# Construction of inclusions with vanishing generalized polarization tensors
by imperfect interfaces††thanks: This study was supported by National Research
Foundation of Korea (NRF) grant funded by the Korean government (MSIT)
(NRF-2021R1A2C1011804).

Doosung Choi 33footnotemark: 3 Department of Mathematics, Louisiana State
University, Baton Rouge, LA 70803, USA (dchoi@lsu.edu). Mikyoung Lim
Department of Mathematical Sciences, Korea Advanced Institute of Science and
Technology, Daejeon 34141, Republic of Korea (mklim@kaist.ac.kr).

(August 29, 2024)

###### Abstract

We investigate the problem of planar conductivity inclusion with imperfect
interface conditions. We assume that the inclusion is simply connected. The
presence of the inclusion causes a perturbation in the incident background
field. This perturbation admits a multipole expansion of which coefficients we
call the generalized polarization tensors (GPTs), extending the previous
terminology for inclusions with perfect interfaces. We derive explicit matrix
expressions for the GPTs in terms of the incident field, material parameters,
and geometry of the inclusion. As an application, we construct GPT-vanishing
structures of general shape that result in negligible perturbations for all
uniform incident fields. The structure consists of a simply connected core
with an imperfect interface. We provide numerical examples of GPT-vanishing
structures obtained by our proposed scheme.

Mathematics Subject Classification. 35J05; 74B05; 65B99

Keywords. Geometric Multipole Expansion; Grunsky Coefficient; Imperfect
Interface; GPT-vanishing structure

##  1 Introduction

We consider the planar conductivity (or anti-plane elasticity) problem, where
an inclusion is inserted into an infinite homogeneous background. The
inclusion, having a different conductivity from that of the background,
generally induces perturbations to the background fields. However, certain
types of inclusions do not cause any perturbation to the uniform background
fields, known as neutral inclusions. Neutral inclusions have been extensively
studied due to their possible applications in designing invisibility cloaking
structures with metamaterials in various contexts, including acoustics,
elasticity, electromagnetics, and microwaves [1, 5, 30, 32, 44, 45, 46]. Most
results on neutral inclusions deal with perfect boundary conditions. On the
other hand, this paper addresses neutral inclusions with imperfect interfaces.
More precisely, we provide a construction scheme for neutral inclusions of
general shape with imperfect interfaces conditions where the core has finite
low conductivity.

Well-known examples of neutral inclusions (with perfect interfaces) that do
not disturb incident uniform fields are coated disks and spheres with
isotropic material parameters [18, 19, 20, 23, 35, 36]. Coated ellipses and
ellipsoids exhibit neutrality to uniform fields of all directions when the
conductivities of the core, shell, and matrix are appropriately selected
(potentially anisotropic) [16, 29, 33, 40, 39]. It is worth noting that these
are the only shapes that can achieve the neutral property for all uniform
fields [25, 26, 34]. Non-elliptical coated inclusions can achieve neutrality
for a single uniform field [21, 34] (see also [31]).

One can design neutral inclusions using the concept of the generalized
polarization tensors (GPTs), which are the coefficients of the multipole
expansion for the perturbed potential caused by the inclusion; this concept
extends the polarization tensor, introduced by Schiffer and Szegö [38], to
higher orders. GPT-vanishing structures–inclusions with vanishing values for
the leading-order terms of the GPTs–are neutral inclusions in the asymptotic
sense. Multi-coated concentric disks or balls have been reported to be GPT-
vanishing structures [4, 43] (see also [3, 5] for results in the context of
Helmholtz and Maxwell’s equations). GPT-vanishing structures of general shape
were also obtained using a shape optimization approach [15, 22, 28] (see also
[11]).

Imperfect interfaces are characterized by discontinuity in either the flux or
the potential, in contrast to perfectly bonding boundaries of which both the
flux and potential are continuous. The interface parameter, possibly a non-
constant function, accounts for the discontinuity of the flux or the potential
on the interface. In [37], Ru investigated neutral inclusions for the two-
dimensional elastic problem with imperfect interfaces when the inclusion is
stiffer than the background medium. For the conductivity problem, Benveniste
and Miloh found neutral inclusions with imperfect interfaces for a single
uniform field [8]. In [27], Kang and Li constructed PT-vanishing structures of
general shape with imperfect interfaces, assuming that the core is simply
connected and perfectly conducting.

In this paper, we propose a new construction scheme of neutral inclusions (of
general shape) for the planar conductivity problem with imperfect interfaces.
The inclusions have discontinuity in the potential, where the core have the
arbitrary constant conductivity. Different from [27], the constant potential
condition in the core is no longer assumed. Consequently, the construction of
the interface parameter satisfying the neutrality condition becomes
challenging. To address this problem, we find GPT-vanishing structures by
employing the concept of the Faber polynomials polarization tensors (FPTs)
(see [11]). The FPTs are linear combinations of GPTs with coefficients given
by Faber polynomials associated with the inclusion. We also refer to [13, 14]
for details of Faber polynomials. The concept of FPTs has been successfully
applied to design semi-neutral inclusions [11] (see also [9, 10, 24]). In this
paper, we derive an explicit matrix expression of the FPTs for an inclusion
with imperfect interface conditions and, by using the matrix expression,
determine the interface parameter that vanishes several leading terms of the
FPTs.

The rest of this paper is organized as follows. In Section 2, we present the
problem formulation. In Section 3, we introduce Faber polynomials based on
conformal mappings and define the FPTs using integral formulation. Section 4
focuses on deriving the matrix expressions for the FPTs. The construction
scheme for inclusions with vanishing GPTs (or FPTs) by imperfect interfaces is
described in Section 5. We also present numerical visualizations in Section 5.

##  2 Problem formulation

We let $D$ be a planar, simply connected domain with an analytic boundary. The
core region $D$ has the constant conductivity $\sigma_{c}$ and the exterior
region $\mathbb{R}^{2}\setminus D$ is occupied by another homogeneous
isotropic material with the constant conductivity $\sigma_{m}>0$. We consider
the potential problem with imperfect interface condition:

$\begin{cases}\displaystyle\nabla\cdot{\sigma}\nabla u=0\qquad&\mbox{in
}\mathbb{\mathbb{R}}^{2},\\\ \displaystyle
p\big{(}u\big{|}^{+}-u\big{|}^{-}\big{)}=\sigma_{m}\frac{\partial
u}{\partial\nu}\Big{|}^{+}\qquad&\mbox{on }\partial D,\\\\[5.69054pt]
\displaystyle\sigma_{m}\frac{\partial
u}{\partial\nu}\Big{|}^{+}=\sigma_{c}\frac{\partial
u}{\partial\nu}\Big{|}^{-}\qquad&\mbox{on }\partial D,\\\\[5.69054pt]
\displaystyle(u-H)(x)=O({|x|^{-1}})\qquad&\mbox{as }|x|\to\infty,\end{cases}$
(2.1)

where $p$ is the interface parameter that is a real-valued function on
$\partial\Omega$ and $H$ is an arbitrary entire harmonic function. The
conductivity distribution ${\sigma}$ has the form

${\sigma}=\sigma_{c}\chi_{D}+\sigma_{m}\chi_{\mathbb{R}^{2}\setminus\overline{D}},$

where $\chi$ is the characteristic function.

###  2.1 Generalized polarization tensors

For a Lipschitz domain $D$, we define the single and double layer potentials
with a density $\varphi\in L^{2}(\partial D)$,

$\displaystyle\mathcal{S}_{\partial D}[\varphi](x)$
$\displaystyle=\int_{\partial D}\Gamma(x-y)\varphi(y)\,d\sigma(y),\quad
x\in\mathbb{R}^{2},$ $\displaystyle\mathcal{D}_{\partial D}[\varphi](x)$
$\displaystyle=\int_{\partial
D}\frac{\partial}{\partial\nu_{y}}\Gamma(x-y)\varphi(y)\,d\sigma(y),\quad
x\in\mathbb{R}^{2}\setminus\partial D,$

where $\Gamma$ is the fundamental solution to the Laplacian, i.e.,
$\Gamma(x)=\frac{1}{2\pi}\ln|x|$, and $\nu_{y}$ denotes the outward unit
normal vector on $\partial D$. We also define the so-called Neumann–Poincaré
(NP) operator

$\mathcal{K}_{\partial D}[\varphi](x)=p.v.\,\int_{\partial
D}\frac{\partial}{\partial\nu_{y}}\Gamma(x-y)\varphi(y)\,d\sigma(y),\quad
x\in\partial D,$

and its $L^{2}$-adjoint operator

$\displaystyle\mathcal{K}^{*}_{\partial D}[\varphi](x)=p.v.\,\int_{\partial
D}\frac{\partial}{\partial\nu_{x}}\Gamma(x-y)\varphi(y)\,d\sigma(y),\quad
x\in\partial D.$

Here, $p.v.$ stands for the Cauchy principal value. The operator
$\mathcal{K}^{*}_{\partial D}$ is bounded on $L^{2}(\partial D)$. We identify
$x=(x_{1},x_{2})\in\mathbb{R}^{2}$ with
$z=x_{1}+\mathrm{i}x_{2}\in\mathbb{C}$. We set

$\mathcal{S}_{\partial D}[\varphi](z):=\mathcal{S}_{\partial D}[\varphi](x).$

Likewise, we define $\mathcal{D}_{\partial D}[\varphi](z)$ and
$\mathcal{K}^{*}_{\partial D}[\varphi](z)$.

On the interface $\partial D$, the layer potentials satisfy the jump relations

$\displaystyle\mathcal{S}_{\partial D}[\varphi]\Big{|}^{+}$
$\displaystyle=\mathcal{S}_{\partial D}[\varphi]\Big{|}^{-},$
$\displaystyle\frac{\partial}{\partial\nu}\mathcal{S}_{\partial
D}[\varphi]\Big{|}^{\pm}$
$\displaystyle=\left(\pm\frac{1}{2}I+\mathcal{K}^{*}_{\partial
D}\right)[\varphi],$ $\displaystyle\mathcal{D}_{\partial
D}[\varphi]\Big{|}^{\pm}$
$\displaystyle=\left(\mp\frac{1}{2}I+\mathcal{K}_{\partial
D}\right)[\varphi],$
$\displaystyle\frac{\partial}{\partial\nu}\mathcal{D}_{\partial
D}[\varphi]\Big{|}^{+}$
$\displaystyle=\frac{\partial}{\partial\nu}\mathcal{D}_{\partial
D}[\varphi]\Big{|}^{-}.$

Here, $I$ denotes the identity operator on $L^{2}(\partial D)$.

Let $\Re[\cdot]$ and $\Im[\cdot]$ denote the real and imaginary part of a
complex number, respectively. The solution $u$ of the system (2.1) can be
represented by

$u(z)=H(z)+\left(\mathcal{S}_{\partial D}\,p-\tau\mathcal{D}_{\partial
D}\right)[\varphi](z),\quad(\Re(z),\Im(z))\in\mathbb{R}^{2}\setminus{\partial
D}$ (2.2)

with $\lambda=\frac{\sigma_{c}+\sigma_{m}}{2(\sigma_{c}-\sigma_{m})}$ and
$\tau=\frac{\sigma_{c}\sigma_{m}}{\sigma_{c}-\sigma_{m}},$ where $p$ denotes
the multiplication operator by $p$ (by abuse of the notation) and the density
function $\varphi$ is the solution to the integral equation

$\left(\left(\lambda I-\mathcal{K}^{*}_{\partial
D}\right)p+\tau\frac{\partial}{\partial\nu}\mathcal{D}_{\partial
D}\right)\left[\varphi\right]=\frac{\partial H}{\partial\nu}.$ (2.3)

This representation was introduced in [27] for the case $\sigma_{c}=\infty$
(that is, $\tau=1$) with arbitrary smooth function $p>0$. Following the proof
in [27], one can show that the operator $\left(\lambda
I-\mathcal{K}^{*}_{\partial
D}\right)p+\tau\frac{\partial}{\partial\nu}\mathcal{D}_{\partial D}$ is
invertible for $0<\sigma\neq 1<\infty$. By the jump relations, the solution
(2.2) meets all conditions of the system (2.1).

######  Definition 1.

Set $Z_{n}(\eta)=\eta^{n}$ for each $n\in\mathbb{N}$. For $m,n\in\mathbb{N}$,
we define

$\displaystyle\mathbb{N}_{mn}^{(1)}(D,p,\sigma_{c},\sigma_{m})$
$\displaystyle=\int_{\partial
D}\bigg{(}Z_{n}(\eta)\,p(\eta)-\tau\frac{\partial
Z_{n}(\eta)}{\partial\nu}\bigg{)}\bigg{(}(\lambda I-\mathcal{K}^{*}_{\partial
D})p+\tau\frac{\partial\mathcal{D}_{\partial
D}}{\partial\nu}\bigg{)}^{-1}\left[\frac{\partial
Z_{m}}{\partial\nu}\right](\eta)\,d\sigma(\eta),$
$\displaystyle\mathbb{N}_{mn}^{(2)}(D,p,\sigma_{c},\sigma_{m})$
$\displaystyle=\int_{\partial
D}\bigg{(}Z_{n}(\eta)\,p(\eta)-\tau\frac{\partial
Z_{n}(\eta)}{\partial\nu}\bigg{)}\bigg{(}(\lambda I-\mathcal{K}^{*}_{\partial
D})p+\tau\frac{\partial\mathcal{D}_{\partial
D}}{\partial\nu}\bigg{)}^{-1}\left[\frac{\partial\overline{Z_{m}}}{\partial\nu}\right](\eta)\,d\sigma(\eta),$

where $\lambda=\frac{\sigma_{c}+\sigma_{m}}{2(\sigma_{c}-\sigma_{m})}$ and
$\tau=\frac{\sigma_{c}\sigma_{m}}{\sigma_{c}-\sigma_{m}}$. We call
$\mathbb{N}_{mn}^{(1)}$ and $\mathbb{N}_{mn}^{(2)}$ the (complex) generalized
polarization tensors (GPTs) corresponding to the inclusion $D$ with the
interface function $p$ and the conductivities $\sigma_{c}$, $\sigma_{m}$.

The GPTs generalize the polarization tensor (PT) that was introduced by
Schiffer and Szegö [38]. The concept of the PT, the GPTs for $m=n=1$, is
applicable to the potential problem with imperfect interface condition.

###  2.2 Multipole expansion

Using this Taylor expansion, the solution expression (2.2) and the definition
of the GPTs, one can show the following multipole expansion for the inclusion
problem with imperfect interface (refer to [2, 6, 7, 42] for the perfect
interface problem).

######  Proposition 2.1.

Let $u$ be the solution to (2.1) for
$H(z)=\Re\left[\sum_{m=1}^{\infty}\alpha_{m}z^{m}\right]$ with complex
coefficients $\alpha_{m}$. Let $\mathbb{N}_{mn}^{(1)}$ and
$\mathbb{N}_{mn}^{(2)}$ be given by Definition 1. We have

$\displaystyle
u(z)=H(z)-\Re\left[\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\frac{1}{4\pi
n}\left(\alpha_{m}\mathbb{N}_{mn}^{(1)}+\overline{\alpha_{m}}\,\mathbb{N}_{mn}^{(2)}\right){z^{-n}}\right],\quad|z|>\sup\left\\{|y|:y\in
D\right\\}.$

###### Proof.

By the Taylor expansion of the logarithmic function, we get as $|z|\to\infty$,

$\Gamma(z-\eta)=\frac{1}{2\pi}\Re\left[\ln(z-\eta)\right]=\frac{1}{2\pi}\Re\bigg{[}\ln
z-\sum_{n=1}^{\infty}\frac{1}{n}\frac{\eta^{n}}{z^{n}}\bigg{]}=\Gamma(z)-\sum_{n=1}^{\infty}\frac{1}{2\pi
n}\Re\left[Z_{n}(\eta)\,z^{-n}\right].$

Using this expansion and (2.2), the solution $u$ satisfies

$\displaystyle u(z)$ $\displaystyle=H(z)+\mathcal{S}_{\partial
D}[p\varphi](z)-\tau\mathcal{D}_{\partial D}[\varphi](z)$
$\displaystyle=H(z)+\int_{\partial
D}\left[\Gamma(z-\eta)\,p(\eta)-\tau\frac{\partial\Gamma(z-\eta)}{\partial\nu_{\eta}}\right]\varphi(\eta)\,d\sigma(\eta)$
$\displaystyle=H(z)+\int_{\partial
D}\Gamma(z)\,p(\eta)\,\varphi(\eta)\,d\sigma(\eta)$
$\displaystyle\quad-\sum_{n=1}^{\infty}\frac{z^{-n}}{2\pi n}\int_{\partial
D}\Re\left[Z_{n}(\eta)\,p(\eta)-\tau\frac{\partial
Z_{n}(\eta)}{\partial\nu_{\eta}}\right]\varphi(\eta)\,d\sigma(\eta)$
$\displaystyle=H(z)-\sum_{n=1}^{\infty}\frac{z^{-n}}{2\pi n}\int_{\partial
D}\Re\left[Z_{n}(\eta)\,p(\eta)-\tau\frac{\partial
Z_{n}(\eta)}{\partial\nu_{\eta}}\right]\varphi(\eta)\,d\sigma(\eta).$ (2.4)

The last equation holds because $\int_{\partial
D}p(\eta)\,\varphi(\eta)\,d\sigma(\eta)=0$. Due to (2.3), we have

$\varphi=\left(\left(\lambda I-\mathcal{K}^{*}_{\partial
D}\right)p+\tau\frac{\partial}{\partial\nu}\mathcal{D}_{\partial
D}\right)^{-1}\left[\Re\left(\sum_{m=1}^{\infty}\alpha_{m}\frac{\partial
Z_{m}}{\partial\nu}\right)\right],$

and substituting this $\varphi$ into (2.4) leads the desired result. $\Box$

In [27], the interface function $p$ was constructed such that the
corresponding PT, i.e., $\mathbb{N}_{11}^{(1)}$ and $\mathbb{N}_{11}^{(2)}$,
vanishes under the assumption that $\sigma_{c}=\infty$ (that is, the core is
occupied with perfectly conducting material).

In the present paper, we construct the interface function $p$ such that the
GPTs vanish up to some finite orders (including the PT), where the core $D$ is
a given domain with arbitrary constant conductivity $\sigma_{c}$. In view of
Proposition 2.1, the resulting inclusion is a neutral inclusion in the
asymptotic sense. For the construction of $p$, we will apply the complex
potential approach in [11], where the concept of the Faber polynomial
polarization tensors was employed to build semi-neutral inclusions of general
shape.

##  3 Faber polynomial polarization tensors (FPTs)

From the Riemann mapping theorem, there uniquely exist $\gamma>0$ and
conformal mapping $\Psi$ from $\\{w\in\mathbb{C}:|w|>\gamma\\}$ onto
$\mathbb{C}\setminus\overline{D}$ such that

$\Psi(w)=w+a_{0}+\frac{a_{1}}{w}+\frac{a_{2}}{w^{2}}+\cdots.$ (3.1)

The quantity $\gamma$ is called the conformal radius of $D$. One can
numerically compute the numerical computation of $a_{n}$ from a given
parametrization of $\partial\Omega$ (see, for instance, [24]). As an univalent
function, $\Psi$ defines the Faber polynomials $\\{F_{m}\\}_{m=1}^{\infty}$ by
the relation [13]

$\frac{\Psi^{\prime}(w)}{\Psi(w)-z}=\sum_{m=0}^{\infty}\frac{F_{m}(z)}{w^{m+1}},\quad
z\in{\overline{D}},\ |w|>\gamma.$ (3.2)

The Faber polynomials $F_{m}$ are monic polynomials of degree $m$ that are
uniquely determined by the conformal mapping coefficients $\\{a_{n}\\}_{0\leq
n\leq m-1}$ via the recursive relation [12]:

$F_{m+1}(z)=zF_{m}(z)-ma_{m}-\sum_{n=0}^{m}a_{n}F_{m-n}(z),\quad m\geq 0.$
(3.3)

In particular, we have

$\displaystyle F_{0}(z)$ $\displaystyle=1,\quad F_{1}(z)=z-a_{0},\quad
F_{2}(z)=z^{2}-2a_{0}z+a_{0}^{2}-2a_{1},$ $\displaystyle F_{3}(z)$
$\displaystyle=z^{3}-3a_{0}z^{2}+3(a_{0}^{2}-a_{1})z-a_{0}^{3}+3a_{0}a_{1}-3a_{2}.$

For $z=\Psi(w)$ with $|w|>r$, it holds that

$F_{m}(\Psi(w))=w^{m}+\sum_{n=1}^{\infty}c_{mn}{w^{-n}}$ (3.4)

with the so-called Grunsky coefficient $c_{mn}$. Recursive relations for the
Faber polynomial coefficients and the Grunsky coefficients are well-known
(see, for instance, [12, 17]).

We introduce the modified polar coordinates
$(\rho,\theta)\in[\rho_{0},\infty)\times[0,2\pi)$, $\rho_{0}=\ln\gamma$, for
$z\in\mathbb{C}\setminus D$ via the relation

$z=\Psi(w)=\Psi(e^{\rho+i\theta}).$

We denote the scale factor as
$h=\left|\frac{\partial\Psi}{\partial\rho}\right|=\left|\frac{\partial\Psi}{\partial\theta}\right|=e^{\rho}\left|\Psi^{\prime}\right|.$
One can easily see that $d\sigma(z)=h(\rho_{0},\theta)d\theta$ on $\partial
D$. For $z=\Psi(w)\in\partial D$, we have

$\frac{\partial u}{\partial\nu}\Big{|}_{\partial
D}^{\pm}=\frac{1}{h(\rho_{0},\theta)}\frac{\partial}{\partial\rho}u(\Psi(w))\Big{|}_{\rho\rightarrow\rho_{0}^{\pm}}.$
(3.5)

We assume that $\Omega$ has an analytic boundary, that is, $\Psi$ can be
conformally extended to $\\{w\in\mathbb{C}:|w|>\gamma-\delta\\}$ for some
$\delta>0$. Then (3.2) holds with a modified conformal radius and domain. The
Grunsky coefficients satisfy the so-called strong and week Grunsky
inequalities. In particular, it holds that $\left|c_{mn}\right|\leq
2m\gamma^{m+n}.$ The same relation holds with $\gamma-\delta$ instead of
$\gamma$, that is,

$\left|c_{mn}\right|\leq 2m(\gamma-\delta)^{m+n}.$ (3.6)

The Faber polynomials form a basis for complex analytic functions in $D$ [12].
Let $v$ be a complex analytic function in $D$ and continuous in
$\overline{D}$. By applying the Cauchy integral formula to $v$ and applying
(3.2), it follows that

$v(z)=\sum_{m=0}^{\infty}b_{m}F_{m}(z)\quad\mbox{in }D$

with $b_{m}=\frac{1}{2\pi i}\int_{|w|=\gamma}\frac{v(\Psi(w))}{w^{m+1}}\,dw.$
Hence, we have

$|b_{m}|\leq\|v\|_{L^{\infty}(\partial D)}\gamma^{-m}.$ (3.7)

###  3.1 FPTs and FPTs-vanishing structure

Since the Faber polynomials form a basis for complex analytic functions, $H$
is given by

$H(z)=\sum_{m=1}^{\infty}\Re\big{[}\alpha_{m}F_{m}(z)\big{]}$ (3.8)

with some complex coefficients $\alpha_{m}$.

Let $u$ be the solution to the imperfect interface problem (2.1). We can
expand $u\big{|}_{D}$ into the real parts of the Faber polynomials.
Furthermore, as the solution to (2.1) linearly depends on the background
solution $H$, we have

$\displaystyle u(z)$
$\displaystyle=\sum_{m=1}^{\infty}\Re\bigg{[}\sum_{n=1}^{\infty}\beta_{mn}F_{n}(z)\bigg{]}\quad\mbox{in
}D$ (3.9)

with some constants $\beta_{mn}$. As $u-H$ is decaying near infinity and
$\Psi$ is conformal, it holds that for $z=\Psi(w)$,

$\displaystyle
u(z)=H(z)+\sum_{m=1}^{\infty}\Re\bigg{[}\sum_{n=1}^{\infty}s_{mn}w^{-n}\bigg{]}\quad\mbox{in
}\mathbb{R}^{2}\setminus\overline{D}$ (3.10)

with some constants $s_{mn}$. Following [11], we call the expansion in (3.10)
the geometric multipole expansion of $u$. For each $m$, both $\beta_{mn}$ and
$s_{mn}$ are determined by $\alpha_{m}$. In particular, $\beta_{mn}=s_{mn}=0$
if we give $\alpha_{m}=0$ for some $m$ in (3.8). In fact, we can decompose the
coefficient $s_{mn}$ by using some quantities defined as below.

######  Definition 1.

Let $F_{n}$ be the Faber polynomial for each $n\in\mathbb{N}$. For
$m,n\in\mathbb{N}$, we define

$\displaystyle\mathbb{F}_{mn}^{(1)}(D,p,\sigma_{c},\sigma_{m})$
$\displaystyle=\int_{\partial
D}\bigg{(}F_{n}(\eta)\,p(\eta)-\tau\frac{\partial
F_{n}(\eta)}{\partial\nu}\bigg{)}\bigg{(}(\lambda I-\mathcal{K}^{*}_{\partial
D})p+\tau\frac{\partial\mathcal{D}_{\partial
D}}{\partial\nu}\bigg{)}^{-1}\left[\frac{\partial
F_{m}}{\partial\nu}\right](\eta)\,d\sigma(\eta),$
$\displaystyle\mathbb{F}_{mn}^{(2)}(D,p,\sigma_{c},\sigma_{m})$
$\displaystyle=\int_{\partial
D}\bigg{(}F_{n}(\eta)\,p(\eta)-\tau\frac{\partial
F_{n}(\eta)}{\partial\nu}\bigg{)}\bigg{(}(\lambda I-\mathcal{K}^{*}_{\partial
D})p+\tau\frac{\partial\mathcal{D}_{\partial
D}}{\partial\nu}\bigg{)}^{-1}\left[\frac{\partial\overline{F_{m}}}{\partial\nu}\right](\eta)\,d\sigma(\eta),$

where $\lambda=\frac{\sigma_{c}+\sigma_{m}}{2(\sigma_{c}-\sigma_{m})}$ and
$\tau=\frac{\sigma_{c}\sigma_{m}}{\sigma_{c}-\sigma_{m}}$. We call
$\mathbb{F}_{mn}^{(1)}$ and $\mathbb{F}_{mn}^{(2)}$ the Faber polynomial
polarization tensors (FPTs) for an inclusion with an imperfect interface, by
extending the definition for an inclusion with perfect interface conditions in
[11].

For $z=\Psi(w)$, integrating the expansion (3.2) with respect to $w$ yields
that

$\frac{1}{2\pi}\ln(z-\eta)=-\sum_{m=0}^{\infty}\frac{F_{m}(\eta)}{2\pi
m}\frac{1}{w^{m}}+\text{constant},\quad\eta\in{\overline{D}},\ |w|>\gamma.$

By using the linear dependence of $u$ on $H$ and the above series expansion,
we can decompose $s_{mn}$ as

$s_{mn}=-\alpha_{m}\frac{\mathbb{F}_{mn}^{(1)}}{4\pi
n}-\overline{\alpha_{m}}\,\frac{\mathbb{F}_{mn}^{(2)}}{4\pi n},$ (3.11)

where $\mathbb{F}_{mn}^{(1)}$ and $\mathbb{F}_{mn}^{(2)}$ are the FPTs
independent of $\alpha_{m}$, namely,

$\displaystyle
u(z)=H(z)-\sum_{m=1}^{\infty}\Re\bigg{[}\sum_{n=1}^{\infty}\frac{1}{4\pi
n}\left(\alpha_{m}\mathbb{F}_{mn}^{(1)}+\overline{\alpha_{m}}\mathbb{F}_{mn}^{(2)}\right)w^{-n}\bigg{]}\quad\mbox{in
}\mathbb{R}^{2}\setminus\overline{D},$ (3.12)

where $H$ is given by an entire Faber series
$H(z)=\sum_{m=1}^{\infty}\Re\big{[}\alpha_{m}F_{m}(z)\big{]}$. We omit the
derivation of (3.12) because it is very similar with the proof of Proposition
2.1.

The first-order terms of the FPTs correspond to the PTs considered in [27].
More precisely, the polarization tensor $M=\\{m_{ij}\\}_{i,j=1,2}$ admits the
expression

$M=\left(\begin{array}[]{cc}m_{11}&m_{12}\\\
m_{21}&m_{22}\end{array}\right)=\frac{1}{2}\left(\begin{array}[]{cc}\displaystyle\Re\left[\mathbb{F}_{11}^{(1)}+\mathbb{F}_{11}^{(2)}\right]&\displaystyle\Im\left[\mathbb{F}_{11}^{(1)}+\mathbb{F}_{11}^{(2)}\right]\\\\[5.69054pt]
\displaystyle\Im\left[\mathbb{F}_{11}^{(1)}-\mathbb{F}_{11}^{(2)}\right]&\displaystyle\Re\left[\mathbb{F}_{11}^{(2)}-\mathbb{F}_{11}^{(1)}\right]\end{array}\right).$

We can express the Faber polynomial $F_{m}(z)$ as

$F_{m}(z)=\sum_{n=0}^{m}q_{mn}z^{n},$ (3.13)

where for a fixed $m$, the coefficient $q_{mn}$ depends only on
$\\{a_{k}\\}_{0\leq k\leq m-1}$. One can easily obtain recursive formulas for
$q_{mn}$ from (3.3). From (3.13) and the definition of the FPTs, it holds for
each $m,n$ that

$\displaystyle\mathbb{F}_{mn}^{(1)}$
$\displaystyle=\sum_{k=1}^{m}\sum_{l=1}^{n}q_{mk}\,q_{nl}\,\mathbb{N}_{kl}^{(1)},$
(3.14) $\displaystyle\mathbb{F}_{mn}^{(2)}$
$\displaystyle=\sum_{k=1}^{m}\sum_{l=1}^{n}\overline{q_{mk}}\,q_{nl}\,\mathbb{N}_{kl}^{(2)}.$

From the relations of the GPTs and FPTs, we can then obtain matrix
factorizations for the GPTs. Set $Q=(q_{mn})_{m,n=1}^{\infty}$ with $q_{mn}$
given by (3.13). We can rewrite relations (3.14) in matrix form as

$\displaystyle\mathbb{F}^{(1)}$ $\displaystyle=Q\,\mathbb{N}^{(1)}Q^{T},$
(3.15) $\displaystyle\mathbb{F}^{(2)}$
$\displaystyle=\overline{Q}\,\mathbb{N}^{(2)}Q^{T},$

where $\overline{Q}$ and $Q^{T}$ denote the conjugate and transpose matrices
of $Q$, respectively. Using the recursive relation in (3.3), we get that, for
each $m\geq 1$,

$\displaystyle q_{mm}=1,\quad q_{(m+1)m}=-(m+1)a_{0},\quad
q_{mn}=0\quad\mbox{for all }n\geq m+1.$ (3.16)

Indeed,

$Q=\left[\begin{matrix}1&0&0\quad\cdots\\\\[4.2679pt]
-2a_{0}&1&0\quad\cdots\\\\[4.2679pt]
3a_{0}^{2}-3a_{1}&-3a_{0}&1\quad\cdots\\\\[4.2679pt]
\vdots&\vdots&\vdots\quad\ddots\end{matrix}\right],$ (3.17)

and then $Q$ is lower triangular and invertible. Consequently, (3.15) gives
that

$\displaystyle\mathbb{N}^{(1)}$
$\displaystyle=Q^{-1}\,\mathbb{F}^{(1)}\left(Q^{-1}\right)^{T},$ (3.18)
$\displaystyle\mathbb{N}^{(2)}$
$\displaystyle=\overline{Q^{-1}}\,\mathbb{F}^{(2)}\left(Q^{-1}\right)^{T}.$

Indeed, the first terms of CGPTs and FPTs are always identical:

$\mathbb{N}_{11}^{(1)}=\mathbb{F}_{11}^{(1)},\quad\mathbb{N}_{11}^{(2)}=\mathbb{F}_{11}^{(2)}.$

Furthermore, if the center of the conformal mapping in (3.1) is given by $0$,
namely, $a_{0}=0$, the lower diagonal elements $Q$ in (3.17) are all zero. In
this case, CGPTs and FPTs have the same leading terms of order up to 2:

$\mathbb{N}_{mn}^{(1)}=\mathbb{F}_{mn}^{(1)},\quad\mathbb{N}_{mn}^{(2)}=\mathbb{F}_{mn}^{(2)},\quad
1\leq m,n\leq 2.$

Recall that our aim is to construct $p$ such that the corresponding GPTs
vanish up to some finite orders. In view of (3.14) the GPT-vanishing
structures can be equivalently achieved by the FPT-vanishing structure. It is
worth remarking that for the instance of the perfectly bonding interface,
explicit formulas of the FPTs were derived in [11].

##  4 Matrix expressions for the FPTs

Let $H$ be an entire harmonic function given by (3.8) and $u$ be the solution
to (2.1). From (3.9), (3.9), (3.8) and the fact that $\Psi$ admits an analytic
extension for $\rho\in[\rho_{0}-\delta,\rho_{0}]$, we have for $z=\Psi(w)$
that

$\displaystyle u(z)$
$\displaystyle=\begin{cases}\displaystyle\Re\bigg{[}\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\beta_{mn}F_{n}(z)\bigg{]}\quad&\mbox{for
}\rho\in[\rho_{0}-\delta,\rho_{0}],\\\
\displaystyle\Re\bigg{[}\sum_{m=1}^{\infty}\alpha_{m}F_{m}(z)+\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}s_{mn}w^{-n}\bigg{]}\quad&\mbox{for
}\rho>\rho_{0}\end{cases}$
$\displaystyle=\begin{cases}\displaystyle\Re\bigg{[}\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\beta_{mn}w^{n}+\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\sum_{l=1}^{\infty}\beta_{ml}c_{ln}w^{-n}\bigg{]}\quad&\mbox{for
}\rho\in[\rho_{0}-\delta,\rho_{0}],\\\
\displaystyle\Re\bigg{[}\sum_{m=1}^{\infty}\alpha_{m}w^{m}+\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\left(\alpha_{m}c_{mn}+s_{mn}\right)w^{-n}\bigg{]}\quad&\mbox{for
}\rho>\rho_{0}\end{cases}$ (4.1)

Since we only consider the background solution $H$ of finite order, we can
assume finite summation for $m$. We may assume a finite summation for $m$
since only finite order polynomials are considered for $H$. For decaying
properties of $\beta_{mn}$ and $c_{ln}$, we refer to (3.6) and (3.7).

The following lemma is useful to derive equivalent coefficients relations for
a complex series to satisfy a boundary condition.

######  Lemma 4.1.

Let $\sum_{n=-\infty}^{\infty}c_{n}w^{n}$ be a complex analytic function in a
neighborhood of $\\{|w|=\gamma\\}$. Then
$\Re\left[\sum_{n=-\infty}^{\infty}c_{n}w^{n}\right]=0$ on $|w|=\gamma$ if and
only if $c_{-n}+\overline{c_{n}}\gamma^{2n}=0$ for each $n\geq 0$.

###### Proof.

Since $\overline{w}=\gamma^{2}w^{-1}$ on $|w|=\gamma$, we have
$2\Re\left[\sum_{n=-\infty}^{\infty}c_{n}w^{n}\right]=\sum_{n=-\infty}^{\infty}\left(c_{n}w^{n}+\overline{c_{n}w^{n}}\right)=\sum_{n=-\infty}^{\infty}\left(c_{-n}+\overline{c_{n}}\gamma^{2n}\right)w^{-n}.$
This proves the lemma. $\Box$

###  4.1 Equivalent relations for the boundary conditions

Let us define semi-infinite matrices

$\displaystyle\boldsymbol{\alpha}$
$\displaystyle=\big{\\{}\alpha_{m}\delta_{mn}\big{\\}}_{m,n\geq
1},\quad\boldsymbol{\beta}=\big{\\{}\beta_{mn}\big{\\}}_{m,n\geq
1},\quad\boldsymbol{s}=\big{\\{}s_{mn}\big{\\}}_{m,n\geq 1},$ (4.2)
$\displaystyle\mathcal{N}$ $\displaystyle=\big{[}n\delta_{mn}\big{]}_{m,n\geq
1},\quad\gamma^{\tau\mathcal{N}}=\big{\\{}\gamma^{\tau
n}\delta_{mn}\big{\\}}_{m,n\geq 1},\quad C=\big{\\{}c_{mn}\big{\\}}_{m,n\geq
1},$

where $\delta_{mn}$ is the Kronecker-Delta function and $\tau\in\mathbb{R}$.

######  Lemma 4.1.

For $u$ given by (4.1), the condition $\frac{\partial
u}{\partial\nu}\left|{}^{+}=\sigma_{c}\frac{\partial
u}{\partial\nu}\right|^{-}$ is equivalent to the relation between the
coefficients:

$\left(\boldsymbol{\alpha}C+\boldsymbol{s}\right)\gamma^{-2\mathcal{N}}=\overline{\boldsymbol{\alpha}}-\sigma\overline{\boldsymbol{\beta}}+\sigma\boldsymbol{\beta}C\gamma^{-2\mathcal{N}}.$
(4.3)

###### Proof.

In view of (3.5), the Neumann boundary condition $\sigma_{m}\frac{\partial
u}{\partial\nu}\big{|}^{+}=\sigma_{c}\frac{\partial
u}{\partial\nu}\big{|}^{-}$ is equivalent to

$\sigma_{m}\frac{\partial}{\partial\rho}u(\Psi(w))\Big{|}_{\rho\rightarrow\rho_{0}^{+}}=\sigma_{c}\frac{\partial}{\partial\rho}u(\Psi(w))\Big{|}_{\rho\rightarrow\rho_{0}^{-}}.$
(4.4)

From (4.1), we obtain

$\displaystyle\frac{\partial}{\partial\rho}u(\Psi(w))\Big{|}_{\rho\rightarrow\rho_{0}^{+}}$
$\displaystyle=\Re\bigg{[}\sum_{m=1}^{\infty}m\alpha_{m}w^{m}-\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}n\left(\alpha_{m}c_{mn}+s_{mn}\right)w^{-n}\bigg{]}\bigg{|}_{\rho=\rho_{0}},$
$\displaystyle\frac{\partial}{\partial\rho}u(\Psi(w))\Big{|}_{\rho\rightarrow\rho_{0}^{-}}$
$\displaystyle=\Re\bigg{[}\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}n\beta_{mn}w^{n}-\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\sum_{l=1}^{\infty}n\beta_{ml}c_{ln}w^{-n}\Big{)}\bigg{]}\bigg{|}_{\rho=\rho_{0}}$

Hence, (4.4) is equivalent to

$\Re\bigg{[}\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}n\Big{(}\sigma_{m}\alpha_{m}c_{mn}+\sigma_{m}s_{mn}-\sum_{l=1}^{\infty}\sigma_{c}\beta_{ml}c_{ln}\Big{)}w^{-n}+\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}n\left(\sigma_{c}\beta_{mn}-\sigma_{m}\alpha_{n}\delta_{mn}\right)w^{n}\bigg{]}=0.$
(4.5)

It follows from Lemma 4.1 that for each $n$,

$\sum_{m=1}^{\infty}\sigma_{m}\left(\alpha_{m}c_{mn}+s_{mn}\right)-\sum_{m=1}^{\infty}\sum_{l=1}^{\infty}\sigma_{c}\beta_{ml}c_{ln}+\sum_{m=1}^{\infty}\overline{\left(\sigma_{c}\beta_{mn}-\sigma_{m}\alpha_{n}\delta_{mn}\right)}\,\gamma^{2n}=0.$
(4.6)

This result completes the proof. $\Box$

######  Corollary 4.2.

The condition $\sigma_{m}\frac{\partial
u}{\partial\nu}\left|{}^{+}=\sigma_{c}\frac{\partial
u}{\partial\nu}\right|^{-}$ is equivalent to

$\displaystyle\boldsymbol{\beta}$
$\displaystyle=\frac{\sigma_{m}}{\sigma_{c}}\boldsymbol{\alpha}-\frac{\sigma_{m}}{\sigma_{c}}\left(\overline{\boldsymbol{s}}+\boldsymbol{s}\gamma^{-2\mathcal{N}}\overline{C}\right)\left(I-\gamma^{-2\mathcal{N}}C\gamma^{-2\mathcal{N}}\overline{C}\right)^{-1}\gamma^{-2\mathcal{N}},$
(4.7)

###### Proof.

Taking the complex conjugate and multiplying $C\gamma^{-2\mathcal{N}}$ to both
sides of (4.3), we obtain

$\sigma_{c}\boldsymbol{\beta}C\gamma^{-2\mathcal{N}}=\sigma_{c}\overline{\boldsymbol{\beta}C}\gamma^{-2\mathcal{N}}C\gamma^{-2\mathcal{N}}+\sigma_{m}\overline{\left(-\boldsymbol{\alpha}C+\overline{\boldsymbol{\alpha}}\gamma^{2\mathcal{N}}-\boldsymbol{s}\right)}\gamma^{-2\mathcal{N}}C\gamma^{-2\mathcal{N}},$

and substitute this formula into (4.3) to get

$\displaystyle\sigma_{c}\overline{\boldsymbol{\beta}}\left(I-\overline{C}\gamma^{-2\mathcal{N}}C\gamma^{-2\mathcal{N}}\right)=\sigma_{m}\overline{\boldsymbol{\alpha}}\left(I-\overline{C}\gamma^{-2\mathcal{N}}C\gamma^{-2\mathcal{N}}\right)-\sigma_{m}\boldsymbol{s}\gamma^{-2\mathcal{N}}-\sigma_{m}\overline{\boldsymbol{s}}\gamma^{-2\mathcal{N}}C\gamma^{-2\mathcal{N}}.$

This result completes the proof with $G$ given by (4.2). For the invertibility
of $I-\gamma^{-2\mathcal{N}}C\gamma^{-2\mathcal{N}}\overline{C}$, we refer the
reader to [10]. $\Box$

Now, we consider the imperfect interface condition,

$\displaystyle
p(\Psi(w))\left(u(\Psi(w))\big{|}_{\rho\rightarrow\rho_{0}^{+}}-u(\Psi(w))\big{|}_{\rho\rightarrow\rho_{0}^{-}}\right)$
$\displaystyle=\sigma_{m}\frac{\partial u}{\partial\nu}\Big{|}_{\partial
D}^{+}=\frac{\sigma_{m}}{h(\rho_{0},\theta)}\frac{\partial}{\partial\rho}u(\Psi(w))\Big{|}_{\rho\rightarrow\rho_{0}^{+}}.$
(4.8)

If holds from (4.1) that

$\displaystyle
u(\Psi(w))\big{|}_{\rho\rightarrow\rho_{0}^{+}}-u(\Psi(w))\big{|}_{\rho\rightarrow\rho_{0}^{-}}=$
$\displaystyle\,\Re\bigg{[}\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}(\alpha_{n}\delta_{mn}-\beta_{mn})w^{n}$
$\displaystyle\quad+\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\Big{(}\alpha_{m}c_{mn}+s_{mn}-\sum_{l=1}^{\infty}\beta_{ml}c_{ln}\Big{)}w^{-n}\bigg{]}$

and

$\displaystyle\frac{\partial}{\partial\rho}u(\Psi(w))\Big{|}_{\rho\rightarrow\rho_{0}^{+}}$
$\displaystyle=\Re\bigg{[}\sum_{m=1}^{\infty}m\alpha_{m}w^{m}-\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}n(\alpha_{m}c_{mn}+s_{mn})w^{-n}\bigg{]}.$

The boundary condition (4.8) is hence equivalent to

$\displaystyle
p(\Psi(w))\Re\bigg{[}\sum_{m=1}^{\infty}\sum_{n=0}^{\infty}(\boldsymbol{\alpha}-\boldsymbol{\beta})_{mn}w^{n}+\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\big{(}\boldsymbol{\alpha}C+\boldsymbol{s}-\boldsymbol{\beta}C\big{)}_{mn}w^{-n}\bigg{]}$
$\displaystyle=$
$\displaystyle\frac{\sigma_{m}}{h(\rho_{0},\theta)}\Re\bigg{[}\sum_{m=1}^{\infty}m\alpha_{m}w^{m}-\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}n\big{(}\boldsymbol{\alpha}C+\boldsymbol{s}\big{)}_{mn}w^{-n}\bigg{]}\quad\mbox{for
}|w|=\gamma.$

Here and in the following, $[\cdot]_{mn}$ denotes the $(m,n)$–element of a
matrix. By using (4.3) and the fact that $h(\rho_{0},\theta)p(w)$ is real-
valued, we obtain

$\displaystyle\Re\bigg{[}h(\rho_{0},\theta)\,p(\Psi(w))\Big{(}\sum_{m=1}^{\infty}\sum_{n=0}^{\infty}x_{mn}w^{n}+\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}y_{mn}w^{-n}\Big{)}\bigg{]}$
(4.9) $\displaystyle=$
$\displaystyle\sigma_{m}\Re\bigg{[}\sum_{m=1}^{\infty}m\alpha_{m}w^{m}-\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}n\big{(}\boldsymbol{\alpha}C+\boldsymbol{s}\big{)}_{mn}w^{-n}\bigg{]}\quad\mbox{for
}|w|=\gamma,$

where

$\displaystyle X$ $\displaystyle=\boldsymbol{\alpha}-\boldsymbol{\beta},\quad
Y=\boldsymbol{\alpha}C+\boldsymbol{s}-\boldsymbol{\beta}C.$ (4.10)

###  4.2 Derivation of matrix formulas for the FPTs

For notational simplicity, we set $p(\theta)=p(\Psi(w))$ for
$w=e^{\rho_{0}+i\theta}.$ We assume that $h(\rho_{0},\theta)\,p(\theta)$
admits a Laurent series expansion:

$h(\rho_{0},\theta)\,p(\theta)=\sum_{n=-\infty}^{\infty}p_{n}w^{n}\quad\mbox{
with }|w|=\gamma,$ (4.11)

where $p_{n}$’s are complex-valued coefficients. In the numerical computation,
we seek for $p$ with a finite number of nonzero coefficients $p_{n}$. Since
$h(\rho_{0},\theta)\,p(\theta)$ is real-valued, we have
$\Re[ih(\rho_{0},\theta)\,p(\theta)]=0$ and, by Lemma 4.1,
$ip_{-n}+\overline{ip_{n}}\gamma^{2n}=0$ for all $n\in\mathbb{Z}$. It follows
that

$p_{-n}=\overline{p_{n}}\,\gamma^{2n}\quad\mbox{for each }n\in\mathbb{Z}.$
(4.12)

By expanding the left hand side of (4.9) and applying (4.9), we derive

$\displaystyle
h(\rho_{0},\theta)\,p(\theta)\Big{(}\sum_{m=1}^{\infty}\sum_{n=0}^{\infty}x_{mn}w^{n}+\sum_{m=1}^{\infty}\sum_{n=0}^{\infty}y_{mn}w^{-n}\Big{)}$
$\displaystyle=\ $
$\displaystyle\sum_{m=1}^{\infty}\Big{(}\sum_{l=-\infty}^{\infty}p_{l}w^{l}\Big{)}\Big{(}\sum_{n=0}^{\infty}x_{mn}w^{n}+\sum_{n=0}^{\infty}y_{mn}w^{-n}\Big{)}$
$\displaystyle=\ $
$\displaystyle\sum_{m=1}^{\infty}\sum_{n=-\infty}^{\infty}\sum_{l=0}^{\infty}\left(p_{n-l}x_{ml}+p_{n+l}y_{ml}\right)w^{n}$

and, for $|w|=\gamma$,

$\displaystyle\Re\bigg{[}\sum_{m=1}^{\infty}\sum_{n=-\infty}^{\infty}\sum_{l=0}^{\infty}\left(p_{n-l}x_{ml}+p_{n+l}y_{ml}\right)w^{n}\bigg{]}=\sigma_{m}\Re\bigg{[}\sum_{m=1}^{\infty}m\alpha_{m}w^{m}-\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}n\big{(}\boldsymbol{\alpha}C+\boldsymbol{s}\big{)}_{mn}w^{-n}\bigg{]}.$

It holds from Lemma 4.1 that, for $n\geq 0$,

$\displaystyle\sum_{l=0}^{\infty}\Big{(}p_{-n-l}x_{ml}+p_{-n+l}y_{ml}+\overline{(p_{n-l}x_{ml}+p_{n+l}y_{ml})}\gamma^{2n}\Big{)}$
$\displaystyle=\sigma_{m}\left[(\overline{\boldsymbol{\alpha}}\gamma^{2\mathbb{N}}-\boldsymbol{\alpha}C-\boldsymbol{s})\mathcal{N}\right]_{mn},$
(4.13)

where $\mathcal{N}$ is given by (4.2), that is, the diagonal matrix with
$(n,n)$-element $n$. Using (4.12) and (4.13), we obtain that for $m,n\geq 0$,

$\displaystyle\sigma_{m}\big{[}\left(\overline{\boldsymbol{\alpha}}\gamma^{2\mathcal{N}}-\boldsymbol{\alpha}C-\boldsymbol{s}\right)\mathcal{N}\big{]}_{mn}$
$\displaystyle=\sum_{l=0}^{\infty}\Big{(}x_{ml}\gamma^{2l}\overline{p_{l+n}}\gamma^{2n}+\overline{y_{ml}p_{l+n}}\gamma^{2n}+\overline{x_{ml}}\gamma^{2l}p_{l-n}+y_{ml}p_{l-n}\Big{)}$
$\displaystyle=\sum_{l=0}^{\infty}\left[\overline{(\overline{x_{ml}}\gamma^{2l}+y_{ml})}\overline{p_{l+n}}\gamma^{2n}+(\overline{x_{ml}}\gamma^{2l}+y_{ml})p_{l-n}\right],$
(4.14)

We set

$\displaystyle P^{+}$ $\displaystyle=\big{\\{}p_{mn}^{+}\big{\\}}_{m,n\geq
1},\quad p_{mn}^{+}=p_{m+n}\gamma^{m+n},$ (4.15) $\displaystyle P^{-}$
$\displaystyle=\big{\\{}p_{mn}^{-}\big{\\}}_{m,n\geq 1},\quad
p_{mn}^{-}=p_{m-n}\gamma^{m-n}.$

From (4.12), we have
$p_{nm}^{-}=p_{n-m}\gamma^{n-m}=\overline{p_{m-n}}\gamma^{m-n}=\overline{p_{mn}^{-}}$.
In particular, $P^{+}$ is a Hankel matrix and $P^{-}$ is a Hermitian Toeplitz
matrix (see Figure 4.1).

$\left[\begin{array}[]{cccccc}p_{2}\gamma^{2}&p_{3}\gamma^{3}&p_{4}\gamma^{4}&p_{5}\gamma^{5}&p_{6}\gamma^{6}&\cdots\\\\[8.53581pt]
p_{3}\gamma^{3}&p_{4}\gamma^{4}&p_{5}\gamma^{5}&p_{6}\gamma^{6}&p_{7}\gamma^{7}&\cdots\\\\[8.53581pt]
p_{4}\gamma^{4}&p_{5}\gamma^{5}&p_{6}\gamma^{6}&p_{7}\gamma^{7}&p_{8}\gamma^{8}&\cdots\\\\[8.53581pt]
p_{5}\gamma^{5}&p_{6}\gamma^{6}&p_{7}\gamma^{7}&p_{8}\gamma^{8}&p_{9}\gamma^{9}&\cdots\\\\[8.53581pt]
p_{6}\gamma^{6}&p_{7}\gamma^{7}&p_{8}\gamma^{8}&p_{9}\gamma^{9}&p_{10}\gamma^{10}&\cdots\\\\[8.53581pt]
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right]$

$\left[\begin{array}[]{cccccc}p_{0}&\overline{p_{1}}\gamma^{1}&\overline{p_{2}}\gamma^{2}&\overline{p_{3}}\gamma^{3}&\overline{p_{4}}\gamma^{4}&\cdots\\\\[8.53581pt]
p_{1}\gamma^{1}&p_{0}&\overline{p_{1}}\gamma^{1}&\overline{p_{2}}\gamma^{2}&\overline{p_{3}}\gamma^{3}&\cdots\\\\[8.53581pt]
p_{2}\gamma^{2}&p_{1}\gamma^{1}&p_{0}&\overline{p_{1}}\gamma^{1}&\overline{p_{2}}\gamma^{2}&\cdots\\\\[8.53581pt]
p_{3}\gamma^{3}&p_{2}\gamma^{2}&p_{1}\gamma^{1}&p_{0}&\overline{p_{1}}\gamma^{1}&\cdots\\\\[8.53581pt]
p_{4}\gamma^{4}&p_{3}\gamma^{3}&p_{2}\gamma^{2}&p_{1}\gamma^{1}&p_{0}&\cdots\\\\[8.53581pt]
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right]$

Figure 4.1: Hankel matrix $P^{+}$ (left) and Hermitian Toeplitz matrix
$P^{-}$(right)

We express (4.14) in a matrix form as

$\displaystyle\sigma_{m}\left(\overline{\boldsymbol{\alpha}}\gamma^{2\mathcal{N}}-\boldsymbol{\alpha}C-\boldsymbol{s}\right)\mathcal{N}=\overline{\left(\overline{X}\gamma^{2\mathcal{N}}+Y\right)}\gamma^{-\mathcal{N}}\overline{P^{+}}\gamma^{\mathcal{N}}+\left(\overline{X}\gamma^{2\mathcal{N}}+Y\right)\gamma^{-\mathcal{N}}P^{-}\gamma^{\mathcal{N}}.$
(4.16)

The relations (4.7) and (4.10) imply

$\displaystyle\sigma_{c}\left(\overline{X}\gamma^{2\mathcal{N}}+Y\right)$
$\displaystyle=\sigma_{c}\left(\overline{\boldsymbol{\alpha}}\gamma^{2\mathcal{N}}+\boldsymbol{\alpha}C-\overline{\boldsymbol{\beta}}\gamma^{2\mathcal{N}}-\boldsymbol{\beta}C+\boldsymbol{s}\right)$
$\displaystyle=\overline{\boldsymbol{\alpha}}(\sigma_{c}-\sigma_{m})\gamma^{2\mathcal{N}}+\boldsymbol{\alpha}(\sigma_{c}-\sigma_{m})C$
$\displaystyle\quad+\boldsymbol{s}\left[(\sigma_{c}-\sigma_{m})I+2\sigma_{m}(I-\gamma^{-2\mathcal{N}}\overline{C}\gamma^{-2\mathcal{N}}C)^{-1}\right]+2\sigma_{m}\overline{\boldsymbol{s}}\left[(I-\gamma^{-2\mathcal{N}}C\gamma^{-2\mathcal{N}}\overline{C})^{-1}\gamma^{-2\mathcal{N}}C\right].$
(4.17)

Solving (4.16) and (4.17), we obtain

$\displaystyle\boldsymbol{\alpha}A_{1}+\overline{\boldsymbol{\alpha}}\overline{A_{2}}+\overline{\boldsymbol{s}}\overline{B_{1}}+\boldsymbol{s}B_{2}=O,$
(4.18)

where $O$ means the zero matrix and

$\begin{cases}\displaystyle
A_{1}&=(\sigma_{c}-\sigma_{m})\gamma^{\mathcal{N}}\,\overline{P^{+}}\,\gamma^{\mathcal{N}}+(\sigma_{c}-\sigma_{m})C\gamma^{-\mathcal{N}}P^{-}\gamma^{\mathcal{N}}+\sigma_{c}\sigma_{m}C\mathcal{N},\\\\[5.69054pt]
\displaystyle
A_{2}&=(\sigma_{c}-\sigma_{m})\gamma^{\mathcal{N}}\,\overline{P^{-}}\,\gamma^{\mathcal{N}}+(\sigma_{c}-\sigma_{m})C\gamma^{-\mathcal{N}}P^{+}\gamma^{\mathcal{N}}-\sigma_{c}\sigma_{m}\gamma^{2\mathcal{N}}\mathcal{N},\\\\[5.69054pt]
\displaystyle
B_{1}&=\left[(\sigma_{c}-\sigma_{m})I+2\sigma_{m}(I-\gamma^{-2\mathcal{N}}\overline{C}\gamma^{-2\mathcal{N}}C)^{-1}\right]\gamma^{-\mathcal{N}}P^{+}\gamma^{\mathcal{N}}\\\
&\quad+2\sigma_{m}(I-\gamma^{-2\mathcal{N}}\overline{C}\gamma^{-2\mathcal{N}}C)^{-1}\,\gamma^{-2\mathcal{N}}\overline{C}\,\gamma^{-\mathcal{N}}\,\overline{P^{-}}\,\gamma^{\mathcal{N}},\\\\[5.69054pt]
\displaystyle
B_{2}&=\left[(\sigma_{c}-\sigma_{m})I+2\sigma_{m}(I-\gamma^{-2\mathcal{N}}\overline{C}\gamma^{-2\mathcal{N}}C)^{-1}\right]\gamma^{-\mathcal{N}}P^{-}\gamma^{\mathcal{N}}\\\
&\quad+2\sigma_{m}(I-\gamma^{-2\mathcal{N}}\overline{C}\gamma^{-2\mathcal{N}}C)^{-1}\,\gamma^{-2\mathcal{N}}\overline{C}\,\gamma^{-\mathcal{N}}\,\overline{P^{+}}\,\gamma^{\mathcal{N}}+\sigma_{c}\sigma_{m}\,\mathcal{N}.\end{cases}$
(4.19)

If $B_{2}$ is invertible, (4.18) leads us to (by taking the complex conjugate)

$\overline{\boldsymbol{s}}=-\overline{\boldsymbol{\alpha}}\,\overline{A_{1}}\,\overline{B_{2}}^{-1}-\boldsymbol{\alpha}A_{2}\overline{B_{2}}^{-1}-\boldsymbol{s}B_{1}\overline{B_{2}}^{-1}.$

Assuming further the invertibility for $B_{2}-B_{1}B_{2}^{-1}\overline{B_{1}}$
and substituting the above relation into (4.18), we finally arrive to

$\displaystyle\boldsymbol{s}=$
$\displaystyle-\boldsymbol{\alpha}\left[\left(A_{1}-A_{2}\overline{B_{2}}^{-1}\overline{B_{1}}\right)\left(B_{2}-B_{1}\overline{B_{2}}^{-1}\overline{B_{1}}\right)^{-1}\right]$
(4.20)
$\displaystyle-\overline{\boldsymbol{\alpha}}\left[\left(\overline{A_{2}}-\overline{A_{1}}\,\overline{B_{2}}^{-1}\overline{B_{1}}\right)\left(B_{2}-B_{1}\overline{B_{2}}^{-1}\overline{B_{1}}\right)^{-1}\right].$

Recall the FPTs $\mathbb{F}_{mn}^{(1)}$ and $\mathbb{F}_{mn}^{(2)}$ are
defined by (3.11). From (4.20), we obtain the matrix expressions for the FPTs.

######  Lemma 4.1.

Let $A_{1},A_{2},B_{1},B_{2}$ are given by (4.19) (see also (4.2) and (4.15)).
Then, FPTs of the interface problem (2.1) have matrix formulations,

$\displaystyle\mathbb{F}^{(1)}=4\pi\left(A_{1}-A_{2}\overline{B_{2}}^{-1}\overline{B_{1}}\right)\left(B_{2}-B_{1}\overline{B_{2}}^{-1}\overline{B_{1}}\right)^{-1}\mathcal{N},$
$\displaystyle\mathbb{F}^{(2)}=4\pi\left(\overline{A_{2}}-\overline{A_{1}}\,\overline{B_{2}}^{-1}\overline{B_{1}}\right)\left(B_{2}-B_{1}\overline{B_{2}}^{-1}\overline{B_{1}}\right)^{-1}\mathcal{N},$

if $B_{2}$ and $B_{2}-B_{1}B_{2}^{-1}\overline{B_{1}}$ are invertible.

For the numerical computation in Section 5, we use the finite truncations of
the matrices $A_{1},A_{2},B_{1},B_{2}$ with a large matrix dimension. For all
examples in Section 5, the finite truncations of $B_{2}$ and
$B_{2}-B_{1}B_{2}^{-1}\overline{B_{1}}$ are invertible.

As a corollary of Lemma 4.1 and the formula (3.18), we explicitly obtain
decompositions of the CGPTs as follows.

######  Theorem 4.2.

The CGPTs associated with the imperfect interface problem (2.1) admit the
matrix decompositions

$\displaystyle\mathbb{N}^{(1)}$
$\displaystyle=4\pi\,Q^{-1}\left(A_{1}-A_{2}\overline{B_{2}}^{-1}\overline{B_{1}}\right)\left(B_{2}-B_{1}\overline{B_{2}}^{-1}\overline{B_{1}}\right)^{-1}\mathcal{N}\left(Q^{-1}\right)^{T},$
$\displaystyle\mathbb{N}^{(2)}$
$\displaystyle=4\pi\,\overline{Q^{-1}}\left(\overline{A_{2}}-\overline{A_{1}}\,\overline{B_{2}}^{-1}\overline{B_{1}}\right)\left(B_{2}-B_{1}\overline{B_{2}}^{-1}\overline{B_{1}}\right)^{-1}\mathcal{N}\left(Q^{-1}\right)^{T},$

where $Q=(q_{mn})_{m,n=1}^{\infty}$ is the lower triangular matrix of Faber
polynomial coefficients in (3.13).

###  4.3 GPTs for a disk

If $D$ is a disk with the radius $\gamma$, then $C$ is identically the zero
matrix and $Q$ is the identity matrix. Consequently, (4.19) leads

$\begin{cases}\displaystyle
A_{1}&=(\sigma_{c}-\sigma_{m})\gamma^{\mathcal{N}}\,\overline{P^{+}}\,\gamma^{\mathcal{N}},\\\\[5.69054pt]
\displaystyle
A_{2}&=(\sigma_{c}-\sigma_{m})\gamma^{\mathcal{N}}\,\overline{P^{-}}\,\gamma^{\mathcal{N}}-\sigma_{c}\sigma_{m}\gamma^{2\mathcal{N}}\mathcal{N},\\\\[5.69054pt]
\displaystyle
B_{1}&=(\sigma_{c}-\sigma_{m})\gamma^{-\mathcal{N}}P^{+}\gamma^{\mathcal{N}},\\\\[5.69054pt]
\displaystyle
B_{2}&=(\sigma_{c}-\sigma_{m})\gamma^{-\mathcal{N}}P^{-}\gamma^{\mathcal{N}}+\sigma_{c}\sigma_{m}\,\mathcal{N},\end{cases}$

which gives $A_{1}=\gamma^{2\mathcal{N}}\overline{B_{1}}$ and
$A_{2}=\gamma^{2\mathcal{N}}\overline{B_{2}}-2\sigma_{c}\sigma_{m}\gamma^{2\mathcal{N}}\mathcal{N}$.
It then follows from Theorem 4.2 that

$\displaystyle\mathbb{N}_{mn}^{(1)}=8\pi
mn\sigma_{c}\sigma_{m}\gamma^{2m}\left[\overline{B_{2}}^{-1}\overline{B_{1}}\left(B_{2}-B_{1}\overline{B_{2}}^{-1}\overline{B_{1}}\right)^{-1}\right]_{mn},$
$\displaystyle\mathbb{N}_{mn}^{(2)}=4\pi n\gamma^{2m}\delta_{mn}-8\pi
mn\sigma_{c}\sigma_{m}\gamma^{2m}\left[\left(B_{2}-B_{1}\overline{B_{2}}^{-1}\overline{B_{1}}\right)^{-1}\right]_{mn}.$

If we set $p_{n}=0$ for all $n\neq 0$, then $P^{+}=O$ and $P^{-}=p_{0}I$, and
thus,

$B_{1}=O,\quad
B_{2}=(\sigma_{c}-\sigma_{m})p_{0}I+\sigma_{c}\sigma_{m}\mathcal{N}.$

We then get the following corollary.

######  Corollary 4.1.

For a disk $D$, if $p_{n}=0$ for all $n\neq 0$, namely, if the interface
function is constant, then GPTs of the interface problem (2.1) are

$\displaystyle\mathbb{N}_{mn}^{(1)}=0,\quad\mathbb{N}_{mn}^{(2)}=4\pi
n\gamma^{2m}\delta_{mn}\frac{(\sigma_{c}-\sigma_{m})p_{0}-\sigma_{c}\sigma_{m}m}{(\sigma_{c}-\sigma_{m})p_{0}+\sigma_{c}\sigma_{m}m}.$

By this corollary, $\mathbb{N}_{1n}^{(1)}=\mathbb{N}_{1n}^{(2)}=0$ for all
$n\geq 1$ if

$p_{0}=\frac{\sigma_{c}\sigma_{m}}{\sigma_{c}-\sigma_{m}}.$

In other words, by using (4.11) and $h(\rho_{0},\theta)=\gamma$ for the disk
$\Omega$, if we set $p$ as

$p=\frac{1}{\gamma}\frac{\sigma_{c}\sigma_{m}}{\sigma_{c}-\sigma_{m}}$

the inclusion is perfectly neutral and does not perturb the incident field at
all when the incident field is unform [41].

##  5 GPT-vanishing inclusions with imperfect interfaces

The computation in this section is very similiar with the numerical part in
[11]. However, main difference is that we iterate the coefficients of the
interface function instead of conductivites of inclusions. We utilize FPTs to
construct imperfect interfaces that vanish GPTs. Specifically, we determine
the interface condition function $p$ for a given pair $(D,\sigma_{c})$ such
that FPTs vanish up to a finite order. Then, GPTs would also vanish up to the
same finite order.

###  5.1 Numerical scheme

For a given $D$, we use the quantities $\gamma$, $\Psi(w)$, and
$(\rho,\theta)$ as defined in Section 3. The interface function, $p$, and its
coefficient matrices, $P^{+}$ and $P^{-}$, are defined by (4.11), (4.15), and
Figure 4.1. Let us assume that $\sigma_{m}=1$ and $\sigma_{c}$ is fixed. The
incident field is assumed to be a uniform field, which means that we only need
to consider FPTs for $m=1$.

Our objective is to find $\boldsymbol{p}=(p_{0},\dots,p_{N-1})$ such that the
leading terms of the FPTs vanish. To accomplish this, we aim to solve the
following equation:

$\boldsymbol{f}(\boldsymbol{p})=\boldsymbol{0},$ (5.1)

where $\boldsymbol{f}:\mathbb{R}^{N}\to\mathbb{R}^{2N}$ is a nonlinear vector
function mapping $\boldsymbol{p}\longmapsto(f_{1},...,f_{2N})$. Here, $f_{n}$
is given by:

$\displaystyle f_{n}=\begin{cases}&\left|\dfrac{\mathbb{F}^{(1)}_{1n}}{4\pi
n}\right|,\quad 1\leq n\leq N,\\\\[14.22636pt]
&\left|\dfrac{\mathbb{F}^{(2)}_{1n}}{4\pi n}\right|,\quad N+1\leq n\leq
2N,\end{cases}$

In order to achieve $\boldsymbol{p}$ with (5.1), we iterate the
multidimensional Newton method:

$\displaystyle\boldsymbol{p}^{(k+1)}=\boldsymbol{p}^{(k)}-\alpha\,\boldsymbol{J}^{\dagger}\big{[}\boldsymbol{p}^{(k)}\big{]}\,\boldsymbol{f}\big{[}\boldsymbol{p}^{(k)}\big{]},$
(5.2)

where the learning rate is denoted by $\alpha$, while $k$ represents the
iteration number. Additionally, $\boldsymbol{J}^{\dagger}$ refers to the
pseudo-inverse of the Jacobian matrix of $\boldsymbol{f}$. We start with seed
input $\boldsymbol{p}^{(0)}=(p_{0}^{(0)},\dots,p_{N-1}^{(0)})$, with each
coefficient having a magnitude less than 2.

We use Theorem 4.1 to calculate $\boldsymbol{f}(\boldsymbol{p}^{(k)})$,
truncating all semi-infinite matrices to finite matrices with size $100\times
100$. To compute $\boldsymbol{J}^{\dagger}[\boldsymbol{p}^{(k)}]$, we apply
the finite difference method to the partial derivatives of $\boldsymbol{f}$
based on Matlab R2022b. We continue applying the multidimensional Newton
method until the coefficients of the interface function meet the stopping
criterion:

$\frac{|\boldsymbol{p}^{(k+1)}-\boldsymbol{p}^{(k)}|}{|\boldsymbol{p}^{(k)}|}<10^{-15}.$

###  5.2 Examples

We visualize our results on the construction of the GPT-vanishing structures.
We plot the exterior potential in (3.10) induced by the presence of inclusion
based on (3.11) and Theorem 4.1.

We give three different boundary conditions that are perfectly bonding as
(5.3) and imperfectly bonding as (5.4) and (5.5). For perfectly bonding case,
we set $p=\infty$, and hence, the boundary condition

$\sigma_{m}\frac{\partial u}{\partial\nu}\Big{|}^{+}=\sigma_{c}\frac{\partial
u}{\partial\nu}\Big{|}^{-}=p(u\big{|}^{+}-u\big{|}^{-})\qquad\mbox{on
}\partial D$

is equivalent to

$u\big{|}^{+}=u\big{|}^{-},\quad\sigma_{m}\frac{\partial
u}{\partial\nu}\Big{|}^{+}=\sigma_{c}\frac{\partial
u}{\partial\nu}\Big{|}^{-}\qquad\mbox{on }\partial D,$ (5.3)

which are the typical transmission conditions.

For GPTs of order 1 vanishing case, namely, PT-vanishing structure, we give

$P(w)=p_{0}+p_{2}w^{2}+\overline{p_{2}}\gamma^{4}w^{-2}.$ (5.4)

Since $p_{0}$ and $p_{2}$ are free variables, we have the degree of freedom as
3. This is enough to vanish two quantities and we could make the weakly
neutral inclusion satisfying
$\mathbb{N}_{11}^{(1)}=\mathbb{N}_{11}^{(2)}\approx 0$.

For GPTs of order up to 2 vanishing structure, we give

$P(w)=p_{0}+p_{1}w+\overline{p_{1}}\gamma^{2}w^{-1}+p_{2}w^{2}+\overline{p_{2}}\gamma^{4}w^{-2}+p_{3}w^{3}+\overline{p_{3}}\gamma^{6}w^{-3}.$
(5.5)

Now we have four free variables, which enable us to control four quantities
that vanish. Thus, provided that $P$ is given as (5.5), we can make the GPTs
of order up to 2 vanishing structures satisfying
$\mathbb{N}_{11}^{(1)}=\mathbb{N}_{11}^{(2)}=\mathbb{N}_{12}^{(1)}=\mathbb{N}_{12}^{(2)}\approx
0$. This is quite surprising because the solution $u$ in (2.1) could decay
faster than the weakly neutral inclusion as follows:

$u(x)=a\cdot x+O(|x|^{-3}),\quad|x|\to\infty.$

######  Example 1.

We first consider a domain with conformal mapping given by

$\Psi(w)=w+\frac{0.25}{w}+\frac{0.125}{w^{2}}+\frac{0.1}{w^{3}}.$

To construct the PT-vanishing structure, we set $P$ as defined in (5.4), where
$p_{0}=1.5925$ and $p_{2}=-0.7240$. For the GPTs of order up to 2 vanishing
structure, we define $P$ as in (5.5), with $p_{0}=1.7214$, $p_{1}=0.1933$,
$p_{2}=-0.7480$, and $p_{3}=-0.5522$. The field perturbation among three
different boundary conditions is shown in Figure 5.1.

(a) Perfectly bonding

(b) PT-vanishing

(c) GPT-vanishing up to order 2

Figure 5.1: The background field is given by $H(x)=x_{1}$. Colored curves represent contours of the potential $u$. We set $\gamma=1$ and the conductivity of the inclusion is given by $\sigma_{c}=5$. | $\mathbb{N}_{11}^{(1)}$ | $\mathbb{N}_{12}^{(1)}$ | $\mathbb{N}_{11}^{(2)}$ | $\mathbb{N}_{12}^{(2)}$
---|---|---|---|---
Perfectly bonding | 1.1693 | 1.0918 | 7.7190 | $-0.7977$
1st order vanishing | $-2.1799\times 10^{-16}$ | 2.6301 | $3.8367\times 10^{-15}$ | $-0.1007$
2nd order vanishing | $-3.3353\times 10^{-15}$ | $1.7439\times 10^{-16}$ | $2.6159\times 10^{-16}$ | $2.2846\times 10^{-14}$
Table 1: The leading GPTs for the domains in Figure 5.1.

######  Example 2.

We now observe a kite-shaped domain with conformal mapping

$\Psi(w)=w+\frac{0.1}{w}+\frac{0.25}{w^{2}}-\frac{0.05}{w^{3}}+\frac{0.05}{w^{4}}-\frac{0.04}{w^{5}}+\frac{0.02}{w^{6}}.$

To construct the PT-vanishing structure, we selected values for $P$ based on
equation (5.4), where $p_{0}$ is set to $1.4022$ and $p_{2}$ is set to
$-0.2742$. For GPTs up to order 2 vanishing structure, we determined the
values for $P$ using equation (5.5), with $p_{0}$ set to $1.7975$, $p_{1}$ set
to $0.1582$, $p_{2}$ set to $-0.3251$, and $p_{3}$ set to $-1.0352$. Figure
5.2 illustrates the field perturbation under three different boundary
conditions.

(a) Perfectly bonding

(b) PT-vanishing

(c) GPT-vanishing up to order 2

Figure 5.2: The background field is given by $H(x)=x_{1}$ (first row) and $H(x)=x_{2}$ (second row). Colored curves represent contours of the potential $u$. We set $\gamma=1$ and $\sigma_{c}=100$. | $\mathbb{N}_{11}^{(1)}$ | $\mathbb{N}_{12}^{(1)}$ | $\mathbb{N}_{11}^{(2)}$ | $\mathbb{N}_{12}^{(2)}$
---|---|---|---|---
Perfectly bonding | 1.2018 | 5.9817 | 12.2216 | $0.0192$
1st order vanishing | $2.0709\times 10^{-16}$ | 6.2342 | $5.3517\times 10^{-15}$ | $0.0085$
2nd order vanishing | $-1.6295\times 10^{-15}$ | $-6.7032\times 10^{-15}$ | $9.2429\times 10^{-15}$ | $-1.1772\times 10^{-15}$
Table 2: The leading GPTs for the domains in Figure 5.2.

##  6 Conclusion

We propose a novel concept of GPT-vanishing structures for inclusions with
arbitrary conductivity and imperfect interfaces. We show that inclusions of
general shape can achieve nearly neutral behavior for the background field by
carefully selecting a proper imperfect interface parameter. We employ exterior
conformal mappings and FPT techniques to compute the coefficients of the
interface parameter. Furthermore, we provide two distinct examples that
highlight the effect of the vanishing FPTs, demonstrating the practical
implications of our proposed concept.

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