Inverse problems for the anisotropic Maxwell equations
aa r X i v : . [ m a t h . A P ] M a y INVERSE PROBLEMS FOR THE ANISOTROPICMAXWELL EQUATIONS
CARLOS E. KENIG, MIKKO SALO, AND GUNTHER UHLMANN
Abstract.
We prove that the electromagnetic material parameters areuniquely determined by boundary measurements for the time-harmonicMaxwell equations in certain anisotropic settings. We give a uniquenessresult in the inverse problem for Maxwell equations on an admissibleRiemannian manifold, and a uniqueness result for Maxwell equationsin Euclidean space with admissible matrix coefficients. The proofs arebased on a new Fourier analytic construction of complex geometricaloptics solutions on admissible manifolds, and involve a proper notion ofuniqueness for such solutions. Introduction
Let (
M, g ) be a compact Riemannian manifold with smooth boundary ∂M , and assume that dim M = 3. We consider the inverse problem ofrecovering electromagnetic material parameters of the medium ( M, g ) byprobing with time-harmonic electromagnetic fields. The fields in (
M, g ) aredescribed by complex 1-forms E and H (electric and magnetic fields), andthe behavior of the fields is governed by the Maxwell equations in M , (cid:26) ∗ dE = iωµH, ∗ dH = − iωεE. (1.1)Here ω > d is the exterior derivative, and ∗ is theHodge star operator on ( M, g ). The material parameters are given by thecomplex functions ε and µ (permittivity and permeability, respectively). Weassume the following conditions on the parameters: ε, µ ∈ C ∞ ( M ) , (1.2)Re( ε ) >
0, Re( µ ) > M . (1.3)For the inverse problem, we need to describe the electromagnetic fieldmeasurements at the boundary ∂M . Let i : ∂M → M be the canonicalembedding, and consider the tangential trace on k -forms, t : Ω k ( M ) → Ω k ( ∂M ) , η i ∗ η. There is a discrete set of resonant frequencies such that if ω is outside thisset, then for any f in Ω ( ∂M ) the system (1.1) has a unique solution ( E, H )satisfying tE = f (see Theorem A.1). We shall assume that ω > The boundary measurements are given by the admittance mapΛ : Ω ( ∂M ) → Ω ( ∂M ) , tE tH. The inverse problem for time-harmonic Maxwell equations is to recover thematerial parameters ε and µ from the knowledge of the admittance map Λ.In the case of lossy materials, one writes ε = Re( ε ) + iσ/ω where σ ≥ ω = 0) then formallycorresponds to the conductivity equation δ ( σdu ) = 0 . Here δ is the codifferential. In three and higher dimensions, the inverseproblem of determining σ from boundary measurements for the conductivityequation was studied in [4] in a special class of Riemannian manifolds. Definition.
A compact 3-manifold (
M, g ) with smooth boundary ∂M iscalled admissible if ( M, g ) is embedded in (
T, g ) where T = R × M , ( M , g )is a simple 2-manifold, and g = c ( e ⊕ g ) where c is a smooth positive functionand e is the Euclidean metric on R .Simple manifolds are defined as follows: Definition.
A compact manifold ( M , g ) with smooth boundary ∂M iscalled simple if for each p in M the map exp p is a diffeomorphism from aclosed neighborhood of 0 in T p M onto M , and if ∂M is strictly convex(meaning that the second fundamental form of ∂M is positive definite).Admissible manifolds include compact submanifolds of Euclidean space,hyperbolic space, and S minus a point, and also sufficiently small subman-ifolds of conformally flat manifolds. If M is a bounded open set in R withsmooth boundary, equipped with a metric which in some local coordinates x = ( x , x ′ ) has the form g ( x ) = c ( x ) (cid:18) g ( x ′ ) (cid:19) , then ( M, g ) is admissible if g is a simple metric in some sufficiently largeball. Also, admissible manifolds are stable under small perturbations of g .See [4] for more details.We will prove the following result, showing that boundary measurementsfor the Maxwell equations uniquely determine the material parameters inan admissible manifold. Theorem 1.1.
Let (
M, g ) be an admissible manifold, and let ( ε , µ ) and( ε , µ ) be two sets of coefficients satisfying (1.2)–(1.4). If the admittancemaps satisfy Λ = Λ , then ε ≡ ε and µ ≡ µ in M .The second result involves Maxwell equations in a bounded domain Ω in R with smooth boundary. The coefficients ε, µ are assumed to be smoothpositive definite symmetric (1 , g ε and g µ , which are Riemannian metrics in Ω describingpropagation of waves with different polarizations. We shall assume that thevelocity of wave propagation is independent of polarization, which amountsto the property that ε and µ are in the same conformal class [12]. NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 3
The Maxwell equations in Ω can be written as (cid:26) ∇ × ~E = iωµ ~H, ∇ × ~H = − iωε ~E, (1.5)where ~E and ~H are complex vector fields and ω > ~E tan | ∂ Ω = ~f , (1.6)where ~f is a smooth tangential vector field on ∂ Ω and ~E tan | ∂ Ω is the tan-gential part of ~E | ∂ Ω . Under the above assumptions, there is a discrete set ofresonant frequencies outside which the boundary problem for Maxwell equa-tions has a unique smooth solution ( ~E, ~H ) (see Section 7). The admittancemap is given by Λ : ~E tan | ∂ Ω ~H tan | ∂ Ω . The next result considers the inverse problem of recovering the electromag-netic parameters from Λ.
Theorem 1.2.
Let ε j and µ j be smooth symmetric positive definite (1 , ω > j be the corresponding admittancemaps ( j = 1 , g in Ω suchthat ε , µ , ε , and µ are conformal multiples of g − . If the admittancemaps satisfy Λ = Λ , then ε ≡ ε and µ ≡ µ in Ω.To our knowledge, Theorems 1.1 and 1.2 are the first positive resultson the inverse problem for time-harmonic Maxwell equations in anisotropicsettings. For bounded domains in R where g is the Euclidean metric,Theorems 1.1 and 1.2 were proved in [17].There has recently been considerable interest in invisibility cloaking [7],where one looks for anisotropic materials for which uniqueness does not hold.The prescriptions of electromagnetic parameters for cloaking [6] satisfy that ε = µ . Moreover the parameters are singular, so that one of the eigenvaluesis zero at the boundary of the cloaked region. Theorems 1.1 and 1.2 implythat there is no cloaking for materials whose electromagnetic parameterssatisfy the given conditions.Formally, the proofs of Theorems 1.1 and 1.2 follow the Euclidean case.The proof of the uniqueness result in [17] was considerably simplified in [19],and the simplified proof can be described by the following seven steps:1. Reduction of the Maxwell system to a Dirac system, by introducingtwo auxiliary scalar fields Φ and Ψ. A solution X of the Dirac systemgives a solution to the original Maxwell system iff Φ = Ψ = 0.2. Reduction to a rescaled Dirac system ( P − k + W ) Y = 0, where Y is obtained by rescaling the components of X by ε / and µ / .3. Reduction to the Schr¨odinger equation ( − ∆ − k + Q ) Z = 0, whichis possible since ( P − k + W )( P + k − W t ) = − ∆ − k + Q .4. Construction of complex geometrical optics solutions to the equation( − ∆ − k + Q ) Z = 0, which also gives solutions Y = ( P + k − W t ) Z to the Dirac system. NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 4
5. Construction of solutions to the original Maxwell system. This re-quires showing that the scalar fields in Step 1 vanish identically,which follows from a uniqueness result for Z .6. Inserting complex geometrical optics solutions in an integral identity,which allows to recover nonlinear differential expressions involvingthe electromagnetic parameters.7. An application of the unique continuation principle for a semilinearelliptic system to recover the parameters.In [18], it was shown that Steps 1 to 3 above can be carried out also forthe Maxwell equations on a Riemannian manifold ( M, g ). However, Step 4requires complex geometrical optics solutions, and these were only availablefor the Euclidean metric. Therefore, it was not possible to go further in thenon-Euclidean case.A construction of complex geometrical optics solutions for scalar ellipticequations, valid on admissible Riemannian manifolds (
M, g ), was given in[4]. We will combine the ideas in [4] with the scheme outlined above toprove the uniqueness result for the inverse problem for Maxwell equationson admissible manifolds.It will turn out that the main technical obstacle is Step 5, which requiresa uniqueness result for the complex geometrical optics solutions. In [4] theconstruction of solutions is based on Carleman estimates, and there is noconcept of uniqueness for the solutions so obtained. In this article we givea new construction of solutions based on direct Fourier arguments. Thisconstruction comes with a suitable uniqueness result, which can be used tocarry out the proof of the Maxwell result.The main step in the new construction is a counterpart of the basic normestimates of Sylvester-Uhlmann [22]. We outline the idea in a simple case.The estimate is valid in (
T, g ) where T = R × M and g = e ⊕ g , but here( M , g ) can be any compact ( n − e τx ( − ∆ g )( e − τx u ) = f in T, (1.7)with ∆ g the Laplace-Beltrami operator in ( T, g ) and τ a large parameter.In the Sylvester-Uhlmann estimates T = R n and g is the Euclideanmetric, f is in a weighted L space such that h x i δ +1 f ∈ L ( R n ) where − < δ <
0, and one obtains a unique solution u with h x i δ u ∈ L ( R n ).Here h x i = (1 + | x | ) / . In our case we write x for the special Euclidean coordinate in T , and useAgmon-type weighted spaces L δ ( T ) = { f ∈ L ( T ) ; kh x i δ f k L ( T ) < ∞} . The Sobolev space H sδ ( T ) is defined via the norm k u k H sδ ( T ) = kh x i δ u k H s ( T ) ,and H δ, ( T ) is the set { u ∈ H δ ( T ) ; u | R × ∂M = 0 } .The next result is a special case of Proposition 4.1 (since there is nopotential it follows that one may take τ = 1). NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 5
Theorem 1.3.
Let δ > /
2. If | τ | ≥ f ∈ L δ ( T ) there is a unique solution u ∈ H − δ, ( T ) of the equation (1.7). Infact, one has u ∈ H − δ ( T ) and k u k H s − δ ( T ) ≤ C | τ | s − k f k L δ ( T ) , ≤ s ≤ , with C independent of τ .In the Sylvester-Uhlmann result, the proof applies the Fourier transformand one obtains uniqueness by fixing decay at infinity. In our case there is atransversal metric in M , and the Fourier transform or conditions at infinityare not readily available. However, one can ask for decay in the Euclideanvariable and Dirichlet boundary values on ∂M . This makes it possible touse the Fourier transform in x and eigenfunction expansions in M .The proof of Theorem 1.3 is robust in the sense that one can essentiallyreplace the Laplacian in M by any positive operator with a complete setof eigenfunctions. We will need this flexibility in the Maxwell result whenproving similar estimates for the Hodge Laplacian on forms. There is alsoan extra twist in the construction of solutions since one needs a result likeTheorem 1.3 which applies to functions f which may not decay (so one isout of the standard Agmon setting), see Sections 4 and 5 for these moregeneral results.The construction could also be used to develop constructive methods forcertain anisotropic inverse problems. In the Euclidean case, results of thistype were given in [16] for the 3D conductivity equation and in [17] forMaxwell equations.Earlier work on the inverse problem for the Maxwell system in Euclideanspace includes a study of the linearized inverse problem [20], a local unique-ness result [21], and a result for the corresponding inverse scattering prob-lem in the case where µ is constant [2]. As mentioned above, the full inverseproblem was solved in [17], and in [19] the proof was simplified and also areconstruction from measurements based on dipole point sources was given.The paper [18] is a survey and also considers the manifold setting. The in-verse problem for Maxwell equations in chiral media was considered in [15].Boundary determination results are given in [10] and [14]. Finally, [1] givesa partial data result for this problem, based on Isakov’s method [9]. Forresults on inverse problems for the Maxwell equations in time domain, werefer to [12] and the references therein.The structure of the paper is as follows. Section 2 contains notationand identities in Riemannian geometry which will be used throughout thearticle. The reductions of the Maxwell equations to Dirac and Schr¨odingerequations are given in Section 3. The norm estimates and uniqueness resultsrequired for constructing complex geometrical optics solutions are given inSections 4 and 5, and the construction of solutions is taken up in Section 6.In Section 7 we prove Theorems 1.1 and 1.2. There are two appendices, oneon the wellposedness theory of boundary value problems for Maxwell, andone including a unique continuation result for principally diagonal systemsrequired for our results. NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 6
Acknowledgements.
C.K. is partly supported by NSF grant DMS0456583,M.S. is supported in part by the Academy of Finland, and G.U. is partlysupported by NSF and a Walker Family Endowed Professorship.2.
Notation and identities
We will briefly introduce some basic notation and identities in Riemanniangeometry which will be used throughout. We refer to [23] for these facts.In this section let (
M, g ) be a smooth (= C ∞ ) n -dimensional Riemannianmanifold with or without boundary. All manifolds will be assumed to beoriented. We write h v, w i for the g -inner product of tangent vectors, and | v | = h v, v i / for the g -norm. If x = ( x , . . . , x n ) are local coordinates and ∂ j the corresponding vector fields, we write g jk = h ∂ j , ∂ k i for the metric inthese coordinates. The determinant of ( g jk ) is denoted by | g | , and ( g jk ) isthe matrix inverse of ( g jk ).We shall often do computations in normal coordinates. These are coor-dinates x defined in a neighborhood of a point p ∈ M int such that x ( p ) = 0and geodesics through p correspond to rays through the origin in the x coordinates. The metric in these coordinates satisfies g jk (0) = δ jk , ∂ l g jk (0) = 0 . For points p ∈ ∂M we will employ boundary normal coordinates, which arecoordinates y = ( y ′ , y n ) near p so that y ( p ) = 0, y ′ are normal coordinateson ∂M centered at p , and y n ( q ) is the geodesic distance from a point q to ∂M . The metric has the form g ( y ) = (cid:18) g ( y ) 00 1 (cid:19) , g ( y ) = ( g jk ( y )) n − j,k =1 , and g jk (0) = δ jk , ∂ l g jk (0) = 0. We denote by ν the 1-form corresponding tothe outer unit normal vector of ∂M , so that ν = − dy n in boundary normalcoordinates.The Einstein convention of summing over repeated upper and lower in-dices will be used. We convert vector fields to 1-forms and vice versa by themusical isomorphisms, which are given by( X j ∂ j ) ♭ = X k dx k , X k = g jk X j , ( ω k dx k ) ♯ = ω j ∂ j , ω j = g jk ω k . The set of smooth k -forms on M is denoted by Ω k M , and the graded algebraof differential forms is written asΩ M = ⊕ nk =0 Ω k M. The set of k -forms with L or H s coefficients are denoted by L (Ω k M ) and H s (Ω k M ), respectively. Here H s for s ∈ R are the usual Sobolev spaces on M . The inner product and norm are extended to forms and more generallytensors on M in the usual way.Let d : Ω k M → Ω k +1 M be the exterior derivative, and let ∗ : Ω k M → Ω n − k M be the Hodge star operator. We introduce the sesquilinear inner NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 7 product on Ω k M , ( η | ζ ) = Z M h η, ¯ ζ i dV = Z M η ∧ ∗ ¯ ζ. Here dV = ∗ | g | / dx · · · dx n is the volume form. The codifferential δ : Ω k M → Ω k − M is defined as the formal adjoint of d in the inner producton real valued forms, so that( dη | ζ ) = ( η | δζ ) , for η ∈ Ω k − M, ζ ∈ Ω k M compactly supported and real . These operators satisfy the following relations on k -forms in M : ∗∗ = ( − k ( n − k ) , δ = ( − k ( n − k ) − n + k − ∗ d ∗ . If ξ is a 1-form then the interior product i ξ is the formal adjoint of ξ ∧ in theinner product on real valued forms, and on k -forms it has the expression i ξ = ( − n ( k − ∗ ξ ∧ ∗ . The Hodge Laplacian on k -forms is defined by − ∆ = ( d + δ ) = dδ + δd. It satisfies ∆ ∗ = ∗ ∆.The Levi-Civita connection, defined on tensors in M , is denoted by ∇ .We will slightly abuse notation and reserve the expression ∇ f (where f isany function) for the metric gradient of f , defined by ∇ f = ( df ) ♯ = g jk ∂ j f ∂ k . The H and H norms may be expressed invariantly as k f k H ( M ) = k f k L ( M ) + k∇ f k L ( M ) , k f k H ( M ) = k f k H ( M ) + k∇ f k L ( M ) . Here of course k T k L ( M ) = (cid:0)R M | T | dV (cid:1) / for a tensor T .For n = 3, the surface divergence of f ∈ H s (Ω ( ∂M )) is given byDiv( f ) = h d ∂M f, dS i where dS is the volume form on ∂M . A computation in boundary normalcoordinates shows that Div( f ) = −h ν, ∗ du i| ∂M where u ∈ H s +1 / (Ω M ) isany 1-form with tu = f (here s > n = 3, we collect a number of identities which will beuseful for computations. Below let f be a smooth function, α = α j dx j and β = β j dx j and γ = γ j dx j three 1-forms, η a k -form, and ζ an l -form. Forthe Hodge star one has ∗ ∗ η = η, ∗ ( α ∧ ∗ β ) = h α, β i , ∗ ( α ∧ ∗ [ β ∧ γ ]) = h α, γ i β − h α, β i γ, NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 8 and the operators d and δ satisfy δη = ( − k ∗ d ∗ η,δα = −| g | − / ∂ j ( | g | / g jk α k ) ,d ( η ∧ ζ ) = dη ∧ ζ + ( − k η ∧ dζ,δ ( f η ) = f δη + ( − k ∗ df ∧ ∗ η,δ ( α ∧ β ) = ( δα ) β − ( δβ ) α − [ α ♯ , β ♯ ] ♭ . Reduction to Schr¨odinger equation
In this section we present the reductions of the time-harmonic Maxwellsystem to Dirac and Schr¨odinger equations, which corresponds to Steps 1to 3 in the introduction. This mostly follows [18] and [19] although withdifferent notations. We will also give a reduction to the case where thecoefficients are constant near the boundary, namely, ε = ε and µ = µ near ∂M for some constants ε , µ > . (3.1)It is well known that the Maxwell system (1.1) is not elliptic as it iswritten. We perform an elliptization by adding the constituent equations,obtained from (1.1) by applying d ∗ to both equations: (cid:26) d ( µ ∗ H ) = 0 ,d ( ε ∗ E ) = 0 . (3.2)Adding two equations requires adding two unknowns, which will be thescalar fields Φ and Ψ. The choice for how to couple Φ and Ψ into the largersystem obtained from (1.1), (3.2) was motivated in [19] by dimensionalityarguments. The end result is the following system: D ∗ E + Dα ∧ ∗ E − ωµ ∗ Φ = 0 , ∗ D Ψ + DE − ωµ ∗ H + ∗ Dα ∧ Ψ = 0 ,D ∗ H + Dβ ∧ ∗ H − ωε ∗ Ψ = 0 , ∗ D Φ − DH + ∗ Dβ ∧ Φ − ωε ∗ E = 0 . (3.3)Here we have written D = i d and α = log ε , β = log µ . We will also write D ∗ = − i δ for the formal adjoint of D in the sesquilinear inner product onforms.We wish to express (3.3) as an equation for the graded differential form X = Φ + E + ∗ H + ∗ Ψ, written in vector notation as X = (cid:0) Φ ∗ H ∗ Ψ E (cid:1) t . Note that we have grouped the even and odd degree forms together. Thiswill result in a block structure for the equation. Now, taking Hodge star ofthe first and last equations in (3.3) results in the system( P + V ) X = 0 (3.4) NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 9 where P and V are given in matrix notation by P = D ∗ D ∗ DDD D ∗ , V = − ωµ ∗ Dα ∧ ∗− ωµ ∗ Dα ∧ ∗ Dβ ∧ − ωεDβ ∧ − ωε . This is the first Dirac equation we will use. Note that P is just the self-adjoint Dirac type operator D + D ∗ on Ω M , and that ( E, H ) solves theoriginal Maxwell system (1.1) iff X solves (3.4) with Φ = Ψ = 0.For the reduction to a Schr¨odinger equation, it will be convenient torescale X by X = (cid:18) µ − / ε − / (cid:19) Y, (3.5)where Y = (cid:0) Y Y Y Y (cid:1) t , and Y k is the k -form part of Y ∈ Ω M .Assuming (3.1) for the moment, a direct computation using the identitiesin Section 2 shows that (3.4) is equivalent with the rescaled Dirac equationfor Y : ( P − k + W ) Y = 0 . (3.6)Here W is the potential, with compact support in M int , given by W = − ( κ − k ) + 12 ∗ Dα ∧ ∗∗ Dα ∧ ∗ − Dα ∧ Dβ ∧ Dβ ∧ ∗ Dβ ∧ ∗ , where κ = ω ( εµ ) / , k = ω ( ε µ ) / .We will also need the potential W t , which is the formal transpose of W in the inner product on real valued forms, given by W t = − ( κ − k ) + 12 ∗ Dβ ∧ ∗∗ Dβ ∧ ∗ − Dβ ∧ Dα ∧ Dα ∧ ∗ Dα ∧ ∗ . The adjoint is W ∗ = W t . The following result contains the Schr¨odingerequations, involving the Hodge Laplacian − ∆ = dδ + δd on Ω M , in a formwhich will be convenient below. Lemma 3.1.
We have( P − k + W )( P + k − W t ) = − ∆ − k + Q, ( P + k − W t )( P − k + W ) = − ∆ − k + Q ′ , ( P − k + W ∗ )( P + k − ¯ W ) = − ∆ − k + ˆ Q, NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 10 where Q , Q ′ , and ˆ Q are smooth potentials with compact support in M int , Q = k − κ + 12 ∆ α + h dα, dα i • • • • •• • ∆ β + h dβ, dβ i • • • ,Q ′ = k − κ − ∆ β − h dβ, dβ i • • • • α − h dα, dα i • • • • , and • denote smooth coefficients. Proof.
We give the proof of the first identity, the other ones being analogous.One has( P − k + W )( P + k − W t ) = − ∆ − k + W ( P + k ) − ( P − k ) W t − W W t . The point is to show that the first order term vanishes. We write W as W = − ( κ − k ) + 12 W , where W acts on a graded form X = X + + X − , with X + = X + X and X − = X + X , by W X = ( − Dα ∧ + i Dα ) X − + ( Dβ ∧ − i Dβ ) X + . We will use the identities in Section 2. If u is a 0-form then( W P − P W t ) u = 1 i ( W du − ( d − δ )( uDα ))= − ( − dα ∧ du + ∗ dα ∧ ∗ du − du ∧ dα + u ( δdα ) − ∗ du ∧ ∗ dα )= (∆ α ) u. If u is a 1-form we have( W P − P W t ) u = 1 i ( W ( du − δu ) − ( d − δ )( − Dβ ∧ u + h Dβ, u i ))= − ( dβ ∧ du + ∗ dβ ∧ ∗ du − ( δu ) dβ − dβ ∧ du − d h dβ, u i − δ ( dβ ∧ u )) . The identity δ ( dβ ∧ u ) = ( − ∆ β ) u − ( δu ) dβ − [ ∇ β, u ♯ ] ♭ and a computationin normal coordinates implies that( W P − P W t ) u = 2( ∇ β ) u − (∆ β ) u. Here ( ∇ β ) u denotes the 1-form corresponding to the vector field ( ∇ β ) u ♯ .The computation for 2-forms and 3-forms can be reduced to the previouscases by noting that if u is a k -form, then( d − δ ) ∗ u = ( − k ∗ ( d + δ ) u, ( η ∧ − i η ) ∗ u = ( − k − ∗ ( η ∧ + i η ) u. Thus, if ∗ u is a 2-form then( W P − P W t ) ∗ u = ∗ [ dα ∧ du + ( δu ) dα − ∗ dα ∧ ∗ du − dα ∧ du + d h dα, u i + δ ( dα ∧ u )]= ∗ [2( ∇ α ) u − (∆ α ) u ] . NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 11
Similarly, if ∗ u is a 3-form then( W P − P W t ) ∗ u = (∆ β ) ∗ u. We have P ( f u ) − f P u = ( Df ∧ + i Df ) u for a function f , so W P − P W t = 12 ( W P − P W t ) + Dκ ∧ + i Dκ . This shows that ( P − k + W )( P + k − W t ) = − ∆ − k + Q where Q is anoperator of order 0. Since k ( W + W t ) − W W t = k − κ + 12 κ ( W + W t ) − W W t where W W t X = − [( Dα ∧ − i Dα ) X + + ( Dβ ∧ − i Dβ ) X − ], and since one has( ξ ∧ − i ξ ) u = −h ξ, ξ i u for any k -form u , we obtain the required expressionfor Q . (cid:3) The preceding arguments show how to reduce the original Maxwell systemto Dirac and Schr¨odinger equations. In the next lemma, which is similar to[19, p. 1135], we give a reduction on the level of boundary measurements:if the admittance maps for two Maxwell systems coincide, then one has anintegral identity involving the potentials Q j and solutions of the Schr¨odingerand Dirac systems. Note that Z has to be related to a solution for Maxwell,but Y need not be. This flexibility in the choice of Y will simplify therecovery of coefficients. Lemma 3.2.
Let ( ε , µ ) and ( ε , µ ) be two sets of coefficients satisfying(1.2)–(1.4), and assume that Λ = Λ . After replacing ( M, g ) by a largermanifold (which is admissible if (
M, g ) is), one may assume that ε = ε = ε and µ = µ = µ near ∂M for constants ε , µ >
0, (3.7)and one has the identity (( Q − Q ) Z | Y ) = 0 (3.8)for any smooth graded forms Z j , Y j satisfying the following properties:( P − k + W ) Y = 0 , Y = ( P + k − W t ) Z , ( P − k + W ∗ ) Y = 0 ,Y = Y = 0 . The reduction to the case where (3.7) holds is a consequence of the nextboundary determination result.
Theorem 3.3.
Let (
M, g ) be a compact 3-manifold with smooth boundary,and let ε and µ satisfy (1.2)–(1.4). Given a point on ∂M , the admittancemap Λ uniquely determines the Taylor series of ε and µ at that point inboundary normal coordinates. Proof.
This result was proved in [10], [14] in the case where M is a smoothdomain in R and g is the Euclidean metric (the result was for complex ε andreal µ , but the same proof works also for complex µ ). The argument proceedsby showing that the admittance map is a pseudodifferential operator on ∂M ,and by computing the symbol of Λ in boundary normal coordinates at a NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 12 fixed point p ∈ ∂M . One then proves, by looking at the difference of twoadmittance maps, that the Taylor series of ε and µ are uniquely determinedat p .Fortunately, if ( M, g ) is a Riemannian manifold with boundary, the formof the metric in boundary normal coordinates is exactly the same as in theEuclidean case. This means that the arguments of [10], [14], which weregiven in boundary normal coordinates of a Euclidean domain, carry overwithout changes to establish Theorem 3.3 for any Riemannian manifold(
M, g ). (cid:3) Proof of Lemma 3.2.
We first establish (3.8) in the original manifold M .Note that if (3.7) is not satisfied, we may formally take k = 0 in the precedingarguments and then all conclusions remain valid except that W j and Q j maynot be compactly supported in M int .Let Z j and Y j be as described, and let X be the solution to (3.4), withpotential V , corresponding to Y as in (3.5). Since Λ = Λ , we can find asolution ˜ X of ( P + V ) ˜ X = 0 with t ˜ H = tH and t ˜ E = tE on ∂M . Herewe write X = (cid:0) ∗ H E (cid:1) t , ˜ X = (cid:0) ∗ ˜ H E (cid:1) t . If ˜ Y is the solution to ( P − k + W ) ˜ Y = 0 corresponding to ˜ X as in (3.5),then t ( Y − ˜ Y ) = 0 since ε = ε and µ = µ on ∂M by Theorem 3.3.We wish to argue that ν ∧ ( Y − ˜ Y ) = 0 , i ν ( Y − ˜ Y ) = 0 on ∂M. (3.9)The first part is immediate since ν ∧ η = 0 on ∂M iff tη = 0. For thesecond part we use the surface divergence. The fact that X and ˜ X solvethe Maxwell equations, together with Theorem 3.3, implies that h ν, H − ˜ H i| ∂M = 1 iω h ν, µ − ∗ dE − µ − ∗ d ˜ E i| ∂M = − iωµ Div( t ( E − ˜ E )) = 0 . A similar result is true for E − ˜ E . This proves (3.9) since we have ∗ ν ∧∗ η = i ν η = h ν, η i for any 1-form η .Let us next prove that (( W − W ) Y | Y ) = 0 . We have(( W − W ) Y | Y ) = ( W Y | Y ) − ( Y | W ∗ Y )= ( Y | ( P − k ) Y ) − (( P − k ) Y | Y )= ( Y | ( P − k ) Y ) − (( P − k )( Y − ˜ Y ) | Y ) − (( P − k ) ˜ Y | Y )= ( Y | ( P − k ) Y ) − ( Y − ˜ Y | ( P − k ) Y ) − (( P − k ) ˜ Y | Y ) . NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 13
In the last step, the boundary term arising from the integration by partsvanishes because of (3.9). We obtain(( W − W ) Y | Y ) = ( ˜ Y | ( P − k ) Y ) − (( P − k ) ˜ Y | Y )= − ( ˜ Y | W ∗ Y ) + ( W ˜ Y | Y ) = 0 . Now (3.8) will follow if we can prove that(( W − W ) Y | Y ) = (( Q − Q ) Z | Y ) . To show this, we recall that Q j in Lemma 3.1 has the form Q j = W j ( P + k ) − ( P − k ) W tj − W j W tj . Then(( W − W ) Y | Y ) = (( W − W )( P + k − W t ) Z | Y )= (( Q + ( P − k ) W t ) Z | Y ) − (( Q + ( P − k ) W t + W ( W t − W t )) Z | Y )= (( Q − Q ) Z | Y ) + ( W t Z | ( P − k ) Y ) − ( W t Z | ( P − k ) Y ) − (( W t − W t ) Z | W ∗ Y ) . Here, we used that W t = W t on ∂M by Theorem 3.3 so there are noboundary terms. Now (3.8) follows by using the identity − W ∗ Y = ( P − k ) Y in the last term.Finally, we show how it is possible to arrange that (3.7) holds. Thedefinition of admissible manifolds allows to find (upon enlarging ( M , g ) ifnecessary) a connected admissible manifold ( ˜ M , g ) such that M ⊂⊂ ˜ M ⊂⊂ T. If ˜ M is not required to be admissible then any choice ˜ M ⊃⊃ M will do.By the condition Λ = Λ and by Theorem 3.3, we may extend ε j and µ j smoothly to ˜ M so that ε = ε and µ = µ in ˜ M r M , ε j and µ j have positive real parts in ˜ M , and further for some constants ε , µ one has ε = ε = ε and µ = µ = µ near ∂ ˜ M .Let now Z , Y , Y be smooth graded forms in ˜ M satisfying the conditionsin the statement of the lemma in ˜ M . Since the restrictions to M satisfy thesame conditions in M , we have (3.8) in the set M . However, Q = Q in˜ M r M , so (3.8) remains valid in ˜ M . This proves the lemma upon replacing M with ˜ M . (cid:3) Norm estimates and uniqueness
In this section let ( M , g ) be a compact ( n − T = R × M with metric g = c ( e ⊕ g ), where e is theEuclidean metric on R and c is any smooth positive function in T satisfying c ( x , x ′ ) = 1 when | x | is large . Here and below, we write x for the Euclidean coordinate and x ′ for coor-dinates on M . The Laplace-Beltrami operators on ( T, g ) and ( M , g ) aredenoted by ∆ = ∆ g and ∆ x ′ = ∆ g , respectively. We will use the L space L ( T ) = L ( T, dV g ) and the Sobolev spaces H s ( T ). NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 14 If δ ∈ R , define the weighted norms k u k L δ ( T ) = kh x i δ u k L ( T ) , k u k H sδ ( T ) = kh x i δ u k H s ( T ) . Let L δ ( T ) and H sδ ( T ) be the corresponding spaces. We also consider thespaces H ( T ) = { u ∈ L ( T ) ; u ∈ H ([ − R, R ] × M ) for all R > } and H δ, ( T ) = { u ∈ H δ ( T ) ; u | R × ∂M = 0 } ,H , ( T ) = { u ∈ H ( T ) ; u | R × ∂M = 0 } The construction of complex geometrical optics solutions in [4] and [11]is based on limiting Carleman weights. It is shown in [4] that the function ϕ ( x ) = x is a natural limiting Carleman weight in ( T, g ). We consider theconjugated Helmholtz operator e τϕ ( − ∆ − k + q ) e − τϕ . The following result gives a norm estimate, corresponding to the Carlemanestimate in [4, Theorem 4.1], and a uniqueness result for this operator.
Proposition 4.1.
Let k ≥ δ > /
2, and let q be a potentialsatisfying h x i δ q ∈ L ∞ ( T ). There exists τ ≥ | τ | ≥ τ and τ + k / ∈ Spec( − ∆ x ′ ) , then for any f ∈ L δ ( T ) there is a unique solution u ∈ H − δ, ( T ) of theequation e τx ( − ∆ − k + q ) e − τx u = f in T. (4.1)Further, u ∈ H − δ ( T ), and the solution satisfies the estimates k u k H s − δ ( T ) ≤ C | τ | s − k f k L δ ( T ) , ≤ s ≤ , with C independent of τ and f .For the proof, we first claim that it is enough to consider the case where c ≡
1. To see this, note that if g = c ˜ g where ˜ g = e ⊕ g , one has the identity c n +24 ( − ∆ g − k + q )( c − n − v )= ( − ∆ ˜ g − k + h k (1 − c ) + cq − c n +24 ∆ g ( c − n − ) i ) v. (4.2)Consequently, u solves (4.1) iff v = c n − u solves e τx ( − ∆ ˜ g − k + ˜ q ) e − τx v = c n +24 f. Here ˜ q = k (1 − c ) + cq − c n +24 ∆ g ( c − n − ) is another potential such that h x i δ ˜ q ∈ L ∞ ( T ), since c = 1 for | x | large. This reduction shows thatProposition 4.1 will follow from the special case where c ≡ c ≡ q ≡
0. Then g has the form g ( x ) = (cid:18) g ( x ′ ) (cid:19) , and the equation (4.1) may be written as( − ∂ + 2 τ ∂ − τ − k − ∆ x ′ ) u = f. (4.3) NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 15
We are looking for a solution u with u | R × ∂M = 0. This motivates thepartial eigenfunction expansions along the transversal manifold: u ( x , x ′ ) = ∞ X l =0 ˜ u ( x , l ) φ l ( x ′ ) , f ( x , x ′ ) = ∞ X l =0 ˜ f ( x , l ) φ l ( x ′ )where φ l are the eigenfunctions of − ∆ x ′ on M , satisfying − ∆ x ′ φ l = λ l φ l in M and φ l | ∂M = 0.Inserting the expansions of u and f into (4.3) results in the equations( − ∂ + 2 τ ∂ − τ − k + λ l )˜ u ( x , l ) = ˜ f ( x , l ) . (4.4)These are second order ODE for the partial Fourier coefficients. To solvethem, we will use the following simple result on solutions of linear ODEinvolving Agmon type weights. Lemma 4.2. If µ = a + ib where a, b are real, a = 0, consider the equation u ′ − µu = f in R . (4.5)There is a unique solution u = S µ f ∈ S ′ ( R ) for any f ∈ S ′ ( R ). One has S µ : L δ ( R ) → L δ ( R ) if δ ∈ R , and also the norm estimates k S µ f k L δ ( R ) ≤ C | a | k f k L δ ( R ) , | a | ≥ δ ∈ R , k S µ f k L − δ ( R ) ≤ C k f k L δ ( R ) , a = 0 and δ > / . The constant C only depends on δ . Proof.
We take Fourier transforms in (4.5) and observe that for f ∈ S ′ ( R ),there is a unique solution u = S µ f ∈ S ′ ( R ) given by u = F − n m ( ξ ) ˆ f ( ξ ) o where m ( ξ ) = ( iξ − µ ) − . The condition a = 0 implies that m is a smoothfunction which satisfies k m ( k ) k L ∞ ≤ k ! | a | − ( k +1) , k = 0 , , , . . . . Thus, for any δ ∈ R we have m ˆ f ∈ H δ if ˆ f ∈ H δ , which implies S µ f ∈ L δ .If | a | ≥ k m ˆ f k H δ ≤ C δ | a | − k ˆ f k H δ and k S µ f k L δ ≤ C δ | a | − k f k L δ .We now assume f ∈ L δ for δ > /
2. If a >
0, the solution to (4.5) isgiven by S µ f ( x ) = − Z ∞ x f ( t ) e − µ ( t − x ) dt. This has the estimate | S µ f ( x ) | ≤ Z ∞ x | f ( t ) | dt ≤ (cid:18)Z ∞ x h t i − δ dt (cid:19) / k f k L δ ≤ C δ k f k L δ since δ > /
2. Thus one has k S µ f k L − δ ≤ C δ k f k L δ again since δ > /
2. Asimilar argument gives the result if a < (cid:3) NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 16
Proof of Proposition 4.1.
As argued above, we may assume c ≡
1. Let usalso first take q ≡
0. Then we are looking for solutions to the equation (4.3).Let 0 < λ ≤ λ ≤ . . . be the Dirichlet eigenvalues on − ∆ x ′ in M , andlet φ l ∈ H ( M ) be the corresponding eigenfunctions normalized so that { φ l } ∞ l =1 is an orthonormal basis for L ( M ). If u ( x , · ) ∈ L ( M ) we write˜ u ( x , l ) = Z M u ( x , · ) φ l dV g . For uniqueness, let u ∈ H , ( T ) be a solution of (4.3) with f = 0. Thismeans that for all ψ ∈ C ∞ c ( T int ), Z T u ( − ∂ − τ ∂ − τ − k − ∆ x ′ ) ψ dV = 0 . We choose ψ ( x , x ′ ) = χ ( x ) φ lj ( x ′ ) where χ ∈ C ∞ c ( R ) and φ lj ∈ C ∞ c ( M int0 )with φ lj → φ l in H ( M ) as j → ∞ . Since T = R × M , we have Z R (cid:18)Z M u ( x , · ) φ lj dV g (cid:19) ( − ∂ − τ ∂ − τ − k ) χ ( x ) dx + Z R (cid:18)Z M u ( x , · )( − ∆ x ′ φ lj ) dV g (cid:19) χ ( x ) dx = 0 . One has − ∆ x ′ φ lj → λ l φ l in H − ( M ) as j → ∞ . Since u ( x , · ) ∈ H ( M )for a.e. x , we have the limits as j → ∞ Z M u ( x , · ) φ lj dV g → ˜ u ( x , l ) , Z M u ( x , · )( − ∆ x ′ φ lj ) dV g → λ l ˜ u ( x , l ) , which are valid for a.e. x and for all l . Dominated convergence implies( − ∂ + 2 τ ∂ − τ − k + λ l )˜ u ( x , l ) = 0 in R for all l . By taking Fourier transforms in the x variable we obtain( ξ + 2 iτ ξ − τ − k + λ l )ˆ u ( ξ , l ) = 0 , with ˆ u the Fourier transform of ˜ u with respect to x . The symbol ξ +2 iτ ξ − τ − k + λ l is never zero because of the condition τ + k / ∈ Spec( − ∆ x ′ ).This implies that u ≡ τ >
0, the case with negative τ being analogous. We start by writing (4.4),where l is fixed, in the form − (cid:2) ( ∂ − τ ) − ( λ l − k ) (cid:3) ˜ u = ˜ f . (4.6)This equation can be factored into first order equations, where the factor-ization will depend on the sign of λ l − k .If λ l ≥ k , then (4.6) can be written in the form − ( ∂ − τ + p λ l − k )( ∂ − τ − p λ l − k )˜ u = ˜ f . Since ˜ f ∈ L δ and τ = p λ l − k by assumption, Lemma 4.2 implies thatthere is a solution˜ u ( · , l ) = − S τ + √ λ l − k S τ − √ λ l − k ˜ f ( · , l ) . NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 17
On the other hand, if λ l < k , then (4.6) takes the form − ( ∂ − τ + i p k − λ l )( ∂ − τ − i p k − λ l )˜ u = ˜ f and since τ ≥ u ( · , l ) = − S τ + i √ k − λ l S τ − i √ k − λ l ˜ f ( · , l ) . Lemma 4.2, with the trivial estimate k v k L − δ ≤ k v k L δ , implies that k ˜ u ( · , l ) k L − δ ≤ Cτ − k ˜ f ( · , l ) k L δ , λ l < k ,C ( τ + p λ l − k ) − k ˜ f ( · , l ) k L δ , λ l ≥ k ,C ( λ l − k ) − k ˜ f ( · , l ) k L δ , λ l > k + 4 τ . At this point C only depends on δ .We write, for N ≥ u N ( x , x ′ ) = N X l =1 ˜ u ( x , l ) φ l ( x ′ ) . (4.7)The objective is to show that as N → ∞ , u N converges in H − δ ( T ) to afunction u with u | R × ∂M = 0 and k u k H s − δ ( T ) ≤ Cτ s − k f k L δ ( T ) . If theseproperties hold, then since ˜ u satisfies (4.4) one has for u N ( − ∂ + 2 τ ∂ − τ − k − ∆ x ′ ) u N = N X l =1 ˜ f ( x , l ) φ l ( x ′ ) . Consequently, u will be the required solution of (4.3).Assume that τ ≥ τ ≥ k . If 0 ≤ s ≤
2, the estimates for ˜ u show that ∞ X l =1 λ sl k ˜ u ( · , l ) k L − δ ≤ Ck s τ − X λ l
0, then e τx ( − ∆ − k ) e − τx u = − qu. Since qu ∈ L δ ( T ), the uniqueness result for the free case shows that u = − G τ ( qu ) and k u k L − δ ( T ) ≤ Cτ − k qu k L δ ( T ) . If τ is sufficiently large, thisimplies u ≡
0. For existence of a solution to (4.1), we try u = G τ v where v ∈ L δ ( T ) should satisfy ( I + qG τ ) v = f in T. If τ is sufficiently large then qG τ is an operator on L δ ( T ) with norm ≤ / v = ( I + qG τ ) − f with k v k L δ ( T ) ≤ k f k L δ ( T ) .The required norm estimates follow from the estimates for G τ . (cid:3) Remark. If q ∈ L ∞ ( T ) is compactly supported, the preceding proof showsthat the claims in Proposition 4.1 remain true if one looks for a uniquesolution u ∈ H , ( T ) instead of u ∈ H − δ, ( T ). Definition.
We let G τ : L δ ( T ) → H − δ ∩ H − δ, ( T ) be the solution operatorgiven in Proposition 4.1 in the case q ≡ G τ tofunctions which may not decay in x . The proof of Proposition 4.1 involvesoperators S µ in two cases: where | Re( µ ) | is large and where Re( µ ) may beclose to 0. The latter case is problematic since good estimates may not be NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 19 available if there is no decay in x . However, we only need to apply G τ tofunctions of special form: the behaviour in x can be assumed to be like e iλx where λ >
0. The following result will be sufficient for our purposes.
Proposition 4.3.
Let k ≥ δ > / λ >
0, and supposethat h x i δ q ∈ L ∞ ( T ). There exists τ ≥ λ ) such thatwhenever | τ | ≥ τ and τ + k / ∈ Spec( − ∆ x ′ ) , then for any f = f + f where f ∈ L δ ( T ), f ∈ L − δ ( T ), and F x f ( · , x ′ ) has support in {| ξ | ≥ λ } for a.e. x ′ ∈ M , (4.9)there is a unique solution u ∈ H − δ, ( T ) of the equation e τx ( − ∆ − k + q ) e − τx u = f in T. Further, u ∈ H − δ ( T ), and the solution satisfies the estimates k u k H s − δ ( T ) ≤ C | τ | s − h k f k L δ ( T ) + k f k L − δ ( T ) i , ≤ s ≤ , with C independent of τ and f , f . Proof.
We follow the proof of Proposition 4.1, and may assume c ≡ c n +24 f = [ c n +24 f + χf ] + f where χ is compactly supported in x , so the term in brackets is in L δ ( T )) and τ >
0. Uniqueness is provedsimilarly as in Proposition 4.1, and that result also gives existence if f ≡ f = f .Assume first that q ≡
0. Let τ ≥ τ ≥ k , and recall that G τ is defined by G τ f ( x , x ′ ) = − X λ l
1. Since | m ( k ) λ ( ξ ) | ≤ C k λ − − k , we have k S µ w k L r ≤ C r,λ k w k L r . NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 20
Using the assumption (4.9), the last estimate for S µ , and Lemma 4.2, wehave for k ≤ λ l ≤ τ that k S τ + √ λ l − k S τ − √ λ l − k ˜ f ( · , l ) k L r ≤ C r τ + p λ l − k k S τ − √ λ l − k ˜ f ( · , l ) k L r ≤ C r,λ τ + p λ l − k k ˜ f ( · , l ) k L r . It follows that G τ maps L r ( T ) to L r ( T ) with norm ≤ Cτ − . The proof that G τ maps into H r ∩ H r, ( T ) with the right norm estimates is similar to thecorresponding part in Proposition 4.1.It remains to prove existence when f = f and q is a potential with h x i δ q ∈ L ∞ ( T ). We seek a solution u = G τ v , where v solves( I + qG τ ) v = f. Here f satisfies the support condition (4.9), but solving this equation byNeumann series involves multiplication with q which breaks the supportcondition. However, we obtain a solution v = f + ˜ v if ˜ v satisfies( I + qG τ )˜ v = − qG τ f. The right hand side is in L δ ( T ) since G τ maps f = f into L − δ ( T ). We maythen use the estimate in Proposition 4.1 to show that for large τ there is asolution ˜ v ∈ L δ ( T ) with k ˜ v k L δ ≤ Cτ − k f k L − δ . Thus, we obtain a solutionto the original equation having the form u = G τ f + G τ ˜ v. This satisfies k u k H s − δ ( T ) ≤ Cτ s − k f k L − δ ( T ) . (cid:3) Remark.
As in the remark after the proof of Proposition 4.1, if q ∈ L ∞ ( T )is compactly supported, the claims in Proposition 4.3 remain valid if onelooks for a unique solution u ∈ H , ( T ) instead of u ∈ H − δ, ( T ).5. Norm estimates for differential forms
The purpose in this section is to prove a counterpart of Proposition 4.3which applies to the Hodge Laplacian on differential forms. We will assumethat M and T are as in Section 4. For simplicity, we make the furtherassumptions that M is two dimensional (so that T has dimension 3) andthe conformal factor satisfies c ≡ T = Ω T ⊕ Ω T ⊕ Ω T ⊕ Ω T be the graded algebra of differentialforms. If U is in Ω T , as in Section 3 we use the vector notation U = ( R ∗ S ∗ S R ) t (5.1)where R j , S j ∈ Ω j T ( j = 0 , − ∆ = dδ + δd be the Hodge Laplacian on Ω T , and write − ∆ x ′ = d x ′ δ x ′ + δ x ′ d x ′ for the Hodge Laplacian on Ω M . We will sometimes write − ∆ jx ′ for − ∆ x ′ acting on j -forms. Similarly to Section 4, the proof of normestimates for the conjugated Laplacian will require an orthonormal set of NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 21 eigenvectors for − ∆ x ′ . In the case of 0-forms, we already used the orthonor-mal basis { φ l } ∞ l =1 of L ( M ), where − ∆ x ′ φ l = λ l φ l in M , φ l = 0 on ∂M . Here 0 < λ ≤ λ ≤ . . . are the eigenvalues of the Laplace-Beltrami operator,counted with multiplicity.In the case of 1-forms, one needs to make a choice of boundary condi-tions to fix the orthonormal basis. In view of the applications to Maxwellequations, the relative boundary conditions (see [23, Section 5.9]) will bethe right choice: there exists an orthonormal basis { ψ m } ∞ m =1 of L (Ω M )of real valued forms such that − ∆ x ′ ψ m = µ m ψ m in M , tψ m = t ( δ x ′ ψ m ) = 0 , (5.2)with 0 ≤ µ ≤ µ ≤ . . . the eigenvalues of − ∆ x ′ acting on 1-forms.We define Sobolev spaces with relative boundary values: H s R (Ω T ) = { u ∈ H s (Ω T ) ; u | ∂T = 0 } ( s > / ,H s R (Ω T ) = { u ∈ H s (Ω T ) ; tu = t ( δu ) = 0 on ∂T } ( s > / , and H s R (Ω T ) = { u ∈ H s (Ω T ) ; u has the form (5.1) and R j , S j ∈ H s R (Ω j T ) } . We say that u is in L δ (Ω T ) (respectively H sδ (Ω T )) if h x i δ u ∈ L (Ω T )(respectively h x i δ u ∈ H s (Ω T )). These spaces have the norms k u k L δ (Ω T ) = kh x i δ u k L (Ω T ) , k u k H sδ (Ω T ) = kh x i δ u k H s (Ω T ) . Also, u is in H sδ, R (Ω T ) iff h x i δ u ∈ H s R (Ω T ).We may now state the norm estimates and uniqueness result for gradedforms. Proposition 5.1.
Let k ≥ δ > / λ >
0, and supposethat Q : L − δ (Ω T ) → L δ (Ω T ) is a bounded linear operator. There exists τ ≥ | τ | ≥ τ and τ + k / ∈ Spec( − ∆ x ′ ) ∪ Spec( − ∆ x ′ ) , and whenever F = F + F where F ∈ L δ (Ω T ), F ∈ L − δ (Ω T ), and F isof the form F ( x ) = w ( x ) ˜ F with w a scalar function andsupp( ˆ w ) ⊆ {| ξ | ≥ λ } , ˜ F ∈ L ∞ (Ω T ) with ∇ ∂ ˜ F = 0 , then there is a unique solution U ∈ H − δ, R (Ω T ) of the equation e τx ( − ∆ − k + Q ) e − τx U = F in T. The solution satisfies the estimates k U k H s − δ (Ω T ) ≤ C | τ | s − h k F k L δ (Ω T ) + k F k L − δ (Ω T ) i , ≤ s ≤ , with C independent of τ and F , F . NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 22
Proof.
Assume τ >
0, and first consider the case where Q = 0. If v is a0-form on T , we have already observed that e τx ( − ∆ − k ) e − τx v = ( − ∂ + 2 τ ∂ − τ − k − ∆ x ′ ) v. (5.3)If η is a 1-form in T , we write η = η dx + η ′ where η = h η, dx i and where η ′ = η dx + η dx is a 1-form on M depending on the parameter x . Adirect computation in normal coordinates, using the identities in Section 2,the fact that g ( x , x ′ ) = (cid:16) g ( x ′ ) (cid:17) , and the identity δη = − ∂ η + δ x ′ η ′ ,implies that − ∆( η dx ) = ( − ∆ η ) dx , − ∆ η ′ = − ∆ x ′ η ′ − ( ∂ η ) dx − ( ∂ η ) dx . Thus, replacing η by e − τx η and using that ∇ ∂ η ′ = ( ∂ η ) dx + ( ∂ η ) dx ,we obtain e τx ( − ∆ − k ) e − τx η = (cid:2) ( − ∂ + 2 τ ∂ − τ − k − ∆ x ′ ) η (cid:3) dx ++ ( −∇ ∂ + 2 τ ∇ ∂ − τ − k − ∆ x ′ ) η ′ . (5.4)The formulas (5.3) and (5.4) give explicit expressions for the conjugatedHelmholtz operator acting on 0-forms and 1-forms. Now ∗ commutes with e τx ( − ∆ − k ) e − τx since it commutes with − ∆, so we have a correspondingexpression for e τx ( − ∆ − k ) e − τx acting on graded forms written as (5.1).Let U be as in (5.1), and let F = ( F ∗ G ∗ G F ) t . Write R = R dx + ( R ) ′ , and similarly for S , F , G . The equation e τx ( − ∆ − k ) e − τx U = F in T can be written in terms of components as( − ∂ + 2 τ ∂ − τ − k − ∆ x ′ ) (cid:26) R S = (cid:26) F G , (5.5)( − ∂ + 2 τ ∂ − τ − k − ∆ x ′ ) (cid:26) R S = (cid:26) F G , (5.6)( −∇ ∂ + 2 τ ∇ ∂ − τ − k − ∆ x ′ ) (cid:26) ( R ) ′ ( S ) ′ = (cid:26) ( F ) ′ ( G ) ′ . (5.7)The existence and uniqueness of solutions to (5.5) and (5.6) follows fromProposition 4.3.For the last two equations, we express ( F ) ′ and ( G ) ′ in terms of theeigenvectors (5.2) as( F ) ′ ( x , x ′ ) = ∞ X m =1 ] ( F ) ′ ( x , m ) ψ m ( x ′ ) , ( G ) ′ ( x , x ′ ) = ∞ X m =1 ] ( G ) ′ ( x , m ) ψ m ( x ′ ) . NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 23
We look for ( R ) ′ and ( S ) ′ in a similar form. Then (5.7) is equivalent withthe following equations for the partial Fourier coefficients:( − ∂ + 2 τ ∂ − τ − k + µ m ) ( ] ( R ) ′ ( x , m ) ] ( S ) ′ ( x , m ) = ( ] ( F ) ′ ( x , m ) ] ( G ) ′ ( x , m ) . (5.8)Since τ + k / ∈ Spec( − ∆ x ′ ), we may use the method in Propositions 4.1and 4.3 to solve (5.7).More precisely, if U ∈ H − δ, R (Ω T ) and the right hand sides in (5.7) arezero, then the relative boundary conditions imply that ( − ∆ x ′ ( R ) ′ | ψ m ) = µ m (( R ) ′ | ψ m ) and one obtains for all m ( − ∂ + 2 τ ∂ − τ − k + µ m )(( R ) ′ | ψ m ) = 0 . Thus (( R ) ′ | ψ m ) = 0 for all m , showing that ( R ) ′ = 0. The same argumentapplies to ( S ) ′ . Existence follows by solving (5.8) as in Propositions 4.1and 4.3 and by writing ( R ) ′ and ( S ) ′ in terms of the Fourier coefficients.We have given the proof in the case Q ≡
0. However, the case where Q is bounded operator L − δ ( T ) → L δ ( T ) is completely analogous to thecorresponding parts of Propositions 4.1 and 4.3. (cid:3) Construction of solutions
In this section we present a construction of complex geometrical opticssolutions to the various Schr¨odinger, Dirac, and Maxwell equations whichwere introduced in Section 3.Let ( M , g ) be a simple 2-manifold, and let ( ˜ M , g ) be another simple2-manifold with M ⊆ ˜ M int0 . Write ˜ T = R × ˜ M , T = R × M , and T int = R × M int0 for the various cylinders. Assume that ˜ T is equipped withthe Riemannian metric g ( x , x ′ ) = (cid:18) g ( x ′ ) (cid:19) . (6.1)The following result provides the solutions which will be used in the integralidentity of Lemma 3.2 to recover the coefficients. Part (a) corresponds to asolution for the Maxwell system, and part (b) gives a solution to the Diracsystem. We write − ∆ x ′ for the Hodge Laplacian in ( M , g ). Theorem 6.1.
Let (
M, g ) ⊂⊂ ( T, g ) be a compact manifold with boundary.Assume that ε and µ are coefficients in M satisfying (1.2), (1.3), (3.1). Let p be a point in ˜ M r M , and let ( r, θ ) be polar normal coordinates in ˜ M with center p .There exists τ ≥ τ with | τ | ≥ τ and τ + k / ∈ Spec( − ∆ x ′ ) ∪ Spec( − ∆ x ′ ) , and for any constants s , t ∈ R , the following statements hold:(a) For any constant λ > χ = χ ( θ ), thereexists a solution to ( − ∆ − k + Q ) Z = 0 in M such that one has in M ( P − k + W ) Y = 0 , Y = ( P + k − W t ) Z,Y = Y = 0 , NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 24 where Z has the form Z = e − τ ( x + ir ) | g | − / e iλ ( x + ir ) χ ( θ ) s t ∗ + R (6.2)and k R k L (Ω M ) ≤ C | τ | − where C is independent of τ .(b) There exists a solution to ( P − k + W ∗ ) Y = 0 in M of the form Y = e − τ ( x + ir ) | g | − / s − is dx ∧ drt ∗ it ∗ dx ∧ dr + R (6.3)where k R k L (Ω M ) ≤ C | τ | − with C independent of τ .To prove this, we begin by considering the Schr¨odinger equation in ( T, g ),where k > Q is a smooth potential with compact supportin T int . Following [4, Section 5], we wish to construct a solution to( − ∆ − k + Q ) Z = 0 in T (6.4)by using a WKB ansatz with complex phase function, having the form Z = e − τρ ( A + R ) . (6.5)Here ρ = ϕ + iψ is a complex weight, where ϕ ( x ) = x is the limitingCarleman weight. Also, τ > A ∈ Ω T is an amplitude,and R ∈ H − δ (Ω T ) is a correction term where δ > / l T , d τ = e τρ de − τρ = d − τ dρ ∧ ,δ τ = e τρ δe − τρ = δ + ( − l +1 τ ∗ dρ ∧ ∗ . The conjugated Hodge Laplacian is then given by − ∆ τ = e τρ ( − ∆) e − τρ = d τ δ τ + δ τ d τ . The next result gives explicit expressions for ∆ τ in terms of powers of τ .Here ∇ is the Levi-Civita connection and ∇ ρ is the metric gradient of ρ . Lemma 6.2. If u is a 0-form or 1-form, then∆ τ u = τ h dρ, dρ i u − τ [2 ∇ ∇ ρ u + (∆ ρ ) u ] + ∆ u, ∆ τ ∗ u = ∗ (cid:8) τ h dρ, dρ i u − τ [2 ∇ ∇ ρ u + (∆ ρ ) u ] + ∆ u (cid:9) . Proof.
The first identity for 0-forms is a straightforward computation. If u is a 1-form, we have − ∆ τ u = ( d − τ dρ ∧ )( δu + τ h dρ, u i ) + ( δ − τ ∗ dρ ∧ ∗ )( du − τ u dρ )= − ∆ u + τ [ d h dρ, u i − ( δu ) dρ − δ ( dρ ∧ u ) − ∗ dρ ∧ ∗ du ] − τ h dρ, dρ i u. The identities in Section 2 and a computation in normal coordinates showthat d h dρ, u i − ( δu ) dρ − δ ( dρ ∧ u ) − ∗ dρ ∧ ∗ du = 2 ∇ ∇ ρ u + (∆ ρ ) u. NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 25
This proves the first identity for 1-forms. The Hodge star commutes with∆ τ since it commutes with ∆, and second identity follows. (cid:3) We write A = (cid:0) A ∗ B ∗ B A (cid:1) t where A , B are 0-forms and A , B are 1-forms. Using Lemma 6.2, the WKB construction for solutionsto (6.4) having the form (6.5) results in the following equations in T : h dρ, dρ i = 0 , (6.6)2 ∇ ∇ ρ A j + (∆ ρ ) A j = 0 ( j = 0 , , (6.7)2 ∇ ∇ ρ B j + (∆ ρ ) B j = 0 ( j = 0 , , (6.8) e τρ ( − ∆ − k + Q ) e − τρ R = (∆ + k − Q ) A. (6.9)We follow the construction in [4, Section 5] and employ special coordi-nates to solve these equations. Considering the real and imaginary partsseparately, the first equation (6.6) reads | dψ | = | dϕ | , h dψ, dϕ i = 0 . Recall that ϕ ( x ) = x . Choose a point p ∈ ˜ M r M , and let ( r, θ ) be polarnormal coordinates in ( ˜ M , g ) with center p . Then r is smooth in M , andwe obtain a solution ψ by setting ψ ( x , r, θ ) = r. Note that in the ( x , r, θ ) coordinates one has in Tg ( x , r, θ ) = m ( r, θ ) where m = | g | is smooth. We write ∂ = 12 (cid:18) ∂∂x + i ∂∂r (cid:19) . The following result gives solutions to the transport equations (6.7)–(6.8).
Lemma 6.3.
Assume the above notations.(1) If a is a 0-form, then 2 ∇ ∇ ρ a + (∆ ρ ) a = 0 iff ∂ ( | g | / a ) = 0.(2) If η is a 1-form, then 2 ∇ ∇ ρ η + (∆ ρ ) η = 0 iff η = a dx + a r dr + a θ dθ with ∂ ( | g | / a ) = ∂ ( | g | / a r ) = ∂ ( | g | − / a θ ) = 0. Proof.
We have ρ = x + ir, ∇ ρ = 2 ∂, ∆ ρ = ∂ log | g | . The equation for 0-forms is 4 ∂a + (cid:0) ∂ log | g | (cid:1) a = 0, which proves the firstpart. For the second part, the form of g shows thatΓ l k = 0 , Γ lrk = (cid:26) ∂ r (log | g | ) , k = l = θ, , otherwise . NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 26
Consequently ∇ ∂ dx j = 0 for all j and ∇ ∂ r dx = ∇ ∂ r dr = 0, ∇ ∂ r dθ = − ∂ r (log | g | ) dθ . The result follows by noting that ∇ ∇ ρ ( a dx + a r dr + a θ dθ )= (2 ∂a ) dx + (2 ∂a r ) dr + (2 ∂a θ − i ∂ r (log | g | ) a θ ) dθ where i ∂ r (log | g | ) = ∂ (log | g | ). (cid:3) We are now ready to give the construction of complex geometrical opticssolutions to the Schr¨odinger equation.
Proposition 6.4.
Let ( M , g ) ⊂⊂ ( ˜ M , g ) be two simple 2-manifolds, andconsider the cylinders T = R × M and ˜ T = R × ˜ M equipped with themetric g given by (6.1). Let k ≥ δ > /
2, and let Q bea bounded linear operator L − δ (Ω T ) → L δ (Ω T ). There exists τ ≥ | τ | ≥ τ and τ + k / ∈ Spec( − ∆ x ′ ) ∪ Spec( − ∆ x ′ ) , and if p, λ, a , a , b , b are any parameters such that p is a point in ˜ M r M and λ > , ( r, θ ) are polar normal coordinates in ˜ M with center p,a l , b l are smooth functions in T of the form e iλx w ( x ′ ) , ( ∂ + i∂ r ) a l = ( ∂ + i∂ r ) b l = 0 in T , then the equation ( − ∆ − k + Q ) Z = 0 in T has a unique solution Z = e − τ ( x + ir ) | g | − / a b ∗ dx b ∗ a dx + R (6.10)where R ∈ H − δ, R (Ω T ). The remainder R satisfies k R k L − δ (Ω T ) ≤ C | τ | − with C independent of τ . Proof.
Take ρ = x + ir and A = | g | − / a , A = | g | − / a dx , and also B = | g | − / b , B = | g | − / b dx . It follows from the discussion above thatequations (6.6)–(6.8) are satisfied, and that Z solves ( − ∆ − k + Q ) Z = 0iff e τρ ( − ∆ − k + Q ) e − τρ R = F (6.11)where F = (∆ + k − Q ) A .The form of a l and b l implies that A ∈ L − δ (Ω T ), so QA ∈ L δ (Ω T ),and also that (∆ − k ) A = e iλx ˜ F ( x ′ ) ∈ L − δ (Ω T ) where ˜ F ∈ L ∞ (Ω T ), ∇ ∂ ˜ F = 0. The latter fact follows from the formula for ∆ acting on Ω T computed in the proof of Proposition 5.1. Thus we have a decomposition F = F + F as in Proposition 5.1, and that result shows that there is aunique solution R ∈ H − δ, R (Ω T ) to (6.11) with the required estimate if | τ | is large and outside a discrete set. (cid:3) NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 27
We now prove the main result on complex geometrical optics solutions.For part (a) we need to use the uniqueness of the solutions above to concludethat Y = Y = 0. Part (b) is in fact much easier since it is enough toconstruct solutions to a Dirac equation in M without worrying about thevanishing of scalar parts. Proof of Theorem 6.1(a).
Assume the conditions in Theorem 6.1, and ex-tend ε and µ smoothly as constants into ˜ T . Then Q satisfies the assumptionin Proposition 6.4, and that result guarantees the existence of a solution Z to ( − ∆ − k + Q ) Z = 0 in T of the form (6.10) with R ∈ H − δ, R ( T ) and k R k L − δ (Ω T ) ≤ C | τ | − . Setting Y = ( P + k − W t ) Z , Lemma 3.1 shows that Y solves ( P − k + W ) Y = 0 in T .The main point is to show that Y = Y = 0. For this we use an ideaappearing in [19]. By Lemma 3.1 we have ( − ∆ − k + Q ′ ) Y = 0. Looking atthe 0-form and 3-form parts and using the special form of Q ′ , the equationdecouples and we obtain the following equations in T :( − ∆ − k + q ) Y = 0 , ( − ∆ − k + q ) ∗ Y = 0 , where q and q are smooth potentials with compact support in M int , q = k − κ −
12 ∆ β + 14 h dβ, dβ i ,q = k − κ −
12 ∆ α + 14 h dα, dα i . Now, writing ρ = x + ir and Z = e − τρ ( A + R ), Y has the form Y = (( P + k − W t ) Z ) = e − τρ (cid:18)(cid:18) − i δ τ + k − W t (cid:19) ( A + R ) (cid:19) = e − τρ ( y + r ) . Here we have written y = − i δ τ A + kA ,r = − ( W t ( A + R )) − i δ τ R + kR . Since R ∈ H − δ, R (Ω T ) we see that r ∈ H − δ ( T ) and r | ∂T = 0.We will choose A , A so that y ≡ . (6.12)Then e − iτr r will be a solution in H − δ, ( T ) of the equation e τx ( − ∆ − k + q ) e − τx ( e − iτr r ) = 0 in T, and the uniqueness part in Proposition 4.1 will show that r = 0, so also Y = 0, if | τ | is sufficiently large. To obtain (6.12) we make the choices NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 28 A = | g | − / a , A = | g | − / a dx where a = iks τ e iλ ( x + ir ) χ ( θ ) , (6.13) a = 1 ik δ τ ( a dx ) . (6.14)This is consistent with Proposition 6.4 since ( ∂ + i∂ r ) a = 0 and since a = 1 ik ( − ∂ a + τ a ) = τ − iλik a (6.15)so also ( ∂ + i∂ r ) a = 0, and both a and a are of the form e iλx w ( x ′ ). Now(6.12) holds because y = − i δ τ A + kA = − i δ τ ( | g | − / a dx ) + k | g | − / ik δ τ ( a dx ) = 0 . We have established that Y = 0 given the choices (6.13)–(6.14). Asimilar computation for Y , with b = ikt τ e iλ ( x + ir ) χ ( θ ) ,b = 1 ik δ τ ( b dx ) , shows that Y = 0. Finally, we note that Z is of the form Z = e − τρ [ | g | − / (cid:0) a ∗ b dx ∗ b a dx (cid:1) t + R ]where a = s e iλ ( x + ir ) χ ( θ ) + O ( | τ | − ) by (6.15), and a = O ( | τ | − ). Wehave written O ( | τ | − ) for quantities whose L ( M ) norm is ≤ C | τ | − . Similarexpressions are true for b and b . This shows that Z has the required form(6.2) with k R k L (Ω M ) ≤ C | τ | − .(Note that a j and b j are mildly τ -dependent, but their W l, ∞ ( T ) normsare bounded uniformly in τ which implies that final constant C does notdepend on τ ). (cid:3) Proof of Theorem 6.1(b).
Again, assume the conditions in Theorem 6.1 andextend ε and µ smoothly as constants into ˜ T . Let ˆ Q be the potential inLemma 3.1. We look for a solution to ( − ∆ − k + ˆ Q ) Z = 0 in M of the form Z = e − τ ( x + ir ) | g | − / t ∗ dx s dx + R . Write ρ = x + ir and A = | g | − / (cid:0) t ∗ dx s dx (cid:1) t . Followingthe WKB construction, it is enough to solve e τρ ( − ∆ − k + ˆ Q )( e − τρ R ) = (∆ + k − ˆ Q ) A in M .
Define F ∈ L δ (Ω T ) with F = e − iτr (∆ + k − ˆ Q ) A in M and F = 0 in T r M , and let e − iτr R be a solution provided by Proposition 5.1 of theequation e τx ( − ∆ − k + ˆ Q )( e − τx [ e − iτr R ]) = F in T . This gives the requiredsolution Z in M satisfying k R k L (Ω M ) ≤ C | τ | − and k R k H (Ω M ) ≤ C . NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 29
We set Y = 1 τ ( P + k − ¯ W ) Z. By Lemma 3.1 this satisfies ( P − k + W ∗ ) Y = 0 in M , and Y = e − τρ (cid:18) i d τ − i δ τ + k − ¯ W (cid:19) ( τ − A + τ − R )= e − τρ (cid:20) − | g | − / i dρ ∧ ( s dx + t ∗ dx ) − | g | − / i ∗ dρ ∧ ∗ ( s dx − t ∗ dx ) + O ( | τ | − ) (cid:21) = e − τρ | g | − / is s dx ∧ drit ∗ − t ∗ dx ∧ dr + O ( | τ | − ) . Here O ( | τ | − ) denotes a quantity whose L (Ω M ) norm is ≤ C | τ | − . Theresult follows upon replacing s by − is and t by − it . (cid:3) Recovering the coefficients
We shall use the complex geometrical optics solutions constructed in The-orem 6.1 to prove Theorem 1.1. The first step is a reduction to the case wherethe conformal factor in the metric is equal to one. We write Λ = Λ g,ε,µ forthe admittance map in (
M, g ) with coefficients ε and µ . Lemma 7.1.
Let (
M, g ) be a compact Riemannian 3-manifold with smoothboundary, and let ε and µ satisfy (1.2)–(1.4). If c is any smooth positivefunction on M , then Λ cg,ε,µ = Λ g,c / ε,c / µ . Proof.
Follows by noting that ∗ cg u = c / − k ∗ g u for a k -form u , so thata pair ( E, H ) satisfies (1.1) with metric cg and coefficients ε and µ iff itsatisfies (1.1) with metric g and coefficients c / ε and c / µ . (cid:3) Proof of Theorem 1.1.
According to Lemma 3.2 we may assume that (3.7)holds and the identity (3.8) is valid. By the definition of admissible mani-folds, there are global coordinates x = ( x , x ′ ) such that g has the form g ( x ) = c ( x ) (cid:18) g ( x ′ ) (cid:19) . If Λ g,ε ,µ = Λ g,ε ,µ , then also Λ c − g,c / ε ,c / µ = Λ c − g,c / ε ,c / µ byLemma 7.1. This shows that we may also assume c ≡ τ is outside a discrete set and | τ | is sufficiently large,and if χ ( θ ) is a smooth function and λ > s and t are real numbers,then there exist Z and Y in H (Ω M ) satisfying the conditions in Lemma NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 30 Z = e − τ ( x + ir ) | g | − / e iλ ( x + ir ) χ ( θ ) s t ∗ + R ,Y = e τ ( x − ir ) | g | − / s is dx ∧ drt ∗ − it ∗ dx ∧ dr + R where k R j k L (Ω M ) ≤ C | τ | − with C independent of τ . In the second solu-tion, we used − r instead of r as the solution of the eikonal equation.By Lemma 3.2, these solutions satisfy the identity Z M h ( Q − Q ) Z , ¯ Y i dV = 0 . Letting τ → ∞ outside the discrete set and using the estimates for R j , weobtain in terms of the x = ( x , r, θ ) coordinates that Z M * ( Q − Q ) s t ∗ , s − is dx ∧ drt ∗ it ∗ dx ∧ dr + e iλ ( x + ir ) χ ( θ ) dx = 0 . Let q α and q β be the elements of Q − Q , interpreted as a 8 × , , q α = 12 ∆( α − α ) + 14 h dα , dα i − h dα , dα i − ω ( ε µ − ε µ ) ,q β = 12 ∆( β − β ) + 14 h dβ , dβ i − h dβ , dβ i − ω ( ε µ − ε µ ) . With the two choices ( s , t ) = (1 ,
0) and ( s , t ) = (0 , Q and Q in Lemma 3.1 shows that we obtain the two identities R M e iλ ( x + ir ) χ ( θ ) q α ( x ) dx = 0 , R M e iλ ( x + ir ) χ ( θ ) q β ( x ) dx = 0 . We extend q α and q β to be zero in T r M , where T ⊃⊃ M is as in thedefinition of admissible manifolds. Then the integrals above may be takenover T = R × M . Varying χ ( θ ), it follows that for all θ we have Z ∞ e − λr (cid:20)Z ∞−∞ e iλx q α ( x , r, θ ) dx (cid:21) dr = 0and similarly for q β . Now, since ( r, θ ) are polar normal coordinates in M ,the curves r ( r, θ ) are geodesics in M . Denoting the expression in brack-ets by f α ( r, θ ) and varying the point p in Theorem 6.1 and varying θ , weobtain that Z ∞ f α ( γ ( r )) exp (cid:20) − Z r λ ds (cid:21) dr = 0for all geodesics γ in M which begin and end at points of ∂M . This showsthe vanishing of the geodesic ray transform of the function f α with constant NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 31 attenuation − λ . For more details we refer to [4, Section 7]. In particular,the injectivity result given by Theorem 7.1 in [4] implies that f α ≡ λ which are sufficiently small. Thus Z ∞−∞ e iλx q α ( x , r, θ ) dx = 0for such λ and for all r and θ . Since q α is compactly supported in x , thePaley-Wiener theorem shows that q α ≡ M . We obtain q β ≡ M bythe exact same argument.We have arrived at the following two equations in M : − ∆( α − α ) − h d ( α + α ) , d ( α − α ) i + ω ( ε µ − ε µ ) = 0 , − ∆( β − β ) − h d ( β + β ) , d ( β − β ) i + ω ( ε µ − ε µ ) = 0 . Let u = ( ε /ε ) / and v = ( µ /µ ) / . Then ( α − α ) = log u , and theequations become − ∆(log u ) − ( ε ε ) − / h d ( ε ε ) / , d (log u ) i + ω ( ε µ − ε µ ) = 0 , − ∆(log v ) − ( µ µ ) − / h d ( µ µ ) / , d (log v ) i + ω ( ε µ − ε µ ) = 0 . Multiplying the first equation by ( ε ε ) / and the second by ( µ µ ) / , andusing that δ ( a ∇ w ) = − a ∆ w − h da, dw i , we see that u and v satisfy thesemilinear elliptic system δ ( ε du ) + ω ε µ ( u v − u = 0 ,δ ( µ dv ) + ω ε µ ( u v − v = 0 . The condition (3.7) ensures that one has u = 1 and v = 1 near ∂M . Also,the above equations imply that the pair (˜ u, ˜ v ) = (1 ,
1) is a solution of thesemilinear system in all of M . By Theorem B.1 unique continuation holdsfor this system, and we obtain u ≡ v ≡ M . This proves that ε ≡ ε and µ ≡ µ in M as required. (cid:3) We now prove Theorem 1.2. The treatment below follows [12]. LetΩ ⊆ R be a bounded open set with smooth boundary, and let ε and µ be symmetric positive definite (1 , e . The Maxwell equations (1.5) can be written as (cid:26) curl e ( ~E ) = iωµ ~H, curl e ( ~H ) = − iωε ~E. (7.1)Here curl e ( ~X ) = ( ∗ e d ~X ♭ ) ♯ with the flat and sharp operators taken with respect to e .For the vector fields ~E = ( E , E , E ) and ~H = ( H , H , H ), let E = ~E ♭ = E j dx j and H = ~H ♭ = H j dx j be the corresponding 1-forms. To write(7.1) in a form similar to (1.1), it is enough to find Riemannian metrics g ε and g µ so that ∗ e ( ε ~E ) ♭ = ∗ g ε E, ∗ e ( µ ~H ) ♭ = ∗ g µ H. These conditions will be satisfied if we choose in local coordinates g jkε = 1det( ε ) ε kj , g jkµ = 1det( µ ) µ kj . (7.2) NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 32
Then (7.1) is equivalent with (cid:26) ∗ g µ dE = iωH, ∗ g ε dH = − iωE. (7.3)We now use the assumption that ε and µ are in the same conformal class,so that µ = α ε for some smooth positive function α on Ω. This allows todefine a metric g on Ω by g = α g ε = α − g µ . Since ∗ cg u = c − / ∗ g u for a 2-form u , (7.3) is equivalent with (cid:26) ∗ g dE = iωαH, ∗ g dH = − iωα − E. (7.4)This is of the form (1.1), and further the tangential boundary condition tE = f is of the same form as (1.6). Since (7.4) is equivalent with (1.5),Theorem A.1 implies that for ω outside a discrete set of frequencies thesystem (1.5)–(1.6) is uniquely solvable for a given boundary value ~f . Theadmittance map Λ for (1.5) reduces to the map Λ g,α − ,α .Given this reduction, it is easy to prove the second main result of thepaper. Proof of Theorem 1.2.
Upon interpreting ε and µ as (0 , ε = det( ε ) g − ε , µ = det( µ ) g − µ . From the assumption in the theorem, we know that there is an admissiblemetric g and smooth positive functions c j , ˜ c j on Ω for which g ε j = c j g, g µ j = ˜ c j g. Using these formulas in (7.3), the Maxwell equations in Ω with coefficients ε j and µ j are equivalent with (cid:26) ∗ g dE = iω ˜ c j H, ∗ g dH = − iωc j E. Since the admittance maps for the Maxwell equations in Ω coincide, it followsthat Λ g,c , ˜ c = Λ g,c , ˜ c . From Theorem 1.1 we obtain c = c and ˜ c = ˜ c ,which implies that g ε = g ε and g µ = g µ . By (7.2)1det( ε ) ε = 1det( ε ) ε , µ ) µ = 1det( µ ) µ . Taking determinants gives that det( ε ) = det( ε ) and det( µ ) = det( µ ).Consequently ε ≡ ε and µ ≡ µ . (cid:3) Remark.
Note that in the setting of (1.1), if ε and µ are real valued, thena conformal scaling of the metric would reduce (1.1) to a system of the form(7.4). Therefore, in Sections 3 and 6 it would be enough to consider the casewhere µ = ε − , which would simplify some of the arguments slightly. NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 33
Appendix A. Wellposedness theory
Let (
M, g ) be a compact oriented Riemannian 3-manifold with smoothboundary ∂M . Consider the Maxwell equations (cid:26) ∗ dE = iωµH in M, ∗ dH = − iωεE in M, (A.1)with the tangential boundary condition tE = f on ∂M. (A.2)Here we assume that ε and µ are complex functions in C k ( M ) whosereal parts are positive in M , and ω is a complex number. To describe theboundary condition in more detail, we introduce the Div-spaces H s Div ( M ) = { u ∈ H s Ω ( M ) ; Div( tu ) ∈ H s − / ( ∂M ) } ,T H s Div ( ∂M ) = { f ∈ H s Ω ( ∂M ) ; Div( f ) ∈ H s ( ∂M ) } . There are Hilbert spaces with norms k u k H s Div ( M ) = k u k H s Ω ( M ) + k Div( tu ) k H s − / ( ∂M ) , k f k T H s Div ( ∂M ) = k f k H s Ω ( ∂M ) + k Div( f ) k H s ( ∂M ) . It is easy to see that t ( H s Div ( M )) = T H s − / ( ∂M ) for s > / Theorem A.1.
Let ε, µ ∈ C k ( M ), k ≥
2, be functions with positive realparts. There is a discrete subset Σ of C such that if ω is outside this set,then one has a unique solution ( E, H ) ∈ H k Div ( M ) × H k Div ( M ) of (A.1)–(A.2)given any f ∈ T H k − / ( ∂M ). The solution satisfies k E k H k Div ( M ) + k H k H k Div ( M ) ≤ C k f k T H k − / ( ∂M ) with C independent of f . In particular, if ε, µ ∈ C ∞ ( M ) and f ∈ Ω ( ∂M ),then there is a unique solution ( E, H ) ∈ Ω ( M ) × Ω ( M ).The existence of a solution will be proved by the well-known variationalmethod as in [3], [13]. We proceed to describe this method. The first step isto solve for H in the first line of (A.1) and to substitute this on the secondline, which leads to the second order equation δ ( µ − dE ) − ω εE = 0 . However, this equation does not imply the divergence condition δ ( εE ) = 0which is necessary for solutions of (A.1). To make up for this, we considerthe modified equation δ ( µ − dE ) + sε − dδ ( εE ) − ω εE = 0where s is a positive real number. The condition δ ( εE ) = 0 will follow laterfrom the equation sδ ( ε − d [ δ ( εE )]) − ω δ ( εE ) = 0 , which is obtained by applying δ to the earlier equation.To connect the present situation to boundary value problems for theHodge Laplacian, we write e = εE and note that e should satisfy δ ( µ − d ( ε − e )) + sε − dδe − ω e = 0 . (A.3) NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 34
Taking the L inner product of this with ¯ ε ˜ e for a 1-form ˜ e , and assumingthe relative boundary conditions (see [23, Section 5.9]) te = t ˜ e = 0 and t ( δe ) = t ( δ ˜ e ) = 0or the absolute boundary conditions t ( ∗ e ) = t ( ∗ ˜ e ) = 0 and t ( δ ∗ e ) = t ( δ ∗ ˜ e ) = 0 , we end up with the following bilinear form for solving the Maxwell equations. Definition. If e, ˜ e ∈ H b Ω ( M ) (where b = R or b = A ), we define B ( e, ˜ e ) = ( µ − d ( ε − e ) | d (¯ ε ˜ e )) + s ( δe | δ ˜ e ) . Here we have used the spaces H R Ω ( M ) = { u ∈ H Ω ( M ) ; tu = 0 } ,H A Ω ( M ) = { u ∈ H Ω ( M ) ; t ( ∗ u ) = 0 } . If k ≥ H kR Ω ( M ) = { u ∈ H k Ω ( M ) ; tu = t ( δu ) = 0 } and H kA Ω ( M ) = { u ∈ H k Ω ( M ) ; t ( ∗ u ) = t ( δ ∗ u ) = 0 } .One defines weak solutions of (A.3) in the usual way. The main point isthe following solvability result. Proposition A.2.
Let ε and µ be functions in C ( M ) with positive realparts, and let s be a positive real number. There is a discrete set Σ s in C such that if ω is outside this set, then for any f ∈ ( H b Ω ( M )) ′ the equation δ ( µ − d ( ε − e )) + sε − dδe − ω e = f (A.4)has a unique solution e ∈ H b Ω ( M ) ( b = R or A ). One has k e k H ≤ C k f k ( H b ) ′ . Proof.
Clearly B is a sesquilinear form on H b Ω ( M ) with | B ( e, ˜ e ) | ≤ C k e k H k ˜ e k H . We may write B ( e, e ) = B ( e, e ) + B ( e, e ) where B ( e, e ) = ( µ − de | de ) + s ( δe | δe ) , and B is a sesquilinear form such that | B ( e, e ) | ≤ C k e k L k e k H . It followsthat Re B ( e, e ) ≥ c k de k L + s k δe k L − C k e k L k e k H . We now invoke a Poincar´e inequality for 1-forms with relative or absoluteboundary values: by [23, Section 5.9] one has k u k H ≤ C ( k u k L + k du k L + k δu k L ) , u ∈ H b Ω ( M ) . It follows that Re B ( e, e ) ≥ c k e k H − C k e k L . We have proved that for some C >
0, the sesquilinear form B + d isbounded and coercive on H b Ω ( M ) for any d ∈ L ∞ ( M ) with Re( d ) ≥ C . By the Lax-Milgram theorem there is a bounded linear operator T :( H b Ω ) ′ → H b Ω which maps f to the unique solution e of (A.4) where NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 35 − ω is replaced by the constant C = C / min x ∈ M Re( ε ). Now, e solves(A.4) iff ( I − ( C + ω ) T ) e = T f.
The last equation has a unique solution iff either ( C + ω ) − / ∈ Spec( T ) or ω = − C . The operator T : H b Ω → H b Ω is compact by the compactembedding H b Ω → L Ω , and 0 / ∈ Spec( T ), so Spec( T ) is discrete. Thenthe set Σ s = { ω ∈ C r {± i p C } ; ( C + ω ) − ∈ Spec( T ) } is also discrete and (A.4) is uniquely solvable for any ω / ∈ C r Σ s . (cid:3) Given the last result, higher order regularity for solutions follows in asimilar way as for the Hodge Laplacian (for more details see [23, Proposition9.7] and the results mentioned there).
Proposition A.3.
Let ε and µ be functions in C k ( M ), k ≥
2, with positivereal parts, and let s >
0. If ω / ∈ Σ s , then for any f ∈ H k − Ω ( M ) theequation (A.4) has a unique solution e ∈ H kb Ω ( M ) ( b = R or A ), and k e k H k ≤ C k f k H k − . Further, if ω is any complex number and if e ∈ H b Ω ( M ) solves (A.4) forsome f ∈ H k − Ω ( M ), then e ∈ H kb Ω ( M ).Finally, we connect the above discussion to the Maxwell system and provethe main result. Proof of Theorem A.1.
We take Σ to be the set Σ in Proposition A.2, andassume that ω / ∈ Σ. As a technical preparation, we choose s > ω /s is not an eigenvalue of the operator u δ ( ε − du ) defined on H ( M ).We then have ω / ∈ Σ s , which may be seen as follows: if e ∈ H R Ω ( M )satisfies (A.3), then e ∈ H R Ω ( M ) by Proposition A.3, and applying δ toboth sides of (A.3) shows that u = δe is a solution in H ( M ) of sδ ( ε − du ) − ω u = 0 . By the choice of s we have u = 0, which implies that e satisfies (A.3) with s = 1. Then e = 0 by the assumption ω / ∈ Σ , and therefore ω / ∈ Σ s .For uniqueness, if ( E, H ) ∈ H ( M ) × H ( M ) solve (A.1)–(A.2) with f = 0, then one has δ ( µ − dE ) = ω εE,δ ( εE ) = 0 . It follows that e = εE is in H R Ω ( M ) and δ ( µ − d ( ε − e )) + ε − dδe − ω e = 0 . Proposition A.2 shows that e = 0, which implies E = H = 0.Let us proceed to prove existence of solutions. Given f ∈ T H k − / ( ∂M ),choose E ∈ H k Div ( M ) with tE = f and tδ ( εE ) = 0. A computation inboundary normal coordinates shows that the extension map f E can betaken to be bounded and linear T H k − / ( ∂M ) → H k Div ( M ). NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 36
We let e ∈ H kR Ω ( M ) be the solution, given by Proposition A.3, of δ ( µ − d ( ε − e )) + sε − dδe − ω e = F with F = − δ ( µ − dE ) − sε − dδ ( εE ) + ω εE ∈ H k − Ω ( M ). Now define E = ε − e + E and H = iωµ ∗ dE . With these conventions, E satisfies theequation δ ( µ − dE ) + sε − dδ ( εE ) − ω εE = 0 . (A.5)We now claim that δ ( εE ) = 0 . (A.6)In fact, by taking δ of both sides of (A.5), the function u = δ ( εE ) ∈ H ( M )satisfies sδ ( ε − du ) − ω u = 0, showing that u = 0 by the choice of s .The first equation in (A.1) is satisfied by definition, and also the secondequation is valid since by (A.5) and (A.6) ∗ dH = 1 iω δ ( µ − dE ) = − iω ε − dδ ( εE ) − iωεE = − iωεE. The 1-form E is in H k Div ( M ) and tE = f . The 1-form H is initially in H k − Ω ( M ) by definition. However, a similar argument which was used toestablish (A.3) shows that h = µH satisfies the second order equation δ ( ε − d ( µ − h )) + µ − dδh − ω h = 0 . Also, a computation in boundary normal coordinates gives the followingboundary conditions for h : t ( ∗ h ) = 1 iω t ( dE ) = 1 iω Div( f ) dS,t ( δ ∗ h ) = t ( ∗ d ( µH )) = t ( ∗ dµ ∧ H ) − iωµεf. Since f ∈ T H k − / and H ∈ H k − , one can check by a computation inboundary normal coordinates that there exists h ∈ H k Ω ( M ) for which˜ h = h − h is in H k − A Ω ( M ). Now ˜ h satisfies the equation δ ( ε − d ( µ − ˜ h )) + µ − dδ ˜ h − ω ˜ h = ˜ F for some ˜ F ∈ H k − Ω ( M ). Elliptic regularity (as in Proposition A.3) impliesthat ˜ h ∈ H k Ω ( M ) which is then true for H too. We have H ∈ H k Div ( M )because Div( tH ) = −h ν, ∗ dH i| ∂M = iωε h ν, E i| ∂M ∈ H k − / ( ∂M ) . (cid:3) Remarks.
1. If ε, µ ∈ C ( M ), then the conclusion of Theorem A.1 isvalid also for k = 1. This follows from the above argument uponapproximating f ∈ T H / by smooth tangential fields.2. Theorem A.1 considers the case where ε and µ are independent of ω .In applications the coefficients are often ω -dependent, for instance inlossy materials one writes ε = Re( ε ) + iσ/ω with Re( ε ) > , σ ≥ µ > ω . In the last case, wellposedness inEuclidean domains was shown in [20]. Of course, Theorems 1.1 and1.2 are valid whenever the admittance map is well defined (this isthe content of assumption (1.4)). NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 37
Appendix B. Unique continuation
This section contains a unique continuation result for principally diagonalsystems required in the final recovery of coefficients. The result is well knownand follows from standard scalar Carleman estimates, but since we could notfind a proper reference a proof is included here.
Theorem B.1.
Let (
M, g ) be a compact connected Riemannian manifoldwith boundary, and let α r , β r be Lipschitz continuous functions in M withpositive real parts ( r = 1 , . . . , N ). Consider the operators P r u = 1 α r δ ( β r du ) ,P = diag( P , . . . , P N ) . Let Γ be an open subset of ∂M . If ~u ∈ H ( M ) N satisfies | P ~u ( x ) | ≤ C ( | ~u ( x ) | + |∇ ~u ( x ) | ) for a.e. x ∈ M ,~u | Γ = ∂ ν ~u | Γ = 0 , then ~u ≡ M .More generally, strong unique continuation holds in this setting. We willdeduce Theorem B.1 from the next result which is stated in R n . Theorem B.2.
Let B be a ball in R n with center x , let ( g jk ) nj,k =1 be aLipschitz continuous symmetric positive definite matrix in B , and let α r , β r be Lipschitz continuous functions in B with positive real parts. Considerthe operators P r u = 1 α r ∂ x j ( β r g jk ∂ x k u ) ,P = diag( P , . . . , P N ) . If ~u ∈ H ( B ) N satisfies for all K > | P ~u ( x ) | ≤ C ( | ~u ( x ) | + |∇ ~u ( x ) | ) for a.e. x ∈ B, lim r → r − K Z B ( x ,r ) | ~u ( x ) | dx = 0 , then ~u ≡ B . Proof of Theorem B.1. If ~u is as in Theorem B.1, we fix a point on Γ andtake ˜ M to be a manifold obtained by enlarging M slightly near this point.Extending ~u by zero to ˜ M and extending α r and β r as Lipschitz functions,we see that | P ~u | ≤ C ( | ~u | + |∇ ~u | ) a.e. in ˜ M and ~u = 0 in some open subset.Working in local coordinates and using Theorem B.2 with a connectednessargument proves the result. (cid:3) To prove Theorem B.2, note that we can assume α r ≡
1, and by dif-ferentiation that β r ≡
1. Letting Lu = ∂ x j ( g jk ∂ x k u ), this implies that P = . . . = P N = L . We may also assume that x = 0 and B = B (0 , ~u is real valued. The result is a consequence of the following scalarCarleman estimate. NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 38
Proposition B.3.
Let λ > λ − | ξ | ≤ g jk ξ j ξ k ≤ λ | ξ | in B and P nj,k =1 | g jk ( x ) − g jk ( y ) | ≤ λ | x − y | for x, y ∈ B . There exists 0 < δ < M >
0, only depending on n and λ , and a function w satisfying | x | /M ≤ w ( x ) ≤ M | x | in B, such that for all α ≥ M and all u ∈ C ∞ c ( B (0 , δ ) r { } ) we have Z B ( αw − α |∇ u | + α w − − α u ) dx ≤ M Z B w − α ( Lu ) dx. This estimate is proved in [5, Theorem 2.1], and is also contained in[8] with slightly different hypotheses. Theorem 2.1 in [5] is given in theparabolic setting, but the result above follows by applying the estimatein [5] to v ( x, t ) = θ ( t ) u ( x ) where θ is a cutoff function. One then notesthat when the coefficients are independent of t , the w ( x, t ) constructed in[5] depends only on x , so the inequality (2.1) in [5] yields the lemma byintegrating in t and absorbing the extra term on the right by making α evenlarger.As an immediate corollary to Proposition B.3, for ~u ∈ C ∞ c ( B (0 , δ ) r { } ) N we have that Z B ( αw − α |∇ ~u | + α w − − α | ~u | ) dx ≤ M Z B w − α | P ~u | dx. Once we have the last estimate, after the initial reductions, Theorem B.2follows immediately using the standard Carleman method.
References [1] P. Caro, P. Ola, and M. Salo,
Inverse boundary value problem for Maxwell equationswith local data , preprint (2009), arXiv:0902.4026.[2] D. Colton and L. P¨aiv¨arinta,
The uniqueness of solution to an inverse scatteringproblem for electromagnetic waves , Arch. Rational Mech. Anal. (1992), 59–70.[3] M. Costabel,
A coercive bilinear form for Maxwell’s equations , J. Math. Anal. Appl. (1991) 527–541.[4] D. Dos Santos Ferreira, C. E. Kenig, M. Salo, and G. Uhlmann,
Limiting Carlemanweights and anisotropic inverse problems , Invent. Math. (to appear).[5] L. Escauriaza and S. Vessella,
Optimal three cylinder inequalities for solutions toparabolic equations with Lipschitz leading coefficients , Inverse problems: theory andapplications (Cortona/Pisa, 2002), 79–87, Contemp. Math. , Amer. Math. Soc.,Providence, RI, 2003.[6] A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann,
Full-wave invisibility of activedevices at all frequencies , Comm. Math. Phys. (2007), 749–789.[7] A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann,
Invisibility and inverse prob-lems , Bull. Amer. Math. Soc. (2009), 55–97.[8] L. H¨ormander, Uniqueness theorems for second order elliptic differential equations ,Comm. PDE (1983), 21–64.[9] V. Isakov, On uniqueness in the inverse conductivity problem with local data , InverseProbl. Imaging (2007), 95–105.[10] M. Joshi and S. McDowall, Total determination of material parameters from electro-magnetic boundary information , Pacific J. Math. (2000), 107–129.[11] C. E. Kenig, J. Sj¨ostrand, and G. Uhlmann,
The Calder´on problem with partial data ,Ann. of Math. (2007), 567–591.[12] Y. Kurylev, M. Lassas, and E. Somersalo,
Maxwell’s equations with a polarizationindependent wave velocity: Direct and inverse problems , J. Math. Pures Appl. (2006), 237–270. NVERSE PROBLEMS FOR THE ANISOTROPIC MAXWELL EQUATIONS 39 [13] R. Leis,
Zur Theorie elektromagnetischer Schwingungen in anisotropen inhomogeneneMedien , Math. Z. (1968), 213-224.[14] S. McDowall,
Boundary determination of material parameters from electromagneticboundary information , Inverse Problems (1997), 153–163.[15] S. McDowall, An electromagnetic inverse problem in chiral media , Trans. Amer. Math.Soc. (2000), 2993–3013.[16] A. Nachman,
Reconstructions from boundary measurements , Ann. Math. (1988),531–576.[17] P. Ola, L. P¨aiv¨arinta, and E. Somersalo,
An inverse boundary value problem in elec-trodynamics , Duke Math. J. (1993), 617–653.[18] P. Ola, L. P¨aiv¨arinta, and E. Somersalo, Inverse problems for time harmonic electro-dynamics , Inside out: inverse problems, MSRI publications (2003), 169–191.[19] P. Ola and E. Somersalo, Electromagnetic inverse problems and generalized Sommer-feld potentials , SIAM J. Appl. Math. (1996), 1129–1145.[20] E. Somersalo, D. Isaacson, and M. Cheney, A linearized inverse boundary value prob-lem for Maxwell’s equations , J. Comput. Appl. Math. (1992), 123–136.[21] Z. Sun and G. Uhlmann, An inverse boundary value problem for Maxwell’s equations ,Arch. Rational Mech. Anal. (1992), 71–93.[22] J. Sylvester and G. Uhlmann,
A global uniqueness theorem for an inverse boundaryvalue problem , Ann. of Math. (1987), 153–169.[23] M. E. Taylor, Partial differential equations I: Basic theory. Springer, 1999.
Department of Mathematics, University of Chicago
E-mail address : [email protected] Department of Mathematics and Statistics, University of Helsinki
E-mail address : [email protected] Department of Mathematics, University of Washington
E-mail address ::