# Important Issues on Spectral Properties of a Transmission Eigenvalue Problem

Authors: Besiana Cobani; Aurora Simoni; Ledia Subashi
Journal: International Journal of Differential Equations
 (2021)

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## 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.


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## Body

## 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: letΩ⊂ℝd,d∈2,3 be a collection of bounded simply connected domains with piecewise smooth boundary ∂Ω, and with connected exterior ℝd/Ω¯, and let the bounded and real-valued function ndenote the refractive index, ν the outward unit normal to the boundary ∂Ω, k>0 the wave number, and η a real-valued, positive boundary parameter [7]. We assume that Ω is given. The interior transmission eigenvalue problem corresponding to the scattering problem is to determine the values k∈ℂ such that there exists a nontrivial solution to(1)Δw+k2nw=0,inΩ,(2)Δv+k2v=0,inΩ,(3)w−v=−η∂v∂ν,on∂Ω,(4)∂w∂ν−∂v∂ν=0,on∂Ω.We call such values ofk 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(5)H˜02Ω=u∈H2Ω,such that∂u∂ν=0,on∂Ω.The spaceH˜02Ω is equipped with ·H2Ω norm [2]. Then, we give a more formal presentation of the interior problem as a quadratic form related to k, 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 ∂Ω is of class C2, n∈L∞Ω is real-valued and there exists α such that 0<n∗<1 or n∗>1, where(6)n∗=infx∈Ωnx,n∗=supx∈Ωnx.Furthermore,η∈L∞∂Ω is real-valued such that η>0.The transmission eigenvalue problem is as follows: for given functionsn∈L∞Ω and η∈L∞∂Ω find k>0 and w,v∈L2Ω×L2Ω such that ∂w/∂ν−∂v/∂ν∈H˜02Ω and w,v satisfies (1)–(4).Next, we reformulate this transmission eigenvalue problem as an eigenvalue problem for a fourth-order differential operator. To this end, letu∈H˜02Ω denote the difference w−v. Then, u satisfies(7)Δ+k2nu=−k2n−1v,inΩ.Applying the operatorΔ+k2 to both sides, we have(8)Δ+k21n−1Δ+k2nu=0,inΩ.Note thatn−1 is either strictly positive or strictly negative in Ω thanks to the above assumptions on n. From the relation (5), the function v can be given as v=1/−k2n−1Δ+k2nu, and as a result, we write the boundary condition (3) as(9)u=ηk2∂∂ν1n−1Δ+k2nu,on∂Ω.Multiplying with a test functionφ¯, identity (8), and integrating over Ω, we obtain(10)∫ΩΔ+k21n−1Δ+k2nuφ¯dx=0,∀φ∈H˜02Ω,and using the boundary condition (7) and Green’s identity, we arrive at(11)∫∂Ωφ¯k2uηds−∫Ω1n−1Δu+k2nu·Δφ¯+k2φ¯dx=0.Hence, the transmission eigenvalue problem becomes a quadratic form related tok. This is important to prove the discreteness of transmission eigenvalues [2].We operate on the variational (11) rewritten as(12)∫Ω1n−1Δu+k2nu·Δφ¯+k2φ¯dx−∫∂Ωφ¯k2uηds=0,∀φ∈H˜02Ω,where we group each factor in terms of k as follows:(13)∫Ω1n−1Δu·Δφ¯dx−k2∫∂Ωφ¯uηds+k2∫Ω1n−1Δuφ¯+nuΔφ¯dx+k4∫Ω1n−1nu·φ¯dx=0.By the Riesz representation theorem [9, 11, 12], we define the following operators:(14)T:H˜02Ω⟶H˜02Ω,such that(15)Tu,φ=∫Ω1n−1Δu·Δφ¯dx+∫Ω∇u·∇φ¯dx+∫Ωu·φ¯dx,Tk:H˜02Ω⟶H˜02Ω,by(16)Tku,φ=−∫Ω∇u·∇φ¯dx−∫Ωu·φ¯dx−k2∫∂Ωφ¯uηds+k2∫Ω1n−1Δuφ¯+nuΔφ¯dx+k4∫Ω1n−1nu·φ¯dx.Then, the problem can be written as(17)Tu+Tku=0,∀φ∈H˜02Ω.Next, we prove that the operatorT is invertible, and the operator T−1:H˜02Ω⟶H˜02Ω is bounded. This assertion follows from the coercivity of the following sesquilinear form:(18)a·,·≔∫Ω1n−1Δu·Δφ¯dx+∫Ω∇u·∇φ¯dx+∫Ωu·φ¯dxis coercive, that is proven by the following inequalities:(19)au,φ≥1n∗−1ΔuL2ΩΔφL2Ω+∇uL2Ω∇φL2Ω+uL2ΩφL2Ω≥c∇uH2Ω∇φH2Ω.Theorem 1.
The operatorTK:H˜02Ω⟶H˜02Ω is compact.Proof.
Letu∈H02Ω which is compactly embedded in H2Ω.(20)∇u∈H1Ω,u∈H2Ω,u|∂Ω∈H3/2∂Ω.
Thus,∫Ω∇u·∇φ¯dx contains L2Ω norm of ∇u which is in H1Ω and H1Ω is compactly embedded in L2Ω. Furthermore, ∫Ω∇u·∇φ¯dx contains L2Ω norm of u and H2Ω is compactly embedded in L2Ω. For the third integral, ∫∂Ω1/nu·φ¯ contains L2∂Ω norm of u because u is in H3/2∂Ω which is compactly embedded in L2∂Ω. We have the following relations using Green’s first identity:(21)∫Ω1n−1Δu·φ¯dx=−∫Ω∇u·∇1n−1φ¯dx+∫∂Ω∂u∂ν·1n−1φ¯ds=−∫Ω∇u·∇1n−1φ¯dx.
Hence,(22)∫Ω1n−1Δu·φ¯dx=−∫Ω∇u·∇1n−1φ¯dx,and ∇u is in H1Ω. Thus, the same as above H1Ω is compactly embedded in L2Ω.Finally, from the above, we can write (17)(23)I+T−1Tku=0,∀u∈H02Ω.The operatorTk is an analytic operator valued (it is polynomial in k∈ℂ). The analytic Fredholm theory implies that(24)I+T−1Tkhas an empty kernel for all k∈ℂ, except for possibly a discrete set, if we find one k for which the kernel is empty. Now, in Section 3, we show that k 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.
Letτ⟶Aτ be a continuous mapping from 0,∞ to the set of self-adjoint and positive definite bounded linear operators on U, where U is a separable Hilbert space, and let B be a self-adjoint and nonnegative compact bounded linear operator on U. We assume that there exist two positive constants τ0>0 and τ1>0 such that(1)
Aτ0−τ0B is positive on U.(2)
Aτ1−τ1B is non-positive on a k dimensional subspace Wk of U. Then, each of the equations λjτ=τ for j=1,k¯ has at least one solution in τ0,τ1 where λjτ is the j th eigenvalue (counting multiplicity) of Aτ with respect to B.To use the above theorem, we have to write relation (11) as Aτ−τB=0, where Aτ is a family of positive definite self-adjoint bounded linear operator, B is a nonnegative compact bounded linear operator, defined on appropriate Hilbert spaces. Then, a transmission eigenvalue is the solution of λτ−τ=0, where λτ is an eigenvalue of the generalized eigenvalue problem Aτ−τB=0, where in our case τ:=k2 [13]. In this section, we assume that k is real.Let us first recall the variational formulation (11):(25)∫∂Ωφ¯k2uηds−∫Ω1n−1Δu+k2nu·Δφ¯+k2φ¯dx=0,∀φ∈H˜02Ω.To have the generalized eigenvalue problemAτ−τB=0, we do the following operation in (11):(26)∫∂Ωφ¯k2uηds−∫Ω1n−1Δu+k2n−1+1u·Δφ¯+k2φ¯dx=0,∀φ∈H˜02Ω,∫∂Ωφ¯k2uηds−∫Ω1n−1Δu+k2u·Δφ¯+k2φ¯dx−∫Ωk2uΔφ¯+k2φ¯dx=0,∫∂Ωφ¯k2uηds−∫Ω1n−1Δu+k2u·Δφ¯+k2φ¯dx−k2−∫Ω∇u·∇φ¯dx+∫∂Ωu∂φ∂νds−k4∫Ωu·φ¯dx=0.Sinceφ∈H˜02Ω, then ∫∂Ωu∂φ/∂νds=0; so, the following identity is(27)∫∂Ωφ¯k2uηds−∫Ω1n−1Δu+k2u·Δφ¯+k2φ¯dx+k2∫Ω∇u·∇φ¯dx−k4∫Ωu·φ¯dx=0.Multiplying by−1 both sides, we have the final result(28)−∫∂Ωφ¯k2uηds+∫Ω1n−1Δu+k2u·Δφ¯+k2φ¯dx−k2∫Ω∇u·∇φ¯dx+k4∫Ωu·φ¯dx=0.We define the following sesquilinear forms:(29)Aku,φ=1n−1Δu+k2u,Δφ+k2φΩ+k4u,φΩ,Bu,φ=∇u,∇φΩ+∫∂Ωφ¯uηds.So, ifn∗>1, we can write our TEP as Aku,φ−k2Bu,φ=0, for all φ∈H˜02Ω.Next, we need to prove that we are in the conditions of the theorem, which means that the operatorsAk are self-adjoint positive definite bounded linear operators on H˜02Ω and the operator B is self-adjoint and nonnegative compact bounded on H˜02Ω.Let us first show that operatorAk has the property required. Since n∗>1, then 1/n−1>β>0 almost everywhere in Ω. The following inequalities show that Ak is a coercive sesquilinear form on H˜02Ω×H˜02Ω.(30)Aku,u≥βΔu+k2uL22+k4uL22≥βΔuL2Ω2−2βΔuL2Ω2k2uL2Ω2+β+1k4uL2Ω4.If we take anyϵ between β and β+1, then we have(31)εk2uL2Ω2−βεΔuL2Ω22+β−β2εΔuL2Ω22+β+1−εk4uL2Ω4,for β<ε<β+1. Since the first term of (31) is nonnegative, we obtain(32)Aku,u≥β−β2εΔuL2Ω22+β+1−εk4uL2Ω4.Since foru∈H˜02Ω,∂u/∂ν=0, then we can use Poincare inequality [7, 14] and have(33)∇uL2Ω≤1λ1ΩΔuL2Ω2.The choice ofε guaranty now is(34)Aku,u≥c1ΔuL2Ω2+c2k4uL2Ω2=CkuH2Ω2.From Riesz representation theorem, we define the bounded linear operators(35)Ak:H˜02Ω×H˜02Ω,byAk2u,vH2Ω=Au,v.Sincen and β are real the sesquilinear forms Ak are Hermitian; so, the operators are self-adjoint.Also, we see thatBu,φ=∇u,∇φΩ+∫∂Ωφ¯u/ηds is nonnegative because(36)Bu,u=∇u,∇u¯Ω+∫∂Ωu¯uηds=∫Ω∇u·u¯dx+∫∂Ωu¯uηds=∫Ω∇u2dx+∫∂Ωu2ηds≥0.Since from the assumptionη>0.In Theorem1, we have shown that ∇u is in H1Ω and H1Ω is compactly embedded in L2Ω. Furthermore, ∫∂Ω1/nu·φ¯ contains L2∂Ω norm of u because u is in H3/2∂Ω which is compactly embedded in L2∂Ω; so, the operator(37)Bu,φ:H˜02Ω×H˜02Ω,defined byBu,vH2Ω=ℬu,v,is a nonnegative compact self-adjoint operator. According to the theorem, we have to show that the operator Ak−k2B is positive for some k0 on H˜02Ω. We will consider as above the case when n>1. From assumption η>0, we have the coersivity of operator B.(38)Aku,u−k2Bu,u=∫Ω1n−1Δu+k2u2dx+k2∫Ωu2dx−k2∫Ω∇u2dx+∫∂Ωu21ηds≥∫Ω1n−1Δu+k2u2dx+k2∫Ωu2dx−k2min1,1ηmin∫Ω∇2dx+∫∂Ωu2ds≥C1ΔuL2Ω2+C2k4uL2Ω2−k2α1λ1ΩΔuL2Ω2.SinceC1 does not depend on k, for k small enough, there exists a k0 such that the quantity C1−k2α1/λ1Ω>0. We proved that condition 1 of Theorem 2 is satisfied.Next, we must show that assumption (2) of Theorem2 is also satisfied. We prove this in Theorem 3.Theorem 3.
Assume thatn>1 a.e in Ω, 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 Sj=Sxj,ε≔x∈Rm,m=2,3:x−xj<ε, where xj∈Ω and ε>0.
The number of ballsSj such that Si∩Sj=Φ is denoted by Nε, where ε is sufficiently small to have the inclusion Sj¯⊂Ω. For the following interior transmission problem, there exists infinitely many transmission eigenvalues [16].(39)Δwj+k2n∗wj=0,andΔvj+k2vj=0inSj,(40)wj−vj=0,and∂wj∂ν−∂vj∂ν=0,on∂Sj,where n∗=infnx for x∈Ω. Let uj denote the difference uj=wj−vj∈H02˜Sj and let uj˜ be the extension of uj by zero to Ω and H02Ω=u∈H2Ω,such thatu=0,inΩ,∂u/∂ν=0on∂Ω. Obviously, H02Ω⊂H˜02Ω, so uj˜∈H02˜Ω. uj˜ is orthogonal to ui˜ for all j≠i in H02˜Ω since their support is disjoint. Let denote H˜Nε=spanu1˜,u2˜,…,u˜Nε. H˜Nε forms an Nε-dimensional subspace of H02Ω. Let k=kj be an eigenvalue of (39) and (40) and uj=wj−vj the corresponding eigenfunctions, then(41)0=∫Ω1n∗−1Δuj˜+k2uj˜·Δuj˜¯+k2uj˜¯dx≤∫Sj1n∗−1Δuj˜+k2uj˜·Δuj˜¯+k2uj˜¯dx=∫Sj1nm−1Δuj˜+k2uj˜2dx+k4∫Sjuj˜2dx−k2∫Sj∇uj˜2dx.
We denote bykε the first transmission eigenvalue, which is the same for each of these balls since they have the same radius and refractive index. This eigenvalue kε has the respective eigen functions w˜j,v˜j, and u˜j=w˜j−v˜j. Then, we have(42)Akεuj˜,uj˜−kε2ℬuj˜,uj˜=∫Ω1nm−1Δuj˜+kε2uj˜2dx+kε4∫Ωuj˜2dx−kε2∫Ω∇uj˜2dx−kε2∫∂Ω1ηuj˜2ds≤∫Sj1nm−1Δuj˜+k2uj˜2dx+k4∫Sjuj˜2dx−k2∫Sj∇uj˜2dx=0.
So,Akεu,u−kε2ℬu,u≤0. We are in the conditions of Theorem 2, so they are Nε eigenvalues in the interval 0,kε.


## 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 tok 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.



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