Ali Feizmohammadi
email: a.feizmohammadi@ucl.ac.uk

Yavar Kian
email: yavar.kian@univ-amu.fr

RECOVERY OF NON-SMOOTH COEFFICIENTS APPEARING IN ANISOTROPIC WAVE EQUATIONS

Keywords: Dirichlet to Neumann map, inverse problems, time-dependent coefficients, nonsmooth parameters, simple manifolds, light ray transform, magnetic potential

We study the problem of unique recovery of a non-smooth one-form A and a scalar function q from the Dirichlet to Neumann map, Λ A,q , of a hyperbolic equation on a Riemannian manifold (M, g). We prove uniqueness of the one-form A up to the natural gauge, under weak regularity conditions on A, q and under the assumption that (M, g) is simple. Under an additional regularity assumption, we also derive uniqueness of the scalar function q. The proof is based on the geometric optic construction and inversion of the light ray transform extended as a Fourier Integral Operator to non-smooth parameters and functions.

. Reduction to the light ray transform and the proof of uniqueness 18 4.1. Reduction to the light ray transform of A 1 -A 2 and proof of Theorem 1.1 18 4.2. Reduction to the light ray transform of q 1 -q 2 and proof of Theorem 1.2 20 5. Inversion of the light ray transform 20

Introduction

Let T > 0, and let (M, g) denote a compact connected smooth n-dimensional Riemannian manifold with smooth boundary ∂M . We consider the Lorentzian manifold (M, ḡ) defined as M = (0, T ) × M with the metric ḡ = -(dt) 2 + g. Let div ḡ (resp., ∇ ḡ) denote the divergence operator (resp., gradient operator) on (M, ḡ) and define the Laplace-Beltrami operator associated to (M, ḡ) through ∆ ḡ• = div ḡ∇ ḡ•. In local coordinates (t = x 0 , x 1 , . . . , x n ) = (t, x), we have

∆ ḡ• = n i,j=0 1 |ḡ| ∂ i ( |ḡ|ḡ ij ∂ j •) = (-∂ 2 t + ∆ g )•,
where ∆ g is analogously defined on (M, g). In this paper, we will make the standing assumption that (M, g) is simple, that is to say, it is simply connected, any geodesic in M has no conjugate points and the boundary ∂M is strictly convex in the sense that the second fundamental form is positive for every point on the boundary. Any two points in a simple manifold can be connected through a unique geodesic. We consider a scalar function q and a one-form A on (M, ḡ). In local coordinates, we have

(1.1) A(t, x) = b(t, x) dt + n i=1 a j (t, x) dx j = b(t, x) dt + A(t, x),
where A is a time-dependent one-form on (M, g). Throughout this paper we impose the following regularity assumptions on these coefficients:

(1.2)

A ∈ W 1,1 (0, T ; L 2 (M ; T * M)) ∩ C(M; T * M) div ḡ A ∈ L p 1 (0, T ; L p 2 (M )), q ∈ L p 1 (0, T ; L p 2 (M )).
where p 1 > 1 and p 2 ∈ [n, ∞] \ {2}. We consider the initial boundary value problem (IBVP)

(1.3)    L A,q u := -∆ ḡu + A∇ ḡu + qu = 0, on M, u = f, on (0, T ) × ∂M, u(0, •) = 0, ∂ t u(0, •) = 0 on M, This problem is well-posed for any f ∈ H 1 0 ((0, T ] × ∂M ) (see Section 2.1) and admits a unique solution u in the energy space (1.4) X := C 1 (0, T ; L 2 (M )) ∩ C(0, T ; H 1 (M )).

We define the Dirichlet-to-Neumann (DN in short) map

(1.5) Λ A,q : H 1 0 ((0, T ] × ∂M ) f → ∂ ν u - Aν 2 u | (0,T )×∂M ∈ L 2 ((0, T ) × ∂M )
for equation (1.3). Here ν represents the outward normal unit vector to (0, T ) × ∂M . We refer the reader to Sections 2.1-2.2 for a rigorous presentation of the direct problem (1.3) and this formulation of the DN map. In this paper, we are interested in determining the unknown complex valued coefficients A, q, given the map Λ A,q , up to the natural obstructions for this problem as discussed in [22, Section 1.2].

1.1. Main results. Before stating the main theorem, we need to define the set E ⊂ M where we recover the coefficients. Let us define the domain of influence

D := {(t, x) ∈ M | dist (x, ∂M ) < t < T -dist (x, ∂M )}.
By finite speed of propagation, no information can be obtained about the coefficients A, q from Λ A,q on the set M \ D. Thus, D represents the maximal set where one can, in theory, recover the coefficients. Now, for T > 2 Diam(M ), we start by fixing a subset of D given by

E := {(t, x) ∈ M | D g (x) < t < T -D g (x)},
where D g (x) denotes the length of the longest geodesic passing through the point x in M . Since (M, g) is simple, this is a well-defined continuous function on M . With the definition of E complete, we can state the main results in our paper as follows:

Theorem 1.1. Suppose T > 2 Diam(M ) and that (M, g) is a simple Riemannian manifold. Let A 1 , A 2 denote one-forms and q 1 , q 2 denote scalar functions satisfying (1.2) and such that

(1.6) supp (A 1 -A 2 ) ⊂ E and supp (q 1 -q 2 ) ⊂ E.

Then the condition

(1.7) Λ A 1 ,q 1 = Λ A 2 ,q 2
implies that there exists ψ ∈ C 1 (M) with ψ| ∂M = 0 such that (1.8)

A 1 = A 2 + dψ ∀ (t, x) ∈ M,
where d denotes the exterior derivative on M.

Theorem 1.2. Let the hypothesis of Theorem 1.1 be fulfilled and assume additionally that (1.9)

q 1 -q 2 ∈ L p 1 (0, T ; L ∞ (M )), div ḡ(A 1 -A 2 ) ∈ L p 1 (0, T ; L ∞ (M ))
holds. Then the condition Λ A 1 ,q 1 = Λ A 2 ,q 2 implies that there exists ψ ∈ C 1 0 (M) with ∆ ḡψ ∈ L p 1 (0, T ; L ∞ (M )) such that (1.10)

A 1 = A 2 + dψ, q 1 = q 2 + 1 2 ∆ ḡψ - 1 2 A 2 ∇ ḡψ - 1 4 ∇ ḡψ, ∇ ḡψ ḡ ∀ (t, x) ∈ M.
The proof of Theorems 1.1-1.2 rely in part on the inversion of the light ray transform of one-forms and scalar functions over M under the hypothesis (1.6) and the regularity conditions (1.2). This has already been accomplished for C 1 one-forms and continuous scalar functions in [START_REF] Feizmohammadi | Recovery of time dependent coefficients from boundary data for hyperbolic equations[END_REF], but some additional analysis is required here as we are working with a wider regularity class for the coefficients A and q. Let us briefly recall the notion of the light ray transform here. We denote by SM ⊂ T M the unit sphere bundle of (M, g), and by γ(•; x, v) the geodesic with the initial data (x, v) ∈ SM . For all (x, v) ∈ SM int , we define the exit times τ ± (x, v) = inf{r > 0 : γ(±r; x, v) ∈ ∂M } and note that since (M, g) is simple, we have τ ± (x, v) < Diam(M ). Define

∂ ± SM = {(x, v) ∈ SM | x ∈ ∂M ± v, ν(y) g > 0}.
All geodesics in M int can be parametrized by γ(•; x, v), (x, v) ∈ ∂ -SM . The geodesic ray transform on (M, g) is defined for f ∈ C ∞ (M ) by

If (x, v) = τ + (x,v) 0 f (γ(r; x, v))dr, (x, v) ∈ ∂ -SM.
Next, we consider the Lorentzian manifold R × M with metric ḡ = -(dt) 2 + g. Recall that a curve β in R × M is called a null geodesic (also called light rays) if (1.11) ∇ ḡβ β = 0 and β, β ḡ = 0.

One can use the product structure of the Lorentzian manifold R × M to see that the null geodesics β can be parametrized as

β(r; s, x, v) = (r + s, γ(r; x, v)) ∀(s, x, v) ∈ R × ∂ -SM.
Thus, we can identify null geodesics

β through β(•; s, x, v) with (s, x, v) ∈ R × ∂ -SM over their maximal intervals [0, τ + (x, v)]. We define the light ray transform on R × M that is defined for f ∈ C ∞ (R × M ) by Lf (s, x, v) = τ + (x,v) 0 f (r + s, γ(r; x, v)) dr, ∀(s, x, v) ∈ R × ∂ -SM.
Similarly, we define the light ray transform corresponding to smooth one-forms B, through the expression LB (s, y, v) := L(B β) (s, y, v). We will sometimes use the short hand notation L β f, L β B in place of the above notation. In Section 2.4, we will show that L β f is a Fourier Integral Operator and that the domain of definition can be extended to L p spaces. We will prove the following Proposition in Section 5, that is a key step in proving Theorems 1.1-1.2. The proof of statement (ii) will be identical to that of statement (ii) in [22, Proposition 1.4], with the only difference being that B ∈ C(M; T * M) here as opposed to C 1 (M; T * M). Reproducing the exact same analysis as in the proof there shows that one obtains existence of a ψ ∈ C 1 (M) with ψ| ∂M = 0 such that (ii) holds and therefore for the sake of brevity we omit this proof. We will however prove statement (i) in Section 5.

Proposition 1.3. Let f ∈ L 1 (0, T ; L 2 (M ))
1.2. Previous literature. Historically, uniqueness results for the recovery of coefficients can be divided into two categories, based on whether or not the geometry and coefficients are dependent on time. The time-independent case has been studied extensively and one can outline at least three general methods for the recovery of the coefficients in this case. The first approach, stemming from the seminal works [START_REF] Belishev | An approach to multidimensional inverse problems for the wave equation[END_REF][START_REF] Belishev | To the reconstruction of a Riemannian manifold via its spectral data (BC-method)[END_REF], relies on the so-called Boundary Control (BC) method together with Tataru's sharp unique continuation theorem [START_REF] Tataru | Unique continuation for solutions to PDE; between Hörmanders theorem and Holmgrens theorem[END_REF]. This method yields recovery of time-independent coefficients under very weak assumptions on the transversal manifold (M, g). We refer the reader to [START_REF] Kian | Application of the boundary control method to partial data Borg-Levinson inverse spectral problem[END_REF] for an introduction to the BC method and to the recent paper [START_REF] Kurylev | Inverse problems for the connection Laplacian[END_REF] for an example of a state of the art result and finally to [START_REF] Belishev | Recent progress in the boundary control method[END_REF][START_REF] Katchalov | Inverse boundary spectral problems[END_REF] for review. The stability results are in general double logarithmic ( [START_REF] Bosi | Stability of the unique continuation for the wave operator via Tataru inequality and applications[END_REF]) although in [START_REF] Liu | A Lipschitz stable reconstruction formula for the inverse problem for the wave equation[END_REF] a stronger low-frequency stability estimate was obtained by using ideas from the BC method. Tataru's unique continuation theorem fails when the time-dependence of the metric or the coefficients is not real analytic [START_REF] Alinhac | Non-unicité du probléme de Cauchy[END_REF][START_REF] Alinhac | A non uniqueness result for operators of principal type[END_REF], and therefore adaptations based on the BC method fails beyond this scenario. We refer the reader to [START_REF] Eskin | Inverse hyperbolic problems with time-dependent coefficients[END_REF] for recovery of coefficients when the time-dependence is real analytic. An alternative approach in deriving uniqueness results in the time-independent category started from the seminal work [START_REF] Bukhgeim | Global uniqueness of a class of multidimensional inverse problem[END_REF], where Carleman estimates were used for the first time in the context of inverse problems. Proofs based on Carleman estimates tend to yield stronger stability estimates compared to BC method. Methods based on using the classical geometric optic solutions to the wave equation have also been quite fruitful in deriving uniqueness results in time-independent category (see for example [START_REF] Bellassoued | Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map[END_REF][START_REF] Bellassoued | Lipschitz stability for a hyperbolic inverse problem by finite local boundary data[END_REF][START_REF] Stefanov | Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media[END_REF][START_REF] Stefanov | Stable determination of the hyperbolic Dirichlet-to-Neumann map for generic simple metrics[END_REF]).

In the time-dependent category, apart from [START_REF] Eskin | Inverse hyperbolic problems with time-dependent coefficients[END_REF] mentioned above, most of the results are concerned with wave equations with constant coefficient principal part. In [START_REF] Stefanov | Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials[END_REF], the author used geometric optic solutions for the wave equation with constant principal terms and an unknown zeroth order term to prove uniqueness by showing that the boundary data determines the light ray transform of the unknown scalar function in Minkowski space and subsequently inverting this transform. We refer the reader to [START_REF] Aicha | Stability estimate for hyperbolic inverse problem with time-dependent coefficient[END_REF][START_REF] Isakov | Completness of products of solutions and some inverse problems for PDE[END_REF][START_REF] Kian | Unique determination of a time-dependent potential for wave equations from partial data[END_REF][START_REF] Kian | Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data[END_REF][START_REF] Ramm | An inverse problem of the wave equation[END_REF][START_REF] Salazar | Determination of time-dependent coefficients for a hyperbolic inverse problem[END_REF] for similar results in this category.

Literature dealing with uniqueness results for the case of a wave equation with timedependent first and zeroth order coefficients on a Riemannian manifold, where the timedependence is non-analytic, is sparse. We refer to [START_REF] Kian | Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations[END_REF][START_REF] Waters | Stable determination of X-ray transforms of time dependent potentials from partial boundary data[END_REF] for the study of recovering a time-dependent zeroth order term appearing in the wave equation. In the recent paper [START_REF] Stefanov | The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds[END_REF], the authors used Fourier Integral Operators to show that a micro-local formulation of the Dirichlet to Neumann map Λ A,q uniquely determines the light ray transforms of the one-form A and scalar function q. There, it was assumed that the coefficients are in some C k space with k large enough. It was recently proved in [START_REF] Feizmohammadi | Recovery of time dependent coefficients from boundary data for hyperbolic equations[END_REF] that if the one-form is C 1 smooth and the scalar function is continuous, then one can use the classical Gaussian beam construction to uniquely obtain the light ray transforms of A and q from the knowledge of Λ A,q . The inversion of the light ray transform was also proved for the first time under the assumption that the geodesic ray transform is injective on the transversal manifold (M, g) and that the coefficients are known for some explicit lengths of time near t = 0 and t = T . In this paper, we generalize the result obtained in [START_REF] Feizmohammadi | Recovery of time dependent coefficients from boundary data for hyperbolic equations[END_REF] to the case of non-smooth coefficients. We prove that if (M, g) is simple, the Dirichlet to Neumann map uniquely determines the light ray transform of the non-smooth coefficients and subsequently show the inversion of the light ray transform as a Fourier Integral Operator under the additional assumption that the coefficients are known on a slightly larger set compared to the sharp domain D where no information can be obtained about the coefficients. This generalization and the difficulties therein are discussed in more detail in the subsequent section. 1.3. Comments about our results. We discuss some of the main novelties of our result, both by previewing some of the technical challenges and also by motivating the study of non-smooth coefficients in their own right. The technical difficulties are three-fold. One difficulty stems from the study of the forward problem and the need for sharper energy estimates for the determination of the correction terms appearing in the formal geometric optic ansatz. Another key difficulty stems from the one-form A, as any lack of smoothness in the one-form appears at the level of the principal term corresponding to the geometric optic ansatz, thus making the task of a meaningful geometric optic solution to the wave equation and the reduction to the light ray transform of the coefficients more challenging. Finally, let us remark that in [START_REF] Feizmohammadi | Recovery of time dependent coefficients from boundary data for hyperbolic equations[END_REF] the inversion of the light ray transform was proved for smooth coefficients. We generalize this inversion method to non-smooth functions by extending the notion of the light ray transform and the inversion method, in a distributional sense, to non-smooth coefficients.

Aside from the technical challenges, it should be remarked that the recovery of nonsmooth coefficients is a well-motivated question in its own right as it can be associated with the determination of various unstable phenomenon which can not be modeled by smooth parameters. For elliptic equations, this topic has received a lot of attention these last few decades (see [START_REF] Astala | Calderón's inverse conductivity problem in the plane[END_REF][START_REF] Caro | Global Uniqueness for The Calderón Problem with Lipschitz Conductivities[END_REF][START_REF] Haberman | Uniqueness in Calderón's problem for conductivities with unbounded gradient[END_REF][START_REF] Haberman | Uniqueness in Calderón's problem with Lipschitz conductivities[END_REF][START_REF] Krupchyk | Uniqueness in an inverse boundary problem for a magnetic Schrodinger operator with a bounded magnetic potential[END_REF]). However, only few authors have addressed this issue for hyperbolic equations. Concerning the recovery of time dependent coefficients, [START_REF] Hu | Determination of singular time-dependent coefficients for wave equations from full and partial data, to appear in Inverse Problem and Imaging[END_REF] seems to be the only paper addressing this issue. The result of [START_REF] Hu | Determination of singular time-dependent coefficients for wave equations from full and partial data, to appear in Inverse Problem and Imaging[END_REF] concerns the recovery of a zeroth order coefficient on a flat Lorentzian manifold with the Minkowski metric. In Theorem 1.1 and 1.2, we prove, for what seems to be the first time, the extension of this work to the recovery of non-smooth first and zeroth order coefficients appearing in a hyperbolic equation associated with a more general Lorentzian manifold.

Let us observe also that our inverse problem is intricately connected with the recovery of nonlinear terms appearing in hyperbolic equations. Indeed, following the strategy set by [START_REF] Caro | Determination of convection terms and quasi-linearities appearing in diffusion equations[END_REF][START_REF] Choulli | Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term[END_REF][START_REF] Isakov | On uniqueness in inverse problems for semilinear parabolic equations[END_REF][START_REF] Isakov | Uniqueness of recovery of some quasilinear Partial differential equations[END_REF] for parabolic equations, through a linearization procedure initially introduced by [START_REF] Isakov | On uniqueness in inverse problems for semilinear parabolic equations[END_REF], one can reduce the problem of determining coefficients appearing in a non-linear problem to the recovery of time-dependent coefficients appearing in a linear equation. In [START_REF] Kian | On the determination of nonlinear terms appearing in semilinear hyperbolic equations[END_REF] the author proved the extension of this approach to semi-linear hyperbolic equations. Note that in this procedure, the time-dependent coefficient under consideration depends explicitly on solutions of the nonlinear equation. Therefore, following the analysis of [START_REF] Kian | On the determination of nonlinear terms appearing in semilinear hyperbolic equations[END_REF], the recovery of non-smooth coefficients can be seen as an important step in the more difficult problem of determining quasi-linear terms appearing in nonlinear hyperbolic equations.

1.4. Outline of the paper. This paper is organized as follows. In Section 2, we start by considering the direct problem (1.3) and rigorously justify the definition (1.5) also deriving a key boundary integral identity (see Lemma 2.3). Moreover, we discuss smooth approximations of the coefficients A, q and also extend the notion of the light ray transform to L p functions. Section 3 is concerned with the construction of geometric optic solutions to (1.3) concentrating on maximal null geodesics in the set D. In Section 4 we prove Theorems 1.1-1.2 by applying the geometric optic construction and Proposition 1.3. Finally, Section 5 is concerned with the proof of statement (i) in Proposition 1.3. As explained in Section 1.1 statement (ii) follows analogously to statement (ii) in [START_REF] Feizmohammadi | Recovery of time dependent coefficients from boundary data for hyperbolic equations[END_REF]Proposition 1.4].

Preliminaries

Direct problem.

In this section we study the wave equation (1.3) and show that for A, q satisfying (1.2) and each f ∈ H 1 0 ((0, T ] × ∂M ) it admits a unique solution u in energy space (1.4). We will repeatedly use the Sobolev embedding theorem as follows.

(2.1)

f 1 f 2 L p 1 (0,T ;L 2 (M )) f 1 L p 1 (0,T ;L p 2 (M )) f 2 C(0,T ;H 1 (M )) .
This estimate holds since

H 1 (M ) ⊂ L 2n n-2 (M ) for n > 2 and H 1 (M ) ⊂ L p (M ) for n = 2 and any p ∈ [1, ∞).
In order to study the IBVP given by (1.3), we start by considering the following IBVP (2.2)

   -∆ ḡv + A∇ ḡv + qv = F, (t, x) ∈ M, v(0, x) = v 0 (x), ∂ t v(0, x) = v 1 (x), x ∈ M v(t, x) = 0, (t, x) ∈ (0, T ) × ∂M.
We have the following well-posedness result for this IBVP.

Proposition 2.1. Let p 1 ∈ (1, +∞) and p 2 ∈ [n, +∞) \ {2}. For q ∈ L p 1 (0, T ; L p 2 (M )), A ∈ L ∞ (M; T * M) and F ∈ L p 1 (0, T ; L 2 (M )), problem (2.2) admits a unique solution v in the space (2.3) X 0 := C([0, T ]; H 1 0 (M )) ∩ C 1 ([0, T ]; L 2 (M )) satisfying ∂ ν v ∈ L 2 ((0, T ) × ∂M ) and the estimate (2.4) ∂ ν v L 2 ((0,T )×∂M ) + v X 0 C( v 0 H 1 (M ) + v 1 L 2 (M ) + F L p 1 (0,T ;L 2 (M )) ),
with C depending only on p 1 , p 2 , n, T , M and any N q

L p 1 (0,T ;L p 2 (M )) + A L ∞ (M) .
Proof. We will prove this result by following the approach developed in [27, Proposition 2.1].

Our first goal is to show that for any v ∈ W 2,∞ (0, T ;

H 1 0 (M )) solving (2.
2), the a priori estimate (2.4) holds true. Then, applying [44, Theorem 8.1, Chapter 3], [44, Remark 8.2, Chapter 3] and [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF]Theorem 8.3,Chapter 3], the proof will be completed. We introduce the energy E(t) at time t ∈ [0, T ] given by

E(t) := M |∂ t v(t, x)| 2 + |∇ g v(t, x)| 2 dV g (x).
Multiplying (2.2) by ∂ t v, taking the real part and integrating by parts we get (2.5)

E(t) = -2R t 0 M [A(s, x)∇ ḡv(s, x) + q(s, x)v(s, x)]∂ t v(s, x)dV g (x)ds + 2R t 0 M F (s, x)∂ t v(s, x)dV g (x)ds.
Repeating the arguments of [27, Proposition 2.1] we get (2.6)

| t 0 M q(s, x)v(s, x)∂ t v(s, x)dV g (x)ds| + | t 0 M F (s, x)∂ t v(s, x)dV g (x)ds| F 2 L p 1 (0,T ;L 2 (M )) + C t 0 E(s) p 1 p 1 -1 ds p 1 -1 p 1
, where C depends only on T , M , p 1 , p 2 , n and any N q L p 1 (0,T ;L p 2 (M )) . In the same way, we obtain (2.7)

| t 0 M [A(s, x)∇ ḡv(s, x)]∂ t v(s, x)dV g (x)ds| A L ∞ (M) t 0 E(s)ds A L ∞ (M) t 1 p 1 t 0 E(s) p 1 p 1 -1 ds p 1 -1 p 1 A L ∞ (M) T 1 p 1 t 0 E(s) p 1 p 1 -1 ds p 1 -1 p 1 .
Combining (2.6)-(2.7) with (2.5), we obtain

E(t) F 2 L p 1 (0,T ;L 2 (M )) + C t 0 E(s) p 1 p 1 -1 ds p 1 -1 p 1
, where C depends only on T , M , p 1 , p 2 , n and any N q L p 1 (0,T ;L p 2 (M )) + A L ∞ (M) . Using this last estimate we can deduce that (2.2) admits a unique solution v in the space (2.3) satisfying

(2.8) v X 0 C( v 0 H 1 (M ) + v 1 L 2 (M ) + F L p 1 (0,T ;L 2 (M )) )
by applying arguments similar to the end of the proof of [27, Proposition 2.1]. Therefore the proof of the proposition will be completed if we show that ∂ ν v ∈ L 2 ((0, T ) × ∂M ) and that the estimate (2.9)

∂ ν v L 2 ((0,T )×∂M ) C( v 0 H 1 (M ) + v 1 L 2 (M ) + F L p 1 (0,T ;L 2 (M )) ) is fulfilled. For this purpose, notice that v solves    -∆ ḡv(t, x) = F v (t, x), (t, x) ∈ M, v(0, x) = v 0 (x), ∂ t v(0, x) = v 1 (x), x ∈ M v(t, x) = 0, (t, x) ∈ (0, T ) × ∂M, with F v = -A∇ ḡv -qv + F . Applying the Sobolev embedding theorem we deduce that F v ∈ L 1 (0, T ; L 2 (M )). Then, applying [32, Lemma 2.39] we deduce that ∂ ν v ∈ L 2 ((0, T ) × ∂M ) and ∂ ν v L 2 ((0,T )×∂M ) C( F v L 1 (0,T ;L 2 (M )) + v 0 H 1 (M ) + v 1 L 2 (M ) ) C( v C([0,T ];H 1 (M )) + v C 1 ([0,T ];L 2 (M )) + F L p 1 (0,T ;L 2 (M )) ).
Combining this with (2.8) we deduce (2.9) and this completes the proof of the proposition.

We can use Proposition 2.3 to show that equation (1.3) admits a unique solution u in energy space (1.4). Recall the following classical IBVP:

(2.10)

   -∆ ḡw = 0, (t, x) ∈ M, w(0, x) = 0, ∂ t w(0, x) = 0, x ∈ M w(t, x) = f, (t, x) ∈ (0, T ) × ∂M.
According to [32, Theorem 2.30] (see also [START_REF] Lasiecka | Non homogeneous boundary value problems for second order hyperbolic operators[END_REF]), this equation admits a unique solution w in the energy space (1.4). We now return to (1.3) and note that we have u = w + v, where w solves (2.10) with boundary term f , and v solves (2.2) with F := -A∇ ḡw -qw.

As A, q satisfy (1.2) and since w is in the energy space (1.4), it is immediate that F ∈ L p 1 (0, T ; L 2 (M )). Thus, Proposition 2.1 applies to show that u is in the energy space (1.4), with ∂ ν u ∈ L 2 ((0, T ) × ∂M ), and we have that

∂ ν u L 2 ((0,T )×∂M ) + u X C f H 1 0 ((0,T ]×∂M )
. Using this estimate we can define the DN map as the bounded operator from H 1 0 ((0, T ]×∂M ) to L 2 ((0, T ) × ∂M ) defined by

Λ A,q f = ∂ ν u - Aν 2 u | (0,T )×∂M
for u the solution of (1.3).

We have the following lemma that will be used in Section 3.

Lemma 2.2. Let F ∈ L p 1 (0, T ; L 2 (M )) and suppose u is the unique solution to (2.2) subject to u 0 = u 1 = 0. Then the following estimate holds:

u C(0,T ;L 2 (M )) C t 0 F (s) ds L p 1 (0,T ;L 2 (M ))
.

Proof. We set v(t, x) := t 0 u(s, x) ds and note that v solves (2.11)

   -∆ ḡv = H, (t, x) ∈ M, v(0, x) = 0, ∂ t v(0, x) = 0, x ∈ M v(t, x) = 0, (t, x) ∈ (0, T ) × ∂M, with H := - t 0 A(s, x)∇ ḡu(s, x) ds - t 0 q(s, x)u(s, x)ds + t 0 F (s, x) ds. Since u is in the energy space (1.4), we deduce v ∈ C 2 ([0, T ]; L 2 (M )) ∩ C 1 ([0, T ]; H 1 (M )). In addition, since qu ∈ L p 1 (0, T ; L 2 (M )) (see (2.1)) and A ∈ L ∞ (M; T * M), we deduce that H ∈ W 1,p 1 (0, T ; L 2 (M )) ⊂ L 2 (M)
and that v solves the elliptic boundary value problem

-∆ g v = E, (t, x) ∈ (0, T ) × M, v(t, x) = 0, (t, x) ∈ (0, T ) × ∂M, with E = -∂ 2 t v+H ∈ L 2 (M).
Then, from the elliptic regularity of solutions of this boundary value problem, we get v ∈ L 2 (0, T ; H 2 (M )) and it follows that v ∈ H 2 (M). We define the energy E(t) at time t associated with v and given by

E(t) := M |∂ t v| 2 (t, x) + |∇ g v| 2 g (t, x) dV g (x) M |u| 2 dV g (x).
Multiplying (2.11) by ∂ t v and taking the real part, we find (2.12)

E(t) = -2R t 0 M s 0 q(τ, x)u(τ, x) dτ ∂ t v(s, x) dV g (x) ds -2R t 0 M s 0 A(τ, x)∇ ḡu(τ, x) dτ ∂ t v(s, x) dV g (x) ds + 2R t 0 M s 0 F (τ, x) dτ ∂ t v(s, x) dV g (x) ds .
Repeating some arguments of [27, Lemma 3.1], we find (2.13)

| t 0 M s 0 q(τ, x)u(τ, x) dτ ∂ t v(s, x) dV g (x) ds| C q 2 L p 1 (0,T ;L p 2 (M )) t 0 E(τ ) p 1 (p 1 -1) dτ (p 1 -1) p 1 + E(t) 5 , (2.14)
| t 0 M s 0 F (τ, x) dτ ∂ t v(s, x) dV g (x) ds| F * 2 L p 1 (0,T ;L 2 (M )) + T p 1 -1 p 1 t 0 E(τ ) p 1 p 1 -1 dτ p 1 -1 p 1
, where F * (t, x) := t 0 F (s, x)ds and C > 0 depends on M and T . In the same way, using the fact that div ḡA ∈ L p 1 (0, T ; L p 2 (M )) and v ∈ C 1 ([0, T ]; H 1 0 (M )), we get (2.15)

t 0 M s 0 A∇ ḡu(τ, x)dτ ∂ t v(s, x) dV g (x) ds = - t 0 s 0 M (div ḡA)u ∂ t v(s, x) dV g (x) dτ ds - t 0 M s 0 u A(τ, x)∂ t ∇ ḡv(s, x) dV g (x) dτ ds - t 0 M b(s, x)|∂ t v(s, x)| 2 dV g (x) ds.
Repeating the arguments of (2.13), we find (2.16)

| t 0 s 0 M (div ḡA)u(τ, x)∂ t v(s, x) dV g (x) dτ ds + t 0 M b(s, x)|∂ t v(s, x)| 2 dV g (x) ds| C t 0 E(τ ) p 1 (p 1 -1) dτ (p 1 -1) p 1 + E(t) 5 ,
with C depending on T , M , div ḡA L p 1 (0,T ;L p 2 (M )) and b L ∞ (M)) . Moreover, applying Fubini's theorem, we have

t 0 M s 0 u(τ, x)A(τ, x)∂ t ∇ ḡv(s, x) dV g (x) dτ ds = M t 0 uA(τ, x) t τ ∂ t ∇ ḡv(s, x) ds dτ dV g (x) = M t 0 uA(τ, x)∇ ḡv(t, x) dV g (x) dτ - M t 0 uA(τ, x)∇ ḡv(τ, x) dτ dV g (x).
It follows that

| t 0 M s 0 uA(τ, x)∂ t ∇ ḡv(s, x) dV g (x) dτ ds| A L ∞ (M) t 0 E(τ ) 1 2 dτ E(t) 1 2 + A L ∞ (M) t 0 E(τ ) dτ 5 A 2 L ∞ (M) t 0 E(τ ) 1 2 dτ 2 + E(t) 5 + A L ∞ (M) t 0 E(τ ) dτ 5 A 2 L ∞ (M) T t 0 E(τ ) dτ + E(t) 5 + A L ∞ (M) t 0 E(τ ) dτ (5 A 2 L ∞ (M) T + A L ∞ (M) ) t 0 E(τ ) dτ + E(t) 5 .
Applying Hölder's inequality, we get

| t 0 M s 0 uA(τ, x)∂ t ∇ ḡv(s, x) dV g (x) dτ ds| (5 A 2 L ∞ (M) T + A L ∞ (M) )T 1 p 1 t 0 E(τ ) p 1 (p 1 -1) dτ (p 1 -1) p 1 + E(t) 5 .
Combining this with (2.16), we deduce that (2.17)

| t 0 M s 0 uA(τ, x)∂ t ∇ ḡv(s, x) dV g (x) dτ ds| C t 0 E(τ ) p 1 (p 1 -1) dτ (p 1 -1) p 1 + 2E(t) 5 ,
with C depending only on T and A L ∞ (M) . We deduce that there exists C depending on T , M , q L p 1 (0,T ;L p 2 (M )) , A L ∞ (M) and div ḡA L p 1 (0,T ;L p 2 (M )) such that

E(t) 4E(t) 5 + C t 0 E(τ ) p 1 p 1 -1 dτ p 1 -1 p 1 + F * 2 L p 1 (0,T ;L 2 (M ))
and therefore

E(t) 5C t 0 E(τ ) p 1 p 1 -1 dτ p 1 -1 p 1 + 5 F * 2 L p 1 (0,T ;L 2 (M )) .
Applying the Gronwall inequality yields

E(t) p 1 p 1 -1 c 1 F * 2p 1 p 1 -1 L p 1 (0,T ;L 2 (M )) e c 2 t c 1 F * 2p 1 p 1 -1 L p 1 (0,T ;L 2 (M )) e c 2 T , t ∈ (0, T ),
where c 1 depends only on p 1 and c 2 on C and p 1 .

2.2. Dirichlet to Neumann map. In this section we will derive a representation formula involving the DN map (Lemma 2.3) and also recall some invariance properties of the DN map (Lemma 2.4).

Let us consider the following problem (2.18)

   L * A,q v = -∆ ḡv -A∇ ḡv + (q -div ḡA)v = 0, on M, v = h, on (0, T ) × ∂M, v(T, •) = 0, ∂ t v(T, •) = 0 on M.
Here, the differential operator L * A,q represents the formal adjoint of L A,q . Repeating the arguments of the previous section we can prove that, for each h ∈ H 1 0 ([0, T ) × ∂M ), this problem admits a unique solution v in energy space (1.4), with ∂ ν v ∈ L 2 ((0, T ) × ∂M ), satisfying the estimate

∂ ν v L 2 ((0,T )×∂M ) + v X C h H 1 0 ([0,T )×∂M )
. Therefore, we can define the DN map associated with (2.18) as follows

(2.19) Λ * A,q h = ∂ ν v + Aν 2 v | (0,T )×∂M .
It is straightforward to show that (2.20)

< Λ A,q f, h >=< f, Λ * A,q h > ∀(f, h) ∈ H 1 0 ((0, T ] × ∂M ) × H 1 0 ([0, T ) × ∂M ), where f 1 , f 2 := (0,T )×∂M f 1 f 2 dV ḡ.
Using this equality together with Green's identity, we can derive the following classical representation formula. Lemma 2.3. Let A 1 , A 2 , q 1 , q 2 satisfy (1.2). Given any f 1 ∈ H 1 0 ((0, T ] × ∂M ) and f 2 ∈ H 1 0 ([0, T ) × ∂M ), the following identity holds:

(2.21) (Λ A 1 ,q 1 -Λ A 2 ,q 2 )f 1 , f 2 = M u 2 A∇ ḡu 1 -u 1 A∇ ḡu 2 2 + (q - 1 2 div ḡA)u 1 u 2 dV ḡ,
where A := A 1 -A 2 , q := q 1 -q 2 , u 1 solves (1.3) with A = A 1 , q = q 1 and lateral boundary term f 1 while u 2 solves (2.18) with A = A 2 , q = q 2 and lateral boundary term f 2 .

We also have the following lemma regarding the gauge equivalence of the Dirichlet to Neumann map: Lemma 2.4. Let A, q satisfy (1.2). Suppose ψ ∈ C 1 (M) vanishes on (0, T ) × ∂M and satisfies ∆ ḡψ ∈ L p 1 (0, T ; L p 2 (M )). Then

Λ A,q = Λ Ã,q , where (2.22) Ã = A + dψ and q = q + 1 2 ∆ ḡψ - 1 2 A∇ ḡψ - 1 4 ∇ ḡψ, ∇ ḡψ ḡ.
Proof. We start by observing that if u solves differential equation (1.3) with coefficients A, q and a lateral boundary condition f , then ũ = e 1 2 ψ u solves equation (1.3) with coefficients Ã, q and the same lateral boundary condition f . Then it follows that

Λ Ã,q f = ∂ ν ũ - Ãν 2 ũ | (0,T )×∂M = ∂ ν u + ∂ ν ψ 2 u - Ãν 2 u | (0,T )×∂M = ∂ ν u - Aν 2 u | (0,T )×∂M = Λ A,q f.

2.3.

Smooth approximation of the coefficients A and q. The goal of this section is to show that given one-forms A k , k = 1, 2 satisfying (1.2), it is possible to find smooth approximations A k,ρ that are defined in a slightly larger manifold M and such that (2.24) holds. Let M ⊂ M int ⊂ M int denote a small artificial extension of the simple manifold M , so that M , M are also simple manifolds and define M = R × M . We first consider the Sobolev extension of A k , k = 1, 2 to the larger manifold R × M such that the extension belongs to W 1,1 (R; L 2 (M )) ∩ C(R × M ) and then extend this extended one-form to R × M by setting it equal to zero on R × ( M \ M ). The scalar functions q k are extended to R × M by setting them equal to zero outside of M. Let p ∈ M \ M . As M is simple, there exists a global coordinate chart on a neighborhood of M given by (y 1 , . . . , y n ). Indeed one such coordinate system would be the polar normal coordinates around a point p ∈ M \ M (see for example [49, Chaper 9, Lemma 15]). We then consider the coordinate chart (t, y 1 , . . . , y n ) on a neighborhood of M in R × M and note that using this chart we can easily define smooth approximations of the coefficients A k , q k . Indeed, let ρ > 0 and define the smooth function

ζ ρ : M → R through ζ ρ (t, y) = ρ n+1 4 χ(ρ 1 4 t 2 + (y 1 ) 2 + . . . + (y n ) 2 )
where χ : R → R is a non-negative smooth function satisfying χ(t) = 1 for |t| < 1 4 and χ = 0 for |t| > 1 2 and χ L 1 (R) = 1. We define the smooth approximations A k,ρ of the coefficients A k through the expressions

(2.23) A k,ρ (t, x) := (A k * ζ ρ )(t, x) = b k,ρ dt + A k,ρ ∀(t, x) ∈ M k = 1, 2.
and note that in view of (1.2), the following estimates hold for k = 1, 2:

(2.24)

lim ρ→∞ A k,ρ -A k W 1,1 (0,T ;L 2 (M )) + A k,ρ -A k L p (M) = 0 ∀ p ∈ [1, ∞), A k,ρ W k,∞ (M) ρ k 4 ∀k ∈ N * .
Additionally, since A ∈ C(M; T * M) we can write

(2.25) lim ρ→∞ A k,ρ -A C((0,T )×Ωρ) = 0, k = 1, 2
where

Ω ρ = {x ∈ M | dist (x, ∂M ) ρ -1 4 }.

2.4.

Light ray transform as a Fourier Integral Operator. The main goal of this section is to extend the notion of L β over scalar functions in L p Sobolev spaces. This extension is based on showing that L is a Fourier Integral Operator. We will assume through out this section that (M, ḡ) and (M, g) are as discussed in the introduction and that M ⊂ M int with M as in Section 2.3. We start with the notion of light ray transform L of scalar functions over null geodesics in R × M showing that it has a unique continuous extension as an operator from E (R × M ) to D (R × ∂ -S M ). This would naturally show that the light ray transform L β of scalar functions over null geodesics on M has a continuous extension from L 1 (0, T ;

L 2 (M )) to D (R × ∂ -SM ) as L 1 (0, T ; L 2 (M )) ⊂ E (R × M ).
We will now show that the kernel of L is locally represented by an oscillatory integral. It suffices to consider f ∈ C ∞ c (R × M ) that is supported in a coordinate neighborhood and work in local coordinates on M . Let us also extend the geodesics γ(

•; y, v), (y, v) ∈ ∂ -S M , as functions from R to M so that γ(s; y, v) / ∈ supp (f ) for s / ∈ [0, τ + (x, v)]. Then in local coordinates Lf (s, y, v) = R f (r + s, γ(r; y, v)) dr = R R n f (t, x)δ(x -γ(t -s; y, v)) dx dt,
and writing ϕ(x, t; s, y, v; ξ) = ξ(x -γ(t -s; y, v)) it holds that δ(x -γ(t -s; y, v)) = R n e iϕ(x,t;s,y,v;ξ) dξ.

Moreover, ϕ is an operator phase function in the sense of [START_REF] Hörmander | Fourier integral operators[END_REF]Def. 1.4.4]. Indeed for fixed (s, y, v) it clearly has no critical points when ξ = 0. That the same is true for fixed (t, x) follows from the next lemma. Lemma 2.5. Let (y 0 , v 0 ) ∈ ∂ -S M and r 0 ∈ (0, τ + (y 0 , v 0 )), and consider a small neighborhood U of (r 0 , y 0 , v 0 ) in R×∂ -S M . Then γ(r; y, v) as a map from U to M has surjective differential at (r 0 , y 0 , v 0 ).

Proof. Write x 0 = γ(r 0 ; y 0 , v 0 ), w 0 = γ(r 0 ; y 0 , v 0 ) and let ξ * ∈ T x 0 M . Choose a path α in M such that α(0) = x 0 and α(0) = ξ * . Consider -w 0 in local coordinates as a vector in all

T α(ε) M for small ε > 0. As γ(r 0 ; x 0 , -w 0 ) = y 0 , γ(r 0 ; x 0 , -w 0 ) = -v 0 / ∈ T y 0 (∂ M ),
it follows from the implicit function theorem that there is unique r(ε) near r 0 such that γ(r(ε); α(ε), -w 0 ) ∈ ∂ M . Writing

y(ε) = γ(r(ε); α(ε), -w 0 ), v(ε) = -γ(r(ε); α(ε), -w 0 ),
we have that γ(r(ε); y(ε), v(ε)) = α(ε). Hence the differential of the map γ takes vectors ( ṙ(0), ẏ(0), v(0)) to ξ * = α(0).

As ϕ is an operator phase function, the light ray transform L has a unique continuous extension as an operator from E (R × M ) to D (R × ∂ -S M ) by [START_REF] Hörmander | Fourier integral operators[END_REF]Th. 1.4.1].

Geometric Optics

Throughout this section we consider one-forms A 1 , A 2 and scalar functions q 1 , q 2 to satisfy regularity conditions given by (1.2) and consider their extensions to the manifold M and their smooth approximations on the manifold M as outlined in Section 2.3. We consider a fixed null geodesic β ⊂ D parametrized with respect to the time variable. The projection of this null-geodesic on M is denoted by the (Riemannian) unit speed geodesic γ(•; y, v) defined over its maximal domain I = [0, τ + (y, v)]. We extend γ to M and let the interval Î = [-δ-, τ + (y, v) + δ+ ] to denote the maximal domain of definition of γ on the manifold M . Subsequently we can parametrize the extended null geodesic β on M through β(t; s, y, v) = (s + t, γ(r; y, v)) for t ∈ Î, where s ∈ R is a constant. We are interested in constructing the so called geometric optic solutions u 1 , u 2 in energy space (1.4), of the problems (3.1)

-∆ ḡu 1 + A 1 ∇ ḡu 1 + q 1 u 1 = 0, (t, x) ∈ M, u 1 (0, x) = ∂ t u 1 (0, x) = 0, x ∈ M, -∆ ḡu 2 -A 2 ∇ ḡu 2 + (q 2 -div ḡA 2 )u 2 = 0, (t, x) ∈ M, u 2 (T, x) = ∂ t u 2 (T, x) = 0, x ∈ M, taking the form (3.2) u 1 (t, x) = e iρΦ(t,x) c 1,ρ (t, x) + R 1,ρ (t, x), (t, x) ∈ M, (3.3) u 2 (t, x) = e -iρΦ(t,x) c 2,ρ (t, x) + R 2,ρ (t, x), (t, x) ∈ M,
with ρ > 1. The phase function Φ and the smooth amplitude functions c j,ρ , j = 1, 2 are constructed in a way that the principal terms e iρΦ c j,ρ are compactly supported near the null geodesic β. The remainder terms R j,ρ asymptotically converge to zero as ρ → ∞.

As we are interested in a particular null geodesic β, we outline a polar normal coordinate system specific to this null geodesic. We start by considering a point p on {0} × γ with p ∈ {0} × ( M \ M ) and construct the polar normal coordinates (t, r, θ) about the point p defined for r > 0 and θ ∈ S p M = {v ∈ T p M | |v| g = 1} through the diffeomorphism (t, x) = (t, exp(rθ)). In this coordinate system the metric ḡ is smooth away from the point p and takes the form

(3.4) ḡ(t, r, θ) = -(dt) 2 + (dr) 2 + g 0 (r, θ),
where g 0 is a Riemannian metric on S p M . As we will be only considering this coordinate system on the manifold M and owing to the fact that M is simple, we can identify θ with a globally defined coordinate system (θ 1 , . . . , θ n-2 ) ∈ R n-2 . This can in fact be done in such a manner that the null geodesic β on M can be represented with coordinates (s+s 0 , s, 0, . . . , 0) with s ∈ Î.

In order to make the analysis simpler, we will introduce a new coordinate system near β denoted by (z 0 , z 1 , . . . , z n ) in terms of the polar normal coordinates (t, r, θ) on M given by (i)

z 0 := 1 √ 2 (t + r), (ii) z 1 := 1 √ 2 (-t + r + s 0 ), (iii
) z j := θ j for j = 2, . . . , n. In this coordinate system, the null geodesic β on M is given by the coordinates (s, 0) with s ∈ (a 0 , b 0 ) for some constants a 0 , b 0 . Furthermore the metric ḡ takes the form

(3.5) ḡ(z) = 2 dz 0 dz 1 + n j,k=2
g jk (z) dz j dz k .

We define a tubular neighborhood around the null geodesic β where the amplitude functions are compactly supported, as follows:

(3.6) V β = {z ∈ M | z 0 ∈ [a 0 , b 0 ], |z | := |z 1 | 2 + . . . + |z n | 2 < δ },
where δ > 0 is sufficiently small that the set V β is disjoint from {0} × M and {T } × M . This can be guaranteed due to the assumption β ⊂ D.

3.1. Construction of the Geometric Optics. We proceed to carry out the construction of the geometric optic solutions to (3.1) in detail. We impose to the remainder term

R k,ρ ∈ C([0, T ]; H 1 0 (M )) ∩ C 1 ([0, T ]; L 2 (M )), k = 1, 2 the following decay property (3.7) lim ρ→+∞ ( R k,ρ C(0,T ;L 2 (M )) + ρ -1 R k,ρ H 1 (M) ) = 0.
To prove the decay of R k,ρ with respect to ρ, given by (3.7), we need to suitably construct Φ, c 1,ρ , c 2,ρ . We write (3.8)

L A 1,ρ ,q 1 (e iρΦ c 1,ρ ) = e iρΦ ρ 2 SΦ -iρT A 1,ρ c 1,ρ + L A 1,ρ ,q 1 c 1,ρ , L * A 2,ρ ,q 2 (e -iρΦ c 2,ρ ) = e -iρΦ ρ 2 SΦ + iρT -A 2,ρ c 1,ρ + L * A 2,ρ ,q 2 c 2,ρ , where (3.9)
SΦ := ∇ ḡΦ, ∇ ḡΦ ḡ and

T A • = 2 ∇ ḡΦ, ∇ ḡ• ḡ + (-A∇ ḡΦ + ∆ ḡΦ) • .
We proceed to determine the phase function Φ(t, x) such that the eikonal equation Existence of global smooth solutions to this equation is not guaranteed in general, but owing to the assumption that the manifold is simple, we can find plenty of such solutions. Indeed for the remainder of this section, we will be working in the z coordinate system defined earlier. Recall that this coordinate system is well-defined on M and the null geodesic β on M is represented by (s, 0) with s ∈ [a 0 , b 0 ]. Recalling the form of the metric from (3.5), we pick

(3.12) Φ(z) = z 1 .
To determine the amplitude functions, we first use (3.5) again to rewrite the transport equations (3.11) as

(3.13) ∂ z 0 c 1,ρ + ∂ z 0 log |g| 4 - (A 1,ρ ) 0 2 c 1,ρ = 0, (3.14) ∂ z 0 c 2,ρ + ∂ z 0 log |g| 4 + (A 2,ρ ) 0 2 c 2,ρ = 0,
where (A k,ρ ) 0 := A k,ρ ∇ ḡΦ for k = 1, 2 and in particular we have

(3.15) (A k,ρ ) 0 | β = A k,ρ β.
We can take c k,ρ as follows:

(3.16) c 1,ρ (z) := |g(z)| -1/4 χ( |z | δ ) exp 1 2 z 0 a 0 [(A 1,ρ ) 0 (s, z )] ds , and
(3.17) c 2,ρ (z) := |g(z)| -1/4 χ( |z | δ ) exp - 1 2 z 0 a 0 [(A 2,ρ ) 0 (s, z )] ds ,
where χ is as defined in Section 2.3 and δ < δ (see (3.6)). It is clear that the amplitude functions c k,ρ are compactly supported in the set V β and as a result

(3.18) c k,ρ (s, x) = ∂ t c k,ρ (s, x) = 0, for k = 1, 2, s ∈ {0, T }, x ∈ M.
With the construction of the phase and amplitude functions completed as above, we let

F 1,ρ = -L A 1 ,q 1 c 1,ρ e iρΦ , F 2,ρ = -L * A 2
,q 2 c 2,ρ e -iρΦ and we recall that (3.10)-(3.11) imply that

(3.19) F 1,ρ = -e iρΦ L A 1,ρ ,q 1 c 1,ρ + iρ(A 1 -A 1,ρ )∇ ḡΦ c 1,ρ , (3.20)
F 2,ρ = -e -iρΦ L * A 2,ρ ,q 2 c 2,ρ + iρ(A 2 -A 2,ρ )∇ ḡΦ c 2,ρ .
We define the expression R j,ρ , j = 1, 2, by the solution of the following IBVP (3.21)

   L A 1 ,q 1 R 1,ρ = F 1,ρ , (t, x) ∈ M, R 1,ρ (0, x) = 0, ∂ t R 1,ρ (0, x) = 0, x ∈ M R 1,ρ (t, x) = 0, (t, x) ∈ (0, T ) × ∂M, (3.22)
   L * A 2 ,q 2 R 2,ρ = F 2,ρ , (t, x) ∈ M, R 2,ρ (T, x) = 0, ∂ t R 2,ρ (T, x) = 0, x ∈ M R 2,ρ (t, x) = 0, (t, x) ∈ (0, T ) × ∂M.
In order to complete the construction of the solutions u 1 , u 2 of (3.1), we only need to check the decay of the expression R k,ρ , k = 1, 2, given by (3.7). According to (2.23)-(2.24), we have

(3.23) c j,ρ W k,∞ (M) C k ρ k with C
F j,ρ L p 1 (0,T ;L 2 (M )) C(ρ 1 2 + ρ A j,ρ -A j L p 1 (0,T ;L 2 (M )) ), j = 1, 2.
Using (2.24) again and the estimate (2.4) it follows that

lim ρ→+∞ ρ -1 R j,ρ H 1 (M) C lim ρ→+∞ ρ -1 F j,ρ L p 1 (0,T ;L 2 (M )) = 0, j = 1, 2.
Therefore, in order to prove (3.7), it only remains to prove that

(3.24) lim ρ→+∞ R j,ρ C(0,T ;L 2 (M )) = 0, j = 1, 2.
Proof of Estimate 3.24. The result for R 1,ρ and R 2,ρ being similar, we will only consider this claim for R 1,ρ . In view of Lemma 2.2, the proof of the estimate will be completed if we show that

F * ,ρ (t, x) = t 0 e iρΦ(τ,x) L A 1,ρ ,q 1 c 1,ρ (τ, x) + iρ(A 1 -A 1,ρ )∇ ḡΦ c 1,ρ (τ, x) dτ = t 0 e iρΦ(τ,x) (L A 1,ρ ,q 1 -q 1 )c 1,ρ (τ, x) + iρ(A 1 -A 1,ρ )∇ ḡΦ c 1,ρ (τ, x) dτ I + t 0 e iρΦ(τ,x) q 1 c 1,ρ (τ, x) dτ II .
To analyze the terms I and II we will integrate by parts in the τ variable and note that by equation (3.12) we have

(3.27) ∂ τ Φ(τ, x) = 1 √ 2 = 0.
For the term I, using the fact that A ∈ W 1,1 (0, T ; L 2 (M )) and equation (3.18), we can integrate by parts, with respect to τ ∈ (0, t), and write (3.28)

√ 2 2 I = -iρ -1 e iρΦ(t,x) [(L A 1 ,q 1 -q 1 )c 1,ρ (t, x) + iρ ((A 1 -A 1,ρ )∇ ḡΦ)) c 1,ρ (t, x)] + iρ -1 t 0 e iρΦ(τ,x) [∂ t A 1 ∇ ḡc 1,ρ (τ, x) + (L A 1 ,q 1 -q 1 )∂ t c 1,ρ (τ, x)] dτ - t 0 e iρΦ(τ,x) [((A 1 -A 1,ρ )∇ ḡΦ)) ∂ t c 1,ρ (τ, x)] dτ - t 0 e iρΦ(τ,x) [((∂ t A 1 -∂ t A 1,ρ )∇ ḡΦ)) c 1,ρ (τ, x)] dτ = S 1 + S 2 + S 3 + S 4 .
For the term S 1 , we can apply (3.23) and (2.24) to write

S 1 L p 1 (0,T ;L 2 (M )) Cρ -1 c 1,ρ W 2,∞ (M) + A 1 -A 1,ρ L p 1 (0,T ;L 2 (M )) c 1,ρ L ∞ (M) C(ρ -1 2 + A 1 -A 1,ρ L p 1 (0,T ;L 2 (M )) ) = o(1).
For the term S 2 we similarly write (3.29)

ρ -1 t 0 e iρΦ(τ,x) [∂ t A 1 ∇ ḡc 1,ρ (τ, x) + (L A 1 ,q 1 -q 1 )∂ t c 1,ρ (τ, x)] dτ L p 1 (0,T ;L 2 (M )) Cρ -1 ( A 1 W 1,1 (0,T ;L 2 (M )) ) c 1,ρ W 2,∞ (M)) Cρ -1 2 ,
with C independent of ρ. For the terms S 3 and S 4 , let us first assume that

A 1 ∈ C 2 ([0, T ]; L 2 (M )).
Then, integrating by parts with respect to τ ∈ (0, t) and applying (3.23), (2.24), we have

t 0 e iρΦ(τ,x) [((A 1 -A 1,ρ )∇ ḡΦ) ∂ t c 1,ρ (τ, •)] dτ L p 1 (0,T ;L 2 (M )) Cρ -1 2 , t 0 e iρΦ(τ,x) [((∂ t A 1 -∂ t A 1,ρ )∇ ḡΦ) c 1,ρ (τ, •)] dτ L p 1 (0,T ;L 2 (M )) Cρ -1 2 ,
with C independent of ρ. Then, applying the density of C 2 ([0, T ]; L 2 (M )) in W 1,1 (0, T ; L 2 (M ), we deduce that lim ρ→∞

S 3 L p 1 (0,T ;L 2 (M )) = lim ρ→∞ S 4 L p 1 (0,T ;L 2 (M )) = 0.
Combining the above estimates we conclude that lim ρ→+∞ I L p 1 (0,T ;L 2 (M )) = 0.

Moreover, in a similar way to the terms S 3 and S 4 , using a density argument combined with (3.23), we have lim

ρ→+∞ II L p 1 (0,T ;L 2 (M )) = 0.
This completes the proof of estimate 3.24.

4. Reduction to the light ray transform and the proof of uniqueness 4.1. Reduction to the light ray transform of A 1 -A 2 and proof of Theorem 1.1. Suppose A j , q j for j = 1, 2 satisfy regularity conditions (1.2) and consider their extensions to M and smooth approximations A j,ρ satisfying (2.24). We assume that Λ A 1 ,q 1 = Λ A 2 ,q 2 and proceed to show that for every β ⊂ R × M the following identity holds:

(4.1) L β A = 0,
where A := A 1 -A 2 . We start by considering a maximal null geodesic β ⊂ D and extend it to M. Define u j in energy space (1.4) to be solutions of (3.1) taking the form (3.2)-(3.3) with the properties described in the previous section. Let

f 1 := u 1 | (0,T )×∂M ∈ H 1 0 ((0, T ] × ∂M ) and f 2 := u 2 | (0,T )×∂M ∈ H 1 0 ([0, T ) × ∂M ). Applying Lemma 2.3 we deduce that: (4.2) 0 = (Λ A 1 ,q 1 -Λ A 2 ,q 2 )f 1 , f 2 = M u 2 A∇ ḡu 1 -u 1 A∇ ḡu 2 2 + (q - 1 2 div ḡA)u 1 u 2 dV ḡ,
where q := q 1 -q 2 . Using the Sobolev embedding (2.1), and the bounds (3.23)-(3.7), we write (4.3)

|ρ -1 M R j,ρ A∇ ḡR k,ρ dV ḡ| ρ -1 A L ∞ (M) R k,ρ L 2 (M) R j,ρ H 1 (M) = o(1), |ρ -1 M e ±iρΦ QR k,ρ c j,ρ dV ḡ| ρ -1 Q L p 1 (0,T ;L p 2 (M )) R k,ρ C(0,T ;L 2 (M )) = o(ρ -1 ), |ρ -1 M QR j,ρ R k,ρ dV ḡ| ρ -1 Q L p 1 (0,T ;L p 2 (M )) R j,ρ C(0,T ;L 2 (M )) R k,ρ C(0,T ;H 1 (M )) = o(1).
for j, k = 1, 2 and Q = q -1 2 div ḡA. Dividing equation (4.2) by ρ, using (3.2)-(3.3) and applying the latter bounds, we observe that lim ρ→∞ M A∇ ḡΦ c 1,ρ c 2,ρ dV ḡ = 0.

Recall from (3.16)-(3.17) that c k,ρ are compactly supported on V β . Recalling that A = 0 outside of (0, T ) × M (both A 1 , A 2 vanish there), and additionally using (2.3), we write

lim ρ→∞ V β A ρ ∇ ḡΦ c 1,ρ c 2,ρ dV ḡ = 0. which reduces to lim ρ→∞ (a 0 ,b 0 )×B(0,δ) (A ρ ) 0 (z 0 , z ) χ( |z | δ ) 2 exp 1 2 z 0 a 0 (A ρ ) 0 (s, z ) ds dz 0 dz = 0,
where (A ρ ) 0 = A∇ ḡΦ. Observing that

(A ρ ) 0 (z 0 , z ) exp 1 2 z 0 a 0 (A ρ ) 0 (s, z ) ds = d dz 0 exp 1 2 z 0 a 0 (A ρ ) 0 (s, z ) ds ,
together with (2.25) and vanishing of A in the exterior of M, we simplify the former equation to obtain

B(0,δ) χ( |z | δ ) 2 exp 1 2 b 0 a 0 (A) 0 (s, z ) ds dz = 0,
Finally, by taking δ → 0, and observing that (A) 0 (s, 0) = A β, we observe that

L β A ∈ 4πiZ.
Note that the above claim holds for any null geodesic β ⊂ D. Recall from the hypothesis of Theorem 1.1 that A is supported on the set E. Thus, we can conclude that the latter equality holds for any null geodesic in R × M . Let β = (s 0 + t, γ(t)) for some s 0 and consider a one-parameter family of null geodesics β s = (s 0 + s + t, γ(t)). Since A is continuous and since L βs = 0 for s large, we conclude that equality (4.1) holds. Applying statement (ii) in Proposition 1.3 completes the proof of Theorem 1.1.

4.2.

Reduction to the light ray transform of q 1 -q 2 and proof of Theorem 1.2. We will assume throughout this section that the additional regularity assumptions (1.9) hold. Applying Theorem 1.1 implies that there exists ψ ∈ C 1 0 (M) such that A 1 = A 2 + dψ. Clearly,

∆ ḡψ = div ḡ(A 1 -A 2 ) ∈ L p 1 (0, T ; L ∞ (M )).
Let us now define Ã2 = A 2 + dψ and q2

= q 2 + 1 2 ∆ ḡψ -1 2 A 2 ∇ ḡψ -1 4 ∇ ḡψ, ∇ ḡψ ḡ. Lemma 2.4 applies to show that (4.4) Λ A 1 ,q 1 = Λ Ã2 ,q 2 = Λ A 1 ,q 2 .
Analogously to the previous section, we start by considering a null geodesic β ⊂ D and extend it to M. Define u j in energy space (1.4) to be solutions of (3.1) corresponding to differential operators L A 1 ,q 1 and L * A 1 ,q 2 , taking the form (3.2)-(3.3) and with the properties described in Section 3. Let

f 1 := u 1 | (0,T )×∂M ∈ H 1 0 ((0, T ] × ∂M ) and f 2 := u 2 | (0,T )×∂M ∈ H 1 0 ([0, T ) × ∂M ).
Applying Lemma 2.3 again, we deduce that:

(4.5) 0 = (Λ A 1 ,q 1 -Λ A 1 ,q 2 )f 1 , f 2 = M qc 1,ρ c 2,ρ dV ḡ,
where q := q 1 -q2 ∈ L p 1 (0, T ; L ∞ (M )). Recall that c 1,ρ , c 2,ρ are supported in the tubular set V β near the null geodesic β. Estimate (3.24) implies that

| M qc k,ρ R j,ρ dV ḡ| q L p 1 (0,T ;L 2 (M )) c k,ρ L ∞ (M) R j,ρ C(0,T ;L 2 (M )) = o(1), | M qR 1,ρ R 2,ρ dV g | q L p 1 (0,T ;L ∞ (M )) R 1 C(0,T ;L 2 (M )) R 2 C(0,T ;L 2 (M )) = o(1)
.

We now use the z coordinate system and note that by taking the limit as ρ → ∞ and using equations (3.16)-(3.17) with the preceding correction term bounds, we have

(a 0 ,b 0 )×B(0,δ) q(z 0 , z )χ( |z | δ ) 2 dz 0 dz = 0.
The arguments in Section 2.4 apply to deduce that

L β q = 0.
Together with statement (i) in Proposition 1.3, we conclude that equation (1.10) holds.

Inversion of the light ray transform

This section is concerned with the proof of Proposition 1.3. Recalling Section 1.1, we will identify maximal null geodesics β ⊂ R × M with triplets (s, x, v) ∈ R × ∂ -SM . Let us first recall the unique inversion of light ray transform on smooth functions. This is reproduced here as some of the arguments are necessary for the extension of the proof to L 1 (0, T ; L 2 (M )) functions. There is a 0 ∈ R such that f = 0 in (-∞, a 0 ) × M . If s < a 0 -T then the non-trapping assumption implies that the light ray β(r) = (r + s, γ(r; y, v)) does not intersect supp (f ) for any (y, v) ∈ ∂ -SM . Now setting U = (-∞, a), with a = a 0 -T -1, and V = ∂ -SM , we can choose χ so that χ = 1 in W and χ = 0 in supp (f ). Then Lf vanishes in (-∞, a) × ∂ -SM . Similarly we can get an upper bound for the support with respect to time. 5.0.3. On partial Fourier transform in time. On a product manifold R × M we define the partial Fourier transform in time by

f (z), ϕ E ×C ∞ (M ) = f, e -izt ⊗ ϕ E ×C ∞ (R×M ) , f ∈ E (R × M ), z ∈ C.
It follows from [START_REF] Hörmander | The analysis of linear partial differential operators I[END_REF]Th. 2.1.3] that z → f (z), ϕ is smooth and for all j = 1, 2, . . . , ∂ j z f (z), ϕ = f, ∂ j z e -izt ⊗ ϕ , ∂ z f (z), ϕ = 0. The latter identity says that z → f (z), ϕ is analytic, and the former implies that the map f → ∂ j z f (z)| z=0 is continuous from E (R × M ) to E (M ). Let a, b ∈ R and consider L 2 ((a, b)×M ) as a subspace of L 2 (R; E) with E = L 2 (M ). Then the above definition of f (z) coincides with the usual definition of the Fourier transform on L 2 (R; E). Let us recall that the Fourier transform on L 2 (R; E) is a unitary isomorphism as E is a Hilbert space, see e.g. the discussion on p. 16 of [START_REF] Lions | Non-homogeneous boundary value problems and applications[END_REF]. It is also easy to see that the map f → ∂ j z f (z)| z=0 is continuous from L 2 ((a, b) × M ) to L 2 (M ). 5.0.4. Geodesic ray transform on L 2 functions. Since ∂M is strictly convex, I extends as a map from L 2 (M ) to L 2 (∂ -SM ) with a suitably chosen measure on ∂ -SM (see for example [START_REF] Salazar | Determination of time-dependent coefficients for a hyperbolic inverse problem[END_REF]Th. 4.2.1]). In what follows, we will therefore assume that I is a map from L 2 (M ) to L 2 (∂ -SM ). and R j has a unique continuous extension as a map from L 2 (M ) to L 2 (∂ -SM ).

5.0.6. The inversion. Let f ∈ L 1 ((0, T ); L 2 (M )) and choose a sequence of functions f j ∈ C ∞ c ((0, T ) × M ) such that f j → f in L 1 ((0, T ); L 2 (M )). Then Lf j → Lf in D (R × ∂ -SM ). As Lf and Lf j are compactly supported in time by Lemma 5.2, also ∂ k z Lf j (0) → ∂ k z Lf (0) in D (∂ -SM ). Furthermore, ∂ k z f j (0) → ∂ k z f (0) in L 2 (M ). Finally, using the L 2 -continuity of I and R k , we see that the identity (5.1), that holds for each f j , holds also for f by passing to the limit.

Recalling that I is injective on L 2 (M ) for simple manifolds (M, g) (see for example [START_REF] Assylbekov | A sharp stability estimate for the geodesic ray transform[END_REF] or [START_REF] Sharafutdinov | Integral geometry of tensor fields[END_REF]), we see that Lf = 0 implies that ∂ k z f (0) = 0, as a function in L 2 (M ), for all k = 0, 1, . . . . For any ϕ ∈ C ∞ c (M ) all the derivatives of the analytic function f (z), ϕ vanish at the origin. Hence f (z), ϕ vanishes identically. Therefore f vanishes as a function in L 2 (R; E) with E = L 2 (M ). We conclude that f = 0.

  and B ∈ C(M; T * M) both vanish on the set M \ E. Then the following statements hold: (i) If L β f = 0 for all maximal null geodesics β ⊂ D, then f ≡ 0. (ii) If L β B = 0 for all maximal null geodesics β ⊂ D, then B ≡ dψ for some ψ ∈ C 1 (M) with ψ| ∂M = 0.

(3. 10 )

 10 SΦ = 0 on M is satisfied. The amplitude functions c k,ρ (t, x) for k = 1, 2 are constructed such that the transport equations (3.11) T A 1,ρ c 1,ρ = 0 and T -A 2,ρ c 2,ρ = 0 on M hold. Let us start with the eikonal equation.

F

  * ,ρ L p 1 (0,T ;L 2 (M )) = 0, where F * ,ρ (t, x) := -t 0 F 1,ρ (s, x) ds. Recall that(3.26)

5.0. 1 .Lemma 5 . 1 .

 151 Inversion of light ray transform for smooth functions. For f ∈ C ∞ c (R × M ), the transform Lf (s, x, v) is compactly supported in s. Inversion of L is based on the following Fourier slicing in timeLf (τ, x, v) = R e -iτ s Lf (s, x, v) ds = τ + (x,v) 0 R e -iτ s f (r + s, γ(r; x, v)) ds dr = τ + (x,v) 0 e iτ r R e -iτ t f (t, γ(r; x, v)) dt dr = τ + (x,v) 0e iτ r f (τ, γ(r; x, ξ)) dr.In particular, Lf (0,x, v) = I( f (0, •))(x, v). Straightforward differentiation gives the following lemma. For f ∈ C ∞ c (R × M ), k = 0, 1, . . . , and (x, v) ∈ ∂ -SM it holds that ∂ k τ Lf (τ, x, v)| τ =0 = I(∂ k τ f (τ, •)| τ =0 )(x, v) + k-1 j=0 k j R k-j (∂ k τ f (τ, •)| τ =0 )(x, v), (5.1)whereR j f (x, v) = τ + (x,v) 0 (ir) j f (γ(r, x, v)) dr, f ∈ C ∞ c (M ). If I is injective then Lf = 0 implies that ∂ k τ f (τ, •)| τ =0= 0 for all k = 0, 1, . . . . As f is compactly supported in t, the Fourier transform f is analytic in τ . Hence f = 0 in this case.5.0.2. A localization property.We have the following natural localization property. Lemma 5.2. Let U ⊂ R and V ⊂ ∂ -SM be open. Define W to be the set of points (t, x) ∈ R × M such that t = r + s and x = γ(r; y, v) for some r∈ [0, τ + (y, v)], s ∈ U and (y, v) ∈ V . Suppose that χ ∈ C ∞ (R × M ) satisfies χ| W = 1. Then Lf | U ×V = L(χf )| U ×V , f ∈ E (R × M ).In particular, for anyf ∈ E (R × M ) there are a, b ∈ R such that the support of Lf is contained in [a, b] × ∂ -SM . Proof. The claimed localization clearly holds when f ∈ C ∞ 0 (R × M ). For a distribution f ∈ E (R × M ) we choose a sequence of functions f j ∈ C ∞ 0 (R × M ) such that f j → f in E (R × M ). ThenLf | U ×V = lim j→∞ Lf j | U ×V = lim j→∞ L(χf j )| U ×V = L(χf )| U ×V .

5.0. 5 .

 5 The remainder operator R j on L 2 functions. Let us consider the operators R j , j = 1, 2, . . . , defined in Lemma 5.1. For f ∈ C ∞ c (M ) it holds that |R j f (x, v)| ≤ L j τ + (x,v) 0 |f (γ(r, x, v))| dr = L j I(|f |)(x, v), (x, v) ∈ ∂ -SM,where L = Diam(M ). ThereforeR j f L 2 (∂ -SM ) ≤ L j I(|f |) L 2 (∂ -SM ) ≤ C f L 2 (M ) ,

  and C k independent of ρ. Combining this with (3.19)-(3.20), we find

, k ∈ N * , c j,ρ W 1,1 (0,T ;L 2 (M )) C,

Acknowledgments 22 References 23

Acknowledgments

A.F acknowledges the support from the EPSRC grant EP/P01593X/1 and Y.K. acknowledges the support from the Agence Nationale de la Recherche grant ANR-17-CE40-0029. The authors would also like thank Lauri Oksanen for helpful discussions and his contributions to parts of this paper.


Wave energy converters

Judicael Aubry, Hamid Ben Ahmed, Bernard Multon, Aurélien Babarit, Alain

19 20


C Bessaguet

E Dantras B '

G Michon

M Chevalier

L Laffont

C Lacabanne

Electrical behavior of a graphene/PEKK and carbon black/PEKK nanocomposites in the vicinity of the percolation threshold

Keywords:

Graphene and carbon black have been dispersed in a high performance thermoplastic polymer, the poly(ether ketone ketone), to improve its electrical conductivity. The dispersion of graphene has a significant influence on the percolation threshold. A simple exfoliation protocol to obtain graphene monolayers has led to a significant decrease of the percolation threshold from 4.2 to 1. 9 vol%. To the best of our knowledge, it is one of the lowest percolation values for unfunctionalized graphene dispersed by melt blending in a high performance thermo plastic matrix. The conductivity value above the percolation threshold (1.2 S•m -i) means that graphene was not degraded during the elaboration process. Below the percolation threshold, Maxwell-Wagner-Sillars phenomenon increases the dielectric permittivity from 2.7 to 210 for PEKK/6 vol% graphene at 180 °C and 1 Hz. Dynamic mechanical analyses have shown that mechanical moduli were not significantly modified by conductive particles until 6 vol%.

Introduction

High performance thermoplastic composites raise a growing interest for structural applications in aeronautical and space industries. In order to improve their electrical properties, the introduction of conductive particles (metal or carbon) within the polymer matrix has been the subject of many studies. Usually, metallic fillers like gold [START_REF] Ramachandran | Conduction mecbanisms in P(VDF-TrFE)/gold nanowire composites: tunnelling and thennally activated hopping process near the percolation threshold[END_REF][START_REF] Salvadori | Electrical con ductivity of gold-implanted alumina nanocomposite, Nue![END_REF] or silver fillers [START_REF] Zheng | Fabrication of nove! Ag nanowires/poly(vinylidene fluoride) nanocomposite film with high dielectric constant[END_REF][START_REF] Rivière | Silver fillers aspect ratio influence on electrical and thermal conductivity in PEEK/ Ag nano composites[END_REF][START_REF] Cortes | High-performance thermoplastic composites poly(ether ketone ketone)/silver nanowires: morphological, mechanical and electrical properties[END_REF] permit to reach high electrical conductivity (a = 10 2 -10 3 S•m-1 ) but their high density implies an increase of the structure weight. Due to their low density, carbon fillers as carbon black [START_REF] Xu | Super high dielectric constant carbon black-filled polymer composites as integral capacitor dielectrics[END_REF], carbon nanotubes [START_REF] Van Durmen | Etude de l'influence de la dispersion de nanotubes de carbone sur les propriétés électriques de composites à matrice PEEK[END_REF][START_REF] Carponcin | Discontinuity of physical properties of carbon nanotube/polymer composites at the percolation threshold[END_REF] or graphene [START_REF] Dhand | A comprehensive review of graphene nanocomposites: research status and trends[END_REF] are of interest in a wide range of applications, where a high electrical conductivity for electrical dissipation (a= 0.1-1 S-m-1 ) is not required [10]. In order to maintain the mechanical properties of the matrix, for low particle content, con duction phenomenon may occur. The critical transition from an in sulating to a conducting behavior is associated with the percolation threshold. It corresponds to the formation of an infinite conducting path within the material. The value of the percolation threshold only de pends on the apparent conductive particle aspect ratio I; [START_REF] Kuilla | Recent advances in gra phene based polymer composites[END_REF][START_REF] Tkalya | Experimental and theoretical study of the in fluence of the state of dispersion of graphene on the percolation threshold of con ductive graphene/polystyrene nanocomposites[END_REF]. Lower percolation thresholds are reached for particles with a high aspect ratio. For example, percolation threshold is observed near 8--10 vol% [START_REF] Medalia | Electrical conduction in carbon black composites[END_REF] with carbon black (1; .,, 1) and lower than 0.4 vol% [START_REF] Carponcin | Evolution of dispersion of carbon nanotubes in polyamide 11 matrix composites as determined by DC conductivity[END_REF] with carbon • Corresponding author.

E-mail address: eric.dantras@univ-tlse3.fr (E. Dantras).

nanotubes (1; .,, 1000-10,000).

In the present work, graphene was used for the elaboration of conductive nanocomposites. Graphene was recently highlighted by Novoselov and Geim in 2004 [START_REF] Novoselov | Electric field effect in atomically thin carbon fihns[END_REF][START_REF] Novoselov | Two-dimensional atomic crystals[END_REF]. Its good electrical properties and high aspect ratio have aroused great interest among the scientific community [START_REF] Singh | Graphene based mate rials: past, present and future[END_REF][START_REF] Bolotin | Ultrahigh electron mobility in suspended graphene[END_REF][START_REF] Nair | Fine structure constant defines visual transperency of graphene[END_REF][START_REF] Seo | Two-dimensional phonon transport in supported graphene[END_REF][START_REF] Lee | Measurement of elastic properties and intrinsic strength of monolayer graphene[END_REF]: high charge mobility (230,000 cm 2 /Vs), low visible light absorption (2.3%), high thermal conductivity (3000 W / mK), high strength (130 GPa), high elastic modulus (1 TPa) and the highest theoretical specific surface area (2600 m 2 /g). A reference sample processed with spherical carbon black was also elaborated to compare with graphene.

The poly(ether ketone ketone) (PEKK) was used as polymer matrix. It is a high performance thermoplastic polymer that belongs to the poly Caryl ether ketone) family. One of the main advantages of PEKK is that the melting point can be modulated, as reported by Gardner [START_REF] Gardner | Structure, crystallization and morphology of poly Caryl ether ketone ketone)[END_REF]. 100% para PEKK and para -iso PEKK show differences in the melting point temperature, whereas the glass transition temperature stays quasi constant. By increasing the proportion of meta isomers, defects in the crystal packing decrease the melting point temperature and enhance the processing conditions.

The processing of nanocomposite and, more specifically, the dis persion of particle is the most important challenge to obtain https:/ /doi.org/10.1016/j.jnoncrysol.2019.02.017 macroscopic homogeneous properties. Due to strong interactions be tween graphene sheets, it makes difficult to disperse them randomly in the polymer matrix. Especially in case of graphene, aggregates could be generated and decrease the apparent aspect ratio [START_REF] Singh | Graphene based mate rials: past, present and future[END_REF][START_REF] Stankovich | Graphene-based composite materials[END_REF][START_REF] Syurik | Graphene network organisation in conductive polymer composites[END_REF]. In order to improve the graphene dispersion in nanocomposites several ex foliation ways are studied. The solvent way, which uses commonly N methyl-2-pyrrolidone [START_REF] Hemandez | High-yield production of graphene by liquid-phase exfoliation of graphite[END_REF] or N,N-dimethylformamide [START_REF] Li | Graphene/ thermoplastic polyurethane nanocomposites: surface modification of graphene through oxidation, polyvinyl pyrrolidone coating and reduction[END_REF], is an easy process that can bring high graphene concentration (higher than 1 mg•mL -1 ) but leads to use solvent. An aqueous way have been pro posed with an anionic surfactant as sodium dodecyl sulfate for example. In this case, high graphene concentration is more difficult to obtain [START_REF] Yang | Understanding graphene production by ionic surfactant exfoliation : a molecular dynamics simulation study[END_REF][START_REF] Wang | Adding ethanol can effectively enhance the graphene concentration in water -surfactant[END_REF][START_REF] Yeon | High-yield graphene exfoliation using sodium dodecyl sulfate accompanied by alcohols as surface-tension-reducing agents in aqueous solution[END_REF].

The manuscript presents the protocol to graphene exfoliation and the processing used for PEKK/graphene and PEKK/carbon black nano composites. The influence of the nature of conductive fillers on di electric properties such as electrical conductivity and dielectric per mittivity and mechanical properties such as storage and loss moduli were studied.

Experimental section

Materials

Poly(ether ketone ketone) PEKK (Arkema) (diameter of powder 20 µm) and Carbon black (CB, glassy carbon from Sigma Aldrich), were supplied in powder, respectively. Graphene (GR from Avanzare) was supplied in ethanol suspension (EtOH) with a concentration of 0.5 wt%. The morphology of GR and CB was investigated by scanning electron microscopy (SEM -FEG JEOL JSM 7800F Prime) using secondary electron detector. Fig. 1 shows the SEM images of CB (a) and "as re ceived" GR (b), respectively.

The CB particles are spherical with a mean diameter of 6.3 µm (d 50 ) determined using a laser granulometer (Fig. la). Fig. lb shows gra phene stack due to the SEM resolution, which does not allow to observe GR monolayers. To analyze precisely the GR nanotexture, images were performed using a JEOL JEM 2100F electron microscope operated at 200 kV. Diffraction patterns were recorded using selected area electron diffraction (SAED) mode with a 150nm aperture.

Fig. 2 is a bright field image of GR and one monolayer is visible in the black circle, which indicates the selected area for the SAED pattern. This pattern is characteristic of one graphene layer (P63/mmc, 6 6 a = 2461 A and c = 6708 A). The wrinkled aspect and the grey contrast differences suggest that GR is also composed by a large amount of GR multilayers classically labelled exfoliated graphene nanoplatelets in the literature [START_REF] Via | Electrical conductivity modeling of carbon black/polycarbonate, carbon nanotube/polycarbonate, and exfoliated gra phite nanoplatelet/polycarbonate composites[END_REF]. This local microscopie observation is not appropriate to characterize the entire GR batch. The number of GR layers modifies the 0 apparent aspect ratio and has a significant influence on nanocomposite electrical properties. The electrical percolation threshold allows us to determine the apparent aspect ratio of these particles.

Nanocomposite processing

In order to manage the distribution of GR in the polymer, a simple protocol was applied. It consists in two steps process: a GR exfoliation followed by a decantation. For exfoliation an aqueous solution with sodium dodecyl sulfate (SDS) as surfactant was chosen [START_REF] Yeon | High-yield graphene exfoliation using sodium dodecyl sulfate accompanied by alcohols as surface-tension-reducing agents in aqueous solution[END_REF]. SDS was solubilized in EtOH at 2.4 mg•mL -l. lt corresponds to the critical mi cellar concentration (CMC) in water. GR suspension was diluted in a solution containing EtOH and SDS following the 3/5 ratio, respectively. Suspension was sonicated for 1 min (power 300 W). After 10 min of decantation, GR suspension was collected to process the nanocompo site.

For the PEKK/GR (GR-NC) and PEKK/CB (CB-NC) nanocomposites processing, a premix powder was used. GR/EtOH or CB/EtOH suspen sions and polymer powder (20 µm in diameter) were mixed.

PEKK polymer powder is in suspension in this mix. Then it was sonicated with ultrasound to promote a homogeneous dispersion. EtOH was evaporated with a rotary evaporator at 80 °C. Nanocomposites were elaborated through melt blending at 360 °C with mechanical mixing using a laboratory twin screw extruder (GR-NCl). GR-NC2, GR NC3 and CB-NC were processed with hot press. GR-NCl was elaborated from "as received" GR without any treatment. GR-NC2 was elaborated from GR exfoliated according to the previously described protocol. GR NC3 was elaborated from GR exfoliated by the commercial supplier. As it has been reported in previous works, this protocol leads to a very good particle dispersion in polymer matrix even for high aspect ratio [START_REF] Ramachandran | Conduction mecbanisms in P(VDF-TrFE)/gold nanowire composites: tunnelling and thennally activated hopping process near the percolation threshold[END_REF][START_REF] Rivière | Silver fillers aspect ratio influence on electrical and thermal conductivity in PEEK/ Ag nano composites[END_REF][START_REF] Cortes | High-performance thermoplastic composites poly(ether ketone ketone)/silver nanowires: morphological, mechanical and electrical properties[END_REF][START_REF] Carponcin | Evolution of dispersion of carbon nanotubes in polyamide 11 matrix composites as determined by DC conductivity[END_REF]. The nanocomposite electrical conductivity study, and more specifically, the percolation threshold value allows us to analyze the dispersion quality at a macroscopic scale. As the percolation threshold theoretical value of spherical particles dispersion is well known, CB-NC was elaborated to check the dispersion protocol.

Dielectric and mechanical analyses

Dynamic electrical conductivity and dielectric permittivity of na nocomposites performed from -150 °C to 200 °C and 10-2 Hz to 10 6 Hz using a Novocontrol Broadband Dielectric Spectrometer (BDS). Disk-shape samples (20 mm in diameter) have been processed by compression molding at 360 °C to determine the bulk electrical con ductivity. In order to promote a 3D conductivity, thick samples were hot pressed (1 mm thickness).

To extract the percolation threshold value the following Eq. ( 1) was used [START_REF] Kirkpatrick | Percolation and conduction[END_REF][START_REF] Stauffer | Introduction to Percolation Theory[END_REF]:

(1)
where a vc is the static nanocomposite conductivity, a 0 the GR pathway conductivity, p the GR content, P c the percolation threshold and t the universal critical exponent that depends on the system dimensionality (1 < t < 1.3 for a 2D network and 1.5 < t < 2 for a 3D network [START_REF] Kirkpatrick | Percolation and conduction[END_REF][START_REF] Stauffer | Introduction to Percolation Theory[END_REF]).

Dynamic mechanical analyses (DMA) were carried out on PEKK, GR-NCl and CB-NC using an ARES Gl TA Instruments apparatus in shear mode, from -130 °C to 270 °C at 3 °C•min -1 . The applied strain and frequency were fixed at 0.1 % and 0.16 Hz (an angular frequency of 1 rad•s-1 ), respectively. Parallelepipedic samples with 40 mm long, 10 mm wide and 0.6 mm thick were studied.

Results & discussion

Study of the dispersion quality by bulle electrical conductivity analysis

Fig. 3 represents the DC conductivity behavior at room temperature as a fonction of GR and CB content: GR-NCl (PEKK/"as received" GR nanocomposites), GR-NC2 (PEKK/lab-exfoliated GR nanocomposites), GR-NC3 (PEKK/supplier exfoliated GR nanocomposites) and CB-NC (PEKK/CB nanocomposites).

For GR-NCl the percolation threshold is determined near 4.2 vol%, the conductivity value above P c reaches 4.9 x 10-4 S•m-1 • The per colation threshold decreases until 3 vol% after the exfoliation/decan tation protocol (GR-NC2) and the conductivity value above P c is 6.5 x 10-2 s-m-1 • For GR-NC3, the percolation threshold is even lower (1. 9 vol%) with the highest conductivity value of 1.2 S•m -l.

In order to compare these results with a reference material, CB nanocomposites were elaborated. The percolation threshold for CB-NC is determined at 16.2 vol%.

The percolation threshold value is highly dependent on the GR dispersion, which is associated with the elaboration process, the GR aspect ratio and its surface modification. Kim et al. [START_REF] Kim | Graphene/polyurethane nanocomposites for im proved gas barrier and electrical conductivity[END_REF] have shown the influence of elaboration process by comparing three technics: in solu tion, in-situ polymerization and melt process. The first and second one lead to the lowest percolation threshold due to the very low viscosity of the mixture [START_REF] Liao | Ultralow percolation graphene/polyurethane acrylate nanocomposites[END_REF]. For example, Stankovich et al. [START_REF] Stankovich | Graphene-based composite materials[END_REF] dispersed GR (chemically modified) in polystyrene (PS) and they reached a percolation threshold near 0.1 vol%. The GR dispersion in the viscous polymer melt is more difficult. The GR fonctionalization is used to generate repulsive interactions between layers. Yang et al. in 2013 [START_REF] Yang | Design and pre paration of graphene/poly(ether ether ketone) composites with excellent electrical conductivity[END_REF] dispersed GR functionalized poly(ethersulfone) (PES) in PEEK. They obtained a percolation threshold of 0.76 vol%. However, the GR sheets fonctionalization leads to structural defects and the GR electrical con ductivity decreases. Thus, conductivity values above the percolation threshold are generally lower in the case of fonctionalized GR [START_REF] Stankovich | Graphene-based composite materials[END_REF][START_REF] Yang | Design and pre paration of graphene/poly(ether ether ketone) composites with excellent electrical conductivity[END_REF] in comparison with unfunctionalized ones [START_REF] Tkalya | Experimental and theoretical study of the in fluence of the state of dispersion of graphene on the percolation threshold of con ductive graphene/polystyrene nanocomposites[END_REF][START_REF] Syurik | Graphene network organisation in conductive polymer composites[END_REF][START_REF] Kuila | Characterization and properties of in situ emulsion polymerized poly(methyl methacrylate)/graphene nanocomposites[END_REF].

GR aggregates decrease the apparent aspect ratio and increase the percolation threshold. This work was consistent with those carried out by Tkalya [START_REF] Tkalya | Experimental and theoretical study of the in fluence of the state of dispersion of graphene on the percolation threshold of con ductive graphene/polystyrene nanocomposites[END_REF], which works on the dispersion of unfonctionalized GR in a poly(styrene) matrix. Their percolation thresholds values (between 2 and 4.5 vol%) or conductivities (between 1 and 10 S•m -l) results were similar to PEKK/GR nanocomposite.

The percolation threshold for CB-NC determined at 16.2 vol% is closed to the theoretical value [START_REF] Scher | Critical density in percolation processes[END_REF]; i.e. perfect sphere randomly dis persed. It is important to note that many factors could affect this ex perimental percolation threshold value [START_REF] Zhang | Carbon based conductive polymer composites[END_REF].

Fig. 4 reports the linear fit of log(a vc ) as a fonction of (p -P c ).

Extracted values of P c , a 0 and twere reported in the Table 1 for GR-NCl, GR-NC2, GR-NC3 and CB-NC.

GR-NC2

Hot press

GR-NC3

Hot press

1.9

CB-NC

Hot press

16.2

2.5 X 10-9 6.4 7.9 X 10-4 3.6 3.1 X 10-1 1.0 6.3 X 10-3 1.6

Universal critical exponent values of GR-NCl and GR-NC2 were 6.4 and 3.6, respectively. These values are higher than the estimated values (1.5 < t < 2 for a 3D network [START_REF] Kirkpatrick | Percolation and conduction[END_REF][START_REF] Stauffer | Introduction to Percolation Theory[END_REF]). In the literature, high t values have already been identified in GR/polymer nanocomposites: 5.8 for PEEK/GR nanocomposites [START_REF] Yang | Design and pre paration of graphene/poly(ether ether ketone) composites with excellent electrical conductivity[END_REF], 3.8 for PVDF/GR nanocomposites [39], 3.2 for ABS/GR and SAN/GR nanocomposites [START_REF] Gao | Graphene networks with low percolation threshold in ABS nanocomposites: selective localization and electrical and rheological properties[END_REF]. Authors attributed this behavior to the inhomogeneity of the GR dispersion. Aggregates or folding limited the electronic transport within the nanocomposite and increased the t value. Thus, it is consistent with GR-NC3 results; i.e. a better dispersion leads to a decrease of t at 1. GR-NCl nanocomposites, elaborated by extrusion from a premix nanocomposite powder, has a very low conductivity above the perco lation threshold (10-3 S•m-1 ). The shear and compression strengths during the extrusion process alter the electrical properties. These ob servations, already reported in the literature, were related to the re duction of exfoliated graphene nanoplatelets aspect ratio [START_REF] Colonna | Effect of processing conditions on the thermal and electrical conductivity of poly (buty lene terephthalate) nanocomposites prepared via ring-opening polymerization[END_REF] and/or re-agglomeration phenomenon [START_REF] Kalaitzidou | A new compounding method for ex foliated graphite-polypropylene nanocomposites with enhanced flexural properties and lower percolation threshold[END_REF]. The conductivity above the per colation threshold is thereby greatly reduced. a 0 is related to the in trinsic conductivity of the pristine GR sheet [START_REF] Li | Graphene/ thermoplastic polyurethane nanocomposites: surface modification of graphene through oxidation, polyvinyl pyrrolidone coating and reduction[END_REF] (generally known at 10 3 S•m -l for carbon particles).

The GR-NC3 nanocomposite has the higher a0• This elaboration protocol (exfoliation/decantation then hot press lamination) promote better GR electrical properties. Finally, as regards CB-NC nanocompo sites, the CB intrinsic conductivity a 0 is 6.3 x 10-3 S-m-1 . It is an in termediate value between GR-NC2 and GR-NC3; t value is 1.6, which corresponds to a 3D network [START_REF] Kirkpatrick | Percolation and conduction[END_REF][START_REF] Stauffer | Introduction to Percolation Theory[END_REF]. For GR-NC (1.2 S•m-1 at 6 vol%) and CB-NC (1.0 S•m -l at 50volo/o), we noticed that the maximum of conductivity above the percolation threshold was similar in both cases. This value was clearly dependent from the particle nature (carbonac eous as comparison with metallic particles).

Influence of graphene on the conductivity frequency and temperature dependence

Fig. 5 represents the evolution of the real part of the electrical conductivity as a function of frequency from -150 °C to 200 °C. Ex periments were carried out for PEKK/home-exfoliated GR (GR-NC2) with 3 and 4 vol%; i.e. in the vicinity of the percolation threshold. Two behaviors are highlighted as a function of frequency: below Pc, con ductivity is dependent on frequency. This behavior is consistent with the Jonscher universal power law for an insulator [START_REF] Jonscher | The "universal" dielectric response[END_REF]. Above Pc , conductivity is not dependent on frequency: it is associated with a conductor behavior.

In both cases conductivity increased with temperature for a wide range of frequency. The real part of the complex conductivity is com posed of two terms: a direct courant conductivity a vc and an alternating courant conductivity a Ac which is frequency dependent (Eq. ( 2)).

(2)

For a critical frequency W c , a change of the electronic conduction phenomenon occurred:

For OJ < OJ c a'(w) = a vc For OJ > OJ c a'(w) = aAc (w) = Aw'

(3) (4)
where A is a pre-exponential constant and s, the power law exponent (0 < s < 1). The inset graph in Fig. 5 represents the evolution of the s 10 °r----------------- exponent as a function of temperature for PEKK/3 vol% GR-NC2 (below Pc). The exponent s was calculated by fitting the experimental data with a power law for each isotherm. Two behaviors were identified. Below -SOC, the s temperature dependence is weak. It means that two elec tronic conduction behaviors exist in the vicinity of the percolation threshold: tunneling effect below -50 °C and hopping activation pro cess above. At low temperature, tunneling effect is not temperature dependent whereas hopping is thermally activated. In both case, elec tronic transport through the conductive path do not require a surface contact between GR or CB particles.

Influence of graphene on the dielectric permittivity in the vicinity of the percolation threshold

Dielectric permittivity has been studied for GR-NCl nanocomposites in order to analyze behaviors within the graduai percolation threshold. Fig. 6(a) presents the evolution of permittivity measured at 1 Hz as a function of GR content for different temperatures: at room temperature, near the glass transition temperature and above 20 °C, 160 °C and 180 °C, respectively. The inset graph in Fig. 6(a) represents the DC conductivity as a fu nction of GR content, each composite relative to the percolation threshold is identified ( * ).

The introduction of graphene leads to an increase of the dielectric permittivity in the vicinity of the percolation threshold. Permittivity increases from 2.7 for the neat PEKK to 83 for PEKK/6 vol% GR at room temperature (the dielectric permittivity value reaches 210 at 180 °C).

Classically, the real part of the dielectric permittivity increases above the dielectric manifestation of the glass transition temperature. But at low frequency, an additional dielectric phenomenon can be ob served. Fig. 6(b) represents the evolution of the real part of the per mittivity measured at room temperature as a function of frequency for different graphene contents. At low frequency, permittivity increases with the graphene fraction. For GR-NCl with 6 vol%, permittivity in creases from 20 at 100 kHz to 83 at 1 Hz. It has been already shown that the introduction of conductive particles into a dielectric polymer matrix increases the dielectric permittivity [START_REF] Fan | Graphene/poly(vinylidene fluoride) composites with high dielectric constant and low percolation threshold[END_REF][START_REF] Wang | Dielectric properties of reduced graphene oxide/polypropylene composites with ultralow percolation threshold[END_REF][START_REF] Al-Saleh | Graphene nanoplatelet-polystyrene nanocomposite: dielectric and charge storage behaviors[END_REF]. It has been associated with a Maxwell -Wagner -Sillars (MWS) polarization: electric charges are trapped at the particle/matrix interface [START_REF] Trihotri | Effect of silver coating on electrical properties of sisal fibre-epoxy composites[END_REF][START_REF] Trihotri | Effect of curing on activation energy and dielectric properties of carbon black-epoxy composites at different temperatures[END_REF]. This dielectric phenomenon could have a specific interest for electronic devices [START_REF] Xu | Super high dielectric constant carbon black-filled polymer composites as integral capacitor dielectrics[END_REF]. -G' C:::J G "

Ta-50

°C - T•••H 1 C: -�--�•-•• -----L -: • . "/ � . .. ♦••O"' O o�<>o .,.
-<>oo,;...�-=t . .

. .

•or�o•t

Tc,; •:,/

,.�"Q cte:

-100 0 100

:oo

CW'

T emper :dure (>C) + -t ,..,,.

Influence of graphene on the mechanical properties below the percolation threshold

The improvement of electrical properties of high performance thermoplastic should be done without altering their mechanical prop erties. The influence of carbon particles (GR-NCl or CB-NC) on the conservative and dissipative moduli has been determined in compar ison with unfilled PEKK. d and G" values were determined at T a -so•c (vitreous plateau), T a (in the viscoelastic relaxation domain) and T a + so•c (rubbery plateau) and reported in Fig. 7. The inset graph in Fig. 7 shows the PEKK thermogram over ail the temperature range. The study of the mechanical relaxations have been extensively detailed in a previous paper [START_REF] Bessaguet | Piezoelectric and mechanical behavior of NaNbO3/PEKK lead-free nanocomposites[END_REF].

Mechanical properties of PEKK and its composites are presented in Fig. 7. Considering the error bars, the particles until 6 vol% do not influence significantly the mechanical moduli even for graphene. Thus, the high performance mechanical properties of the polymer matrix is maintained.

Conclusion

Commercial graphene and carbon black were satisfactory dispersed in a high performance thermostable thermoplastic PEKK polymer in order to improve their electrical properties. Particle aspect ratio has a strong influence on the percolation threshold value. lt decreased from 16.2 vol% with carbon black (ç "" 1) to 4.2 vol% with an "as received" graphene (GR-NCl) and 1.9 vol% after a graphene exfoliation/decan tation process (GR-NC3). Exfoliation/decantation improved the gra phene dispersion by limiting the number of layer and increasing the graphene aspect ratio. Above the percolation threshold, high con ductivity values (1-1.2 S-m-1 ) were obtained for graphene and carbon black. It was one decade higher than the conductivity measured for carbon nanotubes nanocomposites. Graphene pathway also increased dielectric permittivity from 2.6 for PEKK to 83 for the PEKK/6 vol% GR nanocomposite. Finally, until 6 vol% graphene and 4 vol% carbon black, nanocomposite mechanical behaviors are not significantly modified.

Fig. 2 .

 2 Fig. 2. Bright field TEM image of the "as received" graphene. The black circle shows the area selected for the SAED pattern (in insert).

Fig. 1 .

 1 Fig. 1. SEM images of carbon black (a) and the "as received" graphene (b).

Fig. 3 .

 3 Fig. 3. DC conductivity versus conductive particles volume fraction for GR-NCl (□), GR-NC2 (0), GR-NC3 (,6) and CB-NC (e). Solid line is the percolation threshold fit.

8 Fig. 4 .

 84 Fig. 4. Power law fit for GR-NCl (□), GR-NC2(0), GR-NC3 (,6) and CB-NC (e). Solid line is the linear fit.

Fig. 5 .

 5 Fig. 5. Real part of the conductivity versus frequency for PEKK/home-ex foliated graphene (GR-NC2) 3 vol% (solid line) and 4 vol% (dashed line) for isotherms from -150 °C to 200 °C.

Fig. 6 .

 6 Fig. 6. (a) Real part of the dielectric permittivity versus graphene vol% for PEKK and PEKK/"as received" graphene (GR-NCl) at (■) 20 •c, (0) 160 •c and Cà) 180 •c. Inset: DC conductivity versus GR content at room temperature. (b) Real part of the dielectric pennittivity versus frequency for PEKK and PEKK/"as received" graphene (GR-NCl) at room temperature: (■) 2 vol% graphene, (0) 4.5 vol% graphene, Cà) 5 vol% graphene and ('i7) 6 vol% graphene. Solid lines are guide for the eye.

Fig. 7 .

 7 Fig. 7. Storage modulus G' and loss modulus G" of unfilled PEKK, PEKK/2 vol% GR, PEKK/4 vol% GR, PEKK/6 vol% GR and PEKK/4 vol% CB, measured at Ta -so•c, Ta and Ta+ so•c. Inset: PEKK matrix thermogram between -130 •c and 270 •c.

Table 1 Processing methods and percolation law parameters. Processing p, (vol%) GR-NCl

 1

	Extrusion	4.2

Acknowledgments

These results were obtained under the research project "COMPINNOVTP" at the IRT Saint Exupéry. We thank the industrial and academic members of the IRT who supported this project through their contributions, both financial and in terms of specific knowledge: AIRBUS OPERATIONS, ARIANE GROUP, AIRBUS HELICOPTERS, AIRBUS GROUP INNOVATIONS, THALES ALENIA SPACE, CIRIMAT, ISAE, ICA, IMRCP, UPS and CNRS. We also thank the Commissariat Général aux Investissements and the Agence Nationale de la Recherche for their financial support in the Programme d'Investissement d'Avenir (PIA).