Important Issues on Spectral Properties of a Transmission Eigenvalue Problem

Abstract

Nowadays, inverse scattering is an important field of interest for many mathematicians who deal with partial differential equations theory, and the research in inverse scattering is in continuous progress. There are many problems related to scattering by an inhomogeneous media. Here, we study the transmission eigenvalue problem corresponding to a new scattering problem, where boundary conditions differ from any other interior problem studied previously. more specifically, instead of prescribing the difference Cauchy data on the boundary which is the classical form of the problem, we consider the case when the difference of the trace of the fields is proportional to the normal derivative of the field. Typical concerns related to TEP (transmission eigenvalue problem) are Fredholm property and solvability, the discreteness of the transmission eigenvalues, and their existence. In this article, we provide answers for all these concerns in a given interior transmission problem for an inhomogeneous media. We use the variational method and a very important theorem on the existence of transmission eigenvalues to arrive at the conclusion of the existence of the transmission eigenvalues.

1. Introduction

To study thoroughly a problem in inverse scattering for an inhomogeneous media with a conductive boundary, we should deal with its interior transmission problem. Its homogeneous version is referred to as the transmission eigenvalue problem. The TEP (transmission eigenvalue problem) is nonlinear and not self-adjoint [1, 2]. In this paper, the focus is to prove the discreteness and the existence of real transmission eigenvalues of a given problem.

The discreteness of transmission eigenvalues is important to prove the solvability of the interior transmission problem because the latter satisfies the Fredholm Alternative. From a practical point of view the discreteness is needed to guarantee that the reconstruction methods, such as linear sampling, succeed in the reconstruction of the scatterer of an inhomogeneous medium [3, 4]. The importance of the existence of transmission eigenvalues is that they determinate data about the material properties of the inhomogeneous media. Besides the theoretical importance of transmission eigenvalues in connection with uniqueness and reconstruction results in inverse scattering theory, recently they have been used to obtain information about the index of refraction from measured data [5]. This is based on the important result that transmission eigenvalues can be determined from the measured far field data [2, 4]. The interior problem is the key problem in inverse scattering. After studying thoroughly the issue of the well-posedness of the direct problem, [6] which is necessary to continue with the inverse problem, we focus our work in the solution of the interior problem which is the central problem in inverse scattering. In inhomogeneous media, the issue of solving the interior problem is related to the transmission eigenvalues. The question that concerns the most is the existence of transmission eigenvalues [4].

In this paper, we study the discreteness and the existence of real transmission eigenvalues associated with the following scattering problem: let be a collection of bounded simply connected domains with piecewise smooth boundary , and with connected exterior , and let the bounded and real-valued function denote the refractive index, the outward unit normal to the boundary , the wave number, and a real-valued, positive boundary parameter [7]. We assume that is given. The interior transmission eigenvalue problem corresponding to the scattering problem is to determine the values such that there exists a nontrivial solution to

We call such values of the interior transmission eigenvalues. This problem is known to be non-self-adjoint eigenvalue problem [8]. The goal is to show that the set of transmission eigenvalues (real and complex) is at most discrete and then prove the existence of real transmission eigenvalues.

To this end, we first formulate the respective problem in the variational form [3, 9]. For this propose, we operate in the Sobolev space

The space is equipped with norm [2]. Then, we give a more formal presentation of the interior problem as a quadratic form related to , where all the operators present are compact and self-adjoint. From spectral theory, the discreetness follows [2, 10]. The most important part of the study is showing the existence of real transmission eigenvalues. To this end, we rely on the variational formulation [9, 11] of the transmission eigenvalue problem, and we use a very important theorem of [8].

2. The Discreteness of the Transmission Eigenvalues

For analytic consideration, we make the following assumptions [2]: the boundary is of class , is real-valued and there exists such that or , where

Furthermore, is real-valued such that .

The transmission eigenvalue problem is as follows: for given functions and find and such that and satisfies (1)–(4).

Next, we reformulate this transmission eigenvalue problem as an eigenvalue problem for a fourth-order differential operator. To this end, let denote the difference . Then, satisfies

Applying the operator to both sides, we have

Note that is either strictly positive or strictly negative in thanks to the above assumptions on . From the relation (5), the function can be given as , and as a result, we write the boundary condition (3) as

Multiplying with a test function , identity (8), and integrating over , we obtainand using the boundary condition (7) and Green’s identity, we arrive at

Hence, the transmission eigenvalue problem becomes a quadratic form related to . This is important to prove the discreteness of transmission eigenvalues [2].

We operate on the variational (11) rewritten aswhere we group each factor in terms of as follows:

By the Riesz representation theorem [9, 11, 12], we define the following operators:such thatby

Then, the problem can be written as

Next, we prove that the operator is invertible, and the operator is bounded. This assertion follows from the coercivity of the following sesquilinear form:is coercive, that is proven by the following inequalities:

Theorem 1. The operator is compact.

Proof. Let which is compactly embedded in .Thus, contains norm of which is in and is compactly embedded in . Furthermore, contains norm of and is compactly embedded in . For the third integral, contains norm of because is in which is compactly embedded in . We have the following relations using Green’s first identity:Hence,and is in . Thus, the same as above is compactly embedded in .

Finally, from the above, we can write (17)

The operator is an analytic operator valued (it is polynomial in ). The analytic Fredholm theory implies thathas an empty kernel for all , except for possibly a discrete set, if we find one for which the kernel is empty. Now, in Section 3, we show that real and small enough are not transmission eigenvalues. Hence, the set of transmission eigenvalues in the complex plane is discrete with as the only accumulation point.

3. The Existence of Real Transmission Eigenvalues

The study of existence of transmission eigenvalues is based in the following theorem (Cakoni-Haddar) [8].

Theorem 2. Let be a continuous mapping from to the set of self-adjoint and positive definite bounded linear operators on , where is a separable Hilbert space, and let be a self-adjoint and nonnegative compact bounded linear operator on . We assume that there exist two positive constants and such that(1)  is positive on .(2)  is non-positive on a k dimensional subspace of . Then, each of the equations for has at least one solution in where is the th eigenvalue (counting multiplicity) of with respect to .

To use the above theorem, we have to write relation (11) as , where is a family of positive definite self-adjoint bounded linear operator, is a nonnegative compact bounded linear operator, defined on appropriate Hilbert spaces. Then, a transmission eigenvalue is the solution of , where is an eigenvalue of the generalized eigenvalue problem , where in our case [13]. In this section, we assume that is real.

Let us first recall the variational formulation (11):

To have the generalized eigenvalue problem , we do the following operation in (11):

Since , then ; so, the following identity is

Multiplying by both sides, we have the final result

We define the following sesquilinear forms:

So, if , we can write our TEP as , for all .

Next, we need to prove that we are in the conditions of the theorem, which means that the operators are self-adjoint positive definite bounded linear operators on and the operator is self-adjoint and nonnegative compact bounded on .

Let us first show that operator has the property required. Since , then almost everywhere in . The following inequalities show that is a coercive sesquilinear form on .

If we take any between and , then we havefor . Since the first term of (31) is nonnegative, we obtain

Since for , then we can use Poincare inequality [7, 14] and have

The choice of guaranty now is

From Riesz representation theorem, we define the bounded linear operators

Since and are real the sesquilinear forms are Hermitian; so, the operators are self-adjoint.

Also, we see that is nonnegative because

Since from the assumption .

In Theorem 1, we have shown that is in and is compactly embedded in . Furthermore, contains norm of because is in which is compactly embedded in ; so, the operatoris a nonnegative compact self-adjoint operator. According to the theorem, we have to show that the operator is positive for some on . We will consider as above the case when . From assumption , we have the coersivity of operator .

Since does not depend on , for small enough, there exists a such that the quantity . We proved that condition of Theorem 2 is satisfied.

Next, we must show that assumption (2) of Theorem 2 is also satisfied. We prove this in Theorem 3.

Theorem 3. Assume that a.e in , then there exists infinitely many transmission eigenvalues for problem (1)–(4).

Proof. The well-known result for the spherically stratified domain given by Colton et al. [15, 16] is applied to arrive at the existence of T.E. First, let , where and .
The number of balls such that is denoted by , where is sufficiently small to have the inclusion . For the following interior transmission problem, there exists infinitely many transmission eigenvalues [16].where for . Let denote the difference and let be the extension of by zero to and . Obviously, , so . is orthogonal to for all in since their support is disjoint. Let denote . forms an -dimensional subspace of . Let be an eigenvalue of (39) and (40) and the corresponding eigenfunctions, thenWe denote by the first transmission eigenvalue, which is the same for each of these balls since they have the same radius and refractive index. This eigenvalue has the respective eigen functions , and . Then, we haveSo, . We are in the conditions of Theorem 2, so they are eigenvalues in the interval .

4. Conclusions

In this paper, we studied in detail the discreteness and the existence of transmission eigenvalues of the given transmission problem in inhomogeneous media. We first used the variational approach to the problem which allowed us to present the TEP as a quadratic form related to k to prove that the spectrum of the transmission eigenvalue problem is discrete. From a practical point of view, the discreteness is needed to guarantee that the numerical methods such as linear sampling succeed in the reconstruction of the scatterer of an inhomogeneous medium. The most important part of the work is the proof of the existence of the transmission eigenvalues. For this, we used a very important theorem of [8]. The existence of the eigenvalues is very important because they provide information about material properties of the scattering media and can be determined from scattering data and since they provide information about material properties of the scattering object and the transmission eigenvalues can play an important role in a variety of problems in target identification.

Data Availability

All data required for this study are included within this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.