Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map

In this article we seek stability estimates in the inverse problem of determining the potential or the velocity in a wave equation in an anisotropic medium from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the wave equation. We prove in dimension $n\geq 2$ that the knowledge of the Dirichlet-to-Neumann map for the wave equation uniquely determines the electric potential and we prove H\"older-type stability in determining the potential.


Introduction
In this paper, we are interested in the following inverse boundary value problem: on a Riemannian manifold with boundary, determine the potential or the velocity -i.e. the conformal factor within a conformal class of metrics -in a wave equation from the vibrations measured at the boundary. Let (M, g) be a compact Riemannian manifold with boundary ∂M. All manifolds will be assumed smooth (which means C ∞ ) and oriented. We denote by ∆ g the Laplace-Beltrami operator associated to the metric g. In local coordinates, ∆ g is given by (1.1) Here (g jk ) is the inverse of the metric g and det g = det(g jk ). Let us consider the following initial boundary value problem for the wave equation with bounded (real valued) electric potential q ∈ L ∞ (M) u(0, ·) = 0, ∂ t u(0, ·) = 0 in M, where f ∈ H 1 ((0, T ) × ∂M). Denote by ν = ν(x) the outer normal to ∂M at x ∈ ∂M, normalized so that n j,k=1 g jk ν j ν k = 1. We may define the dynamical Dirichlet-to-Neumann map Λ g, q by Λ g, q f = n j,k=1 ν j g jk ∂u ∂x k (0,T )×∂M . (1. 3) It is clear that one cannot hope to uniquely determine the metric g = (g jk ) from the knowledge of the Dirichlet-to-Neumann map Λ g, q . As was noted in [39], the Dirichlet-to-Neumann map is invariant under a gauge transformation of the metric g. Namely, given a diffeomorphism Ψ : M → M such that Ψ| ∂M = Id one has Λ Ψ * g, q = Λ g, q where Ψ * g denotes the pullback of the metric g under Ψ. The inverse problem should therefore be formulated modulo the natural gauge invariance. Nevertheless, when the problem is restricted to a conformal class of metrics, there is no such gauge invariance and the inverse problem now takes the form: knowing Λ cg,q , can one determine the conformal factor c and the potential q?
Belishev and Kurylev gave an affirmative answer in [5] to the general problem of finding a smooth metric from the Dirichlet-to-Neumann map. Their approach is based on the boundary control method introduced by Belishev [4] and uses in an essential way an unique continuation property. Unfortunately it seems unlikely that this method would provide stability estimates even under geometric and topological restrictions. Their method also solves the problem of recovering g through boundary spectral data. The boundary control method gave rise to several refinements of the results of [5]: one can cite for instance [30], [29] and [1].
In this paper, the inverse problem under consideration is whether the knowledge of the Dirichlet-to-Neumann map Λ g, q on the boundary uniquely determines the electric potential q (with a fixed metric g) and whether the knowledge of the Dirichlet-to-Neumann map Λ g = Λ g,0 uniquely determines the conformal factor of the metric g within a conformal class. From the physical viewpoint, our inverse problem consists in determining the properties (e.g. a dispersion term) of an inhomogeneous medium by probing it with disturbances generated on the boundary.
The data are responses of the medium to these disturbances which are measured on the boundary, and the goal is to recover the potential q(x) and the velocity c(x) which describes the property of the medium. Here we assume that the medium is quiet initially, and f is a disturbance which is used to probe the medium. Roughly speaking, the data is ∂ ν u measured on the boundary for different choices of f .
In the Euclidian case (g = e) Rakesh and Symes [35], [34] used complex geometrical optics solutions concentrating near lines with any direction ω ∈ S n−1 to prove that Λ e,q determines q(x) uniquely in the wave equation. In [35], Λ e,q gives equivalent information to the responses on the whole boundary for all the possible input disturbances. Ramm and Sjöstrand [36] extended the results in [35] to the case of a potential q depending both on space x and time t. Isakov [25] considered the simultaneous determination of a potential and a damping coefficient. A key ingredient in the existing results, is the construction of complex geometric optics solutions of the wave equation in the Euclidian case, concentrated along a line, and the relationship between the hyperbolic Dirichlet-to-Neumann map and the X-ray transform plays a crucial role.
Regarding stability estimates, Sun [42] established in the Euclidean case stability estimates for potentials from the Dirichlet-to-Neumann map. In [39] and [41] Stefanov and Uhlmann considered the inverse problem of determining a Riemannian metric on a Riemannian manifold with boundary from the hyperbolic Dirichlet-to-Neumann map associated to solutions of the wave equation (∂ 2 t − ∆ g )u = 0. A Hölder type of conditional stability estimate was proven in [39] for metrics close enough to the Euclidean metric in C k , k ≥ 1 or for generic simple metrics in [41].
Uniqueness properties for local Dirichlet-to-Neumann maps associated with the wave equation are rather well understood (e.g., Belishev [4], Katchlov, Kurylev and Lassas [29], Kurylev and Lassas [30]) but stability for such operators is far from being apprehended. For instance, one may refer to Isakov and Sun [27] where a local Dirichet-to-Neumann map yields a stability result in determining a coefficient in a subdomain. As for results involving a finite number of data in the Dirichlet-to-Neumann map, see Cheng and Nakamura [14], Rakesh [34]. There are quite a few works on Dirichlet-to-Neumann maps, so our references are far from being complete: see also Cardoso and Mendoza [13], Cheng and Yamamoto [15], Eskin [18]- [19]- [20], Hech and Wang [23], Rachele [33], Uhlmann [43] as related papers.
The main goal of this paper is to study the stability of the inverse problem for the dynamical anisotropic wave equation. The approach that we develop is a dynamical approach. Our inverse problem corresponds to a formulation with bound-ary measurements at infinitely many frequencies. On the other hand, the main methodology for formulations of inverse problems involving a measurement at a fixed frequency, is based on L 2 -weighted inequalites called Carleman estimates. For such applications of Carleman inequalities to inverse problems we refer for instance to Bellassoued [6], Isakov [26]. Most papers treat the determination of spatially varying functions by a single measurement. As for observability inequalities by means of Carleman estimates, see [8], [9], [10].
Our proof is inspired by techniques used by Stefanov and Uhlmann [41], and Dos Santos Ferreira, Kenig, Salo and Uhlmann [17]. In the last reference, an uniqueness theorem for an inverse problem for an elliptic equation is proved following ideas which in turn go back to the work of Calderón [12]. The heuristic underlying idea is that one can (at least formally) translate techniques used in solving the elliptic equation ∂ 2 t + ∆ g (which is the prototype of equations studied in [17]) to the case of the wave equation ∂ 2 t − ∆ g by changing t into it. Our problem turns out to be somehow easier because we don't need to construct complex geometrical solutions, but can rely on classical WKB solutions.

Weak solutions of the wave equation
Let (M, g) be a (smooth) compact Riemannian manifold with boundary of dimension n ≥ 2. We refer to [28] for the differential calculus of tensor fields on Riemannian manifolds. If we fix local coordinates x = [x 1 , . . . , x n ] and let ∂ ∂x 1 , . . . , ∂ ∂xn denote the corresponding tangent vector fields, the inner product and the norm on the tangent space T x M are given by If f is a C 1 function on M, we define the gradient of f as the vector field ∇ g f such that X(f ) = ∇ g f, X g for all vector fields X on M. In local coordinates, we have The metric tensor g induces the Riemannian volume dv n g = (det g) 1/2 dx 1 ∧ · · · ∧ dx n . We denote by L 2 (M) the completion of C ∞ (M) with respect to the usual inner product . The normal derivative is given by where ν is the unit outward vector field to ∂M. Moreover, using covariant derivatives (see [22]), it is possible to define coordinate invariant norms in H k (M), k ≥ 0. Let us consider the following initial boundary value problem for the wave equation (1.6) The following result is well known (see [24]).
and the mapping F → ∂ ν v is linear and continuous from H to L 2 ((0, T ) × ∂M). Furthermore, there is a constant C > 0 such that A proof of the following lemma may be found for instance in [31].
There exists an unique solution to the problem (1.2). Furthermore, the map f → ∂ ν u is linear and continuous from Therefore the Dirichlet-to-Neumann map Λ g,q defined by (1.3) is continuous. We denote by Λ g,q its norm in L H 1 ((0, T ) × ∂M); L 2 ((0, T ) × ∂M) . Our last remark concerns the fact that when q is real valued, the Dirichlet-to-Neumann map is self-adjoint; more precisely, we have Λ * g,q = Λ g,q . This simple fact will be proven in section 2. We denote the Dirichlet-to-Neumann map when there is no potential in the wave equation.

Statement of the main results
In this section we state the main stability results. Let us begin by introducing an admissible class of manifolds for which we can prove uniqueness and stability results in our inverse problem. For this we need the notion of simple manifolds [41].

Definition 1
We say that the Riemannian manifold (M, g) (or more shortly that the metric g) is simple, if ∂M is strictly convex with respect to g, and for any x ∈ M, the exponential map exp x : exp −1 x (M) −→ M is a diffeomorphism. Note that if (M, g) is simple, one can extend (M, g) into another simple manifold M 1 such that M ⋐ M 1 .
Let us now introduce the admissible set of potentials q and the admissible set of conformal factors c. Let M 0 > 0, k ≥ 1 and ε > 0 be given. Set and The main results of this paper can be stated as follows.
Theorem 1 Let (M, g) be a simple Riemannian compact manifold with boundary of dimension n ≥ 2, let T > Diam g (M), there exist constants C > 0 and κ 1 ∈ (0, 1) such that for any real valued potentials q 1 , q 2 ∈ Q(M 0 ) such that q 1 = q 2 on the boundary ∂M, we have where C depends on M, T , M 0 , n, and s.
As a corollary of Theorem 1, we obtain the following uniqueness result.
Corollary 1 Let (M, g) be a simple Riemannian compact manifold with boundary of dimension n ≥ 2, let T > Diam g (M), let q 1 , q 2 ∈ Q(M 0 ) be real valued potentials such that q 1 = q 2 on ∂M. Then Λ g,q 1 = Λ g,q 2 implies q 1 = q 2 everywhere in M.
Theorem 2 Let (M, g) be a simple Riemannian compact manifold with boundary of dimension n ≥ 2, let T > Diam g (M), there exist k ≥ 1, ε > 0, 0 < κ 2 < 1 and C > 0 such that for any c ∈ C (M 0 , k, ε) with c = 1 near the boundary ∂M, the following estimate holds true where C depends on (M, g), M 0 , n, ε, k and s.
As a corollary of Theorem 2, we obtain the following uniqueness result.
Corollary 2 Let (M, g) be a simple Riemannian compact manifold with boundary of dimension n ≥ 2, let T > Diam g (M), there exist k ≥ 1, ε > 0, such that for any c ∈ C (M 0 , k, ε) with c = 1 near the boundary ∂M, we have Λ cg = Λ g implies c = 1 everywhere in M.

Spectral inverse problem
For q ∈ Q(M 0 ) and q ≥ 0, we denote by A q the unbounded operator . The spectrum of A q consists of a sequence of eigenvalues, counted according to their multiplicities: with lim k→∞ λ k,q = ∞. The corresponding eigenfunctions are denoted by (φ k,q ). We may assume that this sequence forms an orthonormal basis of L 2 (M).
In the sequel C denotes a generic positive constant depending only on M and M 0 (M 0 is given by (1.10)). Since φ k,q is the solution of the following boundary value problem (1.14) Therefore On the other hand, by Weyl's asymptotics, there exists a positive constant C ≥ 1 such that Here C can be chosen uniformly with respect to q provided 0 ≤ q(x) ≤ M for x ∈ M. Therefore we have We fix r such that n/2 + 1 < r ≤ n + 1 and it follows that We recall that ℓ 1 is the Banach space of real-valued sequences such that the corresponding series is absolutely convergent. This space is equipped with its natural norm.
Let ω = (ω k ) be the sequence given by ω k = k −2r/n for each k ≥ 1. We introduce the following Banach spaces and ℓ 1 ω (C) = y = (y k ) k ; y k ∈ C, k ≥ 1, and (ω k |y k |) k ∈ ℓ 1 . The natural norms on those spaces are We will apply Theorem 1 to prove the following result.
Theorem 3 Let (M, g) be a simple Riemannian compact manifold with boundary of dimension n ≥ 2. There exist C > 0 and κ 3 ∈ (0, 1) such that the following estimate holds for any non-negative q 1 , q 2 ∈ Q(M 0 ) which are equal on the boundary ∂M, where is assumed to be small and Theorem 3 is an extension of a result in [16] which is itself a variant of a theorem in [2]- [7]. To the best of our knowledge, [2] is the first result in the literature concerned with stability estimates for multidimensional inverse spectral problems.
The outline of the paper is as follows. In section 2 and 3 we collect some of the formulas needed in the paper. In section 4 we construct special geometrical optics solutions to the wave equation. In section 5 and 6, we establish stability estimates for related integrals over geodesics crossing M and prove our main results. In section 7 we prove Theorem 3.

Preliminaries
In this section we collect formulas needed in the rest of this paper. We denote by divX the divergence of a vector field X ∈ H 1 (T M) on M, i.e. in local coordinates, Then if f ∈ H 1 (M) and w ∈ H 2 (M), the following identity holds , we denote by u 1 , respectively by u 2 , the solutions to (1.2) with potential q and Dirichlet datum f 1 , respectivelyq and Dirichlet datum f 2 . By Green's formula, we have This shows that Λ * g,q = Λ g,q .
In particular, this implies that Λ g,q is selfadjoint when q is real-valued (and therefore Λ g ). From now on, we will suppose the potential to be real-valued. For x ∈ M and θ ∈ T x M we denote by γ x,θ the unique geodesic starting at the point x in the direction θ. We denote the sphere bundle and co-sphere bundle of M. The exponential map exp x : (2.5) A compact Riemannian manifold (M, g) with boundary is a convex non-trapping manifold, if it satisfies two conditions: (a) the boundary ∂M is strictly convex, i.e. the second fundamental form of the boundary is positive definite at every boundary point, (b) for every point x ∈ M and every vector θ ∈ T x M, θ = 0, the maximal geodesic γ x,θ (t) satisfying the initial conditions The second condition is equivalent to all geodesics having finite length in M. An important subclass of convex non-trapping manifold are simple manifolds. Recall that a compact Riemannian manifold (M, g) which is simple satisfies the following properties (a) the boundary is strictly convex, (b) there are no conjugate points on any geodesic.
A simple ndimensional Riemannian manifold is diffeomorphic to a closed ball in R n , and any pair of points on the manifold can be joined by an unique minimizing geodesic.
In the rest of this article, C will be a generic constant which might change from one line to another, but which only depends on the quantities allowed in the statement of the theorems (namely the quantities involved in the sets Q, C , the manifold (M, g), the dimension n, the final time T and the Hölder exponents κ j ).

The geodesical ray transform
We introduce the submanifolds of inner and outer vectors of SM where ν is the unit outer normal to the boundary. Note that ∂ + SM and ∂ − SM are compact manifolds with the same boundary S(∂M), and Let C ∞ (∂ + SM) be the space of smooth functions on the manifold ∂ + SM. The ray transform (also called geodesic X-ray transform) on a convex non-trapping manifold M is the linear operator The right-hand side of (3.3) is a smooth function on ∂ + SM because the integration bound τ + (x, θ) is a smooth function on ∂ + SM, see Lemma 4.1.1 of [38]. The ray transform on a convex non-trapping manifold M can be extended to a bounded operator for every integer k ≥ 1, see Theorem 4.2.1 of [38].
The Riemannian scalar product on T x M induces a volume form on S x M denoted by dω x (θ) and given by We introduce the volume form dv 2n−1 g on the manifold SM g is the Riemannnian volume form on M. By Liouville's theorem, the form dv 2n−1 g is preserved by the geodesic flow. The corresponding volume form on the boundary ∂SM = {(x, θ) ∈ SM, x ∈ ∂M} is given by This Hilbert space is endowed with the scalar product given by (3.5) The ray transform I is a bounded operator from L 2 (M) into L 2 µ (∂ + SM) and its adjoint I * : where ψ * is the extension of the function ψ from ∂ + SM to SM constant on every orbit of the geodesic flow, i.e.
Let (M, g) be a simple metric, we assume, as we may, that (M, g) extends smoothly into a simple manifold such that for any f ∈ L 2 (M), see Theorem 3 in [40]. If V is an open set of the simple Riemannian manifold (M 1 , g), the normal operator I * I is an elliptic pseudodifferential operator of order −1 on V whose principal symbol is a multiple of |ξ| g (see [32,40]). Therefore there exists a constant C k > 0 such that for all f ∈ H k (V ) compactly supported in V

Geometrical optics solutions
We will now construct geometrical optics solutions of the wave equation. We extend the manifold (M, g) into a simple manifold M 2 ⋑ M and consider a simple manifold (M 1 , g) such that M 2 ⋑ M 1 . The potentials q 1 , q 2 may also be extended to M 2 and their H 1 (M 1 ) norms may be bounded by M 0 . Since q 1 and q 2 coincide on the boundary, their extension outside M can be taken the same so that Let us assume for a moment that there exist a function ψ ∈ C 2 (M) which satisfies the eikonal equation and a function a ∈ H 1 (R, H 2 (M)) which solves the transport equation with initial or final data We also introduce the norm · * given by The constant C depends only on T and M (that is C does not depend on a and λ).
Proof . We set then the estimates (4.7) holds. We shall prove the estimate for κ = 0, and the κ = T case may be handled in a similar way. We have Taking into account (4.1) and (4.2), the right-hand side of (4.10) becomes where k 0 ∈ H 1 (0, T ; L 2 (M)) and satisfies Since the coefficient q does not depend on t, the function solves the mixed hyperbolic problem (4.9) with right-hand side Integrating by parts with respect to s, we conclude that By Lemma 1.1, we find v λ ∈ C 1 (0, T ; L 2 (M)) ∩ C(0, T ; H 1 0 (M)) (4.12) and Since k L 2 ((0,T )×M) ≤ C a * , using again the energy estimates for the problem (4.9), we obtain (4.14) The proof is complete.

Remark 1
In the construction of geometrical optics solutions, it is not necessary to assume that the potential is time independent. In the case where the potential q is also time dependent, one can proceed along the following lines. With the same notations, w λ satisfies the equation If one uses Lemma 1.1 on the interval [0, τ ] one gets and Gronwall's inequality allows to conclude We now proceed to construct a phase function ψ solution to the eikonal equation (4.1) and an amplitude function a solution to the transport equation (4.2).
In these coordinates (which depend on the choice of y) the metric takes the form g(r, θ) = dr 2 + g 0 (r, θ) where g 0 (r, θ) is a smooth positive definite metric on S y M 1 . For any function u compactly supported in M, we set for r > 0 and θ ∈ S y M 1 u(r, θ) = u(exp y (rθ)) where we have extended u by 0 outside M. To solve the eikonal equation (4.1) it is enough to take ψ(x) = d g (x, y). Then by the simplicity assumption, since y ∈ M 2 \M, we have ψ ∈ C ∞ (M) and ψ(r, θ) = r = d g (x, y). We now proceed to the transport equation (4.2). Recall that if f (r) is any function of the geodesic distance r, then where α = α(r, θ) denotes the square of the volume element in geodesic polar coordinates. The transport equation ( Let φ ∈ C ∞ 0 (R) and b ∈ H 2 (∂ + SM), we choose a of the form A simple calculation shows that ∂ a ∂t (t, r, θ) = α −1/4 φ ′ (t − r)b(y, θ).  If we assume that supp φ ⊂ (0, ε 0 ), with ε 0 > 0 small enough so that then for any x = exp y (rθ) ∈ M, it is easy to see that a(t, r, θ) = 0 if t ≤ 0 and t ≥ T .

Remark 2
If T > Diam g (M) + 4ε 0 and cg is ε-close to g, then we also have T > Diam cg (M) + 3ε 0 .

Stability estimate for the electric potential
In this section, we complete the proof of Theorem 1. We are going to use the geometrical optics solutions constructed in the previous section; this will provide information on the geodesic ray transform of the difference of electric potentials.

Preliminary estimates
The main purpose of this section is to present a preliminary estimate, which relates the difference of the potentials to the Dirichlet-to-Neumann map. As before, we let q 1 , q 2 ∈ Q(M 0 ) be real valued potentials. We set Recall that we have extended q 1 , q 2 as H 1 (M 2 ) in such a way that q = 0 on M 2 \ M.

Lemma 5.1
There exists C > 0 such that for any a 1 , a 2 ∈ H 1 (R, H 2 (M)) satisfying the transport equation (4.2) with initial data (4.3) the following estimate holds true: for any sufficiently large λ > 0.
Let us denote by f λ the function Let v denote the solution of the following initial boundary value problem Since q(x)u 2 ∈ L 1 (0, T ; L 2 (M)) by Lemma 1.1, we deduce that w ∈ C 1 (0, T ; L 2 (M)) ∩ C(0, T ; H 1 (M)).

End of the proof of the stability estimate
Let us now complete the proof of the stability estimate in Theorem 1. Using Lemma 5.2, for any y ∈ ∂M 1 and b ∈ H 2 (∂ + SM) we have Integrating with respect to y ∈ ∂M 1 we obtain Now we choose b(y, θ) = I (I * I(q)) (y, θ).
Taking into account (3.8) and (3.4), we obtain By interpolation, it follows that Using (3.7), we deduce that This completes the proof of Theorem 1.

Remark 3
In the proof of Theorem 1, we have used the time independence of the potential at two stages: 1. In the construction of the remainder term v λ (t, x) in the proof of Lemma 4.1.
But as was noted in Remark 2, this restriction may be bypassed.
2. In equation (5.14) to get rid of the φ 2 (t−r) term and obtain the ray transform. The adaptation to the time dependent case does not seem to be straightforward.
We leave the case of a time dependent potential as an open problem.

Stability estimate for the conformal factor
We shall use the following notations. Let c ∈ C (M 0 , k, ε), we denote Then the following holds 3) The first step in our analysis is the following result.
Proof . We multiply both hand sides of the first equation (6.4) by u 2 ; integrating by parts in time and using Green's formula (2.3)-(2.4) in (M, g), we obtain Another integration by parts in time and an application of Green's formula in (M, cg) yield Taking into account the facts that Λ cg is self-adjoint, that c = 1 on ∂M and This completes the proof of the Lemma.

Modified geometrical optics solutions
As in the case of potentials, we extend the manifold (M, g) into a simple manifold M 2 ⋑ M so that M 2 ⋑ M 1 ⋑ M with (M 1 , g) simple. We extend the conformal factor c by 1 outside the manifold M; its C k (M 1 ) norms may also be bounded by M 0 . Let ψ 1 , ψ 2 be two phase functions solving the eikonal equation with respect respectively to the metrics g and cg.

Lemma 6.2
Let c ∈ C (M 0 , k, ε) be such that c = 1 near the boundary ∂M. Then the equation has a solution of the form when λ is large enough, which satisfies The constant C depends only on T and M (that is C does not depend on a, λ and ε).
Proof . We set To prove our Lemma it would be enough to show that if v solves then the estimates (6.16) holds. We have Taking into account (6.9) and (6.10), the right-hand side of (6.20) becomes Since k j ∈ L 1 (0, T ; L 2 (M)), by Lemma 1.1, we deduce that Moreover, we have k L 2 ((0,T )×M) ≤ C 1 λ a 2 * + ε a 2 * + ελ 2 a 2 * (6.25) and by using again the energy estimates for the problem (6.18)-(6.19), we obtain This ends the proof of Lemma 6.2.

Lemma 6.3
There exists C > 0 such that for any a 1 , a 2 ∈ H 1 (R, H 2 (M)) satisfying the transport equation (6.10) with (4.3) the following estimate holds true for any sufficiently large λ.
Proof . Following Lemma 6.2 let u 2 be a solution to the problem (∂ 2 t − ∆ cg )u = 0 of the form where v 2,λ satisfies (6.16) and a 3 satisfies (6.12). Thanks to Lemma 4.1 let u 1 be a solution to the (∂ 2 t − ∆ g )u = 0 of the form where v 1,λ satisfies (4.4). Then Let us compute the first term in the right hand side of (6.6). We have Thus, we have from (6.12), (6.16) and (4.4) the following identity where On the other hand, we have (6.32) and the second term in the right side of (6.6) becomes From (6.12), (6.16) and (4.4), we have Taking into account (6.6), (6.30) and (6.33), we deduce that In view of (6.34) and (6.31), we obtain This completes the proof.

End of the proof of Theorem 2
This subsection is devoted to the end of the proof of Theorem 2. We need the following known result (see [41] proposition 4.1).
Then there exists C > 0 such that with some 0 < µ < 1 depending only on the dimension n.
From this lemma, we can derive the following estimate Corollary 3 There exists a constant C > 0 such that the following estimate holds true with some 0 < µ < 1 depending on n only.
Proof . Linearizing near g, we get, as in [21] d g (x, y) − d cg (x, y) = 1 2 I(̺)(x, y) + R(̺)(x, y), ∀x, y ∈ ∂M, where, with some abuse of notation, I(̺)(x, y) stands for I(̺)(x, θ) with θ = exp −1 x (y)/ exp −1 x (y) . The remainder term R(̺)(x, y) is nonlinear and satisfies the estimate (see [21]) with C > 0 uniform in c if 0 < ε ≪ 1. By Lemma 6.5, we have Apply I * to both sides, and use the estimate Since ̺ vanishes outside M with all derivatives and I * I is a pseudodifferential operator of order −1, we have for all integers m. Using interpolation, we get Therefore, (6.44) and (6.43) imply This completes the proof.

Proof of Theorem 3
Let q ∈ L ∞ (M), we first define the elliptic Dirichlet-to-Neumann map; let σ(A q ) = {λ k,q } be the spectrum of A q and ρ(A q ) = C \ σ(A q ) be the resolvent set of A q . From well known results (e.g., [31]), for any z ∈ ρ(A q ) and h ∈ H 3/2 (∂M), the nonhomogeneous boundary value problem has an unique solution in u q,h ∈ H 2 (M) and the Dirichlet-to-Neumann map defines a bounded operator from H 3/2 (∂M) to H 1/2 (∂M). We fix T > Diam g M and consider the following function space and the operator where ·, · denotes the L 2 (∂M)-scalar product. Then R g,q defines a bounded operator from H 1 to H 2 = L 2 (0, T ; H s (∂M)).
We will need in the sequel the following three lemmas. Their proof can be found in [2] or can be deduced easily from the results in this reference (see also [16]). We fix 0 ≤ s < 1 2 .

Lemma 7.
3 For each f ∈ H 1 , we have where Λ ♯ g,q is the restriction of Λ g,q to H 1 .
Moreover Λ ♯ g,q is a linear and continuous map from  On the other hand, since ∆ g v = v 1 , by the elliptic regularity, we get v L 2 (0,T ;H 2 (M)) ≤ C v 1 L 2 (0,T ;L 2 (M)) + f H 1 .
Thus, for u = v + w, we have            We shall denote by Λ ♯ g,q L(H 1 ,H 2 ) the operator norm of Λ ♯ g,q . Lemma 7.4 Let (M, g) be a simple Riemannian compact manifold with boundary of dimension n ≥ 2, let T > Diam g (M), there exist constants C > 0 and κ ∈ (0, 1) such that for any real valued potentials q 1 , q 2 ∈ Q(M 0 ) such that q 1 = q 2 on the boundary ∂M, we have where C depends on M, T , M 0 , n, and s.
Thus, we can complete the proof of (7.8) in the same way as in section 5.2.
Proof of Theorem 3. From (7.10) and Lemma 7.2, we obtain P (j) (0) s ≤ C |z| −j− 1−2s 4 + |z| n−j+1 ǫ µ 1 , and then In particular where µ 2 ∈ (0, 1). Let R g,q be defined as in Lemma 7.3. We can proceed as in the proof of Lemma 7.5 to prove R g,q 1 − R g,q 2 L(H 1 ,H 2 ) ≤ Cǫ µ 3 . From identity (7.1), estimates (7.19) and (7.18), we deduce provided that ǫ is sufficiently small. To finish, we only need to remark that the traces of the geometrical optics solutions constructed in section 4 in fact satisfy u 1 | ∂M , u 2 | ∂M ∈ H 1 so that in the proof of Theorem 1 the right-hand side of (1.12) may be replaced by . This completes the proof of Theorem 3.