On an inverse problem in electromagnetism with local data: stability and uniqueness

In this paper we prove a stable determination of the coefficients of the time-harmonic Maxwell equations from local boundary data. The argument --due to Isakov-- requires some restrictions on the domain.


Introduction
Let Ω be a bounded Lipschitz domain in the three-dimensional euclidean space. Assume the medium, modeled by Ω, to be non-homogeneous and isotropic. Suppose the electromagnetic properties of Ω to be described by the electric permittivity ε, the magnetic permeability µ and the electric conductivity σ. Let E, H denote the electric and magnetic fields, respectively. The time-harmonic Maxwell equations at frequency ω > 0 read dH + iωε * E = σ * E, dE − iωµ * H = 0, whenever the total electric current density is given by σ * E. Writing γ = ε + iσ/ω, the time-harmonic Maxwell equations can be expressed as It is known that this system may present positive resonant frequencies even when the domain is assumed to be of class C 2 (see [23]) or σ is assumed to vanish (see [15]). This means that, for some positive frequencies, the system may have non-trivial solutions of (1) with zero boundary conditions.
In these notes, we shall study the inverse boundary value problem of determining in a stable manner the coefficients µ, ε, σ by local boundary measurements. When setting this problem, the possible existence of resonant frequencies makes natural the use of the restricted Cauchy data set as a model of non-invasive measurements, instead of using either the restricted admittance or impedance maps. Cauchy data sets have been used successfully in [4], [21], [22] and [6].
In order to quantify the proximity of the restricted Cauchy data sets we introduce a pseudo-metric distance which was already used in [6].
Definition 1 Let µ 1 , γ 1 and µ 2 , γ 2 be two pairs of coefficients. Consider ω a positive frequency and let C j Γ denote C(µ j , γ j ; Γ). Let us define the pseudometric distance between the restricted Cauchy data sets C 1 Γ and C 2 Γ as In order to state our result, we need stable determination of the problem on the boundary. Since this has not been proven yet, we shall introduce the following definitions.
Definition 2 Given two constants M, s such that 0 < M, 0 < s < 1/2, we shall say that the pair of coefficients µ, γ is admissible if they satisfy the following conditions.
Definition 3 Let M, s be the constants given in Definition 2 and let ω be a positive frequency. We shall say that a pair µ, γ is in the class of B-stable coefficients on Γ at frequency ω if µ, γ is an admissible pair and there exists a modulus of continuity B such that, for any other admissible pairμ,γ, one has Here C Γ ,C Γ are the Cauchy data sets associated to the pairs µ, γ andμ,γ, respectively. The first idea in our argument is to construct special solutions vanishing on ∂Ω \ Γ, the inaccessible part of the boundary. In [10] Isakov proposed a reflection argument which allows to construct solutions for the conductivity equation with the desired behavior on the boundary. This argument was extended in [5] to the time-harmonic Maxwell equation.
In order to carry out Isakov's approach it seems to be necessary to assume some geometrical restrictions about the domain, namely, the inaccessible part is supposed to be either part of a plane or part of a sphere. Despite this restriction, the method allows to prove the following result from local boundary data.
Theorem 1 Let U be either a suitable partially flat domain or a suitable partially spherical domain and let ω be a positive frequency. Consider µ 1 , γ 1 and µ 2 , γ 2 any two pairs in the class of B-stable coefficients on Γ at frequency ω, with B satisfying |r| ≤ B(|r|) for all |r| < 1. Assume that ∂ N µ j = ∂ N γ j = 0 on ∂U \ Γ with j = 1, 2. Then, there exists a constant C = C(M) such that the following estimate holds for some constant λ such that 0 < λ < s 2 /3. Here C 1 Γ , C 2 Γ are the restricted Cauchy data sets associated to the pairs µ 1 , γ 1 and µ 2 , γ 2 , respectively.
The exact meaning of suitable is explained in Subsection 1.4.
As in the inverse conductivity problem, it should be possible to prove that any admissible pair is in the class of Hölder-stable coefficients on Γ for any frequency ω, that is, with B(|r|) = |r| α for 0 < α < 1. Note that we have obtained the same kind of stability as in the global data case (see [6]).
From the point of view of applications it might be useful to suppose the coefficients to be equal on the accessible part of the boundary. In this particular case we get the following corollary.
Corollary 1 Let U be either a suitable partially flat domain or a suitable partially spherical domain and let ω be a positive frequency. Consider µ 1 , γ 1 and µ 2 , γ 2 any two pairs of admissible coefficients. Assume that and ∂ N µ k = ∂ N γ k = 0 on ∂U \ Γ with j = 1, 2, 3 and k = 1, 2. Then, there exists a constant C = C(M) such that the following estimate holds for some constant λ such that 0 < λ < s 2 /3.
Furthermore, if we follow the proof of Theorem 1 one can state the following uniqueness result.
Theorem 2 Let U be either a suitable partially flat domain or a suitable partially spherical domain and let ω be a positive frequency. Consider µ 1 , γ 1 and µ 2 , γ 2 in C 1,1 (U) such that As in the inverse conductivity problem, it should be possible to prove that the coefficients are equal on the accessible part of the boundary Γ whenever C 1 Γ = C 2 Γ . The problem of determining the electromagnetic coefficients by data taken on the entire boundary has been studied by several authors. The unique recovery of C 3 -coefficients γ and µ from boundary data was proved in [18], and later simplified in [20]. Boundary determination results were given in [12] in the case that the boundary is smooth. The more general chiral media was studied in [16]. For a slightly more general approach and more background information, see also the review article [19].
The inverse problem of determining the electromagnetic coefficients from partial data has been much less considered. As far as the author knows the only work in that direction is [5].
Two different approaches have been used to attack the inverse conductivity problem from partial boundary data. The first one was proposed in [4] and generalized in [14]. In [7], this method was used to give a log-logstable determination in the framework of [4]. In this approach there are not any strong geometrical restriction about the domain but the partial measurements have to be taken in the whole boundary. Getting an optimal stability (i. e. a stability with a log-type modulus of continuity) in the context of [4] may be difficult and the stability for [14] is an open question. The second approach for partial data was proposed in [10] and the optimal stable determination was stated in [8]. As we have already mentioned, this argument requires a strong restriction on the domain. However, the measurements are localized on the accessible part of the boundary and it is possible to get the optimal stable determination. These two facts are very important from the point of view of applications. For instance, Alessandrini and Vessella proved in [2] that a logarithmic estimate yields Lipschitz stability for some finite dimensional spaces of conductivities.
These two approaches have been extended to systems. In [22] Salo and Tzou followed the spirit of [14] to prove uniqueness in the context of Dirac's equation. Isakov's argument was extended in [5] to Maxwell's equations. The proof given here takes some ideas from [13] and it turns out to be more convenient and useful for us than the proof given in [5]. In fact, it avoids the long computations made there to prove the thesis of Theorem 2 and it allows to relax the hypothesis about the domain and the smoothness of the coefficients. In [5] the domain was assumed to be of class C 1,1 and the coefficients were assumed to be C 4 . Besides, a technical hypothesis about the extension of the coefficients had to be supposed.
An overview of the paper is the following. It has three sections. In the first one, we give some preliminaries about the functional spaces used. In the second section we prove our results when U is partially flat. To achieve this, we use a reflection argument to construct special solutions vanishing on the inaccessible part of the boundary. In the third section we connect the flat case with the spherical one by means of the Kelvin transform.
Acknowledgement. This paper is part of the author's doctoral dissertation and it has been written under the supervision of Alberto Ruiz. The author would like to thank him for his support and dedication. The author would also like to thank Petri Ola and Mikko Salo for their invitation to inverse problems in electromagnetism. The author was economically supported by Ministerio de Ciencia e Innovación de España, MTM2008-02568-C02-01.

Preliminaries
Let E be the three-dimensional euclidean point space and let its tangent bundle be denoted by T E. Let T E be the module of smooth vector fields over the real smooth functions C ∞ (E; R) and define The elements of X E will be called complex vector fields. Let the bundle of alternating tensors be denoted by Λ k T E with k = 0, 1, 2, 3. Let A k E be the vector space of differential k-forms and define The elements of Λ k E will be called complex k-forms. Recall that 0-forms are smooth functions by definition. As it is usual, d and ∧ denote the exterior derivative operator and the exterior product of forms, respectively.
The euclidean metric e induces a volume element denoted by dV , a distance denoted by d e and a point-wise inner product denoted by ω, η for any ω, η ∈ A k E with k = 0, 1, 2, 3. Recall that the Hodge star operator is the unique bundle map * : Moreover, * * ω = ω. Let us define |η| 2 = η, η .
The formal adjoint of d will be denoted by δ and it can be expressed by for η ∈ A k . Let us define the laplacian on k-forms as −∆ := δd + dδ.
We also recall that we can identify vectors and 1-forms by means of the metric, that is, If u ∈ T E, its corresponding 1-form will be denoted by u ♭ . However if the difference is clear by the context it will be denoted by u. On the other hand, if v ∈ A 1 E, its corresponding vector field will be denoted by v ♯ . As before, this notation will be used whenever the context is not clear.

The functional spaces
Along these notes we shall say that a domain Ω is Lipschitz if its boundary ∂Ω is locally the graph of a Lipschitz function. Additionally, N denotes the outward unit vector normal to ∂Ω and ν := N ♭ is its corresponding 1-form.
In order to perform the proofs of the results stated in the introduction we require some standard functional spaces: H s (E), H s (Ω), H s 0 (Ω) with s ∈ R denote the potential Sobolev spaces based in L 2 ; W 2,∞ (Ω) stands for the Sobolev space with two derivaties in L ∞ ; B s (∂Ω) with 0 < |s| < 1 denotes the Besov spaces B s p,q (∂Ω) with p = q = 2. A quite complete description of these spaces can be found in [11].
Additionally, when working with Maxwell's equations other non-standard Sobolev and Besov spaces turn to be useful. Those are mainly, H (Ω; div), H (Ω; curl) and T H(∂Ω). The first one corresponds to the fields or 1-forms in L 2 with divergence in L 2 . In this space, it makes sense the normal traces as elements of B −1/2 (∂Ω). The space H (Ω; curl) corresponds to the fields or 1-forms in L 2 whose curl is in L 2 . The tangential traces of elements of H (Ω; curl) make sense as elements of B −1/2 (∂Ω; C 3 ) or B −1/2 (∂Ω; Λ 1 T E). The space T H(∂Ω) is defined as the space of tangential traces of H (Ω; curl) and we have that if w is a vector field, or if w is a 1-form. Here Div stands for the surface divergence. Recall that the surface divergence of w ∈ T H(∂Ω) makes sense, and it can be defined as an element of B 1/2 (∂Ω) in the following way: Finally, we recall some key points. For any u ∈ H (Ω; div) and any g ∈ where u is a vector field, f ∈ H 1 (Ω) and f | ∂Ω = g. We also have where u is a 1-form, f ∈ H 1 (Ω) and f | ∂Ω = g. The maps for vector fields, and for 1-forms. A detailed exposition of the collected facts can be seen in [17] and [6].

Some remarks on the boundary
Along this section, Ω denotes any bounded Lipschitz domain, ∂Ω denotes its boundary and Γ stands for a proper non-empty open subset of ∂Ω. Moreover, | Γ and | Γ denote the restriction to Γ and Γ, respectively.
Proof: (a) was proven in [6]. (b) and (c) follow easily from (a) taking g ∈ C 0,s+ǫ (∂Ω) an extension of g| Γ such that Regarding to extensions from a closed subset of E, see [24].
On the other hand, define Lemma 7 Let N be the outward unit vector normal to ∂Ω and let ν be its associated 1-form. Then Proof: Here we prove the first identity. The second one follows by the correspondence between vector fields and 1-forms.
Let l : T H(Γ) → C be a bounded linear functional, we can construct another functionall : T H(∂Ω) → C defined byl(w) = l(w| Γ ), for any w ∈ T H(∂Ω). Sincel is linear, bounded and It is clear that supp N ×z ⊂ Γ, hence z ∈ T H 0 (Γ) and Conversely, given z ∈ T H 0 (Γ) we can define l : Therefore, l is a bounded linear operator with norm Lemma 8 There exists a positive constant C such that: (a) For any w ∈ T H(∂Ω) and any f ∈ C 0,1 (∂Ω), one has that (b) For any w ∈ T H 0 (∂Ω) and any f : one has that Proof: (a) was proven in [6]. (b) follows easily from (a) taking and extensionf of f | Γ such thatf ∈ C 0,1 (∂Ω) and satisfying Again, regarding to extensions from a closed subset of E, see [24].

Lemma 9
The following items hold: Proof: It is easy to check both items.
(a) Div w is well-defined and belongs to B −1/2 (∂Ω). It remains to prove that supp Div w ⊂ Γ. In order to verify this last point, we just need to have in mind the following facts: The estimate is now immediate using either (3) or (5).
(b) By an analogous argument to the one given in (a), we have that if z| Γ = z| Γ then (Divz)| Γ = (Div z)| Γ . Hence the identity follows. The estimate is a consequence of the identity and (3) or (5).

Maxwell's system as a Schödinger equation
In this section we shall transform Maxwell's equations into a Schrödingertype equation. The idea of this transformation was already introduced in [20].
There is a well known process which allows us to transform Maxwell's equations into a Schrödinger-type equation. In order to do so we require some extra smoothness of the coefficients, namely µ, γ ∈ C 1,1 (Ω). The first step in this process is to augment the Maxwell system with two scalar equations: The information coded in these scalar equations was already present in the initial system. In order to check this, it is enough to take * d in each equation in (1). Next, we introduce a new system inspired in the four mentioned equations. This new system reads as The new terms preserve the physical units of measure of the original four equations. Choosing euclidean coordinates, the new system -called henceforth augmented system-can be written in vector field notation as it follows  In a much more compact manner we shall express the augmented system as Note that E, H is a solution for Maxwell's equations, if and only if, X t = h H t e E t is a solution for the augmented system and the scalar fields e, h vanish. The next step is to rescale the augmented system, that is with κ = ωµ 1/2 γ 1/2 . We shall call (P + W )Y = 0 the rescaled system. The advantage of rescaling is that where Q, Q ′ ,Q are zeroth-order terms. Here W t denotes the transposed of W and W * stands for W t . No first order terms appear in (20), (21) and (22), giving as a result a Schrödinger-type equation. Mind Note that if Z is a solution for (20) in Ω, then Y = (P − W t )Z is a solution for the rescaled system in Ω, hence is a solution for the augmented system. In the same manner, ifẐ is a solution for (22), thenŶ = (P − W )Ẑ is a solution for (P + W * )Ŷ = 0 in Ω. For later uses, with . In order to make as concise as possible the presentation of our proofs, we introduce some additional notation. Let Y, Z be in the form In the first identity we are assuming f j , g j ∈ C ∞ (Ω) and u j , v j ∈ X E| Ω with j = 1, 2, while in the second identity f j , g j ∈ C ∞ (∂Ω) and u j , v j ∈ X E| ∂Ω with j = 1, 2. The following integration by parts holds Here, when A is a (possibly complex) vector field we denote Finally, for elements Y in the form given above we define, for |s| > 0, On the other hand, we define, for 0 < |s| < 1,

About the geometry of U
In order to make precise the geometrical restrictions assumed in Theorem 1, Corollary 1 and Theorem 2, we give the following definitions.

Definition 10
We shall say that a bounded Lipschitz domain U ⊂ E is partially flat if there exists a plane q ⊂ E and some euclidean coordinates E such that, We shall say that a bounded Lipschitz domain U ⊂ E is partially spherical if there exist a point Q 0 ∈ E, r 0 > 0 and some euclidean coordinates E such that (i) Q 0 = y 0 and U ⊂ B(y 0 ; r 0 ) := {y ∈ R 3 : |y − y 0 | < r 0 }, (ii) Γ 0 := int S(y 0 ;r 0 ) (∂U ∩ S(y 0 ; r 0 )) = ∅ where S(y 0 ; r 0 ) := ∂B(y 0 ; r 0 ), In the two previous cases, we denote Γ := ∂U \ Γ 0 .

Definition 11
We shall say that a partially flat domain U is suitable if its symmetric extension with respect to q -that is Ω := U ∪ Γ 0 ∪ R(U)is also Lipschitz. Here R denotes the reflection with respect to q and it is defined In addition, we shall say that a partially spherical domain U is suitable if its inversion with respect to S(0; 2r 0 ) -that is Ω := K(U)is a suitable partially flat domain. Here K denotes the inversion with respect to S(0; 2r 0 ) and it is defined as y −→ r 2 1 /|y| 2 y with r 1 = 2r 0 .
We have to restrict ourselves to these suitable domains because we need to make an extension of the coefficients preserving their smoothness (see Subsection 2.3).

The domain U is partially flat
Along this section we assume U to be a suitable partially flat domain and we follow the notation in Definition 10 and Definition 11.
Consider the system in Ω. The push-forward of the reflection map R reads Let f be a smooth function and u, u ′ two vector fields on E. Let g, v, v ′ denote the function and the vector fields given by . It is a straight forward computation to check that On the other hand, let a be a smooth function defined in {x ∈ R 3 : x 3 < 0} and setã, the extension of a to E, defined asã(

Lemma 12
Given such that E, H ∈ H (Ω; curl) is a solution of (28) in Ω, one has that E − E, H −Ḣ, withĖ ; is also a solution of (28) in Ω satisfying Proof: Let E, H be a solution of (28) in Ω. It is an immediate consequence of (31) and the definition ofμ,γ in Ω thatĖ,Ḣ is also a solution for (28) in Ω. Further, from the weak definition of tangential trace one can derive that N × (E −Ė)| Γ 0 = 0. Indeed, let w ∈ B 1/2 (∂U; C 3 ) such that supp w ⊂ Γ 0 and consider v ∈ H 1 (U; C 3 ) such that v| ∂U = w, then Here we have used (31) twice, and the fact that u, defined as v in U and as R * v R in R(U), belongs to H 1 (Ω; C 3 ) and u| ∂Ω = 0.
Proof: It is easy to check, using (36) that with E 2 , H 2 ∈ H (U; curl) an arbitrary solution of in U and satisfying supp N×E 2 ⊂ Γ. Since (P +W * 2 )Y 2 = 0 and (P +W 2 )L = 0 in U, one has that (L |P Y 2 ) U = (P L |Y 2 ) U , hence On the other hand, we have, using (7) and (9) , that Furthermore, from the Maxwell's equations one deduces that for j = 1, 2. Hence, Let us denote N × E j = T j , N × H j | Γ = S j , then by using the appropriate dualities, the boundary conditions (34), (35) and the estimates (12), (13), (15), (16) and (17) we get This estimate holds for all (T 2 , S 2 ) ∈ C 2 Γ , since E 2 , H 2 was chosen to be an arbitrary solution of (38) in U satisfying supp N×E 2 ⊂ Γ. Finally, the wanted estimate is a consequence of Definition 1.
Proposition 15 Let γ 1 , µ 1 and γ 2 , µ 2 be in the class of B-stable coefficients on Γ at frequency ω. Then, there exists a constant C(M) such that, for any

as in Lemma 14 and any
Here Q j is the matrix (23) associates toμ j ,γ j with j = 1, 2.
Proof: From (23) one has In order to get the penultimate identity, we used twice that (P + W * 2 )Y 2 = 0, while to get the last one, we used that Y 1 = (P −W t 1 )Z 1 and that (P +W 1 )Y 1 = 0.
It is a straight forward computation to check the next estimate Here, as usually, the norm of L ∞ (Γ; C 3 ) is for any vector field w. It is a routine computation to check that, on one hand and on the other hand, With all these estimates and Lemma 14 in mind, we get , hence we deduce the estimate given in the statement.

Recalling the existence of special solutions
Let B(O; ρ) be the open ball centered at the origin O with radius ρ > 0 and such that Ω ⊂ B(O; ρ). Sometimes B(O; ρ) will be denoted by B to simplify the notation. Let ε 0 and µ 0 denote the electric and magnetic constants, respectively. Extend the coefficientsμ j ,γ j defined in Ω to functions in E -still denoted byμ j ,γ j -, preserving their smoothness and in such a way that µ j − µ 0 ,γ j − ε 0 have compact support in B(O; ρ) (regarding to extensions see [24]). Note two simple facts. Firstly, the extensions still satisfy the a priori bound and the a priori ellipticity condition in E. Secondly, the extensions of the matrices (24), (25) In the following, we state two propositions which were proven in [6]. Their proofs are based on ideas from [25], [3], [20] and [13].
Proposition 16 Let δ be a constant such that −1 < δ < 0 and let ζ ∈ C 3 be such that ζ · ζ = ω 2 ε 0 µ 0 with |ζ| > C(δ, ρ) j=1,2 Then, there exists a Z = e iζ·x (L + R) solution of (−∆I 8 + Q)Z = 0 in E, with Z| Ω ∈ H 2 (Ω; Y), for A, B constant complex vector fields, and R satisfying Furthermore, Y = (P − W t )Z is solution for (P + W )Y = 0 in E and it reads The norm in the proposition is Again Y is meanless, it just stands to remark the form of the elements for which the norms are taken. .
Choosing B 1 = B 2 = 0 and A 1 , A 2 such that one gets, when τ becomes large, that where the implicit constant is C(ρ, Ω, M). In addition, by Lemma 18 with and (41), (42) one has one gets, when τ becomes large, that where the implicit constant is C(ρ, Ω, M). Again, by Lemma 18 and (41), (42) one has where 1 Ω is the indicator function of Ω. By Proposition 15 and the properties of the special solutions, there exist three constants c = c(Ω), C = C(ρ, Ω, M) and C ′ = C ′ (ρ, M) such that, for any τ ≥ C ′ one has Note that, for R ≥ 1, one has Lemma 19 One has that Changing to cylindrical coordinates it is enough to study (1 + r 2 + t 2 ) −1 r(r 2 + t 2 ) r 2 + t 2 + (r 2 + t 2 )r 2 + 4(τ 2 + k 2 )r 2 dt dr.
On the other hand, the a priori bound was chosen to have for 0 < s < 1/2. Finally, interpolation theory ensures the existence of two constants C ′ = C ′ (ρ, M) and C = C(ρ, Ω, M, ω) such that, for any d ≤ C ′ , the following estimate holds The idea now is to transfer this estimate from f, g to the difference of the coefficientsμ 1 −μ 2 andγ 1 −γ 2 . This can be accomplished by using the following Carleman estimate.
There exists a positive constant C(Ω) such that, for all h ≤ 1 and any function φ smooth enough, the following estimate holds ∈ Ω. The constant here depends on the distance from x 0 to Ω and on the diameter of Ω. A Carleman estimate of this type can be found in [9].
A simple computation give: Note that, thanks to the a priori bound, we have the following differential inequalities: In order to simplify the notation, we shall write φ 1 =γ .
The constants above depend on the a priori bounds M. With these inequalities and estimate (43), we obtain

The domain U is partially spherical
Along this section we assume U to be a suitable partially spherical domain and we follow the notation in Definition 10 and Definition 11. Furthermore, n and ν will denote the outward unit normal forms of U and Ω, respectively. The basic idea in this section is to use the Kelvin transform K to generalize our result on partially flat domain to the case of partially spherical domain. To achieve this, we study the behavior of Maxwell's equations and the distance δ C under K.
Note that K = K −1 and K is a conformal transformation from (Ω, e) onto (U, e): where K * denotes the pull-back of K. LetẼ = K * E,H = K * H,μ = K * µ, andγ = K * γ. The following is the transformation law for Maxwell's equations under the Kelvin transform. in Ω.
Here η is k-form and c is an arbitrary positive smooth function.