1 Introduction

We consider the field perturbation due to the presence of an elastic or electrical inclusion in a homogeneous background \(\mathbb {R}^2\). An elastic or electrical inclusion with different material parameters from that of the background induces a perturbation in the applied background field. This boundary integral formulation leads us to the classical multipole expansion for the field perturbation, whose coefficients are the so-called generalized polarization tensors (GPTs) [4, 5]. The classical multipole expansion holds in a region away from the inclusion, but generally does not near the boundary of the inclusion. Consequently, it cannot be employed to find the solution to the transmission problem; it provides a solution only when the inclusion is a circular or spherical domain. For a small inclusion, asymptotic expansions of electric and magnetic fields that are valid uniformly in space have been studied by many authors. We refer to [9, 59], where the asymptotic expansions were derived for scattering problems with multiple small inclusions embedded in a smooth domain. The leading-order term involves the polarization tensors that were introduced by Schiffer and Szegö [52]. In [8], high-order asymptotic expansions of the electric and magnetic fields, which are uniform in space, were derived by using the method of matched asymptotic expansions. However, it is difficult to extend this approach to higher orders.

Complex analysis techniques have been used to study various inclusion problems in two dimensions [3, 12, 17, 33]. For any simply connected region, there uniquely exists an exterior conformal mapping. The Faber polynomials are then defined depending on the exterior conformal mapping [21], where they form a basis for analytic functions in the region [18]. Recently, Jung and Lim [33] obtained series expansions of the layer potential operators based on geometric function theory; we refer the reader to [16, 32] for their applications.

In this paper, to overcome the limitation of the classical multipole expansion that it does not hold near the inclusion for an inclusion of general shape, we propose a new concept of geometric multipole expansion for a simply connected or multi-coated inclusion of general shape by using the exterior conformal mapping and the Faber polynomials associated with the inclusion. Multi-coated inclusions are assumed to be the images of concentric disks via the conformal mapping. Expansion coefficients are defined related to the Faber polynomials and are therefore named the Faber polynomial polarization tensors (FPTs), so that the proposed expansion is of simple form. The FPTs, which coincide with the GPTs for the circular inclusion case, are linear combinations of the GPTs with weights determined by the Faber polynomials. Unlike the classical multipole expansion, the geometric multipole expansion holds in the whole exterior of the inclusion, and, consequently, one can solve the transmission problem using this expansion. We further provide a matrix expression for the FPTs in terms of the material parameters and the exterior conformal mapping of the inclusion; hence, one can reformulate the transmission problem as a semi-infinite matrices problem. In particular, for the simply connected inclusion with extreme conductivity, the FPTs admit simple formulas in terms of the conformal mapping associated with the inclusion.

Many important phenomena in composites such as plasmon resonance, neutral inclusion and invisibility cloaking for multilayer circular and spherical inclusions were analyzed based on the series solution related to the classical multipole expansion. An investigation of the conductivity interface problem with a multi-coated inclusion of general shape based on the geometric series expansion and the FPTs is now possible, which may lead to a new understanding of composites. For instance, the authors of the present paper successfully applied the concept of the FPTs for analytical shape recovery of a simply connected conductivity inclusion [15].

Coated disks and spheres are well-known examples of neutral inclusions, that is, structures not disturbing the applied uniform field [25,26,27, 30, 47, 48]. Appropriately coated ellipses and ellipsoids, possibly with anisotropic conductivity, are neutral to all uniform exterior fields [23, 39, 45, 53, 54], and they are the only shapes for which coated inclusions have the uniform field property [35, 36, 46]. The idea of a neutral inclusion has been widely studied for invisible cloaking using metamaterials. For instance, Zhou et al. designed multi-coated spheres that are invisible to acoustic, elastic, and electromagnetic waves [61,62,63]. After that, Landy and Smith [42] experimentally characterized the neutral inclusions with microwaves. For the case of Maxwell’s equations, Liu et al. [44] proposed nearly non-scattering wave sets. Alù and Engheta [2] and Ammari et al. [7] also constructed multi-coated neutral inclusions. The GPT-vanishing structures are concentric disks or balls whose values of the leading-order GPTs are negligible [6, 60]. One can interpret them as multi-coated neutral inclusions. It is worth remarking that inclusions of general shape that cancel the first-order GPTs have been constructed [22, 37].

As an application of the FPTs, for a given core of general shape, we construct multi-coated inclusions that show relatively negligible field perturbations for low-order polynomial loadings. We call such a structure a semi-neutral inclusion. The coating layers of this inclusion are images of concentric circles via the exterior conformal mapping of the core. The FPTs can be divided into two groups \(\mathbb {F}^{(1)}\) and \(\mathbb {F}^{(2)}\) (see Theorem 4.2 in Sect. 4). The first mainly depends on the shape of the inclusion, and the second depends more on the material parameters. For concentric disks, \(\mathbb {F}^{(1)}=0\) due to the symmetry of the shape [6]. Hence, the GPT-vanishing structures obtained in [6] are in fact the \(\mathbb {F}^{(2)}\)-vanishing structures of concentric multi-coated disks. In general, \(\mathbb {F}^{(2)}\) shows a larger magnitude compared to \(\mathbb {F}^{(1)}\), and \(\mathbb {F}^{(2)}\) significantly contributes to the field perturbation. We numerically find semi-neutral inclusions such that \(\mathbb {F}^{(2)}\)-terms of leading orders vanish by a simple optimization procedure.

The paper is organized as follows. In Sect. 2, we review the classical multipole expansion and outline the Faber polynomials and the Grunsky coefficients. In Sect. 3, we define the geometric multipole expansion and the FPTs. We then derive the matrix expressions of the FPTs for a simply connected domain in Sect. 4. Section 5 is devoted to formulating the conductivity interface problem with a multi-coated inclusion based on the geometric series expansion and to deriving matrix formulas of the FPTs for a multi-coated structure. By using this formula, we then construct semi-neutral inclusions and show numerical examples in Sect. 6. The paper ends with a conclusion in Sect. 7.

2 Preliminary

2.1 Classical multipole expansion

Let D be a bounded and simply connected domain in \(\mathbb {R}^2\) with Lipschitz boundary. We assume that D has the constant conductivity \(\sigma _0>0\) and is embedded in the background with the constant conductivity \(\sigma _m\). For simplicity, we assume \(\sigma _m=1\). We consider the resulting conductivity (or anti-plane elasticity) transmission problem in two dimensions:

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \nabla \cdot \sigma \nabla u=0\quad &{}\text{ in } \mathbb {R}^2, \\ \displaystyle u(x) - H(x) =O({|x|^{-1}})\quad &{}\text{ as } |x| \rightarrow \infty \end{array}\right. } \end{aligned}$$
(2.1)

with the conductivity distribution given by \(\sigma = \sigma _0\, \chi (D)+\chi (\mathbb {R}^2\setminus \overline{D})\) and an entire harmonic function H. Here, \(\chi \) indicates the characteristic function. It holds that

$$\begin{aligned} u\big |^+=u\big |^- ,\quad \frac{\partial u}{\partial \nu }\Big |^+= \sigma _0 \frac{\partial u}{\partial \nu }\Big |^-\quad \text{ on } \partial D. \end{aligned}$$
(2.2)

The symbols \(+\) and − indicate the limits from the exterior and interior of \(\partial D\), respectively.

For \(\varphi \in L^2(\partial D)\), we define

$$\begin{aligned} \displaystyle&\mathcal {S}_{\partial D}[\varphi ](x)=\int \limits _{\partial D}\Gamma (x-\tilde{x})\varphi (\tilde{x})\, d\sigma (\tilde{x}),\quad x\in \mathbb {R}^d,\\ \displaystyle&\mathcal {K}_{\partial D}^*[\varphi ](x)=p.v.\,\frac{1}{2\pi }\int \limits _{\partial D}\frac{\left\langle x-\tilde{x},\nu _x\right\rangle }{|x-\tilde{x}|^2}\varphi (\tilde{x})\, d\sigma (\tilde{x}),\quad x\in \partial D, \end{aligned}$$

where \(\Gamma (x)\) is the fundamental solution to the Laplacian, i.e., \(\Gamma (x)=\frac{1}{2\pi }\ln |x|\), p.v. stands for the Cauchy principal value, and \(\nu _x\) is the outward unit normal vector to \(\partial D\) at x. We call \(\mathcal {S}_{\partial D}[\varphi ]\) and \(\mathcal {K}^*_{\partial D}\) the single-layer potential and the Neumann–Poincaré (NP) operator, respectively. On \(\partial D\), the following jump relation holds:

$$\begin{aligned} \begin{aligned} \mathcal {S}_{\partial D}[\varphi ]\big |^{+}&=\mathcal {S}_{\partial D}[\varphi ]\big |^{-},\\ \frac{\partial }{\partial \nu }\mathcal {S}_{\partial D}[\varphi ]\Big |^{\pm }&=\left( \pm \frac{1}{2}I+\mathcal {K}^*_{\partial D}\right) [\varphi ]. \end{aligned} \end{aligned}$$
(2.3)

The \(L^2\) adjoint of \(\mathcal {K}^*_{\partial D}\) is

$$\begin{aligned} \displaystyle \mathcal {K}_{\partial D}[\varphi ](x)=p.v.\,\frac{1}{2\pi }\int \limits _{\partial D}\frac{\left\langle \tilde{x}-{x},\nu _{\tilde{x}}\right\rangle }{|x-\tilde{x}|^2}\varphi (\tilde{x})\, d\sigma (\tilde{x}),\quad x\in \partial D. \end{aligned}$$

We also call \(\mathcal {K}_{\partial D}\) the NP operator by an abuse of terminology. The operator \(\mathcal {K}^*_{\partial D}\) can be extended to act on the Sobolev space \(H^{-1/2}(\partial D)\) by using its \(L^2\) pairing with \(\mathcal {K}_{\partial D}\). We identify \(x=(x_1,x_2)\) in \(\mathbb {R}^2\) with the complex variable \(z=x_1+ix_2\) in \(\mathbb {C}\). We denote \(\mathcal {S}_{\partial D}[\varphi ](z):=\mathcal {S}_{\partial D}[\varphi ](x)\) and similarly for the NP operators.

From (2.2) and (2.3), the solution u to (2.1) admits the single-layer potential ansatz:

$$\begin{aligned} u(x)=H(x)+\mathcal {S}_{\partial D}[\varphi ](x),\quad x\in \mathbb {R}^2, \end{aligned}$$
(2.4)

where

$$\begin{aligned} \varphi =(\lambda I-\mathcal {K}_{\partial D}^*)^{-1}\left[ \nu \cdot \nabla H\right] \quad \text{ with } \lambda = \frac{\sigma _0+1}{2(\sigma _0-1)}. \end{aligned}$$
(2.5)

The operator \(\lambda I-\mathcal {K}_{\partial D}^*\) is invertible on \(L^2_0(\partial D)\) (or \(H_0^{-1/2}(\partial D)\)) for \(|\lambda |\ge 1/2\) [19, 38, 58] (see also [19, 20] for the stability results).

The operator \(\mathcal {K}^*_{\partial D}\) is symmetric in \(L^2(\partial D)\) only for a disk or a ball [43]. However, the NP operators can be symmetrized using Plemelj’s symmetrization principle (see [10, 34, 40]). We refer the reader to [28, 29] for the numerical computation with high precision and to [5] and references therein for more properties of the NP operators and their applications.

For a multi-index \(\alpha =(\alpha _1,\alpha _2)\in \mathbb {N}\times \mathbb {N}\), we set \(x^{\alpha }=x_1^{\alpha _1}x_2^{\alpha _2}\) and \(|\alpha |=\alpha _1+\alpha _2\). Applying the Taylor series method, the integral formula (2.4) leads to the multipole expansion [5]:

$$\begin{aligned} u(x) = H(x)+\sum _{|\alpha |,|\beta |=1}^\infty \frac{(-1)^{|\beta |}}{\alpha !\beta !}\partial ^\beta \Gamma (x)M_{\alpha \beta }(D,\sigma _0)\partial ^\alpha H(0), \quad |x|\gg 1, \end{aligned}$$
(2.6)

with

$$\begin{aligned} M_{\alpha \beta }(D,\sigma _0) = \int \limits _{\partial D} y^\beta \left( \lambda I - \mathcal {K}^*_{\partial \Omega }\right) ^{-1}\left[ \frac{\partial x^\alpha }{\partial \nu }\right] (y) \, d\sigma (y). \end{aligned}$$
(2.7)

The terms \(M_{\alpha \beta }(D,\sigma _0)\) are the so-called generalized polarization tensors (GPTs) corresponding to the inclusion D with the conductivity \(\sigma _0\).

Now, we identify \(x=(x_1,x_2)\) in \(\mathbb {R}^2\) with \(z=x_1+ix_2\) in \(\mathbb {C}\) and define the GPTs in complex form.

Definition 2.1

([4]) Let \(\lambda =\frac{\sigma _0+1}{2(\sigma _0-1)}\), and, for each \(n\in \mathbb {N}\), \(P_n(z)=z^n\). For \(m,n\in \mathbb {N}\), we define

$$\begin{aligned} \mathbb {N}_{mn}^{(1)}(D,\sigma _0)&=\int \limits _{\partial D} P_n(z) \left( \lambda I-\mathcal {K}^*_{\partial D}\right) ^{-1}\left[ \frac{\partial P_m }{\partial \nu } \right] (z) \,d\sigma (z),\\ \mathbb {N}_{mn}^{(2)}(D,\sigma _0)&=\int \limits _{\partial D} P_n(z) \left( \lambda I-\mathcal {K}^*_{\partial D}\right) ^{-1}\left[ \frac{\partial \overline{P_m}}{\partial \nu }\right] (z) \,d\sigma (z). \end{aligned}$$

We call \(\mathbb {N}_{mn}^{(1)}\) and \(\mathbb {N}_{mn}^{(2)}\) the complex generalized polarization tensors (CGPTs) corresponding to the inclusion D with the conductivity \(\sigma _0\).

The CGPTs are complex-valued linear combinations of the GPTs, where the expansion coefficients are determined by the Taylor series coefficients of \(z^n\). We refer the reader to [4, 5] for more properties of the CGPTs.

From the expansion of the complex logarithm

$$\begin{aligned} \log (z-\xi )=\log z-\sum _{n=1}^\infty \frac{1}{n}\, {{\xi }^n}{z^{-n}}\quad \text{ for } |z|>|\xi |, \end{aligned}$$

by taking the real part of the expansion, the fundamental solution to the Laplacian satisfies

$$\begin{aligned} \frac{1}{2\pi }\ln |z-\xi |&=\frac{1}{2\pi }\ln |z|-\sum _{n=1}^\infty \frac{1}{4\pi n}\left( {\xi ^n}{z^{-n}} + \overline{{\xi ^n}}\, \overline{z^{-n}}\right) \quad \text{ for } |z|>|\xi |. \end{aligned}$$
(2.8)

A real-valued entire harmonic function H(x) admits the expansion

$$\begin{aligned} H(x) =\alpha _0+ \sum _{m=1}^\infty \left( \alpha _m z^m+\overline{\alpha _m}\, \overline{ z^m}\right) \end{aligned}$$
(2.9)

with some complex coefficients \(\alpha _m\). Using (2.8), we can expand (2.4) into complex functions.

Theorem 2.1

([4]) For an entire harmonic function H given by (2.9), the solution u to (2.1) satisfies that, for \(|z|>\sup \left\{ |y|:y\in D\right\} \),

$$\begin{aligned} u(z)=H(z)-\sum _{n=1}^\infty \sum _{m=1}^\infty \frac{1}{4\pi n}\Bigg [ \left( \alpha _m \mathbb {N}_{mn}^{(1)}+\overline{\alpha _m}\, \mathbb {N}_{mn}^{(2)} \right) {z^{-n}} + \left( \overline{\alpha _m }\, \overline{\mathbb {N}_{mn}^{(1)}} + \alpha _m \overline{\mathbb {N}_{mn}^{(2)}}\right) \overline{z^{-n}}\, \Bigg ]. \end{aligned}$$

2.2 Faber polynomials and Grunsky coefficients

From the Riemann mapping theorem, there uniquely exist a real number \(\gamma >0\) and a complex function \(\Psi (w)\) that conformally maps the region \(\{w\in \mathbb {C}:|w|> \gamma \}\) onto \(\mathbb {C}\setminus \overline{D}\) and satisfies \(\Psi (\infty )=\infty \) and \(\Psi '(\infty )=1\). We set \(\rho _0=\ln \gamma \). The function \(\Psi (w)\) admits the following Laurent series expansion:

$$\begin{aligned} \Psi (w)=w+a_0+\sum _{k=1}^{\infty }a_kw^{-k} \end{aligned}$$
(2.10)

for some complex coefficients \(a_n\). We call \(\gamma \) the conformal radius of D. The exterior conformal mapping extends to \(\partial D\) as a homeomorphism by the Caratheodory extension theorem [13]. Thus, it defines an orthogonal curvilinear coordinate system \((\rho ,\theta )\in [\gamma ,\infty )\times [0,2\pi )\) for each z in \(\mathbb {C}\setminus D\) via the relation

$$\begin{aligned} z=\Psi (\rho ,\theta ):=\Psi (e^{\rho +\textrm{i}\theta }). \end{aligned}$$

The scale factors with respect to \(\rho \) and \(\theta \) coincide with each other. We denote them by

$$\begin{aligned} h(\rho , \theta ) := \left| \frac{\partial \Psi }{\partial \rho }\right| = \left| \frac{\partial \Psi }{\partial \theta }\right| . \end{aligned}$$

The length element on \(\partial D\) is given by \(d\sigma (z)=h(\rho _0,\theta )d\theta \), and for a function \(v(z)=(v\circ \Psi )(\rho ,\theta )\) defined in the exterior of D, it holds that

$$\begin{aligned} \frac{\partial v}{\partial \nu }\Big |_{\partial D}^{+}(z)=\frac{1}{h(\rho _0,\theta )}\frac{\partial }{\partial \rho }v\left( \Psi (\rho ,\theta )\right) \Big |_{\rho \rightarrow \rho _0^+}. \end{aligned}$$

If we further assume that D is a \(C^{1,\alpha }\) domain for some \(0<\alpha <1\), then, by the Kellogg–Warschawski theorem [51], \(\Psi '\) can be continuously extended to the boundary.

As a univalent function, \(\Psi \) defines the so-called Faber polynomials \(\{F_m(z)\}_{m=1}^\infty \), first introduced by G. Faber [21], by the relation

$$\begin{aligned} \frac{w\Psi '(w)}{\Psi (w)-z}=\sum _{m=0}^\infty \frac{F_m(z)}{w^{m}},\quad z\in {\overline{D}},\ |w|>\gamma . \end{aligned}$$
(2.11)

This provides explicit expressions for \(F_m\) in terms of \(a_n\). For example, \(F_0(z)=1,\ F_1(z)=z-a_0,\ F_2(z)=z^2-2a_0 z+a_0^2-2a_1.\) For each m, \(F_m(z)\) is a polynomial of degree m. From (2.11), it holds that for \({z}=\Psi (w)\in \mathbb {C}\setminus \overline{D}\) and \(\xi \in D\),

$$\begin{aligned} \log ({z}-\xi )=\log w-\sum _{m=1}^\infty \frac{1}{m}F_m(\xi )w^{-m} \end{aligned}$$
(2.12)

with a proper branch cut. The expansion (2.12) sheds new light to understand the solution of the transmission problem (2.1) and the NP operator [32, 33].

For each m, \(F_m(\Psi (w))\) has only one nonnegative-order term \(w^m\) and satisfies

$$\begin{aligned} F_m(\Psi (w))=w^m+\sum _{n=1}^{\infty }c_{mn}{w^{-n}},\quad |w|>\gamma . \end{aligned}$$
(2.13)

The coefficients \(c_{mn}\) are called the Grunsky coefficients. The Grunsky identity holds for all \(m,n\in \mathbb {N}\): \(n c_{mn}=m c_{nm}\). The following recursive formula holds:

$$\begin{aligned} c_{m(n+1)} = c_{(m+1)n} - a_{m+n} + \sum _{s=1}^{m-1} a_{m-s}c_{sn} - \sum _{s=1}^{n-1} a_{n-s}c_{ms}, \quad m,n\ge 1 \end{aligned}$$
(2.14)

with initial values \(c_{1n} = a_n\) and \(c_{n1} = na_n\), \(n\ge 1\). The Faber polynomials form a basis for complex analytic functions in D [18, 55]. We refer the reader to [24, 50] for further details on the Faber polynomials and the Grunsky coefficients.

We denote by C the Grunsky matrix and by G its symmetrization, that is,

$$\begin{aligned} \begin{aligned} C&=\big (c_{mn}\big )_{m,n=1}^\infty , \\ G&=\big (g_{mn}\big )_{m,n=1}^\infty \quad \text{ with } g_{mn}=\sqrt{\frac{n}{m}} \frac{c_{mn}}{\gamma ^{m+n}}. \end{aligned} \end{aligned}$$
(2.15)

From the Grunsky identity, it holds that the symmetry relation \(g_{mn}=g_{nm}\) for all \(n,m\in \mathbb {N}.\) We can express G as

$$\begin{aligned} G = \mathbb {N}^{-\frac{1}{2}} \gamma ^{-\mathbb {N}} C \gamma ^{-\mathbb {N}} \mathbb {N}^{\frac{1}{2}}, \end{aligned}$$
(2.16)

where \(\gamma ^{\pm k\mathbb {N}}\) and \(\mathbb {N}^{\pm \frac{1}{2}}\) denote the semi-infinite diagonal matrices whose (n, n)-entries are \(\gamma ^{\pm kn}\) and \(n^{\pm \frac{1}{2}}\), respectively. One can interpret the matrix G as a linear operator from \(l^2(\mathbb {C})\) to \(l^2(\mathbb {C})\) defined by

$$\begin{aligned} (v_m) \longmapsto (w_m) \text{ with } w_m = \sum _{k=1}^\infty g_{mk} v_k, \end{aligned}$$

where \(l^2(\mathbb {C})\) denotes the vector space of the complex sequence \((v_m)\) satisfying \(\sum _{m=1}^\infty |v_m|^2<\infty \). According to [50, Theorems 9.12-13], it holds that, for some constant \(\kappa \in [0,1)\),

$$\begin{aligned} \left\Vert {G} \right\Vert _{l^2\rightarrow l^2} \le \kappa <1 \end{aligned}$$
(2.17)

if \(\partial D\) is quasiconformal (refer to [1] and [51, Chapter 5.4] for the characterization of a quasiconformal curve). We refer the reader to [11, 41, 56] for more properties of quasiconformality.

3 Geometric multipole expansion

We introduce the new concept of geometric multipole expansion and the GPTs for the simply connected and multi-coated cases.

3.1 Simply connected case

As for the classical multipole expansion (Theorem 2.1), we consider the transmission problem with a conductivity inclusion in two dimensions:

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \nabla \cdot \sigma \nabla u=0\quad &{}\text{ in } \mathbb {R}^2, \\ \displaystyle u(x) - H(x) =O({|x|^{-1}})\quad &{}\text{ as } |x| \rightarrow \infty , \end{array}\right. } \end{aligned}$$
(3.1)

where the conductivity distribution is given by

$$\begin{aligned} \sigma = \sigma _0\, \chi (D)+\chi (\mathbb {R}^2\setminus \overline{D}). \end{aligned}$$
(3.2)

Differently from the classical multipole expansion that holds in the exterior of a disk containing the inclusion, the geometric multipole expansion holds in the whole exterior region of the inclusion. This convenient property results in a series solution method for the transmission problem with an inclusion of general shape, as will be discussed in Sect. 5.1.

Assume that D is a simply connected inclusion. Let \(\Psi (w)\) be the associated conformal mapping as in Sect. 2.2. Differently from (2.8), we now expand the logarithm function in terms of the exterior conformal mapping associated with the inclusion, which leads to a series expression that holds in the whole exterior region of the inclusion. More precisely, from the expansion of the complex logarithm into the Faber polynomials (2.12), it holds that for \(z=\Psi (w)\in \mathbb {C}\setminus \overline{D}\) and \(\xi \in \overline{D}\),

$$\begin{aligned} \frac{1}{2\pi }\ln |z-\xi |&=\frac{1}{2\pi }\ln |w|-\sum _{n=1}^\infty \frac{1}{4\pi n}\left( F_n(\xi )w^{-n} + \overline{F_n(\xi )}\,\overline{ w^{-n}}\right) . \end{aligned}$$
(3.3)

Indeed, (3.3) converges uniformly with respect to \(|w|>\gamma \) and uniformly with respect to \(\xi \) belonging to any fixed compact F in the domain \(\overline{D}\) (see [21, 57]). Also, an entire real harmonic function H satisfies

$$\begin{aligned} H(x) =\alpha _0+ \sum _{m=1}^\infty \left( \alpha _m F_m(z)+\overline{\alpha _m}\,\overline{ F_m(z)}\right) \end{aligned}$$
(3.4)

for some complex coefficients \(\alpha _n\), where the series converges uniformly on any given compact domain [31]. We modify the GPTs by replacing \(z^n\) with the Faber polynomials as follows.

Definition 3.1

Let \(F_n\) be the Faber polynomials of D. For \(m,n\in \mathbb {N}\), we define

$$\begin{aligned} \mathbb {F}_{mn}^{(1)}(D,\sigma _0)=\int \limits _{\partial D}F_n(z)\left( \lambda I-\mathcal {K}^*_{\partial D}\right) ^{-1}\left[ \frac{\partial F_m}{\partial \nu }\right] (z)\,d\sigma (z),\\ \mathbb {F}_{mn}^{(2)}(D,\sigma _0)=\int \limits _{\partial D}F_n(z)\left( \lambda I-\mathcal {K}^*_{\partial D}\right) ^{-1}\left[ \frac{\partial \overline{F_m}}{\partial \nu }\right] (z)\,d\sigma (z) \end{aligned}$$

with \(\lambda =\frac{\sigma _0+1}{2(\sigma _0-1)}\). We call \(\mathbb {F}_{mn}^{(1)}\) and \(\mathbb {F}_{mn}^{(2)}\) the Faber polynomial polarization tensors (FPTs) corresponding to the domain D with the conductivity \(\sigma _0\).

Remark 3.1

For a disk centered at the origin, the Faber polynomials are \(z^n\) and, thus, (2.8) and (2.9) are in fact expansions into the Faber polynomials (and their complex conjugates) corresponding to a disk centered at the origin. Hence, the FPTs and GPTs for a disk centered at the origin are identical.

From (3.3) and (3.4), we can express the single-layer potential in terms of the exterior conformal mapping as follows. First, it holds that for \(z=\Psi (w)\in \mathbb {C}\setminus \overline{D}\) with \(|w|>\gamma \),

$$\begin{aligned} \mathcal {S}_{\partial D}[\varphi ](z)&=\frac{1}{2\pi } \int \limits _{\partial D}\ln |z-\xi | \varphi (\xi )\, d\sigma (\xi )\nonumber \\&=- \int \limits _{\partial D} \sum _{n=1}^\infty \frac{1}{4\pi n} \left( F_n(\xi )w^{-n} + \overline{F_n(\xi )}\,\overline{ w^{-n}}\right) \varphi (\xi )\, d\sigma (\xi ). \end{aligned}$$
(3.5)

Second, the density function \(\varphi \), given by (2.5), satisfies

$$\begin{aligned} \varphi =(\lambda I-\mathcal {K}_{\partial D}^*)^{-1}\left[ \,\sum _{m=1}^\infty \left( \alpha _m \frac{\partial F_m}{\partial \nu }+\overline{\alpha _m}\,\frac{\partial \overline{ F_m}}{\partial \nu }\right) \, \right] . \end{aligned}$$
(3.6)

The infinite series expansions in both (3.5) and (3.6) are uniformly convergent for \(\xi \in \partial D\) (with z fixed). Since \((\lambda I-\mathcal {K}_{\partial D}^*)^{-1}\) is a bounded operator, we then can exchange the order of integral and summation in (3.5) and get the desired expansion.

Theorem 3.1

(Geometric multipole expansion) For an entire harmonic function H given by (2.9), the solution u to (2.1) satisfies that, for \(z=\Psi (w)\in \mathbb {C}\setminus \overline{D}\),

$$\begin{aligned} u(z)=H(z)-\sum _{m=1}^\infty \sum _{n=1}^\infty \frac{1}{4\pi n}\Bigg [ \left( \alpha _m \mathbb {F}_{mn}^{(1)}+ \overline{\alpha _m}\, \mathbb {F}_{mn}^{(2)} \right) {w^{-n}} + \left( \overline{\alpha _m}\,\overline{ \mathbb {F}_{mn}^{(1)}} + \alpha _m \overline{\mathbb {F}_{mn}^{(2)}} \right) \overline{w^{-n}}\, \Bigg ]. \end{aligned}$$
(3.7)

3.2 Multi-coated case

We extend the concept of geometric multipole expansion and FPTs to a multi-coated inclusion, namely \(\Omega \), that consists of the core D, also denoted by \(\Omega _0\), and the coating layers \(\Omega _j\) for \(j=1,\dots ,L\). The exterior of \(\Omega \) is denoted by \(\Omega _{L+1}\). We further assume that the coating layers are images of concentric annuli via the exterior conformal mapping associated with the core \(\Omega _0\) (see Fig. 1 for the geometry of the considered inclusion). The conductivities in \(\Omega _j\) are assumed to be positive constants \(\sigma _j\) for \(j=0,1,\dots ,L\) (\(\sigma _{L+1}=1\)). For notational simplicity, we set \( \varvec{\sigma }=(\sigma _0,\dots ,\sigma _{L}). \)

Fig. 1
figure 1

Multi-coated inclusion. Each coating layer is an image of concentric annuli via the exterior conformal mapping associated with the core \(\Omega _0\)

Full size image

We now consider the conductivity interface problem (3.1) with the conductivity distribution

$$\begin{aligned} \sigma = \sum _{j=0}^L \sigma _j \chi (\Omega _j) +\chi (\mathbb {R}^2\setminus \overline{\Omega }) \end{aligned}$$
(3.8)

with a harmonic function H given by (3.4). The solution u satisfies the transmission condition:

$$\begin{aligned} u\big |^+=u\big |^-,\quad \sigma _{j+1}\frac{\partial u}{\partial \nu }\Big |^+=\sigma _j\frac{\partial u}{\partial \nu }\Big |^-\quad \text{ on } \partial \Omega _j \end{aligned}$$
(3.9)

for each \(j=0,1,\dots ,L\).

Since the problem (3.1) is linear with respect to H, we can linearly decompose u into the terms corresponding to \(\alpha _m F_m(z)\) and their conjugates. Since \(\Psi \) is conformal and \(\Psi (w)\) converges to z as \(|w|\rightarrow \infty \), \((u-H)\circ \Psi (w)\) is harmonic in \(\{w:|w|>{r_L}\}\) and decays to zero as \(|w|\rightarrow \infty \). Therefore, \((u-H)\circ \Psi (w)\) expands into \(w^{-n}\) and their harmonic conjugates. We conclude that the solution u to the conductivity interface problem with a multi-coated inclusion, (3.1) and (3.8), satisfies the geometric multipole expansion (3.7) for \(z=\Psi (w)\in \mathbb {C}\setminus \overline{\Omega }\). The expansion coefficients \(\mathbb {F}_{mn}^{(1)}\) and \(\mathbb {F}_{mn}^{(2)}\) depend on \(\Omega \) and \(\sigma _0,\dots ,\sigma _L\).

Definition 3.2

We call \(\mathbb {F}_{mn}^{(1)}\) and \(\mathbb {F}_{mn}^{(2)}\) the FPTs corresponding to the multi-coated inclusion \(\Omega \) with the conductivity \(\varvec{\sigma }=(\sigma _0,\dots ,\sigma _{L})\). We denote the FPTs in matrix form as

$$\begin{aligned} \mathbb {F}^{(1)}(\Omega ,\varvec{\sigma }) = \left( \mathbb {F}^{(1)}_{mn}\right) _{m,n=1}^\infty , \quad \mathbb {F}^{(2)}(\Omega ,\varvec{\sigma }) = \left( \mathbb {F}^{(2)}_{mn}\right) _{m,n=1}^\infty . \end{aligned}$$

4 FPTs for a simply connected inclusion

4.1 Expression in terms of the Grunsky matrix

Let \((\rho ,\theta )\) be the orthogonal coordinate system given by (2.1). We define the density basis functions on \(\partial D\) as

$$\begin{aligned}{} & {} \eta _0(z)=1,\quad \zeta _0(z)=\frac{1}{h(\rho _0,\theta )},\\{} & {} \eta _{m}(z)=|m|^{-\frac{1}{2}}e^{i m \theta },\quad \zeta _{m}(z)=|m|^{\frac{1}{2}}\frac{e^{ i m \theta }}{h(\rho _0,\theta )}\quad \text{ for } m\in \mathbb {Z}\setminus \{0\}. \end{aligned}$$

If D has a \(C^{1,\alpha }\) boundary, then \(\zeta _m\) (resp. \(\eta _m\)) form a basis of \(H^{-1/2}(\partial D)\) (resp. \(H^{1/2}(\partial D)\)) and, furthermore, \(\zeta _m\) and \(\eta _m\) jointly form a bi-orthogonal system for the pair of spaces \(H^{-1/2}(\partial D)\) and \(H^{1/2}(\partial D)\) [33].

Lemma 4.1

([33]) Let D be a bounded and simply connected domain in \(\mathbb {R}^2\) with \(C^{1,\alpha }\) boundary for some \(\alpha >0\). For \(z=\Psi (w)\in \mathbb {C}\setminus \overline{D}\) with \(w=e^{\rho +i\theta }\), we have \( \mathcal {S}_{\partial \Omega }[\zeta _0](z)= \ln \gamma \ \text{ in } \overline{D},\ \ln |w|\ \text{ in } \mathbb {C}\setminus \overline{D}. \) For \(m\in \mathbb {N}\), we have

$$\begin{aligned} \mathcal {S}_{\partial D}[\zeta _m](z)= {\left\{ \begin{array}{ll} \displaystyle -\frac{1}{2\sqrt{m}\gamma ^m}F_m(z)\quad &{}\text {in }\overline{D},\\ \displaystyle -\frac{1}{2\sqrt{m}\gamma ^m}\left( \sum _{n=1}^{\infty }c_{mn}e^{-n(\rho +i\theta )}+\gamma ^{2m}e^{m(-\rho +i\theta )}\right) \quad &{}\text {in } \mathbb {C}\setminus \overline{D}, \end{array}\right. } \end{aligned}$$
(4.1)

where, for any fixed \(\rho _1>\rho _0\), the series converges uniformly for \((\rho ,\theta )\) satisfying \(\rho \ge \rho _1\). For the negative index case, it holds that \(\mathcal {S}_{\partial D}[\zeta _{-m}](z)=\overline{\mathcal {S}_{\partial D}[\zeta _{m}](z)}.\)

Furthermore, the NP operators satisfy \(\mathcal {K}^*_{\partial D}\left[ \zeta _0\right] =\frac{1}{2}\zeta _0\) and \(\mathcal {K}_{\partial D}\left[ 1\right] =\frac{1}{2}\). For \(m\in \mathbb {N}\), we have

$$\begin{aligned}&\mathcal {K}^*_{\partial D}\left[ {\zeta _m}\right] =\frac{1}{2}\sum _{n=1}^{\infty }\frac{\sqrt{n}}{\sqrt{m}}\frac{c_{mn}}{\gamma ^{m+n}}\, {\zeta }_{-n},\quad \mathcal {K}^*_{\partial D}\left[ \zeta _{-m}\right] =\frac{1}{2}\sum _{n=1}^{\infty }\frac{\sqrt{n}}{\sqrt{m}}\frac{\overline{c_{mn}}}{\gamma ^{m+n}}\, \zeta _{n}, \end{aligned}$$
(4.2)
$$\begin{aligned}&\mathcal {K}_{\partial D}\left[ \eta _m\right] =\frac{1}{2}\sum _{n=1}^{\infty }\frac{\sqrt{n}}{\sqrt{m}}\frac{c_{mn}}{\gamma ^{m+n}}\, {\eta _{-n}}, \quad \mathcal {K}_{\partial D} \left[ \eta _{-m}\right] =\frac{1}{2}\sum _{n=1}^{\infty }\frac{\sqrt{n}}{\sqrt{m}}\frac{\overline{c_{mn}}}{\gamma ^{m+n}}\, \eta _{n}. \end{aligned}$$
(4.3)

The infinite series in (4.2) converge in \(H^{-1/2}(\partial D)\), and those in (4.3) converge in \(H^{1/2}(\partial D)\).

The FPTs of D with conductivity \(\sigma _0\) admit the matrix expressions in terms of the Grusnky coefficient \(c_{mn}\) and the Grunsky matrix C associated with D (see (2.13) and (2.15)) as follows.

Theorem 4.2

Let D be a bounded and simply connected domain in \(\mathbb {R}^2\) with \(C^{1,\alpha }\) boundary for some \(\alpha >0\), and \(\lambda =\frac{\sigma _0+1}{2(\sigma _0-1)}\). The FPTs satisfy

$$\begin{aligned} \mathbb {F}_{mn}^{(1)}(D,\sigma _0)&=4\pi n c_{mn}+ 4\pi n \left( \frac{1}{4}-\lambda ^2\right) \left[ C\left( \lambda ^2 I - \frac{\gamma ^{-2\mathbb {N}} \overline{C}\gamma ^{-2\mathbb {N}} C}{4} \right) ^{-1} \right] _{mn}, \\ \mathbb {F}_{mn}^{(2)}(D,\sigma _0)&=8\pi n \lambda \gamma ^{2m}\, \delta _{mn}+ 8\pi n \lambda \gamma ^{2m} \left( \frac{1}{4}-\lambda ^2\right) \left[ \left( \lambda ^2 I - \frac{\gamma ^{-2\mathbb {N}} \overline{C} \gamma ^{-2\mathbb {N}} C}{4} \right) ^{-1} \right] _{mn}, \end{aligned}$$

where \(\delta _{mn}\) is the Kronecker delta function.

Proof. From (2.3) and (4.1), we have

$$\begin{aligned} \frac{\partial F_m}{\partial \nu } =-2 \sqrt{m}\gamma ^m \left( -\frac{1}{2}I+\mathcal {K}^*_{\partial D}\right) \left[ \zeta _m\right] \quad \text{ on } \partial D. \end{aligned}$$

From (4.1) and (4.3), it holds in the \(H^{1/2}(\partial D)\) sense that

$$\begin{aligned} \mathcal {S}_{\partial D}\left[ \zeta _m\right]&=-\left( \frac{1}{2}I+\mathcal {K}_{\partial D}\right) \left[ \eta _m\right] \quad \text{ on } \partial D. \end{aligned}$$

Hence, the FPTs become

$$\begin{aligned} \mathbb {F}_{mn}^{(1)}&=4\sqrt{mn}\, \gamma ^{m+n}\int \limits _{\partial D}\Big (\frac{1}{2}I+\mathcal {K}_{\partial D}\Big )[\eta _n]\left[ \big (\lambda I-\mathcal {K}^*_{\partial D}\big )^{-1}\Big (\frac{1}{2}I-\mathcal {K}^*_{\partial D}\Big )[\zeta _m]\right] d\sigma \\&=4\sqrt{mn}\, \gamma ^{m+n}\int \limits _{\partial D}\eta _n \left[ \Big (\frac{1}{2}I+\mathcal {K}^*_{\partial D}\Big )\big (\lambda I-\mathcal {K}^*_{\partial D}\big )^{-1}\Big (\frac{1}{2}I-\mathcal {K}^*_{\partial D}\Big )[\zeta _m]\right] d\sigma \\&=4\sqrt{mn}\, \gamma ^{m+n}\int \limits _{\partial D}\eta _n \left[ \lambda I+\mathcal {K}^*_{\partial D}+\Big (\frac{1}{4}-\lambda ^2\Big )\left( \lambda I-\mathcal {K}^*_{\partial D}\right) ^{-1}\right] [\zeta _m]\, d\sigma \end{aligned}$$

and

$$\begin{aligned} \mathbb {F}_{mn}^{(2)}=4\sqrt{mn}\, \gamma ^{m+n}\int \limits _{\partial D}\eta _n \left[ \lambda I+\mathcal {K}^*_{\partial D}+\Big (\frac{1}{4}-\lambda ^2\Big )\left( \lambda I-\mathcal {K}^*_{\partial D}\right) ^{-1}\right] [\zeta _{-m}]\, d\sigma . \end{aligned}$$

From the fact that \(d\sigma (z)=h(\rho _0,\theta )d\theta \), one can easily find that

$$\begin{aligned} \int \limits _{\partial D}\eta _n\zeta _m d\sigma =0,\ \int \limits _{\partial D}\eta _n{\zeta _{-m}} d\sigma = 2\pi \, \delta _{mn}\quad \text{ for } \text{ all } m,n\in \mathbb {N}. \end{aligned}$$
(4.4)

Then, by using (4.2) and (4.4), we have

$$\begin{aligned} \mathbb {F}_{mn}^{(1)}(D,\sigma _0)&=4\pi n c_{mn}+\left( \frac{1}{4}-\lambda ^2\right) A_{m,n}, \end{aligned}$$
(4.5)
$$\begin{aligned} \mathbb {F}_{mn}^{(2)}(D,\sigma _0)&=8\pi n \lambda \gamma ^{2m}\, \delta _{mn}+ \left( \frac{1}{4}-\lambda ^2\right) A_{-m,n} \end{aligned}$$
(4.6)

with

$$\begin{aligned} A_{\pm m, n}:=4\sqrt{mn}\,\gamma ^{m+n} \int \limits _{\partial D}\eta _n \left( \lambda I-\mathcal {K}^*_{\partial D}\right) ^{-1}\left[ \zeta _{\pm m}\right] \, d\sigma . \end{aligned}$$
(4.7)

In the remainder of the proof, we derive an explicit expression for \(A_{\pm m, n}\).

Since \(\zeta _m\) form a basis of \(H^{-1/2}(\partial D)\), we can expand \(\left( \lambda I-\mathcal {K}^*_{\partial D}\right) ^{-1}\left[ \zeta _{\pm m}\right] \) as

$$\begin{aligned} \left( \lambda I-\mathcal {K}^*_{\partial D}\right) ^{-1}\left[ \zeta _{m}\right]&= \sum _{n=1}^\infty \left( x_{mn} \zeta _n + y_{mn} \zeta _{-n} \right) , \end{aligned}$$
(4.8)
$$\begin{aligned} \left( \lambda I-\mathcal {K}^*_{\partial D}\right) ^{-1}\left[ \zeta _{-m}\right]&= \sum _{n=1}^\infty \left( \, \overline{x_{mn}} \zeta _{-n} + \overline{y_{mn}} \zeta _{n}\right) \end{aligned}$$
(4.9)

for some expansion coefficients \(x_{mn}\) and \(y_{mn}\). Applying \((\lambda I-\mathcal {K}^*_{\partial D})\) on both sides of (4.8), from (4.2), we obtain

$$\begin{aligned} \zeta _{m}&= \sum _{n=1}^\infty x_{mn} \left( \lambda \zeta _n -\frac{1}{2}\sum _{k=1}^{\infty }\, g_{nk} {\zeta }_{-k} \right) + \sum _{n=1}^\infty y_{mn} \left( \lambda \zeta _{-n} -\frac{1}{2}\sum _{k=1}^{\infty }\, \overline{g_{nk}}\, \zeta _{k}\right) , \end{aligned}$$
(4.10)

where \(g_{kn}\) is the symmetrized Grunsky coefficient. Rearranging (4.10), we have

$$\begin{aligned}&\sum _{n=1}^\infty \left( \lambda x_{mn} -\delta _{mn}\right) \zeta _n + \sum _{n=1}^\infty \lambda y_{mn} \zeta _{-n} =\sum _{n=1}^\infty \sum _{k=1}^{\infty } \frac{y_{mk}}{2}\, \overline{g_{kn}} \, \zeta _{n} + \sum _{n=1}^\infty \sum _{k=1}^{\infty } \frac{x_{mk}}{2}\, g_{kn}\, {\zeta }_{-n}. \end{aligned}$$

This implies that

$$\begin{aligned} \lambda \, x_{mn}&= \delta _{mn}+\frac{1}{2} \sum _{k=1}^{\infty } y_{mk}\, \overline{g_{kn}}, \end{aligned}$$
(4.11)
$$\begin{aligned} \lambda \, y_{mn}&= \frac{1}{2} \sum _{k=1}^{\infty } x_{mk}\, g_{kn}. \end{aligned}$$
(4.12)

By combining (4.11) and (4.12), we have the matrix expressions for \(X=(x_{mn})_{m,n=1}^\infty \) and \(Y=(y_{mn})_{m,n=1}^\infty \) as follows:

$$\begin{aligned} X&= \lambda \left( \lambda ^2 I - \frac{G \overline{G}}{4} \right) ^{-1}, \end{aligned}$$
(4.13)
$$\begin{aligned} Y&= \frac{1}{2} G \left( \lambda ^2 I - \frac{\overline{G} G}{4} \right) ^{-1}. \end{aligned}$$
(4.14)

Assuming that \(\partial D\) is \(C^{1,\alpha }\), \(\partial D\) is quasiconformal (see [1] and [51, Chapter 5.4]). Hence, we have (2.17) and, thus, the inverse matrices in (4.13) and (4.14) exist.

By applying (4.4), (4.7), (4.8), and (4.14), we obtain

$$\begin{aligned} A_{m, n}&= 4\sqrt{mn}\,\gamma ^{m+n} \int \limits _{\partial D}\eta _n \left( \lambda I-\mathcal {K}^*_{\partial D}\right) ^{-1}\left[ \zeta _{m}\right] \, d\sigma \nonumber \\&= 8\pi \sqrt{mn}\,\gamma ^{m+n} y_{mn} \nonumber \\&= 4\pi n \left[ C\left( \lambda ^2 I - \frac{\gamma ^{-2\mathbb {N}} \overline{C}\gamma ^{-2\mathbb {N}} C}{4} \right) ^{-1} \right] _{mn}. \end{aligned}$$
(4.15)

We also use (2.16). Similarly, from (4.4), (4.7), (4.9), and (4.13), it holds that

$$\begin{aligned} A_{-m, n}&= 4\sqrt{mn}\,\gamma ^{m+n} \int \limits _{\partial D}\eta _n \left( \lambda I-\mathcal {K}^*_{\partial D}\right) ^{-1}\left[ \zeta _{-m}\right] \, d\sigma \nonumber \\&= 8\pi \sqrt{mn}\,\gamma ^{m+n} \, \overline{x_{mn}} \nonumber \\&= 8\pi n \lambda \gamma ^{2m} \left[ \left( \lambda ^2 I - \frac{\gamma ^{-2\mathbb {N}} \overline{C} \gamma ^{-2\mathbb {N}} C}{4} \right) ^{-1} \right] _{mn}. \end{aligned}$$
(4.16)

From (4.5), (4.6), (4.15), and (4.16), we prove the proposition. \(\square \)

4.2 Inclusion with extreme conductivity

For the insulating or perfect conducting case (i.e., \(\sigma _0=\infty \) or 0), we have \(\lambda =\pm 1/2\). It is evident from Theorem 4.2 that the following relations hold.

Corollary 4.3

Let D be a bounded, simply connected, and \(C^{1,\alpha }\) domain in \(\mathbb {R}^2\) with the conductivity \(\sigma _0=\infty \) or 0. Then, the corresponding FPTs are

$$\begin{aligned} \mathbb {F}_{mn}^{(1)}(D,\sigma _0)&=4\pi n c_{mn},\\ \mathbb {F}_{mn}^{(2)}(D,\sigma _0)&=\pm 4\pi n \gamma ^{2m}\, \delta _{mn}, \end{aligned}$$

where \(+\) and − correspond to \(\sigma _0=\infty \) and \(\sigma _0=0\), respectively.

Plugging \(m=n=1\) into Corollary 4.3 with \(\sigma _0=\infty \) or 0, we arrive at the relation

$$\begin{aligned} \mathbb {F}_{11}^{(1)}(D,\sigma _0)&=4\pi c_{11}=4\pi a_1, \end{aligned}$$
(4.17)
$$\begin{aligned} \mathbb {F}_{11}^{(2)}(D,\sigma _0)&= \pm 4\pi \gamma ^2. \end{aligned}$$
(4.18)

Since \(F_1(z)=z-a_0\), the first-order FPTs coincide with those of the CGPTs. Hence, it holds that

$$\begin{aligned} \mathbb {F}_{11}^{(1)}&=\mathbb {N}_{11}^{(1)}=m_{11}-m_{22}+i(m_{12}+m_{21}),\\ \mathbb {F}_{11}^{(2)}&=\mathbb {N}_{11}^{(2)}=m_{11}+m_{22}, \end{aligned}$$

where the \(2\times 2\) symmetric matrix

$$\begin{aligned} M=\left[ \begin{array}{cc} m_{11} &{} m_{12}\\ m_{21} &{} m_{22} \end{array}\right] \end{aligned}$$

denotes the polarization tensor (PT) associated with D. In other words, \(m_{11}=M_{(1,0)(1,0)},\ m_{12}=M_{(1,0)(0,1)},\ m_{21}=M_{(0,1)(1,0)}\), and \(m_{22}=M_{(0,1)(0,1)}\), following the definition (2.7). We then obtain the following lemma from (4.17) and (4.18).

Lemma 4.4

Let D be a bounded and simply connected \(C^{1,\alpha }\) domain in \(\mathbb {R}^2\) with the conductivity \(\sigma _0=\infty \) or 0. Then, the corresponding PT satisfies

$$\begin{aligned} M =2\pi \, \Bigg [ \begin{array}{cc} \displaystyle \pm \gamma ^2 +\Re \{a_1\}&{} \Im \{a_1\} \\ \displaystyle \Im \{a_1\} &{} \pm \gamma ^2 -\Re \{a_1\} \end{array} \Bigg ]. \end{aligned}$$

It is worth remarking that Lemma 4.4 was extended to a domain with Lipschitz boundary in [14].

Corollary 4.5

Under the same assumption as in Lemma 4.4, we have

$$\begin{aligned} {\text {Tr}}(M^{-1})&= \pm \left( \pi \gamma ^2- \frac{\pi |a_1|^2}{\gamma ^2}\right) ^{-1}. \end{aligned}$$

Proof. We have \(|a_1|<\gamma ^2\). Indeed, the area of the domain D given by the exterior conformal mapping (2.10) with the conformal radius \(\gamma >0\) is

$$\begin{aligned} 0<|D|=\pi \gamma ^2 -\pi \sum _{k=1}^\infty \frac{k|a_k|^2}{\gamma ^{2k}}. \end{aligned}$$
(4.19)

Let \(\lambda _\alpha \) and \(\lambda _\beta \) be the eigenvalues of the PT. It then holds from Lemma 4.4 that

$$\begin{aligned}&\lambda _\alpha +\lambda _\beta =\text{ Tr }(M)=\pm 4\pi \gamma ^2,\\&\lambda _\alpha \lambda _\beta =\text{ det }(M)=4\pi ^2\left( \gamma ^4-|a_1|^2\right) . \end{aligned}$$

Thus, we have

$$\begin{aligned} {\text {Tr}}(M^{-1})=\frac{1}{\lambda _\alpha }+\frac{1}{\lambda _\beta } =\pm \frac{1}{\pi }\frac{\gamma ^2}{\gamma ^4-|a_1|^2}. \end{aligned}$$

\(\square \)

The Pólya–Szegö conjecture asserts that, for an inclusion D with unit area, \(\left| {{\text {Tr}}}(M^{-1})\right| \) has a minimum value if and only if D is a disk or an ellipse; this conjecture was proved for the general conductivity case in [49]. Corollary 4.5 leads to a simple alternative proof for the insulating or perfecting conducting case in two dimensions. It is evident from Corollary 4.5 and (4.19) that

$$\begin{aligned} |D|\left| {{\text {Tr}}}(M^{-1})\right|&= \left( \pi \gamma ^2 -\pi \sum _{k=1}^\infty \frac{k|a_k|^2}{\gamma ^{2k}}\right) \frac{1}{\pi \gamma ^2- \frac{\pi |a_1|^2}{\gamma ^2}}\le 1. \end{aligned}$$
(4.20)

The equality holds in (4.20) if and only if \(a_k=0\) for all \(k\ge 2\); equivalently, D is a disk or ellipse.

5 Series solution to the conductivity interface problem with a multi-coated inclusion

In this section, we obtain a series solution to the conductivity interface problem with a multi-coated inclusion by using the Faber polynomials and the concept of the FPTs. The basis functions are from the exterior conformal mapping associated with the core of the inclusion, and the expansion coefficients are from a matrix equation involving the Grunsky matrix. The benefit of this method is that, similar to the concentric disks case, one can solve the interface problem in a series form for a multi-coated structure of general shape.

As in Sect. 3, we let a multi-coated inclusion \(\Omega \), which consists of the core D, also denoted by \(\Omega _0\), and the coating layers \(\Omega _j\) for \(j=1,\dots ,L\), be embedded in an infinite medium (see Fig. 1). We define \(\gamma \), \(\Psi (w)\), and \((\rho ,\theta )\) associated with the core as in Sect. 2.2.

We assume that \(\Psi \) can be conformally extended to \(\{w\in \mathbb {C}:|w|>\gamma -\delta \}\) for some \(\delta >0\). We further assume that the coating layers are images of concentric annuli via the mapping \(\Psi \). In other words, we have

$$\begin{aligned}&\Omega _j=\big \{\Psi (\rho ,\theta )\,:\, r_{j-1}< e^\rho < r_j\big \},\quad j=1,2,\dots ,L, \end{aligned}$$
(5.1)
$$\begin{aligned}&\Omega _{L+1}=\big \{\Psi (\rho ,\theta )\,:\, e^\rho > r_L\big \} \end{aligned}$$
(5.2)

for some constants \(r_j\) satisfying \(\gamma =r_0<r_1<\cdots<r_L<\infty .\)

5.1 Expression of the solution in terms of FPTs

Fix \(m\in \mathbb {N}\). Let \(u_m\) be the solution to (3.1) and (3.8) with H given by

$$\begin{aligned} H_m(x) = \alpha _m F_m(z) + \overline{\alpha _m}\,\overline{F_m(z)}, \end{aligned}$$

where \(\alpha _m\) is some complex number.

Since \(u_m\) is harmonic in each \(\Omega _j\), \(j=0,\dots ,L+1\), and \((u_m-H_m)(z)\) decays to zero as \(|z|\rightarrow \infty \), we expand \(u_m\) as

$$\begin{aligned} u_m(x) = {\left\{ \begin{array}{ll} \displaystyle \sum _{n=1}^\infty \left( b_{mn} F_n(z) + \overline{b_{mn}}\,\overline{F_n(z)} \right) \quad &{}\text{ in } {\Omega _0},\\ \displaystyle \sum _{n=1}^\infty \left( \, \beta _{mn}^{j,1}\, w^n +\overline{\beta _{mn}^{j,2}}\, w^{-n}+ \overline{\beta _{mn}^{j,1}}\,\overline{w^n} + {\beta _{mn}^{j,2}}\,\overline{w^{-n}}\,\right) \quad &{}\text{ in } \Omega _j,\quad j=1,\dots ,L,\\ \displaystyle \alpha _m F_m(z) + \overline{\alpha _m}\,\overline{F_m(z)}+\sum _{n=1}^\infty \left( \, {s_{mn}} w^{-n} + \overline{s_{mn}}\,\overline{w^{-n}}\,\right) \quad &{}\text{ in } \Omega _{L+1} \end{array}\right. } \end{aligned}$$
(5.3)

with some complex coefficients \(b_{mn}\), \(s_{mn}\), \(\beta _{mn}^{j,1}\), \(\beta _{mn}^{j,2}\), where \(z=x_1+ix_2=\Psi (w)\) for \((x_1,x_1)\in \mathbb {R}^2\setminus \overline{\Omega _0}\). We can further rewrite the z-dependent terms in (5.3), that is, in \(\Omega _0\) and \(\Omega _{L+1}\), into \(w^{\pm n}\) and \(\overline{w^{\pm n}}\). Since \(\Psi \) can be conformally extended to \(\{w\in \mathbb {C}:|w|>\gamma -\delta \}\) for some \(\delta >0\), in \(\Omega _0\), we can expand \(u_m\) into \(w^{\pm n}\) for \(|w|\in (\gamma -\delta ,\gamma )\); see the discussion at the end of Sect. 2.2. For \(z\in \Omega _{L+1}\) (i.e., \(|w|>\rho _L\)), we just apply (2.13). It then follows that

$$\begin{aligned} u_m(x)&= \displaystyle \sum _{n=1}^\infty \left( \, \beta _{mn}^{0,1}\, w^n + \overline{\beta _{mn}^{0,2}}\, w^{-n} + \overline{\beta _{mn}^{0,1}}\,\overline{w^n}+ {\beta _{mn}^{0,2}}\,\overline{w^{-n}}\,\right) \quad \text{ for } |w|\in (\gamma -\delta ,\gamma ),\\ u_m(x)&=\displaystyle \sum _{n=1}^\infty \left( \,\beta _{mn}^{L+1,1}\, w^n +\overline{\beta _{mn}^{L+1,2}}\, w^{-n} +\overline{\beta _{mn}^{L+1,1}}\,\overline{w^n}+{ \beta _{mn}^{L+1,2}}\,\overline{w^{-n}}\,\right) \quad \text{ in } \Omega _{L+1} \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} \beta _{mn}^{0,1}:=b_{mn},\quad&{\beta _{mn}^{0,2}}:=\sum _{k=1}^\infty \overline{b_{mk}}\,\overline{c_{kn}},\quad \beta _{mn}^{L+1,1} := \alpha _m \delta _{mn},\quad&\beta _{mn}^{L+1,2}:=\overline{\alpha _m}\,\overline{c_{mn}}+\overline{s_{mn}}. \end{aligned} \end{aligned}$$
(5.4)

Applying (5.3) and (5.4) to the transmission condition (3.9), we obtain that, for each \(j=0,1,\dots ,L\),

$$\begin{aligned} \begin{aligned} \displaystyle \beta _{mn}^{j+1,1}\, r_j^n + {\beta _{mn}^{j+1,2}}\,r_j^{-n}&=\displaystyle \beta _{mn}^{j,1}\, r_j^n + {\beta _{mn}^{j,2}}\,r_j^{-n},\\ \displaystyle \sigma _{j+1} \left( \beta _{mn}^{j+1,1}\, r_j^n -{\beta _{mn}^{j+1,2}}\, r_j^{-n}\right)&=\displaystyle \sigma _j\left( \beta _{mn}^{j,1}\, r_j^n -{\beta _{mn}^{j,2}}\,r_j^{-n}\right) . \end{aligned} \end{aligned}$$

In other words, we have

$$\begin{aligned} \left[ \begin{array}{c} \displaystyle \beta _{mn}^{j,1} \\ \displaystyle {\beta _{mn}^{j,2}} \end{array} \right] =T_{n}^j \begin{bmatrix} \displaystyle \beta _{mn}^{j+1,1} \\ \displaystyle {\beta _{mn}^{j+1,2}} \end{bmatrix} \quad \text{ for } j=0,1,\dots ,L \end{aligned}$$
(5.5)

with

$$\begin{aligned} T_{n}^j=\frac{1}{2}\left( 1-\frac{\sigma _{j+1}}{\sigma _j}\right) \begin{bmatrix} \displaystyle 2\lambda _j &{} \displaystyle r_j^{-2n}\\ \displaystyle r_j^{2n} &{} \displaystyle 2\lambda _j \end{bmatrix} ,\quad \lambda _j = \frac{\sigma _{j} + \sigma _{j+1}}{2(\sigma _{j} - \sigma _{j+1})}. \end{aligned}$$
(5.6)

It then directly follows from (5.4) and (5.5) that

$$\begin{aligned} \begin{bmatrix} \displaystyle \beta _{mn}^{0,1} \\ \displaystyle {\beta _{mn}^{0,2}} \end{bmatrix} =\begin{bmatrix} d_n^{(1)}&{} d_n^{(2)}\\ d_n^{(3)} &{} d_n^{(4)} \end{bmatrix} \begin{bmatrix} \displaystyle \beta _{mn}^{L+1,1} \\ \displaystyle {\beta _{mn}^{L+1,2}} \end{bmatrix} \quad \text{ with } \begin{bmatrix} d_n^{(1)}&{} d_n^{(2)}\\ d_n^{(3)} &{} d_n^{(4)} \end{bmatrix} :=T_{n}^0 T_{n}^1\cdots T_{n}^L. \end{aligned}$$
(5.7)

Note that \(d_n^{(1)},\dots ,d_n^{(4)}\) are real-valued constants that are independent of \(\alpha _m\) and m. From (5.4) and (5.7), it holds that

$$\begin{aligned} \alpha _m\, \delta _{mn} \, d_n^{(1)} + \left( \overline{\alpha _m}\,\overline{c_{mn}}+ \overline{s_{mn}} \right) d_n^{(2)}&= b_{mn}, \end{aligned}$$
(5.8)
$$\begin{aligned} \alpha _m\, \delta _{mn}\, d_n^{(3)} + \left( \overline{\alpha _m}\,\overline{c_{mn}}+ \overline{s_{mn}} \right) d_n^{(4)}&= \sum _{k=1}^\infty \overline{b_{mk}}\, \overline{c_{kn}}. \end{aligned}$$
(5.9)

As \(d_n^{(1)},\dots ,d_n^{(4)}\) are real-valued, it follows by plugging (5.8) into (5.9) that

$$\begin{aligned} \alpha _m\, \delta _{mn}\, d_n^{(3)} + (\overline{\alpha _m}\,\overline{c_{mn}}+ \overline{s_{mn}}) d_n^{(4)}&= \sum _{k=1}^\infty \overline{b_{mk}} \overline{c_{kn}} \nonumber \\&= \sum _{k=1}^\infty \left( \overline{\alpha _m}\, \delta _{mk}\,d_k^{(1)} \, \overline{c_{kn}} + \left( \alpha _m \,c_{mk} + s_{mk}\right) \, d_k^{(2)} \overline{c_{kn}} \right) . \end{aligned}$$
(5.10)

From (5.4) and (5.5), one can express the coefficients of the solution (5.3) in terms of \(s_{mn}\). Since (3.1) is linear with respect to the background field H, we can express \(s_{mn}\) in (5.3) as, fixing m and n, \(s_{mn}=\alpha _m \, t_1 +\overline{\alpha _m}\, t_2\) for some complex values \(t_1\) and \(t_2\) independent of \(\alpha _m\). In view of the geometric multipole expansion (3.7), we have

$$\begin{aligned} s_{mn} = - \alpha _m \frac{\mathbb {F}_{mn}^{(1)}}{4\pi n} - \overline{\alpha _m}\, \frac{\mathbb {F}_{mn}^{(2)}}{4\pi n}. \end{aligned}$$
(5.11)

Hence, one can get the FPTs from \(s_{mn}\), and vice versa. In Sect. 5.3, from (5.10), we will deduce the matrix expressions for the FPTs, which involve the diagonal matrix with the diagonal entries \(d_n^{(j)}\), that is,

$$\begin{aligned} D_j:=\left( d_n^{(j)} \delta _{mn}\right) _{m,n=1}^\infty ,\quad 1\le j\le 4. \end{aligned}$$
(5.12)

5.2 Asymptotic behavior of \(d_n^{(2)}\) and \(d_n^{(4)}\) as \(n\rightarrow \infty \)

For notational convenience, we denote the multiplication of the scaling constant in (5.6) by

$$\begin{aligned} \tau _L := \frac{1}{2^{L+1}}\left( 1-\frac{\sigma _1}{\sigma _0}\right) \left( 1-\frac{\sigma _2}{\sigma _1}\right) \cdots \left( 1-\frac{\sigma _{L+1}}{\sigma _L}\right) = \frac{1}{\left( 2\lambda _0+1 \right) \left( 2\lambda _1+1 \right) \cdots \left( 2\lambda _L+1 \right) }. \end{aligned}$$

It follows from determinants of the \(2\times 2\)-matrix in (5.6) and (5.7) that

$$\begin{aligned} d_n^{(1)} d_n^{(4)} - d_n^{(2)} d_n^{(3)} = \tau _L^2 \left( 4\lambda _0^2-1 \right) \left( 4\lambda _1^2-1 \right) \cdots \left( 4\lambda _L^2-1 \right) = \frac{\sigma _1 \sigma _2 \cdots \sigma _{L+1}}{\sigma _0 \sigma _1 \cdots \sigma _L} = \frac{\sigma _{L+1}}{\sigma _0}. \end{aligned}$$
(5.13)

For the case \(L=1\), we obtain

$$\begin{aligned} \begin{aligned} d_n^{(1)}&= \tau _1 \left( 4\lambda _0\lambda _1 + r_0^{-2n} r_1^{2n}\right) = \tau _1 r_0^{-2n} r_1^{2n} + o\left( r_0^{-2n} r_1^{2n}\right) ,\\[2mm] d_n^{(2)}&= \tau _1 \left( 2\lambda _0 r_1^{-2n} + 2 \lambda _1 r_0^{-2n} \right) = 2 \tau _1 \lambda _1 r_0^{-2n} + o\left( r_0^{-2n}\right) ,\\[2mm] d_n^{(3)}&= \tau _1 \left( 2\lambda _0 r_1^{2n} + 2 \lambda _1 r_0^{2n} \right) = 2 \tau _1 \lambda _0 r_1^{2n} + o\left( r_1^{2n}\right) ,\\[2mm] d_n^{(4)}&= \tau _1 \left( 4\lambda _0\lambda _1 + r_0^{2n} r_1^{-2n} \right) = 4 \tau _1 \lambda _0\lambda _1 + o(1), \end{aligned} \end{aligned}$$

where \(o(\cdot )\) denotes the standard little-\(\mathrm o\) notation. If \(L=2\), we have

$$\begin{aligned} d_n^{(1)}&= \tau _2 \left( 8\lambda _0 \lambda _1 \lambda _2 + 2\lambda _0 r_1^{-2n}r_2^{2n} + 2 \lambda _1 r_0^{-2n} r_2^{2n} + 2 \lambda _2 r_0^{-2n} r_1^{2n} \right) = 2 \tau _2 \lambda _1 r_0^{-2n} r_2^{2n} + o\left( r_0^{-2n} r_2^{2n}\right) ,\\[2mm] d_n^{(2)}&= \tau _2 \left( 4\lambda _0\lambda _1 r_2^{-2n} + 4\lambda _0 \lambda _2 r_1^{-2n} + 4 \lambda _1 \lambda _2 r_0^{-2n} + r_0^{-2n} r_1^{2n} r_2^{-2n}\right) = 4 \tau _2 \lambda _1 \lambda _2 r_0^{-2n} + o\left( r_0^{-2n}\right) ,\\[2mm] d_n^{(3)}&= \tau _2 \left( 4\lambda _0 \lambda _1 r_2^{2n} + 4 \lambda _0 \lambda _2 r_1^{2n} + 4 \lambda _1 \lambda _2 r_0^{2n} + r_0^{2n} r_1^{-2n}r_2^{2n}\right) = 4 \tau _2 \lambda _0 \lambda _1 r_2^{2n} + o\left( r_2^{2n}\right) ,\\[2mm] d_n^{(4)}&= \tau _2 \left( 8\lambda _0 \lambda _1 \lambda _2 + 2\lambda _0 r_1^{2n} r_2^{-2n} + 2 \lambda _1 r_0^{2n} r_2^{-2n} + 2 \lambda _2 r_0^{2n} r_1^{-2n} \right) = 8 \tau _2 \lambda _0 \lambda _1 \lambda _2 + o(1). \end{aligned}$$

For general L, one can show the following.

Lemma 5.1

For each fixed L, \(d_n^{(1)},\dots ,d_n^{(4)}\) satisfy the asymptotic behavior as \(n\rightarrow \infty \):

$$\begin{aligned} d_n^{(1)}&= 2^{L-1 } \tau _L \lambda _1 \lambda _2 \cdots \lambda _{L-1} r_0^{-2n} r_L^{2n} + o(r_0^{-2n} r_L^{2n}), \\[2mm] d_n^{(2)}&= 2^L \tau _L \lambda _1 \lambda _2 \cdots \lambda _L r_0^{-2n} + o(r_0^{-2n}),\\[2mm] d_n^{(3)}&= 2^L \tau _L \lambda _0 \lambda _1 \cdots \lambda _{L-1} r_L^{2n} + o(r_L^{2n}), \\[2mm] d_n^{(4)}&= 2^{L+1} \tau _L \lambda _0 \lambda _1 \cdots \lambda _L + o(1). \end{aligned}$$

By Lemma 5.1, we have \(d_n^{(4)}\ne 0\) for sufficiently large n. Furthermore, it holds for \(n\gg 1\) that

$$\begin{aligned} I-D_2 \overline{C} D_4^{-1} D_2 {C} D_4^{-1} \approx \lambda _0^{-2} \left( \lambda _0^2 I - \frac{r_0^{-2\mathbb {N}}\overline{ C} r_0^{-2\mathbb {N}} {C}}{4} \right) , \end{aligned}$$
(5.14)

where C denotes the Grunsky matrix (see (2.15)). Note that the right-hand side of (5.14) is invertible from (2.17). For all examples in Sect. 6.2, the \(100\times 100\)-dimensional truncated matrices of \(D_4\) and \(D_4 - D_2 \overline{C} D_4^{-1} D_2 {C}\) are invertible.

From now on, we assume that \(d_n^{(4)}\ne 0\) for all n so that \(D_4\) is invertible. We also assume that \(D_4 - D_2 \overline{C} D_4^{-1} D_2 {C}\) is invertible.

5.3 Matrix expression for FPTs using the Grunsky matrix

We set the two semi-infinite matrices \(\varvec{\alpha }\) and S as

$$\begin{aligned} \varvec{\alpha }=\big (\alpha _m \delta _{mn}\big )_{m,n=1}^\infty ,\quad S=\big (s_{mn}\big )_{m,n=1}^\infty . \end{aligned}$$

It then follows from (5.10) that

$$\begin{aligned}{} & {} \displaystyle \varvec{\alpha } D_3 + (\overline{\varvec{\alpha }}\, \overline{C} + \overline{S}) D_4=\displaystyle \overline{\varvec{\alpha }}\, D_1 \overline{C} + \left( \varvec{\alpha } C+ S \right) D_2 \overline{C}, \end{aligned}$$
(5.15)
$$\begin{aligned}{} & {} \displaystyle \varvec{\alpha } C + S = \displaystyle \varvec{\alpha } D_1 C D_4^{-1} + \left( \overline{\varvec{\alpha }} \overline{C}+ \overline{S} \right) D_2 C D_4^{-1} - \overline{\varvec{\alpha }} D_3 D_4^{-1}. \end{aligned}$$
(5.16)

Here, (5.16) is a direct consequence of (5.15) by taking complex conjugates. Substituting (5.16) into (5.15), we obtain

$$\begin{aligned} \left( \overline{\varvec{\alpha }}\, \overline{C} + \overline{S}\right) \left( D_4-D_2 C D_4^{-1} D_2 \overline{C}\right) = \overline{\varvec{\alpha }}\, D_1 \overline{C} + \varvec{\alpha } D_1 C D_4^{-1} D_2 \overline{C} - \overline{\varvec{\alpha }}\, D_3 D_4^{-1} D_2 \overline{C} - \varvec{\alpha } D_3 \end{aligned}$$

and, thus,

$$\begin{aligned} \overline{S} = \left( \overline{\varvec{\alpha }}\, D_1 \overline{C} + \varvec{\alpha } D_1 C D_4^{-1} D_2 \overline{C} - \overline{\varvec{\alpha }}\, D_3 D_4^{-1} D_2 \overline{C} - \varvec{\alpha } D_3 \right) \left( D_4-D_2 C D_4^{-1} D_2 \overline{C}\right) ^{-1} - \overline{\varvec{\alpha }} \,\overline{C}. \end{aligned}$$

Applying (5.11), we then derive

$$\begin{aligned}&-\left( \overline{\varvec{\alpha }}\, \overline{\mathbb {F}^{(1)}} + \varvec{\alpha }\, \overline{\mathbb {F}^{(2)}}\right) (4\pi \mathbb {N})^{-1} \\&\quad = \left( \,\overline{\varvec{\alpha }}\, D_1 \overline{C} + \varvec{\alpha } D_1 C D_4^{-1} D_2 \overline{C} - \overline{\varvec{\alpha }} \, D_3 D_4^{-1} D_2 \overline{C} - \varvec{\alpha } D_3 \right) \left( D_4-D_2 C D_4^{-1} D_2 \,\overline{C}\,\right) ^{-1} - \overline{\varvec{\alpha }}\, \overline{C}, \end{aligned}$$

or equivalently,

$$\begin{aligned}&\varvec{\alpha } \, \left( \left( D_3-D_1 C D_4^{-1} D_2 \,\overline{C}\,\right) \left( D_4-D_2 C D_4^{-1} D_2 \,\overline{C}\,\right) ^{-1} - \overline{\mathbb {F}^{(2)}} \left( 4\pi \mathbb {N}\right) ^{-1}\, \right) \nonumber \\&\quad =\, \overline{\varvec{\alpha }} \, \left( \left( D_1 D_4 - D_2 D_3\right) \, D_4^{-1}\, \overline{C} \, \left( D_4-D_2 C D_4^{-1} D_2 \overline{C}\right) ^{-1} - \overline{C} + \overline{\mathbb {F}^{(1)}} \left( 4\pi \mathbb {N}\right) ^{-1} \right) . \end{aligned}$$
(5.17)

Since \(\varvec{\alpha }\) is diagonal and corresponds to an arbitrary entire function H, both sides of (5.17) vanish. Furthermore, from (5.13), the diagonal entry of \( D_1 D_4 - D_2 D_3\) is as follows: for each \(m\in \mathbb {N}\),

$$\begin{aligned} \big [ D_1 D_4 - D_2 D_3 \big ]_{mm} = d_m^{(1)}d_m^{(4)} - d_m^{(2)} d_m^{(3)} = \frac{\sigma _{L+1}}{\sigma _0}. \end{aligned}$$

Hence, we have the following theorem.

Theorem 5.2

We set C and \(D_j\) as in (2.15) and (5.12), respectively. The FPTs of the multi-coated structure \(\Omega \) given at the beginning of Sect. 5 have the following matrix expressions, given that \(D_4\) and \(D_4 - D_2 \overline{C} D_4^{-1} D_2 C\) are invertible: for each \(m,n\in \mathbb {N}\),

$$\begin{aligned} \mathbb {F}_{mn}^{(1)}(\Omega ,\varvec{\sigma })&= 4\pi n \, c_{mn} - 4\pi n \, \sigma _0^{-1} \sigma _{L+1} \left[ D_4^{-1} C \, \left( D_4-D_2\, \overline{C} \, D_4^{-1} D_2 \, C \right) ^{-1} \right] _{mn},\\ \mathbb {F}_{mn}^{(2)}(\Omega ,\varvec{\sigma })&= 4\pi n \left[ \left( D_3-D_1\, \overline{C}\, D_4^{-1} D_2\, C \right) \left( D_4-D_2 \,\overline{C} \, D_4^{-1} D_2\, C \right) ^{-1}\right] _{mn}. \end{aligned}$$

5.4 FPTs for special cases

5.4.1 Multi-coated inclusion with an elliptic core

When \(\Omega _0\) is a disk, \(c_{mn}=0\) for all \(m,n\in \mathbb {N}\). Thus, C is identical to the zero matrix. Hence, we have the following corollary.

Corollary 5.3

([6]) If \(\Omega _0\) is a disk, then the FPTs of the multi-coated inclusion \(\Omega \) satisfy

$$\begin{aligned} \mathbb {F}_{mn}^{(1)}(\Omega ,\varvec{\sigma })=0,\quad \mathbb {F}_{mn}^{(2)}(\Omega ,\varvec{\sigma }) = 4\pi m \, \frac{d_m^{(3)}}{d_m^{(4)}}\delta _{mn}\quad \text{ for } \text{ all } m,n\in \mathbb {N}. \end{aligned}$$

For the case when \(\Psi (w)=w+a_0+\frac{a_1}{w}\), one can easily derive from (2.14) that

$$\begin{aligned} c_{mn}=\delta _{mn}\, a_1^n\quad \text{ for } \text{ all } m,n\in \mathbb {N}. \end{aligned}$$
(5.18)

Hence, C and FPTs have diagonal matrix forms.

Corollary 5.4

If \(\Omega _0\) is an ellipse, the FPTs of the multi-coated ellipse \(\Omega \) are diagonal matrices:

$$\begin{aligned} \mathbb {F}_{mn}^{(1)}(\Omega ,\varvec{\sigma })&= 4\pi m \, a_1^m \delta _{mn}- \frac{4\pi m \, a_1^m \sigma _0^{-1} \sigma _{L+1} }{ \big (d_m^{(4)}\big )^2 - \big (d_m^{(2)}\big )^2 |a_1|^{2m}}\delta _{mn},\\ \mathbb {F}_{mn}^{(2)}(\Omega ,\varvec{\sigma })&= 4\pi m \,\frac{d_m^{(3)} d_m^{(4)}-d_m^{(1)} d_m^{(2)} |a_1|^{2m}}{ \big (d_m^{(4)}\big )^2 - \big (d_m^{(2)}\big )^2 |a_1|^{2m}}\delta _{mn}\qquad \text{ for } \text{ all } m,n\in \mathbb {N}. \end{aligned}$$

In particular, for \(L=1\), it holds that, for each \(m\in \mathbb {N}\),

$$\begin{aligned} \mathbb {F}_{mm}^{(1)}(\Omega ,\varvec{\sigma })&=\displaystyle 4\pi m \, a_1^m - \frac{4\pi m \, a_1^m (4\lambda _0^2-1)(4\lambda _1^2-1)}{\left( 4\lambda _0\lambda _1 + \frac{r_0^{2m}}{r_1^{2m}} \right) ^2 - 4 \left( \frac{\lambda _0}{r_1^{2m}} + \frac{\lambda _1}{r_0^{2m}}\right) ^2 |a_1|^{2m} },\\[2mm] \mathbb {F}_{mm}^{(2)}(\Omega ,\varvec{\sigma })&= \displaystyle 8\pi m \frac{ \left( 4\lambda _0\lambda _1 + \frac{r_0^{2m}}{r_1^{2m}} \right) \left( \lambda _0 r_1^{2m} + \lambda _1 r_0^{2m}\right) - \left( 4\lambda _0\lambda _1 + \frac{r_1^{2m}}{r_0^{2m}} \right) \left( \frac{\lambda _0}{r_1^{2m}} + \frac{\lambda _1}{r_0^{2m}} \right) |a_1|^{2m} }{\left( 4\lambda _0\lambda _1 + \frac{r_0^{2m}}{r_1^{2m}} \right) ^2 - 4 \left( \frac{\lambda _0}{r_1^{2m}} + \frac{\lambda _1}{r_0^{2m}} \right) ^2 |a_1|^{2m}}. \end{aligned}$$

Remark 5.1

For the simply connected inclusion (that is, \(L=0\)), we get

$$\begin{aligned} \begin{bmatrix} \displaystyle d_n^{(1)} &{} \displaystyle d_n^{(2)}\\ \displaystyle d_n^{(3)} &{} \displaystyle d_n^{(4)} \end{bmatrix} = \frac{1}{2\lambda +1} \begin{bmatrix} \displaystyle 2\lambda &{} \displaystyle \gamma ^{-2n}\\ \displaystyle \gamma ^{2n} &{} \displaystyle 2\lambda \end{bmatrix} \quad \text{ with } \lambda = \frac{\sigma _0+1}{2(\sigma _0-1)} \end{aligned}$$

and, hence,

$$\begin{aligned} D_1 = D_4 = \left( \frac{2\lambda }{2\lambda +1}\right) I, \qquad D_2 = \left( \frac{1}{2\lambda +1}\right) \gamma ^{-2\mathbb {N}}, \qquad D_3 = \left( \frac{1}{2\lambda +1}\right) \gamma ^{2\mathbb {N}}. \end{aligned}$$

Applying the above relations to Theorem 5.2, we obtain the same results as Theorem 4.2.

Fig. 2
figure 2

Domains with rotational symmetry of orders 2, 3, and 5

Full size image

5.4.2 Multi-coated inclusion with rotational symmetry

Let us consider some properties of FPTs for a multi-coated structure with rotational symmetry. Let N be a positive integer. A domain is said to have rotational symmetry of order N if it is invariant under the rotation by \(2\pi /N\) (see Fig. 2). Then, the exterior conformal mapping \(\Psi \) corresponding to the domain satisfies

$$\begin{aligned} \Psi (w) = e^{\frac{2\pi i}{N}}\Psi \left( e^{-\frac{2\pi i}{N}}w\right) \quad \text{ on } |w |= \gamma , \end{aligned}$$

which is equivalent to

$$\begin{aligned} a_n = 0 \quad \text{ for } n+1\not \equiv 0 \ (\text{ mod } N). \end{aligned}$$

Here, \(m\not \equiv 0\) (mod N) means that \(m=kN+r\) for some \(k\in \mathbb {N}\) and \(r=1,\dots ,N-1\). It follows by induction using (2.14) that the Grunsky coefficients satisfy

$$\begin{aligned} c_{mn} = 0 \quad \text{ for } m+n \not \equiv 0 \ (\text{ mod } N). \end{aligned}$$
(5.19)

Let us define two collections of semi-infinite matrices. The first is a set of diagonally striped infinite matrices

$$\begin{aligned} \mathcal {S}^+_N := \left\{ S = (S_{mn})_{m,n\in \mathbb {N}} : S_{mn} = 0 \quad \text{ for } m-n \not \equiv 0 \ (\text{ mod } N) \right\} , \end{aligned}$$

and the second is a set of anti-diagonally striped infinite matrices

$$\begin{aligned} \mathcal {S}^-_N :=\left\{ S= (S_{mn})_{m,n\in \mathbb {N}} : S_{mn} = 0 \quad \text{ for } m+n \not \equiv 0 \ (\text{ mod } N)\right\} . \end{aligned}$$

For instance, the following matrices A and B belong to \(\mathcal {S}^+_3\) and \(\mathcal {S}^-_3\), respectively:

$$\begin{aligned} A= \left[ \begin{array}{ccccccc} * &{} \ 0 &{}\ 0 &{}\ * &{}\ 0 &{}\ 0 &{} \cdots \\ 0 &{} \ * &{}\ 0 &{}\ 0 &{}\ * &{}\ 0 &{} \cdots \\ 0 &{}\ 0 &{}\ * &{}\ 0 &{}\ 0 &{}\ * &{} \cdots \\ * &{}\ 0 &{}\ 0 &{}\ * &{}\ 0 &{}\ 0 &{} \cdots \\ 0 &{}\ * &{}\ 0 &{}\ 0 &{}\ * &{}\ 0 &{} \cdots \\ 0 &{}\ 0 &{}\ * &{}\ 0 &{}\ 0 &{}\ * &{} \cdots \\ \vdots &{}\ \vdots &{}\ \vdots &{}\ \vdots &{}\ \vdots &{}\ \vdots &{}\ \ddots \end{array} \right] , \quad B= \left[ \begin{array}{ccccccc} 0 &{}\ * &{}\ 0 &{}\ 0 &{}\ * &{}\ 0 &{} \cdots \\ * &{}\ 0 &{}\ 0 &{}\ * &{}\ 0 &{}\ 0 &{} \cdots \\ 0 &{}\ 0 &{}\ * &{}\ 0 &{}\ 0 &{}\ * &{} \cdots \\ 0 &{}\ * &{}\ 0 &{}\ 0 &{}\ * &{}\ 0 &{} \cdots \\ * &{}\ 0 &{}\ 0 &{}\ * &{}\ 0 &{}\ 0 &{} \cdots \\ 0 &{}\ 0 &{}\ * &{}\ 0 &{}\ 0 &{}\ * &{} \cdots \\ \vdots &{}\ \vdots &{}\ \vdots &{}\ \vdots &{}\ \vdots &{}\ \vdots &{}\ \ddots \end{array} \right] , \end{aligned}$$

where \(*\) represents elements that can be nonzero. One can easily find that the following product rules hold.

Lemma 5.5

For \(U^+,V^+\in \mathcal {S}^+_N\) and \(U^-,V^-\in \mathcal {S}^-_N\), we have

$$\begin{aligned} \begin{aligned}&\displaystyle U^+V^+,\, U^-V^- \in \mathcal {S}^+_N,\\&\displaystyle U^+U^-,\, U^-U^+ \in \mathcal {S}^-_N. \end{aligned} \end{aligned}$$

Proposition 5.6

Let \(\Omega \) be a multi-coated inclusion given at the beginning of this section. If the core \(\Omega _0\) has a rotational symmetry of order N, it follows that

$$\begin{aligned}&\mathbb {F}_{mn}^{(1)}(\Omega , \varvec{\sigma }) = 0 \quad \text{ for } m+n \not \equiv 0 \ (\text{ mod } N),\\&\mathbb {F}_{mn}^{(2)}(\Omega , \varvec{\sigma }) = 0 \quad \text{ for } m-n \not \equiv 0 \ (\text{ mod } N). \end{aligned}$$

Proof. From (5.19), we have

$$\begin{aligned} C \in \mathcal {S}^-_N. \end{aligned}$$
(5.20)

Since \(D_j\) in Theorem 5.2 is a diagonal matrix, we have \(D_j, D^{-1}_j \in \mathcal {S}^+_N\). It follows from Lemma 5.5 and (5.20) that

$$\begin{aligned} \begin{aligned} D_3-D_1\, \overline{C}\, D_4^{-1} D_2\, C&\in \mathcal {S}^+_N, \\ D_4-D_2 \,\overline{C} \, D_4^{-1} D_2\, C&\in \mathcal {S}^+_N, \end{aligned} \end{aligned}$$
(5.21)

and \( D_2 \, C \, D_4^{-1} \in \mathcal {S}^-_N. \) Hence, we have

$$\begin{aligned} (D_2 \,\overline{C} \, D_4^{-1} D_2\, C \, D_4^{-1})^k \in \mathcal {S}^+_N. \end{aligned}$$

Thus, the geometric series on \(\left( I - D_2 \,\overline{C} \, D_4^{-1} D_2\, C \, D_4^{-1} \right) ^{-1}\) implies that

$$\begin{aligned} \left( D_4-D_2 \,\overline{C} \, D_4^{-1} D_2\, C \right) ^{-1}&= D_4^{-1} \left( I - D_2 \,\overline{C} \, D_4^{-1} D_2\, C \, D_4^{-1} \right) ^{-1} \nonumber \\&= D_4^{-1} \sum _{k=0}^{\infty } \left( D_2 \,\overline{C} \, D_4^{-1} D_2\, C \, D_4^{-1} \right) ^{k}\in \mathcal {S}^+_N. \end{aligned}$$
(5.22)

By using (5.21), (5.22), and Theorem 5.2, we finally obtain that

$$\begin{aligned} \left[ \frac{\mathbb {F}_{mn}^{(1)}(\Omega ,\varvec{\sigma })}{4\pi n} \right] _{m,n=1}^\infty&= C - \sigma _0^{-1} \sigma _{L+1} D_4^{-1} C \, \left( D_4-D_2\, \overline{C} \, D_4^{-1} D_2 \, C \right) ^{-1} \in \mathcal {S}^-_N,\\ \left[ \frac{\mathbb {F}_{mn}^{(2)}(\Omega ,\varvec{\sigma })}{4\pi n} \right] _{m,n=1}^\infty&= \left( D_3-D_1\, \overline{C}\, D_4^{-1} D_2\, C \right) \left( D_4-D_2 \,\overline{C} \, D_4^{-1} D_2\, C \right) ^{-1} \in \mathcal {S}^+_N, \end{aligned}$$

which completes the proof. \(\square \)

6 Construction of semi-neutral inclusions

In this section, as an application of FPTs, we construct semi-neutral inclusions that are layered structures, whose layers, except for the core, are images of concentric annuli via the exterior conformal mapping of the core (see Definition 6.1). For such a multi-coated inclusion, we mainly consider the material parameters in the layers, unlike the approach in [22], in which the shapes of coating layers were determined by a shape optimization procedure.

For the concentric disks, \(\mathbb {F}^{(1)}=0\) [6]. In view of the expressions in Theorem 5.2 with the Grunsky matrix C associated with the core \(\Omega _0\), one can deduce that \(\mathbb {F}^{(1)}\) significantly depends on the shape of \(\Omega _0\). Hence, one cannot find a neutral inclusion of the considered multilayered geometry except the concentric disks. Instead, we can obtain semi-neutral inclusions with the core of general shape that are not perfectly neutral but show relatively negligible field perturbations for low-order polynomial loadings.

For the simply connected domain of general shape, \(\mathbb {F}^{(1)}\) is not zero but is still significantly smaller than \(\mathbb {F}^{(2)}\), and, hence, \(\mathbb {F}^{(2)}\) mainly contributes to the field perturbation. For a given core \(\Omega _0\) of general shape, one can construct layered structures that show relatively small field perturbations. We choose the number of layers and determine the appropriate conductivity values in the coating layers such that the multi-coated inclusions satisfy the following condition.

Definition 6.1

(Semi-neutral inclusion) Let \(\Omega \) be a multi-coated inclusion with the core \(\Omega _0\) and L layers of coating, as in Sect. 3.2. We say that \(\Omega \) is a semi-neutral inclusion provided that, for some \(N_1, N_2\in \mathbb {N}\), the second FPTs are negligible for low-order terms, that is,

$$\begin{aligned} \frac{\mathbb {F}^{(2)}_{mn}(\Omega ,\varvec{\sigma })}{4\pi n} \approx 0\quad \text{ for } \, m\le N_1 \text{ and } n\le N_2. \end{aligned}$$

6.1 Numerical scheme

For a given \(\Omega _0\), we denote \(\gamma \), \(\Psi (w)\), and \((\rho ,\theta )\) as in Sect. 2.2. We set \(\Omega _j\) (\(j=1,\dots ,L+1\)) as in (5.1) and (5.2). We assume that \(\sigma _{L+1}=1\) and that the conductivity \(\sigma _0\) is fixed.

We find \(\varvec{\sigma } = (\sigma _1,\dots ,\sigma _L)\) such that the condition Definition 6.1 holds. In other words, we look for \(\varvec{\sigma }\) that satisfies the equation

$$\begin{aligned} \varvec{f}(\varvec{\sigma })\approx 0, \end{aligned}$$
(6.1)

where \(\varvec{f}:(\mathbb {R}^+)^{L}\rightarrow \mathbb {R}^{N_1N_2}\) is a nonlinear vector-valued function \(\varvec{\sigma }\longmapsto (f_1,...,f_{N_1N_2})\) given by

$$\begin{aligned} f_l = \left| \frac{\mathbb {F}^{(2)}_{mn}}{4\pi n}\right| ,\quad l=N_2(m-1)+n, \end{aligned}$$

for \(1 \le m \le N_1\), \(1\le n \le N_2\). To find \(\varvec{\sigma }\) satisfying (6.1), we use the multivariate Newton’s method:

$$\begin{aligned} \varvec{\sigma }^{(k+1)} = \varvec{\sigma }^{(k)} - \alpha \, \varvec{J}^{\dagger }\big [\varvec{\sigma }^{(k)}\big ]\, \varvec{f}\big [\varvec{\sigma }^{(k)}\big ], \end{aligned}$$
(6.2)

where \(\alpha \) is a constant in (0, 1), and k indicates the iteration step, and \(\varvec{J}^\dagger \) denotes the pseudo-inverse of the Jacobian matrix of \(\varvec{f}\). We give the initial guess \(\varvec{\sigma }^{(0)}\) as a proper alternating series.

We calculate \(\varvec{f}(\varvec{\sigma }^{(k)})\) by Theorem 5.2, where we truncate the semi-infinite matrices \(C, D_2, D_4\) to be \(100\times 100\) matrices. To obtain \(\varvec{J}^{\dagger }[\varvec{\sigma }^{(k)}]\), we use the finite difference approximation of the partial derivatives of \(\varvec{f}\). All the computations in this paper are performed by MATLAB R2020a software. We iterate the multivariate Newton’s method until the conductivity distribution satisfies the stopping condition:

$$\begin{aligned} \frac{|\varvec{\sigma }^{(k+1)}-\varvec{\sigma }^{(k)}|}{|\varvec{\sigma }^{(k)}|} < 10^{-15}. \end{aligned}$$

6.2 Numerical examples

6.2.1 Elliptical coated inclusions

Let \(\Omega \) be a multi-coated inclusion of the form (5.1) with \(\Omega _0\) given by \(\Psi (w)= w+\frac{0.25}{w}\) with \(\gamma =1\). The corresponding Grunsky matrix is diagonal (see (5.18)), and thus, \(\mathbb {F}^{(1)}\) and \(\mathbb {F}^{(2)}\) are diagonal by Theorem 5.2.

Example 6.1

We construct semi-neutral inclusions with \(L=1,2\). For \(L=1\), we set \(r_1=1.1\) and \((N_1,N_2) = (1,1)\); for \(L=2\), we set \((r_1,r_2)=(1.1,\, 1.2)\) and \((N_1,N_2) = (2,2)\). The conductivities in the core and the background are set to be \(\sigma _0=5\) and \(\sigma _{L+1}=1\). We set the initial guess for the iteration (6.2) as \(\varvec{\sigma }^{(0)}=(\sigma _1^{(0)},\dots ,\sigma _L^{(0)})\) with \(\sigma _j^{(0)} = 2^{(-1)^j} \quad \text{ for } j=1,\dots ,L\). We then choose the conductivity values in the coating layers by following the numerical scheme in Sect. 6.1. The resulting values are \(\sigma _1=0.1216\) for the 1-coated ellipse (i.e., \(L=1\)), and \((\sigma _1,\sigma _2)=(0.0782,\, 3.6782)\) for the 2-coated ellipse (i.e., \(L=2\)), respectively.

Figure 3 indicates the potential perturbation due to the uncoated (first column), 1-coated (second column), and 2-coated (third column) inclusions, where the background field is either \(H(x)=x_1\) or \(H(x)=x_2\). Colored curves represent level curves of the perturbed potential u, where we compute u based on the boundary integral formulation for the conductivity transmission problem with the Nyström discretization; we refer the reader to [4, Chapter 17] for the details and computation codes. Coated ellipses exhibit much smaller perturbations than the uncoated ellipse. Table 1 shows the first two diagonal elements of FPTs and verifies that the obtained coated ellipses are semi-neutral inclusions with \(L=1,2\).

Fig. 3
figure 3

Semi-neutral coated ellipses exhibit much smaller perturbations than the uncoated one

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Table 1 FPTs for the ellipse and coated ellipses in Fig. 3. Since the core is an ellipse, \(\mathbb {F}_{mn}^{(1)}=\mathbb {F}^{(2)}_{mn}=0\) for \(m\ne n\). The values of the first FPTs, \(\mathbb {F}^{(1)}_{mn}\), of the coated ellipses are similar to those corresponding to the core, i.e., the uncoated inclusion. The second FPTs, \(\mathbb {F}^{(2)}_{mn}\), vanish for \(m,n\le N\) with \(N=1\) for the 1-coated inclusion and \(N=2\) for the 2-coated inclusion
Full size table

6.2.2 Non-elliptical shapes

In this subsection, we provide two examples of non-elliptical shapes. It is necessary to find the conductivity distribution that makes a coated inclusion semi-neutral. As in Example 6.1, the uniform background field is given by \(H(x)=x_1\) or \(H(x)=x_2\).

Example 6.2

Let us observe a kite-shaped domain with exterior conformal mapping,

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

There is no rotational symmetry for this domain so that the vanishing property in Proposition 5.6 does not hold. Figure 4 illustrates the potential perturbation u in the background field H. Table 2 shows the low-order FPTs. Each coated kite is semi-neutral because its second FPT is significantly smaller than the uncoated one.

In Fig. 4, the background field is given by \(H(x)=x_1\). Colored curves represent contours of u. We commonly set \(\gamma = 1\) and \(\sigma _0 = 100\). We set the initial guess for the iteration (6.2) as \(\varvec{\sigma }^{(0)}=(\sigma _1^{(0)},\dots ,\sigma _L^{(0)})\) with \(\sigma _j^{(0)} =2^{(-1)^j}\) for \(j=1,\dots ,L\), except that we set \(\sigma _j^{(0)} = 10^{(-1)^j}\) instead of \(\sigma _j^{(0)} = 2^{(-1)^j}\) for the 3-coated kite in Fig. 4.

For the 1-coated kite, \(r_1 = 1.1\), \(\sigma _1=0.0964\), and \((N_1,N_2) = (1,1)\). For the 2-coated kite, \((r_1,r_2) = (1.1,\,1.2)\), \((\sigma _1, \sigma _2)= (4.3533\times 10^{-6}, \,11.5212)\), and \((N_1,N_2) = (1,2)\). For the 3-coated kite, \((r_1,r_2,r_3) = (1.1,\,1.2,\,1.3)\), \((\sigma _1, \sigma _2, \sigma _3)= (5.8354\times 10^{-6}, \,17.0950, \, 0.2358)\), and \((N_1,N_2) = (2,2)\).

Fig. 4
figure 4

Semi-neutral coated kites show much smaller perturbations than the uncoated one

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Table 2 Low-order FPTs of the uncoated and coated kites in Fig. 4. As we coat the inclusion, the first FPTs are almost invariant, but the second FPTs decrease significantly
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Example 6.3

In this example, we assume that the core \(\Omega _0\) is given by the conformal mapping

$$\Psi (w) = w + \frac{0.2}{w^4}.$$

Since the resulting star-shape domain has rotational symmetry of order 5, the associated FPTs show the following periodicity from Proposition 5.6:

$$\begin{aligned}&\mathbb {F}_{mn}^{(1)} = 0 \quad \text{ for } m+n \not \equiv 0 \ (\text{ mod } 5),\\&\mathbb {F}_{mn}^{(2)} = 0 \quad \text{ for } m-n \not \equiv 0 \ (\text{ mod } 5). \end{aligned}$$

Then, \(\mathbb {F}_{mn}^{(1)}=0\) for all \(m,n\le 2\), and \(\mathbb {F}_{mn}^{(2)}=0\) for all \(m\ne n\) with \(m,n \le 5\). Hence, we focus only on vanishing the nonzero leading terms of FPTs. Figure 5 illustrates contours of the potential perturbation for a given background field H. Table 3 compares the leading FPTs of domains in Fig. 5.

In Fig. 5, the background field is given by \(H(x)=x_2\). Colored curves represent contours of u. We commonly set \(\gamma = 1\) and \(\sigma _0 = 0.25\). The initial guess for the iteration (6.2) is given by \(\varvec{\sigma }^{(0)}=(\sigma _1^{(0)},\dots ,\sigma _L^{(0)})\) with \(\sigma _j^{(0)} = 10^{(-1)^{j+1}} \quad \text{ for } j=1,\dots ,L\).

For the 1-coated star, \(r_1 = 1.1\), \(\sigma _1= 7.0224\), and \((N_1,N_2)=(1,1)\). For the 2-coated star, \((r_1,r_2) = (1.1,\,1.2)\), \((\sigma _1, \sigma _2)= (11.4177, \,0.2763)\), and \((N_1,N_2)=(2,2)\). For the 3-coated star, \((r_1,r_2,r_3) = (1.1,\,1.2,\,1.3)\), \((\sigma _1, \sigma _2, \sigma _3)= (540.1332,\, 0.0595,\, 4.2144)\), and \((N_1,N_2)=(2,6)\).

Fig. 5
figure 5

Semi-neutral coated stars have quite smaller perturbations than the uncoated star

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Table 3 Leading terms of the FPTs for the uncoated and coated stars in Fig. 5. As we coat the inclusion, the second FPTs vanish
Full size table

7 Conclusion

By modifying the classical multipole expansion and using conformal mapping, we presented the new concept of geometric multipole expansion for the conductivity interface problem in two dimensions with a simply connected or multi-coated inclusion, where the coating layers are the images of the concentric annulus via the conformal mapping of the core. Unlike the classical expansion, the geometric multipole expansion is applicable to the whole exterior of the inclusion. We then provided a series solution method for the conductivity interface problem with a multi-coated inclusion based on the proposed expansion, where the expansion coefficients, the FPTs \(\mathbb {F}^{(1)}\) and \(\mathbb {F}^{(2)}\), are expressed in terms of the geometry and material parameters of the inclusion. As an application, by generalizing the concept of the GPTs-vanishing structure, we proposed a new concept of semi-neutral inclusions that show relatively negligible field perturbations for low-order polynomial loadings. We numerically constructed semi-neutral inclusions by finding \(\mathbb {F}^{(2)}\)-vanishing structures for leading orders by a simple optimization procedure. The geometric multipole expansion and the FPTs will lead to further applications to inclusion problems and composite materials.