Electromagnetic interior transmission eigenvalue problem for inhomogeneous media containing obstacles and its applications to near cloaking Jingzhi Li∗ Xiaofei Li† Hongyu Liu† Yuliang Wang‡ 7 1 January 20, 2017 0 2 n a J Abstract 9 This paper is concerned with the invisibility cloaking in electromagnetic wave scattering 1 from a new perspective. We are especially interested in achieving the invisibility cloaking by ] completely regular and isotropic mediums. Our study is based on an interior transmission h eigenvalue problem. We propose a cloaking scheme that takes a three-layer structure including p a cloaked region, a lossy layer and a cloaking shell. The target medium in the cloaked region - h can be arbitrary but regular, whereas the mediums in the lossy layer and the cloaking shell are t both regular and isotropic. We establish that there exists an infinite set of incident waves such a m that the cloaking device is nearly-invisible under the corresponding wave interrogation. The set of waves is generated from the Maxwell-Herglotz approximation of the associated interior [ transmission eigenfunctions. We provide the mathematical design of the cloaking device and 1 sharply quantify the cloaking performance. v 1 Key words. electromagnetic scattering, invisibility cloaking, interior transmission eigenvalues 0 3 5 1 Introduction 0 . 1 0 Invisibility cloaking has received significant attentions in recent years in the scientific community 7 due to its practical importance; see [1–6,13–15,18,19,21] and the references therein for the relevant 1 mathematical literature. The crucial idea is to coat a target object with a layer of artificially : v engineered material with desired properties so that the electromagnetic waves pass through the i X device without creating any shadow at the other end; namely, invisibility cloaking is achieved. r Invisibility cloaking could find striking applications in many areas of science and technology such a as radar and sonar, medical imaging, earthquake science and, energy science and engineering, to name just a few. Generally speaking, a region of space is said to be cloaked if its contents, together with the cloak, are invisible to a particular class of wave measurements. In the literature, most of the existing works are concerned with the design of certain artificial mechanisms of controlling wave propagationsothatinvisibilityisachievedindependentofthesourceofthedetectingwaves; thatis, for any generic wave fields that one uses to impinge on the cloaking device, there will be invisibility ∗Faculty of Science, South University of Science and Technology of China, 518055, Shenzhen, P. R. China (lijz@sustc.edu.cn, lixf@sustc.edu.cn). †Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR (hongyu.liuip@gmail.com, jadelightking@qq.com). 1 effect produced. In this paper, we shall develop a novel cloaking scheme where the invisibility is only achieved with respect to detecting waves from a particular set. In doing so, one can achieve the invisibility cloaking by completely regular and isotropic mediums. Next, we first present the mathematical setup and then discuss the main results of the current study. Considerthetime-harmonicelectromagnetic(EM)wavescatteringinahomogeneousspacewith the presence of an inhomogeneous scatterer. Let us first characterize the optical properties of an EM medium with the electric permittivity (cid:15), magnetic permeability µ, and electric conductivity σ. We recall that M3×3 is the space of real-valued symmetric matrices and that, for any Lipschitz sym domain Ω R3, we say that γ is a tensor in Ω satisfying the uniform ellipticity condition if ⊂ γ L∞(Ω;M3×3) and there exists 0 < c < 1 such that sym 0 ∈ c ξ 2 γ(x)ξ ξ c−1 ξ 2 for a.e. x Ω and every ξ R3. 0| | ≤ · ≤ 0 | | ∈ ∈ c shall be referred to as the ellipticity constant of the tensor γ. It is assumed that both (cid:15)(x) and 0 µ(x), x R3, belong to L∞(Ω;M3×3), and are uniform elliptic with constant c R ; whereas it sym 0 + ∈ ∈ is also assumed that σ L∞(Ω;M3×3) satisfies sym ∈ 0 σ(x)ξ ξ λ ξ 2 for a.e. x R3 and every ξ R3, 0 ≤ · ≤ | | ∈ ∈ where λ R+. Denote (Ω;(cid:15),µ,σ) the medium Ω associated with (cid:15),µ,σ, and it is said to be 0 ∈ regular if the material parameters fulfill the conditions described above. Moreover, γ(x), x Ω, ∈ is said to be isotropic if there exists α(x) L∞(Ω;R) such that γ(x) = α(x) I , where I 3×3 3×3 ∈ · signifies the 3 3 identity matrix. Suppose (Ω;(cid:15),µ,σ) is located in an isotropic and homogeneous × background/matrix medium whose material parameters are given by (cid:15)(x) = I , µ(x) = I , σ(x) = 0, for x R3 Ω¯. 3×3 3×3 ∈ \ Let ω R+ denote an EM wavenumber, corresponding to a certain EM spectrum. Consider ∈ the EM radiation in this frequency regime in the space (R3;(cid:15),µ,σ) = (Ω;(cid:15),µ,σ) (R3 Ω¯;I ,I ,0). 3×3 3×3 ∧ \ Let (Ei,Hi) be a pair of entire electric and magnetic fields, modeling the illumination source. They verify the time-harmonic Maxwell equations, curlEi iωHi = 0, curlHi+iωEi = 0 in R3. (1) − Thepresenceoftheinhomogeneousscatterer(Ω;(cid:15),µ,σ)interruptsthepropagationoftheEMwaves Ei and Hi, leading to the so-called wave scattering. We let Es and Hs denote, respectively, the scattered electric and magnetic fields. Define E := Ei+Es, H := Hi+Hs, to be the total electric and magnetic fields, respectively. Then the EM scattering is governed by the following Maxwell system  curlE(x) iωµ(x)H(x) = 0, x R3,   − ∈ curlH(x)+iω(cid:15)(x)E(x) = σ(x)E(x), x R3, (2) lim (cid:0)µ1/2(x)Hs(x) x x (cid:15)1/2(x)Es(x)(cid:1) = 0. ∈ |x|→+∞ × −| | 2 The last limit in (2) is known as the Silver-Mu¨ller radiation condition. The Maxwell system (2) is well-posed and there exists a unique pair of solutions (E,H) H(curl,R3). Here and also in what ∈ follows, for any open set Ω R3, we make use of the following Sobolev spaces: ⊂ H(curl,Ω) := (cid:8)u L2(Ω)3 curlu L2(Ω)3(cid:9), ∈ | ∈ H (curl,Ω) := u H(curl,Ω) : ν u = 0, ν curlu = 0 on ∂Ω , 0 { ∈ × × } (3) H2(curl,Ω) := u H(curl,Ω) : curlu H(curl,Ω) , { ∈ ∈ } H2(curl,Ω) := u H (curl,Ω) : curlu H(curl,Ω) , 0 0 { ∈ ∈ } endowed with the scalar product (u,v) = (u,v) +(curlu,curlv) , H(curl,Ω) L2(Ω) L2(Ω) (u,v) = (u,v) +(curlu,curlv) , H2(curl,Ω) H(curl,Ω) H(curl,Ω) and the corresponding norms and . Moreover, define (cid:107)·(cid:107)H(curl,Ω) (cid:107)·(cid:107)H2(curl,Ω) TH−1/2(∂Ω) := (cid:8)U TH−1/2(∂Ω) : DivU H−1/2(∂Ω)(cid:9), Div ∈ ∈ where Div is the surface divergence operator on ∂Ω, THs(∂Ω) is the subspace of all those V ∈ (Hs(∂Ω))3 which are orthogonal to ν, and Hs( ) is the usual L2-based Sobolev space of order · s R. ∈ For the solutions to (2), we have that as x + [11,23]: | | → ∞ eiω|x| (cid:18) 1 (cid:19) Es(x) = E (xˆ)+O , x ∞ x 2 | | | | eiω|x| (cid:18) 1 (cid:19) Hs(x) = H (xˆ)+O , x ∞ x 2 | | | | where xˆ := x/ x S2, x R3 0 . E and H are, respectively, referred to as the electric and ∞ ∞ | | ∈ ∈ \{ } magnetic far-field patterns, and they satisfy H (xˆ) = xˆ E (xˆ) and xˆ E (xˆ) = xˆ H (xˆ) = 0, xˆ S2. ∞ ∞ ∞ ∞ × · · ∀ ∈ The medium (Ω;(cid:15),µ,σ) is said to be invisible under the electromagnetic wave interrogation by (Ei,Hi) if E (xˆ,(Ei,Hi),(Ω;(cid:15),µ,σ)) 0 and H (xˆ,(Ei,Hi),(Ω;(cid:15),µ,σ)) 0, xˆ S2. (4) ∞ ∞ ≡ ≡ ∀ ∈ In the current work, we shall consider the cloaking technique in achieving the invisibility. Let D (cid:98) Ω be a bounded Lipschitz domain. Consider a cloaking device of the following form (R3;(cid:15),µ,σ) = (D;(cid:15) ,µ ,σ ) (Ω D¯;(cid:15) ,µ ,σ ) (R3 Ω¯;I ,I ,0), (5) c c c m m m 3×3 3×3 ∧ \ ∧ \ where (D;(cid:15) ,µ ,σ ) denotes the target object being cloaked, and (Ω D¯;(cid:15) ,µ ,σ ) denotes the c c c m m m \ cloaking shell medium (see Figure 1). For a practical cloaking device of the form (5), there are several crucial ingredients that one should incorporate into the design: The target object (D;(cid:15) ,µ ,σ ) can be allowed to be arbitrary (but regular). That is, the c c c • cloaking device should not be object-dependent. In what follows, this issue shall be referred to as the target independence for a cloaking device. 3 (D;(cid:15) ,µ ,σ ) c c c (Ω D¯;(cid:15) ,µ ,σ ) m m m \ Figure 1: A two-layer cloaking device. The cloaking medium (Ω D¯;(cid:15) ,µ ,σ ) should be feasible for construction and fabrication. m m m • Indeed, it would be the \most practically feasible if (Ω D¯;(cid:15) ,µ ,σ ) is uniformly elliptic m m m \ with fixed constants and isotropic as well. In what follows, this issue shall be referred to as the practical feasibility for a cloaking device. For an ideal cloaking device, one can expect the invisibility performance (4). However, in • practice, especially in order to fulfill the above two requirements, one can relax the ideal cloaking requirement (4) to be E (xˆ,(Ei,Hi),(Ω;(cid:15),µ,σ)) 1 and H (xˆ,(Ei,Hi),(Ω;(cid:15),µ,σ)) 1, ∞ ∞ | | (cid:28) | | (cid:28) xˆ S2 and (Ei,Hi) , where is a set of incident waves consisting of entire solutions ∀ ∈ ∀ ∈ H H to the Maxwell equation (1). That is, near-invisibility can be achieved for scattering measurements made with interrogating waves from the set . In what follows, this issue shall be H referred to as the relaxation and approximation for a cloaking device. In this paper, we shall develop a cloaking scheme that addresses all of the issues discussed above. Our study connects to a so-called interior transmission eigenvalue problem associated with the Maxwell system as follows,  curlEm(x)−iωµmHm(x) = 0 in Ω\D¯, curlHm(x)+iω(cid:15)mEm(x) = 0 in Ω\D¯,  curlE (x) iωH (x) = 0 in Ω, 0 0 − (6) curlH0(x)+iωE0(x) = 0 in Ω,   ν ×Em = 0 on ∂D,  ν E = ν E , ν H = ν H on ∂Ω. m 0 m 0 × × × × Concerning the problem (6), some remarks are in order. The interior transmission eigenvalue problemisassociatedwithanisotropicEMmedium(Ω D¯;(cid:15) ,µ ,0)thatcontainsaPEC(perfectly m m \ electric conducting) obstacle D. A similar interior transmission eigenvalue problem arising from the acoustic scattering governed by the Helmholtz system associated with an isotropic acoustic medium containing an impenetrable obstacle was considered in [7]. The corresponding result was applied to the invisibility cloaking study for acoustic waves in [19]. In the current article, we shall extend those studies to the much more technical and complicated Maxwell system governing the electromagnetic scattering. We shall first prove the discreteness and existence of the interior transmissioneigenvaluesofthesystem(6). Toourbestknowledge,thoseresultsarenewtoliterature on the study of interior transmission eigenvalue problems. Then we shall apply the obtained results to the invisibility cloaking study. The first result we can show concerning the invisibility cloaking is as follows. 4 (Σ;(cid:15) ,µ ,σ ) a a a (D Σ¯;(cid:15),µ,σ) l l l \ (Ω D¯;(cid:15) ,µ ,0) m m \ Figure 2: A three-layer cloaking device. Proposition 1.1. Consider the EM configuration (R3;(cid:15),µ,σ) in (5) with σ 0 and (D;(cid:15) ,µ ,σ ) m c c c ≡ replaced to be a PEC obstacle. Let ω R+ be an interior transmission eigenvalue associated with (Ω D¯;(cid:15) ,µ ,0), and (E ,H ), (E ,∈H ) be a corresponding pairs of eigenfunctions of (6). For m m m m 0 0 \ any sufficiently small ε R+, by the denseness property of Maxwell-Herglotz functions (cf. (44)), there exists (Eg,Hg) su∈ch that ω ω Eg E < ε, Hg H < ε. (7) ω 0 H(curl,Ω) ω 0 H(curl,Ω) (cid:107) − (cid:107) (cid:107) − (cid:107) Consider the scattering problem (2) by taking the incident electric and magnetic wave fields Ei = Eg, Hi = Hg, ω ω then there hold E (xˆ,(Eg,Hg),(Ω D¯;(cid:15) ,µ ,0)) Cε, xˆ S2, (8) ∞ ω ω m m | \ | ≤ ∀ ∈ and H (xˆ,(Eg,Hg),(Ω D¯;(cid:15) ,µ ,0)) Cε, xˆ S2, (9) ∞ ω ω m m | \ | ≤ ∀ ∈ where C is a positive constant independent of ε and (Eg,Hg). ω ω Proposition 1.1 is kind of a folk-telling result in the literature on the study of interior transmission eigenvalues, and it shall be needed in our cloaking study. Motivated by the two-layer cloaking device we then consider a three-layer cloaking device. Let Σ (cid:98) D be bounded Lipschitz domain such that D Σ¯ is connected and consider an EM medium \ configuration as follows (see Figure 2), (R3;(cid:15),µ,σ) = (Σ;(cid:15) ,µ ,σ ) (D Σ¯;(cid:15) ,µ ,σ ) (Ω D¯;(cid:15) ,µ ,0) (R3 Ω¯;I ,I ,0), (10) a a a l l l m m 3×3 3×3 ∧ \ ∧ \ ∧ \ where (Ω D¯;(cid:15) ,µ ,0) is isotropic and the lossy layer (D Σ¯;(cid:15) ,µ ,σ ) is chosen to be m m l l l \ \ (cid:15) = α τ−1 I , µ = α τ I , σ = α τ−1 I , l 1 3×3 l 2 3×3 l 3 3×3 · · · where τ R+ is an asymptotically small parameter, and α ,α ,α are constants in R+. The target 1 2 3 ∈ medium (Σ;(cid:15) ,µ ,σ ) in the cloaked region Σ can be arbitrary but regular. a a a For such a cloaking construction, our main theorem is as follows: Theorem 1.2. Let (R3;(cid:15),µ,σ) be described in (10). Let ω and (Eg,Hg) be the same as those ω ω in Proposition 1.1. Consider the scattering system (2) corresponding to (R3;(cid:15),µ,σ) with Ei = Eg, Hi = Hg. Then xˆ S2 we have ω ω ∀ ∈ E (xˆ,(Eg,Hg),(D Σ¯;(cid:15) ,µ ,σ ) (Ω D¯;(cid:15) ,µ ,0)) ∞ ω ω l l l m m | \ ∧ \ | (cid:16) (cid:17) C ε+τ1/2 Eg +τ1/2 Hg , (11) ω H(curl,Ω) ω H(curl,Ω) ≤ (cid:107) (cid:107) (cid:107) (cid:107) 5 H (xˆ,(Eg,Hg),(D Σ¯;(cid:15) ,µ ,σ ) (Ω D¯;(cid:15) ,µ ,0)) ∞ ω ω l l l m m | \ ∧ \ | (cid:16) (cid:17) C ε+τ1/2 Eg +τ1/2 Hg , (12) ω H(curl,Ω) ω H(curl,Ω) ≤ (cid:107) (cid:107) (cid:107) (cid:107) where C is positive constant independent of ε,τ,(Eg,Hg) and (cid:15) ,µ ,σ . ω ω a a a ByTheorem1.2, thecloakinglayer(D Σ¯;(cid:15) ,µ ,σ ) (Ω D¯;(cid:15) ,µ ,0)makesanarbitraryobject l l l m m located in the cloaked region Σ nearly inv\isible to the∧wav\e interrogation by (Eg,Hg). ω ω Therestofthepaperisorganizedasfollows. Insection2, weintroducetheinteriortransmission eigenvalue problem for inhomogeneous media containing obstacles, and derive its variational form. In section 3, we investigate the spectral property of the interior transmission eigenvalue problem. Weprovethediscretenessandtheexistenceoftheinteriortransmissioneigenvalues. Insection4,we introduce the Maxwell-Herglotz approximation. In section 5, we consider the isotropic invisibility cloaking, and establish the near-invisibility results. This paper is ended with a short discussion. 2 Interior transmission eigenvalue problem for inhomogeneous media containing obstacles 2.1 Physical background of the interior transmission eigenvalue problem Let us consider the EM configuration (5). We assume that σ = 0 and first consider the case m that (D;(cid:15) ,µ ,σ ) is replaced by a PEC obstacle. Then the scattering problem is described by the c c c following Maxwell system for (E,H) H(curl,R3 D¯) ∈ \  curlE(x)−iωµmH(x) = 0 in Ω\D¯, curlH(x)+iω(cid:15)mE(x) = 0 in Ω\D¯, curlE(x) iωH(x) = 0 in R3 Ω¯, − \ (13) curlH(x)+iωE(x) = 0 in R3 Ω¯,  \  ν E = 0 on ∂D,  × E Ei,H Hi satisfy the Silver-Mu¨ller radiation condition, − − where ν is the unit outward normal vector to boundary ∂D. By straightforward manipulations, we can also write the Maxwell system (13) in terms of E H2(curl,R3 D¯) as follows, ∈ \  curlcurlE(x) ω2n(x)E(x) = 0 in Ω D¯,  − \ curlcurlE(x) ω2E(x) = 0 in R3 Ω¯, − \ (14) ν E(x) = 0 on ∂D,  × E Ei satisfies the Silver-Mu¨ller radiation condition, − where n(x) := µ (x)(cid:15) (x) > 0. Let Es = E Ei and E∞ signify the corresponding scattered m m − wave field and the far-field pattern, respectively. If perfect invisibility is achieved for the scattering system (14), namely, E (xˆ,(Ei,Hi),(Ω;(cid:15),µ,σ)) 0, then one has that ∞ ≡ Es(x) = 0 for x R3 Ω¯. (15) ∈ \ By using the standard transmission conditions across ∂Ω for the solution E and H, one has ν E = ν E and ν H = ν H on ∂Ω, (16) − + − + × | × | × | × | 6 where ν E (x) := lim ν E(x hν(x)), x ∂Ω, ± × | h→+0 × ± ∈ ν H (x) := lim ν H(x hν(x)), x ∂Ω. ± × | h→+0 × ± ∈ The second condition in (16) can also be written as ν µ−1curlE = ν curlE on ∂Ω. m − + × | × | We assume that the the magnetic permeability µ = I in Ω D¯. The above condition becomes m 3×3 \ ν curlE = ν curlE on ∂Ω. − + × | × | Applying(15)to(16)andbysettingE (x) := E(x)forx Ω D¯ andE (x) := Ei(x)forx Ω, m 0 ∈ \ ∈ one can readily show that if perfect invisibility is achieved, then the following interior transmission eigenvalue problem  curlcurlEm−ω2n(x)Em = 0 in Ω\D¯, curlcurlE ω2E = 0 in Ω, 0 0 − (17) ν ×Em = 0 on ∂D,  ν E = ν E , ν curlE = ν curlE on ∂Ω m 0 m 0 × × × × has nontrivial solutions, where n(x) = (cid:15) . Note that the interior transmission eigenvalue problem m (17) is non-self adjoint. The equivalent interior transmission eigenvalue system of (17) for the pair (E,H) is (6). 2.2 Variational formulation of the interior transmission eigenvalue problem In this subsection we are going to derive the variational form of the interior transmission problem (17). Let E˜ =: E E in Ω D¯. By the first and second identities in (17) we have that E˜ satisfies m 0 − \ curlcurlE˜ ω2nE˜ = ω2(n 1)E in Ω D¯. (18) 0 − − \ We also get the boundary conditions ν E˜ = 0, ν curlE˜ = 0 on ∂Ω, × × and ν E˜ = ν E on ∂D. 0 × − × Then the interior transmission problem (17) can be reformulated in terms of E˜ and E as follows: 0  curlcurlE˜ −ω2nE˜ = ω2(n−1)E0 in Ω\D¯, curlcurlE ω2E = 0 in Ω, 0 0 − ν ×E˜ = −ν ×E0 on ∂D, ν E˜ = 0, ν curlE˜ = 0 on ∂Ω, × × With continuity of the data ν E and ν curlE across ∂D, we have from (18) that 0 0 × × (cid:16) 1 (cid:17)(cid:12) ν (n 1)−1(curl curl ω2n)E˜ (cid:12) = ν E on ∂D, (19) × ω2 − − (cid:12)+ × 0|− 7 and (cid:16) 1 (cid:17)(cid:12) ν curl (n 1)−1(curl curl ω2n)E˜ (cid:12) = ν curlE on ∂D. (20) × ω2 − − (cid:12)+ × 0|− Multiplying both sides of (18) by (n 1)−1 and applying operator (curl curl ω2) on both sides, we get a fourth order equation for E˜ in−Ω D¯: − \ (curl curl ω2)(n 1)−1(curl curl ω2n)E˜ = 0 in Ω D¯. − − − \ Note that E˜ is only defined in Ω D¯. We define \ (cid:40) E˜ in Ω D¯, E = \ E in D. 0 − Then E W, where W is defined by ∈ (cid:110) (cid:111) W := E H (curl,Ω) : curlcurlE L2(Ω D¯) , 0 Ω\D¯ ∈ | ∈ \ which is equipped with the norm E 2 = E 2 + curlcurlE 2 . (21) (cid:107) (cid:107)W (cid:107) (cid:107)H(curl,Ω) (cid:107) (cid:107)L2(Ω\D¯) Now we are able to formulate a fourth order system of E W as follows, ∈  (curl curl ω2)(n 1)−1(curl curl ω2n)E = 0 in Ω D¯,  − − − \ curlcurlE ω2E = 0 in D,  −  ν ×E(cid:18)|+1 = ν ×E|− (cid:19)(cid:12) on ∂D, ν (n 1)−1(curl curl ω2n)E (cid:12) = ν E on ∂D, (22)  × ω2 − − (cid:12)+ − × |−  (cid:18) 1 (cid:19)(cid:12) ν ×curl ω2(n−1)−1(curl curl−ω2n)E (cid:12)(cid:12)+ = −ν ×curlE|− on ∂D,   ν E = 0, ν curlE = 0 on ∂Ω. × × Take a test vector function Φ W. Multiplying the first equation in (22) by Φ¯ we have ∈ (cid:90) 0 = (curl curl ω2)(n 1)−1(curl curl ω2n)E Φ¯ dx. (23) Ω\D¯ − − − · Denote Ψ = (n 1)−1(curl curl ω2n)E in Ω D¯. (24) − − \ With the aid of the vector identity curlcurlE = ∆E+ divE, (25) − ∇ by Green’s second vector theorem, together with the boundary conditions in (22), equation (23) becomes (cid:90) (cid:90) (cid:0) (cid:1) 0 = Ψ (curl curl ω2)Φdx ν Ψ curlΦ curlΨ ν Φ ds. (26) Ω\D¯ · − − ∂D × · − · × By (24), the fourth and fifth transmission conditions in (22) can be written in terms of ν Ψ = ω2ν E , ν curlΨ = ω2ν curlE on ∂D. + − + − × | − × | × | − × | 8 Use the above boundary conditions, (26) becomes (cid:90) (cid:90) 0 = (n 1)−1(curlcurlE ω2E) (curlcurlΦ ω2Φ)dx ω2E curlcurlΦdx Ω\D¯ − − · − − Ω\D¯ · (cid:90) (cid:90) + ω4E Φ¯ dx+ω2 (cid:0)ν E curlΦ+ν curlE Φ¯(cid:1)ds. − − Ω\D¯ · ∂D × | · × | · By Green’s first vector theorem, and with the aid of the vector identity (25), we obtain (cid:90) 0 = (n 1)−1(curlcurlE ω2E) (curlcurlΦ ω2Φ)dx Ω\D¯ − − · − (cid:90) (cid:90) ω2 curlE curlΦdx+ω4 E Φ¯ dx. − · · Ω Ω Therefore, the variational formulation of the interior transmission problem (22) becomes: find E W such that ∈ (cid:90) (n 1)−1(curlcurlE ω2E) (curlcurlΦ ω2Φ)dx Ω\D¯ − − · − (cid:90) (cid:90) ω2 curlE curlΦdx+ω4 E Φ¯ dx = 0, (27) − · · Ω Ω forallΦ W. Bytakingappropriatetestfunctionitiseasytoseethatasolutionofthevariational ∈ problem(27)definesaweeksolutionto(22)andthereforetotheinteriortransmissionproblem(17). 3 Spectral property of the interior transmission eigenvalue problem In this section, we investigate the spectral property of the interior transmission eigenvalue problem (17). We prove the discreteness and the existence of the interior transmission eigenvalues by considering n 1 < 0 in Ω D¯ and n 1 > 0 in Ω D¯, respectively. − \ − \ 3.1 Discreteness of the spectrum when n 1 < 0 − Theorem 3.1. Assume that n(x) 1 < 0 in Ω D¯. And moreover, n(x) = (cid:15) (x) satisfies m − \ n(x)/n(x) 1. Then the set of transmission eigenvalues is discrete. L∞(Ω\D¯) (cid:107)∇ (cid:107) (cid:28) Proof. Suppose n(x) 1 < 0 in Ω D¯. We define two sesquilinear forms on W W: − \ × (cid:90) (E,Φ) = (n 1)−1(curlcurlE ω2E) (curlcurlΦ ω2Φ)dx ω A − Ω\D¯ − − · − (cid:90) (cid:90) +ω2 curlE curlΦdx+ω4 E Φ¯ dx · · Ω Ω and (cid:90) (E,Φ) = 2 E Φ¯ dx, B · Ω whereW isdefinedby(21). Thenthevariationalform(27)oftheinteriortransmissionisequivalent to Find E W such that ∈ 9 (E,Φ) ω4 (E,Φ) = 0 for all Φ W. ω A − B ∈ By the Riesz representation theorem there exist two bounded linear operators A : W W and ω → B : W W such that → (A E,Φ) := (E,Φ) and (BE,Φ) := (E,Φ). ω W ω W A B By Lemma 3.2 and 3.3 in the sequel we have A is coercive and B is compact. Hence the ω operator A ω4B is Fredholm with index zero. The transmission eigenvalues are the values of ω − ω > 0 for which I ω4A−1B has a nontrivial kernel. To apply the analytic Fredholm theorem, it ω − remains to show that I ω4A−1B or A ω4B is injective for at least one ω. ω ω − − For all E W we have that ∈ (E,E) ω4 (E,E) ω A − B (cid:90) = (1 n)−1 curlcurlE ω2E 2dx+ω2 curlE 2 ω4 E 2 . (28) Ω\D¯ − | − | (cid:107) (cid:107)L2(Ω)− (cid:107) (cid:107)L2(Ω) The Poincaré inequality gives us that: (cid:16) (cid:17) E 2 C curlE 2 + divE 2 , (29) (cid:107) (cid:107)L2(Ω) ≤ (cid:107) (cid:107)L2(Ω) (cid:107) (cid:107)L2(Ω) for all E satisfies ν E = 0 on ∂Ω, where constant C is independent of E (see [22]). × We observe from (17) that divE = 0 in D, then, for all E W we have from (28) that ∈ (E,E) ω4 (E,E) ω2(1 ω2C) curlE 2 ω4C divE 2 . (30) Aω − B ≥ − (cid:107) (cid:107)L2(Ω)− (cid:107) (cid:107)L2(Ω\D¯) Since E = E E and divE = 0 in Ω D¯, we have divE = divE . From (17) we can see that m 0 0 m div(nE ) = 0 −in Ω D¯. Due to the identi\ty div(nE ) = n(divE )+ n E in Ω D¯, we have m m m m \ ∇ · \ divE = divE n(x)/n(x) E . L2(Ω\D¯) m L2(Ω\D¯) L∞(Ω\D¯) m L2(Ω\D¯) (cid:107) (cid:107) (cid:107) (cid:107) ≤ (cid:107)∇ (cid:107) (cid:107) (cid:107) If n(x)/n(x) 1, then it follows from (30) that (E,E) ω2 (E,E) > 0 for all ω > 0 L2(Ω\D¯) ω (cid:107)∇ (cid:112)(cid:107) (cid:28) A − B such that ω < 1/C. Hence ω2 is injective for such ω and the analytical Fredholm theorem ω A − B implies that the set of transmission eigenvalues is discrete. Lemma 3.2. The operator A is coercive. ω Proof. Taking specific Φ = E W, we have ∈ (cid:90) (A E,E) = (1 n)−1 curlcurlE ω2E 2dx+ω2 curlE 2 +ω4 E 2 ω W Ω\D¯ − | − | (cid:107) (cid:107)L2(Ω) (cid:107) (cid:107)L2(Ω) (cid:90) = (1 n)−1(cid:0) curlcurlE 2 2Re ω2E curlcurlE +ω4 E 2(cid:1)dx Ω\D¯ − | | − { · } | | +ω2 curlE 2 +ω4 E 2 . (31) (cid:107) (cid:107)L2(Ω) (cid:107) (cid:107)L2(Ω) Denote n = inf n(x), and n+ = sup n(x). We assume 0 < n < n(x) < n+ < 1 for − x∈Ω\D¯ x∈Ω\D¯ − x Ω D¯. Set γ = 1 . Using the equality ∈ \ 1−n− (cid:16) γ (cid:17)2 (cid:18) γ2(cid:19) γX2 2γXY +(1+γ)Y2 = δ Y X + γ X2+(1+γ δ)Y2, (32) − − δ − δ − 10

Electromagnetic interior transmission eigenvalue problem for inhomogeneous media containing obstacles and its applications to near cloaking

Similar Electromagnetic interior transmission eigenvalue problem for inhomogeneous media containing obstacles and its applications to near cloaking