The Poisson embedding approach to the Calderón problem

We introduce a new approach to the anisotropic Calderón problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large class of Calderón type inverse problems for quasilinear equations in the real analytic case. The approach also leads to a new proof of the result of Lassas et al. (Annales de l’ ENS 34(5):771–787, 2001) solving the Calderón problem on real analytic Riemannian manifolds. The proof uses the Poisson embedding to determine the harmonic functions in the manifold up to a harmonic morphism. The method also involves various Runge approximation results for linear elliptic equations.


Introduction
The anisotropic Calderón problem consists in determining a conductivity matrix of a medium, up to a change of coordinates fixing the boundary, from electrical voltage and current measurements on the boundary. In dimensions n ≥ 3 this problem may be written geometrically as the determination of a Riemannian metric on a compact manifold with boundary from Dirichlet and Neumann data of harmonic functions. More precisely, if (M, g) is a compact oriented Riemannian manifold with smooth boundary, we consider the Dirichlet problem for the Laplace-Beltrami operator g , Communicated by Y. Giga. g u = 0 in M, u| ∂ M = f and denote u = u f . We define the Dirichlet-to-Neumann map (DN map) where ∂ ν is the normal derivative on ∂ M. One has the coordinate invariance g = φ * g for any diffeomorphism φ : M → M fixing the boundary.
It is a long-standing conjecture [30] that if two Riemannian manifolds (M 1 , g 1 ) and (M 2 , g 2 ) with mutual boundary have the same DN maps, then there is a boundary fixing isometry between the manifolds. In this paper we give a candidate for this isometry: The points x 1 ∈ M 1 and x 2 ∈ M 2 are to be identified if and only if for every f ∈ C ∞ (∂ M) the harmonic extensions u 1 f and u 2 f of f to (M 1 , g 1 ) and (M 2 , g 2 ) satisfy For general Riemannian manifolds such an identification does not exist. However, if such an identification exists, we show that it induces a mapping M 1 → M 2 , and that this mapping is the boundary fixing isometry required for the solution of the Calderón problem. The Calderón problem thus reduces to showing that the equality of DN maps implies that the above identification exists. In this case we say that the manifolds can be identified by their harmonic functions.
We introduce a tool for studying the existence of this identification. This will be an embedding P of a Riemannian manifold into the linear dual of the space of smooth functions on its boundary. If (M, g) is a compact Riemannian manifold with C ∞ boundary and x ∈ M, then the value of P at x is a linear functional given by the formula where u f ∈ C ∞ (M) solves the Dirichlet problem We call the mapping P the Poisson embedding, and it will be the main object of study of this paper. If P 1 (M 1 ) ⊂ P 2 (M 2 ), then is a well-defined map M 1 → M 2 , and we prove that it is a boundary fixing isometry if n ≥ 3. If n = 2, it is a conformal mapping. The condition P 1 (M 1 ) ⊂ P 2 (M 2 ) is equivalent to existence of the identification (1).
There are different points of view to the Poisson embedding: 1. The Poisson formula for solutions of the Dirichlet problem gives that where G(x, y) is the Green function for g in M and ∂ ν y G(x, · ) is the Poisson kernel. Thus P identifies the point x in M with the Poisson kernel ∂ ν y G(x, · ) on ∂ M (hence the name Poisson embedding). 2. One also has the formula where ω x is the harmonic measure for g at x. Thus P identifies points in M with measures on ∂ M; points of ∂ M are identified with the corresponding Dirac measures, and points in M int are identified with C ∞ functions since dω x = ∂ ν y G(x, · ) d S in this case. 3. For x ∈ M int one has G(x, · ) = 0 on ∂ M, and thus the knowledge of ∂ ν y G(x, · )| ∂ M determines the Green function G(x, · ) in M by elliptic unique continuation. Thus, instead of identifying points of M with the corresponding Green functions in M as in [28], we use the normal derivatives of the Green functions on ∂ M. This change of point of view allows one to work on the boundary, which is natural since the measurements are given on ∂ M.
The problem of finding the isometry, or conformal mapping if n = 2, from the knowledge of the DN map is known as the geometric Calderón problem. This problem has been solved in [28] in the following cases: Theorem Let (M, g) be a compact connected C ∞ Riemannian manifold with C ∞ boundary.
(a) If n = 2, the DN map g determines the conformal class of (M, g). As a first application of our techniques we give new proofs of these results. The proofs consist of three steps.
(1) The first step is to determine the harmonic functions near the boundary, using a standard boundary determination result [30] and real-analyticity. (2) The second step uses a unique continuation argument for harmonic functions where the manifold and harmonic functions are continued simultaneously. The use of harmonic coordinates (see e.g. [9]) and the Runge approximation property are key ingredients in this step.
(3) Finally, we show that we can read the metric (conformal metric if n = 2) from the knowledge of harmonic functions.
The works [27,28,30] study the Calderón problem on real-analytic manifolds, and an analogous result for Einstein manifolds (which are real-analytic in the interior) is proved in [20]. See [26] for a recent result for a conformal Calderón problem. It remains a major open problem to remove the real-analyticity condition in dimensions ≥ 3; for recent progress in the case of conformally transversally anisotropic manifolds see [10,12] and also [11,21] for the linearized problem. Other interesting approaches for related problems may be found in [2] and [5], and counterexamples for disjoint data are given in [7].
The inverse problems may not be uniquely solvable even when the metric is a priori known to be real analytic. Indeed, the above theorem proven in [28] does not hold for non-compact manifolds in the two-dimensional case. It is shown in [27] that there are a compact and a non-compact complete, two-dimensional manifold for which the boundary measurements are the same. This counterexample was obtained using a blow-up map. Analogous non-uniqueness results have been studied in the invisibility cloaking, where an arbitrary object is hidden from measurements by coating it with a material that corresponds to a degenerate Riemannian metric [8,[15][16][17][18].
Our main tool for studying the Poisson embedding and constructing the metric from harmonic functions is the Runge approximation property. This property allows one to approximate local solutions to an elliptic equation by global solutions. In particular this implies that harmonic functions separate points and have prescribed Taylor expansions modulo natural constraints. There are many results of this type in the literature, see e.g. [4,29,31,34]. We require specific approximation results for uniformly elliptic operators, and for completeness they will be given in Appendix A together with proofs.

An inverse problem for quasilinear equations
As a new result we prove a determination result for Calderón type inverse problems for quasilinear equations on manifolds (M, g) with boundary ∂ M, including models that are both anisotropic and nonlinear. We consider the equation where f ∈ C ∞ c (W ), W ⊂ M is a fixed open set, and Q is a uniformly elliptic quasilinear operator of the form Here and below we use Einstein summation rule, which means that repeated indices are always summed over 1, . . . , n. The source-to-solution mapping S : C ∞ c (W ) → C ∞ (W ) for this problem is defined as where u f solves (3). We assume that (M, g) and the matrix valued function A and the function B are real analytic. In this case we show that the source-to-solution mapping, even for small data, determines the manifold and the coefficients A and B up to a diffeomorphism and a built in "gauge symmetry" of the problem: Theorem 1.1 Let (M 1 , g 1 ) and (M 2 , g 2 ) be compact connected real analytic Riemannian manifolds with mutual boundary and assume that Q j , j = 1, 2, are quasilinear uniformly elliptic operators of the form (4) satisfying (22)- (27). Assume that the coefficients A j , B j are real analytic in all their arguments.
Let W j ⊂ M int j , j = 1, 2, be open sets so that there is a diffeomorphism φ : W 1 → W 2 , and assume that the local source-to-solution maps S j for Q j on W j agree, The mapping J satisfies In Theorem 1.1, the maps A, B 1 and J * B 2 have (x, c, σ ) ∈ M 1 × R × T * x M 1 as their argument, and (g) k ab and (J * g) k ab refers to the Christoffel symbols of the metrics g and J * g respectively.
The reason why B can not be determined independently of A and , as presented in (5), is due to the fact that the covariant Hessian in the definition of Q contains first order terms.
The proof proceeds by linearizing the problem and using a slightly modified Poisson embedding for the linearized equation. Using the linearization we can first use the Poisson embedding approach to construct the manifold, but not yet the coefficients A and B. The source-to-solution mapping determines the coefficients in the measurement set W . Since the manifold is now known, the proof is completed by determining the coefficients on the whole manifold by analytic continuation from the set W . The linearization method goes back to [24] and has been used in various inverse problems for nonlinear equations, including anisotropic problems [22,37]. We refer to [35,36] for further references.

Further aspects of Poisson embedding
The governing principle of this paper is that instead of trying to find the metric in the anisotropic Calderón problem directly, one can focus on finding the harmonic functions. This principle is implemented by Poisson embedding and by the fact, which we prove, that the metric can be determined from the knowledge of harmonic functions. We will now discuss in more detail some aspects the Poisson embedding approach.
If J is an isometry (M 1 , g 1 ) → (M 2 , g 2 ), then J can be locally represented in various coordinate charts. Useful coordinate charts include boundary normal coordinates and harmonic coordinates [9,39]. In the study of the Calderón problem, boundary normal coordinates have been used to locally identify real analytic manifolds near boundary points by showing that in boundary normal coordinates the metrics g 1 and g 2 agree. See e.g. [20,[26][27][28]30]. However, local representations do not yield a global candidate for the isometry J required for solving the Calderón problem. In contrast, the Poisson embeddings P j are globally defined objects, and they yield a candidate P −1 2 • P 1 for the isometry (in fact we prove that if P −1 2 • P 1 is well defined, then it gives the required isometry). The representation formula P −1 2 • P 1 also gives uniqueness of the boundary fixing isometry directly if it exists.
In [27] the authors introduce an embedding of real analytic Riemannian manifolds by using Green functions to study the Calderón problem. Their approach involves an analytic continuation argument based on the implicit function theorem applied to the embedding. In contrast, the analogous step for Poisson embedding can be done simply by using harmonic coordinates. This feature also emphasizes the role of choosing suitable coordinates in the study of the anisotropic Calderón problem. Moreover, the recovery of the metric using Poisson embedding is an elementary linear algebra argument that yields a representation of the metric in terms of harmonic functions (see Proposition 4.1). In [27] an asymptotic expansion of Green functions near the diagonal is used.
The basic principle of Poisson embedding is to control the points on a manifold by the values of solutions to Dirichlet problems on the manifold. This principle generalizes to nonlinear equations, to linear systems or to nonlocal operators where Green functions might not be easily accessible. The Poisson embedding also generalizes directly to less regular, say piecewise smooth or C k regular, settings.
Finally, we mention that ideas related to the Poisson embedding have been used in other fields as well. In [19] an embedding by finitely many harmonic functions is used to embed an open Riemannian manifold into a higher dimensional Euclidean space. Their embedding is similar to the Poisson embedding, in the sense that the Poisson embedding parametrizes the manifold by the data of all, instead of a finite number, of harmonic functions on the manifold. Another related result is that when the mapping P −1 2 • P 1 : M 1 → M 2 exists, it is necessarily a harmonic morphism. The study of harmonic morphisms, which are mappings that preserve harmonic functions, has applications in the study of minimal surfaces and in mathematical physics [1]. In Sect. 4 we give a new proof of the characterization of harmonic morphisms as homotheties [1, Corollary 3.5.2] based on harmonic coordinates.

Outline of the paper
In Sect. 2 we introduce the Poisson embedding and its basic properties, in a way that does not directly involve the Calderón problem. In Sect. 3 we determine the harmonic functions from the knowledge of the DN map on real analytic manifolds by using the Poisson embedding. From the knowledge of harmonic functions, we then determine the metric in Sect. 4, which gives a new proof of the main result of [30] in dimension n ≥ 3. Section 5 gives a new proof of the two-dimensional result of [30]. In Sect. 6 we use the Poisson embedding approach and linearization to prove Theorem 1.1, yielding uniqueness in the inverse problem for quasilinear equations. Appendix A is independent of the rest of the paper and contains Runge approximation results.

Poisson embedding
We begin by introducing the Poisson embedding of a Riemannian manifold (M, g) with boundary. Throughout this section, we assume that M is connected and M and g are C ∞ . In particular, we do not require real analyticity in the definition of the Poisson embedding. We will solve Dirichlet problems with boundary values supported on a nonempty open set of the boundary ∂ M. The domain of the Poisson embedding will be where u f (x) solves the Dirichlet problem We call P the Poisson embedding due to the connection with the representation formula for the solution u f (x) in terms of the Poisson kernel where x ∈ M and y ∈ ∂ M, as Here G(x, y) is the Dirichlet Green's function of (M, g) and d S g ∂ M is the induced metric on the boundary. Thus P(x) can be identified with the distribution ∂ ν y G(x, · ) on . In fact, if x ∈ M int then ∂ ν y G(x, · ) is in C ∞ ( ), and if x ∈ then ∂ ν y G(x, · ) = δ x ( · ) so P(x) is always a measure on .
We have the following basic properties of P. Below, for s ∈ R, we write H s (∂ M) for the standard L 2 based Sobolev space on ∂ M, and H s ( ) is defined by restriction, The Fréchet derivative of P at x is a linear mapping given by where u f solves g u f = 0 in M and In the proposition · refers to the canonical pairing of vectors and covectors on M , that is, We omit the proof of Proposition 2.1, which is just an application of standard estimates for solutions of elliptic equations. However, let us formally calculate where the formula for the derivative of P comes from. Let x ∈ M and V ∈ T x M . By definition V is given by a path γ : Let us next show that the mapping P is indeed an embedding, which means a C ∞ injective mapping with injective Fréchet derivative. The main tool that we encounter here for the first time is Runge approximation, which allows one to approximate locally defined harmonic functions by global harmonic functions. It is known since [29,31] that approximation results of this type follow by duality from the unique continuation principle. We have devoted Appendix A to various Runge approximation results. We will mostly use the following consequence, whose proof may also be found in Appendix A. (a) If x ∈ M , y ∈ M and x = y, there is f ∈ C ∞ c ( ) such that Proof Let x 1 , x 2 ∈ M and assume that P(x 1 ) = P(x 2 ). That is, for all boundary value functions f ∈ C ∞ c ( ), we have We need to show that x 1 = x 2 . We argue by contradiction and suppose that This is a contradiction and we must have x 1 = x 2 . Thus P is injective.
To show that the differential of P is injective, let x ∈ M and V ∈ T x M , and assume that D P x V = 0. By the formula (6) for the differential D P x , we have for all boundary value functions f ∈ C ∞ c ( ). To conclude that V = 0, we use Proposition 2.2 again and choose f ∈ C ∞ c ( ) such that ∇u f (x) = V . Here ∇u is the Riemannian gradient. Thus the condition (7) yields showing that V = 0. Thus the differential of P is injective on T M .

Composition of Poisson embeddings
Let (M 1 , g 1 ) and (M 2 , g 2 ) be compact manifolds with mutual boundary ∂ M. One of the aims of this paper is to give a candidate for the isometry that one hopes to construct in the anisotropic Calderón problem. This candidate is We will see that, whenever this mapping is well defined from M 1 to M 2 , it is exactly the mapping that one seeks in the Calderón problem. In dimensions n ≥ 3, it is an isometry. In dimension 2, it is a conformal mapping. It also fixes the boundary if we consider full data problem, or the part ⊂ ∂ M, if we consider partial data problem with measurements made on .
We now begin to study the composition P −1 2 • P 1 . We include considerations related to partial data problem on the part of the boundary. For this purpose we denote throughout this section Since we already know that the Poisson embeddings P j , j = 1, 2, are injective, we know that the mapping is well defined and bijective if the image sets of P j : (M j , g j ) → D ( ) agree: Thus solving the Calderón problem would reduce to verifying this condition from the knowledge that the DN maps of (M 1 , g 1 ) and (M 2 , g 2 ) agree. The next lemma considers the smoothness properties of the mapping J assuming it is defined on some open set of M 1 .

Lemma 2.4
Let (M j , g j ), j = 1, 2, be compact manifolds with mutual boundary ∂ M. Assume that Proof We first note that, writing u j f (x) = P j (x) f , for any f ∈ C ∞ c ( ) one has This follows from the computation P 1 (x) = P 2 (P −1 2 • P 1 (x)) = P 2 • J (x). Now, to show that J (B) ⊂ M 2 , we argue by contradiction and assume that there is some x ∈ B ⊂ M 1 with J (x) ∈ ∂ M\ . But then by (8) which is impossible by Proposition 2.2. Thus J (B) ⊂ M 2 . We next prove that J : B → M 2 is continuous (this perhaps surprisingly uses compactness of M 2 ). Let x ∈ B. If J would not be continuous at x, there would be ε > 0 and a sequence ( . By the compactness of M 2 , passing to a subsequence (still denoted by (x l )), we may assume that J (x l ) converges to y ∈ M 2 . We have d(y, J (x)) ≥ ε. Now, using Proposition 2.2 in M 2 and the fact that J (x) ∈ M 2 , we can find f ∈ C ∞ c ( ) such that the harmonic function The formula (8) shows that Since harmonic functions are continuous, taking the limit l → ∞ yields which is a contradiction. Thus J is continuous. We will next show that J : B → M 2 is C ∞ . This follows an idea from [39] related to smoothness of Riemannian isometries. Fix x ∈ B, and choose harmonic coordinates . This can be done by Since the harmonic functions u j f k , j = 1, 2, are smooth, the smoothness of J near x follows.
It remains to show that the differential of J is invertible on B. Since J : B → M 2 is injective, the claim will then follow from the inverse function theorem.
Together with (6) this gives: The left hand side is equal to du 1 f · X . Thus for any f ∈ C ∞ c ( ) we have the equation This equation will be used again later on and we name it "the equation of injectivity". (This equation can be interpreted as the infinite dimensional counterpart of the chain rule for the composition P 2 • (P −1 2 • P 1 ).) Assume that (D J (x))X = 0 and choose by Proposition 2.2 a harmonic function u 1 f so that the Riemannian gradient satisfies ∇u 1 Thus X = 0. It follows that D J (x) is injective at x and since the manifolds are of the same dimension, it is invertible. This proves the claim.
with dim(M) ≥ 3, the knowledge of the DN-map determines the Riemannian manifold up to isometry. Throughout this section we will assume that the manifolds M j and metrics g j , j = 1, 2, are real analytic, and n = dim(M j ) ≥ 3. We continue to denote M j = M int j ∪ . We first show that near any boundary point there exist coordinates in which the coordinate representations of harmonic functions, corresponding to a common boundary value f , agree. This follows from boundary determination [30] and unique continuation. Assume also that is real analytic. Then, for any p ∈ there are boundary normal coordinates ψ j , j = 1, 2, defined on neighborhoods U j ⊂ M j of p, such that ψ 1 and ψ 2 agree on and such that for any boundary function f ∈ C ∞ c ( ) we have Here the functions u with boundary value f . Proof Let p ∈ ⊂ ∂ M and let ψ j , j = 1, 2, be boundary normal coordinates near p on manifolds (M j , g j ), respectively, so that ψ 1 | = ψ 2 | . Then by the boundary determination result in [30], we have that in these coordinates, the jets of the Riemannian metrics g j agree. Since g j and are real analytic, it follows that ψ j are real analytic coordinate charts, and thus the coordinate representations ψ −1 * j g j of g j agree near with the same Cauchy data (since the DN-maps agree) Thus by elliptic unique continuation [25, Theorem 3.3.1], we havẽ as claimed.
Proof By Lemma 3.1, in its notation, we have that for any p ∈ there exist U j ⊂ M j , j = 1, 2, such that for all f ∈ C ∞ c ( ), we have Thus for We have now shown that knowledge of the DN map on determines harmonic functions near . We proceed in the real analytic case to determine the harmonic functions globally. From this knowledge we will then determine Riemannian manifolds up to isometry in Sect. 4.
In the following result a (global) harmonic morphism means a mapping that preserves solutions to the Dirichlet problem. Precisely, a C ∞ mapping H : Here u 1 f and u 2 f are the solutions to the Dirichlet problems for g 1 and g 2 as usual. For more details on harmonic morphisms, we refer to [1]. (In [1] a harmonic morphism is a mapping that also preserves local harmonic functions instead of just global harmonic functions, but our results will show that there is no difference at least when dim(M 1 ) = dim(M 2 ).) Then we have P 1 (M 1 ) = P 2 (M 2 ) and is a diffeomorphic global harmonic morphism extending F.
Proof Redefine B to be the largest connected open subset of M 1 such that P 1 (B) ⊂ P 2 (M 2 ). By our assumption B is nonempty. We will show that B is closed and thus B = M 1 . Then P 1 (M 1 ) ⊂ P 2 (M 2 ), and from Lemma 2.4 it will follow that It is also a harmonic morphism, since for any f ∈ C ∞ c ( ) and x ∈ M 1 we would have This would prove the claim. We argue by contradiction and assume that B is not closed. Then there is a sequence We actually have that This cannot be true by Proposition 2.2, so indeed x 2 must be in M 2 . Let be harmonic coordinates on a neighborhood 2 ⊂ M 2 of x 2 , as in Lemma 2.4.
Here u 2 f l , l = 1, . . . , n, are global harmonic functions on M 2 with boundary values is well defined by Lemma 2.4. Now comes the punch line: we set We will prove shortly that V is a coordinate system in some neighborhood 1 ⊂ M 1 of x 1 . The map U −1 • V will then give us a local identification of neighborhoods Next we note that we have at p k the equation of injectivity (10) This holds for all X k ∈ T p k M 1 . Since near x 1 on B, we have J = U −1 • V , see (9), we can substitute this to the equation of injectivity yielding is a continuous (in fact smooth) matrix field even though we still do not know if it is invertible. Taking the limit as k → ∞ in (13) gives Thus V is also a coordinate system near x 1 as claimed.
Let us continue on the main line of the proof of the proposition. Let 1 ⊂ M 1 be a neighborhood of x 1 so that V is a coordinate system in 1 , and redefine 1 , if necessary, so that V ( 1 ) ⊂ U ( 2 ). On B we have by (8) What we will show is that Note that we require (15) to hold in 1 , not only in B ∩ 1 . In particular we will have Since 1 in this case extends B to a neighborhood of the point x 1 ∈ ∂ B\B, this will give a contradiction and prove the theorem since M 1 is connected. So far we have not used the assumption of real analyticity, but we use it now to prove (15).
and V and U are local C ω diffeomorphisms in 1 ∩ M int 1 and 2 ∩ M int 2 , respectively. Thus we have where V ( 1 ∩ M int 1 ) ⊂ R n . Since on B ∩ 1 , we have by (11) and (14) that By the real analyticity, stated in (16), this holds on the whole V ( 1 ). This means that the coordinate representations of u 1 f and u 2 f in the (harmonic) coordinates V and U agree.
Given any f ∈ C ∞ c ( ), we have the chain of equivalences Since we have proven the latter, and since the boundary function f was arbitrary, we have proven (15). Consequently we have Thus B extends to a neighborhood of the point x 1 ∈ ∂ B\B, which gives a contradiction. We have now proved that B is closed. Since M 1 is connected, we conclude that Inverting the role of (M 1 , g 1 ) and (M 2 , g 2 ), and replacing F and J in the statement of the theorem by F −1 and J −1 , shows that P 1 (M 1 ) = P 2 (M 2 ) and that J is surjective onto M 2 . Consequently J : M 1 → M 2 is diffeomorphism by Lemma 2.4. Since we already showed that J is a global harmonic morphism at the beginning of the proof in (11), the claim follows.
Combining Lemma 3.2 and Theorem 3.3, we have proved the following statement. Proof We only need to show that J is real analytic in M 1 . In the interior this follows from the representation (9) in terms of harmonic coordinates, which are real analytic in the interior. Near points of this follows from the statement of Lemma 3.1, which implies that near points of one has J = ψ −1 2 • ψ 1 where ψ j are real analytic boundary normal coordinates.

Recovery of the Riemannian metric from a harmonic morphism
In the previous section we showed that the Poisson embedding can be used to determine the manifold up to a global harmonic morphism from the knowledge of the DN map in the real analytic case, n ≥ 3. To give a new proof for the Calderón problem in the real analytic case, we need to show that a global harmonic morphism in this case is an isometry. Throughout this section, we assume that (M j , g j ), j = 1, 2, are compact connected and C ∞ smooth, n ≥ 3.
We show that if J = P −1 2 • P 1 : M 1 → M 2 is defined, and thus is a global harmonic morphism, it is then a homothety, when n ≥ 3. If we additionally assume that the DN-maps of (M 1 , g 1 ) and (M 2 , g 2 ) agree on ⊂ ∂ M, then boundary determination implies that λ = 1.
It is known that a mapping between Riemannian manifolds having the same dimension n ≥ 3 that pulls back local harmonic functions to local harmonic functions is in fact a homothety [13], see also [1,Cor. 3.5.2]. Our definition of harmonic morphisms assumes that the mapping pulls back global harmonic functions to global harmonic functions. Our condition is seemingly slightly different, but it follows from the next result that, for manifolds having the same dimension, these conditions are equivalent.
We give a proof that a global harmonic morphism is in fact a homothety when n ≥ 3 by using harmonic coordinates. This seems to give a new proof for the result in the local case as well. Proof Let x ∈ M 1 and let U = (u 2 f 1 , . . . , u 2 f n ) be an n-tuple of global harmonic functions that define harmonic coordinates on 2 In the coordinates V and U the coordinate representations of harmonic functions In any g j -harmonic coordinates, j = 1, 2, the Laplace-Beltrami operator is particularly simple: Here we abuse notation and denote by g 1 and g 2 the coordinate representations For any symmetric matrix (H i j ) such that g i j 2 (z)H i j = 0 we may find by Runge approximation a global g 2 -harmonic function u = u f with f ∈ C ∞ c ( ) whose Hessian at z is (H i j ) in the g 2 -harmonic coordinates U . This is proved in Proposition A.5 (note that Hess g (w) corresponds to (∂ jk w) in harmonic coordinates). Thus we have that since the coordinate representations of the harmonic functions v = v f and u = u f , and thus their Hessians, agree in the coordinates V and U .
Since H can be any symmetric matrix with g i j 2 (z)H i j = 0, the above means that g −1 1 and g −1 2 have the same orthocomplement at z with respect to the Hilbert-Schmidt inner product in the space of symmetric matrices. Thus g −1 for some nonzero real number λ(z). Due to the positive definiteness of g j , j = 1, 2, we have that λ(z) > 0. The argument above can be repeated for all z ∈ , and we have where λ is a positive smooth function (λ is smooth since g 1 and g 2 are). We next show that the function λ is constant in . For this we use the fact that the coordinates, where (18) holds, are harmonic. This is equivalent to saying that the contracted Christoffel symbols g ab j c ab (g j ) vanish. By lowering the index, this means that . By taking the contracted Christoffel symbol of the Eq. 18, we have Here we have used the following simple computation for the conformal scaling of the contracted Christoffel symbols: Since n ≥ 3, Eq. 19 shows that λ is constant in . Recalling that g 1 (z) and g 2 (z) were the coordinate representations of g 1 and g 2 in coordinates V and U , with V = J * U , we have g 1 = λJ * g 2 on 1 . Since this identity holds near an arbitrary point x ∈ M 1 and since M 1 is connected, we have proved the claim.

Remark 4.2
We remark that in the setting of the proof above we can by Eq. 17 actually express (a multiple of) g −1 j at z in terms of Hessians of solutions u j f at z for some − 1, be a basis for the orthocomplement {g j (z) −1 } ⊥ in the space of symmetric matrices equipped with the Hilbert-Schmidt inner product. By the Runge approximation of Proposition A.5, we may find f k so that H j k = Hess g 1 (u 1 f k (z)) = Hess g 2 (u 2 f k (z)). Then we have Here * is the Hodge star operator in the space of symmetric matrices defined with respect to the Hilbert-Schmidt inner product and ∧ is the wedge product in that space. Equation (20) holds because it follows from the definition of the Hodge star that the right hand side is orthogonal to each H k j . Thus the right hand side has the same orthocomplement as g −1 j (x 0 ) has.
Another remark is that since a homothety maps harmonic functions to harmonic functions, we have that a mapping between same dimensional Riemannian manifolds is global harmonic morphism if and only if it is a local harmonic morphism as defined in [1, Definition 4.1.1].
Next we show that if the DN maps agree, the homothety constant λ is 1. Proof By Proposition 4.1, we know that J is homothety and that Fix a point p ∈ , and use boundary determination (see the proof of Lemma 3.1) to deduce that , the first equation and the fact that J | = Id imply that while the second equation gives that Thus λ = 1.
Combining the results so far, we have achieved a new proof of the uniqueness in the Calderón problem in the real analytic case when n ≥ 3 [28]: Theorem Let (M 1 , g 1 ) and (M 2 , g 2 ) be compact real analytic Riemannian manifolds, n ≥ 3, with mutual boundary whose DN maps agree on an open set ⊂ ∂ M. Assume also that is real analytic. Then there is a real analytic diffeomorphism J : M 1 → M 2 such that g 1 = J * g 2 and J | = I d.

Uniqueness in the 2D Calderón problem
In this section we use the Poisson embedding technique to give a new proof of uniqueness in the Calderón problem in dimension 2. This result is also due to [28]. In this section we assume (M 1 , g 1 ) and (M 2 , g 2 ) are compact, connected C ∞ Riemannian manifolds with mutual boundary ∂ M. Note that it is not required that the manifolds are real analytic. Here λ is a smooth positive function in M 1 , λ| = 1, and J | = Id.
The proof relies on the fact that on two-dimensional manifolds there exist isothermal coordinates near any point, i.e. coordinates (u 1 , u 2 ) such that du 1 = * du 2 , see [38,Section 5.10]. In these coordinates the metric looks like g jk = cδ jk for some positive function c. Isothermal coordinates are also harmonic coordinates in dimension 2. We will use both of these facts.
We first prove local determination of harmonic functions near a boundary point, and then extend local determination to global determination. These are analogues of Lemma 3.2 and Theorem 3.3. After this, a two-dimensional version of Proposition 4.1 determines the metric up to a conformal mapping.
For the determination of harmonic functions near a boundary point, we note that in isothermal coordinates a g-harmonic function actually satisfies the Laplace equation in a subset R 2 . We show that the boundary determination result [30] of the metric in boundary normal coordinates implies determination of the metric on the boundary also in isothermal coordinates. Determination of harmonic functions near the boundary then follows from unique continuation for harmonic functions on R 2 .
The determination of harmonic functions near the boundary in isothermal coordinates involves some technicalities. These are consequences of the fact that the boundary determination result of [30], that we rely on, is given in boundary normal coordinates instead of isothermal coordinates. We address the technicalities in the next lemma. The proof of the lemma follows from the usual construction of isothermal coordinates [38,Section 5.10] and by referring to the boundary determination result [30, p. 1106]. We omit the actual proof. (M 1 , g 1 ) and (M 2 , g 2 )

Moreover, there exists an open set B
Proof Let f ∈ C ∞ c ( ) and let u j f , j = 1, 2, be the corresponding harmonic functions on (M j , g j ). Let U j be the isothermal coordinates of the previous lemma. Then the Cauchy data of u j f agree on˜ ⊂ R 2 in coordinates U j . In isothermal coordinates, which are always also harmonic coordinates in dimension 2, the Laplace-Beltrami equation for u j f reads Thus we see that u j f • U −1 j satisfy the same Euclidean Laplace equation with the same Cauchy data locally on a smooth mutual part of the boundary of the domain. By elliptic unique continuation, see e.g. [25,Theorem 3.3.1], and by setting F = U −1 2 • U 1 we obtain the claim with B replaced by U −1 1 (U 1 ( 1 ) ∩ U 2 ( 2 )). We can then enlarge B as in Lemma 3.2 to conclude the proof.
We record the following: Proposition 5.4 Let (M 1 , g 1 ) and (M 2 , g 2 ) be two-dimensional C ∞ Riemannian manifolds with mutual boundary. Let J : M 1 → M 2 be a locally diffeomorphic C ∞ global harmonic morphism. Then J is conformal, The proof is identical to that of Proposition 4.1 except that we cannot deduce that λ(x) is constant by the argument using Eq. (19). We omit the proof.
We will now prove global determination of harmonic functions.
Then we have P 1 (M 1 ) = P 2 (M 2 ) and Proof We proceed as in the proof of Theorem 3.3 to which we refer the reader for more details. Let us recall the notation and some facts from there. We redefine ∅ = B ⊂ M 1 to be the largest open connected set such that P 1 (B) ⊂ P 2 (M 2 ). The task is to show that B is closed. We argue by contradiction and assume that it is not. Then the points x 1 ∈ ∂ B\B and x 2 ∈ M 2 are limits of sequences ( p k ) ⊂ B and (J ( We construct isothermal coordinates on neighborhoods 1 of x 1 ∈ M 1 and 2 of x 2 ∈ M 2 as follows. Let us first choose by Runge approximation a boundary function f ∈ C ∞ c ( ) such that Let us denote by u 1 a harmonic conjugate of u 2 f near x 2 . This is a function solving near x 2 the equation where * g 2 is the Hodge star of g 2 . A local solution u 1 exists since the right hand side is closed, because u 2 f is harmonic. We may assume u 1 (x 2 ) = 0. Let us denote Then U is an isothermal coordinate system on a neighborhood 2 of x 2 . Likewise, let v 1 be a harmonic conjugate of u 1 f near x 1 with v 1 (x 1 ) = 0. Thus v 1 solves We set These are isothermal coordinates on a neighborhood 1 of x 1 . The fact that the Jacobian of V is invertible at x 1 follows from Since Hodge star is conformally invariant when operating on 1-forms in dimension 2, the above is (The point here is that U −1 • V is defined on the whole 1 and gives us a good candidate for a local extension of J onto the whole 1 .) Let f ∈ C ∞ c ( ). To conclude the proof, we will show that on V ( 1 ), we have Since  B). Since the coordinates in question are isothermal and harmonic (where a (g j ) = 0) we have that .
Since these functions agree on the open set V ( 1 ∩ B) they agree everywhere on V ( 1 ) by unique continuation. This shows that we may indeed extend J to B ∪ 1 , which gives a contradiction and concludes the proof.

On determining the coefficients of quasilinear elliptic operators from source-to-solution mapping
In this section we apply the Poisson embedding technique for a Calderón type inverse problem for second order quasilinear elliptic operators on Riemannian manifolds. Let (M, g) be a Riemannian manifold with boundary. The quasilinear operators we study are assumed to have the coordinates representation Here we assume that A ab (x, c, σ ) ∈ T 2 x M for given (x, c, σ ) ∈ M × R × T * x M is a 2- tensor field and B(x, c, σ ) is a function. The covariant derivative ∇ is determined by g as usual. We consider A and B as mappings and where π (A(x, c, σ )) = x. Here π is the canonical projection T 2 0 (M) → M for 2-contravariant tensors. The notation M × R ⊗ T * M refers to the subset We can also think of M × R ⊗ T * M as tensor product of the trivial line bundle M × R and the vector bundle T * M. We assume that Q is quasilinear elliptic, which means that for all (x, c, σ ) ∈ M × R ⊗ T * M and ξ ∈ T * x M we have We assume that the coefficients A and B are C ∞ smooth and in the main theorem of this section, Theorem 6.2, we assume that they are real analytic. We will assume throughout that 0 is a solution, i.e. Q(u) = 0, which is equivalent with the condition The linearization of Q at u = 0 is the operator where in the last term we have identified T * x M with R n . We will also assume that, for some fixed α with 0 < α < 1, It follows from these assumptions and from the implicit function theorem in Banach spaces that there are unique small solutions of Q(u) = f , u| ∂ M = 0 when f is small. We record these facts in the following lemma. We omit the proof.
Assume that L is the linearization of Q at u = 0 given in (26), and assume that (27) holds.
There are constants C, ε, δ > 0 such that whenever f C α (M) ≤ ε, the equation Operators of the above form appear e.g. in the study of minimal surfaces or prescribed scalar curvature questions (Yamabe problem), see [14,38] for more information.
We describe the inverse problem we are about to study. Let Q, ε, δ be as in Proposition 6.1. We assume that we know the source-to-solution mapping of Q on an open subset W of M. The source-to-solution mapping is defined as where u is the unique solution to Note that indeed u ∈ C ∞ (M), using Schauder estimates for linear elliptic equations and the fact that u ∈ C 2,α (M). The aim of the inverse problem is to determine the coefficients A and B of Q up to a diffeomorphism and possible other symmetries of the problem. When the coefficients of Q are real analytic, our main theorem shows that in this case there is only one additional symmetry, which we describe next. We note that there is a simple transformation between coefficient B and the Christoffel symbols k ab contracted by A ab that leaves the source-to-solution mapping intact: Assume that u solves Then u also solvesQ and whereB is defined as Here∇ and˜ k ab denote the Levi-Civita connection and Christoffel symbols of some other Riemannian metricg on M. Therefore the source-to-solution mapping defined with respect toQ coincides with the source-to-solution map S of Q. Note that even though Christoffel symbols does not constitute a tensor field, the difference of two Christoffel symbols˜ k ab − k ab is a tensor field. It follows that we can not make the symmetry vanish by choosing a suitable coordinate system. This symmetry will be called the gauge symmetry of the inverse problem.
The gauge symmetry is an obstruction for finding B, A and k ab independently of each other in the general case. However, in some cases when we have extra information about the coefficients A and B, the gauge symmetry vanishes. We give examples of conditions when this happens in Corollary 6.3.
We remark that if the coefficients are not real analytic, other symmetries in the inverse problem can appear. An easy example is the standard Laplace-Beltrami operator in dimension 2 where one can scale the metric by a positive function that is constant 1 on the measurement set W without affecting the source-to-solution mapping. Another similar example is given by the conformal Laplacian in dimensions n ≥ 3 [26].
Our main theorem of this section is the following determination result. (M 1 , g 1 ) and (M 2 , g 2 ) be compact connected real analytic Riemannian manifolds with mutual boundary and assume that Q j , j = 1, 2, are quasilinear operators of the form (21) having coefficients A j , B j satisfying (22)- (27). Moreover, assume that A j and B j are real analytic in all their arguments. Let W j , j = 1, 2, be open subsets of M j , and assume that there is a diffeomorphism φ : W 1 → W 2 so that the source-to-solutions maps S j for Q j agree in the sense that

Theorem 6.2 Let
The mapping J satisfies commutes when f is small. We describe our strategy for proving the theorem. By the arguments in the preceding sections and by the fact that solutions to quasilinear equations with real analytic coefficients are real analytic [33], it would be natural to define a mapping analogous to the Poisson embedding for the quasilinear elliptic operator Q and then use tools analogous to those we built around the Poisson embedding. However, as far as we know, Runge approximation for quasilinear operators is not known. This prevents us of using this natural approach for the moment.
Instead we do the following. We linearize the source-to-solution mapping (at the mutual solution 0) that yields a linear Calderón type inverse problem for a linear second order elliptic operator whose source-to-solution map is known. For this linearized problem we use the Poisson embedding technique modified slightly to deal with the source-to-solution map instead of the DN map. In this way we will find the manifold up to a real analytic diffeomorphism. This is the first step. The modified Poisson embedding is given in Definition 6.1.
The second step is the following. We will see that knowing the source-to-solution map on the open set W determines the coefficients A and B on W , up to the gauge symmetry described in (32). In this step we read the coefficients A and B in W from the solutions, which is similar to the argument in Proposition 4.1.
Since we have determined the manifold up to a real analytic diffeomorphism, we can view the coefficients A and B on a single fixed manifold and use standard real analytic unique continuation there. This determines the coefficients of the quasilinear operator on the whole manifold up to a diffeomorphism and the gauge symmetry.
As already mentioned, with some suitable extra information about the coefficients B and A the gauge symmetry vanishes and we can determine A and B independently. Corollary 6.3 Assume the conditions and notation in Theorem 6.2, and assume also one of the following: (1) A 1 (x, c, σ ) (or A 2 (x, c, σ )) is s-homogeneous in the σ -variable and B 1 (x, c, σ ) and B 2 (x, c, σ ) are s -homogeneous with s = s + 1 for all x ∈ W 1 and c ∈ R; or is an isometry.
Then we have

and also
A ab Note that the corollary does not claim that we can find the Riemannian metrics g 1 and g 2 up to J . An example satisfying the first condition is the nonlinear Schrödinger operator An example satisfying the second condition is the case, where one knows the Riemannian metric and the manifold (φ = Id) on a measurement set W .

Linearized problem
Let us first linearize the source-to-solution map of the quasilinear problem. This yields a Calderón type inverse problem for the linearized equation. We record the following result. We leave out the proof of the result and refer to [22] for a proof of a similar result. (29). Then, for any f ∈ C ∞ c (W ),

Proposition 6.4 Let (M, g) and Q be as in Proposition 6.1. Let W ⊂ M be open, and let S be the source-to-solution map defined in
where L is given in (26).
We have now seen that linearizing the source-to-solution mapping S of a quasilinear equation leads to a Calderón type inverse problem for a linear equation. Next we show that in the real analytic case the source-to-solution mapping of the linearized problem determines the manifold up to a real analytic diffeomorphism. Precisely we will prove: Theorem 6.5 Let (M 1 , g 1 ) and (M 2 , g 2 ) be compact real analytic manifolds with mutual boundary. Let L 1 and L 2 be second order uniformly elliptic partial differential operators on M 1 and M 2 , respectively, of the general form where A ab j and B a j are the components of a 2-tensor field A and vector field B j , and C is a function. Assume that L j are injective on C ∞ (M j ) ∩ H 1 0 (M j ), and assume that A, B and C are real analytic up to boundary.
Let W j ⊂ M j be open sets, and let φ : W 1 → W 2 be a diffeomorphism so the source-to-solutions maps S L j of L j satisfy Then there is a real analytic diffeomorphism J : In the theorem u 1 f •φ and u 2 f are the solutions to We prove the theorem by modifying the Poisson embedding technique to suit the study of source-to-solution map instead of the DN map. The arguments are very similar to those that we have used in the previous sections. We keep the exposition short. In the definition D (W ) means [C ∞ c (W )] * as usual. The reason why we consider R to be defined only in the interior of M is because we assume that the boundary values of the solutions u f vanish on the boundary. Thus we have no control on the points on the boundary by using solutions u f . Even though R is not the Poisson embedding of the previous section, we use the same name for R, and we note that R is related to the linear elliptic operator L.
The basic properties of the Poisson embedding R are as follows.
, and · refers to the canonical pairing of vectors and covectors on M.
In the statement, we are not claiming that the continuation of R onto M is injective on ∂ M. We omit the proof of Proposition 6.6.
To prove that R is an embedding, we use the following analogue of Proposition 2.2 which follows from a suitable Runge approximation result. Its proof is in Appendix A. Proposition 6.7 Let (M, g) be a compact manifold with boundary, and let L be a second order uniformly elliptic differential operator on M which is injective on Proof The injectivity of R follows from Proposition 6.7(a). Let x ∈ M int and V ∈ T x M.

be a nonempty open subset of M, and denote by u f the solution of Lu
. By Proposition 6.7(b) one can find f ∈ C ∞ c (W ) so that du f (x) · V = 0, unless V = 0. This shows injectivity of the differential.
We construct next local coordinate systems from solutions u f to Lu f = f , u f | ∂ M = 0. We call these coordinates solution coordinates. These coordinates are constructed by Runge approximating local solutions to Lu = 0. Proof Let x 0 ∈ M int . If x 0 ∈ W , we redefine W as a smaller open set such that x 0 / ∈ W . By Proposition 6.7(b) we may find f 1 , . . . , f n ∈ C ∞ c (W ) such that the Jacobian matrix of U = (u f 1 , . . . , u f n ) is the identity matrix at x 0 . Thus U is a coordinate system in some neighborhood of x 0 , and by shrinking if necessary we may assume that f j = 0 in .
If the coefficients of L are real analytic, the coordinate system U is real analytic on by elliptic regularity, since f j | = 0.
The next lemma is an analogue of Lemma 2.4, where we considered the composition of Poisson embeddings. If W 1 and W 2 are open subsets of M 1 and M 2 and φ : W 1 → W 2 is a diffeomorphism, we define a mapping φ * R 2 : with x ∈ M int 2 and f ∈ C ∞ c (W 1 ). From Propositions 6.6 and 6.8, it follows that φ * R 2 is a C k embedding M Proof We prove the continuity of J differently than we did in the corresponding situation in Lemma (2.4). It follows from Proposition 6.6 that we may continue φ * R 2 by zero to a continuous mapping M 2 → H −s−1 (W 1 ). Let E ⊂ M int 2 be closed in M 2 . It follows that φ * R 2 (E) is closed in H −s−1 (W 1 ) by continuity and by compactness of M 2 . Since φ * R 2 is injective on M int 2 , we have that I := (φ * R 2 ) −1 is defined as a mapping φ * R 2 (M int 2 ) → M int 2 and we have that I −1 (E) equals φ * R 2 (E), which is closed. Thus I is continuous and consequently J is continuous B → J (B) by Proposition 6.6.
Let x 0 ∈ M int 1 and let U = (u 2 f 1 , . . . , u 2 f n ) be solution coordinates of Lemma 6.9 on a neighborhood 2 In case x 0 ∈ W 1 , we may assume that φ( and It follows from Proposition 6.7 that D J (x 0 ) is invertible. Since J is also injective by Proposition 6.8, it follows that J : We prove next the main result of this subsection.
Proof of Theorem 6.5 Assume that the source-to-solutions maps agree: By shrinking W j if necessary, we may assume that W j are small geodesic balls. Redefine B to be the largest connected open set of M int 1 such that To see this, let f ∈ C ∞ c (W 1 ) and x ∈ W int 1 . We have by using definitions and (35) that Note that φ(x) ∈ M int 2 since otherwise we have u 1 , which by Proposition 6.7 leads to a contradiction to the fact that x ∈ M int 1 . We conclude that B = ∅. We show that B is closed in M int 1 and thus ) and thus, by the definition (34), By (36), we directly also have that To show that B is closed in M int 1 , we argue by contradiction and let p k → x 1 ∈ ∂ B\B, with p k ∈ B. Then x 1 ∈ M int 1 . By passing to a subsequence, we have x 2 := lim k J ( p k ) ∈ M 2 . We have that x 2 ∈ M int 2 ; otherwise using (37) with x = p k and taking the limit as k → ∞ would imply that , which would contradict Proposition 6.7(a). Let be solution coordinates as in Lemma 6.9 on a neighborhood 2 ⊂ M int 2 of the point x 2 . Here f j ∈ C ∞ c (W 2 ), j = 1, . . . , n, and the limit x 2 is found by first passing to a subsequence.
We set We have J = U −1 • V on B near x 1 by (37). We have that V is a solution coordinate system on a neighborhood 1 ⊂ M int 1 of x 1 , and the "equation of injectivity" holds for X ∈ T x 1 M 1 . These facts can be proved similarly following the argument in Theorem 3.3. We redefine the sets 1 and 2 , if necessary, so that we have V ( 1 ) = U ( 2 ). We next show that

Local determination of the coefficients
We determine the coefficients of a quasilinear elliptic operator on open sets where the source-to-solution mapping of the operator is known. Precisely, we prove the following: Proposition 6.11 Let (M 1 , g 1 ) and (M 2 , g 2 ) be compact connected Riemannian manifolds with mutual boundary and assume that Q j , j = 1, 2, are quasilinear operators of the form (21) having coefficients A j , B j satisfying (22)- (27). Let W j , j = 1, 2, be open subsets of M j , and assume that there is a diffeomorphism φ : W 1 → W 2 so that the source-to-solutions maps S j for Q j agree in the sense that 1 be an open set. Then there is δ > 0 such that for all x ∈ W and (c, σ ) ∈ T x W , with |c| + |σ | g 1 ≤ δ, we have and To prove the proposition, we begin with the following observation.
Lemma 6.12 Assume the conditions and notation in Proposition 6.11. Let x ∈ W int 1 and let U 2 be coordinates on a neighborhood 2 HereQ j are the coordinate representations of Q j in coordinates U j , j = 1, 2.
Proof Let x ∈ W int 1 and let U 1 and U 2 be coordinate systems as in the statement of the lemma. Let ε j , δ j > 0, j = 1, 2, be as in the Definitions (28)-(30) of the source-tosolution mapping of Q j . Set ε = min(ε 1 , ε 2 ). Let δ > 0 be such Such a δ can be found since B 2 (x, 0, 0) = 0 by Assumption (25), since Q 2 : C 2,α (W 2 ) → C α (W 2 ) is continuous and since composing a compactly supported function with C ∞ diffeomorphism is continuous on Hölder spaces. Let has a unique solution u 2 F with u 2 F C 2,α (M 2 ) ≤ δ 2 by Proposition 6.1. Let us extend v • U 2 by zero onto M 2 . By the condition B 2 (x, 0, 0) = 0 and we have Since v • U 2 C 2,α (M 2 ) ≤ δ 2 , by uniqueness we have that By the definition of δ we have that Since the source-to-solutions mappings agree, we have By using φ| 1 = U −1 2 • U 1 it follows that We also have on W 2 that Using the coordinate invariance of Q j , we have by (43)-(45) that The lemma tells us that we can use any test function v with small enough C 2,α norm to solve for the coefficients ofQ j from the equationQ 1 v =Q 2 v in the coordinates U 1 and U 2 . The local determination result of Proposition 6.11 is a consequence of this observation. Its proof is similar to that of Proposition 4.1 where we used harmonic functions to solve for (a multiple of) the metric in the Calderón problem.
Proof of Proposition 6.11 Let x 0 ∈ W int 1 and let U 2 be coordinates on a neighborhood 2 ⊂⊂ W int 2 of φ(x 0 ) ∈ W int 2 and let U 1 = φ * U 2 be coordinates on a neighborhood 1 = φ −1 ( 2 ) of x 0 . Denote = U 2 ( 2 ) = U 1 ( 1 ). By the Lemma 6.12 above, we have that there is δ > 0 such that for all v ∈ C ∞ c ( ) with v C 2,α ( ) ≤ δ we haveQ HereQ j are the coordinate representations of Q j in coordinates U j , j = 1, 2.
There is δ > 0 such that for all y ∈ and (c, Here A is the Hilbert-Schmidt norm of matrices and δ is independent of y ∈ . Let y ∈ and let (c, σ ) ∈ R × T * y with |c| + |σ | g 2 ≤ δ. Let A be a matrix with A ≤ δ and let v = v (y,c,σ,A) be the function defined in (47). Since v C 2,α ( ) < δ by (48), Eq. 46 implies that for x ∈ . Here k j,ab are the Christoffel symbols of g j , and A j and B j are understood as their coordinate representations in coordinates U j .
Let us first choose as the matrix A a one with A ≤ δ and that satisfies have Here ⊥ is defined with respect to the Hilbert-Schmidt inner product of matrices. We have v C 2,α ( ) ≤ 2δ and thus Eq. 49 holds. It follows that We next choose the matrix A as where ρ > 0 is a number small enough so that A ≤ δ. It follows from (49) and by using (50) that where · is the Hilbert-Schmidt norm of matrices. Thus which combined with (50), yields Equations (51) and (52) hold for all (y, c, σ ) ∈ × R ⊗ T * with |c| + |σ | g 2 ≤ δ. Recall that (51) and (52) are equations for the coordinate representations of A 1 , B 1 and A 2 , B 2 in coordinates U 1 and U 2 respectively, where U 1 = φ * U 2 . Thus we have, by redefining δ if necessary, that for (x, c, σ ) ∈ (φ −1 ) × R ⊗ T * (φ −1 ) with |c| + |σ | g 1 ≤ δ. Let W be an open set compactly contained in W int 1 . Since we may carry out the argument above on a neighborhood of any point x 0 ∈ W int 1 , we have the claim by compactness for some δ > 0.

Global determination of the coefficients
We prove the main theorem of this section.
Proof of Theorem 6.2 By Theorem 6.5 we know that there is a real analytic diffeomor- It follows that J * A 2 and J * B 2 , which are given by and are real analytic for all x ∈ M int 1 and (c, σ ) ∈ R × T * x M int 1 . By Proposition 6.11 we have that there is a non-empty open set W ⊂ W int 1 and δ > 0 such that for all (c, σ ) ∈ R × T * x W with |c| + |σ | g 1 ≤ δ, we have c, σ ) and where A = A 1 = J * A 2 . By real analyticity, we have (54) for all x ∈ M int 1 and Proof of Corollary 6.3 (1) Let (x, c, σ ) ∈ M int 1 ×R⊗ T * M int 1 . Assume that A 1 (x, c, σ ) is s-homogeneous in the σ -variable and that B 1 and B 2 are s -homogeneous with s = s + 1. It follows from (53) that J * B 2 is s -homogeneous in the σ -variable. By Theorem 6.2 we know that J * A 2 = A 1 is s + 1 homogeneous in the σ -variable. Thus is s -homogeneous, s = s + 1, in the σ -variable, we must have (2) Assume that φ is an isometry. Since J | W int ab is a real analytic tensor field, which vanishes on W int 1 , it vanishes on M int Proof Let r ∈ R, and consider the map  (N ) ; A * v = 0}, and the range of T r is given by In particular, the kernel and cokernel are independent of r . Writing E for the orthocomplement of N * in H −s+2 (N ) we get that the operator T : E → A * (H −s+2 (N )), T u = A * u is continuous and bijective, hence invertible by the open mapping theorem, and u = T −1 F is the unique minimal norm solution of A * u = F. We will also write u = (A * ) −1 F. This proves the first statement. Assume now that F| U ≡ 0, let V ⊂⊂ U and let m ≥ 0. Let χ, χ 1 ∈ C ∞ c (U ) satisfy χ = 1 near V and χ 1 = 1 near supp(χ ). Let also Q ∈ −2 (N ) be a parametrix for A * , so that using that F| U ≡ 0. Using the pseudolocal property of Q and the fact that R is smoothing, we get the estimate We will also need the following simple lemma. (N , g) be a closed manifold, let A be a second order uniformly elliptic differential operator on N , let N be the kernel of A, and let {ψ 1 , . . . , ψ m } be an L 2 (N )-orthonormal basis of N . Let also V ⊂ N be any nonempty open set. There exist η 1 , . . . , η m ∈ C ∞ c (V ) such that
Since the matrix S is invertible, T is surjective.
By the Hahn-Banach theorem, it is enough to show that F(u) = 0 for all u ∈ S.
Let (M, g) ⊂⊂ (M 1 , g) ⊂⊂ (N , g), where M 1 is a compact manifold with boundary and N is a closed connected manifold. We extend the coefficients of L to N without destroying the ellipticity to define an elliptic second order operator A in N (thus Aw| M = L(w| M ) for w ∈ C ∞ (N )). Also, defineF ∈ H −s (N ) bŷ We defineF ∈ H −s (N ) with supp(F) ⊂ U ∪ V bỹ It will be convenient to choose smooth approximationsF j =F j + η j , whereF j ∈ C ∞ c (M int ) has support near U andF j →F in H −s (N ) and η j ∈ C ∞ c (V ) is defined as in (55) but withF replaced byF j . ThenF j →F in H −s (N ) as j → ∞. We let v j ∈ C ∞ (N ) be the minimal norm solution of In particular, one has A * v = 0 near ∂ M, so v is C ∞ near ∂ M by elliptic regularity. We will next solve In the last expression, E(P f ) is supported in M int 1 and A * v j | M 1 vanishes outside a neighborhood of U . Thus this expression can be understood as an integral over M.
Integrating by parts, we get Here ∂ A * ν is the normal derivative associated with A * , and we used that ϕ j | ∂ M = 0. Now, if W is a small neighborhood of ∂ M, the higher regularity estimate in Lemma A.3 implies that v j → v in H m (W ) for any m ≥ 0. Thus one has ∂ A * ν ϕ j → ∂ A * ν ϕ in H m (∂ M) for any m ≥ 0, and consequently Since the previous result holds for all f ∈ C ∞ c ( ), we see that ϕ solves Also, ϕ is smooth in M\U . The unique continuation principle implies that ϕ| M\U = 0.
Thus ϕ may be identified with an element of H −s+2 Using that ψ j ∈ C ∞ c (U ) and Lu = 0 in U , we may integrate by parts in U and obtain that F(u) = 0 for all u ∈ S. This concludes the proof.
As a consequence of the Runge approximation property we can find global harmonic functions with prescribed 2-jet at a given point, up to the natural restrictions given by the equation g u = 0. The existence of a local harmonic function with this property is found e.g. in [1, Lemma A.1.1]), but we give the details for completeness. Thus, assume that p ∈ M int . The proof will be given in four steps.
Step 1. First we find a local g( p)-harmonic function with prescribed 2-jet at p. Let x = (x 1 , . . . , x n ) be normal coordinates in a small geodesic ball U centered at p. In these coordinates, let ξ 0 = (ξ 0 ) j dx j | p and H 0 = H jk dx j ⊗ dx k | p . Define the function u 0 : U → R, u 0 (exp p (x j ∂ j | p )) = a 0 + (ξ 0 ) j x j + 1 2 H jk x j x k .
Clearly u 0 ∈ C ∞ (U ), u 0 ( p) = a 0 and du 0 ( p) = ξ 0 . The Hessian, computed in normal coordinates, is given by since l jk | p = 0 in normal coordinates. Since H 0 is trace free it follows that g( p) u 0 = n j=1 H j j = 0 in U , i.e. u 0 is harmonic in U with respect to the metric g( p) with coefficients frozen at p.
We can make the 2-jet at p exactly equal to (0, 0, H 0 ) as follows. Consider the operator Step 3. We will next find a local g-harmonic function near p with 2-jet (a 0 , ξ 0 , H 0 ). In fact, the argument in Step 2 with the choice u 0 (x) = (ξ 0 ) j x j and scalingũ(x) = ε −1 u(εx) leads to local harmonic functions with 1-jet at p first arbitrarily close to (0, ξ 0 ), and then exactly equal to (0, ξ 0 ) as in the end of Step 2. Adding one of the functions obtained in Step 2 yields a local harmonic function with 2-jet (0, ξ 0 , H 0 ), and adding a constant gives the 2-jet (a 0 , ξ 0 , H 0 ).
Step 4. Finally, to find a global harmonic function with prescribed 2-jet at p, consider the operator where u f is the harmonic function in M with u| ∂ M = f . Given any (a 0 , ξ 0 , H 0 ) ∈ R × T * p M × (S 2 tf ) p M, Step 3 shows that there is a local harmonic function u 0 in a small geodesic ball U = B ε ( p) having 2-jet (a 0 , ξ 0 , H 0 ) at p. Now Proposition A.2 implies that there is a sequence ( f j ) ⊂ C ∞ c ( ) such that u f j | U → u 0 in H n/2+3 (U ), hence u f j → u 0 in C 2 (U ) by Sobolev embedding. This shows that T (C ∞ c ( )) is a dense subspace of R × T * p M × (S 2 tf ) p M, but since the last space is finite dimensional T has to be surjective. with supp(η) ⊂ V , where V ⊂ N \M 1 , andF is L 2 -orthogonal to the kernel of A. Moreover we may findF j ∈ C ∞ c (M int ∪ V ) withF j →F in H −s (N ), and using the same procedure as in the proof of Proposition A.2 we can representF andF j as Similarly, in M we have In the last expression, E(K f ) is supported in M int 1 and A * v j | M 1 vanishes outside a neighborhood of U ⊂ M int . Thus this expression can be understood as an integral over M. Integrating by parts, we get 0 = lim (L * ϕ j , K f ) M = lim(ϕ j , L K f ) M = lim(ϕ j , f ) W .
Here we used that K f and ϕ j vanish on ∂ M and thus the boundary terms vanish. The higher regularity estimate in Lemma A.3 implies that ϕ j → ϕ in H m (W ) for any m ≥ 0, and consequently 0 = (ϕ, f ) W .
Since the previous result holds for all f ∈ C ∞ c (W ), we see that ϕ solves Also, ϕ is smooth in the connected set M\U . The unique continuation principle implies that ϕ| M\U = 0. Thus ϕ ∈ H −s+2 U (N ). As in the proof of Proposition A.2, we have ψ j ∈ C ∞ c (U ), ψ j → ϕ in H −s+2 (N ). If E denotes a bounded extension operator from H s (U ) to H s (N ) with supp(Ew) ⊂ M int for any w ∈ H s (U ), we have If u ∈ S, we can integrate by parts to see that F(u) = 0.
As a consequence, we obtain an analogue of Proposition A.1 where boundary sources are replaced by interior sources.