ELECTROMAGNETIC INTERIOR TRANSMISSION EIGENVALUE PROBLEM FOR AN INHOMOGENEOUS MEDIUM WITH A CONDUCTIVE BOUNDARY

. The interior transmission eigenvalue problem plays a basic role in the study of inverse scattering problems for an inhomogeneous medium. In this paper, we consider the electromagnetic interior transmission eigenvalue problem for an inhomogeneous medium with conductive boundary. Our main focus is to understand the associated eigenvalue problem, more speciﬁcally to prove the transmission eigenvalues form a discrete set and show that they exist by employing a variety of variational techniques under various assumptions on the index of refraction.


1.
Introduction. In recent years, the interior transmission eigenvalue problem has become an important area of research in inverse scattering theory. Although simply stated, the transmission eigenvalue problem is a non-selfadjoint eigenvalue problem that can not be covered by the standard theory of eigenvalue problems for elliptic equations. Our particular interest is the fact that transmission eigenvalues carry information about the material properties of the scattering object. Besides, the transmission eigenvalues can be determined by the measured scattering data, see [5,17,19,22,29]. For a connection of the interior transmission problem with the scattering problem we refer the readers to [4,10,11,14,16,21].
Up to now, some progress has been made in the study of transmission eigenvalue problem. Regarding the transmission eigenvalue problem for Helmholtz equation, several papers have appeared that address both the question of discreteness and existence of transmission eigenvalues, for details, we refer to the monographs [1,6,7,9,12,24,25,27,28] and the references therein. Similarly, some research has been made about the transmission eigenvalues for Maxwell's equations and plays an important role in application [18,26]. In [2,3], discreteness of electromagnetic transmission eigenvalues has been proved under the condition that the magnetic and electric permittivity does not change sign near the boundary. It is known [8] that the transmission eigenvalues for Maxwell's equations form a discrete set without finite accumulation point and there exists an infinite set of real transmission eigenvalues under some assumptions for the index of refraction. In the current article [23], they extend the result [7] to the much more technical and complicated Maxwell's system governing the electromagnetic scattering, and prove the discreteness and existence 1388 YUEBIN HAO of the interior transmission eigenvalues. For more related works, we refer to the monographs [13,20] and the references therein.
In the current paper, the underlying scattering problem is the scattering of electromagnetic waves by a non-magnetic material situated in homogenous background, which in terms of the electric field reads: where D ⊂ R 3 be a collection of bounded simply-connected domains with C ∞smooth boundary ∂D and ν denote the outward unit normal to the boundary ∂D. Let n(x) denote the refractive index, k be the wave number and η be a boundary parameter. Besides, E := E i +E s be the total electric field, where E i be the incident electric field and E s be the scattered electric field. E + (E − ) denote the limit of E on the surface ∂D from the exterior (interior) of ∂D, that is ). The Silver-Müller radiation condition is satisfied uniformly with respect tox = x/|x|.
We assume that D is given. The interior transmission eigenvalue problem corresponding to (1) is to determine k > 0 such that there exists a nontrivial solution to If the above problem (2) has a nontrivial solution, then k is called the interior transmission eigenvalues. In this paper, we will consider the case where η is realvalued and positive. We shall first prove the discreteness and existence of the interior transmission eigenvalues of the system (2). To our best knowledge, those results are new to literature in the study of electromagnetic interior transmission eigenvalue problems.
The rest of the paper is organized as follows. In section 2, we define the interior transmission eigenvalue problem in the appropriate Sobolev spaces and derive its variational form. In section 3, we investigate the spectral properties of the interior transmission eigenvalue problem. We prove the discreteness and the existence of the interior transmission eigenvalues, provided that the real-valued index of refraction n := n(x) in the medium satisfies 0 < n < 1. Our approach does not work if n > 1. In section 4, we obtain monotonicity results for the transmission eigenvalues with respect to the material parameters n and η.

Preliminaries and variational formulation.
2.1. Preliminaries. In order to formulate our transmission eigenvalue problem more precisely, we need to introduce some related spaces.
Let D ⊂ R 3 be defined as above. We denote by (·, ·) D the L 2 (D) 3  and the corresponding norm · H 2 (curl,D) . The electromagnetic interior transmission eigenvalue problem reads as follows: for given functions n ∈ C 1 (D) and η ∈ L ∞ (∂D), find k > 0 and E 1 , (2). For analytical considerations we put the following hypotheses on n, η. (

Variational formulation.
In this subsection we are going to derive the variational form of the interior transmission problem (2).
By the first and second identities in (2) we have that E satisfies We also get the boundary conditions and By the boundary conditions (5) and (6), together with Green's second vector theorem, we have where F ∈ H 2 0 (curl, D) be a test vector function and we have used the equation

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According to (3) and (4), we obtain Therefore, Furthermore, by Green's first vector theorem, we obtain Therefore, the variational formulation of the interior transmission problem (3)-(6) becomes: find E ∈ H 2 0 (curl, D) satisfies (8) for all F ∈ H 2 0 (curl, D). The functions E 2 and E 1 are related to E through Definition 2.1. The values of k > 0 are said to be a transmission eigenvalue if the electromagnetic interior transmission problem (2) has a nontrivial solution . If k > 0 is a transmission eigenvalue, we call the solution E ∈ H 2 0 (curl, D) of (8) the corresponding eigenfunction.
3. Spectral property of the electromagnetic interior transmission eigenvalue problem. In this section, we investigate the spectral properties of the interior transmission eigenvalue problem (2). We will prove the discreteness and the existence of interior transmission eigenvalue by considering 0 < n < 1 in D. Next, we denote by n * = inf x∈D n(x) and n * = sup x∈D n(x).

Discreteness of the transmission eigenvalues. Let us define the following sesquilinear forms
Then the interior transmission problem in the variational form now consists of finding E ∈ H 2 0 (curl, D) such that Using the Riesz representation theorem we define two bounded linear operators  1 a.e. in D. Then the set of transmission eigenvalues is at most discrete. Moreover, the only accumulation point for the set of transmission eigenvalues is +∞.
We will give two lemmas before proving the main theorem and assume that 0 < n * < n < n * < 1.
Lemma 3.2. The operator A k is coercive.
Proof. Taking specifically F = E ∈ H 2 0 (curl, D), we have Setting γ = 1 1−n * and using the equality and arbitrary µ > 0, then (9) becomes , where γ < µ < γ + 1. For such an µ, we conclude that there exists a constant C > 0 such that , for all E ∈ H 2 0 (curl, D), which proves that A k is coercive. Lemma 3.3. The operator B is compact on H 2 0 (curl, D). Proof. Let E n ∈ H 2 0 (curl, D) be a bounded sequence. We can extract a subsequence, that we abusively denote by E n , that converges weakly to some E 0 ∈ H 2 0 (curl, D). For E ∈ H 2 0 (curl, D) and by the identity div(n E) = ndiv E + ∇n · E, we have div(n E) 2 under the hypotheses (H 1 ) on n. From (2) we can see that div(nE 1 ) = 0 and divE 2 = 0 in D. Due to the identity E = E 1 − E 2 and div(nE 1 ) = ndivE 1 + ∇n · E 1 in D. Then, we have According to (10) and (11), we obtain div(n E) 2 L 2 (D) < +∞ under the assumption that ∇n(x)/n(x) L ∞ (D) 1. Then, it is easy to check that H 2 0 (curl, D) is a subspace of H(curl, divn, D) (see Appendix B in [15]), under the boundary condition ν × E = 0 and the embedding of H(curl, divn, D) into L 2 (D) is compact, (see Proposition B.2. in [15]).
From the definition of B and the Cauchy-Schwarz inequality, we have . One deduces that B E n converges strongly to B E 0 , we have the corresponding conclusions. Now, we complete the proof of Theorem 3.1.
Proof of Theorem 3.1. To prove the discreteness of transmission eigenvalues we use the analytic Fredholm theory ( [14]). By Lemma 3.2 and 3.3, we know that the operator A k − k 4 B is Fredholm with index zero. The transmission eigenvalues are the values of k > 0 for which I − k 4 A −1 k B has a nontrivial kernel. To apply the analytic Fredholm theorem, it remains to show that I − k 4 A −1 k B or A k − k 4 B is injective for at least one k. To this end, we recall the Poincaré inequality for all E satisfies ν × E = 0 on ∂D, where constant C 0 is independent of E (see [26]). According to (11) and (12), we have .
is injective for such k and the analytical Fredholm theory implies that the set of transmission eigenvalues is discrete and from the analyticity with +∞ and the only possible accumulation point.

3.2.
Existence of the transmission eigenvalues. In this subsection, we want to prove the existence of the electromagnetic interior transmission eigenvalues. If we consider the generalized eigenvalue problem which is known to have an infinite sequence of eigenvalues λ j (k), j ∈ N, then the transmission eigenvalues are the solutions λ j (k) = k 4 of (13), j ∈ N, under the assumption n − 1 < 0. We prove the existence of infinitely many transmission eigenvalues using [9, Theorem 2.3]. We recall this key result in the following lemma. to the set of self-adjoint positive definite bounded linear operators on the Hilbert space U and assume that B is a self-adjoint non-negative compact linear operator on U . We assume that there exists two positive constants τ 0 and τ 1 such that 1. A τ0 − τ 0 B is positive on U . 2. A τ1 − τ 1 B is non-positive on a m dimensional subspace of U . Then each of the equations λ j (τ ) = τ for j = 1, ..., m, has at least one solution in [τ 0 , τ 1 ] where λ j (τ ) = τ is the j th eigenvalue (counting multiplicity) of A τ with respect to B, i.e., ker(A τ − λ j (τ )B) = 0.
The existence of the interior eigenvalues is given in the following theorem.
Proof. In the proof of Theorem 3.1 we have shown that for k 2 0 < 1 C0 , where C 0 is given in (12) D). Hence, the first assumption in Lemma 3.4 is satisfied. Now we try to find k 1 such that A k1 − k 4 1 B is non-positive in a subspace of H 2 0 (curl, D). Let B j r = B(x j , r) := {x ∈ R 3 : |x − x j | < r}, j = 1, ..., M (r) and r > 0. Define M (r) as the number of disjoint balls B j r , i.e., B j r ∩ B i r = 0, with r small enough such that B j r ⊂ D. We denote by k 1 the first transmission eigenvalue corresponding to the interior transmission problem for B j r for all j = 1, ..., M (r) with index of refraction n * which is known to exist [14]. Let E j ∈ H 2 0 (curl, B j r ), j = 1, ..., M (r) be the corresponding eigenvector which satisfies for all F ∈ H 2 0 (curl, B j r ). We denote by E * j ∈ H 2 0 (curl, D) the extension of E j by zero to the whole of D and we define a M (r)-dimensional subspace of H 2 0 (curl, D) By Green's first vector theorem and ν × E j = 0, we have Substituting (16) into (15), together with (14), we have Thus, the second assumption in Lemma 3.4 is satisfied. we conclude that there exists M (r) transmission eigenvalues in 1 C0 , k 1 . Letting r → 0, we have that M (r) → ∞ and thus we can now deduce that there exists an infinite set of transmission eigenvalues.
4. Monotonicity of the transmission eigenvalues. For this section we turn our attention to proving that the first transmission eigenvalue can be used to determine information about the material parameters n and η. To this end, we will show that the first transmission eigenvalue is a monotonic function with respect to the functions n and η. From the monotonicity we will obtain a uniqueness result for a homogeneous refractive index and homogeneous conductive boundary parameter. Recall that the transmission eigenvalues satisfy λ j (k; n, η) − k 4 (n, η) = 0, f or 0 < n < 1. (17) and the first transmission eigenvalue is the smallest root of (17) for λ 1 (k; n, η). Notice that λ 1 (k; n, η) satisfies for E = 0 λ 1 (k; n, η) = min It is clear that λ 1 (k; n, η) is a continuous function of k ∈ (0, ∞). Notice that the minimizers of (18) are the eigenfunctions corresponding to λ 1 (k; n, η). We will denote the first transmission eigenvalue by k 1 (n, η).
By the proof of the previous result we have the following uniqueness result for a homogeneous media and homogeneous boundary parameter η from the strict monotonicity of the first transmission eigenvalue. Corollary 1. 1. If it is known that 0 < n < 1 is a constant refractive index with η known and fixed, then n is uniquely determined by the first transmission eigenvalue. 2. If 0 < n < 1 is known and fixed with η a constant, then the first transmission eigenvalue uniquely determines η.