Abstract and Applied Analysis

Volume 2012 (2012), Article ID 306272, 21 pages

doi:10.1155/2012/306272

Research Article

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

Fenglong Qu

School of Mathematics and Information Science, Yantai University, Yantai, Shandong 264005, China

Received 26 August 2012; Accepted 6 December 2012

Academic Editor: Yong Hong Wu

Copyright © 2012 Fenglong Qu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 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.

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 in [2]. Variational methods for the homogeneous isotropic medium problem were proposed in [1], 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 [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 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.

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 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 [27]), 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 [27]).

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

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 [27]) 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,

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 , 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 [27] 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 [28])

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

Define

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

where is a magnetic dipole defined in (3.41), and is a solution of the problem

Here satisfies the Silver-Müler radiation condition at infinity. Noting that satisfies the variational equation (3.15) in Lemma 3.2 with and , it follows that both and satisfy the relationship (3.14), then we obtain

For case (i), by the fact that and the singularities of the magnetic dipole defined in (3.41), it can be proved that as , 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 . By choosing and using the similar analysis as in the proof above, one can prove that . Finally, we obtain that . For other cases, due to the singularities of , a contradiction also arises in (3.46) as . This proves the theorem.

Theorem 3.4. Assume with parameters and the scattered fields satisfy for all , then we have on .

Proof. From the proof of Theorem 3.3, it follows that there exists two singular solutions of the conductive boundary problem with respect to the obstacle for some . By Lemma 3.2 and the identity , it can be checked that

The singularities of ensure that . 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).

References

F. Hettlich, “Uniqueness of the inverse conductive scattering problem for time-harmonic electromagnetic waves,” SIAM Journal on Applied Mathematics, vol. 56, no. 2, pp. 588–601, 1996.

T. S. Angell and A. Kirsch, “The conductive boundary condition for Maxwell's equations,” SIAM Journal on Applied Mathematics, vol. 52, no. 6, pp. 1597–1610, 1992.

G. Vasseur and P. Weidelt, “Bimodal electromagnetic induction in nonuniform thin sheets with an application of the northern Pyrenean induction anomaly,” Geophysical Journal of the Royal Astronomical Society, vol. 51, pp. 669–690, 1977.

F. Cakoni, D. Colton, and P. Monk, “The determination of the surface conductivity of a partially coated dielectric,” SIAM Journal on Applied Mathematics, vol. 65, no. 3, pp. 767–789, 2005.

F. Cakoni, D. Colton, and P. Monk, “The inverse electromagnetic scattering problem for a partially coated dielectric,” Journal of Computational and Applied Mathematics, vol. 204, no. 2, pp. 256–267, 2007.

F. Cakoni, D. Colton, and P. Monk, “The electromagnetic inverse-scattering problem for partly coated Lipschitz domains,” Proceedings of the Royal Society of Edinburgh A, vol. 134, no. 4, pp. 661–682, 2004.

F. Cakoni and H. Haddar, “Identification of partially coated anisotropic buried objects using electromagnetic Cauchy data,” Journal of Integral Equations and Applications, vol. 19, no. 3, pp. 359–389, 2007.

V. Isakov, “On uniqueness of recovery of a discontinuous conductivity coefficient,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 865–877, 1988.

V. Isakov, “On uniqueness in the inverse transmission scattering problem,” Communications in Partial Differential Equations, vol. 15, no. 11, pp. 1565–1587, 1990.

A. Kirsch and R. Kress, “Uniqueness in inverse obstacle scattering,” Inverse Problems, vol. 9, no. 2, pp. 285–299, 1993.

T. Gerlach and R. Kress, “Uniqueness in inverse obstacle scattering with conductive boundary condition,” Inverse Problems, vol. 12, no. 5, pp. 619–625, 1996.

D. Mitrea and M. Mitrea, “Uniqueness for inverse conductivity and transmission problems in the class of Lipschitz domains,” Communications in Partial Differential Equations, vol. 23, no. 7-8, pp. 1419–1448, 1998.

N. Valdivia, “Uniqueness in inverse obstacle scattering with conductive boundary conditions,” Applicable Analysis, vol. 83, no. 8, pp. 825–851, 2004.

M. Di Cristo and J. Sun, “An inverse scattering problem for a partially coated buried obstacle,” Inverse Problems, vol. 22, no. 6, pp. 2331–2350, 2006.

M. Di Cristo and J. Sun, “The determination of the support and surface conductivity of a partially coated buried object,” Inverse Problems, vol. 23, no. 3, pp. 1161–1179, 2007.

G. Yan and P. Y. H. Pang, “Inverse obstacle scattering with two scatterers,” Applicable Analysis, vol. 70, no. 1-2, pp. 35–43, 1998.

X. Liu and B. Zhang, “Inverse scattering by an inhomogeneous penetrable obstacle in a piecewise homogeneous medium,” Acta Mathematica Scientia B, vol. 32, no. 4, pp. 1281–1297, 2012.

X. Liu, B. Zhang, and G. Hu, “Uniqueness in the inverse scattering problem in a piecewise homogeneous medium,” Inverse Problems, vol. 26, no. 1, p. 015002, 14, 2010.

X. Liu and B. Zhang, “Direct and inverse obstacle scattering problems in a piecewise homogeneous medium,” SIAM Journal on Applied Mathematics, vol. 70, no. 8, pp. 3105–3120, 2010.

P. Hähner, “A uniqueness theorem for a transmission problem in inverse electromagnetic scattering,” Inverse Problems, vol. 9, no. 6, pp. 667–678, 1993.

X. Liu, B. Zhang, and J. Yang, “The inverse electromagnetic scattering problem in a piecewise homogeneous medium,” Inverse Problems, vol. 26, no. 12, p. 125001, 19, 2010.

D. Colton, R. Kress, and P. Monk, “Inverse scattering from an orthotropic medium,” Journal of Computational and Applied Mathematics, vol. 81, no. 2, pp. 269–298, 1997.

M. Piana, “On uniqueness for anisotropic inhomogeneous inverse scattering problems,” Inverse Problems, vol. 14, no. 6, pp. 1565–1579, 1998.

P. Hähner, “On the uniqueness of the shape of a penetrable, anisotropic obstacle,” Journal of Computational and Applied Mathematics, vol. 116, no. 1, pp. 167–180, 2000.

F. Cakoni and D. Colton, “A uniqueness theorem for an inverse electromagnetic scattering problem in inhomogeneous anisotropic media,” Proceedings of the Edinburgh Mathematical Society II, vol. 46, no. 2, pp. 293–314, 2003.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93, Springer, New York, NY, USA, 2nd edition, 1998.

P. Monk, Finite Element Methods for Maxwell’s Equations, Oxford University Press, Oxford, UK, 2003.

M. Cessenat, Mathematical Methods in Electromagnetism, vol. 41, World Scientific Publishing, Singapore, 1996.