1. Introduction
One of the most prominent models in the theory of composites, the coated disk (or sphere) assemblage, represents a composite structure comprising circular inclusions of one phase embedded within a matrix of another phase. This model was first introduced by Hashin [
1] and developed in [
2] (Chapter 7) and [
3,
4,
5]. The problem can be approached as a neutral inclusion problem, wherein researchers sought a specific non-zero solution to illustrate the cloaking phenomenon, the process of concealing an object from detection. This perspective played a pivotal role in the advancement of the metamaterial theory [
6,
7], a rapidly evolving domain in modern physics. In the quasistatic limit, the electromagnetic wave propagation equations are simplified to standard conductivity equations, albeit with complex local fields and complex-valued conductivity (or permittivity). Veselago first observed the negative permittivity values [
8]. Complex permittivity values emerge due to energy loss in certain materials. The real part of permittivity represents the traditional dielectric constant, while the imaginary part characterizes energy dissipation, also known as loss permittivity [
2,
9]. Although the number of pure substances in nature is finite, their diverse compositions give rise to an infinite variety of materials at the macroscale. Consequently, the classical theory of composites covers a broad range of material property constants, including complex-valued parameters for metamaterials that adhere to some bounds [
2]. In previous studies, analytical formulas and simulations of local fields were employed to demonstrate the physical attributes of the structures examined. The current work extends this exploration by linking such problems to a classical spectral framework.
Spectral boundary value problems for PDEs are classic analysis topics [
10,
11,
12,
13]. We omit an extended study on spectral problems for ODEs and instead refer to one of the latest papers [
14]. To present the significance of the new result obtained in this paper, first, we outline three types of spectral problems and describe the third type related to a new physical trend concerning metamaterials.
Spectral problems involving a spectral parameter
that is a control parameter of a partial differential equation have been sufficiently well studied. Equation
in a domain
D with the Dirichlet boundary condition
on
is one of the fundamental spectral problems. The main keys of the theory for this problem in a Riemannian manifold
D are presented in [
15]. A recent review of this problem can be found in [
10,
11,
12,
13], and the works cited therein.
The study of spectral boundary value problems with the spectral parameter in the boundary condition begins with the Steklov eigenvalue problem for the Laplace equation
in
D with the boundary condition
on
, where
denotes the normal derivative. Recent results on this problem can be found in [
16,
17,
18,
19], and the works cited therein. The present paper is devoted to the third type of spectral problem when, instead of a boundary condition, the following linear conjugation condition is given on a smooth curve
L, dividing the plane
into two domains
and
[
20,
21,
22,
23]:
where the signs ± correspond to the limit values of the function
u harmonic in
and continuously differentiable in the closures of
. Here, it is convenient to consider the extended complex plane
and assume that
. It should be noted that the domain
may not be connected, and the constants
may be different in the different components of
L. The conjugation condition is called the perfect contact or transmission condition in the theory of composites.
Schiffer [
24] considered an eigenvalue problem for a double-layer potential on a set of curves and showed that this problem can be reduced to the spectral problem (
1). The spectral parameter
was supposed to be the same for all components of the curve.
The present paper is devoted to studying a coated composite with complex-valued conductivity (permittivity) phases. This physical problem is stated as a spectral
-linear problem. The problem is reduced to an iterative functional equation [
25]. It is established that the theory of neutral inclusion related to metamaterials may be considered a new type of spectral problem with the spectral parameter
in the conjugation condition (
1).
2. Functional Spaces
Let
denote the classic Hardy space of analytic functions in the unit disk
[
26], and let
be the corresponding space on the disk
. The Hilbert space
is endowed with the norm
Introduce a sequence space
isomorphic to
as follows. Let a function
belong to
. Then, the coefficients
of the function (
3) can be considered as elements of a discrete space. The sets of these coefficients
form the space
isomorphic to
.
The isomorphism
has to be checked on the level of the scalar product since the considered spaces are Hilbert. Consider the radial limit of
on the circle
existing for almost every argument
:
The scalar product
in the Hardy space
coincides with the scalar product
in the space
given by the integral
Transform the integral (
6)
By substituting the Taylor series of functions
and
, we find that
Find
and substitute the first expressions of (
8) and (
9) into (
7). The obtained integral can be calculated by the residue theorem, by the coefficient in
of the integrand
Consequently, the norms in the considered spaces can be written in the form
3. Spectral Problem
Let
,
and
be domains in the extended complex plane
(see
Figure 1). We are looking for functions
,
and
that are harmonic in
,
, and
D, respectively, and belonging to the Sobolev spaces
[
27] in the corresponding domains.
Consider the following conductivity problem. Let the domains
,
, and
D be occupied by the materials of conductivities
and
, and the normalized conductivity
, respectively. The perfect contact between the components of the considered composite has the form [
21]
where
are the outward normal derivatives to the circles. It is supposed that
The boundary value problem in Equations (
12)–(
14) can be stated for the spectral parameter
or
. We will demonstrate below its equivalence for a complex boundary value problem in relation to Equations (
12)–(
14).
Following [
21], we introduce complex potentials analytically in the considered domain:
where
are the imaginary parts of the considered analytic functions. Let
denote a tangent derivative, where
s is a natural length parameter of arcs. The Riemann–Cauchy conditions on the circles
and
have the form
Then, the second Equations (
12) and (
13) become
By integrating (
18), we get
where
are arbitrary constants.
We then introduce the contrast parameters
The first real Equation (
12) and the real Equation (
19) for
can be written in the first of one complex equation. The same concerns the second pair of real equations for
. The obtained complex equations can be written in the form of the
–linear problem:
where
are constants. The function
is bounded at infinity.
We introduce the complex velocity
and
The boundary conditions (
21) can be differentiated along the curves
. It was shown in [
20,
21] that
where the unit outward normal vectors
are expressed in terms of complex values.
4. Functional Equation
In the present section, the boundary value problem is reduced to a functional equation that will be solved.
The normal vector can be written as follows:
Substitute (
25) into (
24) and write the resulting equation in the extended form:
The function
near infinity is represented by the series
The spectral problem for the complex velocity can be stated as follows. It is required to find
and constants such as
and
that the boundary value problem given by Equations (
26) and (
29) has a non-zero solution. One of the parameters
and
can be fixed. For example,
is fixed, and
has to be found. In the case of the unknown
, we arrive at the Schiffer spectral problem [
24].
Introduce the analytic function separately in
,
, and
, and ensure that it is separately continuous in the closures of the considered domains:
Calculate the jump of
across
using the relation
,
according the first relation (
26). Therefore,
is analytically continued through the circle
by the principle of analytic continuation. In the same way, one can check that
is analytically continued through the circle
. Find
using Equation (
29). By Liouville’s theorem,
everywhere in the extended complex plane. Writing
in the extended form (
30) in
and
, we obtain the system of functional equations
Substituting
instead of
z in the second Equation (
32), we obtain
Substituting Equation (
33) into the first Equation (
32) yields the functional equation
The unknown function
is analytic in the disk
and continuous in
. One can consider the functional equation in the wider Hardy space
. The result does not depend on the space since the function
is analytically continued to the complex plane
by Equation (
34).
4.1. Case of Equal Contrast Parameters
In the present section, consider the case
. Introduce the operator
acting on
in the written part of Equation (
34). This is a shift operator inside the domain of analyticity. According to Shapiro [
28], the shift operator is a compact operator in
. Thus, we arrive at the eigenvalue problem for the operator
A:
with the spectral parameter
.
Represent the unknown function in the form of its Taylor series as follows:
Substitute it into Equation (
36) and take the coefficients in the same power of
z:
Equation (
38) has a countable number of non-zero solutions:
The corresponding solutions of the functional Equation (
36) have the form
Theorem 1. The compact operator A defined by Equation (35) is self-adjoint in . Its simple eigenvalues are given by Equation (39) and the eigenfunctions in (40). The relation (39) can be written in the form Proof. Find an operator
adjoint to the operator
A by the definition of the scalar product (
5)
It will be shown that
.
We have
Using (
35), calculate
Equate the latter expression to
Hence, . Therefore, the operator A is self-adjoint.
It follows from the properties of the compact self-adjoint operators in the Hilbert space that the spectrum of the operator
A consists of the eigenvalues in (
39) satisfying the inequality
One can see that
, as
, corresponds to the general theory. The eigenvalues given by (
40) are simple since the monomials
are orthogonal with respect to the inner product (
5), and form a basis for
.
The theorem is proved. □
4.2. Case of Different Contrast Parameters
In the present section, the parameter is fixed and is considered as a spectral parameter.
Introduce the operator
acting on
in the right part of (
34).
Theorem 2. The compact operator B defined by Equation (47) is self-adjoint in . Its eigenvalues are given bythe corresponding eigenfunctions . The relation (48) can be written in the form Proof. Substitute the series (
37) into Equation (
34)
where
. In Equation (
50), the coefficients with the same powers
z are taken as
This yields Equation (
48) and
, as
.
The operator is self-conjugate. It is proven in a similar way to Theorem 1 for the operator A.
The theorem is proved. □
Consider the particular case when
. Then,
A numerical example of the vector field for
is presented in
Figure 2 and
Figure 3. It is worth noting that the domains
and
are occupied by conducting and absorbing materials, respectively. This serves as the physical explanation of the non-vanishing field in the free-source medium. The vector field is expressed by the complex velocity. The gradient (
22) can be calculated by the formula
and
4.3. Hashin–Shtrikman Assemblage
The Hashin–Shtrikman assemblage is not governed by the spectral problem considered above. The condition (
29) means that the external flux at infinity is absent. Replace Equation (
29) by the condition of the non-vanishing external flux parallel to the
-axes:
Then, we arrive at the Hashin–Shtrikman problem. Its solution is then written in terms of complex potentials in the corresponding domains:
The analog of spectral relation (
49) has the form
Here,
and 1 are the conductivities of the core and the coating, respectively. In the framework of the Hashin–Shtrikman model,
yields the effective conductivity of neutral non-overlapping, coated inclusions distributed in the plane. It can be written in the form of the Hashin–Shtrikman bound and, equivalently, as the Clausius–Mossotti (Maxwell) approximation [
2] (Chapter 7). It is worth noting that condition (
57) can be fulfilled for positive conductivities
and
.
5. Conclusions and Discussion
The coated inclusions in the theory of composites distinguish many exceptional properties. In the present paper, we consider a composite, displayed in
Figure 1, with complex-valued conductivities under the absence of any sources and sinks in the whole plane, including infinity. It is established that a non-zero flux takes place under the conditions (
49) or (
41) on the contrast parameters
and
expressed through the conductivity of components by Equation (
20). The relations (
49) and (
41) cannot be fulfilled for the traditional positive conductivities
and
and can hold only for negative and complex values. Such a case takes place for metamaterials [
6,
8].
A boundary value problem is stated with the spectral parameter on the boundary. Complex potentials are introduced in the considered domain through analytic functions. Next, the boundary value problem is reduced to a spectral problem for a functional equation that has been solved. Equal and different contrast parameters are considered. Using the classic theory of self-adjoint operators in the Hilbert space, the complete set of eigenvalues and eigenfunctions modeling special fields are written without external sources.
The classic spectral problem with the parameter
in the partial differential equation, e.g., Laplace’s equation, models wave processes for the special frequencies when an initial mode holds in a medium. In the considered model, Laplace’s equation is not perturbed, which corresponds to a stationary process. However, the wave mode is hidden in the imaginary parts of the material constants
and
[
9]. Hence, the main physical mechanism, wave mode, is the main reason for the occurrence of unusual phenomena in materials. It is interesting to study the general geometric configurations to check this hypothesis.
The Hashin–Shtrikman assemblage discussed in
Section 4.3 is an example of neutral inclusions [
2,
3] with the classical case of positive material constants
and
. It is worth noting that the complex velocity is bounded at infinity, e.g., Equation (
55). At the same time, the complex potential
defined by Equation (
16) has a pole at infinity. This observation distinguishes the neutral inclusion from others.
The results obtained highlight the significance of the eigenvalue problem in metamaterial theory. This approach helps refine the range of specific metamaterial constants, contributing to a more precise characterization.