Geometric multipole expansion and its application to semi-neutral inclusions of general shape

We consider the conductivity problem with a simply connected or multi-coated inclusion in two dimensions. The potential perturbation due to an inclusion admits a classical multipole expansion whose coefficients are the so-called generalized polarization tensors (GPTs). The GPTs have been fundamental building blocks in conductivity inclusion problems. In this paper, we present a new concept of geometric multipole expansion and its expansion coefficients, named the Faber polynomial polarization tensors (FPTs), using the conformal mapping and the Faber polynomials associated with the inclusion. The proposed expansion leads us to a series solution method for a simply connected or multi-coated inclusion of general shape, while the classical expansion leads us to a series solution only for a single- or multilayer circular inclusion. We also provide matrix expressions for the FPTs using the Grunsky matrix of the inclusion. 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. As an application of the concept of the FPTs, we construct semi-neutral inclusions of general shape that show relatively negligible field perturbations for low-order polynomial loadings. These inclusions are of the multilayer structure whose material parameters are determined such that some coefficients of geometric multipole expansion vanish.


Introduction
We consider the field perturbation due to the presence of an elastic or electrical inclusion in a homogeneous background R 2 . An elastic or electrical inclusion with different material parameters from that of the background induces a perturbation on the applied background field. For this conductivity transmission problem, one can find the solution using a single-layer potential ansatz, where the density function involves the so-called Neumann-Poincaré operator. This boundary integral formulation provides us the classical multipole expansion of the field perturbation, whose coefficients are the so-called generalized polarization tensors (GPTs) [4,5]. The classical multipole expansion holds in a far-field region, but, in general, it does not hold near the boundary of the inclusion. Consequently, the classical multipole expansion 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.
In this paper, to overcome the limitation of the classical multipole expansion, we propose a geometric multipole expansion applicable to solving the conductivity transmission problem with an inclusion of general shape. We assume that the inclusion is either a simply connected or multilayered domain whose layers are enclosed by images of concentric circles via the exterior structure. By using this formula, we then construct semi-neutral inclusions and show numerical examples in Section 6.

Layer potential technique for the conductivity transmission problem
Let D be a bounded and simply connected domain in R 2 with Lipschitz boundary. We assume that D has the constant conductivity σ 0 > 0 and is embedded in the background with the constant conductivity σ m . For simplicity, we assume σ m = 1. We consider the resulting conductivity (or anti-plane elasticity) transmission problem in two dimensions: with the conductivity distribution given by σ = σ 0 χ(D) + χ(R 2 \ D) and an entire harmonic function H. Here, χ indicates the characteristic function. It holds that The symbols + and − indicate the limits from the exterior and interior of ∂D, respectively. For ϕ ∈ L 2 (∂D), we define where Γ(x) is the fundamental solution to the Laplacian, i.e., Γ(x) = 1 2π ln |x|, p.v. stands for the Cauchy principal value, and ν x is the outward unit normal vector to ∂D at x. We call S ∂D [ϕ] and K * ∂D the single-layer potential and the Neumann-Poincaré (NP) operator, respectively. On ∂D, the following jump relation holds: (2. 3) The L 2 adjoint of K * ∂D is x − x, νx |x −x| 2 ϕ(x) dσ(x), x ∈ ∂D.
We also call K ∂D the NP operator by an abuse of terminology. The operator K * ∂D can be extended to act on the Sobolev space H −1/2 (∂D) by using its L 2 pairing with K ∂D . We identify x = (x 1 , x 2 ) in R 2 with the complex variable z = x 1 + ix 2 in C. We denote S ∂D [ϕ](z) := S ∂D [ϕ](x) and similarly for the NP operators.
The operator K * ∂D is symmetric in L 2 (∂D) only for a disk or a ball [41]. However, the NP operators can be symmetrized using Plemelj's symmetrization principle [38]: We denote by H is the duality pairing between the Sobolev spaces H −1/2 (∂D) and H 1/2 (∂D).
The operator K * ∂D is self-adjoint in H 1/2 0 (∂D) equipped with a new inner product that involves the single-layer potential [8,32,38].

Exterior conformal mapping and associated orthogonal coordinates
From the Riemann mapping theorem, there uniquely exist a real number γ > 0 and a complex function Ψ(w) that conformally maps the region {w ∈ C : |w| > γ} onto C \ D and satisfies Ψ(∞) = ∞ and Ψ (∞) = 1. We set ρ 0 = ln γ. The function Ψ(w) admits the following Laurent series expansion: for some complex coefficients a n . We call γ the conformal radius of D. From the well-known Bieberbach conjecture, it holds that assuming that the area of D is positive. From the Caratheodory extension theorem [11], Ψ(ρ, θ) extends to the boundary of D as a homeomorphism. The conformal mapping Ψ defines an orthogonal curvilinear coordinate system (ρ, The scale factors with respect to ρ and θ coincide with each other. We denote them by The length element on ∂D is given by dσ(z) = h(ρ 0 , θ)dθ, and for a function v(z) = (v • Ψ)(ρ, θ) defined in the exterior of D, it holds that ∂v ∂ν If we further assume that D is a C 1,α domain for some 0 < α < 1, then, by the Kellogg-Warschawski theorem [46], Ψ can be continuously extended to the boundary.
As a univalent function, Ψ defines the so-called Faber polynomials {F m (z)} ∞ m=1 , which were first introduced by G. Faber [19] and have been extensively studied in various areas. They are defined by the relation This provides explicit expressions for F m in terms of a n . For example, F 0 (z) = 1, The Faber polynomials form a basis for complex analytic functions in D [16]. An essential property of the Faber polynomials is that F m (Ψ(w)) is the addition of w m and negative order terms. In other words, where the coefficients c mn are called the Grunsky coefficients. It holds the Grunsky identity: nc mn = mc nm for all m, n ∈ N. One can obtain the Grunsky coefficients from the exterior conformal mapping by the recursive formula: a m−s c sn − n−1 s=1 a n−s c ms , m, n ≥ 1 (2.10) with initial values c 1n = a n and c n1 = na n , n ≥ 1. The complex logarithm admits the following expansion [16,19,31]: with a proper branch cut. The expansion (2.11) sheds new light to understand the solution of the transmission problem (2.1) and the NP operator [30,31]. The Grunsky coefficients satisfy the so-called strong Grunsky inequalities [16,22]: let N be a positive integer and λ 1 , λ 2 , . . . , λ N be complex numbers that are not all zero, then we have It then follows from (2.9) that for |w| > γ, (2.14) Let v be a complex analytic function in D R := D ∪ {Ψ(w) : γ ≤ |w| < R} for some R > γ. Fix any r ∈ (γ, R). Then, (2.8) holds also for z ∈ D r and |w| > r. By applying the Cauchy integral formula to v and by applying (2.8), it follows that Here, b m is independent of choice of R and |b m | ≤ M (R ) −m for some constant M . From (2.14) and the maximum principle for complex analytic functions, (2.15) uniformly and absolutely converges for z ∈ D r . Furthermore, (2.9) and (2.14) imply that converges uniformly and absolutely for z ∈ {Ψ(w) : r 1 ≤ |w| ≤ r 2 } for any γ < r 1 < r 2 < R. In particular, we can change the order of summation in (2.16).

Series expansions of layer potential operators using Faber polynomials
In this subsection, we review the series expansions of the single-layer potential and the NP operator that were developed in [31] using the exterior conformal mapping and the Faber polynomials associated with the inclusion. We set the density basis functions: for z = Ψ(ρ, θ) ∈ ∂D, If D has a C 1,α boundary, then ζ m (resp. η m ) form a basis of H −1/2 (∂D) (resp. H 1/2 (∂D)) [31]. Furthermore, ζ m and η m jointly form a bi-orthogonal system for the pair of spaces H −1/2 (∂D) and H 1/2 (∂D). In particular, it holds that for any f ∈ H 1/2 (∂D) and g ∈ H −1/2 (∂D), Theorem 2.1 ( [31]). Let D be a bounded and simply connected domain in R 2 with C 1,α boundary for some α > 0. For z = Ψ(w) ∈ C \ D with w = e ρ+iθ , the single-layer potential satisfies and, for m ∈ N, c mn e −n(ρ+iθ) + γ 2m e m(−ρ+iθ) in C \ D. (2.18) The series converges uniformly for all (ρ, θ) such that ρ ≥ ρ 1 for any fixed ρ 1 > ρ 0 . For the density functions with negative index, it holds that Furthermore, the NP operators satisfy (2.20) The infinite series in (2.19) converge in H −1/2 (∂D), and those in (2.20) converge in H 1/2 (∂D).
We have from (2.17) that where the second equality follows from the continuity of the single-layer potential. From (2.18) and (2.20), we then have the following relation in H 1/2 (∂D) sense: 3 Classical and geometric multipole expansions

Classical multipole expansion and CGPTs
For a multi-index α = (α 1 , α 2 ) ∈ N × N, we set x α = x α 1 1 x α 2 2 and |α| = α 1 + α 2 . Applying the Taylor series method, the integral formula (2.4) leads to the multipole expansion [5]: The terms M αβ (D, σ 0 ) are the so-called generalized polarization tensors (GPTs) corresponding to the inclusion D with the conductivity σ 0 . Now, we identify x = (x 1 , x 2 ) in R 2 with z = x 1 + ix 2 in C and define the GPTs in complex form: , and, for each n ∈ N, P n (z) = z n . For m, n ∈ N, we define We call N (1) mn and N (2) mn the complex generalized polarization tensors (CGPTs) corresponding to the inclusion D with the conductivity σ 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 by taking the real part of the expansion, the fundamental solution to the Laplacian satisfies A real-valued entire harmonic function H(x) admits the expansion with some complex coefficients α m . Using (3.3), we can expand (2.4) into complex functions: For an entire harmonic function H given by (3.4), the solution u to (2.1) satisfies that, for |z| > sup {|y| : y ∈ D},

Geometric multipole expansion and FPTs
If D is a disk centered at the origin, the associated Faber polynomials are z n . Hence, (3.3) is in fact an expansion into the Faber polynomials (and its complex conjugates) corresponding to the disk. For an inclusion D of general shape, the complex logarithm admits the expansion (2.11). Using (2.11), we can generalize (3.3): for z = Ψ(w) ∈ C \ D andz ∈ D, Indeed, (3.5) converges uniformly with respect to |w| > γ and uniformly with respect toz belonging to any fixed compact F in the domain D [19,50]. Also, for an entire real harmonic function H, we have for some complex coefficients α n . Moreover, (3.6) converges uniformly on any given compact domain [29]. As one of the main contribution of this paper, we now generalize the concept of CGPTs and the classical multipole expansion (3.1) by using the Faber polynomials as follows.
2(σ 0 −1) and F n be the Faber polynomials of D. For m, n ∈ N, we define We call F (1) mn and F (2) mn the Faber polynomial polarization tensors (FPTs) corresponding to the domain D with the conductivity σ 0 .
Let us find an expansion of the single layer potential in (2.4) where ϕ is obtained from (2.5) and (3.6) that Both the infinite series in (3.7) and (3.8) are uniformly convergent forz ∈ ∂D (with z fixed). Since (λI − K * ∂D ) −1 is a bounded operator, we then can exchange the order of integral and summation in (3.7) and get the desired expansion: Theorem 3.2 (Geometric multipole expansion). For an entire harmonic function H given by (3.4), the solution u to (2.1) satisfies that, for z = Ψ(w) ∈ C \ D with |w| > γ, We emphasize that the geometric multipole expansion in Theorem 3.2 holds in the whole exterior region of D, differently from the classical multipole expansion in Theorem 3.1.

Grunsky matrix C and its symmetrization G
We denote by C the Grunsky matrix We then denote by G the symmetrization of the Grunsky matrix, i.e., From the Grunsky identity, g mn satisfy the symmetry relation: g mn = g nm for all positive integers n and m. Let l 2 (C) denote the vector space of the complex sequence (v m ) satisfying ∞ m=1 |v m | 2 < ∞. We interpret the matrix G as a linear operator from l 2 (C) to l 2 (C) defined by The inequality (4.2) and the symmetricity of g mk imply According to [45,, it holds for some constant κ ∈ [0, 1) that since ∂Ω is quasiconformal; we refer the reader to [1,9,39,49] for more properties of quasiconformality.
We can express G in terms of C as where γ ±kN and N ± 1 2 denote the semi-infinite diagonal matrices whose (n, n)-entries are γ ±kn and n ± 1 2 , respectively.

FPTs in terms of the Grunsky matrix
Theorem 4.1. Let D be a bounded and simply connected domain in R 2 with C 1,α boundary for some α > 0, and λ = σ 0 +1 2(σ 0 −1) . The FPTs satisfy where δ mn is the Kronecker delta function.
Proof. From (2.3) and (2.18), we have Applying (2.21), the FPTs becomes From the fact that dσ(z) = h(ρ 0 , θ)dθ, one can easily find that Then, by using (2.19) and (4.5), we have In the remaining of the proof, we derive explicit expression for A ±m,n .

Polarization tensor of an inclusion with extreme conductivity
Plugging m = n = 1 into Corollary 4.2 with σ 0 = ∞ or 0, we arrive to the relation It is worth remaking that Lemma 4.3 was extended to a domain with Lipschitz boundary in [12].
Proof. Let λ α and λ β be the eigenvalues of the PT. It then holds from Lemma 4.3 that From (2.7), we have γ 4 − |a 1 | 2 = 0. Thus, we have The Pólya-Szegö conjecture asserts that Tr(M −1 ) for an inclusion D with unit area has a minimum value if and only if D is a disk or an ellipse; this conjecture was proved for general conductivity case in [44]. Corollary 4.4 leads a simple alternative proof for the insulating or perfecting conducting case in two dimensions. Indeed, the area of the domain D given by the exterior conformal mapping (2.6) with the conformal radius γ > 0 is It is then straightforward to see that |a 1 | < γ 2 and The equality holds in (4.21) if and only if a k = 0 for all k ≥ 2, equivalently, D is a disk or ellipse.

An ellipse case
For the case when Ψ(w) = w + a 0 + a 1 w , one can easily derive from (2.10) that c mn = δ mn a n 1 for all m, n ∈ N, (4.22) and thus, Hence This matrix is invertible, by (2.7), for |λ| ≥ 1 2 . Theorem 4.1 leads to the following results.
Lemma 4.5. For an ellipse given by Ψ(w) = w + a 0 + a 1 w with some γ > 0, it holds that and, for each m ∈ N,

Geometric multipole expansion and FPTs
We consider the conductivity interface problem, for a given entire harmonic function H, with the conductivity distribution The solution u satisfies the boundary condition, for each j = 0, 1, . . . , L, Note that (u − H) • Ψ(w) is harmonic in {w : |w| > r L } and decays to zero as |w| → ∞. As (5.3) is linear with respect to H(x), for H given by the solution u to (5.3) admits the geometric multipole expansion: for z = Ψ(w) ∈ C \ Ω, with some complex coefficients F mn the FPTs corresponding to the multi-coated inclusion Ω with the conductivity σ. We denote the FPTs in matrices: We set z = x 1 + ix 2 = Ψ(w) for x ∈ R 2 \ Ω 0 . Since u m is harmonic in each Ω j , j = 0, . . . , L + 1, and (u m − H m )(x) decays to zero as |x| → ∞, we can express u m as
It then directly follows that Note that d (1) n , . . . , d (4) n are real-valued constants that are independent of α m and m. Since (5.3) is linear with respect to the background field H, we can express s mn in (5.6) as, fixing m and n, s mn = α m t 1 + α m t 2 for some complex values t 1 and t 2 independent of α m . In view of the expansion (5.5), we then have From (5.7) and (5.10), we obtain As d (1) n , . . . , d (4) n are real-valued, it follows by plugging (5.12) into (5.13) that k c kn + (α m c mk + s mk ) d (2) k c kn . (5.14) (2) n and d (4) n For notational convenience, we denote the multiplication of the scaling constant in (5.9) by

Asymptotic behavior of d
It follows from determinants of the 2 × 2-matrix in (5.9) and (5.10) that For the case L = 1, we obtain where o(·) denotes the standard little-o notation. If L = 2, we have For general L, one can show the following.
n , . . . , d (4) n satisfy the asymptotic behavior as n → ∞: We denote by D j the diagonal matrix with the diagonal entries d (j) n , that is, By Lemma 5.1, we have d (4) n = 0 for sufficiently large n. Furthermore, it holds for n 1 that where C denotes the Grunsky matrix (see (4.1)). Note that the right-hand side of (5.17) is invertible from (4.16). For all examples in Subsection 6.2, the 100 × 100 dimensional truncated matrices of D 4 and D 4 − D 2 CD −1 4 D 2 C are invertible. From now on, we assume that d (4) n = 0 for all n so that D 4 is invertible. We also assume that

Matrix formulation for FPTs
We set the two semi-infinite matrices: It then follows from (5.14) that Here, (5.19) is a direct consequence of (5.18) by taking complex conjugates. Substituting (5.19) into (5.18), we obtain Now, applying (5.11), we derive or equivalently, Since α is diagonal and corresponds to an arbitrary entire function H, both sides of (5.20) vanish. Furthermore, from (5.15), the diagonal entry of D 1 D 4 − D 2 D 3 is as follows: for each m ∈ N, Hence, we have the following theorem.
Theorem 5.2. We set C and D j as in (4.1) and (5.16), respectively. The FPTs of the multicoated structure Ω given as at the beginning of Section 5 have the following matrix expressions, given that D 4 and D 4 − D 2 CD −1 4 D 2 C are invertible: for each m, n ∈ N, When Ω 0 is a disk, the fact that a 1 = 0 and (4.22) imply c mn = 0 for all m, n ∈ N. Thus, C is identical to the zero matrix. Hence, we have the following corollary.

Corollary 5.3 ([6]).
If Ω 0 is a disk, then the FPTs of the multi-coated inclusion Ω satisfy δ mn for all m, n ∈ N.
The Grunsky coefficient of an ellipse satisfy (4.22), which means C and FPTs have diagonal matrix forms.
If Ω 0 is an ellipse, the FPTs of the multi-coated ellipse Ω are diagonal matrices: δ mn for all m, n ∈ N.
In particular, for L = 1, it holds that, for each m ∈ N, Remark 5.1. For the simply connected inclusion (that is, L = 0), we get and hence, Applying above relations to Theorem 5.2, we obtain the same results as Theorem 4.1.

Domains 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 by 2π/N (see Figure 5.2). Then, the exterior conformal mapping Ψ corresponding to the domain satisfies which is equivalent to a n = 0 for n + 1 ≡ 0 (mod N ).
Here, m ≡ 0 (mod N ) means that m = kN + r for some k ∈ N and r = 1, . . . , N − 1. It follows by induction using (2.10) that the Grunsky coefficients satisfy For instance, the following matrices A and B belong to S + 3 and S − 3 , respectively: where * represents elements that can be nonzero. One can easily find that the following product rules hold: Let Ω be a multi-coated inclusion given as at the beginning of this section. If the core Ω 0 has a rotational symmetry of order N , it follows that Since D j in Theorem 5.2 is a diagonal matrix, we have D j , D −1 j ∈ S + N . It follows from Lemma 5.5 and (5.22) that and Thus, the geometric series on By using (5.23), (5.24), and Theorem 5.2, we finally obtain that which complete the proof. 2

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 below). For such a multi-coated inclusion, we consider mainly the material parameters in the layers, differently from the approach in [20], in which the shapes of coating layers were determined by a shape optimization procedure. For the concentric disks, F (1) = 0 [6]. In view of the expressions in Theorem 5.2 with the Grunsky matrix C associated with the core Ω 0 , one can deduce that F (1) significantly depends on the shape of Ω 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, F (1) is not zero but still significantly smaller than F (2) , and hence, F (2) mainly contributes in the field perturbation. For a given core Ω 0 of general shape, one can construct layered structures that show relatively small field perturbation. We choose the number of layers and determine appropriate conductivity values in the coating layers such that the multi-coated inclusions satisfy the following condition.

Definition 6.1 (Semi-neutral inclusion).
Let Ω be a multi-coated inclusion with the core Ω 0 and L layers of coating, as in Section 5. We say that Ω is a semi-neutral inclusion provided that, for some N 1 , N 2 ∈ N, the second FPTs are negligible for low-order terms, that is, 4πn ≈ 0 for m ≤ N 1 and n ≤ N 2 .

Numerical scheme
For a given Ω 0 , we denote γ, Ψ(w), and (ρ, θ) as in Subsection 2.2. We set Ω j (j = 1, . . . , L + 1) as in (5.1) and (5.2). We assume that σ L+1 = 1 and that the conductivity σ 0 is fixed. We find σ = (σ 1 , . . . , σ L ) such that the condition Definition 6.1 holds. In other words, we look for σ that satisfies the equation To find σ satisfying (6.1), we use the multivariate Newton's method: where α is a constant in (0, 1), and k indicates the iteration step, and J † denotes the pseudoinverse of the Jacobian matrix of f . We give the initial guess σ (0) as a proper alternating series. We calculate f (σ (k) ) by Theorem 5.2, where we truncate the semi-infinite matrices C, D 2 , D 4 to be 100 × 100 matrices. To obtain J † [σ (k) ], we use the finite difference approximation of the partial derivatives of 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:
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 σ 0 = 5 and σ L+1 = 1. We set the initial guess for the iteration (6.2) as σ (0) = (σ We then choose the conductivity values in the coating layers by following the numerical scheme in Subsection 6.1. The resulting values are σ 1 = 0.1216 for the 1-coated ellipse (i.e., L = 1), and (σ 1 , σ 2 ) = (0.0782, 3.6782) for the 2-coated ellipse (i.e., L = 2), respectively. Figure 6.1 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 6.1 shows the first two diagonal elements of FPTs, and verifies that the obtained coated ellipses are semi-neutral inclusions with L = 1, 2.
mn , of coated ellipses are similar to those corresponding to the core, i.e., the uncoated inclusion. The second FPTs, F mn , vanish for m, n ≤ N with N = 1 for the 1-coated inclusion and N = 2 for the 2-coated inclusion.

Non-elliptical shapes
In this subsection, we provide two examples of non-elliptical shape. It is necessary to find the conductivity distribution that makes a coated inclusion semi-neutral. As Example 1, the uniform background field is given by H(x) = x 1 or H(x) = x 2 . There is no rotational symmetry for this domain so that the vanishing property in Proposition 5.6 does not hold. Figure 6.2 illustrates the potential perturbation u on the background field H. Table 6.2 shows the low-order FPTs. Each coated kite is semi-neutral because its second FPT is significantly smaller than the uncoated one. In Figure 6.2, the background field is given by H(x) = x 1 . Colored curves represent contours of u. We commonly set γ = 1 and σ 0 = 100. We set the initial guess for the iteration (6.2) as σ (0) = (σ    Table 6.2: Low-order FPTs of the uncoated and coated kites in Figure 6.2. As we coat the inclusion, the first FPT is almost invariant, but the second FPTs decrease significantly. Example 6.3. In this example, we assume that the core Ω 0 is given by the conformal mapping Ψ(w) = w + 0.2 w 4 .
Then, F mn = 0 for all m, n ≤ 2, and F (2) mn = 0 for all m = n with m, n ≤ 5. Hence, we focus only on vanishing the nonzero leading terms of FPTs. Figure 6.3 that illustrates contours of the potential perturbation for a given background field H. Table 6.3 compares the leading FPTs of domains in Figure 6.3.
In Figure 6.3, the background field is given by H(x) = x 2 . Colored curves represent contours of u. We commonly set γ = 1 and σ 0 = 0.25. The initial guess for the iteration (6.2) is given by σ (0) = (σ   Table 6.3: The leading terms of the FPTs for the uncoated and coated stars in Figure 6.3. As we coat the inclusion, the second FPTs vanish.