E Buchoud

EDF S Blairon

EDF Jean-Marie M Henault

A Girard

Valeriu Jerome I Mars

EDF G D'urso (edf

Parametric Inversion of Brillouin spectra using

Introduction

Geotechnical structures as canals or dykes suffer from long term effects as internal erosion could provoke water leakage or embedded cavities. To ensure their durability and stability, the French electric utility company EDF (Electricité de France) chose an innovative structural health monitoring policy to perform strain measurements. It consists in using Distributed Fiber Optic Sensors (DOFS) in addition to conventional sensors. DOFS, which have gained interests even if many challenges are still opened as referenced by [START_REF] Lanticq | Distributed Optical fiber sensors for structural health monitoring: upcoming challenges, chapter 9[END_REF], are composed by an opto-electronic device linked to an optical fibre cable, the sensing element. A frequency spanned light impulsion is launched into the medium and the backscatter light recorded by the device allows measuring strain and/or temperature profiles along the optical fiber. Three scattering phenomena are observed: the Raman (temperature sensitive), Brillouin and Rayleigh effects (both strain and temperature sensitive).The Brillouin Optical Time Domain Analyzer (BOTDA) records several Brillouin spectra matrices, centered in the Brillouin frequencies and depending on distance along the optical fiber. As these frequencies are linked directly to the strain imposed to the optical fiber, industrial BOTDA devices estimate a strain profile typically with a spatial sampling of 40cm over 25km [START_REF] Bao | Experimental and theorical studies on a distributed temperature sensor based on Brillouin scattering[END_REF]. However, previous work [START_REF] Buchoud | Detection of ground movement using the shape of Brillouin spectrum[END_REF] highlights the fact that firstly, the spectra could be distorted by non-uniform strain distribution along the fiber and, secondly, that the spatial resolution is limited. We propose in this article article an alternative algorithm to perform and reach a centimeter spatial resolution strain estimation from the measured Brillouin matrices. Thus, the detection and localization of strain gradients into structures will be upgraded. To fulfil such goal, a model proposed in Buchoud et al., 2013 explains the link between the hidden information inside Brillouin spectra and the local Brillouin frequencies. A parametric inverse methodology is proposed to estimate these local frequencies. A laboratory test with an optical fiber embedded into a one-to-one scale soil structure exposed by [START_REF] Blairon | Truly distributed optical fiber extensometers for geotechnical structure monitoring (dikes and underground repository): Influence of sensor external coating, 8 th international symposium of Field Measurement in Geomecanics[END_REF] enabled to validate our algorithm. First, a strain profile with 3cm spatial resolution provided by a Rayleigh device is considered here as the reference profile. Secondly, a classical BOTDA device records the Brillouin matrix with a 40cm spatial sampling. We show that from those spectra, the proposed algorithm enables to take advantage of the hidden information contained into the distorted spectra to improve the accuracy of the strain measurement and the spatial sampling form 40cm to 5cm.

The BOTDA principle and model

The BOTDA is based on the stimulated Brillouin scattering (SBS): a light pulse is launched into an optical fibre. The travel time is related to localization z along the fiber, while the duration of the impulsion is related to the area interrogated into the fiber, called the integration base w. For example, for 1ns light pulse duration, 1m of the sensing medium is interrogated. The light frequency v is spanned and a Brillouin spectrum is acquired at z. The measurement operation is repeated at different localizations z spaced by a spatial sampling Δz, allowing to obtain a matrix of Brillouin spectra . For each z, an industrial BOTDA device automatically finds the global Brillouin frequency v B (z), determined by the best fitting the Brillouin spectrum, in terms of L2 norm, with a pseudo-Voigt function. From a state (without strain for instance) to another one (with strain), the relative Brillouin shift Δv B (z) between these two states is linearly related to the strain and temperature conditions along the fiber optic [START_REF] Bao | Experimental and theorical studies on a distributed temperature sensor based on Brillouin scattering[END_REF] as:

(1)

where C T and C ε are about 1MHz/°C and 0.05 MHz/µε for a standard single-mode optical fiber with an operating wavelength of 1550nm. In our study, since the temperature T(z) is constant for all measurements, the strain (z) can be directly deduced from (1).

However, as the BOTDA integrates over 1 meter at each z, it is possible to mix non-uniform local strain within w [START_REF] Buchoud | Enhancement of an optical fiber sensor: source separation based on Brillouin spectrum[END_REF] inducing a broadening and asymmetric Brillouin spectrum. In this previous work, the assumption was made that the measured spectrum at z k is the combination of local stimulated Brillouin spectra S e (v) within w centered on local Brillouin frequencies :

(2) with x a new spatial sampling. A first method proved the possibility of enhancement of the spatial sampling [START_REF] Buchoud | Enhancement of an optical fiber sensor: source separation based on Brillouin spectrum[END_REF]. However, this method missed some local strain variations that is crucial for the soil displacement quantification. Thus, we propose here method based on inverse method to improve the accuracy and the spatial sampling from the given matrix which is easily implemented.

Parametric inverse methodology

The model expressed in (2) can be rewritten in a classical parametric direct model as:

(3)

with H the direct operator described in (2) and the error linked to the model. is the local Brillouin frequencies profile depending on the new spatial sampling Δx and has to be estimated. The principal idea of this methodology is the minimization of the error defined by a cost function C depending on the estimation of :

(4)

The first term is the L2 norm between the data matrix and its estimation, while the second term is a second term related is added to ensure the continuity of the local Brillouin frequencies . The parameter µ is the associated regularization term to be chosen. Note that if too much regularization is imposed, then the data and estimation would not fit and the first term would be too large. On the other hand, if µ is too small, the solution will be dominated by the contributions of data errors [START_REF] Hansen | The L-curve and its use in the numerical treatment of inverse problems[END_REF]. The parameter N represents the number of local Brillouin frequencies to be estimated and is the division of the sensing length by the chosen Δx. A trade-off between these two parameters can be controlled by the L-curve methodology. It is commonly used in geosciences [START_REF] Chen | Application of Cesaro mean and the Lcurve for the deconvolution problem[END_REF] to optimize the regularization parameter µ, known as the L-curve criterion. For several values of µ, the L-curve represents the balance between the two terms in log scale. Then µ is found when the best trade-off between good data estimation and the a priori is guaranteed.

Application on laboratory experiment

A laboratory experiment has been carried out to study the distributed strain sensing chain in temperature controlled conditions (see details in [START_REF] Blairon | Truly distributed optical fiber extensometers for geotechnical structure monitoring (dikes and underground repository): Influence of sensor external coating, 8 th international symposium of Field Measurement in Geomecanics[END_REF]. The configuration is presented in Figure 1.a. Two interrogators, Rayleigh scattering [START_REF] Frogatt | High-spatial-resolution distributed strain measurement in optical fiber with Rayleigh scatter[END_REF] and an industrial BOTDA devices are connected to one cable composed by several optical fibres. The cable is inside a sand soil on two depths h: 0.5 m and 1m above a 1m large metal plate. An underground cavity is generated using a hydraulic controlled actuator that moves the plate. For both interrogators, two state measurements are recorded: before (initial state S 0 ) and after (final state S 1 ) the actuator displacement. The Rayleigh scattering device provides a relative strain reference profile with a 3cm spatial sampling. The profiles for the two depths are shown in Figure 1.b-c. in red curves. The displacement signature is composed by two traction zones (positive strain referenced at -0.5m and 0.7m in distance) Amsterdam RAI, The Netherlands, 16-19 June 2014 and a compressive zone (negative strain at 0.1 m). For the depth h=0.5m, over 1m near the centre of the plate, the strain is going from +200µm/m to -400µm/m, whereas at h=1m, as it is far from the origin of the displacements, the strain gradient is less important from +45µm/m to -270µm/m. In the same time, Brillouin matrices were recorded by the BOTDA device with a spatial sampling Δz=40cm and integration base w=1m. Figure 1.d shown in red stars and blue crosses the relative Brillouin shift Δv B (z) estimated by the BOTDA device for the two state, S 0 and S 1 . A relative strain profile can be estimated by using Eq. ( 1), the results for the two depths being plotted in Figure 1.b-c. in green curves. As these profiles have a resolution of Δz=40cm, we linearly interpolate them in order to compare with the reference ones. As we can see, the displacement signature (positive and negative points) is retrieved. The extreme values are undervalued: for example, at h=0.5m, instead having -400µm/m, the algorithm of the BOTDA device estimates -250µm/m. In addition, due to w=1m and Δz=40cm, the center of the plate and the minimum localization do not correspond. The Means Square Errors (MSE) between the strain reference and the strain profile estimated by the algorithm of the industrial BOTDA, shown in green in figure 1. e-f., highlight these limitations. As our final goal is the displacements measurement into the structure, a precise strain is a crucial need. Thus, the proposed method is applied on the Brillouin matrices. The new spatial sampling Δx is fixed at 5cm. The calculation time increases if Δx is small and the sensing length is important. The first step is the estimation of the stimulated Brillouin spectrum S e (v), which can be found where strain is uniform within w. The regularization parameter is then optimized thanks the L-curve method in order to have the best ratio between measured/modelled date balance and continuity of frequencies. As the global frequencies profiles shown in Figures 1.b-c. are quite similar, µ=10 is founded for both depths. Relative Brillouin shifts with the resolution Δx are then obtained and shown for the two state, S 0 and S 1 at h=1m in Figure 1.d. in continuous lines. The strain profiles estimated using Eq. ( 1) are shown in Figures 1.b-c. in blue curves. At h=1m, our estimation and the strain reference are mixed up as well as at h=0.5m, the negative part of the strain signature being well estimated. The minimum is better localized and its estimated value is closer however still undervalued. The MSE shown in blue in figure 1. e-f. decreases especially in high strain gradient areas. The proposed algorithm overcomes the difficult challenge to provide a precise strain profile with a centimetre spatial sampling from the measurement of Brillouin matrices with industrial 40-cm resolution BOTDA devices. h=1m Amsterdam RAI, The Netherlands, 16-19 June 2014 c. f. Figure 2. a). Scheme of the experiment: two cable at two depths h=1m and 0.5m from the metal plate. b and c) Strain profiles provided by the Rayleigh device (red -refernce); the BOTDA algorithm (greeninterpolated from 40 cm spatial rezolution) and the inverse method (bluewith a new spatial rezolution Δx = 5 cm) for the two depths h=0.5m (b) and h=1m (c). d). Estimation of the Brillouin frequencies at h=1m: provided by the BOTDA (+) state S 0 ; (*) state S 1 ; and the inverse method : (-) S 0 ; (-) S 1 . e and f) MSE between the strain reference profile (Rayleigh device) and the strain profile estimated by the algorithm of the industrial BOTDA (green) respectively the strain profile estimated by our inverse method (blue) for the two depths h=0.5m (e) and h=1m (f).

Conclusion

The classical BOTDA provides a strain measurement profile with a 40cm spatial sampling. Based on previous works, the interpretation of the distortion of the Brillouin spectrum relies in a direct model to retrieve local Brillouin frequencies into the measured Brillouin spectra. A parametric inverse methodology is proposed to estimate those local Brillouin frequencies depending on distance with a new spatial sampling chosen by the user. The best trade-off between data reconstruction and the a priori on the continuity of the Brillouin profile is found thanks the L-curve methodology. On controlled conditions, it has been tested on real data acquired by an industrial BOTDA. Instead having a 40cm spatial sampling, the strain is 5cm resolute. In term of accuracy, our strain estimation is similar to the strain reference profile (provided by a Rayleigh scattering device). Whereas the BOTDA algorithm does not take into account of the hidden information into the Brillouin spectrum and thus we enlighten its lack of accuracy. The challenge of combining centimetre spatial sampling over kilometers with high strain accuracy has been successfully performed thanks our parametric inverse methodology. The next challenge is the estimation of the displacements into the structure from the strain measurements.


François Hamel

Grégoire Nadin

Lionel Roques

A viscosity solution method for the spreading speed formula in slowly varying media

In this paper, we consider reaction-diffusion-advection equations in slowly periodically oscillating media. We prove the existence of and we give explicit expressions of the asymptotic spreading speeds of invasion of the unstable state 0 in any direction, when the period of the invaded medium becomes infinitely large. The limiting spreading speeds involve families of 1-periodic Hamilton-Jacobi equations. In the case of one-dimensional reaction-diffusion equations, we analyze the relative effects of small perturbations of the diffusion and the reaction coefficients, and we compare the spreading speeds in slowly oscillating media to the homogenized spreading speeds in rapidly oscillating media.

Introduction

In this paper, we are concerned with spreading properties of the solutions of reactiondiffusion equations of the type

∂u ∂t = div(a L (x)∇u) + b L (x) • ∇u + f L (x, u), t > 0, x = (x 1 , . . . , x N ) ∈ R N (1)
in slowly oscillating periodic media. That is, we assume that

a L (x) = a x L , b L (x) = b x L , f L (x, s) = f x L , s 1
for all x ∈ R N and s ≥ 0, where the diffusion, drift and reaction coefficients a = (a ij ) 1≤i,j≤N : R N → S N (R), b = (b i ) 1≤i≤N : R N → R N and f (•, s) : R N → R are 1-periodic, for short, that is they are 1-periodic in all variables x 1 , . . . , x N . The notation S N (R) stands for the set of N × N symmetric matrices with real entries. We establish asymptotic limits as L → +∞ of some fundamental spreading quantities associated with the large-time behavior of the nonnegative solutions. Throughout the paper, the function f is assumed to be such that f (x, 0) = 0 [START_REF] Barles | Discontinuous solutions of deterministic optimal stopping time problems[END_REF] for all x ∈ R N , that is 0 is a steady state for equation [START_REF] Aronson | Multidimensional non-linear diffusion arising in population genetics[END_REF]. Under some conditions to be detailed later, the state 0 will be unstable and equation ( 1) also admits a unique positive stable steady state. One of the main questions addressed in this paper is to know at which speed the positive steady state invades the state 0 when the underlying medium is slowly varying, that is when L is large.

In homogeneous media, the study of the spreading properties of such equations has begun with the pioneering papers by Fisher [START_REF] Fisher | The wave of advance of advantageous genes[END_REF] and Kolmogorov, Petrovsky and Piskunov [START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF] in dimension 1, and later by Aronson and Weinberger [START_REF] Aronson | Multidimensional non-linear diffusion arising in population genetics[END_REF] in any dimension. Since then, these equations have found applications in various domains such as combustion [START_REF] Williams | Combustion Theory[END_REF][START_REF] Xin | Front propagation in heterogeneous media[END_REF] and population ecology [START_REF] Okubo | Diffusion and Ecological Problems -Modern Perspectives[END_REF][START_REF] Shigesada | Biological Invasions: Theory and Practice[END_REF]. But since the environment is generally far from being homogeneous, it is therefore natural to try to understand the effects of the spatial heterogeneities on the spreading properties of the solutions of equations of the type [START_REF] Aronson | Multidimensional non-linear diffusion arising in population genetics[END_REF]. This issue has been investigated in a recent but large literature, in periodic media (see e.g. [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF][START_REF] Freidlin | On wave front propagation in periodic media[END_REF][START_REF] Freidlin | Limit-theorems for large deviations and reaction-diffusion equations[END_REF][START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF][START_REF] Majda | Large-scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF] for nonnegative and compactly supported initial conditions, see also [START_REF] Nadin | Travelling fronts in space-time periodic media[END_REF][START_REF] Nolen | Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF] for front-like initial data), in random stationary ergodic environments (see [START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF][START_REF] Lions | Homogenization of "viscous" Hamilton-Jacobi equations in stationary ergodic media[END_REF][START_REF] Nolen | Asymptotic spreading of KPP reactive fronts in incompressible spacetime random flows[END_REF][START_REF] Nolen | KPP fronts in 1D random drift[END_REF]), as well as in general heterogeneous media (see [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF][START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF][START_REF] Berestycki | The speed of propagation for KPP type problems. II -General domains[END_REF][START_REF] Berestycki | Asymptotic spreading for general heterogeneous KPP equations[END_REF]).

The spreading speeds of invasion of the steady state 0 are characterized as follows. Assuming that all terms in (1) are L-periodic in x and that the reaction term is differentiable with respect to its second variable at 0 and satisfies the KPP assumption

f L (•, s) ≤ ∂f L ∂s (•, 0)×s for all s ≥ 0, (3)
and under further assumptions on a L , b L and f L (see the comment before Proposition 2.1 below), it is known [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF][START_REF] Freidlin | On wave front propagation in periodic media[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF] that, given any nonzero compactly supported and nonnegative initial condition u 0 (x) and given any direction e of the unit sphere S N -1 , the solution u(t, x) of the Cauchy problem (1) with initial condition u 0 propagates in the direction e with a positive spreading speed w * L (e) > 0 in the sense that

   lim t→+∞ u(t, x + c t e) = 0 for all c > w * L (e), lim inf t→+∞ u(t, x + c t e) > 0 for all 0 ≤ c < w * L (e), locally uniformly in x ∈ R N . ( 4
)
When problem (1) admits a (unique) positive periodic steady state p L , the spreading speed w * L (e) in a given direction e can be expressed in terms of the minimal speeds of pulsating fronts connecting 0 to p L and propagating in the directions having a positive inner product with e, see [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF]. In the particular one-dimensional case, the spreading speeds in the directions +1 and -1 are then equal to the minimal speeds of pulsating fronts connecting 0 to p L and propagating in these directions. For further existence, uniqueness and stability results of KPP pulsating fronts in one-dimensional or higher-dimensional periodic media, we refer to [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF][START_REF] Berestycki | The speed of propagation for KPP type problems. I -Periodic framework[END_REF][START_REF] Berestycki | Analysis of the periodically fragmented environment model: II -Biological invasions and pulsating travelling fronts[END_REF][START_REF] Hamel | Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity[END_REF][START_REF] Hamel | Uniqueness and stability properties of monostable pulsating fronts[END_REF][START_REF] Nadin | Travelling fronts in space-time periodic media[END_REF][START_REF] Nolen | Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds[END_REF].

On the one hand, the homogenization limit of rapidly oscillating media, corresponding to small values of L, has been well studied. In this case, an explicit formula has been established for w * 0 (±1) = lim L→0 w * L (±1) when N = 1 and the drift b is constant, or more generally when b is divergence-free in higher dimensions [START_REF] Smaily | Homogenization and influence of fragmentation in a biological invasion model[END_REF][START_REF] Kinezaki | Spatial dynamics of invasion in sinusoidally varying environments[END_REF][START_REF] Nadin | The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator[END_REF] (see also [START_REF] Caffarelli | Homogenization and flame propagation in periodic excitable media: the asymptotic speed of propagation[END_REF][START_REF] Heinze | Homogenization of flame fronts[END_REF][START_REF] Heinze | Variational principles for propagation speeds in inhomogeneous media[END_REF] for other nonlinearities). The dependence of the spreading speed with respect to the parameter L has also been studied and it is also known that, for all directions e, the spreading speeds w * L (e) are nondecreasing with respect to L, when b is divergence-free (see [START_REF] Nadin | The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator[END_REF] and numerical results in [START_REF] Kinezaki | Modeling biological invasions into periodically fragmented environments[END_REF][START_REF] Shigesada | Biological Invasions: Theory and Practice[END_REF]).

On the other hand, for equations of the type (1) with KPP nonlinearities f L , the question of the asymptotics of the spreading speeds w * L (e) in slowly oscillating media, that is as L → +∞, had only been investigated in a recent work [START_REF] Hamel | Spreading speeds in slowly oscillating environments[END_REF], in the one-dimensional case, for constant values of a L and spatially piecewise-constant reaction terms f L . In the general heterogeneous framework investigated here, proving the existence of a limit for the spreading speeds w * L (e) and obtaining a formula for w * ∞ (e) := lim L→∞ w * L (e) in any direction e was still an open problem. This question is solved in the present paper. Furthermore, in the one-dimensional case, we prove a formula for w * ∞ (±1), which is expressed in terms of some integral quantities involving the underlying coefficients.

To our knowledge, apart from the above-mentioned reference [START_REF] Hamel | Spreading speeds in slowly oscillating environments[END_REF], comparable problems have only been studied in some papers by Freidlin [START_REF] Freidlin | Limit-theorems for large deviations and reaction-diffusion equations[END_REF] and Evans and Souganidis [START_REF] Evans | A PDE approach to geometric optics for certain semilinear parabolic equations[END_REF]. These papers were concerned with the limiting Cauchy problems obtained after rescaling, when the space and time variables are of order L and when the period L becomes infinitely large. However, the representation formula they provided for the limiting spreading speeds w * L (e), based on game theory techniques, is quite complicated (we also refer to [START_REF] Gärtner | Bistable reaction-diffusion equations and excitable media[END_REF][START_REF] Xin | Front propagation in heterogeneous media[END_REF] for spreading properties of bistable equations in slowly oscillating media).

As a matter of fact, our new results about the limiting spreading speeds have interesting consequences regarding the effects of spatial heterogeneities. Firstly, when the functions L → w * L (e) are non-decreasing -this is true if b is divergence-free [START_REF] Nadin | The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator[END_REF]-the limiting spreading speeds w * ∞ (e) provide upper bounds for the spreading speeds w * L (e) in L-periodic media, which are more precise than earlier results given in [START_REF] Berestycki | The speed of propagation for KPP type problems. I -Periodic framework[END_REF], and the quantities w * ∞ (e)w * 0 (e) give a measure of the maximal effect of the medium oscillations. Secondly, the formula for w * ∞ (e) which we prove in this paper for general equations (1) sheds light on the effect of diffusion compared to that of reaction in heterogeneous media. Namely, it shows that the spreading speeds can be more sensitive to perturbations of the diffusion coefficient than to perturbations of the reaction term. Thirdly, spreading properties of equation [START_REF] Aronson | Multidimensional non-linear diffusion arising in population genetics[END_REF] in slowly oscillating media are related to the spreading properties of this equation in the limit of large reaction terms. We obtain the exact asymptotics of the spreading speeds in periodic media with large amplitude reaction terms as a by-product of our formula for w * ∞ (e).

To obtain this formula for w * ∞ (e), we use the variational characterization which was first derived in [START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF] and later generalized in [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF] in more general media, that is

w * L (e) = min λ∈R N , λ•e>0 k L (λ) λ • e ,
where, for λ ∈ R N , k L (λ) denotes the periodic principal eigenvalue of an elliptic operator defined below. The core of our proof relies on a viscosity solution method together with several qualitative and quantitative properties of the function (λ, L) → k L (λ).

2 Hypotheses and main results

Assumptions

Before stating our main results, let us make more precise the assumptions and notations used throughout this paper. One is given a symmetric matrix field a = (a ij ) 1≤i,j≤N of class W 1,∞ (R N ; S N (R)) (in particular, the coefficients a ij can be assumed to be continuous), a continuous advection field b : R N → R N and a continuous growth term c : R N → R, which are 1-periodic in x, that is

a(x + k) = a(x), b(x + k) = b(x), c(x + k) = c(x) for all k ∈ Z N and x ∈ R N . (5)
The case when the coefficients are L i -periodic with respect to each variable x i , with L i > 0, can be reduced to the above assumption ( 5), after rescaling the variables. Throughout the paper, we assume that the matrices a(x) are uniformly positive definite, that is, there

exists ν > 0 such that a(x) ≥ ν I N for all x ∈ R N (6)
in the sense of symmetric matrices, where I N denotes the identity matrix of size N × N.

We also assume that

M := max x∈R N c(x) - b(x) a(x) -1 b(x) 4 > 0, (7)
where, for any vectors ξ = (ξ 1 , . . . , ξ N ) and

ξ ′ = (ξ ′ 1 , . . . , ξ ′ N ) in R N and any symmetric matrix A = (A ij ) 1≤i,j≤N , the quantity ξAξ ′ denotes ξAξ ′ = 1≤i,j≤N ξ i A ij ξ ′ j .
This last condition (7) will guarantee the linear unstability of the zero state and the existence and positivity of the spreading speeds in slowly oscillating media, as defined below (see Proposition 2.1). Notice in particular that, if b = 0 in R N , then condition ( 7) is fulfilled if and only if max R N c > 0.

For any L > 0, we define the L-periodic rescaled coefficients a L , b L and c L by:

a L (x) = a x L , b L (x) = b x L , c L (x) = c x L for all x ∈ R N .
Then, because of (5-6), there exists, for each λ ∈ R N and L > 0, a periodic principal eigenvalue k L (λ) which is uniquely defined by the existence of a function

φ L,λ ∈ 1≤p<+∞ W 2,p (R N ) such that          div(a L ∇φ L,λ ) -2 λ a L ∇φ L,λ + b L • ∇φ L,λ +(λ a L λ -div(a L λ) -b L • λ + c L ) φ L,λ = k L (λ) φ L,λ in R N , φ L,λ is L-periodic, φ L,λ > 0 in R N , (8)
see e.g. [START_REF] Pinsky | Second order elliptic operator with periodic coefficients: criticality theory, perturbations, and positive harmonic functions[END_REF]. The maps λ → k L (λ) are known to be convex (see [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF]).

For any L > 0 and any unit direction e ∈ S N -1 , we then define

w * L (e) := inf λ∈R N , λ•e>0 k L (λ) λ • e . (9)
In this general setting, the quantities w * L (e) defined in formula ( 9) are a priori not directly associated with an equation of the type (1) and with some spreading speeds in the sense of (4). However, the numbers w * L (e) can really be viewed as spreading speeds under some reasonable assumptions that we give now. Namely, in addition to [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF][START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF], assume here that

f : R N × [0, +∞) → R, (x, s) → f (x, s) is continuous, 1-periodic in x, that c := ∂f ∂s (•, 0
) is continuous and that f satisfies ( 2) and ( 3). If k L (λ) > 0 for all λ ∈ R N , then the following holds: for any nonzero, nonnegative and compactly supported function u 0 ∈ L ∞ (R N ), the solution u of (1) with initial datum u 0 propagates in the sense of (4) with the spreading speed w * L (e) > 0 given by formula (9), in any direction e ∈ S N -1 , see [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF]. This is why we always say in this paper that the quantities w * L (e) represent the spreading speeds of the solutions of some reaction-diffusionadvection equations of the type [START_REF] Aronson | Multidimensional non-linear diffusion arising in population genetics[END_REF].

The following proposition ensures that the aforementioned positivity condition on the real numbers k L (λ) and the positivity of the spreading speeds w * L (e) are fulfilled, at least when L is large enough. Proposition 2.1 Under the above assumptions (5-7), there is L 0 > 0 such that, for all L ≥ L 0 , there holds min λ∈R N k L (λ) > 0 and, for any direction e ∈ S N -1 , the infimum in (9) is reached and positive.

Convergence of the speed when L → +∞ and computation of the limit in dimension 1

When b is divergence-free, the functions L → k L (λ) are nondecreasing (see [START_REF] Nadin | The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator[END_REF]), whence the limit w * ∞ (e) := lim L→∞ w * L (e) is well-defined in R for any direction e. In the general case considered here, the main goal of the present work is twofold: firstly, proving the existence of the limit lim L→∞ w * L (e) and, secondly, computing this limiting value. This is the purpose of the following theorems.

Theorem 2.2 Under assumptions (5-7), for any unit vector e ∈ S N -1 , the limit w * ∞ (e) = lim L→+∞ w * L (e) exists and is a positive real number. Furthermore,

w * ∞ (e) = min λ∈R N , λ•e>0 k ∞ (λ) λ • e , (10)
where the function k ∞ : R N → R is defined in Proposition 3.1 below.

In dimension N = 1, the quantities k ∞ (λ) can be given in terms of explicit integrals involving the underlying coefficients.

Theorem 2.3 Under assumptions (5-7), in dimension N = 1, there holds

w * ∞ (±1) = min λ≥j ± (M ), λ>0 j -1 ± (λ) λ , (11)
where the functions j ± : [M, +∞) → [j ± (M), +∞) are defined by

j ± (k) = 1 0 k -c(x) a(x) + b 2 (x) 4 a 2 (x) dx ± 1 0 b(x) 2 a(x) dx (12)
for all k ≥ M. 1 In particular, in dimension N = 1, when a = 1 and b = 0, we get

j ± (k) = 1 0 k -c(x) dx
and (11) corresponds to the formula conjectured by Hamel, Fayard and Roques in [START_REF] Hamel | Spreading speeds in slowly oscillating environments[END_REF]. Note that, in that case, and when c is a step function, it has been shown in [START_REF] Hamel | Spreading speeds in slowly oscillating environments[END_REF] that [START_REF] Caffarelli | Homogenization and flame propagation in periodic excitable media: the asymptotic speed of propagation[END_REF] leads to an even more explicit formula. More generally speaking, it is noteworthy that formula [START_REF] Caffarelli | Homogenization and flame propagation in periodic excitable media: the asymptotic speed of propagation[END_REF] allows for easy numerical computations of the (equal) limiting spreading speeds w * ∞ (±1). We conjecture that Theorem 2.2 can be extended to the case of space-time periodic coefficients. Assume that a = a(t, x), b = b(t, x) and f = f (t, x, s) are 1-periodic with respect to t and x, and consider the equation

∂u ∂t = div(a L (t, x)∇u) + b L (t, x) • ∇u + f L (t, x, u), t > 0, x = (x 1 , . . . , x N ) ∈ R N (13)
with slow variations in space and in time

a L (t, x) = a t L , x L , b L (t, x) = b t L , x L , f L (t, x, s) = f t L , x L , s .
The existence of pulsating traveling fronts for equation [START_REF] Evans | Periodic homogenization of certain nonlinear partial differential equations[END_REF] has been proved by Nolen, Rudd and Xin [START_REF] Nolen | Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds[END_REF] when f and a do not depend on (t, x), f (0) = f (1) = 0, f (s) > 0 if s ∈ (0, 1) and b is incompressible. This result has been extended by the second author [START_REF] Nadin | Travelling fronts in space-time periodic media[END_REF] when a and f are space-time periodic and b is a general space-time periodic vector field. When f satisfies the KPP assumption of the type (3) with time-dependence, the first two authors, together with Berestycki, proved in [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF] the existence of a spreading speed w * L (e) associated with [START_REF] Evans | Periodic homogenization of certain nonlinear partial differential equations[END_REF] so that (4) holds. A formula like (9) still holds, except that k L (λ) is not associated with an elliptic operator anymore but with a parabolic operator with time periodicity conditions (see [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF]). Then, it should be possible to extend the result of Theorem 2.2 and Proposition 3.1 below, but now k ∞ (λ) is uniquely defined by the existence of a space-time periodic viscosity solution

Z ∈ C 0 (R × R N ) of -∂ t Z + (∇Z -λ) a (∇Z -λ) + b • (∇Z -λ) + c = k ∞ (λ). ( 14
)
The dependence of k L (λ) and k ∞ (λ) with respect to the coefficients (a, b, c) is a topical subject (see [START_REF] Nadin | Some dependence results between the spreading speed and the coefficients of the space-time periodic Fisher-KPP equation[END_REF][START_REF] Nolen | Reaction-diffusion front speeds in spatially-temporally periodic shear flows[END_REF] for a few results concerning k L (λ)). Nolen and Xin conjectured in [START_REF] Nolen | Reaction-diffusion front speeds in spatially-temporally periodic shear flows[END_REF] that temporal variations should slow down the propagation. This is still an open question.

A partial answer is given by our result together with numerical simulations carried out by Khouider, Bourlioux and Majda [START_REF] Khouider | Parameterizing the burning speed enhancement by small-scale periodic flows: I. Unsteady shears, flame residence time and bending[END_REF], who observed that when a = I N , c does not depend on (t, x) and b(t, x 1 , x 2 , ..., x N ) = b(ωt, x 2 , ..., x N )e 1 is a shear flow in the direction e 1 = (1, 0, . . . , 0) with temporal frequency ω, then the spreading speed w * ∞ (e 1 ) is decreasing with respect to ω.

The case of homogeneous reaction terms with no advection

In another particular case, we immediately get a simpler formula for w * ∞ (±1) in dimension N = 1.

Corollary 2.4 Under the conditions of Theorem

2.3, if b = 0 and c = c 0 > 0 is constant, then w * ∞ (±1) = 2 < √ a > H √ c 0 , (15)
where

< √ a > H = 1 0 dx a(x) -1
denotes the harmonic mean of √ a.

Notice that, under the conditions of Corollary 2.4, it is also known that

w * L (±1) → 2 √ < a > H √ c 0 as L → 0 + , (16)
see [START_REF] Smaily | Homogenization and influence of fragmentation in a biological invasion model[END_REF]. More generally speaking, if the arithmetic mean <c> A = Observe now that, because of Cauchy-Schwarz inequality, the right-hand side of ( 15) is (strictly) larger than the right-hand side of ( 16) as soon as the diffusion rate a is not constant (this is coherent with the fact that the spreading speeds w * L (±1) are non-decreasing with respect to L in this case, see [START_REF] Nadin | The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator[END_REF]). Furthermore, the ratio < √ a > H / √ < a > H ≥ 1 then measures the relative increase of the spreading speeds from a completely homogenized medium to the infinitely slowly oscillating one.

1 0 c(x) dx is positive, then lim L→0 + w * L (±1) = 2 √ < a > H √ < c > A .

Respective effects of perturbations of the reaction term and the diffusion term on the speed

The above results have further interesting consequences regarding the respective effects of localized perturbations of the reaction and diffusion terms on the spreading speeds. To make the dependence more transparent, let us use the notations k L,a,b,c (λ), w * L,a,b,c (e) and w * ∞,a,b,c (e) [START_REF] Freidlin | Limit-theorems for large deviations and reaction-diffusion equations[END_REF] for the quantities defined in ( 8), ( 9) and [START_REF] Berestycki | Asymptotic spreading for general heterogeneous KPP equations[END_REF].

For the sake of simplicity of the presentation, assume that the dimension is N = 1 and that b = 0. Assume that a and c fulfill (5-6) and c has a positive average (whence min λ∈R k L,a,0,c (λ) > 0 and w * L,a,0,c (±1) > 0 for all L > 0). Then, consider a real number α > 2 and a family of smooth 1-periodic positive functions (ρ ε ) 0<ε<1/2 such that, on each period,

     0 < ρ ε ≤ 1, 0 < ρ ε ≤ ε α in an interval of length ε, ρ ε = 1 in an interval of length 1 -2ε, (18)
see Fig. 1.

Consider now the case when the diffusion coefficient is multiplied by the perturbation ρ ε , that is when a is replaced with ρ ε × a: it follows from the maximum principle and Corollary 2.4 together with the monotonicity of the maps L → w * L,ρεa,0,c (±1) (because b = 0), that the upper bound holds

0 < w * L,ρεa,0,c (±1) ≤ w * L,ρεa,0,Mc (±1) ≤ w * ∞,ρεa,0,Mc (±1) ≤ 2 ε α/2-1 M a M c ,
where M a = max R a and M c = max R c. As a consequence,

∀ L > 0, w * L,ρεa,0,c (±1) → 0 as ε → 0. ( 19
)
On the other hand, when the growth rate c is multiplied by the perturbation ρ ε (that is when the nonlinearity f L (x, u) is replaced with ρ ε (x/L) × f L (x, u) in equation [START_REF] Aronson | Multidimensional non-linear diffusion arising in population genetics[END_REF], where b L = 0, f satisfies ( 2) and ( 3) and c denotes c = ∂f ∂s (•, 0)), then the associated spreading speeds satisfy (see [START_REF] Smaily | Homogenization and influence of fragmentation in a biological invasion model[END_REF][START_REF] Nadin | The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator[END_REF]), for all L > 0,

w * L,a,0,ρεc (±1) ≥ lim L ′ →0 w * L ′ ,a,0,ρεc (±1) = 2 √ < a > H √ < ρ ε c > A .
Thus, it follows from ( 18) that

∀ L > 0, lim inf ε→0 w * L,a,0,ρεc (±1) ≥ 2 √ < a > H √ < c > A > 0. ( 20
)
In short, formulas [START_REF] Gärtner | Bistable reaction-diffusion equations and excitable media[END_REF] and [START_REF] Hamel | Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity[END_REF] show that small perturbations of the diffusion coefficient, in the sense above, can dramatically reduce the spreading speeds whereas similar perturbations of the reaction term have a limited effect.

Large amplitude of the reaction and advection terms

Another consequence of Theorem 2.2 is concerned with the determination of the asymptotic dependence with respect to the drift and reaction terms in the large amplitude case. In particular, consider the one-dimensional symmetric equation

∂u ∂t = ∂ ∂x a L (x) ∂u ∂x + B f L (x, u), (21)
where f : R × [0, +∞) × R is continuous, 1-periodic in x and satisfies (2) and (3). Assume that c = ∂f ∂s (•, 0) is continuous. If min R a > 0 and M = max R c > 0, then Corollary 2.5 gives an equivalent asymptotics for the spreading speeds w * L,a,0,B c (±1) of the solutions of ( 21) in the limit of large amplitudes B of the reaction term. Namely, for any L > 0, we have

w * L,a,0,B c (±1) ∼ min λ≥j(M ) j -1 (λ) λ × √ B as B → +∞, where j(k) = 1 0 k-c(x)
a(x) dx for all k ≥ M (notice that the coefficient min λ≥j(M ) j -1 (λ)/λ is necessarily positive). This result improves Theorem 1.3 of [START_REF] Berestycki | Analysis of the periodically fragmented environment model: II -Biological invasions and pulsating travelling fronts[END_REF] in the one-dimensional case.

3 Convergence of the periodic principal eigenvalues as L → +∞

The aim of this section is to prove the existence of the limit of the principal eigenvalues k L (λ), defined in (8), as L → +∞ and to express it explicitly in dimension 1. We shall especially use the Hopf-Cole transformation of the eigenfunctions and the stability theorem for Hamilton-Jacobi equations with small viscosity.

Proposition 3.1 Under assumptions (5-6), for all λ ∈ R N , the limit

k ∞ (λ) = lim L→+∞ k L (λ)
exists and it is the unique real number k ∞ (λ) such that there exists a continuous 1-periodic viscosity solution Z of the cell problem

(∇Z -λ) a (∇Z -λ) + b • (∇Z -λ) + c = k ∞ (λ). (22)
Furthermore,

k ∞ (λ) ≥ M = max x∈R N c(x) - b(x) a(x) -1 b(x) 4 ( 23
)
for all λ ∈ R N . Lastly, the function λ → k ∞ (λ) is convex and the convergence k L (λ) → k ∞ (λ) as L → +∞ is locally uniform with respect to λ ∈ R N .
Proof. Let λ be any fixed vector in R N and denote by |λ| its euclidean norm. First, it follows from ( 6) and ( 8) that

ν |λ| 2 -L -1 div(a λ) ∞ -b ∞ |λ| -c ∞ ≤ k L (λ) ≤ max R N (λ a λ) + L -1 div(a λ) ∞ + b ∞ |λ| + c ∞ (24)
for all L > 0. Therefore, the family (k L (λ)) L≥1 is bounded. There exist then a real number k ∞ (λ) and a sequence (L n ) n∈N → +∞ such that

k Ln (λ) → k ∞ (λ) as n → +∞.
For each n ∈ N, let φ Ln,λ ∈ 1≤p<+∞ W 2,p (R N ) be the principal L n -periodic eigenvalue of ( 8) with L = L n . Up to normalization, one can assume that φ Ln,λ (0) = 1. The function φ Ln,λ is positive in R N . Define

Z n (x) = ln φ Ln,λ (L n x) L n for all x ∈ R N . The 1-periodic function Z n ∈ 1≤p<+∞ W 2,p (R N ) satisfies Z n (0) = 0 and L -1 n div(a∇Z n ) + ∇Z n a∇Z n -2 λ a∇Z n + b • ∇Z n + λ a λ -L -1 n div(aλ) -b • λ + c = k Ln (λ).
On the other hand, since the functions φ Ln,λ satisfy ( 8) with coefficients which are uniformly bounded (since L n → +∞) and fulfill [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF], it follows from the standard interior elliptic estimates that there exists a constant C 1 > 0 such that

|∇φ Ln,λ (x)| ≤ C 1 × max |y-x|≤1
φ Ln,λ (y) for all n ∈ N and x ∈ R N . But, since the functions φ Ln,λ are all positive, the Harnack inequality yields the existence of a constant

C 2 > 0 such that max |y-x|≤1 φ Ln,λ (y) ≤ C 2 × min |y-x|≤1 φ Ln,λ (y) ≤ C 2 × φ Ln,λ (x)
for all n ∈ N and x ∈ R N . Therefore, the fields ∇φ Ln,λ /φ Ln,λ are bounded in L ∞ (R N ) uniformly with respect to n ∈ N. In other words, the sequence ( ∇Z n ∞ ) n∈N is bounded. As Z n is 1-periodic and Z n (0) = 0 for all n ∈ N, the sequence (Z n ) n∈N is then bounded in W 1,∞ (R N ). Ascoli's theorem implies that, up to extraction of a subsequence, the sequence (Z n ) n∈N converges uniformly in R N to a Lipschitz-continuous 1-periodic function Z. The stability theorem for Hamilton-Jacobi equations (see [START_REF] Barles | Discontinuous solutions of deterministic optimal stopping time problems[END_REF][START_REF] Barles | Exit time problems in optimal control and vanishing viscosity method[END_REF]) gives that Z is a viscosity solution of [START_REF] Hamel | Uniqueness and stability properties of monostable pulsating fronts[END_REF].

The uniqueness of the real number k ∞ (λ) such that ( 22) admits a periodic viscosity solution follows from a classical theorem by Lions, Papanicolaou and Varadhan (unpublished manuscript, see also [START_REF] Evans | Periodic homogenization of certain nonlinear partial differential equations[END_REF]). As a consequence, the whole family (k L (λ)) L>0 converges to k ∞ (λ) as L → +∞.

By construction, since the function Z is Lipschitz-continuous, it follows from Rademacher's theorem that equation ( 22) is satisfied almost everywhere in R N . Therefore, [START_REF] Heinze | Homogenization of flame fronts[END_REF] holds immediately, since the fields a, b and c are continuous in R N . Lastly, since all functions λ → k L (λ) were convex, the function λ → k ∞ (λ) is convex too, whence continuous. Because of [START_REF] Heinze | Variational principles for propagation speeds in inhomogeneous media[END_REF] and the convexity of k L , the slopes

λ → lim t→0 + k L (λ + tξ) -k L (λ) t
are locally bounded, uniformly with respect to L ≥ 1 and ξ ∈ S N -1 . As a consequence, the limit k ∞ (λ) = lim L→+∞ k L (λ) is locally uniform. The proof of Proposition 3.1 is thereby complete.

An immediate consequence of Proposition 3.1 is the Proof of Proposition 2.1. In addition to [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF][START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF], assume here that (7) holds, that is M > 0. It follows from ( 24) that there exists r > 0 such that

∀ L ≥ 1, ∀ |λ| ≥ r, k L (λ) ≥ ν |λ| 2 2 . ( 25
)
On the other hand, since the convergence lim

L→+∞ k L (λ) = k ∞ (λ) ≥ M > 0 is locally uniform with respect to λ ∈ R N , there is L 0 ≥ 1 such that ∀ L ≥ L 0 , ∀ |λ| ≤ r, k L (λ) ≥ M 2 . (26)
In particular, for any L ≥ L 0 , the infimum over R N of the continuous function k L : λ → k L (λ) is reached and positive.

Let now e be any unit vector of S N -1 . It follows straightforwardly from ( 25) and ( 26) that, for any L ≥ L 0 , the quantity w * L (e) defined in ( 9) is a minimum and is positive.

Lastly, in dimension N = 1, we can now give an explicit expression of the limit k ∞ (λ) for each λ ∈ R, in terms of some integral quantities. be the functions given as in [START_REF] Smaily | Homogenization and influence of fragmentation in a biological invasion model[END_REF]. In dimension 1, there holds, for all λ ∈ R,

k ∞ (λ) =      the unique k ≥ M such that λ = -j -(k) if λ ≤ -j -(M), M if -j -(M) < λ < j + (M), the unique k ≥ M such that λ = j + (k) if λ ≥ j + (M). (27)
Proof. First, let λ ≥ j + (M). Since the function j + is one-to-one and onto from [M, +∞) to [j + (M), +∞), there exists a unique real number k ≥ M such that:

λ = j + (k) = 1 0 k -c(x) a(x) + b 2 (x) 4 a 2 (x) dx + 1 0 b(x) 2 a(x) dx. (28)
Define, for all x ∈ R,

Z(x) = λ x - x 0 k -c(y) a(y) + b 2 (y) 4 a 2 (y) dy - x 0 b(y) 2 a(y) dy. a Z ′ -λ + b 2 a 2 = k -c + b 2 4 a ,
almost everywhere in R and finally Z satisfies [START_REF] Hamel | Uniqueness and stability properties of monostable pulsating fronts[END_REF]. Thus the uniqueness of k

∞ (λ) yields k = k ∞ (λ). Similarly, one can check that, if λ ≤ -j -(M), k ∞ (λ) is the unique real num- ber k ∈ [M, +∞) such that λ = -j -(k).
In particular, there holds

k ∞ (-j -(M)) = k ∞ (j + (M)) = M.
But since the function k ∞ is convex and bounded from below by M (from Proposition 3.1), one concludes that k

∞ (λ) = M if λ ∈ [-j -(M), j + (M)],
which completes the proof of Proposition 3.2.

Proof of Theorems 2.2 and 2.3 and their corollaries

This section is devoted to the proof of the main results announced in Section 2, making use of the propositions and estimates of the previous section. We start with the proof of Theorem 2.2 for the general higher-dimensional case N ≥ 1.

Proof of Theorem 2.2. In addition to (5-6), we assume that (7) holds, that is M > 0.

Let e ∈ S N -1 be any unit vector of R N . We recall that, for each L > 0, the spreading speed w * L (e) is defined as

w * L (e) = inf λ∈R N , λ•e>0 k L (λ) λ • e . (29)
From Proposition 2.1, there exists L 0 > 0 such that k L (0) > 0 and the infimum in the above formula is reached and positive. Let (L n ) n∈N be any sequence such that L n ≥ L 0 for all n ∈ N, and L n → +∞ as n → +∞. For each n ∈ N, let then λ n ∈ R N be such that λ n • e > 0 and

w * Ln (e) = k Ln (λ n ) λ n • e > 0.
Taking λ = κ e in [START_REF] Heinze | Variational principles for propagation speeds in inhomogeneous media[END_REF] for any κ > 0, and using the definition (29), we get

w * L (e) ≤ 2 c ∞ max R N
(e a e) + L -1 div(a e) ∞ + b ∞ for all L > 0. Therefore, the sequence (w Ln (e)) n∈N is bounded and then converges, up to extraction of a sequence, to a nonnegative real number w * ∞ (e). Next, taking λ = λ n in [START_REF] Heinze | Variational principles for propagation speeds in inhomogeneous media[END_REF], one gets

ν |λ n | 2 -L -1 n div(a λ n ) ∞ -|λ n | b ∞ -c ∞ ≤ k Ln (λ n ) = w *
Ln (e)×(λ n • e) ≤ w * Ln (e)×|λ n | for all n ∈ N. Therefore, the sequence (λ n ) n∈N is bounded too and then converges, up to extraction of another sequence, to a vector λ ∞ ∈ R N such that λ ∞ • e ≥ 0. On the other hand, since, from Proposition 3.1, the convergence of the family of functions (k L ) L>0 to the function k ∞ as L → +∞ is locally uniform, it follows that

k ∞ (λ ∞ ) = w * ∞ (e) × (λ ∞ • e).
Since k ∞ (λ) ≥ M > 0 for all λ ∈ R N from Proposition 3.1 and assumption [START_REF] Berestycki | The speed of propagation for KPP type problems. I -Periodic framework[END_REF], one concludes that w * ∞ (e) and λ ∞ • e are both positive, and Finally, the function λ → k ∞ (λ)/(λ • e) reaches its minimum at λ ∞ over the set of λ ∈ R N such that λ • e > 0, and

w * ∞ (e) = k ∞ (λ ∞ ) λ ∞ • e .
w * ∞ (e) = min λ∈R N , λ•e>0 k ∞ (λ) λ • e . (30)
In particular, from the uniqueness of the limit, one concludes that the whole family (w * L (e)) L>0 converges to w * ∞ (e) as L → +∞. The proof of Theorem 2.2 is thereby complete.

The proof of Theorem 2.3 in dimension N = 1 is based on Theorem 2.2 and on the expression of k ∞ (λ) given in Proposition 3.2.

Proof of Theorem 2.3. We only prove the formula for w * ∞ (+1), the case of w * ∞ (-1) could be treated similarly. Two cases have to be considered, according to the sign of j + (M). Let us first assume that j + (M) > 0. From Proposition 3.1, there holds

Figure 1 :

 1 Figure 1: (left) Schematic plot of a perturbation function ρ ε ; (right) A perturbed function x → ρ ε (x) (1 + sin 2 (πx)), with ε = 10 -2 and α = 3 in the definition of ρ ε .

Corollary 2 . 5

 25 Under the assumptions of Theorem 2.2 and notations[START_REF] Freidlin | Limit-theorems for large deviations and reaction-diffusion equations[END_REF], there holds, for any L > 0 and for any unit vector e ∈ S N -1 , ,b,c (e) = w * ∞,a,b,c (e).

Proposition 3 . 2

 32 Under the assumptions and notations of Proposition 3.1, let j ± : [M, +∞) → [j ± (M), +∞)

For

  all λ ∈ R N such that λ • e > 0, one hasw * Ln (e) ≤ k Ln (λ) λ • e for all n ∈ N, whence w * ∞ (e) ≤ k ∞ (λ) λ • e .

  λ ∈ (0, j + (M)), while k ∞ (λ) λ = j -1 (λ) λ for all λ ≥ j + (M),from Proposition 3.2. It follows then from (30) that the function λ → j -1 + (λ)/λ has a minimum on the interval [j + (M), +∞), and thatw * ∞ (+1) = min λ≥j + (M ) j -1 + (λ) λ ,

Notice that the values j ± (M ) may not be positive. Indeed, for instance, if the functions a, b and c are all constants and fulfill (6) and (7), then j ± (M ) = ±b/(2 a) and j ± (M ) are then of the sign of ±b.

The authors are supported by the French "Agence Nationale de la Recherche" within the projects ColonSGS (first and third authors), PREFERED, and URTICLIM (third author). The first author is also indebted to the Alexander von Humboldt Foundation for its support.

which is the desired conclusion [START_REF] Caffarelli | Homogenization and flame propagation in periodic excitable media: the asymptotic speed of propagation[END_REF] in the case j + (M) > 0.

Lastly, if j + (M) ≤ 0, then k ∞ (λ) = j -1 + (λ) for all λ > 0 from Proposition 3.2, and (11) follows immediately from [START_REF] Majda | Large-scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales[END_REF]. That completes the proof of Theorem 2.3.

The proofs of Corollaries 2.4 and 2.5 then follow easily from Theorems 2.3 and 2.2 respectively.

Proof of Corollary 2.4. In dimension N = 1, when b = 0 and c 0 > 0 is constant, then M = c 0 , j ± (M) = 0 and

c 0 for all λ ≥ 0 and the conclusion (15) follows from Theorem 2.3.

Lastly, Corollary 2.5 is just a consequence of Theorem 2.2 and of a rescaling argument.

Proof of Corollary 2.5. For each L > 0 and λ ∈ R N , we call k L,a,b,c (λ) and φ L,a,b,c,λ the (unique) principal eigenvalue and eigenfunction of ( 8) and we also use the notation w * L,a,b,c (e) for the quantities defined in [START_REF] Berestycki | Analysis of the periodically fragmented environment model: II -Biological invasions and pulsating travelling fronts[END_REF]. Now, for any L > 0, B > 0 and λ ∈ R N , the function ψ defined by