Complex Boundary Value Problems of Nonlinear Differential Equations: Theory, Computational Methods, and ApplicationsView this Special Issue
Uniqueness in Inverse Electromagnetic Conductive Scattering by Penetrable and Inhomogeneous Obstacles with a Lipschitz Boundary
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 from the knowledge of the near field data, assuming that the incident fields are electric dipoles located on a large sphere with polarization . Our results extend those in the paper by F. Hettlich (1996) to the case of inhomogeneous Lipschitz obstacles.
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 . In , 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  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 in . Variational methods for the homogeneous isotropic medium problem were proposed in , under the assumption that the bounded domain with boundary in the class , and some additional conditions on . 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 .
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  using potential theory for the impenetrable obstacle case with Neumann boundary conditions. By the same technique, the authors in  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  improved the previous results to the case of Lipschitz domains. In , 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,  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  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 , a different method was used in  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 , 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 .
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 be a bounded penetrable, inhomogeneous, isotropic domain with a Lipschitz boundary denoted by and covered with a thin layer of high conductivity. Assume that the domain is imbedded in a homogeneous background medium. Define and with being the wave number, where and are the refractive index of the domain and the background medium, respectively. Assume that with for all and is a complex constant with . Assume further that with is a complex-valued function describing the surface impedance of the coating. The incident field is considered to be an electric dipole located at on a large sphere with polarization given by Denote by the free space Green tensor of the background medium and define which satisfies where is the Dirac delta function. Note that can be written as where is the scattered electric field due to the background medium and the electric dipole .
In order to formulate precisely the scattering problem, recall the following Sobolev spaces: where denotes the exterior unit normal to . If is unbounded, we denote by the space of functions for any compact set ⊂⊂ . Introduce the space where . Then the scattering problem can be formulated as follows. Given , find the field and the scattered field such that and the scattered field is required to satisfy the Silver-Müller radiation condition uniformly in , where .
We first have the following uniqueness result for the above scattering problem.
Proof. To prove the theorem, it is enough to consider the case whence . Taking the dot product of (2.6) with over and of (2.7) with over with , respectively, and integrating by parts, we obtain by using the conductive conditions (2.8) and (2.9) that
where is the corresponding scattered magnetic field. Taking the complex conjugate of both sides of (2.11) and using the fact that , and are nonnegative gives
An application of the Rellich lemma yields that in (see [26, Theorem 6.10]). This, together with the unique continuation principle, implies that in . From the trace theorem, it follows that on . Thus, taking the imaginary part of (2.11) and using the assumption that for all , we have that in .
Introduce the electric-to-magnetic Calderon operator (see ), which maps the electric field boundary data on the surface of a large ball to the magnetic boundary data on , where satisfies Then the scattering problem (2.6)–(2.10) can be reformulated in the following mixed conductive boundary value problem (MOCKUP) over a bounded domain: where .
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 are the vector spherical harmonics of order given by for and . Here, as usual, denotes the surface gradient on the surface of the unit sphere .
For given by , the operator can be defined by where and is the spherical Hankel function.
If in (2.23), we will obtain another operator . Properties of and are collected in the following lemma (for a proof see ).
Lemma 2.2. The operator is negative definite in the sense that for any with . Furthermore, is compact, where
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 where . Then multiplying (2.17) and (2.18) by test function , using formally integration by parts and using the conductive boundary conditions on , we can derive the following equivalent variational formulation for (CBP). Find such that where is the incident magnetic field and
We rewrite (2.29) as the problem of finding such that where the sesquilinear form is defined by Here denotes the scalar product, and denotes the 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 then we seek such that The variational problem (2.34) can be rewritten as where we define Here we have used to write the tangential component of the gradient of in terms of the tangential gradient on the sphere . By Lemma 2.2, it follows that is negative definite, then we obtain that is a coercive sesquilinear form on . Further by Lax-Milgram theorem, it is easy to see that gives rise to a bijective operator. Since , still by Lemma 2.2, we know that 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: Choosing , it is easy to see that By the definition of the operator , if is the weak solution of the problem then we have where Furthermore, we can compute that which together with the fact implies Therefore the Rellich lemma ensures us that in . From (2.39), we see that on and then which, together with the fact that , implies . This completes the proof of Lemma 2.3.
Lemma 2.4. The space is compactly imbedded in , where is a ball with .
Proof. Consider a bounded set of functions . Each function can be extended to all of by solving the exterior Maxwell equation Define Since the tangential components of are continuous across , it follows that . By using the properties of the Calderon operator and the conditions in , we see that the following equations hold true Then, by the definition of that and the relationship = on , we immediately have Thus, has a well-defined divergence and in , where Now we choose a cut-off function such that in and is supported in a ball . Then one can use the general compactness theorem (Theorem 4.7 in ) to the sequence and extract a subsequence converging strongly in . This proves the lemma.
From the above definitions of and , we have the following Helmholtz decomposition lemma.
Lemma 2.5. The spaces and are closed subspaces of . The space is the direct sum of the spaces and , that is,
We now look for a solution of the variational problem (2.31) in the form , where and is the unique solution of (2.34). We observe that for all by the definition of . Hence the problem of determining is equivalent to the problem of determining such that From Chapter 10.3.2 in  we know that for where the operator is a compact operator from into and the operator satisfies . We now split the sesquilinear form into with The sesquilinear form is obviously bounded and a direct computation verifies that with some constant .
Hence by Lax-Milgram theorem, gives rise to a bijective operator and by the compact embedding of in and the fact that is a compact operator from into , the second term 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 , there exists a unique solution of (CBP) which depends continuously on the incident field .
3. Uniqueness for the Inverse Problem
In this section we will show that the scattering obstacle and the corresponding parameter are uniquely determined from the knowledge of the scattered fields for all , where is the surface of a large ball with . 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 is not an eigenvalue of Maxwell equation for the domain . Then we have(i) the restriction to of is complete in ;(ii) the restriction to of is complete in .
Proof. For simplicity, we only prove statement (ii). Case (i) can be proved similarly.
Let be such that Then it follows that Define By (3.2), it is easy to see that for arbitrary polarization in the tangential plane to at , we have From the definition of (3.3), we immediately have Due to the symmetry of the background Green function, as a function of solves ,. Hence, satisfies the Maxwell's equation in . By (3.6) and the fact that is an arbitrary polarization in the tangential plane to at , we immediately have that .
The uniqueness of the exterior problem implies that in . Thus, the unique continuation principle ensures us that in . By trace theorem, it follows that and on . By the definition of and the jump relations of the vector potential across , it can be checked that satisfies the following equations: Therefore, the uniqueness theorem of the interior problem for Maxwell's equations implies that in . Finally, from the jump relations of the vector potential across , we have which completes the proof.
We now consider two obstacles and with the refractive index and the surface impedance . Let denote the unbounded part of and its open complement. From the proof of Theorem 2.6, it follows that the total field satisfies for any large ball with and all test function , where It is convenient to introduce the following space: where . 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 is not an eigenvalue of Maxwell equation in . Let be a ball with . Let and be the scattered fields with respect to and , respectively, produced by the same incident field . Assume that for all with the radius for a fixed wave number . Then we have Here satisfies the following variational problem: for all , where the coefficients and satisfy that and .
Proof. (i) We first prove that for any fixed , the scattered fields , where is the solution of the following problem:
with the incident field and , , . By Lemma 3.1 and the fact that is not an eigenvalue of Maxwell equation in , it follows that there exists a sequence and such that
Let , then it satisfies the Maxwell equation in . Let , then the well posedness of the problem
and (3.17) imply that
This, together with the fact that , implies (see )
Then by (3.19), (3.20), and the trace theorem, it can be proved that
Denote by and the scattered fields with respect to and produced by the same incident field . By the assumption for all , it is easy to see that . Then by the uniqueness theorem of the exterior scattering problem, it follows that in , which together with the unique continuation principle ensures that in . Now, by (3.21) and the well posedness of the direct problem (2.18), it can be checked that for any compact set , we have
for any fixed .
Therefore, the fact in ensures us that The arbitrarity of implies that for any fixed .
(ii) Next we will show that the identity (3.14) holds. Set , then it follows from (3.11) that for all . Choose two domains with and define a smooth function with in and in . Let , , , it is easy to see that satisfy the assumptions of the lemma. We further assume satisfies (3.15) with respect to , , , so that substituting into the left hand of (3.24) and noting that in yield that Hence substituting into (3.24), it follows from the right hand of (3.24) that We define by for all . By Theorem 2.6, it follows that there exists a unique solution of the problem for all . Choose two domains with and define smooth functions with and . Take in (3.11), it is seen that Equation (3.28) with replaced by yields By (3.26) and (3.27), it can be shown that Taking the difference of (3.29) and (3.30), we have that By (3.28), we can deduce that is a radiating solution of the corresponding Maxwell's equations in , then it can be extended to all of denoted by by solving the exterior Maxwell's equation in with on , which also satisfies the Silver-Müler radiation condition at infinity. By applying the vector Green formula to (3.32), it can be proved that In view of the fact and in , we immediately have Application of the vector Green formula again and noting that both and the extended function satisfy the Silver-Müler radiation condition, it follows that Hence the Stratton-Chu formula combines with (3.34) implies that Since is an arbitrary polarization in the tangential plane to at , we obtain that . By the fact that is a radiating solution of Maxwell's equation in , it follows that in . Hence the unique continuation principle implies that in . Therefore, can be used as a test function for , which satisfies (3.15) with . So that from the left hand of (3.30), we deduce that Thus, it follows from the right hand of (3.30) that . Furthermore, from (3.27) with replaced by , it can be shown that From the definitions of , we observe that which combines (3.38), the definition of the scalar product , and the fact that 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 and be the scattered fields with respect to and , respectively, and the corresponding impedances. Suppose that the assumptions in Lemma 3.2 hold true and is not empty for . If one of the following assumptions holds, then we have . Consider(i);(ii).
Proof. Let us assume that is not included in . Since is connected, we can find a point and a sufficiently small with the following properties:(i);(ii) the points lie in for all , where is the unit normal to at .Denote , the inner part of the domain . We consider the unique solution of the following problem:
Here satisfies the Silver-Müler radiation condition at infinity, and denotes the magnetic dipole defined by
It can be proved that is a solution of Maxwell's equations with homogeneous conductive boundary value conditions on in any domain with and .
Define and .
In view of the above definitions of and , it follows that satisfies the variational equation (3.15) in Lemma 3.2 for the obstacle . The well posedness of the direct problem for (CBP) and the fact that is bounded away from imply that the solution of (3.40) is uniformly bounded in . We now define another singular solution with respect to by