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Neutral inclusions, weakly neutral inclusions, and an over-determined problem for confocal ellipsoids

An inclusion is said to be neutral to uniform fields if upon insertion into a homogenous medium with a uniform field it does not perturb the uniform field at all. It is said to be weakly neutral if it perturbs the uniform field mildly. Such inclusions are of interest in relation to invisibility cloaking and effective medium theory. There have been some attempts lately to construct or to show existence of such inclusions in the form of core-shell structure or a single inclusion with the imperfect bonding parameter attached to its boundary. The purpose of this paper is to review recent progress in such attempts. We also discuss about the over-determined problem for confocal ellipsoids which is closely related with the neutral inclusion, and its equivalent formulation in terms of Newtonian potentials. The main body of this paper consists of reviews on known results, but some new results are also included.


Introduction
This is a survey on recent progress in study on existence and construction of neutral and weakly neutral inclusions, and a related over-determined problem for confocal ellipsoids. The main body of the paper consists of reviews on known results with brief but coherent explanations. However, we include some new results as well.
To explain the problems related to the neutral and weakly neutral inclusion, let us consider the following conductivity problem: for d = 2 or 3, where a is a constant vector so that −a = −∇(a · x) is the background uniform field and σ is a piecewise constant function representing the conductivity distribution. We first consider the problem (CP) when the conductivity distribution σ is given by where D is a simply connected bounded domain in R d whose boundary ∂D is Lipchitz continuous. Here χ(D) denotes the characteristic function of D (χ(R d \ D) likewise), and σ c and σ m are constants representing conductivities of D (the core) and R d \ D (the matrix), respectively. In absence of the inclusion D, the solution to (CP) is nothing but a · x. Thus, if we denote by u the solution to (CP) in presence of the inclusion, u(x) − a · x represents the perturbation occurred by insertion of the inclusion D into the homogeneous medium with the uniform field −a. As we see from Fig. 1, the uniform field is perturbed outside (and inside) the inclusion. It is known (see, e.g., [3]) that the leading order term of the perturbation outside the inclusion can be expressed in terms of the dipolar expansion. In fact, we have the following expansion at infinity: where ω d is the surface area of the unit sphere in R d and M = (M ij ) is the d × d matrix determined by the domain D and the conductivity contrast σ c /σ m . The matrix M is called the polarization (or polarizability) tensor (PT in abbreviation, afterwards) associated with D. The PT is a signature of the existence of the inclusion D and has been effectively used to detect some properties of the inclusion D, for which we refer to [3]. It also plays an important role in the theory of composites and effective medium, for which we refer to [3,31]. If D is a simply connected domain (or a union of simply connected domains), then M is positive-definite if σ c − σ m > 0 and negative-definite if σ c − σ m < 0. In fact, optimal bounds for PT, called the Hashin-Shtrikman bounds, are known, which will be explained in section 2. Therefore, if D is simply connected, then there isx = x/|x| such that a, Mx = 0 and for such x the following holds |u(x) − a · x| ≥ C|x| −d+1 as |x| → ∞ (1. 3) for some C > 0. The dipolar expansion (1.2) shows that in general the solution u to (CP) admits the following: Furthermore, (1.3) shows that the decay rate O(|x| −d+1 ) cannot be replaced by a faster rate, say O(|x| −d ). However, if the inclusion is of a core-shell structure, then the situation can be quite different. Let D be a bounded domain and Ω be a bounded domain containing D so that (D, Ω) becomes a coated structure or a core-shell structure. Suppose that the conductivity distribution is given by (1. 5) In particular, if (D, Ω) is a pair of concentric disks or balls, and if the conductivities σ c , σ s and σ m are scalars and satisfy for d = 2 or 3, where f = |D|/|Ω| (the volume fraction), then the solution to (CP) satisfies namely, the uniform field is not perturbed at all (see Fig. 2). In fact, with the conductivity given by (1.5), the solution u to (CP) is harmonic in R d \ (∂D ∪ ∂Ω), and along the interfaces ∂D and ∂Ω it satisfies the transmission conditions: continuity of the potential and continuity of the flux, namely, where the subscripts + and − indicate the limits from outside and inside D (or Ω), respectively. If D and Ω are concentric disks (or balls), one can use spherical harmonics to find the solution explicitly to satisfy these interface conditions, and show that (1.6) implies (1.7). This easy-to-prove fact was first discovered by Hashin [12], and significance of the discovery lies in its implications. Since insertion of inclusions does not perturb the outside uniform field, the effective conductivity of the assemblage filled with such inclusions of many different scales is the same as σ m (the conductivity of the matrix) satisfying (1.6). It is also proved that such an effective conductivity is one of the Hashin-Shtrikman bounds on the effective conductivity of arbitrary two-phase composites [12,13] (see also [31]).
The inclusion (D, Ω) of core-shell structure (or any other structure), which does not perturb the uniform field −a upon its insertion, that is, satisfying (1.7), is said to be neutral to the field −a. If the inclusion is neutral to all uniform fields, it is said to be neutral to multiple uniform fields. The concentric disks (or balls) satisfying (1.6) is  Figure 2: Neutral inclusion: The uniform field is not perturbed neutral to multiple uniform fields. If σ m is anisotropic (σ c and σ s are scalars), then confocal ellipsoids whose common foci are determined by σ m can be neutral to multiple fields. We include a proof of this fact in section 3. Now the question is if there are coated inclusions other than concentric disks or balls neutral to multiple uniform fields (or confocal ellipses or ellipsoids when σ m is anisotropic), more generally, if we can coat a given domain D of general shape by a shell so that the resulting inclusion (D, Ω) is neutral to multiple uniform fields. The answer is proven to be no in two dimensions. In fact, it has been proved that if (D, Ω) is neutral to multiple uniform fields, then Ω and D are concentric disks if σ m is isotropic, and confocal ellipses if σ m is anisotropic (and the foci are determined by σ m ). This was proved by Milton-Serkov [32] when σ c = 0 or ∞, and by Kang-Lee [15] when σ c is finite. Since these two dimensional results are proved using either the Riemann mapping or existence of harmonic conjugates, the methods of proofs cannot be extended to three dimensions. It is worth mentioning that there are many different shapes of coated inclusions neutral to a single uniform field as shown in two dimensions in [14,32]. In three dimensions, it is proved in [17] that the coated inclusion (D, Ω) being neutral to multiple fields is equivalent to existence of a solution to a certain over-determined problem defined on Ω \ D. It is then proved as a consequence that if σ m is isotropic, then the only inclusions of core-shell structure is a pair of concentric balls. Extension of this result to the anisotropic case has not been proved and is open. We will review on neutral inclusions and the related over-determined problem in section 3. We also include in the same section a proof of their equivalence to a certain formulation in terms of Newtonian potentials.
Other than applications to the theory of composite as explained earlier, there is another interest in neutral inclusions in relation to invisibility cloaking. The neutrality condition (1.7) means that the uniform field is unperturbed at all outside the inclusion, namely, there is no difference of the field with or without the inclusion. It means that the inclusion is invisible from the probe by uniform fields. This was also observed in [24]. Recently, the idea of neutrally coated inclusions has been extended to construct multi-coated circular structures which are neutral not only to uniform fields but also to fields of higher order up to N for a given integer N [4]. It was proved there that the multi-coated structure combined with a transformation optics dramatically enhances the near cloaking of [25].
Since there is no coated inclusion neutral to multiple fields (invisible by uniform fields), we may ask if there are inclusions which are vaguely visible by uniform fields. They are weakly neutral inclusions. In general, the solution to (CP) satisfies the decay condition u(x) − a · x = O(|x| 1−d ) at ∞ and if the inclusion is neutral, then u(x) − a · x ≡ 0 outside the inclusion. This property means that the field outside the inclusion is not perturbed even though the inclusion is inserted. The weakly neutral inclusions perturb the fields mildly: If (1.9) holds for all constant vectors a, then the inclusion is said to be weakly neutral to multiple uniform fields. According to (1.2), in order for (1.9) to hold for all a, the corresponding PT must vanish. Thus weakly neutral inclusions are PT cancelling ones. We now formulate the weakly neutral inclusion problem: Weakly Neutral Inclusion Problem. Given a domain D can we find a domain Ω containing D so that the resulting inclusion of core-shell structure becomes weakly neutral to multiple uniform fields, or equivalently, its PT vanishes.
In section 4, we present two classes of domains which admit coatings so that the resulting inclusions of core-shell structure become weakly neutral to multiple uniform fields. One class is the collection of domains D such that the coefficients b D vanish. Here b D is the leading coefficient of the conformal mapping from the exterior of the unit disk onto the exterior of D (see (4.1)). For such domains we construct the coating explicitly. This is a new result. The other class is that of small perturbations of disks, for which it is proved in [18] that there are coatings so that the resulting inclusions become weakly neutral to multiple uniform fields.
There is yet another way, other than coating, to achieve weak neutrality. It is by introducing imperfect bonding parameter on the boundary of the given domain. We review the result of [16] on this in section 5.
This paper is organized in the following way. Section 2 is to review general properties of the PT, including the Hashin-Shtrikman bounds. In section 3, we discuss about neutral inclusions and related over-determined problem for confocal ellipsoids and an equivalent formulation in terms of the Newtonian potential. We also include a discussion on Neumann ovaloids. Section 4 is to discuss on the weakly neutral inclusion problem. Section 5 is for discussion on the construction of weakly neutral inclusion by imperfect bonding parameters.

Layer potentials and polarization tensors
In this section we represent the PT appearing in the dipolar expansion (1.2) in terms of layer potentials and recall the optimal Hashin-Shtrikman (HS) bounds on traces of the PT and its inverse.

Layer potentials
Let Γ(x) be the fundamental solution to the Laplacian, that is, Γ(x) = 1/(2π) log |x| in two dimensions, and Γ(x) = −(4π|x|) −1 in three dimensions. Let D be a bounded simply connected domain with a Lipschitz continuous boundary. The single layer potential of a function ϕ ∈ H −1/2 (∂D) (the L 2 Sobolev space of order −1/2 on ∂D) is defined by where dS is the line or surface element on ∂D. Let ∂ ν denote the outward normal derivative on ∂D. It is well known (see, for example, [3]) that the following jump relation holds: where I is the identity operator and K * ∂D is the operator defined by Here, , the scalar product in R d . The boundary integral operator K * ∂D is called the Neumann-Poincaré (NP) operator.

Polarization tensors
Let u l , 1 ≤ l ≤ d, be the solution to (CP) when a·x = x l and the conductivity distribution σ is given by (1.1). Then it is known (see, e.g., [3]) that u l can be represented as where ν l is the l-th component of the outward unit normal vector field ν on ∂D. By expanding out the term in this case is given by If the conductivity distribution σ is given by (1.5), the solution u l can be represented as Here we denote the unit outward normal vector fields on ∂D and ∂Ω by ν ∂D and ν ∂Ω , respectively. The numbers λ and µ are given by For unique solvability of the integral equation we refer to [18]. In this case, the PT M = M (D, Ω) = (m ll ′ ) d l,l ′ =1 of the core-shell structure (D, Ω) is given by It is known that M is a symmetric matrix (see, e.g., [3]).

Hashin-Shtrikman bounds
If the conductivity distribution is given by (1.1), then the following optimal bounds on traces of the PT M and its inverse hold: with k = σ c /σ m , and where Tr stands for trace. These bounds are obtained by Lipton [28] under the assumption of periodicity, and by Capdeboscq-Vogelius [7] without assumption of periodicity, and called the Hashin-Shtrikman (HS) bounds after the names of scientists who found optimal bounds of effective properties of two phase composites, as described in the paragraph right after (1.8).
The first one is an upper bound (the green line in Fig. 3) and the second one is a lower one (the pink curve in Fig. 3). The upper bound is never attained by a domain, while the lower bound is attained by ellipses and ellipsoids, and the converse is also true. In fact, it is proved in [20,21] that the simply connected domain whose PT satisfies the lower HS-bounds is an ellipse in two dimensions and an ellipsoid in three dimensions. This is an isoperimetric inequality for PT and a generalized version of the Pólya-Szegö conjecture [35]. The original Pólya-Szegö conjecture asserts that the PT attains its minimal trace on and only on disks or balls. The constant trace lines of the PT are those parallel to the green line in Fig. 3. Thus the minimal trace is attained at the point of tangency of the line parallel to the green line to the pink curve. The generalized version asserts that if eigenvalues of the PT lies on the pink curve, then the domain must be an ellipse or an ellipsoid. The original Pólya-Szegö conjecture is now proved as a simple consequence of the generalized version. See Theorem 3.4 of this paper for more discussion on this. The bounds (2.9) and (2.10) are optimal in the sense that any matrix satisfying (2.9) and (2.10) is actually the PT associated with a domain (see [2,6] for proofs).

Neutral inclusions and an over-determined problem
In this section, the conductivity distribution σ is given by (1.5) with the inclusion (D, Ω) of core-shell structure. We assume that the conductivity of the matrix, σ m , is in general anisotropic, i.e., a symmetric matrix. We review the result saying that if σ m is isotropic, i.e., its eigenvalues are all the same, then the only inclusion of the core-shell structure neutral to multiple uniform fields is concentric balls. We also prove the equivalence of neutral inclusion problems with an over-determined problem for confocal ellipsoids, and an equivalent formulation of the problem using the Newtonian potentials. In relation to these problems, we include a subsection on quadrature domains and Neumann ovaloids.

An over-determined problem for confocal ellipsoids
It was proved in [17] that if (D, Ω) is neutral to multiple uniform fields and σ c > σ s , then the following over-determined problem admits a solution: where A is a symmetric matrix and b is a constant vector, provided that ∂D is connected and R 3 \ D is simply connected. This problem is over-determined since ∇w is prescribed on ∂Ω and ∂D. The matrix A is determined by σ m . If σ m is isotropic for example, so is A.
Let us briefly recall the proof. Suppose, after diagonalization, that Let e j , j = 1, 2, 3, be the standard basis of R 3 and let u j be the solution to (CP) when a = e j . The inclusion (D, Ω) being neutral to multiple uniform fields means that The crux of the proof in [17] lies in proving that v is linear inside D. In fact, it is proved for some constant c 0 and vector b 0 . It is here where the assumption σ c > σ s is needed 1 . It is then shown that ∇v is symmetric, and hence, thanks to the assumption that ∂D is connected and R 3 \D is simply connected, there is a function If σ m is isotropic, so is A as mentioned before. The converse is also true, namely, if (ODP) admits a solution, then (D, Ω) is neutral. For this, see Theorem 3.5 below.

Remark
The assumption that ∂D is connected and R 3 \ D is simply connected in [17,Theorem 1.2] can be replaced with the weaker one that Ω \ D is connected. Indeed, instead of using Stokes' theorem, by combining the formula [17, (2.18) , we see that the function ψ is explicitly given by Hence the function w = w(x) is given by For a domain D in three dimensions and a domain Ω containing D, the assumption that Ω \ D is connected is really more general than that ∂D is connected and R 3 \ D is simply connected. In fact, this general assumption allows us to choose the genus of a closed surface ∂D arbitrarily. If the genus does not equal zero, R 3 \ D is not simply connected, but Ω \ D is connected.
Note that if Ω and D are concentric balls centered at the origin whose respective radii are r e and r i , then the solution w to (ODP) is given by In this case, b = 0 and A = 1 3 (−r 3 e /r 3 i + 1)I which is isotropic. We emphasize that w is radial in this case.
It is shown in [17] that confocal ellipsoids admit a solution to (ODP). In fact, if ∂D is an ellipsoid given by the confocal ellipsoidal coordinate ρ is given by 9) and the confocal ellipsoid ∂Ω is given by ρ = ρ 0 for some ρ 0 > 0. Let and ds, j = 1, 2, 3. (3.11) Then the function w, defined by is a solution of (ODP) with Note that b = 0 and A is anisotropic.
The following problem arises naturally: An over-determined problem for confocal ellipsoids. Prove that if (ODP) admits a solution (in H 1 (Ω \ D)), then Ω and D are confocal ellipsoids (or ellipses) and the common foci (when the volumes are fixed) is determined by the eigenvalues of A.
The two-dimensional problem can be solved using the conformal mapping between Ω \ D and an annulus [1, Theorem 10, p. 255]. If A is isotropic, this problem is solved as the following theorem shows. The case of anisotropic A has not been solved and is open. As an immediate consequence, we have the following Corollary: is the constant vector appearing in (ODP) and ∂ j denotes the partial derivative with respect to x j -variable. Then one can see that ∆A ij w = A ij ∆w = 0 in Ω \ D, A ij w = 0 on ∂Ω, and A ij w = 0 on ∂D provided that A = cI. Hence A ij w = 0 in Ω \ D, which implies that w is radial. Using this fact, one can prove Theorem 3.1. We emphasize that this argument using the angular derivative does not work if A is not isotropic. . This a rather surprising fact that the field inside elliptic or ellipsoidal inclusions is uniform seems to have been known for long time and its proofs go back to Poisson (1826) and Maxwell (1873) (see [30]). The converse is also true as we explain it in the sequel.
For doing so, we need to recall the notion of the Newtonian potential.
The Newtonian potential of the domain D, which we denote by N D , is defined by Usually the Newtonian potential is defined without the averaging factor 1/|D|, but here it is more convenient to define it with the averaging factor. Since ∆N D (x) = 1/|D| for x ∈ D, we have N D = a quadratic part + a harmonic part in D.
If D is an ellipse or an ellipsoid, then the harmonic part of N D is quadratic and so is N D inside D. In fact, this is equivalent to the fact that the field inside elliptic or ellipsoidal inclusions is uniform. Moreover, this property of the Newtonian potential's being a quadratic function inside the domain characterizes the ellipsoid and the ellipse:

Theorem 3.3 Let D be a simply connected bounded domain with the Lipschtiz boundary. If N D is quadratic inside D, then D is an ellipse or an ellipsoid.
This characterization of ellipsoids was proved by Dive in 1931 [9] and by Nikliborc in 1932 [34] (see also [8]). The reason why Dive and Nikliborc considered this problem was to prove the converse of a theorem of Newton. Let D be a simply connected domain whose center of mass is 0 ∈ D and let λD be a dilation of D by λ > 1, i.e., λD = {λx : x ∈ D}. A theorem due to Newton states that if D is an ellipsoid, then the gravitational force induced by the mass λD \ D is zero in D [23]. Dive and Nikliborc independently proved that the converse is true: If the gravitational force induced by the uniform mass on λD \D is zero in D, then D must be an ellipsoid.
The following theorem was proved using the characterization of ellipsoids by Newtonian potentials. (ii) The solution to the conductivity problem (CP) when the conductivity distribution σ is given by (1.1) is linear inside D.

(iii) D is an ellipse in two dimensions and an ellipsoid in three dimensions.
This theorem proves the generalized Pólya-Szegö conjecture explained before. That the linearity of the solution to (CP) when σ is given by (1.1) characterizes ellipsoids is known as the Eshelby's conjecture in the field of composites theory. Actually the Eshelby's conjecture (1961) [10] asserts that the inclusion inside which the field is uniform (or equivalently, the strain is constant) for a uniform loading is an ellipse or an ellipsoid. The corresponding conjecture for the electro-static case is proved by Theorem 3.4. The Eshelby's conjecture (for the elasto-static case) was proved by Sendecyj [37] in two dimensions and by Kang-Milton [21] and Liu [29] in three dimensions.
We now formulate the over-determined problem for the confocal ellipsoids in terms of the Newtonian potential. It is proved in [17]

that the problem (ODP) admits a solution if and only if
(3.14) Now the problem is to show that D and Ω are confocal ellipsoids if (3.14) holds. If Ω and D are confocal ellipsoids, then both N Ω and N D are quadratic polynomials inside D, and so is N Ω − N D . A proof of the fact that N Ω = N D outside Ω can be found in [30, p.61].
In the problem (3.14), the quadratic polynomial inside D determines the common foci of D and Ω. For example, one can see from Theorem 3.1 that if the quadratic polynomial is of the form c|x| 2 + l.o.t, then Ω and D are concentric balls. Now we can show the following theorem.
Theorem 3.5 Suppose that ∂D is connected and R d \ D is simply connected. Consider the following statements: (i) (D, Ω) is neutral to multiple uniform fields for some σ given by (1.5).

(ii) The problem (ODP) admits a solution for some A and b.
(iii) The Newtonian potential formulation (3.14) holds.
The following implications hold to be true: Proof. The first implication was proved in [17] and the proof is briefly reviewed at the beginning of this section. The second implication was proved in the same paper. We prove the third implication, namely, if (3.14) holds, then there are conductivities σ c , σ s and σ m such that (D, Ω) is neutral to multiple uniform fields. Let By a rotation and a translation, if necessary, we may assume that for some constants α 1 , α 2 , α 3 and α. In particular, there is no linear term in the quadratic function. Define u j by where β j 's are defined by (3.2) with the conductivities to be determined later. We claim that u j is the solution to (CP) with (1.7) when a = e j . In fact, we see from the definition (3.16) of w that where the last equality follows from the divergence theorem. Here, f is the volume fraction, namely, f = |D|/|Ω|. Thus, Since the single layer potential is continuous across the boundary, u j is continuous across the interfaces ∂Ω and ∂D. Thanks to the jump relation (2.2) and (3.17), we have, on ∂Ω, Thus, σ m,j ∂ ν u j | + = σ m,j ν j , and σ s ∂ ν u j | − = σ s (β j + 1)ν j , on ∂Ω. Since σ m,j = σ s (β j + 1) by the definition (3.2) of β j , we infer that Similarly, thanks to (2.2), we have from (3.17) and (3.19) that, on ∂D, and hence Thus, we have So if we choose γ and β j (or σ c , σ s and σ m,j ) so that (3.22) holds, then (D, Ω) is neutral to multiple uniform fields.
There is yet another restriction when we solve (3.22) for γ and β j ; σ c , σ s and σ m,j should be positive. This condition can be easily fulfilled. In fact, the following relation follows easily from (3.22): The quantity 2α j γ + f −1 in the above is nonzero since γ can be chosen small as we see shortly. Thus the positivity is achieved if which in turn can be achieved by taking γ so that This completes the proof.
In the course of the proof, we derived the neutrality condition for confocal ellipsoids.
Corollary 3.6 Let D and Ω be confocal ellipsoids whose boundaries are respectively given by (3.8) and (3.9). If the conductivity distribution σ given by (1.5) and (3.1) satisfies then (D, Ω) is neutral to multiple uniform fields. Here f is the volume fraction and α j 's are constants given by (3.17), i.e., Let us look into the neutrality condition (3.22) or (3.23) further. According to (3.16) and (3.17), We then have from (3.22) and hence Writing it in terms of conductivities, we have This is a necessary neutrality condition when σ m is anisotropic. In particular, if σ m is a scalar, namely, σ m,j = σ m , then it is exactly the neutrality condition (1.6) of concentric balls.

Quadrature domains-Neumann ovaloids
Let us look further into the Newtonian potential formulation (3.14) of the problem. The problem is to prove that if it holds, then D and Ω must be confocal ellipsoids. We show that the condition (3.14) in R 3 \ Ω alone does not yield the answer. The condition (3.14) in R 3 \ Ω yields for any g ∈ H −1/2 (∂Ω). By changing the order of integrations, we have where u(x) = S ∂Ω [g](x), x ∈ Ω. Thus (3.26) holds for all H 1 h (Ω) where subscript h means that it is a collection of harmonic functions in Ω. The condition (3.26) does not guarantee that D and Ω are confocal ellipsoids as will be seen in what follows, and the condition (3.14) in D should be utilized.
In fact, an open set Ω ⊂ R d is called a quadrature domain (see, e.g., [38, (4.1)], and also [11,36]) if there exists a distribution µ with a compact support in Ω such that The simplest quadrature domain may be balls: It is well known as the mean value property: where c is the center of the ball. In this case the distribution µ is the point mass (the Dirac delta) multiplied by the volume of Ω. Ellipsoids are also quadrature domains. Let The lower dimensional set F is called the focal ellipsoid of Ω. The following quadrature identity holds (see, e.g., [26,27]): for all u ∈ H 1 h (Ω). Here, x ′ is (x 1 , . . . , x d−1 ). Note that if D and Ω are confocal ellipsoids, then their focal ellipsoids are the same, and hence (3.26) holds.
There is yet another class of domains satisfying (3.26). A domain Ω ⊂ R d is called a Neumann ovaloid if it admits the following quadrature identity where p 1 and p 2 are distinct points in R d and C > 0 is a constant. If C is sufficiently small compared to |p 1 − p 2 |, then a union of two balls of the same radius centered at p 1 and p 2 satisfies the identity (3.30). However, if C is sufficiently large, then there is an axiallysymmetric domain satisfying (3.30). For example, if Ω is the domain in R 2 bounded by the curve where α and ε are some positive constants (see Fig. 5), then it admits a quadrature identity (3.30) with C = |Ω|/2 (see [38, p. 19-20] for a proof). In this case, the relation among |Ω|, α and ǫ is given by |Ω| = π(α 2 + 2ε 2 ). These two-dimensional Neumann ovals were discovered by C. Neumann [33]. The existence and uniqueness of the higher dimensional Neumann ovaloids are known (see [22] and references therein), but there is no known explicit expression except four-dimensional (and two-dimensional) ones, to the best of our knowledge. We refer to a recent paper [22] for an explicit parametrization of a four-dimensional Neumann ovaloid. If (D, Ω) are Neumann ovaloids with same foci, then (3.26) holds.

Weakly neutral inclusions
We now consider the weakly neutral inclusion problem presented at the end of Introduction, namely, the problem of coating a given domain of general shape by another domain so that the resulting inclusion of core-shell structure satisfies the weak neutrality condition (1.9), namely, its polarization tensor vanishes. In this section we present two classes of domains for which the weakly neutral inclusion problem can be solved. The first one is defined by a conformal mapping from the exterior of the unit disk onto the exterior of the domain, and construction of the coating is explicit. This result is new. The other class are small perturbations of a disk for which existence of a coating is proved. This result is from [18,19]. Note that neutral inclusions are weakly neutral to multiple uniform fields. Thus concentric disks or balls can be realized as weakly neutral inclusions. However, no other examples of weakly neutral inclusions are known.

b D -vanishing domains
Let D be a bounded domain in C = R 2 with the Lipschitz continuous boundary, and let z = Φ(ζ) is the Riemann mapping from |ζ| > 1 (C \ U , where U is the unit disk) onto C \ D. The conformal mapping Φ takes the form By dilating and translating D if necessary, we assume that b −1 = 1 and b 0 = 0, and denote b 1 by b D , so that the Riemann mapping Φ takes the form The domains we consider in this subsection are such that b D = 0. We want to coat D by an another bounded domain Ω so that the PT of the coated structure vanishes. Let the conductivity distribution σ be given by (1.5). Furthermore, we assume that σ c = 0 or ∞. This assumption is required because we use the conformal mapping from C \ ∆ onto C \ D. We assume that σ c = ∞, and the other case can be handled in the same. We also assume σ m = 1 without loss of generality.
Since σ c = ∞, (CP) is now of the form where σ = σ s χ(Ω \ D) + χ(R 2 \ Ω). The constant λ is determined by the condition ∂D ∂ ν u| + = 0. The problem is to find σ s and Ω so that the solution u to (CP) ∞ satisfies the weak neutrality condition (1.9).
Since u(x) − a · x tends to 0 as |x| → ∞ and D, Ω are simply connected, there are functions U m and U s analytic in C \ Ω and Ω \ D, respectively, such that ℜU m = u and ℜU s = u in their respective domains (ℜ stands for the real part). One can see using the Cauchy-Riemann equations that the transmission conditions to be satisfied by u on ∂Ω is equivalent to for some constant c. Moreover, U m admits the following expansion at ∞: where α = a 1 − ia 2 . Thus the weak neutrality condition (1.9) is equivalent to α 1 (α) = 0 for all α (or equivalently, for α = 1, i). With the conformal mapping Φ in (4.1), let V α = U • Φ. Then we have Let U ′ be a simply connected domain containing U defined by Φ(∂U ′ ) = ∂Ω. (4.5) The transmission condition (4.2) is transformed by Φ to If b D = 0, then (4.4) takes the form

Thus (4.3) is fulfilled if and only if
for α = 1, i. Since ℜ(V α ) is the solution to (CP) ∞ with D and Ω replaced by U and U ′ , respectively, (4.7) is satisfied if (U, U ′ ) is a neutral inclusion. Since σ c = ∞ and U is the unit disk, it suffices to take U ′ to be a disk of radius r and the conductivity σ s to satisfy the neutrality condition (1.6), which is in this case In summary, for a domain D such that b D = 0, we take σ and r to satisfy (4.8). Then (D, Ω) where Ω is define by (4.5) is weakly neutral to multiple uniform fields.
We now present two results of numerical experiment. In Fig. 6, the conformal mapping for the domain D and the conductivity σ s are given by Φ(ζ) = ζ + 1 4ζ 2 and σ s = 0.5. (4.9) It shows the domains D and its coating Ω determined by the method described above. It clearly shows that field perturbation with the coating is much weaker than that without it. Fig. 7 is with the conformal mapping Φ(ζ) = ζ + 1 4ζ 3 and σ s = 0.3. (d) Figure 6: The core-shell structure defined by the conformal mapping Φ(ζ) = ζ + 1 4ζ 2 . Field perturbation with the coating ((c) and (d)) is much weaker that with it ((a) and (b)).

Small perturbation of disks
We now review the result from [18] which shows that a small perturbation of a disk allows a coating such that the resulting core-shell structure is a weakly neutral inclusion, namely, a PT-cancelling structure. It is an existence result based on the implicit function theorem, so we do not know how small it can be.
Let D 0 be a disk of radius r i centered at the origin. For a given function h on the unit circle T , the perturbation D h of D 0 is defined to be (4.11) We consider W 2,∞ (T ) (the derivatives up to 2 are bounded) for a class of perturbation functions h.
To define domains for coating, we let Ω 0 be the disk of radius r e centered at the origin such that (D 0 , Ω 0 ) be a neutral inclusions, namely, the radius and the conductivities are chosen so that the neutrality condition (1.6) is satisfied. We then define perturbations of Ω 0 as follows: Here b 0 , b 1 , b 2 are real constants.
If h and b are sufficiently small, then (D h , Ω b ) defines an inclusion of the core-shell structure. Let M (h, b) = M (D h , Ω b ) be the PT of (D h , Ω b ) as defined in (2.8). Since M is symmetric, we may regard M as a three-dimensional vector-valued function. Since the collection of all b of the form (4.13) is of three dimensions, M can be regarded as a mapping from U × V into R 3 , where U and V are some neighborhoods of 0 in W 2,∞ (T ) and R 3 , respectively. Since (D 0 , Ω 0 ) is neutral, we have M (0, 0) = 0. It is then proved (4.14) Then an implicit function theorem is invoked to arrive at the following theorem.
namely, the inclusion (D h , Ω b(h) ) of the core-shell structure is weakly neutral to multiple uniform fields. The mapping h → b(h) is continuous.
Proving (4.14) is quite technical. This two-dimensional theorem has been extended in [19] to three dimensions, which is even more technically complicated, to show that small perturbations of a sphere allow coatings so that the resulting inclusions of the core-shell structure are weakly neutral to multiple uniform fields. For that, the functions b in (4.13) is replaced with where Y 2 j (x) are spherical harmonics of order 2 (there are five linearly independent ones). Then the PT is regarded as a local mapping from W 2,∞ (S) × R 6 (S is the unit sphere) into R 6 , and an analogy to (4.14) is proved.

Weakly neutral inclusions by imperfect bonding
So far we consider neutral or weakly neutral inclusions of the core-shell structure. There is yet another method to achieve neutrality: It is by introducing an imperfect bonding parameter on ∂D. The perfect bonding is characterized by the continuity of the flux and the potential along the interface ∂D as given in (1.8), while the imperfect bonding is characterized by either discontinuity of the potential or that of the flux along the interface. The former one is referred to as the low conductivity (LC) type, while the latter as the high conductivity (HC) type (see, e.g., [5]).
The LC type imperfect interface problem is described as follows: as |x| → ∞. Here, β is the interface parameter of the LC type, which is a non-negative function defined on the interface ∂D.
It is proved in [39] that if D is a disk (or a ball) of radius r and then the solution u to (5.1) satisfies u(x) − a · x ≡ 0 for all x ∈ R d \ Ω, (5.3) in other words, D with β is neutral. See Fig. 8. It is proved in [16], based on the neutrality criterion obtained in [5], that the only neutral inclusions with the imperfect bonding parameters are disks (balls) with constant interface parameters if σ m is isotropic, and ellipses (ellipsoids) if σ m is anisotropic. In the same paper a way to construct an imperfect bonding parameter on the boundary of arbitrary domain has been investigated. For that purpose it is assumed that D is a perfect conductor, meaning that σ c = ∞, which is to use the conformal mapping as in (4.1). Under this assumption, the problem (5.1) in two dimensions becomes the following one: as |x| → ∞.

(5.4)
The following theorem is obtained. It is helpful to mention that the condition (5.5) is imposed, even though the definition (5.6) makes sense without the condition, to guarantee the function β defined by (5.6) being positive, and hence uniqueness of the solution to (5.4). Fig. 9 clearly shows that the field with the imperfect bonding parameter is less perturbed than that without it.