Uniqueness in Inverse Electromagnetic Conductive Scattering by Penetrable and Inhomogeneous Obstacles with a Lipschitz Boundary

Abstract

This paper is concerned with the problem of scattering of time-harmonic electromagnetic waves by a penetrable, inhomogeneous, Lipschitz obstacle covered with a thin layer of high conductivity. The well posedness of the direct problem is established by the variational method. The inverse problem is also considered in this paper. Under certain assumptions, a uniqueness result is obtained for determining the shape and location of the obstacle and the corresponding surface parameter 𝜆(𝑥) from the knowledge of the near field data, assuming that the incident fields are electric dipoles located on a large sphere with polarization 𝑝∈ℝ³. Our results extend those in the paper by F. Hettlich (1996) to the case of inhomogeneous Lipschitz obstacles.

1. Introduction

In this paper we are interested in determining the shape and location of a penetrable, inhomogeneous, isotropic, Lipschitz obstacle surrounded by a piecewise homogeneous, isotropic medium. The obstacle is covered with a thin layer of high conductivity. Such penetrable obstacles lead to conductive boundary conditions; for the precise mathematical description, the reader is referred to [1–3]. In this paper, it is shown that the shape and location of the obstacle and the corresponding surface parameter are uniquely determined from a knowledge of the near field data of the scattered electromagnetic wave at a fixed frequency. To this end, we need a well posedness result for the direct problem.

The well posedness of the Helmholtz equation for a penetrable, inhomogeneous, anisotropic medium has been studied recently in [4]. In [5], the authors provided a proof for the well posedness of the scattering problem for a dielectric that is partially coated by a highly conductive layer in the TM case in 2007.

In the case of exterior Maxwell problem for the partially coated Lipschitz domains, the authors in [6] have established the well posedness of a unique solution by variational methods in 2004. For the homogeneous isotropic medium problem, by means of an integral equation method, Angell and Kirsch proved the existence and uniqueness of the classical solution for Maxwell's equations with conductive boundary conditions assuming 𝜆∈𝐶^0,𝛼(𝜕𝐷) in [2]. Variational methods for the homogeneous isotropic medium problem were proposed in [1], under the assumption that the bounded domain 𝐷⊆ℝ³ with boundary 𝜕𝐷 in the class 𝐶², and some additional conditions on 𝑘,𝑛_𝐷,𝜇,𝜆. It is also shown that the obstacle is uniquely determined by the far field patterns of all incident waves with a fixed wave number. For the inhomogeneous anisotropic media, the well posedness of the direct problem was proved in [7].

The uniqueness result for the inverse medium scattering problem was first provided by Isakov (see [8, 9]), in which it is shown that the shape of a penetrable, inhomogeneous, isotropic medium is uniquely determined by its far field pattern of all incident plane waves. The idea is to construct singular solutions of the boundary value problem with respect to two different scattering obstacles with identical far field patterns. Our uniqueness proof is based on this idea. The idea of Isakov was modified by Kirsh and Kress [10] using potential theory for the impenetrable obstacle case with Neumann boundary conditions. By the same technique, the authors in [11] proved the case of a penetrable obstacle with constant index of refraction. The use of potential theory will require strong smoothness assumptions on the scattering object. Then D. Mitrea and M. Mitrea [12] improved the previous results to the case of Lipschitz domains. In [13], they extended Isakov's approach to the case of a penetrable obstacle for Hemholtz equations.The uniqueness theorem of Helmholtz equations for partially coated buried obstacle problem was shown in [14, 15], assuming that the scattering fields were known with point sources as incident fields.

Recently, uniqueness for the inverse scattering problem in a layered medium has attracted intensive studies. For the sound-soft or sound-hard obstacle case, based on Schiffer's idea, [16] proved a uniqueness result. But their method can not be extended to other boundary conditions. In recent years, by employing the generalized mixed reciprocity relation, it was proved in [17, 18] that both the obstacle and its physical property can be uniquely determined for different boundary conditions. For the inverse acoustic scattering by an impenetrable obstacle in a two-layered medium case, it is shown in [19] that interface is uniquely determined from the far field pattern. Unfortunately, this method can not be extended to the electromagnetic case, but using ideas in [20], a different method was used in [21] to establish such a uniqueness result for the electromagnetic case.

There are also some uniqueness results for partial differential equations with constant coefficients by integral equation methods. (see [22, 23]). However, integral equation methods are not well tailored for partial differential equations having inhomogeneous coefficients of the highest derivatives. Consequently, in [24], the author brought together the variational approach and the idea from [8, 9] to provide a uniqueness proof of Helmholtz equations with inhomogeneous coefficients for a penetrable, anisotropic obstacle. Their method depends on a regularity theorem for the direct problem and the well posedness of the interior transmission problem related to the direct problem. This idea has been extended to the case of electromagnetic scattering problem for anisotropic media in [25].

The outline of this paper is as follows. In Section 2, besides the formulation of the direct scattering problem in a penetrable, inhomogeneous, Lipschitz domain, we also provide a proof of the well posedness for the direct problem by using a variational method. The uniqueness result for the inverse problem will be shown in Section 3.

2. The Direct Problem

Let 𝐷⊂ℝ³ be a bounded penetrable, inhomogeneous, isotropic domain with a Lipschitz boundary 𝜕𝐷 denoted by Γ and covered with a thin layer of high conductivity. Assume that the domain 𝐷 is imbedded in a homogeneous background medium. Define 𝑘²_𝐷=𝑘²𝑛_𝐷(𝑥) and 𝑘²_𝑏=𝑘²𝑛_𝑏 with 𝑘>0 being the wave number, where 𝑛_𝐷(𝑥) and 𝑛_𝑏 are the refractive index of the domain 𝐷 and the background medium, respectively. Assume that 𝑛_𝐷∈𝐶(𝐷) with Re[𝑛_𝐷(𝑥)]>0,Im[𝑛_𝐷(𝑥)]>0 for all 𝑥∈𝐷 and 𝑛_𝑏 is a complex constant with Im(𝑛_𝑏)≥0. Assume further that 𝜆∈𝐿^∞(Γ) with Re[𝜆(𝑥)]≥0 is a complex-valued function describing the surface impedance of the coating. The incident field is considered to be an electric dipole located at 𝑥₀ on a large sphere 𝑆_𝑅0={𝑥∈ℝ³∶|𝑥|=𝑅₀} with polarization 𝑝∈ℝ³ given by 𝐸_𝑒𝑥,𝑥₀,𝑝,𝑘_𝑏=𝑖𝑘_𝑏curl_𝑥curl_𝑥𝑝𝑒^{𝑖𝑘𝑏|𝑥−𝑥0|}4𝜋𝑥−𝑥₀. (2.1) Denote by 𝐺(𝑥,𝑥₀) the free space Green tensor of the background medium and define 𝐸^𝑖(𝑥)=𝐸^𝑖(𝑥,𝑥₀,𝑝)=𝑝𝐺(𝑥,𝑥₀) which satisfies curlcurl𝐸^𝑖(𝑥)−𝑘²_𝑏𝐸^𝑖(𝑥)=𝑝𝛿𝑥−𝑥₀inℝ³, (2.2) where 𝛿 is the Dirac delta function. Note that 𝐸^𝑖(𝑥) can be written as 𝐸^𝑖(𝑥)=𝐸_𝑒𝑥,𝑥₀,𝑝,𝑘_𝑏+𝐸^𝑠_𝑏(𝑥), (2.3) where 𝐸^𝑠_𝑏(𝑥) is the scattered electric field due to the background medium and the electric dipole 𝐸_𝑒(𝑥,𝑥₀,𝑝,𝑘_𝑏).

In order to formulate precisely the scattering problem, recall the following Sobolev spaces: 𝐻(curl,𝐷)=𝑢∈𝐿²(𝐷)³,curl𝑢∈𝐿²(𝐷)³,𝐻(div,𝐷)=𝑢∈𝐿²(𝐷)³,div𝑢∈𝐿²(𝐷)³,𝐿²_𝑡(𝜕𝐷)=𝑢∈𝐿²(𝜕𝐷)³,𝐻,𝜈⋅𝑢=0on𝜕𝐷_−1/2div(𝜕𝐷)=𝑢∈𝐻^−1/2(𝜕𝐷)³,𝜈⋅𝑢=0,div_𝜕𝐷𝑢∈𝐻^−1/2(,𝐻𝜕𝐷)_−1/2curl(𝜕𝐷)=𝑢∈𝐻^−1/2(𝜕𝐷)³,𝜈⋅𝑢=0,curl_𝜕𝐷𝑢∈𝐻^−1/2(,𝐻𝜕𝐷)₀(curl,𝐷)=𝑢∈𝐿²(𝐷)³,curl𝑢∈𝐿²(𝐷)³,𝜈×𝑢=0on𝜕𝐷, (2.4) where 𝜈 denotes the exterior unit normal to 𝜕𝐷. If 𝐷 is unbounded, we denote by 𝐻_loc(curl,𝐷) the space of functions 𝑢∈𝐻(curl,𝐾) for any compact set 𝐾  ⊂⊂  𝐷. Introduce the space 𝑋=𝑢∈𝐻_loccurl,ℝ³,𝑢_𝑇Γ∈𝐿²_𝑡(Γ), (2.5) where 𝑢_𝑇=𝜈×𝑢×𝜈. Then the scattering problem can be formulated as follows. Given 𝐸^𝑖, find the field 𝑉 and the scattered field 𝐸^𝑠 such that curlcurl𝑉−𝑘²_𝐷𝑉=0,in𝐷,curlcurl𝐸^𝑠−𝑘²_𝑏𝐸^𝑠=0,in𝐷_𝑒,𝜈×𝐸^𝑠−𝜈×𝑉=−𝜈×𝐸^𝑖,onΓ,𝜈×curl𝐸^𝑠−𝜈×curl𝑉=𝑖𝑘𝜆𝐸^𝑠_𝑇+𝑖𝑘𝜆𝐸^𝑖_𝑇−𝜈×curl𝐸^𝑖,onΓ, (2.6)(2.7)(2.8)(2.9) and the scattered field 𝐸^𝑠=𝐸−𝐸^𝑖 is required to satisfy the Silver-Müller radiation condition lim_𝑟→∞𝑟curl𝐸^𝑠×𝑥−𝑖𝑘𝐸^𝑠=0 (2.10) uniformly in 𝑥=𝑥/|𝑥|, where 𝑟=|𝑥|.

We first have the following uniqueness result for the above scattering problem.

Theorem 2.1. The scattering problem (2.6)–(2.9) has at most one solution.

Proof. To prove the theorem, it is enough to consider the case 𝐸^𝑖=0 whence 𝐸=𝐸^𝑠. Taking the dot product of (2.6) with 𝑉 over 𝐷 and of (2.7) with 𝐸 over 𝐷_𝑅=𝐷_𝑒∩𝐵_𝑅 with 𝐷_𝑒=ℝ³⧵𝐷, respectively, and integrating by parts, we obtain by using the conductive conditions (2.8) and (2.9) that _𝐷|curl𝑉|²−𝑘²_𝐷|𝑉|²𝑑𝑥+_𝐷𝑅|curl𝐸|²−𝑘²_𝑏|𝐸|²𝑑𝑥−𝑖𝑘_Γ𝜆𝐸_𝑇²𝑑𝑠+𝑖𝑘_𝑆𝑅𝜈×𝐻⋅𝐸_𝑇𝑑𝑠=0, (2.11) where 𝐻=(1/𝑖𝑘)curl𝐸 is the corresponding scattered magnetic field. Taking the complex conjugate of both sides of (2.11) and using the fact that Im(𝑛_𝐷),Im(𝑛_𝑏), and Re(𝜆) are nonnegative gives Re_𝑆𝑅𝜈×𝐸⋅𝐻𝑑𝑠=−𝑘⁻¹_𝐷Im𝑛_𝐷|𝑉|²𝑑𝑥−𝑘⁻¹_𝐷𝑅Im𝑛_𝑏|𝐸|²−𝑑𝑥_ΓRe(𝜆)𝐸_𝑇²𝑑𝑠≤0. (2.12) An application of the Rellich lemma yields that 𝐸=0 in ℝ³⧵𝐵_𝑅 (see [26, Theorem 6.10]). This, together with the unique continuation principle, implies that 𝐸=0 in ℝ³⧵𝐷. From the trace theorem, it follows that 𝜈×𝐸=0 on Γ. Thus, taking the imaginary part of (2.11) and using the assumption that Im[𝑛_𝐷(𝑥)]>0 for all 𝑥∈𝐷, we have that 𝑉=0 in 𝐷.
Introduce the electric-to-magnetic Calderon operator 𝐺_𝑒 (see [27]), which maps the electric field boundary data 𝜑 on the surface of a large ball 𝐵_𝑅={𝑥∈ℝ³∶|𝑥|<𝑅} to the magnetic boundary data 𝑥×𝐻 on 𝑆_𝑅=𝜕𝐵_𝑅, where (𝐸,𝐻) satisfies curl𝐸−𝑖𝑘𝐻=0,inℝ³⧵𝐵_𝑅,curl𝐻+𝑖𝑘𝐸=0,inℝ³⧵𝐵_𝑅,𝑥×𝐸=𝜑,on𝑆_𝑅,lim_𝑟→∞(𝐻×𝑥−𝑟𝐸)=0. (2.13)(2.14)(2.15)(2.16) Then the scattering problem (2.6)–(2.10) can be reformulated in the following mixed conductive boundary value problem (MOCKUP) over a bounded domain: curlcurl𝑉−𝑘²_𝐷𝑉=0,in𝐷,curlcurl𝐸^𝑠−𝑘²_𝑏𝐸^𝑠=0,in𝐷_𝑅,𝜈×𝐸^𝑠−𝜈×𝑉=−𝜈×𝐸^𝑖,onΓ,𝜈×curl𝐸^𝑠−𝜈×curl𝑉=𝑖𝑘𝜆𝐸^𝑠_𝑇+𝑖𝑘𝜆𝐸^𝑖_𝑇−𝜈×curl𝐸^𝑖1,onΓ,𝜈×𝑖𝑘curl𝐸^𝑠=𝐺_𝑒𝜈×𝐸^𝑠,on𝑆_𝑅, (2.17)(2.18)(2.19)(2.20)(2.21) where 𝐷_𝑅=𝐷_𝑒∩𝐵_𝑅.
In the following, we introduce some properties of the Calderon operator that will be frequently used in the rest of this section. The basis functions for tangential fields on a sphere 𝑆_𝑅 are the vector spherical harmonics of order 𝑛 given by 𝑈^𝑚_𝑛=1𝑛()∇𝑛+1^𝜏1𝑌^𝑚_𝑛,𝑉^𝑚_𝑛=𝑥×𝑈^𝑚_𝑛 (2.22) for 𝑛=1,2,… and 𝑚=−𝑛,…,𝑛. Here, as usual, ∇_𝜏1 denotes the surface gradient on the surface of the unit sphere 𝑆₁.
For 𝜑∈𝐻^−1/2_div(𝑆_𝑅) given by 𝜑=∑^∞_𝑛=1∑^𝑛_𝑚=−𝑛𝑎_𝑛,𝑚𝑈^𝑚_𝑛+𝑏_𝑛,𝑚𝑉^𝑚_𝑛, the operator 𝐺𝑒 can be defined by 𝐺𝑒𝜑=^∞𝑛_{𝑛=1𝑚=−𝑛}𝑏−𝑖𝑘𝑅^𝑛,𝑚𝛿_𝑛𝑈^𝑚_𝑛+𝑎_𝑛,𝑚𝛿_𝑛𝑉𝑖𝑘𝑅^𝑚𝑛, (2.23) where 𝛿_𝑛ℎ=𝑘𝑅^𝑛(1)′(𝑘𝑅)ℎ_𝑛⁽¹⁾(𝑘𝑅)+1, (2.24) and ℎ_𝑛⁽¹⁾(𝑘𝑅) is the spherical Hankel function.
If 𝑘=𝑖 in (2.23), we will obtain another operator 𝐺_𝑒. Properties of 𝐺𝑒 and 𝐺_𝑒 are collected in the following lemma (for a proof see [27]).

Lemma 2.2. The operator 𝐺_𝑒 is negative definite in the sense that 𝐺_𝑒𝜑,𝜑×𝑥_𝑆𝑅<0 (2.25) for any 𝜑∈𝐻^−1/2_div(𝑆_𝑅) with 𝜑≠0. Furthermore, 𝐺_𝑒𝜑,𝜑×𝑥_𝑆𝑅≥𝐶𝜑²_{𝐻−1/2div𝑆𝑅},∀𝜑∈𝐻^−1/2_div𝑆_𝑅,𝐺𝑒+𝑖𝑘𝐺_𝑒∶𝐻^−1/2_divdiv,𝑆_𝑅→𝐻^−1/2_div𝑆_𝑅 (2.26) is compact, where 𝐻^−1/2_divdiv,𝑆_𝑅=𝜑=^∞𝑛_{𝑛=1𝑚=−𝑛}𝑏_𝑛,𝑚𝑉^𝑚_𝑛∣^∞𝑛_{𝑛=1𝑚=−𝑛}1()𝑏1+𝑛𝑛+1^𝑛,𝑚2<∞. (2.27)

In the remainder of this paper we will refer to (2.17)–(2.21) as (CBP). Here we will adapt the variational approach used in [6, 27] to prove the existence of a unique solution to our (CBP). Define 𝑋=𝑢∈𝐻_loccurl,𝐵_𝑅,𝑢_𝑇Γ∈𝐿²_𝑡(Γ), (2.28) where 𝐷⊂𝐵_𝑅. Then multiplying (2.17) and (2.18) by test function 𝜙∈𝐻_loc(curl,𝐵_𝑅), using formally integration by parts and using the conductive boundary conditions on Γ, we can derive the following equivalent variational formulation for (CBP). Find 𝑤∈𝑋 such that _𝐷curl𝑤⋅curl𝜙−𝑘²_𝐷𝑤⋅𝜙𝑑𝑥+_𝐷𝑅curl𝑤⋅curl𝜙−𝑘²_𝑏𝑤⋅𝜙𝑑𝑥−𝑖𝑘_Γ𝜆𝑤_𝑇⋅𝜙_𝑇𝑑𝑠+𝑖𝑘_𝑆𝑅𝐺_𝑒(𝜈×𝑤)⋅𝜙_𝑇𝑑𝑠=𝑖𝑘_𝑆𝑅𝐺_𝑒𝜈×𝐸^𝑖−𝜈×𝐻^𝑖⋅𝜙_𝑇𝑑𝑠, (2.29) where 𝐻^𝑖=(1/𝑖𝑘_𝑏)curl𝐸^𝑖 is the incident magnetic field and 𝑤=𝑉,𝑥∈𝐷,𝐸=𝐸^𝑖+𝐸^𝑠,𝑥∈ℝ³⧵𝐷. (2.30)

We rewrite (2.29) as the problem of finding 𝑤∈𝑋 such that 𝐴𝑤,𝜙=𝐵𝜙, (2.31) where the sesquilinear form 𝐴∶𝑋×𝑋→𝐶 is defined by 𝐴𝑤,𝜙=curl𝑤,curl𝜙_𝐷−𝑘²_𝐷𝑤,𝜙_𝐷+curl𝑤,curl𝜙_𝐷𝑅−𝑘²_𝑏𝑤,𝜙_𝐷𝑅−𝑖𝑘𝜆𝑤_𝑇,𝜙_𝑇Γ+𝑖𝑘𝐺_𝑒(𝜈×𝑤),𝜙_𝑇𝑆𝑅,𝐵𝜙=𝑖𝑘𝐺_𝑒𝜈×𝐸^𝑖−𝜈×𝐻^𝑖,𝜙_𝑇𝑆𝑅. (2.32) Here (⋅,⋅)_𝐷 denotes the (𝐿²(𝐷))³ scalar product, and ⟨⋅,⋅⟩_𝜕𝐷 denotes the (𝐿²(𝜕𝐷))³ scalar product. We will use a Helmholtz decomposition to factor out the nullspace of the curl operator and then to prove the existence of a unique solution to (CBP).

Define 𝑆=𝑝∈𝐻¹(𝐷)∩𝐻¹𝐷_𝑅∣𝑝=onΓ, (2.33) then we seek 𝑝∈𝑆 such that 𝐴∇𝑝,∇𝜉=𝐵(∇𝜉),∀𝜉∈𝑆. (2.34) The variational problem (2.34) can be rewritten as 𝐴₁𝑝,𝜉+𝐴₂𝑝,𝜉=𝐵(∇𝜉),∀𝜉∈𝑆, (2.35) where we define 𝐴₁𝑝,𝜉=−𝑘²_𝐷∇𝑝,∇𝜉_𝐷−𝑘²_𝑏∇𝑝,∇𝜉_𝐷𝑅+𝑘²𝐺_𝑒𝜈×∇𝑝,∇_𝜏𝜉_𝑆𝑅,𝐴₂𝑝,𝜉=𝑖𝑘𝐺𝑒+𝑖𝑘𝐺_𝑒𝜈×∇𝑝,∇_𝜏𝜉_𝑆𝑅. (2.36) Here we have used ∇_𝜏𝜉 to write the tangential component of the gradient of 𝜉 in terms of the tangential gradient on the sphere 𝑆_𝑅. By Lemma 2.2, it follows that 𝐺_𝑒 is negative definite, then we obtain that 𝐴₁(𝑝,𝜉) is a coercive sesquilinear form on 𝑆×𝑆. Further by Lax-Milgram theorem, it is easy to see that 𝐴₁(𝑝,𝜉) gives rise to a bijective operator. Since 𝜈×∇𝑝∈𝐻^−1/2_div(div,𝑆_𝑅), still by Lemma 2.2, we know that 𝐴₂(𝑝,𝜉) gives rise to a compact operator. In order to apply the Fredholm alternative to the variational problem (2.34), we need to prove the following uniqueness lemma.

Lemma 2.3. The variational problem (2.34) has at most one solution.

Proof. It suffices to consider the following equation: −𝑘²_𝐷∇𝑝,∇𝜉_𝐷−𝑘²_𝑏∇𝑝,∇𝜉_𝐷𝑅+𝑖𝑘𝐺_𝑒𝜈×∇𝑝,∇_𝜏𝜉_𝑆𝑅=0,∀𝜉∈𝑆. (2.37) Choosing 𝜉=𝑝, it is easy to see that 𝑖𝑘𝐺_𝑒𝜈×∇𝑝,∇_𝜏𝑝_𝑆𝑅=𝑘²_𝐷∇𝑝,∇𝑝_𝐷+𝑘²_𝑏∇𝑝,∇𝑝_𝐷𝑅. (2.38) By the definition of the operator 𝐺_𝑒, if 𝐸^𝑠∈𝐻_loc(curl,ℝ³⧵𝐵) is the weak solution of the problem curlcurl𝐸^𝑠−𝑘²𝐸^𝑠=0,inℝ³⧵𝐵,𝜈×𝐸^𝑠=𝜈×∇𝑝,on𝑆_𝑅,lim_𝑟→∞curl𝐸^𝑠×𝑥−𝑖𝑘𝑟𝐸^𝑠=0, (2.39) then we have 𝐺_𝑒𝜈×∇𝑝=𝜈×𝐻^𝑠,on𝑆_𝑅, (2.40) where 𝐻^𝑠=1𝑖𝑘curl𝐸^𝑠. (2.41) Furthermore, we can compute that _𝑆𝑅𝐻^𝑠⋅𝜈×𝐸^𝑠𝑑𝑠=−𝜈×𝐻^𝑠,∇_𝜏𝑝_𝑆𝑅=−𝐺𝑒𝜈×∇𝑝,∇_𝜏𝑝_𝑆𝑅1=−𝑘𝑖𝑘^2𝐷∇𝑝,∇𝑝_𝐷−1𝑘𝑖𝑘^2𝑏∇𝑝,∇𝑝_𝐷𝑅, (2.42) which together with the fact Im𝑛_𝐷≥0,Im𝑛_𝑏≥0 implies Re_𝑆𝑅𝜈×𝐸^𝑠⋅𝐻^𝑠𝑑𝑠=Re_𝑆𝑅𝐻^𝑠⋅𝜈×𝐸^𝑠1𝑑𝑠=−𝑘Im𝑘²_𝐷∇𝑝,∇𝑝_𝐷−1𝑘Im𝑘²_𝑏∇𝑝,∇𝑝_𝐷𝑅≤0. (2.43) Therefore the Rellich lemma ensures us that 𝐸^𝑠=0 in ℝ³⧵𝐵. From (2.39), we see that ∇_𝜏𝑝=0 on 𝑆_𝑅 and then (𝑘²_𝐷∇𝑝,∇𝑝)_𝐷+(𝑘²_𝑏∇𝑝,∇𝑝)_𝐷𝑅=0 which, together with the fact that 𝑝|_Γ=0, implies 𝑝=0. This completes the proof of Lemma 2.3.

Lemma 2.3 together with the Fredholm alternative implies that there exits a unique solution 𝑝₀∈𝑆 of the variational problem (2.34).

Lemma 2.4. The space 𝑋₀={𝑢∈𝑋∣𝐴(𝑢,∇𝜉)=0,∀𝜉∈𝑆} (2.44) is compactly imbedded in (𝐿²(𝐵))³, where 𝐵 is a ball with 𝐷⊂𝐵⊂𝐵_𝑅.

Proof. Consider a bounded set of functions {𝑢_𝑗}^∞_𝑗=1⊂𝑋₀. Each function 𝑢_𝑗∈𝑋₀ can be extended to all of ℝ³ by solving the exterior Maxwell equation ∇×∇×𝑣_𝑗−𝑘²_𝑏𝑣_𝑗=0,inℝ³⧵𝐵,𝜈×𝑣_𝑗=𝜈×𝑢_𝑗,on𝑆_𝑅,lim_|𝑥|→∞|𝑥|∇×𝑣_𝑗×𝜈−𝑖𝑘𝑣_𝑗=0. (2.45) Define 𝑢^𝑒_𝑗=𝑢_𝑗𝑣,if𝑥∈𝐵,_𝑗,if𝑥∈ℝ³⧵𝐵. (2.46) Since the tangential components of 𝑢^𝑒_𝑗 are continuous across 𝑆_𝑅, it follows that 𝑢^𝑒_𝑗∈𝐻_loc(curl,ℝ³). By using the properties of the Calderon operator 𝐺_𝑒 and the conditions in 𝑋₀, we see that the following equations hold true div𝑘²_𝐷𝑢𝑘=0,in𝐷,div²_𝑏𝑢=0,in𝐷_𝑅,𝑘²_𝑏𝜈⋅𝑢=−𝑖𝑘∇_𝜏⋅𝐺_𝑒(𝜈×𝑢),on𝑆_𝑅. (2.47) Then, by the definition of 𝐺_𝑒 that 𝐺𝑒(𝜈×𝑢_𝑗)=(1/𝑖𝑘)𝜈×curl𝑣_𝑗 and the relationship ∇_𝜏⋅(𝜈×𝑣_𝑗) = −𝜈⋅curl𝑣_𝑗 on 𝑆_𝑅, we immediately have 𝑘²_𝑏𝜈⋅𝑢_𝑗=−𝑖𝑘∇_𝜏⋅𝐺_𝑒𝜈×𝑢_𝑗=−∇_𝜏⋅𝜈×curl𝑣_𝑗=𝜈⋅curlcurl𝑣_𝑗=𝑘²_𝑏𝜈⋅𝑣_𝑗,on𝑆_𝑅. (2.48) Thus, 𝑢^𝑒_𝑗 has a well-defined divergence and ∇⋅(𝑘²𝑢^𝑒_𝑗)=0 in ℝ³⧵Γ, where 𝑘²=𝑘²_𝐷𝑘,𝑥∈𝐷,_2𝑏,𝑥∈ℝ³⧵𝐷. (2.49) Now we choose a cut-off function 𝜒∈𝐶^∞₀(ℝ³) such that 𝜒=1 in 𝐵 and 𝜒 is supported in a ball 𝐵_𝑅1⊃𝐵. Then one can use the general compactness theorem (Theorem 4.7 in [27]) to the sequence {𝜒𝑢^𝑒_𝑗} and extract a subsequence converging strongly in (𝐿²(𝐵))³. This proves the lemma.

From the above definitions of 𝑆 and 𝑋₀, we have the following Helmholtz decomposition lemma.

Lemma 2.5. The spaces ∇𝑆 and 𝑋₀ are closed subspaces of 𝑋. The space 𝑋 is the direct sum of the spaces ∇𝑆 and 𝑋₀, that is, 𝑋=𝑋₀⊕∇𝑆. (2.50)

The proof of this Helmholtz decomposition Lemma is entirely classical (see [27, 28]).

We now look for a solution of the variational problem (2.31) in the form 𝑤=𝑤₀+∇𝑝₀, where 𝑤₀∈𝑋₀ and 𝑝₀∈𝑆 is the unique solution of (2.34). We observe that 𝐴(𝑤₀,∇𝜉)=0 for all 𝜉∈𝑆 by the definition of 𝑋₀. Hence the problem of determining 𝑤∈𝑋 is equivalent to the problem of determining 𝑤₀∈𝑋₀ such that 𝐴𝑤₀,𝜙₀=𝐵𝜙₀−𝐴∇𝑝₀,𝜙₀,∀𝜙₀∈𝑋₀. (2.51) From Chapter 10.3.2 in [27] we know that for 𝜑∈𝐻^−1/2_div(𝑆_𝑅)𝐺_𝑒𝜑=𝐺_1𝑒𝜑+𝐺_2𝑒𝜑, (2.52) where the operator 𝐺_1𝑒 is a compact operator from 𝑋₀ into 𝐻^−1/2_div(𝑆_𝑅) and the operator 𝐺_2𝑒 satisfies 𝑖𝑘⟨𝐺_2𝑒(𝜈×𝜑),𝜑_𝑇⟩_𝑆𝑅≥0. We now split the sesquilinear form 𝐴(⋅,⋅) into 𝐴=𝑎+𝑏 with 𝑎𝑤₀,𝜙₀=curl𝑤₀,curl𝜙_0𝐷+𝑤₀,𝜙_0𝐷+curl𝑤₀,curl𝜙_0𝐷𝑅+𝑤₀,𝜙_0𝐷𝑅−𝑖𝑘⟨𝜆𝑤₀)_𝑇,𝜙_0𝑇Γ+𝑖𝑘𝐺_2𝑒𝜈×𝑤₀,𝜙_{0𝑇𝑆𝑅},𝑏𝑤₀,𝜙₀=−𝑘²_𝐷+1𝑤₀,𝜙_0𝐷−𝑘²_𝑏+1𝑤₀,𝜙_0𝐷𝑅+𝑖𝑘⟨𝐺_1𝑒(𝜈×𝑤₀),𝜙₀)_𝑇𝑆𝑅. (2.53)(2.54) The sesquilinear form 𝑎(⋅,⋅) is obviously bounded and a direct computation verifies that 𝑎𝑤₀,𝑤₀⩾𝛼𝑤₀²_𝑋,∀𝑤₀∈𝑋₀, (2.55) with some constant 𝛼>0.

Hence by Lax-Milgram theorem, 𝑎(⋅,⋅) gives rise to a bijective operator and by the compact embedding of 𝑋₀ in (𝐿²(𝐵))³ and the fact that 𝐺_1𝑒 is a compact operator from 𝑋₀ into 𝐻^−1/2_div(𝑆_𝑅), the second term 𝑏(⋅,⋅) gives rise to a compact operator. Then a standard argument implies that the Fredholm alternative can be applied. Finally, the uniqueness theorem yields the existence result. We summarize the above analysis in the following theorem.

Theorem 2.6. For any incident field 𝐸^𝑖, there exists a unique solution 𝑤∈𝑋 of (CBP) which depends continuously on the incident field 𝐸^𝑖.

3. Uniqueness for the Inverse Problem

In this section we will show that the scattering obstacle 𝐷 and the corresponding parameter 𝜆 are uniquely determined from the knowledge of the scattered fields 𝐸^𝑠_1,𝑥0(𝑥,𝑥₀)∣_{𝑥∈𝜕Ω} for all 𝑥₀∈𝑆_𝑅0, where 𝑆_𝑅0 is the surface of a large ball 𝐵_𝑅0 with 𝐷⊂Ω⊂𝐵_𝑅0. By some properties of the scattered fields, we can derive a relationship between them, then constructing special singular solutions which satisfy the relationship. Finally, we can obtain the uniqueness result by using the singularities of the singular solutions that we constructed.

Lemma 3.1. Assume that 𝑘²_𝑏 is not an eigenvalue of Maxwell equation for the domain Ω₀. Then we have(i) the restriction to 𝜕Ω₀ of {𝜈×𝑝(𝑥₀)𝐺(𝑥,𝑥₀)∶𝑥₀∈𝑆_𝑅0} is complete in 𝐻^−1/2_div(𝜕Ω₀);(ii) the restriction to 𝜕Ω₀ of {𝜈×curl_𝑥𝑝(𝑥₀)𝐺(𝑥,𝑥₀)−𝑖𝑘𝜆𝑝(𝑥₀)𝐺(𝑥,𝑥₀)_𝑇∶𝑥₀∈𝑆_𝑅0} is complete in 𝐿²_𝑡(𝜕Ω₀).

Proof. For simplicity, we only prove statement (ii). Case (i) can be proved similarly.
Let 𝑔∈𝐿²_𝑡(𝜕Ω₀) be such that _𝜕Ω0𝜈(𝑥)×curl_𝑥𝑝𝑥₀𝐺𝑥,𝑥₀−𝑖𝑘𝜆𝑝𝑥₀𝐺𝑥,𝑥_0𝑇⋅𝑔(𝑥)𝑑𝑠(𝑥)=0. (3.1) Then it follows that _𝜕Ω0curl_𝑥0𝑔(𝑥)×𝜈(𝑥)𝐺𝑥,𝑥₀−𝑖𝑘𝜆𝑔(𝑥)𝐺𝑥,𝑥₀𝑑𝑠(𝑥)⋅𝑝𝑥₀=0. (3.2) Define 𝐹^∗₁𝑥₀=_𝜕Ω0curl_𝑥0𝑔(𝑥)×𝜈(𝑥)𝐺𝑥,𝑥₀(𝑥),𝐹𝑑𝑠_∗2𝑥₀=−_𝜕Ω0𝑖𝑘𝜆𝑔(𝑥)𝐺𝑥,𝑥₀(𝑥),𝐹𝑑𝑠_∗𝑥₀=𝐹^∗₁𝑥₀+𝐹^∗₂𝑥₀. (3.3)(3.4)(3.5) By (3.2), it is easy to see that for arbitrary polarization 𝑝(𝑥₀) in the tangential plane to 𝑆_𝑅0 at 𝑥₀, we have 𝐹^∗𝑥₀⋅𝑝𝑥₀=0,∀𝑥₀∈𝑆_𝑅0. (3.6) From the definition of (3.3), we immediately have curl_𝑥0curl_𝑥0𝐹^∗₁𝑥₀−𝑘²_𝑏𝐹^∗₁𝑥₀=0,∀𝑥₀∈ℝ³⧵𝜕Ω₀. (3.7) Due to the symmetry of the background Green function, 𝐹^∗₂(𝑥₀) as a function of 𝑥₀ solves curl_𝑥0curl_𝑥0𝐹^∗₂(𝑥₀)−𝑘²_𝑏𝐹^∗₂(𝑥₀)=0,∀𝑥₀∈ℝ³⧵𝜕Ω₀. Hence, 𝐹^∗(𝑥₀) satisfies the Maxwell's equation in ℝ³⧵𝜕Ω₀. By (3.6) and the fact that 𝑝(𝑥₀) is an arbitrary polarization in the tangential plane to 𝑆_𝑅0 at 𝑥₀, we immediately have that 𝜈×𝐹^∗(𝑥₀)|_𝑆𝑅0=0.
The uniqueness of the exterior problem curl_𝑥0curl_𝑥0𝐹^∗𝑥₀−𝑘²_𝑏𝐹^∗𝑥₀=0,inℝ³⧵𝐵_𝑅0,𝜈×𝐹^∗𝑥₀=0,on𝑆_𝑅0,lim_{|𝑥0|→∞}𝑥₀curl_𝑥0𝐹^∗𝑥₀×𝜈𝑥₀−𝑖𝑘𝐹^∗𝑥₀=0 (3.8) implies that 𝐹^∗(𝑥₀)=0 in ℝ³⧵𝐵_𝑅0. Thus, the unique continuation principle ensures us that 𝐹^∗(𝑥₀)=0 in Ω^𝑒₀=ℝ³⧵Ω₀. By trace theorem, it follows that 𝜈×𝐹^∗(𝑥₀)=0 and 𝜈×curl_𝑥0𝐹^∗(𝑥₀)=0 on 𝜕Ω₀. By the definition of 𝐹^∗(𝑥₀) and the jump relations of the vector potential across 𝜕Ω₀, it can be checked that 𝐹^∗(𝑥₀) satisfies the following equations: curl_𝑥0curl_𝑥0𝐹^∗𝑥₀−𝑘²_𝑏𝐹^∗𝑥₀=0,inΩ₀,𝜈×curl_𝑥0𝐹^∗𝑥₀−𝑖𝑘𝜆𝜈×𝐹^∗𝑥₀=0,on𝜕Ω₀. (3.9) Therefore, the uniqueness theorem of the interior problem for Maxwell's equations implies that 𝐹^∗(𝑥₀)=0 in Ω₀. Finally, from the jump relations of the vector potential across 𝜕Ω₀, we have 0=[𝜈×𝐹^∗(𝑥₀)×𝜈_𝜕Ω0=−𝑔𝑥₀, (3.10) which completes the proof.

We now consider two obstacles 𝐷₁ and 𝐷₂ with the refractive index 𝑛_𝐷𝑗 and the surface impedance 𝜆_𝑗,𝑗=1,2. Let 𝑈 denote the unbounded part of ℝ³⧵(𝐷₁∪𝐷₂) and 𝐷₀=ℝ³⧵𝑈 its open complement. From the proof of Theorem 2.6, it follows that the total field 𝑤_𝑗,𝑗=1,2 satisfies curl𝑤_𝑗,curl𝜙_𝐵𝑅−𝑘²_𝑗∗𝑤_𝑗,𝜙_𝐵𝑅−𝑖𝑘𝜆_𝑗𝑤_𝑗𝑇,𝜙_𝑇Γ𝑗=−𝜈×curl𝑤_𝑗,𝜙_𝑆𝑅, (3.11) for any large ball 𝐵_𝑅 with 𝐷⊂𝐵_𝑅 and all test function 𝜙∈𝑋, where 𝑘²_𝑗∗=𝑘²_𝐷𝑗,𝑥∈𝐷_𝑗;𝑘²_𝑏,𝑥∈ℝ³⧵𝐷_𝑗. (3.12) It is convenient to introduce the following space: 𝑋_𝑗=𝐸∈𝐻curl,𝐵_𝑅,𝐸_𝑇Γ𝑗∈𝐿²_𝑡Γ_𝑗,𝑗=1,2, (3.13) where 𝐷_𝑗⊂𝐵_𝑅. The relationship derived in the following lemma plays a central role in the proof of the main result in this section.

Lemma 3.2. Assume that 𝑘²_𝑏 is not an eigenvalue of Maxwell equation in Ω. Let 𝐵_𝑅⊂ℝ³ be a ball with 𝐷₀⊂Ω⊂𝐵_𝑅. Let 𝐸^𝑠_1,𝑥0(𝑥,𝑥₀) and 𝐸^𝑠_2,𝑥0(𝑥,𝑥₀) be the scattered fields with respect to 𝐷₁ and 𝐷₂, respectively, produced by the same incident field 𝑝𝐺(⋅,𝑥₀). Assume that 𝜈×𝐸^𝑠_1,𝑥0|_𝜕Ω=𝜈×𝐸^𝑠_2,𝑥0|_𝜕Ω for all 𝑥₀∈𝑆_𝑅0 with the radius 𝑅₀>𝑅 for a fixed wave number 𝑘. Then we have 𝑘²𝑛_𝐷1−𝑛_𝑏𝑉₁,𝑉_2𝐷1+𝑖𝑘𝜆₁𝑉_1𝑇,𝑉_2𝑇Γ1=𝑘²𝑛_𝐷2−𝑛_𝑏𝑉₁,𝑉_2𝐷2+𝑖𝑘𝜆₂𝑉_1𝑇,𝑉_2𝑇Γ2. (3.14) Here 𝑉_𝑗∈𝑋₁∩𝑋₂ satisfies the following variational problem: 𝑎_𝑗curl𝑉_𝑗,curl𝑉_𝐵𝑅−𝜅_𝑗𝑉_𝑗,𝑉_𝐵𝑅−𝑖𝑘𝜆_𝑗𝑉_𝑗𝑇,𝑉_𝑇Γ𝑗=0, (3.15) for all 𝑉∈𝐻₀(curl,𝐵_𝑅), where the coefficients 𝑎_𝑗,𝜅_𝑗∈𝐿^∞(𝐵_𝑅) and 𝜆_𝑗∈𝐿^∞(Γ_𝑗) satisfy that 𝑎_𝑗|_𝐷0=1,𝜅_𝑗|_𝐷0=(𝑘²_𝑗∗)|_𝐷0 and 𝜆_𝑗|_{Γ𝑗∩𝐷0}=𝜆_𝑗|_Γ𝑗,𝑗=1,2.

Proof. (i) We first prove that for any fixed 𝑧∈𝑈, the scattered fields 𝐸^𝑠_1,𝑧(𝑥,𝑧)=𝐸^𝑠_2,𝑧(𝑥,𝑧), where (𝑉_𝑗,𝑧,𝐸^𝑠_𝑗,𝑧),𝑗=1,2 is the solution of the following problem: curlcurl𝑉−𝑘²_𝐷𝑗𝑉=0,in𝐷_𝑗,curlcurl𝐸^𝑠−𝑘²_𝑏𝐸^𝑠=0,inℝ³⧵𝐷_𝑗,𝜈×𝑉−𝐸^𝑠=𝑓,onΓ_𝑗,𝜈×curl𝑉−𝐸^𝑠=−𝑖𝑘𝜆_𝑗𝐸^𝑠_𝑇+𝑟+ℎ,onΓ_𝑗, (3.16) with the incident field 𝐸^𝑖=𝑝𝐺(⋅,𝑧) and 𝑓=𝜈×𝐸^𝑖, ℎ=𝜈×curl𝐸^𝑖, 𝑟=−𝑖𝑘𝜆_𝑗𝐸^𝑖_𝑇. By Lemma 3.1 and the fact that 𝑘²_𝑏 is not an eigenvalue of Maxwell equation in Ω, it follows that there exists a sequence 𝑎_𝑛∈ℝ and 𝑥₀^(𝑛)∈𝑆_𝑅0 such that 𝜈×𝑝𝐺(⋅,𝑧)−_𝑛𝑎_𝑛𝜈×𝑝𝐺⋅,𝑥₀^(𝑛)_{𝐻−1/2div(𝜕Ω)}<𝜀,∀𝜀>0. (3.17) Let 𝐸(𝑥)=𝑝𝐺(⋅,𝑧)−∑_𝑛𝑎_𝑛𝑝𝐺(⋅,𝑥₀^(𝑛)), then it satisfies the Maxwell equation curlcurl𝐸−𝑘²_𝑏𝐸=0 in Ω⧵{𝑧}. Let 𝑓=(𝜈×𝐸)|_𝜕Ω, then the well posedness of the problem curlcurl𝐸−𝑘²_𝑏𝐸=0,inΩ⧵{𝑧},𝜈×𝐸=𝑓,on𝜕Ω (3.18) and (3.17) imply that ‖𝐸‖_{𝐻(curl,Ω)}≤𝐶₁‖𝜈×𝐸‖_𝐻^−1/2_div(𝜕Ω)<𝐶₁𝜀,∀𝜀>0. (3.19) This, together with the fact that div𝐸=0, implies (see [28]) ‖𝐸‖_𝐻¹_(Ω)≤𝐶₂‖𝐸‖_{𝐻(curl,Ω)}+‖𝐸‖_𝐻(div,Ω)+‖𝜈×𝐸‖_𝐻^−1/2_div(𝜕Ω)≤𝐶₂𝜀,∀𝜀>0. (3.20) Then by (3.19), (3.20), and the trace theorem, it can be proved that ‖𝜈×𝐸‖_𝐻^−1/2_div(Γ𝑗)+‖𝜈×curl𝐸‖_𝐿²_𝑡(Γ𝑗)+𝜈×curl𝐸−𝑖𝑘𝜆_𝑗𝐸_{𝑇𝐿2𝑡(Γ𝑗)}≤𝐶₃‖𝐸‖_{𝐻(curl,𝐷𝑗)}+𝐸_{𝑇𝐿2𝑡(Γ𝑗)}≤𝐶₄‖𝐸‖_{𝐻(curl,𝐷𝑗)}+‖𝐸‖_𝐻¹_(𝐷𝑗)≤𝐶₅‖𝐸‖_{𝐻(curl,Ω)}+‖𝐸‖_𝐻¹_(Ω)≤𝐶₅𝜀,𝑗=1,2,∀𝜀>0. (3.21) Denote by 𝐸^𝑠₁ and 𝐸^𝑠₂ the scattered fields with respect to 𝐷₁ and 𝐷₂ produced by the same incident field ∑_𝑛𝑎_𝑛𝑝𝐺(⋅,𝑥₀^(𝑛)). By the assumption 𝜈×𝐸^𝑠_1,𝑥0|_𝜕Ω=𝜈×𝐸^𝑠_2,𝑥0|_𝜕Ω for all 𝑥₀∈𝑆_𝑅0, it is easy to see that 𝜈×𝐸^𝑠₁|_𝜕Ω=𝜈×𝐸^𝑠₂|_𝜕Ω. Then by the uniqueness theorem of the exterior scattering problem, it follows that 𝐸^𝑠₁=𝐸^𝑠₂ in ℝ³⧵Ω, which together with the unique continuation principle ensures that 𝐸^𝑠₁=𝐸^𝑠₂ in 𝑈. Now, by (3.21) and the well posedness of the direct problem (2.18), it can be checked that for any compact set 𝐾⊂⊂𝑈, we have 𝐸^𝑠_1,𝑧(𝑥,𝑧)−𝐸^𝑠₁(𝑥)_{𝐻(curl,𝐾)}𝐸<𝜀,∀𝜀>0,_𝑠1,𝑧(𝑥,𝑧)−𝐸^𝑠₂(𝑥)_{𝐻(curl,𝐾)}<𝜀,∀𝜀>0, (3.22) for any fixed 𝑧∈𝐾.
Therefore, the fact 𝐸^𝑠₁=𝐸^𝑠₂ in 𝑈 ensures us that 𝐸^𝑠_1,𝑧(𝑥,𝑧)−𝐸^𝑠_2,𝑧(𝑥,𝑧)_{𝐻(curl,𝐾)}<𝜀,∀𝜀>0. (3.23) The arbitrarity of 𝜀 implies that 𝐸^𝑠_1,𝑧(𝑥,𝑧)=𝐸^𝑠_2,𝑧(𝑥,𝑧) for any fixed 𝑧∈𝐾.
(ii) Next we will show that the identity (3.14) holds. Set 𝑤^∗=𝑤₂−𝑤₁, then it follows from (3.11) that curl𝑤^∗,curl𝑉_𝐵𝑅−𝑘²_2∗𝑤^∗,𝑉_𝐵𝑅−𝑖𝑘𝜆₂𝑤^∗_𝑇,𝑉_𝑇Γ2=−𝑘²_1∗−𝑘²_2∗𝑤₁,𝑉_𝐵𝑅−𝑖𝑘𝜆₁𝑤_1𝑇,𝑉_𝑇Γ1+𝑖𝑘𝜆₂𝑤_1𝑇,𝑉_𝑇Γ2 (3.24) for all 𝑉∈𝑋^∗={𝑉∈𝑋₁∩𝑋₂,𝑉_𝑇|_𝑆𝑅=0}. Choose two domains Ω₁,Ω₂⊂𝐵_𝑅 with 𝐷₀⊂Ω₁⊂Ω₂ and define a smooth function 𝜓∈𝐶^∞(ℝ³) with 𝜓=1 in Ω₁ and 𝜓=0 in ℝ³⧵Ω₂. Let 𝑎₂=𝜓, 𝜅₂=𝜓𝑘²_2∗, 𝜆₂=𝜆₂, it is easy to see that 𝑎₂,𝜅₂,𝜆₂ satisfy the assumptions of the lemma. We further assume 𝑉₂ satisfies (3.15) with respect to 𝑎₂=𝜓, 𝜅₂=𝜓𝑘²_2∗, 𝜆₂=𝜆₂, so that substituting 𝑉=𝜓𝑉₂ into the left hand of (3.24) and noting that 𝑤^∗=0 in 𝑈 yield that curl𝑤^∗,curl𝜓𝑉_2𝐵𝑅−𝑘²_2∗𝑤^∗,𝜓𝑉_2𝐵𝑅−𝑖𝑘𝜆₂𝑤^∗_𝑇,𝜓𝑉_2𝑇Γ2=−𝑎₂curl𝑉₂,curl𝑤^∗_𝐵𝑅−𝜅₂𝑉₂,𝑤^∗_𝐵𝑅−𝑖𝑘𝜆₂𝑉_2𝑇,𝑤^∗_𝑇Γ2=0. (3.25) Hence substituting 𝑉=𝜓𝑉₂ into (3.24), it follows from the right hand of (3.24) that −𝑘²_1∗−𝑘²_2∗𝑤₁,𝑉_2𝐵𝑅−𝑖𝑘𝜆₁𝑤_1𝑇,𝑉_2𝑇Γ1+𝑖𝑘𝜆₂𝑤_1𝑇,𝑉_2𝑇Γ2=0. (3.26) We define 𝑓∈𝑋₁ by 𝑓,𝜙_𝑋1=−𝑘²_1∗−𝑘²_2∗𝑉₂,𝜙_𝐵𝑅−𝑖𝑘𝜆₁𝑉_2𝑇,𝜙_𝑇Γ1+𝑖𝑘𝜆₂𝑉_2𝑇,𝜙_𝑇Γ2, (3.27) for all 𝜙∈𝑋₁. By Theorem 2.6, it follows that there exists a unique solution 𝑤₀∈𝑋₁ of the problem curl𝑤₀,curl𝜙_𝐵𝑅−𝑘²_1∗𝑤₀,𝜙_𝐵𝑅−𝑖𝑘𝜆₁𝑤_0𝑇,𝜙_𝑇Γ1+𝑖𝑘𝐺𝑒𝜈×𝑤₀,𝜙_𝑇𝑆𝑅=𝑓,𝜙_𝑋1, (3.28) for all 𝜙∈𝑋₁. Choose two domains Ω_𝑒,Ω_𝑖⊆𝐵_𝑅 with Ω⊆Ω_𝑒,𝐷₀⊂Ω_𝑖⊂Ω and define smooth functions 𝜓_𝑒,𝜓_𝑖∈𝐶^∞(ℝ³) with 𝜓_𝑒=1,𝑥∈Ω,𝜓_𝑒=0,𝑥∈ℝ³⧵Ω_𝑒 and 𝜓_𝑖=1,𝑥∈Ω_𝑖,𝜓_𝑖=0,𝑥∈ℝ³⧵Ω. Take 𝑗=1,𝜙=𝜓_𝑒𝑤₀ in (3.11), it is seen that curl𝑤₁,curl𝜓_𝑒𝑤_0𝐵𝑅−𝑘²_1∗𝑤₁,𝜓_𝑒𝑤_0𝐵𝑅−𝑖𝑘𝜆₁𝑤_1𝑇,𝑤_0𝑇Γ1=0. (3.29) Equation (3.28) with 𝜙 replaced by 𝜓_𝑖𝑤₁ yields curl𝑤₀,curl𝜓_𝑖𝑤_1𝐵𝑅−𝑘²_1∗𝑤₀,𝜓_𝑖𝑤_1𝐵𝑅−𝑖𝑘𝜆₁𝑤_0𝑇,𝑤_1𝑇Γ1=𝑓,𝜓_𝑖𝑤_1𝑋1. (3.30) By (3.26) and (3.27), it can be shown that 𝑓,𝜓_𝑖𝑤_1𝑋1=−𝑘²_1∗−𝑘²_2∗𝑉₂,𝑤_1𝐵𝑅−𝑖𝑘𝜆₁𝑉_2𝑇,𝑤_1𝑇Γ1+𝑖𝑘𝜆₂𝑉_2𝑇,𝑤_1𝑇Γ2=0. (3.31) Taking the difference of (3.29) and (3.30), we have that 0=_Ω⧵Ω𝑖curl𝑤₀⋅curl𝜓_𝑖𝑤₁−𝑤₁−𝑘²_𝑏𝑤₀⋅𝜓_𝑖𝑤₁−𝑤₁−𝑑𝑥_Ω𝑒⧵Ωcurl𝜓_𝑒𝑤₀⋅curl𝑤₁−𝑘²_𝑏𝜓_𝑒𝑤₀⋅𝑤₁𝑑𝑥. (3.32) By (3.28), we can deduce that 𝑤₀ is a radiating solution of the corresponding Maxwell's equations in 𝐵_𝑅, then it can be extended to all of ℝ³ denoted by 𝑤^𝑒₀ by solving the exterior Maxwell's equation in ℝ³⧵𝐵_𝑅 with 𝜈×𝑤^𝑒₀=𝜈×𝑤₀ on 𝑆_𝑅, which also satisfies the Silver-Müler radiation condition at infinity. By applying the vector Green formula to (3.32), it can be proved that 0=_𝜕Ω𝑤₁⋅𝜈×curl𝑤^𝑒₀−𝑤^𝑒₀⋅𝜈×curl𝑤₁𝑑𝑠. (3.33) In view of the fact 𝑤₁=𝐸₁=𝐸^𝑠₁+𝐸^𝑖 and 𝐸^𝑖=𝑝𝐺(⋅,𝑥₀) in ℝ³⧵𝐷₁, we immediately have 0=_𝜕Ω𝐸^𝑠₁⋅𝜈×curl𝑤^𝑒₀−𝑤^𝑒₀⋅𝜈×curl𝐸^𝑠₁𝑑𝑠+_𝜕Ω𝜈×curl𝑤^𝑒₀⋅𝑝𝐺⋅,𝑥₀−𝑤^𝑒₀⋅𝜈×curl𝑝𝐺⋅,𝑥₀𝑑𝑠. (3.34) Application of the vector Green formula again and noting that both 𝐸^𝑠₁ and the extended function 𝑤^𝑒₀ satisfy the Silver-Müler radiation condition, it follows that _𝜕Ω𝐸^𝑠₁⋅𝜈×curl𝑤^𝑒₀−𝑤^𝑒₀⋅𝜈×curl𝐸^𝑠₁𝑑𝑠=_𝑆𝑅𝐸^𝑠₁⋅𝜈×curl𝑤^𝑒₀−𝑤^𝑒₀⋅𝜈×curl𝐸^𝑠₁𝑑𝑠=0. (3.35) Hence the Stratton-Chu formula combines with (3.34) implies that 0=𝑝𝑥₀⋅𝑤^𝑒₀𝑥₀,∀𝑥₀∈𝑆_𝑅0,∀𝑝𝑥₀∈ℝ³. (3.36) Since 𝑝(𝑥₀) is an arbitrary polarization in the tangential plane to 𝑆_𝑅0 at 𝑥₀, we obtain that 𝜈×𝑤^𝑒₀(𝑥₀)|_𝑆𝑅0=0. By the fact that 𝑤^𝑒₀ is a radiating solution of Maxwell's equation in ℝ³⧵𝐵_𝑅0, it follows that 𝑤^𝑒₀=0 in ℝ³⧵𝐵_𝑅0. Hence the unique continuation principle implies that 𝑤^𝑒₀=0 in ℝ³⧵𝑈. Therefore, 𝑤^𝑒₀ can be used as a test function for 𝑉₁, which satisfies (3.15) with 𝑎₁=𝜓_𝑖,𝜅₁=𝜓_𝑖𝑘²_1∗,𝜆₁=𝜆₁. So that from the left hand of (3.30), we deduce that curl𝑤^𝑒₀,curl𝜓_𝑖𝑉_1𝐵𝑅−𝑘²_1∗𝑤^𝑒₀,𝜓_𝑖𝑉_1𝐵𝑅−𝑖𝑘𝜆₁𝑤^𝑒_0𝑇,𝑉_1𝑇Γ1=𝑎₁curl𝑉₁,curl𝑤^𝑒_0𝐵𝑅−𝜅₁𝑉₁,𝑤^𝑒_0𝐵𝑅−𝑖𝑘𝜆₁𝑉_1𝑇,𝑤^𝑒_0𝑇Γ1=0. (3.37) Thus, it follows from the right hand of (3.30) that (𝑓,𝜓_𝑖𝑉₁)_𝑋1=0. Furthermore, from (3.27) with 𝜙 replaced by 𝜓_𝑖𝑉₁, it can be shown that −𝑘²_1∗−𝑘²_2∗𝑉₂,𝑉_1𝐵𝑅−𝑖𝑘⟨𝜆₁𝑉_1𝑇,𝑉_2𝑇Γ1+𝑖𝑘⟨𝜆₂𝑉_2𝑇,𝑉₁)_𝑇Γ2=0. (3.38) From the definitions of 𝑘²_𝑗∗, we observe that 𝑘²_1∗−𝑘²_2∗=𝑘²𝑛_𝐷1−𝑘²𝑛_𝑏,𝑥∈𝐷₁,𝑘²𝑛_𝐷2−𝑘²𝑛_𝑏,𝑥∈𝐷₂,0,𝑥∈𝐵_𝑅⧵𝐷₁∪𝐷₂, (3.39) which combines (3.38), the definition of the scalar product (⋅,⋅)_𝐷, and the fact that ((𝑘²_1∗−𝑘²_2∗)𝑉₂,𝑉₁)_𝐵𝑅=𝑘²((𝑛_𝐷1−𝑛_𝑏)𝑉₂,𝑉₁)_𝐷1−𝑘²((𝑛_𝐷2−𝑛_𝑏)𝑉₂,𝑉₁)_𝐷2 implies that (3.14) holds. This ends the proof of this lemma.

The main result of this section is contained in the following theorem.

Theorem 3.3. Let 𝐸^𝑠₁ and 𝐸^𝑠₂ be the scattered fields with respect to 𝐷₁ and 𝐷₂, respectively, and 𝜆₁,𝜆₂ the corresponding impedances. Suppose that the assumptions in Lemma 3.2 hold true and Γ_𝑗∩(𝐷_𝑖⧵𝐷_𝑗) is not empty for 𝑖,𝑗=1,2,𝑖≠𝑗. If one of the following assumptions holds, then we have 𝐷₁=𝐷₂. Consider(i)Re𝜆_𝑗≥𝛿>0;(ii)Im𝜆_𝑗≥𝛿>0orIm𝜆_𝑗≤−𝛿<0.

Proof. Let us assume that 𝐷₁ is not included in 𝐷₂. Since 𝐷^𝑒₂=ℝ³⧵𝐷₂ is connected, we can find a point 𝑧∈Γ₁⧵𝐷₂ and a sufficiently small 𝜀>0 with the following properties:(i)𝐵_2𝜀(𝑧)∩𝐷₂=∅;(ii) the points 𝑧_𝑛=𝑧+(𝜀/𝑛)𝜈(𝑧) lie in 𝐵_2𝜀(𝑧) for all 𝑛∈𝑁, where 𝜈(𝑧) is the unit normal to Γ₁ at 𝑧.Denote 𝐷=(𝐷₁⧵𝐷₂)^𝑜, the inner part of the domain 𝐷₁⧵𝐷₂. We consider the unique solution of the following problem: curlcurl𝑉−𝑘²_𝐷𝑉=0,in𝐷,curlcurl𝐸^𝑠−𝑘²_𝑏𝐸^𝑠=0,inℝ³⧵𝐸𝐷,𝜈×^𝑠−𝑉=−𝜈×𝐸_𝑚⋅,𝑧_𝑛𝐸,on𝜕𝐷,𝜈×curl^𝑠−𝑉=𝑖𝑘𝜆₁𝐸^𝑠_𝑇+𝐸_𝑚⋅,𝑧_𝑛−𝜈×curl𝐸_𝑚⋅,𝑧_𝑛,onΓ₁∩𝜕𝐷. (3.40) Here 𝐸^𝑠 satisfies the Silver-Müler radiation condition at infinity, and 𝐸_𝑚 denotes the magnetic dipole defined by 𝐸_𝑚𝑥,𝑧_𝑛=curl_𝑥𝜈(𝑧)𝑒^{𝑖𝑘𝑏|𝑥−𝑧𝑛|}𝑥−𝑧_𝑛. (3.41) Define 𝐹₁𝑥,𝑧_𝑛=𝑉(𝑥)𝐸,if𝑥∈𝐷,_𝑠(𝑥)+𝐸_𝑚𝑥,𝑧_𝑛,if𝑥∈ℝ³⧵𝐷∪𝑧_𝑛. (3.42) It can be proved that 𝐹₁(𝑥,𝑧_𝑛) is a solution of Maxwell's equations with homogeneous conductive boundary value conditions on 𝜕𝐷 in any domain Ω⊂ℝ³ with 𝐷₀⊂Ω and 𝑧_𝑛⊈Ω.
Define 𝜅₁(𝑥)=𝑘²_1∗(𝑥)if𝑥∈ℝ³⧵𝐷₁,𝑘⧵𝐷_2𝑏if𝑥∈𝐷₁⧵𝐷, (3.43) and 𝜆₁(𝑥)=𝜆₁.
In view of the above definitions of 𝜅₁(𝑥) and 𝜆₁(𝑥), it follows that 𝐹₁(𝑥,𝑧_𝑛) satisfies the variational equation (3.15) in Lemma 3.2 for the obstacle 𝐷. The well posedness of the direct problem for (CBP) and the fact that 𝑧_𝑛 is bounded away from 𝜕𝐷 imply that the solution (𝑉,𝐸^𝑠) of (3.40) is uniformly bounded in 𝑋. We now define another singular solution with respect to 𝐷₂ by 𝐹₂𝑥,𝑧_𝑛=𝑉(𝑥),if𝑥∈𝐷₂,𝐸^𝑠(𝑥)+𝐸_𝑚𝑥,𝑧_𝑛,if𝑥∈ℝ³⧵𝐷₂∪𝑧_𝑛, (3.44) where 𝐸_𝑚 is a magnetic dipole defined in (3.41), and (𝑉,𝐸^𝑠) is a solution of the problem curlcurl𝑉−𝑘²_𝐷2𝑉=0,in𝐷₂,curlcurl𝐸^𝑠−𝑘²_𝑏𝐸^𝑠=0,inℝ³⧵𝐷₂∪𝑧_𝑛,𝜈×𝐸^𝑠−𝑉=−𝜈×𝐸_𝑚⋅,𝑧_𝑛𝐸,on𝜕𝐷,𝜈×curl^𝑠−𝑉=𝑖𝑘𝜆₂𝐸^𝑠_𝑇+𝐸_𝑚⋅,𝑧_𝑛𝑇−𝜈×curl𝐸_𝑚⋅,𝑧_𝑛,onΓ₂. (3.45) Here 𝐸^𝑠 satisfies the Silver-Müler radiation condition at infinity. Noting that 𝐹₂(𝑥,𝑧_𝑛) satisfies the variational equation (3.15) in Lemma 3.2 with 𝜅₂(𝑥)=𝑘²_2∗ and 𝜆₂=𝜆₂, it follows that both 𝐹₁(𝑥,𝑧_𝑛) and 𝐹₂(𝑥,𝑧_𝑛) satisfy the relationship (3.14), then we obtain 𝑘²𝑛_𝐷−𝑛_𝑏𝐹₁,𝐹_2𝐷+𝑖𝑘𝜆₁𝐹_1𝑇,𝐹_{2𝑇𝜕𝐷}=𝑘²𝑛_𝐷2−𝑛_𝑏𝐹₁,𝐹_2𝐷2+𝑖𝑘𝜆₂𝐹_2𝑇,𝐹_1𝑇Γ2. (3.46) For case (i), by the fact that 𝑧∈Γ₁⧵𝐷₂ and the singularities of the magnetic dipole 𝐸_𝑚 defined in (3.41), it can be proved that |𝑘²((𝑛_𝐷−𝑛_𝑏)𝐹₁,𝐹₂)_𝐷+𝑖𝑘⟨𝜆₁(𝐹₁)_𝑇,(𝐹₂)_𝑇⟩_𝜕𝐷|→∞ as 𝑛→∞, this, together with the fact that the other terms in the right hand of (3.46) are bounded, leads to a contradiction. Hence we have 𝐷₁⊂𝐷₂. By choosing 𝑧∈Γ₂⧵𝐷₁ and using the similar analysis as in the proof above, one can prove that 𝐷₂⊂𝐷₁. Finally, we obtain that 𝐷₁=𝐷₂. For other cases, due to the singularities of 𝐹_𝑗(𝑗=1,2), a contradiction also arises in (3.46) as 𝑛→∞. This proves the theorem.

Theorem 3.4. Assume 𝐷₁=𝐷₂ with parameters 𝜆_𝑗∈𝐶(𝜕𝐷_𝑗) and the scattered fields 𝐸^𝑠_𝑗,𝑥0,(𝑗=1,2) satisfy 𝜈×𝐸^𝑠_1,𝑥0|_𝜕Ω=𝜈×𝐸^𝑠_2,𝑥0|_𝜕Ω for all 𝑥₀∈𝑆_𝑅0, then we have 𝜆₁=𝜆₂ on 𝜕𝐷.

Proof. From the proof of Theorem 3.3, it follows that there exists two singular solutions 𝐹₁,𝐹₂ of the conductive boundary problem with respect to the obstacle 𝐷 for some 𝑧∈𝜕𝐷. By Lemma 3.2 and the identity 𝐷₁=𝐷₂, it can be checked that _𝐷𝑘²_1∗−𝑘²_2∗𝐹₁𝑥,𝑧_𝑛⋅𝐹₂𝑥,𝑧_𝑛𝑑𝑥+𝑖𝑘_𝜕𝐷𝜆₁−𝜆₂𝐹₁𝑥,𝑧_𝑛⋅𝐹₂𝑥,𝑧_𝑛𝑑𝑠(𝑥)=0. (3.47) The singularities of 𝐹₁,𝐹₂ ensure that 𝜆₁=𝜆₂. This completes the proof of the theorem.

Acknowledgments

This work was supported by the NNSF of China under Grant (11201402, 11201266) and Tianyuan Youth Foundations for Mathematics of NNSFC (Grant no. 11026098, 11026150).