Inverse problems for Maxwell’s equations in a slab
with partial boundary data\@footnotetextDate: July 20, 2019
We consider two inverse boundary value problems for the time-harmonic Maxwell equations in an infinite slab. Assuming that tangential boundary data for the electric and magnetic fields at a fixed frequency is available either on subsets of one boundary hyperplane, or on subsets of different boundary hyperplanes, we show that the electromagnetic material parameters, the conductivity, electric permittivity, and magnetic permeability, are uniquely determined by these partial measurements.
In this work, we investigate the unique determination of the electromagnetic material properties of an object from surface measurements. Inverse boundary value problems of this kind arise in many physical situations where one wishes to determine certain properties of a body from measurements taken on its surface, or at a distance from it. In many applications, the situation is described by a partial differential equation, and the inverse problem mathematically amounts to reconstructing parameters of the equation from boundary data of solutions to the equation. Since the seminal paper of A.P. Calderón [C1980] formulating the inverse problem for the conductivity equation on a bounded domain, and the subsequent work by Sylvester and Uhlmann [SU1987] showing unique solvability of this problem for a smooth conductivity, many advances have been made in this field. The method developed in [SU1987] of constructing special exponentially growing solutions to the equation, called complex geometrical optics solutions (CGO solutions), has proved to be applicable to many different situations, and it remains the standard method for showing unique solvability of such inverse boundary value problems. A survey of advances made in the field can be found in [U2009].
This method is also the basis for the study of the inverse boundary value problem for Maxwell’s equations. This problem was first formulated on a bounded domain in [SIC1992], where the linearization of the problem was investigated. Ola, Päivärinta, and Somersalo [OPS1993] first proved global uniqueness of the solution to the inverse problem with full data for smooth parameters; an alternative proof was later given in [OS1996], by relating Maxwell’s equations to a vector Schrödinger equation. Caro and Zhou [CZ2014] reduced the required regularity to continuously differentiable parameters, and the author recently showed that uniqueness also holds for Lipschitz parameters [P2018].
In recent years, partial data inverse problems, in which the boundary data is available only on a subset of the boundary, have become a focus of attention. This is due to their practical importance, since in applications, part of the surface of the object of interest may not be available for measurements, or it may simply be too costly to perform measurements on the whole surface. Mathematically, the consequence of partial boundary data is that the integral identity that is the starting point for the uniqueness proof will involve boundary integrals with unknown functions. Two main methods have been found to be effective in dealing with such problems; on the one hand, the use of Carleman estimates to control the size of solutions on inaccessible parts of the boundary, on the other hand, reflection methods used to construct solutions that vanish on the inaccessible part, assuming this part of the boundary has a suitable geometry.
Carleman estimates were first used in a partial data inverse problem for the conductivity and Schrödinger equations by Bukhgeim and Uhlmann [BU2002]. Kenig, Sjöstrand, and Uhlmann [KSU2007] introduced Carleman estimates with nonlinear weight functions which allowed them to significantly improve on the previous result. The method has since been extended to different inverse problems, such as for Schrödinger operators with magnetic potential [DKSU2007], and for Maxwell’s equations on a manifold [COST2017].
Isakov [I2007] introduced the reflection method and showed unique determination of the conductivity from Dirichlet and Neumann data given on some subset of the boundary, assuming that the inaccessible part of the boundary is part of a plane or of a sphere. This restriction allows to reflect solutions in such a way across the inaccessible part that one obtains solutions with vanishing boundary values. Caro, Ola, and Somersalo [COS2009] applied the reflection argument in studying a partial data inverse problem for Maxwell’s equations on a bounded domain.
The geometry of a slab, an infinite domain bounded by two parallel hyperplanes, has been of considerable interest in view of applications in modeling waveguides and in medical imaging, among others. Consequently, many different types of inverse problems have been studied in this setting. The question of identifying an object embedded in a homogeneous slab was considered in [I2001, SW2006], inverse scattering problems in acoustics and optics have been studied for example in [DM2006, ERY2008], and [MS2001] considered an inverse problem in optical diffusion tomography.
There are also a number of recent results on partial data problems in a slab for the conductivity and Schrödinger equations. Due to the geometry of the boundary, reflection arguments are a powerful tool in this scenario; however, Carleman estimates have been employed as well. Li and Uhlmann [LU2010] studied inverse problems for the conductivity and scalar Schrödinger equations in two different cases, namely with partial Dirichlet and Neumann data given (i) on the same boundary hyperplane or (ii) on opposite boundary hyperplanes. The authors use the Carleman estimate derived in [BU2002] to show that the boundary integrals are negligible in case (i); in case (ii), a reflection argument is employed. In [KLU2012], the inverse problem for a Schrödinger operator with a magnetic potential was studied in the same setting, using reflection arguments to construct the necessary CGO solutions; in [Li2012, Li2012opp] a matrix Schrödinger operator was considered in each of the cases, using a reflection argument and a combination of Carleman estimates and reflection arguments, respectively.
In this paper, we want to investigate each of these two scenarios in the partial data inverse problem for Maxwell’s equations in the slab. We show unique solvability in these cases by relating them to the bounded domain setting investigated in [COS2009], and employing arguments similar to those used there, as well as arguments used in [KLU2012]. We now formulate the problems we will be considering.
We define the slab , with , and denote the boundary planes and . We fix a frequency , and assume that the magnetic permeability , electric permittivity , and conductivity are Lipschitz functions in such that outside of a compact set, . We set and consider the time-harmonic Maxwell equations on with boundary data as follows:
where is a compactly supported function in the space of tangential vector fields on
The admissibility pertains to guaranteeing uniqueness of the solution on the unbounded domain by prescribing a suitable radiation condition for the fields as , and this is made precise in Definition A.1 in the appendix, where the well-posedness of Maxwell’s equations in this setting is discussed. We show there that the problem (1.1a)-(1.1c) has a unique solution under the conditions stated above, as well as the two following assumptions.
Assumption 2: We assume that is such that for all .
We consider the two cases described above: on the one hand, knowledge of the tangential boundary components and on (parts of) opposite boundary hyperplanes, and on the other hand, knowledge of these boundary values on the same boundary hyperplane. Consequently, the partial Cauchy data set will be of the form
where , in the first case, and in the latter case, with ,
Our goal is to prove the following theorems.
Theorem 1.1. Let be the slab defined above, and let , such that outside of a compact set , . Assume that is a domain. Assume furthermore that
and that there exist extensions of the parameters to (of the same name) that belong to and are invariant under reflection across the plane . Let such that . Then if for a fixed frequency we have , then , , and in .
Theorem 1.2. Let be the slab defined above, and let , such that outside of a compact set , . Assume that is a domain. Assume furthermore that
and that there exist extensions of the parameters to (of the same name) that belong to such that are invariant under reflection across the plane , and are invariant under reflection across the plane . Let such that . Then if for a fixed frequency we have , then , , and in .
This paper is organized as follows. In Section 2, we will relate the Maxwell system to a vector Schrödinger equation, following an approach originally presented in [OS1996], with some modifications introduced in [COS2009]. At the end of the section we review the construction of CGO solutions, based on the original method by [SU1987]. In Section 3, we prove Theorem 1. We first derive an integral formula for pairs of solutions vanishing on the inaccessible hyperplane . We then employ a Runge-type approximation argument to show that it suffices to construct solutions on a bounded domain. We reflect CGO solutions across to satisfy the vanishing boundary conditions, and then use these solutions in the integral formula. In the process, products of reflected and non-reflected solutions appear, whose asymptotics need to be studied carefully. The resulting asymptotic expressions will be the same as were obtained in [COS2009], so that we can refer to that work to finish the uniqueness proof. Finally, in Section 4, we prove Theorem 1. We derive a suitable integral formula involving solutions that vanish on different hyperplanes, then construct these solutions by reflecting across respectively , and use them in the integral formula. In addition to products of reflected and non-reflected solutions, we now also obtain products of solutions reflected across different planes. In order to handle these terms, we adapt an argument used in [KLU2012], choosing the phase vectors for the CGO solutions suitably so that the products involving reflected solutions exhibit exponential decay. The non-vanishing terms in the limit will again be of the same form as in [COS2009].
In the appendix, we discuss the necessary conditions to guarantee well-posedness of the direct problem.
2 Transformation of Maxwell’s equations to an elliptic system
We first modify the Maxwell system, following the approach introduced in [OS1996] and adapted with a slightly different scaling in [COS2009]. Note that (1.1a) implies
where , and similarly, with ,
We add these two equations to the Maxwell system, and add some suitable terms introducing two scalar potentials and to rewrite the system as
Note that any solution to (2.1) that has vanishing first and last components is a solution to Maxwell’s equations. In accordance with this block notation, we will from now on write 8-vectors with being scalar functions, and being 3-vectors (corresponding to the magnetic and electric fields, respectively).
For scalars and we introduce the notation
We then define, with ,
and note that if , then
The motivation for this rescaling is the following crucial result, which provides a relationship between the augmented Maxwell system and a matrix Schrödinger equation.
Lemma 2.1. [COS2009, Lemma 1.1] The operators defined above satisfy
where and are zeroth order matrix potentials, given by
and has the same shape as with and interchanged.
For further details concerning the rescaling and properties of the operators we refer to [COS2009].
2.1 Review of construction of CGO solutions
We briefly summarize the construction of CGO solutions to Maxwell’s equations by using the factoring of the Schrödinger equation shown in Lemma 2. The construction follows the classical method developed in [SU1987] and adapted for Maxwell’s equations in [OS1996]; see also [COS2009, Section 2] for more details.
We define weighted and Sobolev spaces with by the following norms:
In the scalar case, it has been shown that if is compactly supported, and is such that and is sufficiently large, then for any there is a unique solution to
and satisfies the estimate
We use the analog of this result in the vector case. Recall our extension of the parameters such that and belong to . We set and define , and note that is thus compactly supported. The first operator from Lemma 2 can then be written as
We first obtain solutions to this second order equation. Let be any fixed nonzero vector and let be a vector such that and
where is a parameter controlling the size of . Let be a vector that is independent of and bounded with respect to . Then we have the following existence result for CGO solutions to the Schrödinger equation in with potential .
Proposition 2.2. [COS2009, Proposition 2.1] Let , and let such that . There exists a CGO solution to
that is of the form
such that for ,
This solution satisfies the following asymptotics as : if we denote
From the factoring (2.2) it follows that the function
then satisfies the first order equation
we can write as
with the asymptotics
for all and bounded open subsets of . In order for this function to further yield solutions to Maxwell’s equations, we need . The next result gives a condition on under which this is the case.
Lemma 2.3. [COS2009, Lemma 2.2] If
then if is sufficiently large,
We will use the following choice of for , which was introduced in [OS1996] and also used in [COS2009]:
3 Proof of Theorem 1
The proof of Theorem 1 is divided into several steps. Using the assumption on the Cauchy data sets, we first derive an integral formula in the slab involving the unknown parameters as well as solutions to Maxwell’s equations with vanishing boundary conditions on , which is analogous to the integral formula obtained in [COS2009] on a bounded domain. Since the CGO solutions that we want to use grow at infinity, we then need a Runge-type approximation result that allows us to consider the integral formula over a bounded domain. Next, we will reflect the CGO solutions constructed in Section 2.1 across to achieve the desired boundary conditions. Finally, we plug these solutions into the integral formula and perform the limit . The resulting asymptotic expressions are the same as those obtained in [COS2009], so that we refer to this work to evaluate the limits and obtain partial differential equations for the unknown parameters. A unique continuation result from [COS2009] then shows that the parameters are in fact equal.
3.1 Integral identity
We start by deriving an integral formula for solutions to the augmented Maxwell system. Recall the assumptions of Theorem 1: we suppose that we can extend the two sets of parameters to functions in all of in such a fashion that the parameters are constant outside the compact set , so that we have
Denote , and let be an admissible solution to
with on and compactly supported on . Let solve
Then satisfies on , and
Since and have the same tangential boundary values on , it further follows from that on , and hence on . We proceed to show that in . Note that since outside all material parameters are constant,
and it follows from (3.1) that the nonzero components of satisfy Maxwell’s equations with constant parameters in , with partial boundary condition on , where . Writing out the equations for the components of , we have
and eliminating either of the functions shows that we have for
Also, on , and similarly for . This implies that on that part of the boundary. Furthermore, given their relationship through Maxwell’s equations, the components of are (up to some constants) the normal derivatives of those of on the boundary. So satisfies a homogeneous Helmholtz equation with zero Dirichlet and Neumann boundary condition on a subset of the boundary, and unique continuation now implies in . By symmetry, the same follows for . In particular, we find that on .
Define , where denotes the complex conjugate, and let solve
Using the integration by parts formula for a bounded domain ,
as well as the fact that if both and have vanishing first and third components,
using the boundary values of and the boundary condition for . Summarizing, we have shown the following:
Proposition 3.1. Let be an admissible solution to
in , with on and compactly supported on . Furthermore, let be a solution to
3.2 Restricting to a bounded domain - Runge approximation
We introduce the function spaces
With this notation, the integral identity holds for and . The functions that we will construct to use in the integral formula have exponential growth at infinity, so we want to restrict ourselves to considering solutions on the bounded domain only. To facilitate this, we show the following density result.
Lemma 3.2. is dense in with respect to the norm.
Proof. Suppose that this density does not hold for (for , the argument is analogous). Then the Hahn-Banach Theorem gives the existence of a function with in such that for all ,
but for some ,
We want to replace the function by , with suitable , so that we can integrate by parts. To this end, let be the admissible (in the sense of Definition A.1) solutions to the following nonhomogeneous Maxwell equations with parameters and and zero tangential boundary condition for ,
Then the second and forth components of are equal to those of . Since , the first and third components of are not relevant and we can replace by in the integral.
We now integrate by parts, using the identity
as well as (3.4). We thus obtain for all
Since can be an arbitrary smooth function on , we find that must vanish on .
We proceed to show that in . Since and the parameters are constant outside , it follows that and satisfy the homogeneous Maxwell equations in . We further have the boundary conditions and on . The same unique continuation argument that was used in the derivation of the integral formula for the auxiliary function applies and yields in . In particular, we may conclude that on . This implies for ,
Now the integral over vanishes because on ; the first term vanishes on by the boundary condition for , and the second term vanishes on by the boundary condition for , and on , since we saw above that on . Thus, all the boundary terms vanish and we arrive at a contradiction, proving the density of in .
3.3 Constructing CGO solutions that vanish on
In Section 2.1 we recalled the construction of CGO solutions to Maxwell’s equations. The solutions to be used in the integral formula (3.5) need to vanish on . This is now achieved by suitably reflecting them across this plane as was also done in [COS2009].
We start by picking the complex vectors and ; our choice is the same as in [COS2009]. For a fixed vector with , we define the unit vectors and as
These vectors satisfy and for . Now we set
where is a parameter controlling the size of . Note that , and , and as becomes large, we have
We further set . With these choices of vectors, let be the CGO solutions for with complex phase vector as constructed in Proposition 2.1 and Lemma 2.1, and let be the CGO solutions for with phase . Recall that these are global solutions to the respective equations. As in Section 2 we now denote and , and perform a reflection of these solutions in such a way that the resulting functions also solve Maxwell’s equations. To this end, we denote the reflection across in Cartesian coordinates by