Determining an unbounded potential from Cauchy data in admissible geometries
aa r X i v : . [ m a t h . A P ] A p r DETERMINING AN UNBOUNDED POTENTIALFROM CAUCHY DATA IN ADMISSIBLEGEOMETRIES
DAVID DOS SANTOS FERREIRA, CARLOS E. KENIG, AND MIKKO SALO
Abstract.
In [4] anisotropic inverse problems were consideredin certain admissible geometries, that is, on compact Riemannianmanifolds with boundary which are conformally embedded in aproduct of the Euclidean line and a simple manifold. In particular,it was proved that a bounded smooth potential in a Schr¨odingerequation was uniquely determined by the Dirichlet-to-Neumannmap in dimensions n ≥
3. In this article we extend this result tothe case of unbounded potentials, namely those in L n/ . In theprocess, we derive L p Carleman estimates with limiting Carlemanweights similar to the Euclidean estimates of Jerison-Kenig [10]and Kenig-Ruiz-Sogge [12].
Contents
1. Introduction 12. L p Carleman estimates 53. Complex geometrical optics 114. Recovering the potential 155. Attenuated ray transform 16Appendix A. Wellposedness 19Appendix B. Normal operator 21References 241.
Introduction
In this paper we consider the problem of proving L p estimates forlimiting Carleman weights on Riemannian manifolds, and the relatedinverse problem of recovering an L n/ potential from the Dirichlet-to-Neumann map (DN map) related to the Schr¨odinger equation. Themain motivation comes from the inverse conductivity problem posed by Date : March 30, 2011. Calder´on [1]. This problem asks to determine the conductivity functionof a medium from electrical measurements made on its boundary.In mathematical terms, if Ω ⋐ R n is the medium of interest hav-ing a positive conductivity coefficient γ , in the Calder´on problem oneconsiders the conductivity equation ∇ · γ ∇ u = 0 in Ωand defines the DN map byΛ γ : u | ∂ Ω γ ∂u∂ν (cid:12)(cid:12)(cid:12) ∂ Ω . This operator maps the voltage at the boundary to the current givenby γ times the normal derivative, which encodes the electrical mea-surements at the boundary. The inverse problem of Calder´on asks todetermine γ from the knowledge of Λ γ . This problem has been exten-sively studied and we refer to [30] for a recent survey.The anisotropic Calder´on problem considers the case where the con-ductivity γ is a symmetric positive definite matrix instead of a scalarfunction. This corresponds to situations where the electrical propertiesof the medium depend on direction. The problem is open in general indimensions n ≥
3, see [4] for known results and more details. Follow-ing [18] the problem may be recast as the determination of the metric g on a compact Riemannian manifold ( M, g ) with boundary from thecorresponding DN map. In [4] progress was made on the anisotropicCalder´on problem in the following class of conformal smooth manifolds.
Definition.
A compact Riemannian manifold ( M, g ) , with dimension n ≥ and with boundary ∂M , is called admissible if M ⋐ R × M forsome ( n − -dimensional simple manifold ( M , g ) , and if g = c ( e ⊕ g ) where e is the Euclidean metric on R and c is a smooth positive functionon M . Here, a compact manifold ( M , g ) with boundary is simple if for any p ∈ M the exponential map exp p with its maximal domain of definitionis a diffeomorphism onto M , and if ∂M is strictly convex (that is, thesecond fundamental form of ∂M ֒ → M is positive definite).In [4] it was proved that a Riemannian metric in a conformal classof admissible geometries is uniquely determined by the DN map. Thiswas obtained as a corollary of a result for the Schr¨odinger equation ina fixed admissible manifold, stating that a bounded smooth potential q is determined by the corresponding DN map. In [4] all coefficients wereassumed infinitely differentiable. In this paper we relax this require-ment and show that a complex potential q ∈ L n/ ( M ) is determinedby the DN map. To state the main result, assume that (
M, g ) is a compact Riemann-ian manifold with smooth boundary ∂M , and let ∆ g be the Laplace-Beltrami operator. Given a complex function q ∈ L n/ ( M ), where n ≥ M , we consider the Dirichletproblem ( − ∆ g + q ) u = 0 in M, u | ∂M = f. We assume throughout that 0 is not a Dirichlet eigenvalue for thisproblem, and then standard arguments (see Appendix A) show thatthere is a well-defined DN mapΛ g,q : H / ( ∂M ) → H − / ( ∂M ) , f ∂ ν u | ∂M . The following uniqueness theorem is the main result for the inverseproblem. (The assumption q ∈ L n/ may be considered optimal in thecontext of the standard wellposedness theory for the Dirichlet problemwith L p potentials, and also for the strong unique continuation principleto hold [10].) Theorem 1.1.
Let ( M, g ) be admissible and let q , q be complex func-tions in L n/ ( M ) . If Λ g,q = Λ g,q , then q = q . In the case where M is a bounded domain in R n and g is the Eu-clidean metric, this result is due to Lavine and Nachman [17] followingthe earlier result of Jerison and Kenig for q j ∈ L n/ ε ( M ) for some ε > q j in a Fefferman-Phong class with small norm). As mentioned above,if q is a smooth function on an admissible manifold M this result wasproved in [4] by using L Carleman estimates. In fact, smoothness of q is not essential, and by inspecting the proof of [4] the uniqueness resultcan be extended to bounded continuous q (with the complex geomet-rical optics construction in the proof going through for q ∈ L n ( M )).However, the proof for q ∈ L n/ requires to replace the L Carlemanestimates in [4] with corresponding L p Carleman estimates.The other main result in this paper is a L p Carleman estimate forlimiting Carleman weights on Riemannian manifolds. The concept oflimiting Carleman weights was introduced in [14] as part of a generalprocedure for producing special complex geometrical optics solutions toelliptic equations, with applications to inverse problems. We refer to [4]for a precise definition and more careful analysis of limiting Carlemanweights, also on Riemannian manifolds. For present purposes it issufficient to mention that the existence of a limiting Carleman weighton (
M, g ) in dimensions n ≥ R n , n ≥
3, include the linear weight ϕ ( x ) = x and logarithmic weight ϕ ( x ) = log | x | .The last two weights are featured in the literature of Carleman esti-mates and unique continuation, in particular in the scale invariant L p Carleman estimates of Kenig-Ruiz-Sogge [12] for the linear weight andof Jerison-Kenig [10] for the logarithmic weight. We prove an analogueof these estimates on more general Riemannian manifolds. Note thatthe existence of a limiting Carleman weight requires at least locally aproduct structure on the manifold, and therefore the following result isstated for the linear weight on a product manifold. The result, in thecase when the manifold ( M , g ) below is the standard n − Theorem 1.2.
Let ( M , g ) be an ( n − -dimensional compact man-ifold without boundary, and equip R × M with the metric g = e ⊕ g where e is the Euclidean metric. The Euclidean coordinate is denotedby x . For any compact interval I ⊆ R there exists a constant C I > such that if | τ | ≥ and τ / ∈ Spec( − ∆ g ) then we have k e τx u k L nn − ( R × M ) ≤ C I k e τx ∆ g u k L nn +2 ( R × M ) when u ∈ C ∞ ( I × M ) . The proof of the L Carleman estimates for limiting Carleman weightsin [4] is based on integration by parts and cannot be used in the L p setting. However, in [13] another proof of the L Carleman estimate isgiven; this proof is based on Fourier analysis and gives an explicit in-verse for the conjugated Laplacian. We will derive the L p bounds fromthis explicit inverse operator. This follows the proof of the L p Carle-man estimate of Jerison-Kenig [10] using Jerison’s approach [9], [26,Section 5.1] based on the spectral cluster estimates of Sogge [26]. Fi-nally, if one allows strongly pseudoconvex Carleman weights then muchstronger estimates are available (see for instance [15, 16]), however forthe applications to inverse problems it seems necessary to restrict tolimiting Carleman weights.
Remark 1.3.
The above theorems are in the setting of (conformal)product manifolds. However, the results also apply to warped products.If f : R → R is a smooth function and ( M , g ) is an ( n − R × e f M is the manifold M = R × M endowed with the metric g ( x , x ′ ) = (cid:18) f ( x ) g ( x ′ ) (cid:19) . We choose coordinates y = η ( x ), y ′ = x ′ for a suitable smooth strictlyincreasing function η . In fact, if η ( t ) = Z t e − f ( s ) d s then η ′ ( t ) − = e f ( t ) and the metric in y coordinates becomes a confor-mal multiple of a product metric, g ( y , y ′ ) = e f ( η − ( y )) (cid:18) g ( y ′ ) (cid:19) . Warped products have a natural limiting Carleman weight ϕ ( y ) = y ,and Theorem 1.1 remains true in conformal multiples of warped prod-ucts whenever ( M , g ) is a simple manifold.The paper is organized as follows. Section 1 is the introduction. InSection 2 we prove the L p Carleman estimate complemented with theusual L Carleman estimates. Section 3 presents the construction ofcomplex geometrical optics solutions for Schr¨odinger equations with L n/ potentials in admissible geometries. The proof of Theorem 1.1 iscontained in Section 4, modulo a uniqueness result for an analogue ofthe attenuated geodesic ray transform acting on unbounded functions.This last result has a different character than the rest of the proof,and it is therefore established separately in Section 5. There are twoappendices concerning the wellposedness of the Dirichlet problem andthe normal operator for the attenuated ray transform. Acknowledgements.
The last named author would like to thankAdrian Nachman for explaining his unpublished argument with RichardLavine [17] which proves a uniqueness result for L n/ potentials in Eu-clidean space. C.K. is supported partly by NSF grant DMS-0968472,and M.S. is supported in part by the Academy of Finland. D. DSF.would like to thank the University of Chicago for its hospitality.2. L p Carleman estimates
The aim of this section is to prove Theorem 1.2, which is an analogueof the L p Carleman estimates obtained in the Euclidean case by Jerisonand Kenig [10] (for logarithmic weights) or by Kenig, Ruiz and Sogge[12] (for linear weights). In fact, we prove a more general result whichimplies Theorem 1.2 by taking f = e τx ∆ g e − τx u for u ∈ C ∞ ( I × M ). The case when ( M , g ) is the standard n − Proposition 2.1.
Let I ⊆ R be a compact interval and ( M , g ) acompact ( n − -dimensional manifold without boundary. Equip N = I × M with the product metric g = e ⊕ g . For | τ | ≥ with τ / ∈ Spec ( − ∆ g ) , there is a linear operator G τ : L ( N ) → H ( N ) such that e τx ( − ∆ g )e − τx G τ v = v for v ∈ L ( N ) ,G τ e τx ( − ∆ g )e − τx v = v for v ∈ C ∞ ( N int ) . This operator satisfies k G τ f k L ( N ) ≤ C | τ | k f k L ( N ) , k G τ f k H ( N ) ≤ C k f k L ( N ) , k G τ f k L nn − ( N ) ≤ C k f k L nn +2 ( N ) , where C is independent of τ (but may depend on I ). Remark 2.2.
In the Euclidean case, L p Carleman estimates with linearweights can be obtained from L p Carleman estimates with pseudocon-vex Carleman weights by scaling. Indeed, suppose that the followingCarleman estimate (cid:13)(cid:13) e τ ( x + x / ε ) u (cid:13)(cid:13) L nn − ( R n ) ≤ C K (cid:13)(cid:13) e τ ( x + x / ε ) ∆ u k L nn +2 ( R n ) , holds for all ε ≤ ε and all u ∈ C ∞ ( K ), then applying this estimate to u µ = u ( µ · ) with µ ≥ u ∈ C ∞ ( K ), one gets (cid:13)(cid:13) e τµ x + τµ x / ε u (cid:13)(cid:13) L nn − ( R n ) ≤ C K (cid:13)(cid:13) e τµ x + τµ x / ε ∆ u (cid:13)(cid:13) L nn +2 ( R n ) . Choosing µ = √ τ , and using the fact that e x / ε ≃ C ε on K , one getsthe Carleman estimate k e µx u k L nn − ( R n ) ≤ C K,ε k e µx ∆ u k L nn +2 ( R n ) , for all u ∈ C ∞ ( K ). However, in the anisotropic case, one has to findanother way.To prepare for the proof of Proposition 2.1 consider the Laplace-Beltrami operator on N , P = ∆ g = ∂ x + ∆ g and the corresponding conjugated operator (by the limiting Carlemanweight x ) e τx P e − τx = ∂ x − τ ∂ x + τ + ∆ g . (2.1) We denote by λ = 0 < λ ≤ λ ≤ . . . the sequence of eigenvalues of − ∆ g on M and ( ψ j ) j ≥ the corresponding sequence of eigenfunctionsforming an orthonormal basis of L ( M ), − ∆ g ψ j = λ j ψ j . We denote by π j : L ( M ) → L ( M ) , u ( u, ψ j ) ψ j the projection onthe linear space spanned by the eigenfunction ψ j so that ∞ X j =0 π j = Id , ∞ X j =0 λ j π j = − ∆ g and by b u ( j ) = Z M u ψ j d V g the corresponding Fourier coefficients of a function u on M . We definethe spectral clusters as χ k = X k ≤ √ λ j Proof of Proposition 2.1. Recall that our main goal is to prove k u k L nn − ( R × M ) ≤ C I k f k L nn +2 ( R × M ) (2.5)when u ∈ C ∞ ( I × M ) and D x u + 2 iτ D x u − ∆ g u − τ u = f (2.6)with D x = − i∂ x . The inverse operator in (2.6) is actually easy towrite down, as was done in [13]. The same procedure appears in [9] and [26, Section 5.1]. Writing f = P ∞ j =0 π j f and similarly for u , theequation formally becomes( D x + 2 iτ D x − τ + λ j ) π j u = π j f for x on the real line and for j ≥ 0. The symbol of the operator onthe left is ξ + 2 iτ ξ − τ + λ j , and this is always nonzero provided that τ = λ j for all j . Thus, if τ / ∈ Spec( − ∆ g ) , an inverse operator may be obtained as˜ G τ f ( x , x ′ ) = ∞ X j =0 Z ∞−∞ m τ (cid:0) x − y , p λ j (cid:1) π j f ( y , x ′ ) d y where m τ ( t, µ ) = 12 π Z ∞−∞ e itη η + 2 iτ η − τ + µ d η, µ > . The operator ˜ G τ is the same as G τ in [13, Section 4], except thatin the present setting { ψ j } is a basis of L ( M ) on a compact man-ifold ( M , g ) without boundary instead of being a basis of Dirichleteigenfunctions on a compact manifold with boundary. Let L δ ( R × M ) = (cid:8) f ∈ L ( R × M ) ; (1 + x ) δ/ f ∈ L ( R × M ) (cid:9) and let H sδ ( R × M ) by the corresponding Sobolev spaces. The proofof [13, Proposition 4.1] goes through for ˜ G τ without changes and showsthat for any fixed δ > / 2, if | τ | ≥ τ / ∈ Spec( − ∆ g ) then theequation e τx ( − ∆ g )e − τx v = f has a unique solution v = ˜ G τ f ∈ H − δ ( R × M ) for any f ∈ L δ ( R × M ).Further, v ∈ H − δ ( R × M ) and k v k H s − δ ( R × M ) ≤ C | τ | s − k f k L δ ( R × M ) , ≤ s ≤ . We define G τ f ( x , x ′ ) = χ ( x ) ˜ G τ f ( x , x ′ )with χ ∈ C ∞ ( R ) which equals 1 on I . It is then clear that all thestatements in the proposition except for the L p estimate follow fromthe results for ˜ G τ explained above.It is sufficient to prove the L p estimate in the case where τ ≥ τ / ∈ Spec( − ∆ g ). We first record a lemma. Lemma 2.3. If τ > , µ > , τ = µ and t ∈ R then | m τ ( t, µ ) | ≤ µ e −| τ − µ || t | . Besides, if τ > then | m τ ( t, | ≤ | t | e − τ | t | . Proof. This follows by writing1( iη − ( τ + µ ))( iη − ( τ − µ )) = 12 µ (cid:20) iη − ( τ + µ ) − iη − ( τ − µ ) (cid:21) and by noting that for α > F − η (cid:26) iη + α (cid:27) ( t ) = (cid:26) , t < − αt , t > , and similarly for α < F − η (cid:26) η + iτ ) (cid:27) ( t ) = (cid:26) t e − τ | t | , t < , t > , and this concludes the proof of the lemma. (cid:3) From the decomposition (2.3), the spectral cluster estimate (2.4),and the fact that spectral clusters are projections ( χ k = χ k ), we getthe following string of estimates k u k L nn − ( M ) = (cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =0 χ k u (cid:13)(cid:13)(cid:13)(cid:13) L nn − ( M ) (2.7) . ∞ X k =0 (1 + k ) − n k χ k u k L ( M ) . Since k χ k u k L ( M ) = (cid:18) X k ≤ √ λ j 0. (Note that when k = 0, a majorant is e − ( τ/ | t | for τ ≥ ∞ X k =0 (1 + k ) − n sup k ≤ √ λ j In this section we will construct the complex geometrical optics so-lutions that will be used to recover an L n/ potential. Throughout thesection, let ( M, g ) be a compact manifold with smooth boundary, andlet ( M, g ) ⋐ ( T, g ) ⋐ ( ˜ T , g ) where T = R × M , ˜ T = R × ˜ M , and g = e ⊕ g , and ( M , g ) ⋐ ( ˜ M , g ) are two ( n − n ≥ M (this follows easily by embedding ( ˜ M , g ) in some compact manifoldwithout boundary and using suitable restrictions and extensions byzero). Proposition 3.1. For | τ | ≥ outside a countable set, there is a linearoperator G τ : L ( M ) → H ( M ) such that e τx ( − ∆ g )e − τx G τ v = v for v ∈ L ( M ) ,G τ e τx ( − ∆ g )e − τx v = v for v ∈ C ∞ ( M int ) . This operator satisfies k G τ f k L ( M ) ≤ C | τ | k f k L ( M ) , k G τ f k H ( M ) ≤ C k f k L ( M ) , k G τ f k L nn − ( M ) ≤ C k f k L nn +2 ( M ) , where C is independent of τ . Let us first construct the required complex geometrical optics solu-tions for the case where no potential is present. This is analogous to[4, Proposition 5.1] for q = 0. Proposition 3.2. Let ω ∈ ˜ M \ M be a fixed point, let λ ∈ R be fixed,and let b ∈ C ∞ ( S n − ) be a function. Write x = ( x , r, θ ) where ( r, θ ) are polar normal coordinates with center ω in ( ˜ M , g ) . For | τ | suffi-ciently large outside a countable set, there exists u ∈ H ( M ) satisfying − ∆ g u = 0 in M,u = e − τx (e − iτr | g | − / e iλ ( x + ir ) b ( θ ) + r ) where r satisfies | τ |k r k L ( M ) + k r k H ( M ) + k r k L nn − ( M ) . . Proof. The claim follows if one can find r satisfyinge τx ( − ∆ g )e − τx r = f with the required norm estimates, where f = e τx ∆ g e − τx (e − iτr e iλ ( x + ir ) b ( θ )) . It is enough to take r = G τ f and to note that k f k L ( M ) = k ∆ g (e iλ ( x + ir ) b ( θ )) k L ( M ) . . The L and H estimates follow from Proposition 3.1. The L nn − esti-mate follows from the H estimate and Sobolev embedding, or alter-natively from the L nn +2 → L nn − estimate for G τ . (cid:3) We next consider the case with a potential q ∈ L n/ ( M ), and try tofind a solution to ( − ∆ g + q ) u = 0 in M of the form u = u + e − τx r . Since − ∆ g u = 0, the function r should satisfy(3.1) e τx ( − ∆ g + q )e − τx r = − q e τx u . Since q is only in L n/ ( M ), here we need to use the L nn +2 → L nn − estimates for G τ . We follow the argument of Lavine and Nachman[17]. It will be convenient to symmetrize the situation slightly. Lateron, the L n functions m j in the next lemma are chosen to be essentially | q | / . Lemma 3.3. Let m , m ∈ L n ( M ) be two fixed functions. Then for | τ | ≥ τ outside a countable set, (3.2) k m G τ m f k L ≤ C k m k L n k m k L n k f k L . Further, (3.3) k m G τ m f k L ( M ) → L ( M ) → as | τ | → ∞ .Proof. The H¨older inequality and Proposition 3.1 imply that k m G τ m f k L ≤ k m k L n k G τ m f k L nn − ≤ C k m k L n k m f k L nn +2 ≤ C k m k L n k m k L n k f k L . Let ε > m j = m ♯j + m ♭j where m ♯j ∈ L ∞ ( M ), k m ♯j k L n ≤ k m j k L n and k m ♭j k L n ≤ ε C max( k m k L n , k m k L n ) . (One can take for instance m ♯j = m j χ {| m j |≤ µ } for large enough µ .) Itfollows from the L estimates for G τ and (3.2) that k m G τ m f k L ≤ k m ♯ G τ m ♯ f k L + k m ♯ G τ m ♭ f k L + k m ♭ G τ m f k L ≤ C k m ♯ k L ∞ k m ♯ k L ∞ | τ | + ε ε ! k f k L . The last expression is bounded by ε k f k L if | τ | is sufficiently large.This proves (3.3). (cid:3) We now finish the construction of complex geometrical optics solu-tions. Proposition 3.4. Assume that q ∈ L n/ ( M ) . Let ω ∈ ˜ M \ M be afixed point, let λ ∈ R be fixed, and let b ∈ C ∞ ( S n − ) be a function.Write x = ( x , r, θ ) where ( r, θ ) are polar normal coordinates with cen-ter ω in ( ˜ M , g ) . For | τ | sufficiently large outside a countable set, thereexists a solution u ∈ H ( M ) of ( − ∆ g + q ) u = 0 in M of the form u = e − τx (e − iτr | g | − / e iλ ( x + ir ) b ( θ ) + ˜ r ) where ˜ r satisfies k ˜ r k L nn − ( M ) . , k ˜ r k L ( M ) → as | τ | → ∞ . Proof. As explained above, we let u be the harmonic function given inProposition 3.2, and look for a solution of the form u = u +e − τx r . Wewrite q ( x ) = | q ( x ) | e iα ( x ) = | q ( x ) | / m ( x ) where m ( x ) = | q ( x ) | / e iα ( x ) .Then | q | / , m ∈ L n ( M ) with L n norms equal to k q k / L n/ .We obtain a solution u provided that (3.1) holds. Trying r in theform r = G τ | q | / v , we see that v should satisfy(Id + mG τ | q | / ) v = − m e τx u . By Lemma 3.3, for | τ | sufficiently large one has k mG τ | q | / k L → L ≤ / 2. One then obtains a solution v = − (Id + mG τ | q | / ) − ( m e τx u ) . Since k m e τx u k L ≤ k m k L n k e τx u k L nn − . 1, it follows that k v k L . 1. Consequently k r k L nn − ≤ C k| q | / v k L nn +2 . . Now u is of the form given in the statement of the proposition, providedthat ˜ r = r + r . This remainder term satisfies k ˜ r k L nn − . k ˜ r k L we fix ε > | q | / = s ♯ + s ♭ where s ♯ ∈ L ∞ ( M ), k s ♯ k L n ≤ k q k / L n/ , and k s ♭ k L n ≤ ε . Then k r k L ≤ k G τ s ♯ v k L + C k G τ s ♭ v k L nn − ≤ (cid:18) C k s ♯ k L ∞ | τ | + C C k s ♭ k L n (cid:19) k v k L . Choosing | τ | sufficiently large, we see that k r k L . ε for | τ | large.Since also k r k L ( M ) . | τ | − , it follows that k ˜ r k L → | τ | → ∞ .Finally, to prove that u ∈ H ( M ), it is enough to consider a compactmanifold ( ˆ M , g ) which is slightly larger than ( M, g ) and extend q byzero outside M , and to perform the above construction of solutions inˆ M . One obtains a solution u ∈ L nn − ( ˆ M ) ⊆ L ( ˆ M ), and ∆ g u = qu ∈ L nn +2 ( ˆ M ) ⊆ H − ( ˆ M ) by Sobolev embedding. Elliptic regularity impliesthat u ∈ H ( ˆ M int ), thus also u ∈ H ( M ). (cid:3) Recovering the potential We are now ready to give the proof of the main uniqueness result. Proof of Theorem 1.1. Assume, as before, that ( M, g ) ⋐ ( T, g ) ⋐ ( ˜ T , g )where T = R × M , ˜ T = R × ˜ M , and ( M , g ) ⋐ ( ˜ M , g ) are two( n − n ≥ 3. Also assume that g = e ⊕ g .From the assumption Λ g,q = Λ g,q , writing q = q − q , we knowfrom Lemma A.1 that(4.1) Z M qu u d V g = 0where u , u ∈ H ( M ) are solutions of ( − ∆ g + q ) u = 0 in M and( − ∆ g + q ) u = 0 in M . By Proposition 3.4, for τ sufficiently largeoutside a countable set there exist solutions u j of the form u = e − τ ( x + ir ) ( | g | − / e iλ ( x + ir ) b ( θ ) + r ) ,u = e τ ( x + ir ) ( | g | − / + r ) . Here λ is a fixed real number, b ∈ C ∞ ( S n − ) is a fixed function, and x = ( x , r, θ ) are coordinates in ˜ T where ( r, θ ) are polar normal coor-dinates in ( ˜ M , g ) with center at a fixed point ω ∈ ˜ M \ M . Also, theremainder terms satisfy k r j k L nn − ( M ) = O (1) , k r j k L ( M ) = o (1)as τ → ∞ .Inserting the solutions in (4.1) and noting that d V g = | g | / d x d r d θ ,we obtain that(4.2) Z M q e iλ ( x + ir ) b ( θ ) d x d r d θ = Z M q ( a r + a r + r r ) d V where a , a are smooth functions in M independent of τ . We showthat the right hand side converges to 0 as τ → ∞ . To do this, fix ε > q = q ♯ + q ♭ where q ♯ ∈ L ∞ ( M ), k q ♯ k L n/ ≤ k q k L n/ , and k q ♭ k L n/ ≤ ε . The right hand side of (4.2) is bounded by (cid:12)(cid:12)(cid:12)(cid:12)Z M q ( a r + a r + r r ) d V (cid:12)(cid:12)(cid:12)(cid:12) . k q ♯ k L ∞ ( k r k L + k r k L + k r k L k r k L )+ k q ♭ k L n/ ( k r k L nn − + k r k L nn − + k r k L nn − k r k L nn − ) . Using the bounds for r j , if τ is sufficiently large then the last quantityis . ε . This shows that the right hand side of (4.2) goes to 0 as τ → ∞ . Extend q by zero into T and interpret the left hand side of (4.2) asan integral over T . Taking the limit as τ → ∞ , we obtain that Z ∞−∞ Z ∞ Z S n − q ( x , r, θ )e iλ ( x + ir ) b ( θ ) d x d r d θ = 0 . This statement is true for all choices of ω ∈ ˜ M \ M , for all real numbers λ , and for all functions b ∈ C ∞ ( S n − ). Since q ∈ L ( M ), Fubini’stheorem shows that q ( · , r, θ ) is in L ( R ) for a.e. ( r, θ ). Consequently(4.3) Z S n − Z ∞ f λ ( r, θ )e − λr b ( θ ) d r d θ = 0where f λ ∈ L ( M ) is the function given by f λ ( r, θ ) = Z ∞−∞ e iλx q ( x , r, θ ) d x . If | λ | is sufficiently small, it follows from Lemma 5.1 below that thevanishing of the integrals (4.3) for all choices ω and b implies that f λ = 0. Since q ( · , r, θ ) is a compactly supported function in L ( R ) fora.e. ( r, θ ), the Paley-Wiener theorem shows that q ( · , r, θ ) = 0 for such( r, θ ). Consequently q = q . (cid:3) Attenuated ray transform It remains to show the following lemma which was used in the proofof Theorem 1.1. Lemma 5.1. Let ( M , g ) be an ( n − -dimensional simple manifold,and let f ∈ L ( M ) . Consider the integrals Z S n − Z τ ( ω,θ )0 f ( r, θ )e − λr b ( θ ) d r d θ where ( r, θ ) are polar normal coordinates in ( M , g ) centered at some ω ∈ ∂M , and τ ( ω, θ ) is the time when the geodesic r ( r, θ ) exits M .If | λ | is sufficiently small, and if these integrals vanish for all ω ∈ ∂M and all b ∈ C ∞ ( S n − ) , then f = 0 . The last result is related to the vanishing of the attenuated geodesicray transform of the function f on M . For the following facts see [3],[20], [22]. To define the ray transform, we consider the unit spherebundle SM = [ x ∈ M S x , S x = (cid:8) ( x, ξ ) ∈ T x M ; | ξ | = 1 (cid:9) . This manifold has boundary ∂ ( SM ) = { ( x, ξ ) ∈ SM ; x ∈ ∂M } which is the union of the sets of inward and outward pointing vectors, ∂ ± ( SM ) = (cid:8) ( x, ξ ) ∈ SM ; ±h ξ, ν i ≤ (cid:9) . Here ν is the outer unit normal vector to ∂M . Note that ∂ + ( SM ) isa manifold whose boundary consists of all the tangential directions { ( x, ξ ) ∈ ∂ ( SM ) ; h ξ, ν i = 0 } . Thus the space C ∞ (( ∂ + ( SM )) int )contains all smooth functions on ∂ + ( SM ) vanishing near tangentialdirections.Denote by t γ ( t, x, ξ ) the unit speed geodesic starting at x indirection ξ , and let τ ( x, ξ ) be the time when this geodesic exits M .Since ( M , g ) is simple, τ ( x, ξ ) is finite for each ( x, ξ ) ∈ SM . We alsowrite ϕ t ( x, ξ ) = ( γ ( t, x, ξ ) , ˙ γ ( t, x, ξ )) for the geodesic flow.The geodesic ray transform, with constant attenuation − λ , acts onfunctions on M by T λ f ( x, ξ ) = Z τ ( x,ξ )0 f ( γ ( t, x, ξ ))e − λt d t, ( x, ξ ) ∈ ∂ + ( SM ) . In Lemma 5.1, if f were a continuous function, one could choose b ( θ )to approximate a delta function at fixed angles θ and obtain that Z τ ( ω,θ )0 f ( r, θ )e − λr d r = 0for any ω ∈ ∂M and any θ ∈ S n − . Since ( r, θ ) are polar normalcoordinates the curves r ( r, θ ) are geodesics in ( M , g ), and thiswould imply that T λ f ( x, ξ ) = 0 for all ( x, ξ ) ∈ ∂ + ( SM ) . One has the following injectivity result from [4, Theorem 7.1]. (If M is two-dimensional the smallness assumption on the attenuationcoefficient was recently removed in [21].) Proposition 5.2. Let ( M , g ) be a simple manifold. There exists ε > such that if λ is a real number with | λ | < ε and if f ∈ C ∞ ( M ) ,then the condition T λ f ( x, ξ ) = 0 for all ( x, ξ ) ∈ ∂ + ( SM ) implies that f = 0 . The previous argument together with Proposition 5.2 proves Lemma5.1 for smooth f . However, this requires well defined restrictions of f to all geodesics and it is not obvious how to do this when f ∈ L .We circumvent this problem by using duality and the ellipticity of thenormal operator T ∗ λ T λ . We will need a few facts. Below we write h ψ ( x, ξ ) = h ( ϕ − τ ( x, − ξ ) ( x, ξ )) , ( x, ξ ) ∈ SM for h ∈ C ∞ ( ∂ + ( SM )), and( h, ˜ h ) L µ ( ∂ + ( SM )) = Z ∂ + ( SM ) h ˜ hµ d( ∂ ( SM ))where µ ( x, ξ ) = −h ξ, ν ( x ) i and d N is the natural Riemannian volumeform on a manifold N . Lemma 5.3. (Santal´o formula) If F : SM → R is continuous then Z SM F d( SM )= Z ∂ + ( SM ) Z τ ( x,ξ )0 F ( ϕ t ( x, ξ )) µ ( x, ξ ) d t d( ∂ ( SM ))( x, ξ ) . Proof. See [3, Lemma A.8]. (cid:3) Lemma 5.4. If f ∈ C ∞ ( M ) and h ∈ C ∞ (( ∂ + ( SM )) int ) then ( T λ f, h ) L µ ( ∂ + ( SM )) = ( f, T ∗ λ h ) L ( M ) where T ∗ λ h ( x ) = Z S x e − λτ ( x, − ξ ) h ψ ( x, ξ ) d S x ( ξ ) , x ∈ M . Proof. By the Santal´o formula( T λ f, h ) L µ ( ∂ + ( SM )) = Z ∂ + ( SM ) Z τ ( x,ξ )0 e − λt f ( γ ( t, x, ξ )) hµ d t d( ∂ ( SM ))= Z ∂ + ( SM ) Z τ ( x,ξ )0 e − λt f ( ϕ t ( x, ξ )) h ψ ( ϕ t ( x, ξ )) µ d t d( ∂ ( SM ))= Z SM e − λτ ( x, − ξ ) f ( x ) h ψ ( x, ξ ) d( SM )= Z M f ( x ) (cid:18)Z S x e − λτ ( x, − ξ ) h ψ ( x, ξ ) d S x ( ξ ) (cid:19) d V ( x ) . This proves Lemma 5.4. (cid:3) Lemma 5.5. T ∗ λ T λ is a self-adjoint elliptic pseudodifferential operatorof order − in M int .Proof. This is contained in [6, Proposition 2], but for completeness wealso include a proof in Appendix B. (cid:3) Proof of Lemma 5.1. The first step is to extend ( M , g ) to a slightlylarger simple manifold and to extend f by zero. In this way we canassume that f is compactly supported in M int0 .We let b also depend on ω and change notations to write the assump-tion in the lemma in the form Z S x Z τ ( x,ξ )0 e − λt f ( γ ( t, x, ξ )) b ( x, ξ ) d t d S x ( ξ ) = 0for all x ∈ ∂M and b ∈ C ∞ (( ∂ + ( SM )) int ). Next we make the choice b ( x, ξ ) = h ( x, ξ ) µ ( x, ξ ) for h ∈ C ∞ (( ∂ + ( SM )) int ) and integrate thelast identity over ∂M to obtain Z ∂ + ( SM ) Z τ ( x,ξ )0 e − λt f ( γ ( t, x, ξ )) h ( x, ξ ) µ d t d( ∂ ( SM )) = 0 . We are now in the same situation as in the proof of Lemma 5.4, andinvoking the Santal´o formula implies Z M f ( x ) T ∗ λ h ( x ) d V ( x ) = 0for all h ∈ C ∞ (( ∂ + ( SM )) int ). Note that the last integral is absolutelyconvergent because f ∈ L ( M ), and also the previous steps are justi-fied by Fubini’s theorem.It remains to choose h = T λ ϕ for ϕ ∈ C ∞ ( M int0 ) to obtain that Z M f ( x ) T ∗ λ T λ ϕ ( x ) d V ( x ) = 0 . Since T ∗ λ T λ is self-adjoint, we have Z M ( T ∗ λ T λ f ( x )) ϕ ( x ) d V ( x ) = 0 . This is valid for all test functions ϕ , so T ∗ λ T λ f = 0. By ellipticity, since f was compactly supported in M int0 , it follows that f ∈ C ∞ ( M int0 ).One can now use the argument for smooth f given above, togetherwith the injectivity result (Proposition 5.2), to conclude the proof that f = 0. (cid:3) Appendix A. Wellposedness Here we recall the standard arguments that show wellposedness ofthe Dirichlet problem for − ∆ g + q on a compact oriented manifold( M, g ) with smooth boundary and with q ∈ L n/ ( M ), n ≥ 3. Considerfirst the inhomogeneous problem for the Schr¨odinger equation,(A.1) ( − ∆ g + q ) u = F in M, u | ∂M = 0 . The bilinear form related to this problem is B ( u, v ) = Z M (cid:0) h d u, d¯ v i + qu ¯ v (cid:1) d V, u, v ∈ H ( M ) , where h · , · i is the complex-linear inner product of 1-forms and d V is the volume form on ( M, g ). By the Sobolev embedding H ( M ) ⊆ L nn − ( M ) and by H¨older’s inequality, B is a bounded bilinear form on H ( M ). Writing q = q ♯ + q ♭ where q ♯ ∈ L ∞ ( M ) and k q ♭ k L n/ ( M ) issmall, we obtain from Poincar´e’s inequality that B ( u, u ) ≥ c k u k H ( M ) − C k u k L ( M ) , u ∈ H ( M ) . This shows that B + C is coercive, and by the Lax-Milgram lemma,compact Sobolev embedding and the Fredholm theorem, the equation(A.1) has a unique solution u ∈ H ( M ) for any F ∈ H − ( M ) if one isoutside a countable set of eigenvalues.We can now consider the Dirichlet problem(A.2) ( − ∆ g + q ) u = 0 in M, u | ∂M = f. We assume that 0 is not a Dirichlet eigenvalue, and it follows from theabove discussion that for any f ∈ H / ( ∂M ) there is a unique solution u ∈ H ( M ). The DN map is formally defined as the mapΛ g,q : H / ( ∂M ) → H − / ( ∂M ) f ∂ ν u | ∂M . More precisely, if f ∈ H / ( ∂M ) we define Λ g,q f weakly as the functionin H − / ( ∂M ) which satisfies Z ∂M (Λ g,q f )¯ h d S = Z M (cid:0) h d u, d¯ v i + qu ¯ v (cid:1) d V where u is the unique solution of (A.2), and v is any extension in H ( M )of h ( v | ∂M = h ). Then Λ g,q is a bounded map H / ( ∂M ) → H − / ( ∂M )again by H¨older and Sobolev embedding.The DN map satisfies in the weak sense Z ∂M (Λ g,q f )¯ h d S = Z ∂M f Λ g, ¯ q h d S. To see this, let u, v ∈ H ( M ) solve ( − ∆ g + q ) u = 0, u | ∂M = f and( − ∆ g + ¯ q ) v = 0, v | ∂M = h . Then Z ∂M (Λ g,q f )¯ h d S = Z M (cid:0) h d u, d¯ v i + qu ¯ v (cid:1) d V = Z M (cid:0) h d v, d¯ u i + ¯ qv ¯ u (cid:1) d V = Z ∂M (Λ g, ¯ q h ) f d S. As a consequence, we have the basic integral identity used in theuniqueness proof. Lemma A.1. If q , q ∈ L n/ ( M ) and Λ g,q = Λ g,q , then Z M ( q − q ) u u d V = 0 for any u j ∈ H ( M ) with ( − ∆ g + q ) u = 0 in M , ( − ∆ g + q ) u = 0 in M .Proof. Follows from the computation0 = Z ∂M (Λ g,q − Λ g,q )( u | ∂M ) u d S = Z ∂M (cid:0) Λ g,q ( u | ∂M ) u − u Λ g, ¯ q (¯ u | ∂M ) (cid:1) d S and the definition of the DN maps. (cid:3) Appendix B. Normal operator The setting is the compact simple Riemannian manifold ( M , g ) ofdimension n − 1. Let T λ be the attenuated ray transform as in Section5. We will prove Lemma 5.5. Write ψ ( x, ξ ) = ϕ − τ ( x, − ξ ) ( x, ξ ) . We compute the normal operator T ∗ λ T λ f for f ∈ C ∞ ( M int0 ) T ∗ λ T λ f ( x )= Z S x e − λτ ( x, − ξ ) ( T λ f ) ψ ( x, ξ ) d S x ( ξ )= Z S x e − λτ ( x, − ξ ) Z τ ( ψ ( x,ξ ))0 e − λt f ( γ ( t, ψ ( x, ξ ))) d t d S x ( ξ )= Z S x e − λτ ( x, − ξ ) Z τ ( x, − ξ )+ τ ( x,ξ )0 e − λt f ( γ ( t, ψ ( x, ξ ))) d t d S x ( ξ ) and using changes of variables we get for the last integral expression T ∗ λ T λ f ( x )= Z S x e − λτ ( x, − ξ ) Z τ ( x,ξ ) − τ ( x, − ξ ) e − λ ( s + τ ( x, − ξ )) f ( γ ( s, x, ξ )) d s d S x ( ξ )= Z S x "Z − τ ( x, − ξ ) + Z τ ( x,ξ )0 e − λ ( s +2 τ ( x, − ξ )) f ( γ ( s, x, ξ )) d s d S x ( ξ )= Z S x Z τ ( x,ξ )0 (cid:2) e − λτ ( x, − ξ ) e − λs + e − λτ ( x,ξ ) e λs (cid:3) f ( γ ( s, x, ξ )) d s d S x ( ξ ) . Changing variables y = exp x ( sξ ) shows that T ∗ λ T λ f ( x ) = Z M K λ ( x, y ) f ( y ) d V ( y )where K λ ( x, y ) = (e − λϕ + ( x,y ) + e − λϕ − ( x,y ) ) d n − g ( x, y ) (cid:18) det g ( x )det g ( y ) (cid:19) | det(exp − x ) ′ ( x, y ) | . with ϕ ± = 2 τ ( x, ∓ grad y d g ( x, y )) ± d g ( x, y ) . The functions ϕ ± are smooth away from the diagonal x = y , and their k -th order derivatives behave as d g ( x, y ) − k . Note that det(exp − ) ′ stands for the Jacobian determinant ofexp − : M × M → R n − ( x, y ) exp − x ( y ) . The kernel of the normal operator is symmetric K λ ( y, x ) = K λ ( x, y )and the singular support of this kernel is the diagonal in M × M .We will now prove that the operator P λ with kernel K λ is actuallya pseudodifferential operator. The first observation in that direction isthat in coordinates d g ( x, y ) = a jk ( x, y )( x j − y j )( x k − y k )(B.1)with a jk ( x, x ) = g jk ( x ). Indeed the square of the distance vanishes atsecond order and its Hessian at x = y is twice the metric. This can beseen from the well known formula ∇ ϕ ( y )( θ, θ ) = ∂ ∂t ϕ (exp y tθ ) (cid:12)(cid:12)(cid:12) t =0 and the fact that if | θ | g = 1 then d g (exp y tθ, y ) = t . To prove that P λ is a pseudodifferential operator in Ψ − ( M int ) we need to show thatfor any couple of cutoff functions ( ψ, χ ) supported in charts of M int ,the operator with kernel˜ K λ ( x, y ) = ψ ( x ) K λ ( x, y ) p det g ( y ) χ ( y )expressed in coordinates , is a pseudodifferential operator on R n − withsymbol in S − . Because of its form and of (B.1), the kernel satisfies | ∂ αx ∂ βy ˜ K λ ( x, x − y ) | ≤ C α | y | − n +2 −| β | (B.2)and has compact support in R n − × R n − .Such operators are pseudodifferential operators and this can easily beseen in the following way: the symbol associated with such an operatoris ˜ p λ ( x, ξ ) = Z ˜ K λ ( x, x − y )e − iy · ξ d y For cutoff functions ψ and χ whose supports don’t intersect, the pre-vious symbol is a Schwartz function because the kernel is a smoothcompactly supported function. So we are only interested in those sym-bols corresponding to kernels ˜ K λ ( x, x − y ) which are supported closeto R n − × { } . In that case, we use a dyadic partition of unity1 = ∞ X µ = −∞ χ (2 − µ z ) , with χ a function supported in an annulus, to decompose the symbolas a sum of terms of the form2 µ ( n − Z e i µ y · ξ χ ( y ) ˜ K λ ( x, x − µ y ) d y. Note that because of the compact support of the kernel, these termsvanish when µ is large, so we are mainly concerned with the termswhere µ is less than some positive integer, say N . Because of the be-haviour (B.2), the rescaled kernel ˜ K λ ( x, x − µ y ) is uniformly boundedby 2 − µ ( n − as well as all its derivatives. Applying the non-stationaryphase when | ξ | ≥ µ ξ is large we get | ˜ p λ ( x, ξ ) | . X µ ≤ N, µ | ξ |≥ µ (2 µ | ξ | ) − N + X µ ≤ N, µ | ξ |≤ µ . (1 + | ξ | ) − By a slight abuse of notations, to lighten the exposition, we don’t write thepullback by the coordinates and think of x and y as variables in R n − . Repeating this argument for the derivatives of this function, we getthat ˜ p λ is a classical symbol of order − p : we have˜ K ( x, y ) = ψ ( x ) det(exp − x ) ′ ( x, y ) d n − g ( x, y ) p det g ( y ) χ ( y )from the previous computation, we see that taking x = y in the non-singular factors, yields error terms whose kernel are less singular by anorder of | x − y | , i.e. errors with symbols of order (1 + | ξ | ) − . Thereforein terms of the principal symbol, it suffices to compute ψ ( x ) χ ( x ) × p det g ( x ) Z e − iy · ξ | g ( x ) y · y | n − d y = c n | g − ( x ) ξ · ξ | − ψ ( x ) χ ( x ) . Finally, these observations show that P has as principal symbol amultiple of | ξ | − g = 1 q g jk ( x ) ξ j ξ k and since the principal symbol of P λ depends smoothly on λ , it doesn’tvanish for λ small enough. This means that for λ small enough, P λ isan elliptic self-adjoint pseudodifferential operator of order − References [1] A. P. Calder´on, On an inverse boundary value problem , Seminar on Numer-ical Analysis and its Applications to Continuum Physics, Soc. Brasileira deMatem´atica, R´ıo de Janeiro, 1980.[2] S. Chanillo, A problem in electrical prospection and a n -dimensional Borg-Levinson theorem , Proc. Amer. Math. Soc. (1990), 761–767.[3] N. S. Dairbekov, G. P. Paternain, P. Stefanov, and G. Uhlmann, The boundaryrigidity problem in the presence of a magnetic field , Adv. Math. (2007),535–609.[4] D. Dos Santos Ferreira, C. E. Kenig, M. Salo, and G. Uhlmann, LimitingCarleman weights and anisotropic inverse problems , Invent. Math. (2009),119–171.[5] D. Dos Santos Ferreira, C. E. Kenig, J. Sj¨ostrand, and G. Uhlmann, Determin-ing a magnetic Schr¨odinger operator from partial Cauchy data , Comm. Math.Phys. (2007), 467–488.[6] B. Frigyik, P. Stefanov, and G. Uhlmann, The X-ray transform for a genericfamily of curves and weights , J. Geom. Anal. (2008), 89–108.[7] C. Guillarmou, A. Sa Barreto, Inverse problems for Einstein manifolds , InverseProbl. Imaging (2009), 1–15.[8] L. H¨ormander, The analysis of linear partial differential operators , SpringerVerlag, 1994. [9] D. Jerison, Carleman inequalities for the Dirac and Laplace operators andunique continuation , Adv. Math. (1986), 118–134.[10] D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigen-values for Schr¨odinger operators , Ann. of Math. (1985), 463–494.[11] A. Katchalov, Y. Kurylev, M. Lassas, Inverse boundary spectral problems ,Monographs and Surveys in Pure and Applied Mathematics 123, ChapmanHall/CRC-press, 2001.[12] C. E. Kenig, A. Ruiz, and C. D. Sogge, Uniform Sobolev inequalities and uniquecontinuation for second order constant coefficient differential operators , DukeMath. J. (1987), 329–347.[13] C. E. Kenig, M. Salo, and G. Uhlmann, Inverse problems for the anisotropicMaxwell equations ,Duke Math. J. (2011), 369–419.[14] C. E. Kenig, J. Sj¨ostrand, and G. Uhlmann, The Calder´on problem with partialdata , Ann. of Math. (2007), 567–591.[15] H. Koch and D. Tataru, Carleman estimates and unique continuation for sec-ond order elliptic equations with nonsmooth coefficients , Comm. Pure Appl.Math. (2001), no. 3, 339–360.[16] H. Koch and D. Tataru, Dispersive estimates for principally normal pseudo-differential operators , Comm. Pure Appl. Math. (2005), 217–284.[17] R. Lavine and A. Nachman, unpublished (personal communication), an-nounced in A. I. Nachman, Inverse scattering at fixed energy , Proceedingsof the Xth Congress on Mathematical Physics, L. Schmdgen (Ed.), Leipzig,Germany, 1991, 434–441, Springer-Verlag.[18] J. Lee, G. Uhlmann, Determining anisotropic real-analytic conductivities byboundary measurements , Comm. Pure Appl. Math., (1989), 1097–1112.[19] A. Nachman, Reconstructions from boundary measurements , Ann. of Math. (1988), 531–576.[20] L. Pestov, G. Uhlmann, Two dimensional compact simple Riemannian mani-folds are boundary distance rigid , Ann. of Math. (2005), 1089–1106.[21] M. Salo, G. Uhlmann, The attenuated ray transform on simple surfaces , J.Diff. Geom. (to appear), arXiv:1004.2323.[22] V. A. Sharafutdinov, Integral geometry of tensor fields , in Inverse and Ill-PosedProblems Series, VSP 1994.[23] Z. Shen, On absolute continuity of the periodic Schr¨odinger operators , IMRN (2001), 1–31.[24] C. D. Sogge, Concerning the L p norm of spectral clusters for second-orderelliptic operators on compact manifolds , J. Funct. Anal. (1988), 123–138.[25] C. D. Sogge, Strong uniqueness theorems for second order elliptic differentialequations , Amer. J. Math. (1990), 943–984.[26] C. D. Sogge, Fourier integrals in classical analysis , Cambridge UniversityPress, 1993.[27] E. M. Stein, Harmonic Analysis: real variable methods, orthogonality and os-cillatory integrals , Princeton University Press, 1993.[28] J. Sylvester, An anisotropic inverse boundary value problem , Comm. PureAppl. Math. (1990), 201–232.[29] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverseboundary value problem , Ann. of Math. (1987), 153–169. [30] G. Uhlmann, Electrical impedance tomography and Calder´on’s problem , InverseProblems (2009), 123011. Universit´e Paris 13, Cnrs, Umr 7539 Laga, 99, avenue Jean-BaptisteCl´ement, F-93430 Villetaneuse, France E-mail address : [email protected] Department of Mathematics, University of Chicago, 5734 Univer-sity Avenue, Chicago, IL 60637-1514, USA E-mail address : [email protected] Department of Mathematics and Statistics, University of Helsinki,PO Box 68, 00014 Helsinki, Finland E-mail address ::