# The Poisson embedding approach to the Calderón problem

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## Abstract

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.

## 1 Introduction

*M*,

*g*) is a compact oriented Riemannian manifold with smooth boundary, we consider the Dirichlet problem for the Laplace–Beltrami operator \(\Delta _g\),

For general Riemannian manifolds such an identification does not exist. However, if such an identification exists, we show that it induces a mapping \(M_1\rightarrow M_2\), 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*.

*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 \(C^\infty \) boundary and \(x\in M\), then the value of

*P*at

*x*is a linear functional given by the formula

*P*the

*Poisson embedding*, and it will be the main object of study of this paper. If \(P_1(M_1) \subset P_2(M_2)\), then

- 1.The Poisson formula for solutions of the Dirichlet problem gives thatwhere$$\begin{aligned} P(x) f = \int _{\partial M} \partial _{\nu _y} G(x,y) f(y) \,dS(y) \end{aligned}$$
*G*(*x*,*y*) is the Green function for \(\Delta _g\) in*M*and \(\partial _{\nu _y} G(x,\,\cdot \,)\) is the Poisson kernel. Thus*P*identifies the point*x*in*M*with the Poisson kernel \(\partial _{\nu _y} G(x,\,\cdot \,)\) on \(\partial M\) (hence the name Poisson embedding). - 2.One also has the formulawhere \(\omega ^x\) is the$$\begin{aligned} P(x) f = \int _{\partial M} f \,d\omega ^x \end{aligned}$$
*harmonic measure*for \(\Delta _g\) at*x*. Thus*P*identifies points in*M*with measures on \(\partial M\); points of \(\partial M\) are identified with the corresponding Dirac measures, and points in \(M^{\mathrm {int}}\) are identified with \(C^{\infty }\) functions since \(d\omega ^x = \partial _{\nu _y} G(x,\,\cdot \,) \,dS\) in this case. - 3.
For \(x \in M^{\mathrm {int}}\) one has \(G(x,\,\cdot \,) = 0\) on \(\partial M\), and thus the knowledge of \(\partial _{\nu _y} G(x,\,\cdot \,)|_{\partial M}\) determines the Green function \(G(x,\,\cdot \,)\) 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 \(\partial M\). This change of point of view allows one to work on the boundary, which is natural since the measurements are given on \(\partial M\).

### Theorem

*M*,

*g*) be a compact connected \(C^{\infty }\) Riemannian manifold with \(C^{\infty }\) boundary.

- (a)
If \(n=2\), the DN map \(\Lambda _g\) determines the conformal class of (

*M*,*g*). - (b)
If \(n \ge 3\) and if

*M*, \(\partial M\) and*g*are real-analytic, then the DN map \(\Lambda _g\) determines (*M*,*g*).

- (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 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.

### 1.1 An inverse problem for quasilinear equations

*M*,

*g*) with boundary \(\partial M\), including models that are both anisotropic and nonlinear. We consider the equation

*Q*is a uniformly elliptic quasilinear operator of the form

*M*,

*g*) and the matrix valued function \(\mathcal {A}\) and the function \(\mathcal {B}\) are real analytic. In this case we show that the source-to-solution mapping, even for small data, determines the manifold and the coefficients \(\mathcal {A}\) and \(\mathcal {B}\) 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 \(\mathcal {A}_j\), \(\mathcal {B}_j\) are real analytic in all their arguments.

*J*satisfies

In Theorem 1.1, the maps \(\mathcal {A}\), \(\mathcal {B}_1\) and \(J^*\mathcal {B}_2\) have \((x,c,\sigma )\in M_1\times \mathbb {R}\times T^*_xM_1\) as their argument, and \(\Gamma (g)_{ab}^k\) and \(\Gamma (J^*g)_{ab}^k\) refers to the Christoffel symbols of the metrics *g* and \(J^*g\) respectively.

The reason why \(\mathcal {B}\) can not be determined independently of \(\mathcal {A}\) and \(\Gamma \), 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 \(\mathcal {A}\) and \(\mathcal {B}\). 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.

### 1.2 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 \((M_1,g_1)\rightarrow (M_2,g_2)\), 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 \(P_2^{-1} \circ P_1\) for the isometry (in fact we prove that if \(P_2^{-1} \circ P_1\) is well defined, then it gives the required isometry). The representation formula \(P_2^{-1}\circ P_1\) 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 \(P_2^{-1}\circ P_1:M_1\rightarrow M_2\) 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.

### 1.3 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 \(n\ge 3\). 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.

## 2 Poisson embedding

*M*,

*g*) with boundary. Throughout this section, we assume that

*M*is connected and

*M*and

*g*are \(C^\infty \). 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 \(\Gamma \) of the boundary \(\partial M\). The domain of the Poisson embedding will be

### Definition 2.1

*Poisson embedding*) Let (

*M*,

*g*) be a compact Riemannian manifold with boundary, and let \(\Gamma \) be a nonempty open subset of \(\partial M\). The

*Poisson embedding*of the manifold

*M*is defined to be the mapping

*P*the Poisson embedding due to the connection with the representation formula for the solution \(u_f(x)\) in terms of the

*Poisson kernel*

*G*(

*x*,

*y*) is the Dirichlet Green’s function of (

*M*,

*g*) and \(dS_{g_{\partial M}}\) is the induced metric on the boundary. Thus

*P*(

*x*) can be identified with the distribution \(\partial _{\nu _y} G(x,\,\cdot \,)\) on \(\Gamma \). In fact, if \(x \in M^{\mathrm {int}}\) then \(\partial _{\nu _y} G(x,\,\cdot \,)\) is in \(C^{\infty }(\Gamma )\), and if \(x \in \Gamma \) then \(\partial _{\nu _y} G(x,\,\cdot \,) = \delta _x(\,\cdot \,)\) so

*P*(

*x*) is always a measure on \(\Gamma \).

We have the following basic properties of *P*. Below, for \(s \in \mathbb {R}\), we write \(H^s(\partial M)\) for the standard \(L^2\) based Sobolev space on \(\partial M\), and \(H^s(\Gamma )\) is defined by restriction, i.e. \(H^s(\Gamma ) = \{ f|_{\Gamma } \,;\, f \in H^s(\partial M) \}\).

### Proposition 2.1

Let (*M*, *g*) be a compact manifold with boundary. For any \(x \in M^\Gamma \), one has \(P(x) \in H^{-s}(\Gamma )\) whenever \(s+1/2>n/2\). The mapping *P* is continuous \(M^\Gamma \rightarrow H^{-s-1}(\Gamma )\) and *k* times Fréchet differentiable considered as a mapping \(M^\Gamma \rightarrow H^{-s-1-k}(\Gamma )\). In particular, \(P:M^\Gamma \rightarrow \mathcal {D}'(\Gamma )\) is \(C^\infty \) smooth in the Fréchet sense.

*P*at

*x*is a linear mapping given by

*M*and \(u_f|_{\partial M} = f \in C^{\infty }_c(\Gamma )\).

In the proposition \(\cdot \) refers to the canonical pairing of vectors and covectors on \(M^\Gamma \), that is, \(du_f(x)\cdot V\equiv [du_f(x)](V)=\partial _au_f(x)V^a\).

*P*comes from. Let \(x\in M^\Gamma \) and \(V\in T_xM^\Gamma \). By definition

*V*is given by a path \(\gamma :[0,1]\rightarrow M^\Gamma \) such that \(\gamma (0)=x\) and \(\frac{d}{dt}\big |_{t=0}\gamma (t)=V\). Let \(f\in C_c^\infty (\Gamma )\). A formal calculation of \(DP_xV\in \mathcal {D}'(\Gamma )\) now gives

*P*is indeed an embedding, which means a \(C^{\infty }\) 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.

### Proposition 2.2

*M*,

*g*) be a compact Riemannian manifold with smooth boundary, and let \(\Gamma \) be a nonempty open subset of \(\partial M\).

- (a)If \(x \in M^{\Gamma }\), \(y \in M\) and \(x \ne y\), there is \(f \in C^{\infty }_c(\Gamma )\) such that$$\begin{aligned} u_f(x) \ne u_f(y). \end{aligned}$$
- (b)If \(x \in M^{\Gamma }\) and \(v \in T_x^* M\), there is \(f \in C^{\infty }_c(\Gamma )\) such that$$\begin{aligned} du_f(x) = v. \end{aligned}$$

### Proposition 2.3

(*P**is an embedding*) Let (*M*, *g*) be a compact manifold with boundary. The mapping \(P:M^\Gamma \rightarrow \mathcal {D}'(\Gamma )\) is a \(C^\infty \) embedding in the sense that it is injective with injective Fréchet derivative on \(TM^\Gamma \).

### Proof

*P*is injective.

*P*is injective, let \(x\in M^\Gamma \) and \(V\in T_xM^\Gamma \), and assume that \(DP_xV=0\). By the formula (6) for the differential \(DP_x\), we have

*P*is injective on \(TM^\Gamma \). \(\square \)

### 2.1 Composition of Poisson embeddings

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

### Proof

We next prove that \(J: B \rightarrow M_2\) is continuous (this perhaps surprisingly uses compactness of \(M_2\)). Let \(x\in B\). If *J* would not be continuous at *x*, there would be \(\varepsilon >0\) and a sequence \((x_l) \subset M_1\) with \(x_l \rightarrow x\) such that \(J(x_{l})\notin B(J(x),\varepsilon )\). By the compactness of \(M_2\), passing to a subsequence (still denoted by \((x_l)\)), we may assume that \(J(x_{l})\) converges to \(y\in M_2\). We have \(d(y,J(x))\ge \varepsilon \).

*J*is continuous.

*J*(

*x*). This can be done by Proposition 2.2 upon choosing \(\{ d u_{f_1}^2(J(x)), \ldots , du_{f_n}^2(J(x)) \}\) linearly independent. Write \(V = (u_{f_1}^1,\ldots , u_{f_n}^2)\). By the formula (8) we have \(V = U \circ J\) in

*B*. Now

*U*is bijective in some neighborhood \(\Omega \subset M_2\) of

*J*(

*x*), and since

*J*is continuous there is a neighborhood

*W*of

*x*with \(J(W) \subset \Omega \). Thus we have

*J*near

*x*follows.

*J*is invertible on

*B*. Since \(J: B \rightarrow M_2\) is injective, the claim will then follow from the inverse function theorem. Let \(f\in C_c^\infty (\Gamma )\), \(x\in B\) and \(X\in T_xM_1\). By (8), we have \(u_f^1=u_f^2\circ J\). Together with (6) this gives:

*DJ*(

*x*) is injective at

*x*and since the manifolds are of the same dimension, it is invertible. This proves the claim. \(\square \)

## 3 Determination of harmonic functions

We have so far acquired the basic properties of the Poisson embedding. We now move on to give a new proof of the fact [28] that for real analytic Riemannian manifolds with \(\dim (M) \ge 3\), 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 \(n = \dim (M_j) \ge 3\). We continue to denote \(M_j^\Gamma = M^{\mathrm {int}}_j \cup \Gamma \).

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.

### Lemma 3.1

*p*, such that \(\psi _1\) and \(\psi _2\) agree on \(\Gamma \) and such that for any boundary function \(f\in C_c^\infty (\Gamma )\) we have

*f*.

### Proof

Let \(p\in \Gamma \subset \partial M\) and let \(\psi _j\), \(j=1,2\), be boundary normal coordinates near *p* on manifolds \((M_j,g_j)\), respectively, so that \(\psi _1|_\Gamma =\psi _2|_\Gamma \). 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 \(\Gamma \) are real analytic, it follows that \(\psi _j\) are real analytic coordinate charts, and thus the coordinate representations \(\psi _j^{-1*}g_j\) of \(g_j\) agree near \(x=\psi _1(p)=\psi _2(p)\in \{ x_n = 0 \}\).

*f*is any \(C^\infty _c(\Gamma )\) boundary function, we have that \(\tilde{u}_f^1=u_f^1\circ \psi _1^{-1}(x)\) and \(\tilde{u}_f^2=u_f^2\circ \psi _2^{-1}(x)\) satisfy the same elliptic equation

### Lemma 3.2

### Proof

*B*. \(\square \)

We have now shown that knowledge of the DN map on \(\Gamma \) determines harmonic functions near \(\Gamma \). 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.

*harmonic morphism*means a mapping that preserves solutions to the Dirichlet problem. Precisely, a \(C^{\infty }\) mapping \(H: (M_1^{\Gamma }, g_1) \rightarrow (M_2^{\Gamma }, g_2)\) is a harmonic morphism, if for any \(f\in C_c^\infty (\Gamma )\) we have

*local*harmonic functions instead of just global harmonic functions, but our results will show that there is no difference at least when \(\dim (M_1)=\dim (M_2)\).)

### Theorem 3.3

*F*.

### Proof

*B*to be the largest connected open subset of \(M_1^{\Gamma }\) such that \(P_1(B)\subset P_2(M_2^\Gamma )\). By our assumption

*B*is nonempty. We will show that

*B*is closed and thus \(B=M_1^{\Gamma }\). Then \(P_1(M_1^\Gamma ) \subset P_2(M_2^\Gamma )\), and from Lemma 2.4 it will follow that

*B*is not closed. Then there is a sequence \((p_k)\) in

*B*with \(p_k \rightarrow x_1\) as \(k\rightarrow \infty \), where \(x_1 \in \partial B {\setminus } B\subset M_1^\Gamma \). After passing to a subsequence, there is \(x_2 \in M_2\) such that

*V*is a coordinate system in some neighborhood \(\Omega _1 \subset M_1^{\Gamma }\) of \(x_1\). The map \(U^{-1} \circ V\) will then give us a local identification of neighborhoods \(\Omega _1\subset M_1^\Gamma \) of \(x_1\) and \(\Omega _2\subset M_2^\Gamma \) of \(x_2\), such that \(\Omega _1\) intersects the complement of the closure of

*B*in \(M_1^\Gamma \) (if nonempty). This will allow us to extend

*B*and reach a contradiction.

*V*is also a harmonic coordinate system around \(x_1\). To see this, first observe that \(V(x_1)=U(x_2)\) since

*B*, we have \(J=U^{-1}\circ V\), see (9), we can substitute this to the equation of injectivity yielding

*TM*as \(k\rightarrow \infty \). Note that \(D(U^{-1}\circ V)\) is a continuous (in fact smooth) matrix field even though we still do not know if it is invertible. Taking the limit as \(k\rightarrow \infty \) in (13) gives

*f*so that \(\nabla u_f^1(x_1)=X\) (by Proposition 2.2) shows that \(X=0\) and consequently that \(D(U^{-1}\circ V)(x_1)\) is invertible. Since

*U*is a local diffeomorphism, it follows that \(DV(x_1)\) is invertible. Thus

*V*is also a coordinate system near \(x_1\) as claimed.

*V*is a coordinate system in \(\Omega _1\), and redefine \(\Omega _1\), if necessary, so that \(V(\Omega _1)\subset U(\Omega _2)\). On

*B*we have by (8)

*B*to a neighborhood of the point \(x_1\in \partial B{\setminus } B\), this will give a contradiction and prove the theorem since \(M_1^\Gamma \) is connected.

*V*and

*U*are local \(C^{\omega }\) diffeomorphisms in \(\Omega _1 \cap M^{\mathrm {int}}_1\) and \(\Omega _2 \cap M^{\mathrm {int}}_2\), respectively. Thus we have

*V*and

*U*agree.

*f*was arbitrary, we have proven (15). Consequently we have

*B*extends to a neighborhood of the point \(x_1\in \partial B{\setminus } B\), which gives a contradiction. We have now proved that

*B*is closed. Since \(M_1^\Gamma \) is connected, we conclude that \(P_1(M_1^\Gamma )\subset P_2(M_2^\Gamma )\). We thus also have \(J(M_1^\Gamma )\subset M_2^\Gamma \).

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 \(P_1(M_1^\Gamma )= P_2(M_2^\Gamma )\) and that *J* is surjective onto \(M_2^\Gamma \). Consequently \(J:M_1^\Gamma \rightarrow M_2^\Gamma \) 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. \(\square \)

Combining Lemma 3.2 and Theorem 3.3, we have proved the following statement.

### Theorem 3.4

Let \((M_1,g_1)\) and \((M_2,g_2)\) be compact real analytic Riemannian manifolds, \(n\ge 3\), with mutual boundary whose DN maps agree on an open set \(\Gamma \subset \partial M\). Assume also that \(\Gamma \) is real analytic. Then there is a diffeomorphic (global) harmonic morphism \(J:M_1^\Gamma \rightarrow M_2^\Gamma \) such that *J* is real analytic in \(M_1^{\Gamma }\) and \(J|_{\Gamma }=\text {Id}\).

### Proof

We only need to show that *J* is real analytic in \(M_1^{\Gamma }\). In the interior this follows from the representation (9) in terms of harmonic coordinates, which are real analytic in the interior. Near points of \(\Gamma \) this follows from the statement of Lemma 3.1, which implies that near points of \(\Gamma \) one has \(J = \psi _2^{-1} \circ \psi _1\) where \(\psi _j\) are real analytic boundary normal coordinates. \(\square \)

## 4 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, \(n\ge 3\). 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 \(C^\infty \) smooth, \(n\ge 3\).

It is known that a mapping between Riemannian manifolds having the same dimension \(n \ge 3\) 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 \(n\ge 3\) by using harmonic coordinates. This seems to give a new proof for the result in the local case as well.

### Proposition 4.1

Let \((M_1,g_1)\) and \((M_2,g_2)\) be \(C^\infty \) Riemannian manifolds having the same dimension \(n\ge 3\) and having a mutual boundary \(\partial M\). Let \(\Gamma \subset \partial M\) be a nonempty open set, and let \(J:(M_1^\Gamma ,g_1)\rightarrow (M_2^\Gamma ,g_2)\) be a locally diffeomorphic \(C^\infty \) global harmonic morphism. Then *J* is a homothety.

### Proof

Let \(x\in M_1^\Gamma \) and let \(U=(u_{f_1}^2,\ldots , u_{f_n}^2)\) be an *n*-tuple of global harmonic functions that define harmonic coordinates on \(\Omega _2\) near \(J(x)\in M_2^\Gamma \), where \(f_k\in C_c^\infty (\Gamma )\). This can be done by Proposition 2.2 upon choosing \(\{ d u_{f_1}^2(J(x)), \ldots , du_{f_n}^2(J(x)) \}\) linearly independent. Since *J* is locally invertible, we have that \(V=J^*U\) is a coordinate system near *x* on \(\Omega _1:=J^{-1}(\Omega _2)\). Since *J* is a global harmonic morphism, the coordinate system *V* is harmonic.

*V*and

*U*the coordinate representations of harmonic functions \(u_f\) and \(v_f=J^*u_f\) agree for arbitrary \(f\in C_c^\infty (\Gamma )\). This is because

*z*is \((H_{ij})\) in the \(g_2\)-harmonic coordinates

*U*. This is proved in Proposition A.5 (note that \(\mathrm {Hess}_g(w)\) corresponds to \((\partial _{jk} w)\) in harmonic coordinates). Thus we have that

*V*and

*U*.

*H*can be any symmetric matrix with \(g_2^{ij}(z)H_{ij}=0\), the above means that \(g_1^{-1}\) and \(g_2^{-1}\) have the same orthocomplement at

*z*with respect to the Hilbert–Schmidt inner product in the space of symmetric matrices. Thus \(g_1^{-1}(z) = \lambda (z)^{-1}g_2^{-1}(z)\) for some nonzero real number \(\lambda (z)\). Due to the positive definiteness of \(g_j\), \(j=1,2\), we have that \(\lambda (z)>0\). The argument above can be repeated for all \(z\in \Omega \), and we have

*V*and

*U*, with \(V=J^*U\), we have \(g_1=\lambda J^*g_2\) on \(\Omega _1\). Since this identity holds near an arbitrary point \(x\in M_1^\Gamma \) and since \(M_1^{\Gamma }\) is connected, we have proved the claim. \(\square \)

### Remark 4.2

We remark that in the setting of the proof above we can by Eq. 17 actually express (a multiple of) \(g_j^{-1}\) at *z* in terms of Hessians of solutions \(u_f^j\) at *z* for some \(f\in C_c^\infty (\Gamma )\), \(j=1,2\). Let \(H_j^k\), \(k=1,\ldots ,m\), \(m=\frac{n(n+1)}{2}-1\), be a basis for the orthocomplement \(\{g_j(z)^{-1}\}^\bot \) 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 \(H_k^j=\text {Hess}_{g_1}(u^1_{f_k}(z))=\text {Hess}_{g_2}(u^2_{f_k}(z))\).

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 \(\lambda \) is 1.

### Proposition 4.3

### Proof

*J*is homothety and that

*p*in \(M_1^{\Gamma }\).

Combining the results so far, we have achieved a new proof of the uniqueness in the Calderón problem in the real analytic case when \(n\ge 3\) [28]:

**Theorem***Let*\((M_1,g_1)\)*and*\((M_2,g_2)\)*be compact real analytic Riemannian manifolds*, \(n\ge 3\), *with mutual boundary whose DN maps agree on an open set*\(\Gamma \subset \partial M\). *Assume also that*\(\Gamma \)*is real analytic. Then there is a real analytic diffeomorphism*\(J:M_1^\Gamma \rightarrow M_2^\Gamma \)*such that*\(g_1 = J^* g_2\)*and*\(J|_{\Gamma }=Id\).

## 5 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 \(C^\infty \) Riemannian manifolds with mutual boundary \(\partial M\). Note that it is not required that the manifolds are real analytic.

### Theorem 5.1

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 \(g_{jk} = c \delta _{jk}\) 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 \(\mathbb {R}^2\). 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 \(\mathbb {R}^2\).

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.

### Lemma 5.2

*p*such that the following statements hold:

- (1)
There is an open subset \(\Gamma _0\) of \(\Gamma \) with \(p \in \Gamma _0\) such that \(U_1(\Gamma _0)=U_2(\Gamma _0)=:\tilde{\Gamma }\subset \mathbb {R}^2\) and \(U_1|_{\Gamma _0} = U_2|_{\Gamma _0}\).

- (2)
If \(f\in C_c^\infty (\Gamma )\), then the Cauchy data of the coordinate representations \(U_1^{-1*}u_f^1\) and \(U_2^{-1*}u_f^2\) agree on \(\tilde{\Gamma }\subset \mathbb {R}^2\).

### Lemma 5.3

*p*such that for \(f\in C_c^\infty (\Gamma )\), we have

### Proof

*B*replaced by \(U_1^{-1}(U_1(\Omega _1)\cap U_2(\Omega _2))\). We can then enlarge

*B*as in Lemma 3.2 to conclude the proof. \(\square \)

We record the following:

### Proposition 5.4

*J*is conformal,

The proof is identical to that of Proposition 4.1 except that we cannot deduce that \(\lambda (x)\) is constant by the argument using Eq. (19). We omit the proof.

We will now prove global determination of harmonic functions.

### Theorem 5.5

*F*.

### 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 \(\emptyset \ne B\subset M_1^\Gamma \) to be the largest open connected set such that \(P_1(B)\subset P_2(M_2^\Gamma )\). The task is to show that *B* is closed. We argue by contradiction and assume that it is not. Then the points \(x_1\in \partial B{\setminus } B\) and \(x_2\in M_2^\Gamma \) are limits of sequences \((p_k)\subset B\) and \((J(p_k))\subset J(B)\). If \(f\in C_c^\infty (\Gamma )\), we have \(u_f^1=u_f^2\circ J\) on *B*.

*U*is an isothermal coordinate system on a neighborhood \(\Omega _2\) of \(x_2\).

*V*is invertible at \(x_1\) follows from

*B*we have

*J*is a \(C^{\infty }\) diffeomorphism in

*B*. By Proposition 5.4 applied with \(M_1^{\Gamma }\) replaced by

*B*and \(M_2^{\Gamma }\) replaced by

*J*(

*B*) (the proof of Proposition 5.4 is really a pointwise argument and applies in this case), we have that

*J*is a conformal mapping on

*B*, \(J^*g_2=\lambda g_1\). Thus we have

*J*onto the whole \(\Omega _1\).)

*B*, for \(f\in C_c^\infty (\Gamma )\), the above holds on the open set \(V(\Omega _1\cap B)\). Since the coordinates in question are isothermal and harmonic (where \(\Gamma ^a(g_j)=0\)) we have that

*J*to \(B \cup \Omega _1\), which gives a contradiction and concludes the proof. \(\square \)

### Proof of Theorem 5.1

By Lemma 5.3, we have that there is \(B\subset M_1^\Gamma \) and a diffeomorphic harmonic morphism \(F:B\rightarrow F(B)\subset M_2^\Gamma \). By Theorem 5.5 the mapping *F* extends to a global harmonic morphism \(J:M_1^\Gamma \rightarrow M_2^\Gamma \). Proposition 5.4 shows that *J* is a conformal mapping. That the implied conformal factor is 1 on \(\Gamma \) follows from calculations in the proof of Proposition 4.3. \(\square \)

## 6 On determining the coefficients of quasilinear elliptic operators from source-to-solution mapping

*M*,

*g*) be a Riemannian manifold with boundary. The quasilinear operators

*g*as usual. We consider \(\mathcal {A}\) and \(\mathcal {B}\) as mappings

*Q*is quasilinear elliptic, which means that for all \((x,c,\sigma )\in M\times \mathbb {R}\otimes T^*M\) and \(\xi \in T_x^*M\) we have

*Q*at \(u=0\) is the operator

*f*is small. We record these facts in the following lemma. We omit the proof.

### Proposition 6.1

*M*,

*g*) be a compact manifold with smooth boundary, let \(\mathcal {A}\), \(\mathcal {B}\) be \(C^{\infty }\) maps satisfying (22)–(25), and let

*L*is the linearization of

*Q*at \(u=0\) given in (26), and assume that (27) holds.

There are constants \(C, \varepsilon , \delta > 0\) such that whenever \(||f ||_{C^{\alpha }(M)} \le \varepsilon \), the equation \(Q(u) = f\) in *M* with \(u|_{\partial M} = 0\) has a solution \(u \in C^{2,\alpha }(M)\) satisfying \(||u ||_{C^{2,\alpha }(M)} \le C ||f ||_{C^{\alpha }(M)}\). If \(u_j\in C^{2,\alpha }(M)\), \(j=1,2\), both solve \(Q(u_j) = f\) in *M* with \(u_j|_{\partial M} = 0\) and \(||u_j ||_{C^{2,\alpha }(M)} \le \delta \), then \(u_1=u_2\).

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.

*source-to-solution mapping*of

*Q*on an open subset

*W*of

*M*. The source-to-solution mapping

*u*is the unique solution to

*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.

*u*solves

*u*also solves

*M*. Therefore the source-to-solution mapping defined with respect to \(\tilde{Q}\) 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 \(\tilde{\Gamma }_{ab}^k-\Gamma _{ab}^k\) 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 \(\mathcal {B}\), \(\mathcal {A}\) and \(\Gamma _{ab}^k\) independently of each other in the general case. However, in some cases when we have extra information about the coefficients \(\mathcal {A}\) and \(\mathcal {B}\), 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 \(n\ge 3\) [26].

Our main theorem of this section is the following determination result.

### Theorem 6.2

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 operators of the form (21) having coefficients \(\mathcal {A}_j\), \(\mathcal {B}_j\) satisfying (22)–(27). Moreover, assume that \(\mathcal {A}_j\) and \(\mathcal {B}_j\) are real analytic in all their arguments.

*J*satisfies

*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 \(\mathcal {A}\) and \(\mathcal {B}\) on *W*, up to the gauge symmetry described in (32). In this step we read the coefficients \(\mathcal {A}\) and \(\mathcal {B}\) 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 \(\mathcal {A}\) and \(\mathcal {B}\) 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 \(\mathcal {B}\) and \(\mathcal {A}\) the gauge symmetry vanishes and we can determine \(\mathcal {A}\) and \(\mathcal {B}\) independently.

### Corollary 6.3

- (1)
\(\mathcal {A}_1(x,c,\sigma )\) (or \(\mathcal {A}_2(x,c,\sigma )\)) is

*s*-homogeneous in the \(\sigma \)-variable and \(\mathcal {B}_1(x,c,\sigma )\) and \(\mathcal {B}_2(x,c,\sigma )\) are \(s'\)-homogeneous with \(s'\ne s+1\) for all \(x\in W_1\) and \(c\in \mathbb {R}\); or - (2)
\(\phi :(W^{\mathrm {int}}_1,g_1|_{W^{\mathrm {int}}_1})\rightarrow (W^{\mathrm {int}}_2,g_2|_{W^{\mathrm {int}}_2})\) is an isometry.

*J*. An example satisfying the first condition is the nonlinear Schrödinger operator

*W*.

### 6.1 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.

### Proposition 6.4

*M*,

*g*) and

*Q*be as in Proposition 6.1. Let \(W \subset M\) be open, and let

*S*be the source-to-solution map defined in (29). Then, for any \(f \in C^{\infty }_c(W)\),

*M*with \(u|_{\partial M} = 0\), 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

*A*and vector field \(B_j\), and

*C*is a function. Assume that \(L_j\) are injective on \(C^{\infty }(M_j) \cap H^1_0(M_j)\), and assume that

*A*,

*B*and

*C*are real analytic up to boundary.

In the theorem \(u_{f\circ \phi }^1\) and \(u_f^2\) are the solutions to \(L_1u_{f\circ \phi }^1=f\circ \phi \) in \(M_1\) and to \(L_2u_{f}^2=f\) in \(M_2\) with \(u_{f\circ \phi }^1|_{\partial M}=u_{f}^2|_{\partial M}=0\).

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.

### Definition 6.1

*Poisson embedding for*

*L*) Let (

*M*,

*g*) be a compact Riemannian manifold with boundary, and let

*W*be an open subset of

*M*. Let

*L*be a second order elliptic differential operator of the form (33) which is injective on \(C^{\infty }(M) \cap H^1_0(M)\). The

*Poisson embedding*

*R*of the manifold

*M*is defined to be the mapping

In the definition \(\mathcal {D}'(W)\) means \([C_c^\infty (W)]^*\) 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.

### Proposition 6.6

Let (*M*, *g*) be smooth compact manifold with boundary. For any \(x \in M^{\mathrm {int}}\), one has \(R(x) \in H^{-s}(W)\) whenever \(s+2>n/2\). The mapping *R* is continuous \(M^{\mathrm {int}}\rightarrow H^{-s-1}(W)\) and *k* times Fréchet differentiable considered as a mapping \(M^{\mathrm {int}}\rightarrow H^{-s-1-k}(W)\). In particular, \(R:M^{\mathrm {int}}\rightarrow \mathcal {D}'(W)\) is \(C^\infty \) smooth in the Fréchet sense. The mapping *R* can be extended continuously to a mapping \(M\rightarrow H^{-s}(W)\) by defining \(R|_{\partial M}=0\).

*R*at \(x\in M^{\mathrm {int}}\) is a linear mapping given by

*M*, \(u_f|_{\partial M}=0\), \(f\in C_c^{\infty }(W)\), and \(\cdot \) 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 \(\partial M\). 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

*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 \(C^{\infty }(M) \cap H^1_0(M)\). Let

*W*be a nonempty open subset of

*M*, and denote by \(u_f\) the solution of \(Lu = f\) in

*M*with \(u|_{\partial M} = 0\).

- (a)If \(x \in M^{\mathrm {int}}\), \(y \in M\) and \(x \ne y\), there is \(f \in C^{\infty }_c(W)\) such that$$\begin{aligned} u_f(x) \ne u_f(y). \end{aligned}$$
- (b)If \(x \in M^{\mathrm {int}}\) and \(v \in T_x^* M\), there is \(f \in C^{\infty }_c(W)\) such that$$\begin{aligned} du_f(x) = v. \end{aligned}$$

### Proposition 6.8

(*R* is an embedding) Let *M*, *L*, *R* be as in Definition 6.1. The mapping \(R:M^{\mathrm {int}}\rightarrow \mathcal {D}'(W)\) is a \(C^\infty \) embedding (and a \(C^k\) embedding \(M^{\mathrm {int}}\rightarrow H^{-s-1-k}(W)\)) in the sense that it is injective with injective Fréchet differential on \(TM^{\mathrm {int}}\).

### Proof

The injectivity of *R* follows from Proposition 6.7(a). Let \(x\in M^{\mathrm {int}}\) and \(V\in T_xM\). Assume that \(0=(DR_xV)f=du_f(x)\cdot V\) for all \(f\in C_c^\infty (W)\). By Proposition 6.7(b) one can find \(f\in C_c^\infty (W)\) so that \(du_f(x)\cdot V\ne 0\), unless \(V=0\). This shows injectivity of the differential. \(\square \)

We construct next local coordinate systems from solutions \(u_f\) to \(Lu_f=f\), \(u_f|_{\partial M}=0\). We call these coordinates *solution coordinates*. These coordinates are constructed by Runge approximating local solutions to \(Lu=0\).

### Lemma 6.9

*M*,

*g*) be a compact Riemannian manifold with boundary. Let

*W*be an open subset of

*M*and let \(x_0\in M^{\mathrm {int}}\). Then there is \(C^\infty \) coordinate system on a neighborhood \(\Omega \) of \(x_0\) of the form \((u_{f_1},\ldots ,u_{f_n})\) where each of the coordinate functions satisfies

*L*are real analytic, then \((u_{f_1},\ldots ,u_{f_n})\) is real analytic on \(\Omega \).

### Proof

Let \(x_0\in M^{\mathrm {int}}\). If \(x_0\in W\), we redefine *W* as a smaller open set such that \(x_0\notin W\). By Proposition 6.7(b) we may find \(f_1, \ldots , f_n \in C^{\infty }_c(W)\) such that the Jacobian matrix of \(U = (u_{f_1},\ldots ,u_{f_n})\) is the identity matrix at \(x_0\). Thus *U* is a coordinate system in some neighborhood \(\Omega \) of \(x_0\), and by shrinking \(\Omega \) if necessary we may assume that \(f_j = 0\) in \(\Omega \).

If the coefficients of *L* are real analytic, the coordinate system *U* is real analytic on \(\Omega \) by elliptic regularity, since \(f_j|_\Omega =0\). \(\square \)

### Lemma 6.10

### 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 \(\phi ^*R_2\) by zero to a continuous mapping \(M_2\rightarrow H^{-s-1}(W_1)\). Let \(E\subset M^{\mathrm {int}}_2\) be closed in \(M_2\). It follows that \(\phi ^*R_2(E)\) is closed in \(H^{-s-1}(W_1)\) by continuity and by compactness of \(M_2\). Since \(\phi ^*R_2\) is injective on \(M^{\mathrm {int}}_2\), we have that \(I:=(\phi ^*R_2)^{-1}\) is defined as a mapping \(\phi ^*R_2(M^{\mathrm {int}}_2)\rightarrow M^{\mathrm {int}}_2\) and we have that \(I^{-1}(E)\) equals \(\phi ^*R_2(E)\), which is closed. Thus *I* is continuous and consequently *J* is continuous \(B\rightarrow J(B)\) by Proposition 6.6.

*U*is invertible, we have that \(J=U^{-1}\circ V\). Since

*J*is continuous, the domain \(\Omega _1:=J^{-1}(\Omega _2)\) of

*V*is an open neighborhood of \(x_0\). We have \(V=(v_{f_1}, \ldots , v_{f_n})\) where \(v_{f_k}\) satisfies

*V*is \(C^\infty \) and consequently \(J=U^{-1}\circ V\) is \(C^\infty \). If \(f\in \mathcal {D}'(W_1)\), we have that

*J*is also injective by Proposition 6.8, it follows that \(J:B\rightarrow J(B)\) is \(C^\infty \) diffeomorphism. \(\square \)

We prove next the main result of this subsection.

### Proof of Theorem 6.5

*B*to be the largest connected open set of \(M^{\mathrm {int}}_1\) such that \(R_1(B)\subset (\phi ^*R_2)(M^{\mathrm {int}}_2)\). We have \(W^{\mathrm {int}}_1 \subset B\). To see this, let \(f\in C_c^\infty (W_1)\) and \(x\in W^{\mathrm {int}}_1\). We have by using definitions and (35) that

*B*is closed in \(M^{\mathrm {int}}_1\) and thus \(B=M^{\mathrm {int}}_1\).

*B*we define

*B*is closed in \(M^{\mathrm {int}}_1\), we argue by contradiction and let \(p_k\rightarrow x_1\in \partial B{\setminus } B\), with \(p_k\in B\). Then \(x_1 \in M^{\mathrm {int}}_1\). By passing to a subsequence, we have \(x_2:=\lim _kJ(p_k)\in M_2\). We have that \(x_2\in M^{\mathrm {int}}_2\); otherwise using (37) with \(x = p_k\) and taking the limit as \(k \rightarrow \infty \) would imply that \(u_{f\circ \phi }^1(x_1)=u_{f}^2(x_2)=0\) for all \(f \in C^{\infty }_c(W_2)\), i.e. \(u_h^1(x_1) = 0\) for all \(h \in C^{\infty }_c(W_1)\), which would contradict Proposition 6.7(a). Let

*B*near \(x_1\) by (37). We have that

*V*is a solution coordinate system on a neighborhood \(\Omega _1\subset M^{\mathrm {int}}_1\) of \(x_1\), and the “equation of injectivity”

*J*and continuity of \(J^{-1}\) we have that

*B*extends to a neighborhood of the point \(x_1\in \partial B{\setminus } B\), which gives a contradiction. Thus

*B*is closed. Since \(M^{\mathrm {int}}_1\) is connected, we conclude that \(B=M^{\mathrm {int}}_1\).

By Lemma 6.10, *J* is \(C^\infty \) diffeomorphism \(M^{\mathrm {int}}_1\rightarrow J(M^{\mathrm {int}}_1)\). Inverting the role of \(M_1\) and \(M_2\), we have \(J(M^{\mathrm {int}}_1)=M^{\mathrm {int}}_2\). Since by Lemma 6.9 we may locally represent *J* as \(U^{-1}\circ V\) near any point, where *U* and *V* are \(C^\omega \) solution coordinates, we have that *J* and \(J^{-1}\) are real analytic. If \(u_f^2\) solves \(L_2u_f^2=f\), where \(f\in C_c^\infty (W_2)\) and \(u_f^2|_{\partial M}=0\), then (37) shows that \(u_{f\circ \phi }^1=J^*u_f^2\). \(\square \)

### 6.2 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 \(\mathcal {A}_j\), \(\mathcal {B}_j\) satisfying (22)–(27).

To prove the proposition, we begin with the following observation.

### Lemma 6.12

Assume the conditions and notation in Proposition 6.11. Let \(x\in W^{\mathrm {int}}_1\) and let \(U_2\) be coordinates on a neighborhood \(\Omega _2\subset \subset W_2\) of \(\phi (x)\in W^{\mathrm {int}}_2\). Let \(U_1=\phi ^*U_2\) be coordinates on a neighborhood \(\phi ^{-1}(\Omega _2)\subset \subset W_1\) of the point *x*.

### Proof

The lemma tells us that we can use any test function *v* with small enough \(C^{2,\alpha }\) norm to solve for the coefficients of \(\tilde{Q}_j\) from the equation \(\tilde{Q}_1v=\tilde{Q}_2v\) 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

*A*is a symmetric \(n\times n\)-matrix. Then we have

*A*be a matrix with \(||A ||\le \delta \) and let \(v=v_{(y,c,\sigma ,A)}\) be the function defined in (47). Since \(||v ||_{C^{2,\alpha }(\Omega )}< \delta '\) by (48), Eq. 46 implies that

*A*a one with \(||A ||\le \delta \) and that satisfies have

*A*as

### 6.3 Global determination of the coefficients

We prove the main theorem of this section.

### Proof of Theorem 6.2

### Proof of Corollary 6.3

(1) Let \((x,c,\sigma )\in M^{\mathrm {int}}_1\times \mathbb {R}\otimes T^*M^{\mathrm {int}}_1\). Assume that \(\mathcal {A}_1(x,c,\sigma )\) is *s*-homogeneous in the \(\sigma \)-variable and that \(\mathcal {B}_1\) and \(\mathcal {B}_2\) are \(s'\)-homogeneous with \(s'\ne s+1\). It follows from (53) that \(J^*\mathcal {B}_2\)

(2) Assume that \(\phi \) is an isometry. Since \(J|_{W^{\mathrm {int}}_1}=\phi |_{W^{\mathrm {int}}_1}\), we have \(\Gamma (g_1)_{ab}^k=\Gamma (\phi ^*g_2)_{ab}^k=\Gamma (J^*g_2)_{ab}^k\) on \(W^{\mathrm {int}}_1\). Since \(\Gamma (g_1)_{ab}^k-\Gamma (J^*g_2)_{ab}^k\) is a real analytic tensor field, which vanishes on \(W^{\mathrm {int}}_1\), it vanishes on \(M^{\mathrm {int}}_1\). Thus \(\Gamma (x,c,\sigma )=0=\mathcal {B}(x,c,\sigma )\) for all \((x,c,\sigma )\in M^{\mathrm {int}}_1\times \mathbb {R}\otimes T^*M^{\mathrm {int}}_1\). \(\square \)

## Notes

### Acknowledgements

Open access funding provided by University of Jyväskylä (JYU). M.L., T.L. and M.S. were supported by the Academy of Finland (Centre of Excellence in Inverse Modelling and Imaging, grant numbers 284715 and 309963). M.S. was also partly supported by the European Research Council under FP7/2007-2013 (ERC StG 307023) and Horizon 2020 (ERC CoG 770924). We would like to thank an anonymous referee for valuable comments.

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