THE PERTURBATION OF TRANSMISSION EIGENVALUES FOR INHOMOGENEOUS MEDIA IN THE PRESENCE OF SMALL PENETRABLE INCLUSIONS

This paper concerns the transmission eigenvalue problem for an inhomogeneous media of compact support containing small penetrable homogeneous inclusions. Assuming that the inhomogeneous background media is known and smooth, we investigate how these small volume inclusions affect the real transmission eigenvalues. Note that for practical applications the real transmission eigenvalues are important since they can be measured from the scattering data. In particular, in addition to proving the convergence rate for the eigenvalues corresponding to the perturbed media as inclusions’ volume goes to zero, we also provide the explicit first correction term in the asymptotic expansion for simple eigenvalues. The correction terms involves the eigenvalues and eigenvectors of the unperturbed known background as well as information about the location, size and refractive index of small inhomogeneities. Thus, our asymptotic formula has the potential to be used to recover information about small inclusions from a knowledge of real transmission eigenvalues.

1. Introduction.The transmission eigenvalue problem, which is a non-selfadjoint and non-linear eigenvalue problem, appears in the study of the scattering problem for inhomogeneous media [7].The corresponding transmission eigenvalues can be determined from the scattering data [3], [13] and provide information about material properties of the scattering media [4], [5].Hence they can play an important role in a variety of inverse problems related to target identification and non-destructive testing [11].The transmission eigenvalue problem is a non-selfadjoint and nonlinear problem that is not covered by the standard theory of eigenvalue problems for elliptic operators.In the past few years transmission eigenvalues have become an important area of research in inverse scattering theory.Since the first proof of existence of transmission eigenvalues in [5] and [16], the interest in the transmission eigenvalue problem has increased, resulting in a number of important advancements in this area.For an updated survey on the topic we refer the reader to [7] (see also the special issue of Inverse Problems Vol. 29, No 10).
This paper is a continuation of the study started in [8].In particular we consider the transmission eigenvalue problem corresponding to an inhomogeneous media perturbed by small volume penetrable inclusions, and investigate the behavior of the corresponding transmission eigenvalues as the volume of perturbation goes to zero.In the current paper the transmission eigenvalue problem is written as a non-linear eigenvalue problem for a compact operator following the fourth order formulation approach developed in [6] and [5].This approach restricts us to study the perturbation of only real eigenvalues, which from practical application point of view is sufficient since the real eigenvalues are measurable from the scattering data.Our discussion allows for the unperturbed media to be inhomogeneous as opposed to [8] where the unperturbed media is assumed homogeneous.On the other hand, in [8], the convergence of the eigenvalues and eigenvectors was obtained for the entire spectrum (complex eigenvalues included).Unfortunately, there is a term missing in the expression for the first order correction in the asymptotic expansion of the eigenvalues calculated in [8].In the current paper, for a simple transmission eigenvalue, we also obtain explicit formulas for the first correction term, hence correcting the formula obtained in [8].Our calculations are based on a nonlinear version [14] of Osborn's theorem [15] which is valid only for simple eigenvalues.For the nonlinear eigenvalue problem, we are finding an eigenvalue λ of T (λ), that is, the operator depends on λ.Simplicity is required when applying Osborn's theorem because the multiplicity introduces other eigenvalues λ i , which are not equal to λ, and hence not nonlinear eigenvalues.Hence the case of multiple eigenvalues is still not completed and of interest since it is known that transmission eigenvalues, in some cases even the first eigenvalue, can have multiplicity greater than one.Of course other approaches that use some existing symmetry, such as J-selfadjointness [12], can be considered to study the perturbation of transmission eigenvalues.Note however that the transmission eigenvalue problem is inherently non-selfadjoint.
The paper is structured as follows.In the next section we formulate the transmission eigenvalue problem as a nonlinear eigenvalue problem for a compact operator and introduce the analytical framework.In Section 3 we prove the main convergence results needed to later obtain asymptotic expressions for the transmission eigenvalues.For the convenience of the reader in Section 4 we recall the nonlinear perturbation formula for the general case proven in [14] and use it to provide an explicit expression with a correction term for the perturbation of real simple transmission eigenvalues.We present the proof of a few technical lemmas in the Appendix.
2. Problem statement.Let our domain D ⊂ R d (d = 2, 3) be bounded with C 2 boundary, and let n 0 ∈ C ∞ (D) be the given smooth background coefficient which becomes constant near the boundary of D. That is, we assume that (1) n 0 − n0 has compact support in D for some constant n0 .
This background n 0 will be perturbed by small volume inhomogeneities of arbitrary smooth shape.For i = 1, . . ., m, we define the bounded shapes B i to be smooth deformations of a ball centered at the origin, so that z i + B i is a small inhomogeneity centered at z i .We also assume that is small enough so that each scaled ball is separated from the others and the boundary, in particular (z i + B i )∩(z j + B j ) = ∅ for i = j.We let W be the union of these inhomogeneities, that is and we define the perturbed squared index of refraction n : (2) where the n i ∈ R are constants.Let H 2 0 (D) denote the Sobolev space given by or, equivalently, the H 2 closure of C ∞ 0 (D) functions.Consider now the interior transmission eigenvalue problem corresponding to a scalar isotropic media with these small inhomogeneities.We wish to find nontrivial w, v Note that the boundary conditions are equivalently stated as w − v ∈ H 2 0 (D).In relation to the scattering problem, the eigenvalue parameter τ := k 2 where k is the wave number proportional to the frequency.The transmission eigenvalue problem is non-selfadjoint and in special cases is known to have complex eigenvalues.However here we limit ourselves to the study of real eigenvalues, which are the only ones that can be measured from scattering data and used to obtain information about the media.Definition 1. Values of τ ∈ R + for which the eigenvalue problem (3)-( 6) has a nontrivial solution are called transmission eigenvalues.
In [6], the transmission eigenvalue problem was shown to posses an infinite set of real transmission eigenvalues assuming that n −1 ≥ α > 0 or 0 ≤ β ≤ 1−n (x) < 1 for α and β independent of .Here we assume that n − 1 ≥ α > 0, and note that all of the following could just as likely be done in the second case.We also note that the existence of real or complex eigenvalues is proven under weaker assumptions in [17].However, our approach makes fundamental use of the framework developed in [6], hence we need the above assumptions.Under our assumption, it is proven in [4], [9], that all transmission eigenvalues τ are uniformly bounded below by τ 0 := λ 1 (D)/α + 1 where λ 1 (D) is the first Dirichlet eigenvalue of −∆ in D. Therefore we may assume that all transmission eigenvalues lie in [τ 0 , +∞), and this assumption plays an important role in our proofs later on.
The goal of this paper is to find an asymptotic expansion with respect to for transmission eigenvalues of the perturbed problem.Since everything in our problem is real valued, throughout the paper, for sake of simplicity, we assume that the involved Sobolev spaces contain real valued functions over real numbers field.
Given our assumptions, the transmission eigenvalue problem (3) is equivalent to the fourth order nonlinear eigenvalue problem [5] which in variational form is stated as follows: find u ∈ H 2 0 (D) such that (8) Following the definitions in [5], we rephrase this in terms of variationaly defined operators.Let us define the bilinear forms on for ≥ 0 and ( 10) Note that due to the restrictions on n , the bilinear forms will be bounded [6], [7].Hence by the Riesz Representation Theorem, we may define operators A τ, , B : for all u, φ ∈ H 2 0 (D).Notice that B is compact due to the compact embedding of H 2 (D) into H 1 (D), and A τ, is invertible since A τ, is coercive for all τ > 0 [6].It is convenient to use the inner product and norm on H 2 0 (D) induced by the bilinear form A τ,0 : (12) (u, φ) and u A := (u, u) A where we note that these norms depend on τ .Here = 0 corresponds to the unperturbed case, i.e. the media with refractive index n 0 .We also denote the adjoint with respect to this inner product by * , that is for an operator T : (T u, φ) A = (u, T * φ) A for u, v ∈ H 2 0 (D).We may now rewrite the variational form (8) of the transmission eigenvalue problem as T (τ ) := A −1 τ, B, so we can write (15) as (17) τ T (τ )u = u.
We have now rephrased the problem as a nonlinear eigenvalue perturbation problem.That is, would like to find τ such that there exists a nontrivial u ∈ H 2 0 (D) satisfying τ T (τ )u = u for > 0. If = 0, then n = n 0 in the definition of T 0 (τ ), and this corresponds to our background problem.Our goal is to find a correction formula for the eigenvalues of the perturbed problem in terms of the eigenvalues and eigenvectors of the background problem.To this end, we will apply Theorem of [14], an application of Osborn's theorem for approximating the eigenvalues of compact operators [15].We can apply the theorem if T (τ ) converges to T 0 (τ ) in norm, and this convergence restricted to the eigenspace of T 0 (τ ) dictates the speed of which τ approaches τ .
For sake of the reader convenience, we state here the main result of this paper which will be proven in Section 4.
Theorem 2.1.Let d = 2, 3, u be a solution to (8) for = 0 (i.e.n := n 0 ), and τ and τ be transmission eigenvalues corresponding to n 0 and n respectively, such that τ is simple.Then for u chosen such that u A = 1, where The proof of this results requires many technical and auxiliary results that will follow.The next section will contain some estimates which will give us these convergence rates and will show that the hypotheses of the theorem are satisfied.Section 4 contains the application of this eigenvalue correction theorem; and in the appendix we prove several technical lemmas used throughout the the paper.
3. Operator convergence.In this section we prove some lemmas which will allow us to successfully apply the nonlinear eigenvalue correction theorem [14].First, we prove convergence in the H 2 0 norm for A −1 τ, f assuming smoothness on f .Next, we introduce a correction term which we will use to improve the convergence rate.We then prove that we have operator norm convergence of A −1 τ, B, and, finally, we derive an asymptotic formula for A −1 τ, f in the inner product of H 2 0 (D).To simplify the analysis, we will assume a single inhomogeneity W = B centered at the origin.The arguments carry over easily to the more general case.

Strong convergence of
where C τ is independent of and Inverse Problems and Imaging Volume 9, No. 3 (2015), 725-748 From the definitions (11) of A τ, and A τ,0 , we have Since we assume that u 0 ∈ C 2,α (D) by the fact that n − n 0 is zero outside of W = B, we have a bound on the first term, and the second For the other terms, we can obtain the desired convergence rate from Sobolev embedding theorem and Thus, we have shown that where . By choosing φ = u − u 0 , the desired result follows from the coercivity of the bilinear form A τ, . Note that the coercivity constant C for A τ, on H 2 0 (D) is independent of and τ provided we assume τ > τ 0 := λ 1 (D)/(α + 1) (see (13) in [6]).
We now construct an appropriate corrector function and use it to improve the estimates for the convergence of A −1 τ, .To do this, we will rescale the problem to one in which the inhomogeneity shape is fixed, similar to what was done in [8].In what follows, we identify n 0 with its constant extension to R d , which is well defined and smooth by the assumption (1).Define for any φ ∈ H 2 0 ( D).The existence and uniqueness of ṽB follows from the Riesz representation theorem.Since the domain D is increasing with , we also define a limiting function v B on all of R d .To do this, we introduce the Sobolev space following [2]: where weights are given by ρ . This space is equipped with the usual H 2 norm with the indicated weights.Then the function As shown in [8], we may choose v B such that it satisfies the following decay conditions at infinity: Remark 3.2.By its definition (30), v B is a weak solution to the partial differential equation: Due to the decay conditions (31), the Laplacian is invertible [2] and we have the identity Recall that u 0 := A −1 τ,0 f as in the previous proof (18).We define the correction ũ(1) to be ũ(1 Notice, this function is in H 2 0 (D) since ṽB (y) ∈ H 2 0 ( D) thus it will act as a correction term in the H 2 0 (D) norm.However, since ṽB depends on , to derive an asymptotic formula we also define the correction term u (1) ∈ H 2 (D) where v B is defined by (30).Note that these two corrections are close to each other due to the lemmas in the appendix, in particular Lemma 5.1.The following lemma and its corollary describe precisely the error when these corrections are introduced.
Proof of Lemma 3.3.Starting from (19), we add and subtract A τ, 2 ũ(1) to the right hand side to have . As in the previous proof, we have the estimate First we will show that We begin by considering .
For the second term in (39), observe that by using Cauchy-Schwarz and ∆ y = 2 ∆ x for y = x/ , where the last line follows from Lemma 5.1.We now use the smoothness of n 0 to replace it with its value at the center of the ball.Since n − n 0 only has support on B, by Taylor's theorem we have for .
We will first bound the term without the gradient of n 0 .For convenience define We calculate After the change of variables y = x/ , To bound the second term, we change variables and use continuity of v to show Estimating the term in (41) with the gradient of n 0 , we obtain Inverse Problems and Imaging Volume 9, No. 3 (2015), 725-748 Therefore, we have Therefore, we compute the bound since v B ∈ L ∞ (R d ) for d = 2, 3. Now, we will estimate the third term of (39) using Cauchy-Schwarz by Proposition 5.2.For the last term we may do similarly to obtain Combining the above inequalities, we may write Using the support of n − n 0 and the definition of A τ, 2 ũ(1) in (37), we have that Recall that by assumption we have u 0 = A −1 τ,0 f ∈ C 2,α (D) for some α > 0 and constant C, so we will be able to estimate the remaining terms of (54).
Similarly, we will use that u 0 is Lipschitz on D to show . By the argument of the past lemmas and since the coercivity of constant associated with A τ, is independent of and τ ≥ τ 0 , we have

Inverse Problems and Imaging
Volume 9, No. 3 (2015), 725-748 We can now combine the previous lemma and the technical lemma in the last section to prove the convergence of BA −1 τ, f .Lemma 3.5.Let d = 2, 3.For A τ, and B defined by (11).
for some α > 0, then Proof.By the definition of the operator B, we have that for . From the previous lemma we have (59) ).We note by Proposition 5.2 the correction term 2 ũ(1) defined by (34) is o( d/2 ).

Norm convergence of A −1
τ, B. We now show operator norm convergence of A −1 τ , B for τ in an open set containing τ .Lemma 3.6.Let d = 2, 3, τ > τ 0 > 0, and let A τ, and B be defined by (11).Then, there exists an α with 0 < α < 1 and an open bounded set U containing τ such that for all τ ∈ U A hence we estimate the term in the right hand side.To this end, we define u = A −1 τ, Bf for ≥ 0 as in Lemma 3.1.We begin by using (20) from the proof of Lemma 3.1, We are immediately able to bound II, III and IV where the estimates follow from ( 22), ( 23), ( 24) and the Sobolev embedding of H 2 0 (D) into C 0,α (D).For the remaining term I, we need to use the fact that u 0 is more regular then |f | due to the presence of compact B. Because Bf solves (63) ∆∆Bf = −∆f, standard elliptic regularity [1] yields that Bf ∈ H 4 (D) for f ∈ H 2 0 (D).Furthermore, by looking at the variational form for A τ,0 , one finds that given u = Bf , A −1 τ,0 u also solves a fourth order elliptic equation: (64) . Therefore, we have the bound (65) . The Uniform Boundedness Principle applied to the set {A −1 τ,0 : τ ∈ U , U compact} yields a bound in H 4 (D) uniform in τ .That is, for some C independent of τ , (66) . Therefore, we may compute a bound for the integral I, Let 1 < p < 2 and let p * be its Sobolev conjugate so that they satisfy 1 Let q be the Hölder dual to p := p * /2 that is 1/q + 1/p = 1.It is important to notice that by our choice of p, 1/q > 0. We may now calculate Bf and (65).This along with (67) and (68) implies (69) . We conclude the proof using (60) and the uniform coercivity of A τ, for τ > τ 0 as in the end of the proof of Lemma 3.1.The bounds derived will hold for any τ ∈ U 4. A correction formula for transmission eigenvalues.Now we turn our attention to the transmission eigenvalue problem.In what follows let u be the solution to normalized with respect to the A-inner product, that is, the A-normalized eigenfunction of the transmission problem for the unperturbed media, and we define u to be Recall that u 0 = T 0 (τ )u = u/τ .The unperturbed eigenfunction u is known to be H 4 (D) H 2 0 (D) due to standard elliptic regularity results [1].We will assume throughout this section that τ is simple, i.e., the eigen-space is one-dimensional.To derive the eigenvalue correction, we will apply a nonlinear eigenvalue correction theorem which was a corollary of [15].For the convenience of the reader, we restate it (Corollary 4.1 of [14]) here: Theorem 4.1 (Nonlinear Eigenvalue Correction [14]).Let X be a Banach space and {T (λ) : X → X} a set of compact operator valued functions of λ which are analytic in a region U of the complex plane, such that T (λ) → T 0 (λ) in norm as → 0 uniformly for λ ∈ U .Let λ 0 = 0, λ 0 ∈ U be a simple nonlinear eigenvalue of T 0 , λ 0 T 0 (λ 0 )φ = φ, define DT 0 (λ 0 ) to be the derivative of T 0 with respect to λ evaluated at λ 0 , and let φ be the normalized eigenfunction and φ * its dual.Then for any small enough there exists λ , a simple nonlinear eigenvalue of T , such that if λ 2 0 DT 0 (λ 0 )φ, φ * = −1 we have the following formula where R(E) is the space spanned by φ and R(E) * is its dual or the space spanned by φ * .
We will apply this theorem using the A-inner product defined in the second section, and thus u will satisfy u A = 1.We note that this inner product depends on τ , and therefore we keep track of how all constants depend on τ .The lemmas from the previous section are sufficient to prove the following: Proof.The first is obvious from Lemma 3.1 because the bilinear form A τ,0 is bounded To derive a finer estimate on the constant, we compute for some C0 , C1 independent of τ .Therefore, combining the previous inequality with the C τ in Lemma 3.1 and computing: 1 τ ) we have the first bound in the statement of the theorem.To prove the faster convergence of the adjoint, notice that for a fixed , the adjoint of T (τ ) (of course with respect to the A-inner product) is (86) T (τ ) * = A −1 τ,0 BA −1 τ, A τ,0 since both A τ, and B are self adjoint with respect to the standard inner product [6].So we may compute using coercivity, D), Lemma 3.5 yields the desired result.The last assertion in the statement of the lemma follows directly from Lemma 3.6 and (84).
Next, we compute an asymptotic expansion for the inner product that appears in the eigenvalue correction theorem.16) for ≥ 0, and u be the solution to the variational problem (81).Then ). (90) Similarly, we can calculate the last term by the same approach The remaining terms follow a similar technique after substituting in the result of Lemma 4.3.The value for (DT 0 (τ )u, u) A in the statement of the theorem is given by (115) and Proposition 5.3.
As discussed in Section 3, the arguments used in the case of a single inhomogeneity carry over to multiple inhomogeneities.In particular, define and for i = 1, . . ., m define the functions ṽBi and v B i as before by (100) We may then define the correction terms ũ(1) and u (1) by ũ where we note that z i is the center of B i .This yields our final theorem.Theorem 4.5.Let d = 2, 3, u be a solution to (81), and T (τ ) defined by ( 16) for ≥ 0. Then for u chosen such that u A = 1, where We conclude by remarking that the formula in Theorem 4.5 can be potentially used to obtain information about the small penetrable inclusion, more specifically the location and refractive index.We note that the transmission eigenvalues for the perturbed media τ can be measured from scattering data, whereas the transmission eigenvalues τ and eigenvectors u for the unperturbed media can be computed since n 0 (x) is known.5. Appendix: Technical lemmas.In this section, we will collect the technical lemmas that are necessary for several of the results in this paper.The first lemma involves the asymptotic behavior of ṽB whereas the second computes the derivative of T 0 (τ ).

Inverse Problems and Imaging
Volume 9, No. 3 (2015), 725-748 5.1.Convergence of ṽB .Our goal is to prove 2 ṽB converges to 0 at the appropriate rate in the L 2 (D) norm.To do so, we will show that 2 (ṽ B (x/ ) − v B (x/ )) → 0 in L 2 (D).Afterwards, we can use that 2 v B itself converges to 0 yielding the desired result.Before we prove this, we restate a lemma from [8].
The proof of this lemma in [8] is for the constant n 0 case, but the identical proof holds for n 0 smooth with property (1).Since v B (x/ ) is not in H 2 0 (D), we require a separate bound on its L 2 norm.Proof.Recall v B is defined by (30).We aim to bound the L 2 norm of 2 (ṽ B (x/ ) − v B (x/ )) by the L 2 norm of its Laplacian, which we know from Lemma 5.1 is o( d/2 ).To do so, we will construct a correction term which allows us to use Poincaré's inequality.We define v 1 to be the solution to ) and in particular v B ∈ C 0 (∂D).Since the domain is C 2 , v 1 is a classical solution and so v 1 ∈ C 2 (D) C 0 (D) [10].From the maximum principle for harmonic functions,