Uniqueness for an inverse problem in electromagnetism with partial data

A uniqueness result for the recovery of the electric and magnetic coefficients in the time-harmonic Maxwell equations from local boundary measurements is proven. No special geometrical condition is imposed on the inaccessible part of the boundary of the domain, apart from imposing that the boundary of the domain is $C^{1,1}$. The coefficients are assumed to coincide on a neighbourhood of the boundary, a natural property in applications.


Introduction
Let µ, ε, σ be positive functions on a nonempty, bounded, open set Ω in R 3 , describing the permeability, permittivity and conductivity, respectively, of an inhomogeneous, isotropic medium modelled by Ω. Let ∂Ω denote the boundary of Ω and N the outward unit vector field normal to the boundary. Consider the electric and magnetic fields, E, H, verifying the so-called time-harmonic Maxwell equations at a frequency ω > 0, namely in Ω, where γ = ε + iσ/ω, i denotes the imaginary unit, and ∇× denotes the curl operator.
The boundary data corresponding to the inverse boundary value problem (IBVP) for the system (1.1) only can be given by a boundary mapping (the Date: October 29, 2014. 1 impedance or admittance map) if ω is not a resonant frequency. The fact that the position of resonant frequencies depends on the unknown coefficients (as it is stated in [53]) motivated Pedro Caro in [16] to consider a Cauchy data set instead of a boundary map as boundary data. Cauchy data sets have been used in [14,56,57,16,17,20].
This work is aimed at the IBVP for the system (1.1) with local boundary measurements established by a Cauchy data set taken just on a part of ∂Ω. More precisely, Definition 1.1 describes the conditions for the domain and the part of its boundary where the measurements are taken and Definition 1.2 (used in [17]) introduces the boundary data for the IBVP studied in this article.  It is known that if the domain Ω is not convex and its boundary is not C 1,1 , Maxwell equations may not admit solutions in H 1 (Ω) even for boundary data in H 1/2 (∂Ω) (see [8,9,58,59,23].) Thus, for a less regular domain (e.g., Lipschitz), some non-standard Sobolev spaces are necessary. Some of them, which will be used in these notes, appear in the following Definition 1.3. Let Ω and Γ be as in Definition 1.1. Let B 1/2 (∂Ω) denote the Besov space B s p,q (∂Ω) with s = 1/2, p = q = 2. These Besov spaces on nonsmooth boundaries of domains are defined, e.g., in [36]. Define B 1/2 (Γ) = {f | Γ : f ∈ B 1/2 (∂Ω)}, with norm g B 1/2 (Γ) = inf{ f B 1/2 (∂Ω) : f | Γ = g}, and . Some properties of these spaces can be found, e.g., in [48], [16], [17].
Next, the class of admissible coefficients for the uniqueness result proven in this article is set.
Let Ω be as in Definition 1.1. Let M > 0. The pair of coefficients µ,γ is admissible if µ, γ ∈ C 1,1 (Ω) ∩ W 2,∞ (Ω), and the following conditions are satisfied: The main result of this work reads as follows.
Let Ω and Γ be as in Definition 1.1 and ω > 0 the time-harmonic frequency. Assume µ j ,γ j (with j = 1, 2) are two pairs of admissible coefficients such that supp( The IBVP for Maxwell equations can be seen as a vector generalization of the inverse conductivity problem of Calderón. In his seminal paper [15], Calderón posed two questions as follows: Firstly, is it possible to uniquely determine the conductivity of an unknown object from boundary measurements? Secondly, in affirmative case, can this conductivity be reconstructed? Here the boundary measurements are set by the Dirichlet-to-Neumann map Λ σ , which for a conductivity σ ∈ L ∞ (Ω) defined on a bounded domain Ω modelling the object, is defined by ∇ · σ∇u = 0 in Ω, u = f on ∂Ω, and (∂u/∂N)| ∂Ω denotes the normal derivative of u on ∂Ω.
Matti Lassas in [42] proved that the boundary measurements of the Calderón problem are a low-frequency limit of the boundary data (impedance map) of the IBVP for time-harmonic Maxwell equations under some restrictions.
Concerning the Calderón problem in the plane, there are three main global uniqueness proofs giving reconstruction D-bar methods based on complex geometrical optics (CGO) solutions: the Schrödinger equation approach for twice differentiable σ by Nachmann [51], the first-order system approach for once differentiable σ by Brown and Uhlmann [12], and the Beltrami equation approach assuming no smoothness (σ ∈ L ∞ (Ω)) by Astala and Päivärinta [4]. The assumption σ ∈ L ∞ (Ω) was the one originally used by Calderón in [15]. Several stability estimates have been proven: [6,7,21,27].
In dimension n ≥ 3 the best known uniqueness result for the Calderón problem is due to Haberman and Tataru [30] for continuously differentiable conductivities. A novel argument of decay in average using Bourgain-type spaces is introduced there. We cite some previous uniqueness results: the foundational [64] for smooth conductivities by Sylvester and Uhlmann, [50] where Nachman presents a reconstruction algorithm, and [10,11,55]. Concerning conditional stability, the best result is by the third author et al. in [18] for C 1,ε conductivities on Lipschitz domains using the method in [30]. A previous stability result was given by Heck in [31]. Roughly speaking, the method introduced by Alessandrini in [1] gives the main guidelines followed by most stability methods for both the scalar and vector problems.
To deal with inverse problems from partial data for scalar elliptic equations, two main approaches are found in the literature in dimension n > 2 (see [3] for n = 2): using Carleman estimates [12,39,32] and using reflection arguments [35,33]. This work applies to the vector case the density argument shown in [2] for the scalar Schrödinger equation by Gunther Uhlmann and Habib Ammari.
The IBVP for stationary Maxwell equations with global data was originally proposed by Somersalo et al. in [61], where the coefficients are supposed to deviate only slightly from constant values. The same year the unique recovery of the parameters from the scattering amplitude for µ constant was presented in [22], and a local uniqueness result for the IBVP from global data was proven in [63]. The first global determination result for the IBVP with general coefficients µ, γ from global boundary measurements was proven by Lassi Päivärinta et al. in [52], assuming C 3 smoothness on µ, γ on C 1,1 domains. The proof is constructive. Later on, the proof was simplified via a relation between Maxwell equations and a matrix Helmholtz equation with a potential in [54]. Boundary determination results appeared in [46] and [37] for smooth boundaries. Chiral media were studied in [47]. Stability from global boundary data was obtained in [16]. Other inverse problems in electromagnetism in settings different to the ones in this paper have been considered in [60,34,41,45,38,20].
The uniqueness and stability issue of the IBVP for Maxwell equations with local boundary data has been little studied. The only works in this direction the authors are aware of are [19], [17], where an extension of Isakov's method in [35] to Maxwell system is performed. Another extension of methods used in the scalar case for partial data to vector systems is in [57], where uniqueness for a Dirac-type system is proven following the ideas of [39].
Our proof uses the CGO solutions given in [16,17] to a matrix Schrödingertype equation related to Maxwell system and a Dirac-type system not related to Maxwell equations. As in Lemma 3.2 of [38] we give an integral identity (Proposition 4.2) involving a solution Z 1 to the Schrödinger-type equation and a solution Y 2 to the Dirac system. In order that such an identity holds for boundary data restricted to Γ, the solutions have to satisfy certain local homogeneous boundary conditions on Γ c . In [17] the solutions with such properties are constructed from the CGO solutions following the reflection principle in [35], arising this way the strong geometrical constraint on Γ c of being plane or part of a sphere.
We manage to avoid this annoying restriction by the density argument given by Lemma 2 in [2] for the scalar Schrödinger equation adapted to a vector Helmholtz equation satisfied by the electric (and magnetic with different coefficients) field related to Z 1 and another matrix Schrödinger-type equation verified by Y 2 . Here, unique continuation principles for the aforementioned vector equations and a stability estimate for the inverse of the Dirac-type operator are required.
The density argument makes the assumption that the boundary is C 1,1 and the coefficients coincide on a neighbourhood of the boundary, the latter being a natural property in applications. The rest of the proof is valid with just Lipschitz boundary.
This article is organized as follows. In Section 2 some key matrix equations are introduced with the novelty with respect to previous works of a Schrödinger-type equation verified by the solutions to the Dirac-type equation (P + W * ) Y = 0 not related to the Maxwell system. The CGO solutions to some of such equations used in this article are recalled in Section 3. Section 4 is devoted to showing an orthogonality identity involving the potentials and solutions to matrix equations corresponding to two couples of admissible coefficients. Two density results which are essential in the proof are presented in Section 5. Finally, our proof of uniqueness is expounded in Section 6 which contains a demonstration of the bounded invertibility of a Dirac-type operator with certain boundary conditions. Throughout this work the following notation is used. Notation. Given a couple of coefficients (µ j , γ j ) for j = 1, 2, denote C j For an expression like U · V , with U, V C m -valued vector fields and m a natural number, · denotes the analytic extension to C m of the Euclidean real-inner product on R m . I m stands for the m×m identity matrix. For a matrix of complex entries U, the expressions U t and U * stand for its transpose and conjugate transpose, U t , respectively. For the domain Ω and complex vector fields U, V ∈ H 1/2 (Ω; C m ), denote where ds denotes the restriction of the Lebesgue measure of R 3 to ∂Ω.
Acknowledgements. This work is supported by the EPSRC project EP/K024078/1. J.M.R. was also supported by the project MTM 2011-02568 Ministerio de Ciencia y Tecnología de España. J.M.R. wishes to thank Pedro Caro for his help over a warm meeting in ICMAT (Madrid, Spain) honouring Alberto Ruiz' 60th birthday, on Propositions 3.1 and 3.2 and uniqueness of Cauchy problems followed from unique continuation properties. The authors also thank Friedrich Gesztesy, Gerd Grubb, Hubert Kalf and William Evans for helpful discussions.

The equations
Here some differential systems related to Maxwell equations are presented. Let us start with the classical Schrödinger-type equation approach by Petri Ola and Erkki Somersalo in [54]. Fix a frequency ω > 0. Assume the coefficients µ,γ to be in C 1,1 (Ω) on a bounded domain with boundary locally described by the graph of a Lipschitz function. Following the notation in [16], write α := log γ, β := log µ, κ := ωµ 1/2 γ 1/2 . E, H ∈ H(Ω; curl) solve Maxwell equations (1.1) with coefficients µ, γ if and only if X = (h H t | e E t ) t solves the so-called augmented system (P +V )X = 0 and the scalar fields e, h vanish, where with D := 1/i∇, ∇ denoting the gradient operator, and ∇· the divergence operator. Further, where × denotes the cross product. Define the terms It can be checked that the expressions W P − P W t and W * P − P W are zerothorder. Therefore, the second-order operators which do not contain first-order terms, are Schrödinger-type. In [16], [17] a further zeroth-order operator Q ′ is considered which it is not used in this work. Note that if (−∆I 8 + Q)Z = 0 and X := diag(µ −1/2 I 4 , γ −1/2 I 4 ) (P − W t )Z then (P + V )X = 0. If additionally the scalar fields in X are identically zero, the vector fields in X give the electromagnetic fields verifying Maxwell equations.
Finally, a new matrix Schrödinger potential is introduced, namely Q in (2.4), satisfying Lemma 2.1 below.
Specify explicitly the (κ, α, β)-dependence of W , W t and W * by writing A straightforward computation gives W * (κ, α, β) = −W t (−κ, α, β). Since the relation among κ, α and β is not involved in the proof of the fact that −P W t +W P is zeroth-order, we deduce that −P W t (−κ, α, β) + W (−κ, α, β)P is zeroth-order. Thus, the matrix operator is zeroth-order. We deduce the following Notation. In the rest of the manuscript, for two pairs of coefficients µ j , γ j with j = 1, 2, we will write Q j , Q j , W j to refer to the zeroth-order matrix operators Q, Q, W defined in (2.3),(2.4),(2.2), respectively, for the case µ = µ j , γ = γ j .

The special solutions
In this section we recall the almost exponentially growing solutions Z, Y constructed in [16] for the systems (−∆I 8 + Q)Z = 0, (P + W * )Y = 0 based on ideas of the papers [64], [10], [54], [38]. Here the coefficients µ j , γ j (j = 1, 2) under Theorem 1.1's conditions have to be considered extended to the whole Euclidean space R 3 . We denote the extended coefficients in the same manner µ j , γ j . The extensions fulfill the properties as follows: 1. They are Whitney type (see [62] for their construction).
where M 8×8 denotes the space of 8 × 8 matrices with complex entries, and Then there exists a solution .

Then there exists a solution
where A 2 ,B 2 are constant complex vector fields, and

An orthogonality identity
This section is aimed at proving an orthogonality identity given by Proposition 4.2 involving solutions on the open set Ω to certain matrix partial differential equations whose traces contain information supported on Γ.
in Ω such that N × E 1 = 0 on Γ c . In addition, suppose that in Ω with f j ∈ H 1 (Ω), u j ∈ H(Ω; curl) and f 1 = N × u 2 = 0 on Γ c . Hence, for any pair E 2 , H 2 in H(Ω; curl) of solutions to in Ω such that N × E 2 | ∂Ω = 0 on Γ c , the following estimate holds: Lemma 4.1 follows from the proof of Lemma 3.3 in [17] by making the solutions Y 1 , Y 2 on Ω play the role of Y 1 , Y 2 on U in Lemma 3.3 from [17]. This is achieved imposing directly to Y 1 , Y 2 the appropriate boundary conditions on ∂Ω, namely the tangential component of the electric field appearing in the structure of Y 1 vanishes on the inaccessible part of the boundary, and concerning Y 2 , the trace of the first component and the tangential component of the second vector field also vanish on the inaccessible part of the boundary. In Lemma 3.3 from [17] such boundary conditions for Y 1 , Y 2 come from a reflection argument using the special geometric conditions assumed to ∂U \ Γ there, which can not be used here.
The proof of Lemma 4.1 uses Lemma 2.2, Lemma 2.4, Lemma 2.5 and Lemma 2.6 in [17]. Lemma 4.1's proof is omitted since, up to these comments, is identical to Lemma 3.3's proof in [17].

Density and unique continuation results
In the remainder of the paper, let Ω and Γ be as in Definition 1.1. If E, H ∈ H(Ω; curl) solve system (1.1) in Ω for certain coefficients µ,γ, then E, H are also solutions to the following second order system: Notation. For known coefficients µ,γ and frequency ω > 0, notation L will refer to the following Helmholtz-type vector second order differential operator defined in the sense of distributions for For Lipschitz continuous functions µ, γ on Ω, assuming µ to be bounded from below, L U is an L 2 vector field if U ∈ H(Ω; curl) and ∇ × (∇ × U) ∈ L 2 (Ω; Note the following integration by parts formula for any E, F ∈ C ∞ (Ω; C 3 ): Next, the density result for the scalar Schrödinger equation given by Lemma 2 in [2] is adapted to Schrödinger-type matrix equations (Proposition 5.1) and the second order operator L (Proposition 5.2).  , W 2 = 0, C = ∅. Part ii) follows from part i), remarking that the boundary conditions on Γ guarantee that the extension of the solution by zero on a neighbourhood Γ ε in R 3 such that Γ ε ∩ Ω = ∅ and Γ ε ∩ Ω is an open subset of Γ with respect to the relative topology on ∂Ω induced by the Euclidean topology of R 3 , satisfies the same equation and maintains the H 2 -regularity on int(( Ω \ Ω ′ ) ∪ Γ ε ). Indeed, the kernel of the trace operator (u| ∂G , (∂u/∂N)| ∂G ) defined for u ∈ H 2 (G), is the closure of C ∞ 0 (G) in H 2 (G) (usually denoted by H 2 0 (G)), for any domain G with C 1,1 boundary ∂G (see [44] or e.g. [29, Theorem 1.5.1.5]). For clarity Figure  1 illustrates the sets Γ ε , Ω, Ω ′ in the plane (although they must be considered in R 3 ). Let Γ ε be a nonempty, open, connected subset of R 3 with Lipschitz boundary such that Γ ε ∩ G = ∅, Γ ε ∩ G is an open subset of Γ ′ with respect to the relative topology on ∂G induced by the Euclidean topology of R 3 . Figure 1 with Ω = G, Γ = Γ ′ illustrates the choice of Γ ε in the plane. The conditions of part ii) guarantee that the extension U of U by zero on Γ ε verifies U ∈ H(G ′ ; curl), Regarding the aforementioned result in [26], note that a counterexample for the stationary Maxwell system with coefficients in the Hölder class C α for every α < 1 is provided in [24] by Demchenko. Proof of Proposition 5.1. Following the lines of Lemma 2's proof in [2], suppose v ∈ K(Ω) satisfies (g|v) Ω ′ = Ω ′ v * g dx = 0 for any g ∈ K(Ω). We are going to prove that v = 0 in Ω.
Proof of Proposition 5.2. Consider the 3 × 3 matrix valued Green's function G given by where x ∈ Ω. Here, L y stands for the operator L acting in the variable y, and N y stands for the outward unit normal vector to the boundary ∂Ω at the point y ∈ ∂Ω. Using the integration by parts formula (5.2) one can check that any E ∈ N (Ω) can be represented as follows for x ∈ Ω: Remark: We deduce that if L U = F in Ω and N × U = 0 on ∂Ω then U(x) = Ω G(x, y) F (y) dy, for x ∈ Ω.
Jochen Brüning and Matthias Lesch in [13] generalize the analysis of Dirac-type operators considered in the well-known paper by Atiyah, Patodi and Singer [5]. Concerning the general Dirac-type operators studied there on compact manifolds with boundary, in [13, Section 1.B] an operator D is introduced acting on sections of a hermitian vector bundle E over an open subset M of a compact oriented Riemannian manifold M such that its boundary N = ∂M is a compact hypersurface in M . The authors call E the vector bundle over M , and E N := E ↾ N. The differential operator D is said to be of Dirac type if it is first order, symmetric and elliptic in L 2 (E) with domain C ∞ 0 (E) verifying that D 2 has scalar principal symbol given by the metric tensor.
Taking M = Ω ′ and E = M ×C 8 the trivial bundle over M, each fiber equipped with the standard hermitian inner product of C 8 , the operator P on Ω ′ defined in (2.1) falls into the category of these Dirac type operators, since P 2 = −∆I 8 , P U, V Ω ′ = U, P V Ω ′ for any U, V ∈ C ∞ 0 (Ω ′ ; C 8 ) and the characteristic form of P , namely Q(λ) = det(Λ(λ)) = −i|λ| 8 , does not vanish for any λ = (λ 1 , λ 2 , λ 3 ) ∈ R 3 \ 0. Here, Λ(λ) denotes the symbol of P given by the matrix form In [13] it is proved that D admits self-adjoint extensions by imposing non-local boundary conditions given by an orthogonal projection π in L 2 (E N ), which is a classical pseudodifferential operator on E N satisfying a certain symmetry property (condition (1.13) in [13]) related to the structure of the operator (see [ The domain D in H 1 (Ω ′ ; C 8 ) with the graph norm associated with P is continuously embedded into H 1 (Ω ′ ; C 8 ). The space H 1 (Ω ′ ; C 8 ) is compactly embedded into L 2 (Ω ′ ; C 8 ). Since W ∈ L ∞ (Ω ′ ; M 8×8 ) for admissible µ, γ, the operator of multiplication by W t , which we write M W t , is bounded and linear in L 2 (Ω ′ ; C 8 ). Therefore, M W t is (P ↾ D)-compact.
Thus, P − W t has also empty essential spectrum and finite-dimensional eigenspaces. If 0 is in the spectrum of P − W t , then 0 must be an eigenvalue with finitely many linearly independent eigen-and associated functions. We can make 0 no longer be an eigenvalue by choosing a new set of boundary conditions which are not satisfied by any of the finitely many linearly independent eigen-and associated functions in the root spaces associated with 0. Let us keep denoting the resultant boundary operator by π so that the condition π(Z| ∂Ω ′ ) = 0 guarantees the existence of a constant C stblty independent of Z such that The argument presented in Subsection 6.1, together with a trick based on an auxiliary system which improves the regularity of (P − W t )Z when Maxwell equations are satisfied, leads to the following Lemma 6.1. For admissible coefficients µ,γ, assume that Additionally, suppose π(Z| ∂Ω ′ ) = 0 for the boundary operator π introduced in Subsection 6.1. Then there exists a constant C only depending on C stblty , M, ω, such that Z L 2 (Ω ′ ;C 8 ) ≤ C E L 2 (Ω ′ ;C 3 ) .
Proof of Lemma 6.1.