator by$\Omega$$+$$\partial\Omega$$\kappa_{\partial\Omega}^{*}$$\partial\Omega$$\nu$$\mathcal{K}_{\partial\Omega}$$\Omega.$$\Omega$$m_{ij}= \int_{\partial\Omega}y^{j}(\lambda I-\mathcal{K}_{\partial\Omega}^{*})^{-1}[\nu_{i}](y)d\sigma(y)$

,

(4.5)

where
$\lambda=\frac{\sigma_{c}-\sigma_{m}}{2(\sigma_{c}-\sigma_{m})}.$(See [8].) So, we may write $m_{ij}=m_{ij}(\lambda)=m_{ij}(\lambda, \Omega)$ . GPTs also can be expressed using the $NP$
operator, but here we restrict the discussion only to the $PT.$
It is known that$- \langle\varphi,S_{\partial\Omega}[\psi]\rangle=-\int_{\partial\Omega}\varphi S_{\partial\Omega}[\psi]d\sigma$(square integrable functions with the mean zero). If we let
is an inner product onequipped with the inner product
Hilbert space$L_{0}^{2}(\partial\Omega)$$L_{0}^{2}(\partial\Omega)$$\langle\varphi, \psi\rangle_{\mathcal{H}}:=-\langle\varphi, S_{\partial\Omega}[\psi]\rangle,$$\mathcal{H}$be the31
thenis self-adjoint on . (See [20] for example.) Since the spectrum oflies in $(-1/2,1/2)$ (see[24] ,
admits the spectral resolution: there is a family of projection operatorson (called aresolution of identity) such that$\mathcal{H}$$\mathcal{K}_{\partial\Omega}^{*}$$\mathcal{K}_{\partial\Omega}^{*}$$)$$\mathcal{K}_{\partial\Omega}^{*}$$\mathcal{E}(t)$$\mathcal{K}_{\partial\Omega}^{*}=\int_{-1/2}^{1/2}td\mathcal{E}(t)$$\mathcal{H}$

.

(4.6)

It then follows from (4.5) that$m_{ij}= \langle y_{j}, (\lambda I-\mathcal{K}_{\partial\Omega}^{*})^{-1}[v_{i}]\rangle=\int_{-1/2}^{1/2}\frac{1}{\lambda-t}d\langle \mathcal{E}(t)[\nu_{i}], y_{j}\rangle.$

So we have$m_{ij}( \lambda, \Omega)=\int_{-1/2}^{1/2}\frac{d\mu_{ij}(t)}{\lambda-t}, \lambda\in \mathbb{C}\backslash [-\frac{1}{2}, \frac{1}{2}]$

.

(4.7)

where
$d\mu_{ij}:=d\langle \mathcal{E}(t)[\nu_{i}], y_{j}\rangle.$

If$\partial\Omega$is$C^{1,\alpha}$

for some $\alpha>0$ , then$\mathcal{K}_{\partial\Omega}^{*}$

is compact on$\mathcal{H}$and the spectral resolution becomes$\mathcal{K}_{\partial\Omega}^{*}=\sum_{j=1}^{\infty}\lambda_{j}\varphi_{j}\otimes\varphi_{j}$

,

where
are eigenvalues ofonare the corresponding (normalized) eigenfunctions. So we have$\lambda_{1},$$\lambda_{2},$$\ldots(|\lambda_{1}|\geq|\lambda_{2}|\geq\ldots)$$\mathcal{K}_{\partial\Omega}^{*}$(4.8)$\mathcal{H}$counting multiplicities, and$d \mu_{ij}=\sum_{k=1}^{\infty}\langle\varphi_{k}, \nu_{i}\rangle_{\mathcal{H}}\langle\varphi_{k}, y_{j}\rangle\delta(t-\lambda_{k})dt$

Here$\delta$

5

Discussions.

$\varphi_{1},$$\varphi_{2},$$\ldots$(4.9)

is the Dirac mass.

Let us now discuss on some problems related to the polarization tensors.
Ifisfor some $\alpha>0$ , then the spectral measure for the $PTd\mu_{ij}$ is a singular measure supported
at the eignevalues of the corresponding $NP$ operator. It is quite interesting to see how this measure looks
like whenis merely Lipschitz. It is particularly interesting to see whether the measure has nonvanishing absolutely continuous part. If domain is a touching disks, then the measure is absolutely
continuous (we will discuss on it in the forthcoming paper). But the domain is not Lipschitz, it has
cusps.
In relation to the question above, it is an outstanding problem to develop the spectral theory of the$NP$ operator. Even though the $NP$ operator
has been studied for long time, not much is known about
its spectrum. Moreover, there is growing interest in the spectrum of the $NP$ operator in relation to theplasmonics. For this we refer to [3, 4] and references therein.
Another challenging problem would be derivation of bounds like (2.8) and (2.9) on $PTM(\lambda)$ whenis complex. We emphasize thatis an analytic function of in$\partial\Omega$$C^{1,\alpha}$$\partial\Omega$$\lambda$$M(\lambda)$$\lambda$$\mathbb{C}\backslash [-1/2,1/2].$

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201-219.

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Jour. Numer. Anal. 42 (2005), 2408-2428.
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