Keywords: career start, career planning process, occupation profiles, job market, agricultural graduates )

Finding a first destination is a challenge for graduates, higher education institutions and employers alike. Graduates look to minimise the risks of an unsuitable first destination and its consequences for their future career and professional development. Institutions seek to optimise society's substantial investment in education by getting their graduates off to a successful career start. Employers try to minimise recruitment costs and to avoid the consequences of mis-recruitment (delays, disruption of existing teams, etc.).

Therefore finding a successful first destination is not only the responsibility of individual students but, through careers guidance, is also part of the educational process for which the institutions are responsible. This type of action requires a set-up that gets students themselves to take matters in hand. The challenge is therefore to get students to see the need to make informed decisions. They need help in being made aware of their abilities, their wishes, their strengths and their weaknesses and they need to be prompted to take the necessary steps and to be pro-active about their education, to give meaning to the skills they acquire and to control the circumstances of their first recruitment.

On the basis of the recent experience of Agrocampus Rennes (France), this paper aims to analyse how a higher education institution may address this issue and may try to respond to the need for a tutorial/training process for the career start ofstudents/young graduates.

For several years now Agrocampus Rennes has conducted an active policy of individual support for its graduates in finding their first destinations. Apart from other existing conventional activities, the paper focusses on two different recent initiatives: the career planning process and the occupation profiles project.

The paper also provides some figures about the first destination of French agricultural students and recalls about the general organisation ofthe French agricultural higher education system.

-An overview of the issues

With the state of the job market being as it is, graduates can no longer rely solely on a degree, however prestigious it may be, to find employment after completing their studies. The sheer variety of degrees and courses, the ever wider spectrum covered by agricultural higher education, the growing importance of specialisation in students' education and training, the competition from other types of courses, the changes in employment sectors and in the positions for which new graduates are recruited make it more difficult than before to get into the job market.

Paradoxically, graduates from our institutions get a good education but, because of their academic career, are not as much in touch with the world of work as young people having been through more vocationaliy-oriented educational courses (with vocational diplomas or first degrees).

Finding a first destination is a challenge for graduates, higher education institutions and employers alike. Graduates look to minimise the risks of an unsuitable first destination and its consequences for their future career and professional development. Institutions seek to optimise society's substantial investment in education by getting their graduates off to a successful career start. Employers try to minimise recruitment costs and to avoid the consequences of mis-recruitment (delays, disruption of existing teams, etc.).

Therefore finding a successful first destination is not only the responsibility of individual students but, through careers guidance, is also part of the educational process for which the institutions are responsible (CEFI, 20A4).

Our universities are responsible for avoiding excessive trial-and-effor or worse still failure because of a poor choice of first destination with the disastrous effects it may have on graduates' self-esteem or on their future career paths.

For all these reasons higher education institutions are fully justified in actively supporting graduates in finding their first destinations and this is arguably as much their responsibility as is their duty to provide a high standard of education. This type of action requires a set-up that gets students themselves to take matters in hand [START_REF] Stephan | Construction du parcours de formation d'ingénieur et le projet professionnel[END_REF]. Not only shouid students be provided with information but they must be made aware that they need this information to make the right decisions. The challenge is therefore to get students to see the need to make informed decisions. They need help in being made aware of their abilities, their wishes, their strengths and their weaknesses and they need to be prompted to take the necessary steps and to be pro-active about their education, to give meaning to the skills they acquire and to control the circumstances of their first recruitment.

On the basis of the recent experience of Agrocampus Rennes (France), this paper aims to analyse how a higher education institution may address this issue and may try to respond to the need for a tutorial/training process for the career start ofstudents/young graduates.

We first set out the general framework of agricultural higher education in France and describe the state of the market for graduate recruitment on the basis of recent survey results. Then we analyse arrangements recently put in place in Rennes ernphasising the role of two pilot schemes.

J 2 -The French agricultural higher education system The French agricultural higher education system is original in that it is part of the école d'ingénieurs system and has a long history of control by the Ministry of Agriculture.

Post-graduate engineerss enjoy a very specific socio-economic position in France. 'In none of the major developed countries, including those which, like the United States and Germany, were the first to promote higher education to train their technical elites, is the distinction between engineer and technician so contrasted, nor the social prestige attached to an engineer's title so high' (Bouffartigue and Gadéa, 1997).

Their education is provided within a specific system known in France and abroad as Grandes Écoles or Écoles d'ingénieurs, which developed historically alongside the conventional university system. The Grandes Écoles d'ingénieurs were first set up as part of the late 18th century Enlightenment and developed with the positivist movement. With the impetus of a centralised, interventionist state they expanded in the 19th century to cater for most sectors of the economy (agriculture included) turning out highly qualified managerial personnel capable of ensuring the modernisation and economic development of a newly industrialising country. This form of education gradually took precedence both in economic spheres and in public administration. Thus engineers have achieved a very high socio-economic standing within French society and in business circles, higher than that of most conventional university graduates. However, they make up less than five per cent of the overall output from the French higher education system.

French agricultural higher education is strongly anchored in this tradition of engineers' education. Unlike in other European countries, there are broadly-based, multidisciplinary curricula with an emphasis on management. This is why economics and management courses are so significant and case studies or work placements feature very prominently, as does foreign language teaching, which is generally mandatory and has gained importance over the last decade.

Education has been dramatically transformed over the last 20 years to take on board changes in agriculture, the rural world and the food chain. The professional outlook of graduates has become much broader, reaching far beyond the bounds of agriculture alone. As in other countries, there is an ongoing debate in the academic community as to whether certain changes are warranted, be it new degree profiles to be developed or the respective standing of fundamental and applied science (Ruffro, 1995).

These factors explain why engineers, whatever their specialist area, fare better than university graduates statistically when it comes to finding employment; they enjoy lower unemployment, faster first recruitment, higher incomes and more dynamic career paths (e.g. Koubi and Mazars, 2003). 4 3 -Characteristics of the first destinations of French agricultural graduates A recent survey conducted by Toulouse Social Sciences University's Student Observatory for the Ministry of Agriculture reported on the first destinations of engineers graduating from the Ministry's institutions in 1998 (analysis of their initial employment and their occupation in 2002,that is four years after graduating) (Boudier, 2003).

Table 1 shows the survey's main findings, separating the three main categories of agricultural educational institutions: public institutions belonging to the group of écoles nationales supérieures d'agronomie (Paris, Montpellier, Rennes, Toulouse, Nancy, Dijon-Food Sciences) (I) other public institutions (Bordeaux, Clermont-Ferrand, Nantes, Dijon-Agricultural Sciences, Angers-Horticulture, etc.) (II) écoles supérieures d'agriculture, private institutions under contract to the state (Angers, Purpan, Beauvais, Lille, Lyon, Val de Reuil) ([I) The survey shows French agricultural science graduates get a good start on the career path. First recruitment is rapid and to good positions insofar as most graduates take up managerial positions. Respondents report this is because of the wide scope of their education, their experience abroad (especially work placements), their foreign language skills, and the career development and employment services of their educational institution. The placement when preparing their master's dissertation is also important in this respect.

The survey does reveal, however, differences by sex and type of institution from which respondents graduated.

Women continue their studies more often than men, who move into working life more quickly. Women are recruited more often on provisional contracts and remain (four years later) less well positioned on the labour market.

The recruitment pattems6 and social backgrounds of students at the three types of educational institutions investigated are somewhat different. Likewise, graduate access to the job market is variable. Graduates from category I institutions tend to go on with their studies and many go into employment unrelated to the food chain, in which they work in managerial positions.

Most graduates from category III institutions go into food chain-related sectors while those from category II favour the public sector at intermediate levels of responsibility.

However, to maintain their lead and to cope with recent difficulties on the job market and with growing competition from other educational courses (e.g. business and management), some institutions have opted to become more actively involved in the career development of their graduates. Source: [START_REF] Boudier | Ecoles supérieures agronomiques : I'insertion des diplômés 1998 -Etude sur le devenir des promotions 1998 des titulaires du titre d'ingénieur au 4è*" trimestre 2002[END_REF] 4 -A structure to promote graduate career starts: the case of

Agrocampus Rennes

For several years now Agrocampus RennesT has conducted an active policy of individual support for its graduates in finding their first destinations:

by formalising an individual career-planning process for each students by providing students with a set of tools and resources as part of a specialised career development service, Thus in addition to the conventional activities included in the students' basic education (work placement, conferences with visiting professionals, compulsory teaching modules for human resource management, etc,), various services are available to students based on their own personal initiatives.

The primary mechanism is to encourage students to think in terms of developing their own carser paths and to ensure they are doing so, to facilitate its formulation and then possibly to help them with the process.

This mostly involves:

group training sessions on recruitment techniques, interviewing and writing a curriculum vitae.

organising an annual one-day recruitment fair attended by some thirty firms or organisations. Students attend presentations by these firms, make direct contacts with them or participate in round-table discussions on the various careers in their specialist areas.

provision of a database of entrance-level vacancies, which is updated weekly and can be consulted on the internet.

access to a specialist library on employment, careers, firms and supervised access to a set of multimedia resources (telephone, fax, computers).

customised individual support with career planning. development of a set of occupation profiles which graduates in our sectors may take up.

The last two initiatives were introduced recently and have been the subject of special focus in recent years. They are worth a closer look.

t This is a new organisation formed by the merger in 2004 of I'École nationale Supérieure Agronomique de Rennes (ENSAR) and l'lnstitut National Supérieur de Formation Agro-alimentaire de Rennes [NSFA).lt turns out some 200 graduates a year at master's level.

t There is also a formalised process for supporting and advising students on their individual education and training (systern ofpersonal tutors among teaching staffand formal trial presentation ofan educational project to a panel oftutors).

4.1 -The career-planning process Definition A service has been in place since 1997 to provide individual support to each final.year student. The service is based on entitlement to an individual interview with a specialist in this area.

Students are free to take advantage of this opportunity if they so wish. They are told of it during their course and actively reminded of it at the beginning of their final year.

The institution secures the assistance of a team of paid advisors (human resource managers, recruitment officers or career guidance offrcers, recruitment consultants) who all share the same concems and apply the same method (preparation and interview form).

Each student wishing to participate is assigned to an advisor with whom the student goes through one or more interviews. The two-hour per student interview credit is generally spread over two or three interviews, depending on how far advanced the student's career-planning process is.

The objective is to move gradually (using the other tools available) from a somewhat hazy idea about a possible future occupation to a specific project and a coherent action plan designed to open the gateway to the desired position. In other words, students move to a position where they are aware of their strengths and of where they stand relative to the environment and are themselves convinced and can convince others that their objectives and ambitions are achievable.

Process

The approach used is similar to the funnel process and involves personal reflection on extemal factors (the economic situation, the state the sectors of activity are in, changes in careers available, the labour market), internal factors (review of hopes and aspirations, evaluation of own potential, definition of objectives, review of skills, etc.) and ends by focusing on an action plan including the sector of activity, type of occupation and specific job, combined with a strategy to achieve it. This strategy includes a time-frame and concrete resorrlces.

Within this process, the confrontation between oneself and the real world is essential. A thorough knowledge and evaluation of both is required. Information about external factors can be investigated by anyone at their own level provided they have the resources to direct their research and to be in a state of personal preparedness that allows it. The interviews are therefore aimed at reinforcing the students' realism and developing their ability to seek out relevant information and then to move in the direction they want to go.

The interview therefore runs through the following stages:

becoming aware of aspirations and objectives, objectivising strengths and weaknesses, formalising a project including the chosen sector and career, defining and planning what steps to take.

Each student receives a personal dossier covering the different stages and presupposing a personal preparatory phase for each stage. The final formulation of the projecf invoiu.r u futt write-up validated by the advisor.

As the process is a voluntary one, the students give an undertaking to see it through to the end. Completion is formalised by a final written document signed by both student and advisor.

Appraisal

This action is now well tried-and-tested and is widely taken up by students. Some 70 per cent take avail of the system. A network of some 40 professionals regularly participate and meet twice yearly to evaluate the action in qualitative and quantitative terms and to take it forward. This action is designed to meet three requirements:

-The occupation profiles

increasing demand from students for greater visibility of the contents of the occupations in question. They need clear and precise information to develop their career plans, to choose their specialist subject for their master's degree and to prepare for job seeking or for recruitment interviews. the need of teaching staff to update course content and curricula and match them to employers' needs and expectations. This information also provides fuel for dialogue with students.

the need for our institution to have a reference and evaluation tool to validate the work experience of professionals (mature students) when awarding qualifications. This is a tool for intemal use. It is not intended as a communication tool to explain occupations to secondary school pupils during careers guidance or to inform potential employers about the courses provided.

Each occupation profile provides four categories of information: mission: general objectives of the occupation within a firm's organisation of work.

activities: list of the main activities characterising the occupation; actions or groups of actions to be performed in this occupation, which may be ranked in hierarchical or chronological order.

competencies: these are the skills needed to successfully carry out the occupation.

e An occupation (métier in French) profile is a generic term for a collection of professional activities describing the common features (usually 60-80 per cent) of a set of positions whose main activities are of a similar kind and where the competencies and know-how involved are relatively uniform, while escaping the conshaints of the diversity of structures and employnent situations.

I Knowledge: acquired in the course of initial or vocational education and training; areas of knowledge required to perform the occupation together with their level based on a rank order (basic, competent, skilled, expert).

Know-how: true potential to do the work that may be nurtured tbrough vocational experience or in-service training. Predisposition: personal qualities, behaviour, affinities, temperament.

I a

The competencies indicated are those which are both specific and relevant to the occupation.

Occupational environment: localisation workers/travelling).

and working conditions (co- To date some 30 occupation profiles have been drawn up covering sectors as varied as animal and plant production, agrifood industry, the environment, and occupations ranging from commercial activity to consultancy via management. A sample occupation profile is appended.

Drawing up the occupation profiles The occupation profiles are developed by a high-resolution analysis in several stages preliminary documentary analysis: company job descriptions, job vacancies, etc.

semi-directed individual interviews with 3-5 professionals currently working in the occupation under analysis and having a degree of control over their employment situation.

validation by a panel of serving professionals and teaching staff to reach final consensus on the contents ofeach occupation profile.

The professionals chosen are a non-representative sample but do provide a grasp of the diversity of situations. The panel discussions are designed to identify the core features of the occupation.

Appraisal

This iterative process (collection of information, synthesis, validation) is time-consuming, demanding and expensive. It entails mobilising specific human resources which are now pooled at national level. Moreover, the process requires regular revision (planned for every 3-5 years) of the profile contents and of the occupations list (mapping) which had to be drawn up beforehand.

In terms of their use, these profiles must provide objective benchmarks against which students can align their ideas about the realities of work. However, there is nothing hard and fast about them as professions are never static. Nor does each professional necessarily exercise all the functions at one and the same time. This is why students using the profiles are advised to discuss them with other students and teaching staff who know these occupations or with their advisor (see above).

Overall, this project is currently at the development stage, and its promoters have been surprised at the interest shown by the firms and organisations contacted. Some professional l0 partners even envisage using this tool to clarify and define the occupations relevant to them, particularly with a view to certification of quality assurance systems in their business.

Conclusion

The experience acquired in Rennes in terms of first destinations of graduates shows that a successful process depends on a number of factors which cannot always readily be brought together:

a firm commitment from the institution's governing bodies, acceptance ofthe operating costs ofsuch a set-up, mobilisation of specific competencies to co-ordinate and operate the system, enhanced awareness of teaching staff, an information system.

The approach must also be open to the outside world and conducted in a network with the world of work (firms and potential employers, alumni associations or specialist employment centreslo).

Table 1 :

 1 First destination of 1998 graduates and their employment situation in 2002

		Type of institution
	û)	ûr)	ûID	Total

6

* graduates having started new courses after completing their agricultural degree course

(obtained in 1998)

  DefinitionAgrocampus Rennes undertook early in 2001 to draw up a list of occupation profilese that graduates might take up in the farming, agrifood and environment sectors. Since 2003 this action has been extended nationwide and Agrocampus Rennes co-ordinates a working party with participants from most French institutions.

A master's type degree: five-year course (300 ECTS credits)

Schematically at the end ofsecondary education (type III) or at bachelor level(type I and II)

Appendix:

A sample occupation profile (in French) Secteur d'activités z PRODUCTIONS VEGETALES sous-secteur : Appui teclmique ù lalïlière réf. fiche : PV-S4F3no4

Ineénieur CoNSEIL(LERE) SpBcnLISE(E)

(en agrono"mie, en productions végëtales, én productions légûnières...)

Mksion

Apporter


Yves Elskens
email: yves.elskens@univ-provence.fr

Nonquasilinear evolution of particle velocity in incoherent waves with random amplitudes

Keywords: weak plasma turbulence, stochastic acceleration, hamiltonian chaos, transport PACS: 05.45.-a, 41.75.-i, 52.35.-g, 84.40.-x, 29.27.-a, 05.65.+b

The one-dimensional motion of N particles in the field of many incoherent waves is revisited numerically. When the wave complex amplitudes are independent, with a gaussian distribution, the quasilinear approximation is found to always overestimate transport and to become accurate in the limit of infinite resonance overlap.

The motion of a particle in the field of many waves is a fundamental process in classical physics [7,8,20], usually discussed in the first chapters of plasma physics textbooks. For modeling purposes, this process can be described by the hamiltonian model

H = v 2 2 + M m=1 A m cos(k m x -ω m t -ϕ m ) (0.1)
where the particle has position x and momentum v (normalizing mass to 1 and writing A m = qE m /k m ). The wave field has M ≫ 1 waves, with a smooth dispersion relation associating a wavenumber k m , a pulsation ω m and a phase velocity v m = ω m /k m to each wave -usually determined by fixed properties of the environment, such as the geometry of the domain where waves propagate (then the k m are discrete). The actual spectrum of the wavefield, given by the complex amplitudes A m e iϕm , is more easily tuned by the experimenter or affected by simple changes in the environment, e.g. excited antennas.

The dynamical system approach to this problem would be to prescribe a single choice for each wave complex amplitude. However it would be quite exceptional to control all waves (though this is e.g. the assumption underlying the standard map with A m = A 0 , ϕ m = ϕ 0 for all waves -see [3] for a discussion).

Because the waves have different frequencies and velocities, it is generally unrealistic to assume their phases to be correlated. Their intensities are more easily observed, but both in nature and in the laboratory the accumulation of statistics on waves often involves only their average power spectra, not the detailed intensity data for each measurement run. Besides, if the waves are excited by a noisy source, as occurs in the self-consistent dynamics of particles and waves [9,[START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF], one cannot expect a systematic reproduction of individual wave data among several experiments (this is the very meaning of a "noisy source").

We thus assume here that these complex amplitudes are random data, and investigate the statistics of the particle motion in the resulting time-dependent random field. This dynamics is a "stochastic acceleration problem" for a "passive particle" in weak plasma turbulence [6,19,21], and its understanding is a prerequisite to a proper analysis of the case where the particle motion feeds back on the wave evolution [START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF].

This hamiltonian generates equations of motion

ẋ = v , (0.2) v = m k m A m sin(k m x -ω m t -ϕ m ) . (0.3)
A simple case for the dispersion relation is

k m = k 0 , ω m = 2π(m -M/2)/T, (0.4)
for some k 0 , T . Then, in the limit M → ∞, the equations of motion yield for A m e iϕm = A 0 (with real A 0 ) the well-known standard map [3]. The case where phases ϕ m are independent random variables uniformly distributed on the circle [0, 2π] while A m = A 0 is given was investigated notably by Cary, Escande, Verga and Bénisti [1,4,[START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF] and occurs in the context of the random phase approximation.

An important observation [5,[START_REF] Escande | [END_REF] on the motion of a particle in the field (0.3) is locality in velocity : the evolution of the particle when it has velocity v depends only weakly on the waves with a Doppler-shifted frequency ω m -k m v much larger than their trapping oscillation frequency 2k m √ A m . In particular, for a two-wave system the resonance overlap parameter

s 1,2 = 2 √ A 1 + 2 √ A 2 |ω 2 /k 2 -ω 1 /k 1 | (0.5)
becomes unity when there exists a velocity u for which

ω 2 /k 2 -u = 2 √ A 2 and u -ω 1 /k 1 = 2 √ A 1 (with k 1 > 0, k 2 > 0, ω 2 > ω 1
). For many waves with overlap parameters s ≫ 1, the relevant phase velocity range for waves influencing the particle is a "resonance box", with a width scaling as

A 2/3 [1,2].
To the extent that the phases and amplitudes of the waves are independent random variables, it is tempting to approximate the acceleration (0.3) by a white noise, with amplitude

σ = E (k 2 m A 2 m ) = k m E A 2 m
, where the relevant mode m is the one with the current particle velocity (E denotes the mathematical expectation, or ensemble average with respect to wave amplitudes and phases). 2This is the core of quasilinear theory [10,17,18,22]. Mathematically, one then interprets (0.2)-(0.3) as a stochastic differential equation, after the necessary reformulation to make sense of divergent series ; this is correct in the case m ∈ Z for dispersion relation (0.4) with gaussian independent complex amplitudes such that E A 2 m = k -2 0 σ 2 . The particle velocity then has a brownian evolution, so that for 0

< t ≤ t ′ < T E (v t -v 0 )(v t ′ -v 0 ) = 2D QL min(t, t ′ ) (0.6)
with the quasilinear diffusion coefficient D QL = σ 2 T /4. In particular,

E ∆v 2 t = σ 2 T 2 t (0.7)
for ∆v t = v t -v 0 . However, the particle evolution for t > T may show a strong correlation to its motion for 0 ≤ t ≤ T because the waves are periodic in time [11,[START_REF] Elskens | [END_REF], and the dispersion relation (0.4) may generate a strong spatial correlation -although it just accounts for the fact that in the limit of a dense spectrum (with a smooth dispersion relation) the relevant waves acting on a particle at any time also have almost the same wavelength. 3In this note we compare numerical properties of the velocity evolution for fixed amplitude with the case of random wave amplitudes : let A m e iϕm = C m + iS m . In the latter case, the amplitudes are drawn independently in such a way that their squares have exponential distributions, with equal expectation

E A 2 m = k -2 0 σ 2 .
Along with uniform distribution of the phases, this is equivalent to gaussian distribution of the complex amplitudes. Indeed, the probability density f ϕ for ϕ is uniform, and the exponential law for A 2 yields for A the density at a > 0

f A (a) = P(a ≤ A < a + da) da = P(a 2 ≤ A 2 < a 2 + 2a da) da = e -a 2 /E A 2 2a E A 2 .
(0.8) The probability density for (C, S) is thus (noting that a 2 = c 2 +s 2 and dc ds = a da dϕ)

f C,S (c, s) = f A (a)f ϕ (ϕ) da dϕ dc ds = 2ae -a 2 /E A 2 2π E A 2 da dϕ dc ds = e -(c 2 +s 2 )/E A 2 π E A 2 (0.9)
which is the density for a gaussian vector (C, S) with zero expectation and

covariance matrix E A 2 /2 0 0 E A 2 /2 .
For the simulations, we let k 0 = 1 and T = 2π so that σ = s 2 /16. Given the value of σ, or equivalently of s, we draw at random N pha = 400 sets of wave data. For each set of wave data, we follow the evolution of N orb = 50 particles, released initially at random points (x, p) in the strip 0 ≤ x ≤ 2π, -0.5 ≤ p ≤ 0.5. Trajectories are computed with a reversible symplectic integrator up to a time, large enough for the motion to exhibit possible departure from quasilinear statistics, but short enough to ensure that the particles remain far from the boundaries of the wave spectrum, which have velocity ±M/2 (where we take M up to 800) : for s = 13, the boundaries are more than 3 standard deviations (which is about 120 as observed in Fig. 1) away from the initial particle velocity. Some trajectories are computed backward from their final point to control numerical accuracy : the accumulated error is about 10 -5 for s = 3.5 for the runs of Fig. 1 and3, though it deteriorates rapidly with increasing s. Our calculations for the equal A case reproduce the findings of Refs [1,4].

Statistical averages discussed below (denoted by • ) are performed over all particle and wave data for the same s value. We focus on the second and fourth moments of particle velocity as functions of time. To the numerical average g 2 (t) = ∆v 2 we fit a linear approximation D 0 + 2D eff t, which defines an effective diffusion coefficient. Such a plot is displayed in Fig. 1. The quasilinear prediction 2D QL t is a straight line. The numerical average g 2 is plotted, along with two lines estimating one standard deviation on either side, viz. Specifically, Fig. 1 displays our results for s = 3.5 and for s = 13. The nonoverlap regime (s 1) allows no large scale transport in velocity, because the particle phase space contains invariant Kolmogorov-Arnol'd-Moser tori. Indeed the case s = 0 is integrable (it corresponds to free particle motion). For increasing overlap parameter value, one expects the transport to become increasingly chaotic, and indeed the particle dynamics typically has a Liapunov exponent scaling like A 1/3 ∼ s 2/3 for large s [START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF]. One may then expect the effective diffusion coefficient to increase towards the quasilinear estimate, D QL , which corresponds to a pure stochastic particle behaviour. However it has been found [1,4,[START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF] that for the equal amplitude, independent phases wave data, the particle velocity undergoes an enhanced transport, with an effective diffusion coefficient up to 2.5 times the quasilinear estimate, and the latter becomes accurate only in the large s limit. It must be noted that this enhanced transport shows up only after time T = 2π, i.e. only on a time scale for which the time periodicity due to the discrete nature of the wave spectrum significantly affects the particle motion -actually the motion for short times is essentially quasilinear thanks to the wave stochasticity [START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF]16]. We also checked that the force correlation function on the particle is essentially zero, except for times near integer multiples of the period T .

Our simulations show that the enhanced transport is not observed for independent wave amplitudes. In the latter case, the velocity variance grows with time at a rate never exceeding the quasilinear value. As shown in Fig. 2 this observation holds for the whole range of overlap parameter values sampled, in agreement with the naive prediction that "increasing chaos" should make dynamics more similar to "pure noise".

The noisiness of the particle evolution is also assessed using a higher-moment test. If the velocity distribution at time t were gaussian, the ratio ∆v 4 /(3g 2 2 ) would equal 1. Fig. 3 displays numerical evidence that for the equal wave amplitude case this ratio remains rather below 1 (the velocity distribution has smaller tails than the gaussian with equal variance), whereas for the gaussian wave data it is somewhat above 1 (the distribution has stronger tails than the gaussian with same variance).

For both wave data types the limiting behaviour for s → ∞ is quasilinear, in agreement with theoretical [1,[START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF]15] and mathematical [11,[START_REF] Elskens | [END_REF] arguments.

These results confirm that random phases only are not sufficient to substantiate the quasilinear approximation for the stochastic acceleration problem. A gaussian wave spectrum may seem closer to the ideal view of white noise for one period T , but over longer times correlations also build up, driving transport away from the quasilinear approximation, especially when the overlap parameter has moderate typical values (1 s 5).

Many fruitful discussions with F. Doveil and D.F. Escande are acknowledged.

g 2 ±Fig. 1 .

 21 Fig.1. Dependence of ∆v 2 on time for random amplitudes (Gauss) and for equal amplitudes (A=cst).

g 2 2 1/ 2 .

 22 The narrowness of the channel so constructed indicates the statistical accuracy of the plotted function g 2 .

Fig. 2 .

 2 Fig. 2. Ratio D eff /D QL as function of overlap parameter s for random amplitudes (Gauss) and for equal amplitudes (A=cst). The lines are guides to the eye.

Fig. 3 .

 3 Fig. 3. Ratio ∆v 4 /(3 ∆v 2 2 ) versus time for random amplitudes (Gauss) and for equal amplitudes (A=cst).

Phases do not appear in σ (nor in s) because the relative phase of two waves ϕ m + ω m t -ϕ n -ω n t varies uniformly over time (hence ϕ m -ϕ n can be absorbed in the choice of the time origin).

There is a large body of literature on the case of incoherent waves with no dispersion relation. Then the sum m becomes a double sum m,n and one varies wavenumbers k n independently from pulsations ω m . This space-time stochastic environment is more noisy that our model and may also be considered to motivate a quasilinear approximation.

 1Supported in part by a CNRS temporary position during this work.


In vitro model system for the study of milk protein secretion

Eric CHANAT, Laboratoire de Biologie Cellulaire et Moléculaire, INRA, 78352 Jouyen-Josas cedex, France.

To study intracellular mechanisms of milk protein traffic to the cell surface and secretory vesicles biogenesis in mammary epithelial cells (MEC), we are establishing a in vitro system using the mouse MEC line HC11. HC11 cells grown on plastic, although heterogeneous, can easily be induced to synthesize ß casein (ßcn) in the presence of insulin, cortisol and PRL. When HC11 cells are grown on porous filters, they expressed 10 times more ßcn. Milk proteins secreted by these cells were essentially found in the ≥ apical medium and identified by western blotting as lactoferrin and γ cn. Only a minor proportion of the cellular ßcn was found in the apical medium. Analysis by SDS-PAGE showed that ßcn from cells and medium migrated as a doublet of lower Mr. than the mature mouse milk ßcn. This increase in electrophoretic mobility suggested that this form had not underwent all post-translational modifications and was blocked in an early compartment of the secretory pathway. Consistent with this hypothesis, we observed by electron microscopy that the rough endoplasmic reticulum of these cells was highly dilated. In some experiments, HC11 cells also expressed αcns and, interestingly enough, the mature form of ßcn was observed in addition to the ßcn doublet. The amount of mature ßcn was found to be proportional to the level of expression of αcns.


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text:
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La forêt paysanne et l'aménagement de I'espace rural.

Introduction

Gerard Balent

To cite this version:

  Gerard Balent. La forêt paysanne et l'aménagement de I'espace rural. Introduction. La forêt paysanne dans l'espace rural : Biodiversité, paysages, produits, 29, INRA, 268 p., 1996, Etudes et Recherches sur les Systèmes Agraires et le Développement, 2-7380-0684-1. hal-02834553


Nils Kuster

Anne-Sophie Bargnoux

Stéphanie Badiou

Anne-Marie Dupuy

Laurence Piéroni

Jean-Paul Cristol

Multilevel qualification of a large set of blood gas analyzers: Which performance goals?

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Mohammad El Smaily
email: mohammad.el-smaily@etu.univ-cezanne.fr

Min-Max formulae for the speeds of pulsating travelling fronts in periodic excitable media

This paper is concerned with some nonlinear propagation phenomena for reaction-advection-diffusion equations in a periodic framework. It deals with travelling wave solutions of the equation

propagating with a speed c. In the case of a "combustion" nonlinearity, the speed c exists and it is unique, while the front u is unique up to a translation in t. We give a min -max and a max -min formula for this speed c. On the other hand, in the case of a "ZFK" or a "KPP" nonlinearity, there exists a minimal speed of propagation c * . In this situation, we give a min -max formula for c * . Finally, we apply this min -max formula to prove a variational formula involving eigenvalue problems for the minimal speed c * in the "KPP" case.

1 Introduction and main results

A description of the periodic framework

The goal of this paper is to give some variational formulae for the speeds of pulsating travelling fronts corresponding to reaction-diffusion-advection equations set in a heterogenous periodic framework. In fact, many works, such as Hamel [START_REF] Hamel | Formules min-max pour les vitesses d'ondes progressives multidimensionnelles[END_REF], Heinze, Papanicolaou, Stevens [START_REF] Heinze | Variational principles for propagation speeds in inhomogeneous media[END_REF], and Volpert, Volpert, Volpert [START_REF] Volpert | Traveling wave solutions of parabolic systems[END_REF] treated this problem in simplified situations and under more strict assumptions. In this paper, we treat the problem in the most general periodic framework. We are concerned with equations of the type

u t = ∇ • (A(z)∇u) + q(z) • ∇u + f (z, u), t ∈ R, z ∈ Ω, ν • A ∇u(t, z) = 0, t ∈ R, z ∈ ∂Ω, (1.1)
where ν(z) is the unit outward normal on ∂Ω at the point z. In this context, let us detail the mathematical description of the heterogeneous setting.

Concerning the domain, let N ≥ 1 be the space dimension, and let d be an integer so that 1 ≤ d ≤ N. For an element z = (x 1 , x 2 , • • • , x d , x d+1 , • • • , x N ) ∈ R N , we denote by x = (x 1 , x 2 , • • • , x d ) and by y = (x d+1 , • • • , x N ) the two tuples so that z = (x, y). Let L 1 , • • • , L d be d positive real numbers, and let Ω be a C 3 non empty connected open subset of R N satisfying

     ∃ R ≥ 0 ; ∀ (x, y) ∈ Ω, |y| ≤ R, ∀ (k 1 , • • • , k d ) ∈ L 1 Z × • • • × L d Z, Ω = Ω + d k=1 k i e i , (1.2)
where (e i ) 1≤i≤N is the canonical basis of R N . As d ≥ 1, one notes that the set Ω satisfying (1.2) is unbounded. We have many archetypes of such a domain. The case of the whole space R N corresponds to d = N, where L 1 , . . . , L N are any positive numbers. The case of the whole space R N with a periodic array of holes can also be considered. The case d = 1 corresponds to domains which have only one unbounded dimension, namely infinite cylinders which may be straight or have oscillating periodic boundaries, and which may or may not have periodic perforations. The case 2 ≤ d ≤ N -1 corresponds to infinite slabs.

In this periodic situation, we give the following definitions:

Definition 1.1 (Periodicity cell) The set

C = { (x, y) ∈ Ω; x 1 ∈ (0, L 1 ), • • • , x d ∈ (0, L d )}
is called the periodicity cell of Ω.

Definition 1.2 (L-periodic fields) A field w : Ω → R N is said to be Lperiodic with respect to x if

w(x 1 + k 1 , • • • , x d + k d , y) = w(x 1 , • • • , x d , y)
almost everywhere in Ω, and for all k = (k

1 , • • • , k d ) ∈ d i=1 L i Z.
Let us now detail the assumptions concerning the coefficients in (1.1). First, the diffusion matrix A(x, y) = (A ij (x, y)) 1≤i,j≤N is a symmetric C 2,δ ( Ω ) (with δ > 0) matrix field satisfying

        
A is L-periodic with respect to x,

∃ 0 < α 1 ≤ α 2 ; ∀(x, y) ∈ Ω, ∀ ξ ∈ R N , α 1 |ξ| 2 ≤ 1≤i,j≤N A ij (x, y)ξ i ξ j ≤ α 2 |ξ| 2 .
(1.3)

The underlying advection q(x, y) = (q 1 (x, y), • • • , q N (x, y)) is a C 1,δ (Ω) (with δ > 0) vector field satisfying:

            
q is L-periodic with respect to x,

∇ • q = 0 in Ω , q • ν = 0 on ∂Ω , ∀ 1 ≤ i ≤ d, C q i dx dy = 0.
(1.4) Lastly, let f = f (x, y, u) be a function defined in Ω × R such that

           f is globally Lipschitz-continuous in Ω × R, ∀ (x, y) ∈ Ω, ∀ s ∈ (-∞, 0] ∪ [1, +∞), f (s, x, y) = 0,
∃ ρ ∈ (0, 1), ∀(x, y) ∈ Ω, ∀ 1 -ρ ≤ s ≤ s ≤ 1, f (x, y, s) ≥ f (x, y, s ). (1.5) One assumes that f is L-periodic with respect to x.

(1.6)

Moreover, the function f is assumed to be of one of the following two types: either

∃ θ ∈ (0, 1), ∀(x, y) ∈ Ω, ∀ s ∈ [0, θ], f (x, y, s) = 0, ∀ s ∈ (θ, 1), ∃ (x, y) ∈ Ω such that f (x, y, s) > 0, (1.7)
or

∃ δ > 0, the restriction of f to Ω × [0, 1] is of class C 1, δ , ∀ s ∈ (0, 1), ∃ (x, y) ∈ Ω such that f (x, y, s) > 0. ( 1
f u (x, y, 0) = lim u→0 + f (x, y, u)/u > 0, (1.9)
with the additional assumption

∀ (x, y, s) ∈ Ω × (0, 1), 0 < f (x, y, s) ≤ f u (x, y, 0) × s, (1.10)
then f is called a "KPP"(for Kolmogorov, Petrovsky, and Piskunov, see [START_REF] Kolmogorov | Étude de l' équation de la diffusion avec croissance de la quantité de matiére et son application a un probléme biologique[END_REF]) nonlinearity.

The simplest examples of "combustion" and "ZFK" nonlinearities are when f (x, y, u) = f (u) where: either (1.11), then it is a homogeneous "combustion" nonlinearity. On the other hand, a nonlinearity f = f (u) that satisfies (1.12) is homogeneous of the "ZFK" type. Moreover, a KPP homogeneous nonlinearity is a function f = f (u) that satisfies (1.12) with the additional assumption ∀ s ∈ (0, 1), 0 < f (s) ≤ f (0) s.

         f is Lipschitz-continuous in R, ∃ θ ∈ (0, 1), f (s) = 0 for all s ∈ (-∞, θ] ∪ [1, +∞), f (s) > 0 for all s ∈ (θ, 1), ∃ ρ ∈ (0, 1 -θ), f is non-increasing on [1 -ρ, 1], (1.11) or          f is defined on R, f ≡ 0 in R \ (0, 1), ∃ δ > 0, the restriction of f on the interval [0, 1] is C 1,δ ([0, 1]), f (0) = f (1) = 0, and f (s) > 0 for all s ∈ (0, 1), ∃ ρ > 0, f is non-increasing on [1 -ρ, 1]. (1.12) If f = f (u) satisfies
(1.13)

As typical examples of nonlinear heterogeneous sources satisfying (1.5-1.6) and either (1.7) or (1.8), one can consider the functions of the type

f (x, y, u) = h(x, y) g(u),
where h is a globally Lipschitz-continuous, positive, bounded, and L-periodic with respect to x function defined in Ω, and g is a function satisfying either (1.11) or (1.12). Definition 1.4 (Pulsating fronts and speed of propagation) Let e = (e 1 , • • • , e d ) be an arbitrarily given unit vector in R d . A function u = u(t, x, y) is called a pulsating travelling front propagating in the direction of -e with an effective speed c = 0, if u is a classical solution of:

                       u t = ∇ • (A(x, y)∇u) + q(x, y) • ∇u + f (x, y, u), t ∈ R, (x, y) ∈ Ω, ν • A ∇u(t, x, y) = 0, t ∈ R, (x, y) ∈ ∂Ω, ∀ k ∈ d i=1 L i Z, ∀ (t, x, y) ∈ R × Ω, u(t + k • e c , x, y) = u(t, x + k, y), lim x•e→-∞
u(t, x, y) = 0, and lim

x•e→+∞ u(t, x, y) = 1, 0 ≤ u ≤ 1, (1.14)
where the above limits hold locally in t and uniformly in y and in the directions of R d which are orthogonal to e .

Several works were concerned with pulsating travelling fronts in periodic media (see [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF], [START_REF] Berestycki | The Speed of Propagation for KPP Type Problems (Periodic Framework)[END_REF], [START_REF] Kinezaki | Modeling biological invasions into periodically fragmented environments[END_REF], [START_REF] Papanicolaou | Reaction-diffusion fronts in periodically layered media[END_REF], [START_REF] Shigesada | Biological invasions: Theory and practice[END_REF], [START_REF] Shigesada | Spatial segregation of interacting species[END_REF], and [START_REF] Xin | Front propagation in heterogeneous media[END_REF]).

In the general periodic framework, we recall two essential known results and then we move to our main results. Theorem 1.5 (Berestycki, Hamel [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF]) Let Ω be a domain satisfying (1.2), let e be any unit vector of R d and let f be a nonlinearity satisfying (1.5-1.6) and (1.7). Assume, furthermore, that A and q satisfy (1.3) and (1.4) respectively. Then, there exists a classical solution (c, u) of (1.14). Moreover, the speed c is positive and unique while the function u = u(t, x, y) is increasing in t and it is unique up to a translation. Precisely, if (c 1 , u 1 ) and (c 2 , u 2 ) are two classical solutions of (1.14), then c 1 = c 2 and there exists h ∈ R such that u 1 (t, x, y) = u 2 (t + h, x, y) for all (t, x, y) ∈ R × Ω.

In a periodic framework, Theorem 1.5 yields the existence of a pulsating travelling front in the case of a "combustion" nonlinearity with an ignition temperature θ. It implies, also, the uniqueness of the speed and of the profile of u. For "ZFK" nonlinearities, we have Theorem 1.6 (Berestycki, Hamel [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF]) Let Ω be a domain satisfying (1.2), let e be any unit vector in R d and let f be a nonlinearity satisfying (1.5-1.6) and (1.8). Assume, furthermore, that A and q satisfy (1.3) and (1.4) respectively. Then, there exists c * Ω,A,q,f (e) > 0 such that the problem (1.14) has no solution (c, u) such that u t > 0 in R × Ω if c < c * Ω,A,q,f (e) while, for each c ≥ c * Ω,A,q,f (e), it has a solution (c, u) such that u is increasing in t.

In fact, the existence and the monotonicity of a solution u * = u * (t, x, y) of (1.14) for c = c * Ω,A,q,f (e) > 0 holds by approaching the "ZFK" nonlinearity f by a sequence of combustion nonlinearities (f θ ) θ such that f θ → f uniformly in R × Ω as θ 0 + (see more details in step 2 of the proof of formula (1.17) below, section 4). It follows, from Theorem 1.5, that for each θ > 0, there exists a solution (c θ , u θ ) of (1.14) with the nonlinearity f θ such that u θ is increasing with respect to t. From parabolic estimates, the functions u θ , converge up to a subsequence, to a function u * in C 2 loc (R × Ω) as θ → 0 + . Moreover, Lemmas 6.1 and 6.2 in [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF] yield the existence of a constant c * (e) = c * Ω,A,q,f (e) > 0 such that c θ c * (e) as θ 0. Hence, the couple (c * (e), u * ) becomes a classical solution of (1.14) with the nonlinearity f and one gets that u * is nondecreasing with respect to t as a limit of the increasing functions u θ . Finally, one applies the strong parabolic maximum principle and Hopf lemma to get that w is positive in R × Ω. In other words, u * is increasing in t ∈ R. Actually, in the "ZFK" case, under the additional nondegeneracy assumption (1.9), it is known that any pulsating front with speed c is increasing in time and c ≥ c * (e) (see [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF]).

The value c * Ω,A,q,f (e) which appears in Theorem 1.6 is called the minimal speed of propagation of the pulsating travelling fronts propagating in the direction -e (satisfying the reaction-advection-diffusion problem (1.14)).

We mention that the uniqueness of the pulsating travelling fronts, up to shifts in time, for each c ≥ c * Ω,A,q,f (e), has been proved recently by Hamel and Roques [START_REF] Hamel | Uniqueness and stability of monostable pulsating travelling fronts[END_REF] for "KPP" nonlinearities. On the other hand, a variational formula for the minimal speed of propagation c * Ω,A,q,f (e), in the case of a KPP nonlinearity, was proved in Berestycki, Hamel, Nadirashvili [START_REF] Berestycki | The Speed of Propagation for KPP Type Problems (Periodic Framework)[END_REF]. This formula involves eigenvalue problems and gives the value of the minimal speed in terms of the domain Ω and in terms of the coefficients A, q, and f appearing in problem (1.14). The asymptotic behaviors and the variations of the minimal speed of propagation, as a function of the diffusion, advection and reaction factors and as a function of the periodicity parameters, were widely studied in Berestycki, Hamel, Nadirashvili [START_REF] Berestycki | Elliptic Eigenvalue Problems with Large Drift and Applications to Nonlinear Propagation Phenomena[END_REF], El Smaily [START_REF] Smaily | Pulsating travelling fronts: Asymptotics and homogenization regimes[END_REF], El Smaily, Hamel, Roques [START_REF] Smaily | Homogenization and influence of fragmentation in a biological invasion model[END_REF], Heinze [START_REF] Heinze | Large convection limits for KPP fronts[END_REF], Ryzhik, Zlatoš [START_REF] Ryzhik | KPP pulsating front speed-up by flows[END_REF], and Zlatoš [START_REF] Zlatoš | Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows[END_REF].

Main results

In the periodic framework, having (in Theorems 1.5 and 1.6) the existence results and some qualitative properties of the pulsating travelling fronts propagating in the direction of a fixed unit vector -e ∈ R d , we search a variational formula for the unique speed of propagation c = c(e) whenever f is of the "combustion" type, and for the minimal speed c * = c * Ω,A,q,f (e) whenever f is of the "ZFK" or the "KPP" type. We will answer the above investigations in the following theorem, but before this, we introduce the following

Notation 1.7 For each function φ = φ(s, x, y) in C 1,δ (R × Ω) (for some δ ∈ [0, 1)), let F [φ] := ∇ x,y •(A∇ x,y φ)+(ẽAẽ)φ ss +∇ x,y •(Aẽφ s )+∂ s (ẽA∇ x,y φ) in D (R×Ω), where ẽ = (e, 0, • • • , 0) ∈ R N and e denotes a unit vector of R d .
The first main result deals with the "combustion" case.

Theorem 1.8 Let e a unit vector of R d . Assume that Ω is a domain satisfying (1.2) and f is a nonlinear source satisfying (1.5) and (1.6). Assume furthermore that A and q satisfy (1.3) and (1.4) respectively. Consider the set of functions

E = ϕ = ϕ(s, x, y), ϕ is of class C 1,µ (R × Ω) for each µ ∈ [0, 1), F [ϕ] ∈ C(R × Ω), ϕ is L-periodic with respect to x, ϕ s > 0 in R × Ω, ϕ(-∞, ., .) = 0, ϕ(+∞, ., .) = 1 uniformly in Ω, and ν • A(∇ x,y ϕ + ẽϕ s ) = 0 on R × ∂Ω} . For each ϕ ∈ E, we define the function Rϕ ∈ C(R × Ω) as, for all (s, x, y) ∈ R × Ω, R ϕ(s, x, y) = F [ϕ](s, x, y) + q • ∇ x,y ϕ(s, x, y) + f (x, y, ϕ) ∂ s ϕ(s, x, y) + q(x, y) • ẽ.
If f is a nonlinearity of "combustion" type satisfying (1.7), then the unique speed c(e) that corresponds to problem (1. Furthermore, the min in (1.15) and the max in (1.16) are attained by, and only by, the function φ(s, x, y) = u s-x•e c(e) , x, y and its shifts φ(s + τ, x, y) for any τ ∈ R, where u is the solution of (1.14) with a speed c(e) (whose existence and uniqueness up to a translation in t follow from Theorem 1.5).

The second result is concerned with "ZFK" nonlinearities.

Theorem 1.9 Under the same notations of Theorem 1.8, if f is a nonlinearity of "ZFK" type satisfying (1.5-1.6) and (1.8), then the minimal speed c * Ω,A,q,f (e) is given by

c * Ω,A,q,f (e) = min ϕ∈E sup (s,x,y)∈R×Ω R ϕ(s, x, y). (1.17)
Furthermore, the min is attained by the function φ * (s, x, y) = u * s-x•e c * (e) , x, y and its shifts φ * (s + τ, x, y) for any τ ∈ R, where u * is any solution of (1.14) propagating with the speed c * (e) = c * Ω,A,q,f (e). In particular, Theorem 1.9 yields that formula (1.17) holds in the "KPP" case (1.10) as well.

Remark 1.10 In Theorem 1.8, the min and the max are attained by, and only by, the pulsating front φ(s, x, y) and its shifts φ(s + τ, x, y) for all τ ∈ R. In Theorem 1.9, the min is achieved by the front φ * (s, x, y) with the speed c * (e) and all its shifts φ * (s + τ, x, y). Actually, if the pulsating front φ * is unique up to shift, then φ * and its shifts are the unique minimizers in formula (1.17). The uniqueness is known in the "KPP" case (see Hamel, Roques [START_REF] Hamel | Uniqueness and stability of monostable pulsating travelling fronts[END_REF]), but it is still open in the general "ZFK" case.

We mention that a max-min formula of the type (1.16) can not hold for the minimal speed c * (e) in the "ZFK" or the "KPP" case. A simple justification is given in section 2.

The variational formulations of the speeds of propagation which are given in Theorems 1.8 and 1.9 are more general than those in Hamel [START_REF] Hamel | Formules min-max pour les vitesses d'ondes progressives multidimensionnelles[END_REF] and Heinze, Papanicolaou, Stevens [START_REF] Heinze | Variational principles for propagation speeds in inhomogeneous media[END_REF]. In Theorems 1.8 and 1.9, we consider nonhomogeneous nonlinearities f = f (x, y, u) and the domain Ω is in the most general periodic situation. However, in [START_REF] Hamel | Formules min-max pour les vitesses d'ondes progressives multidimensionnelles[END_REF], the domain was an infinite cylinder of R N and the advection q was in the form of shear flows. Moreover, in this paper, the nonhomogeneous operator ∇ • (A∇u) replaces the Laplace operator ∆u taken in [START_REF] Hamel | Formules min-max pour les vitesses d'ondes progressives multidimensionnelles[END_REF]. On the other hand, in [START_REF] Heinze | Variational principles for propagation speeds in inhomogeneous media[END_REF], the domain Ω was an infinite cylinder in R N with a bounded cross section. Namely, Ω = R × ω ⊂ R N where the cross section ω is a bounded domain in R N -1 . Moreover, the authors did not consider an advection field in [START_REF] Heinze | Variational principles for propagation speeds in inhomogeneous media[END_REF]. Finally, concerning the nonlinearities, they were depending only on u (i.e f = f (u) and is satisfying either (1.11) or (1.12)) in both of [START_REF] Hamel | Formules min-max pour les vitesses d'ondes progressives multidimensionnelles[END_REF] and [START_REF] Heinze | Variational principles for propagation speeds in inhomogeneous media[END_REF].

Besides the fact that we consider here a wider family of diffusion and reaction coefficients, our assumptions are less strict than those supposed in [START_REF] Heinze | Variational principles for propagation speeds in inhomogeneous media[END_REF] and [START_REF] Volpert | Traveling wave solutions of parabolic systems[END_REF]. Roughly speaking, the authors, in [START_REF] Heinze | Variational principles for propagation speeds in inhomogeneous media[END_REF] and [START_REF] Volpert | Traveling wave solutions of parabolic systems[END_REF], assume a stability condition on the pulsating travelling fronts. We mention that such a stability condition is fulfilled in the homogenous setting; however, it has not been rigorously proved so far that this condition is satisfied in the heterogenous setting. Meanwhile, the assumptions of the present paper only involve the coefficients of the reaction-advection-diffusion equation (1.14), and they can then be checked easily.

Actually, in the "KPP" case, another "simpler" variational formula for the minimal speed c * (e) = c * Ω,A,q,f (e) is known. This known formula involves only the linearized nonlinearity f at u = 0. Namely, it follows from [START_REF] Berestycki | The Speed of Propagation for KPP Type Problems (Periodic Framework)[END_REF] that Theorem 1.11 (Berestycki, Hamel, Nadirashvili [START_REF] Berestycki | The Speed of Propagation for KPP Type Problems (Periodic Framework)[END_REF]) Let e be a fixed unit vector in R d and let ẽ = (e, 0, . . . , 0) ∈ R N . Assume that f is a "KPP" nonlinearity and that Ω, A and q satisfy (1.2), (1.3) and (1.4) respectively. Then, the minimal speed c * (e) of pulsating fronts solving (1.14) and propagating in the direction of e is given by

c * (e) = c * Ω,A,q,f (e) = min λ>0 k(λ) λ , (1.18)
where k(λ) = k Ω,e,A,q,ζ (λ) is the principal eigenvalue of the operator L Ω,e,A,q,ζ,λ which is defined by

L Ω,e,A,q,ζ,λ ψ := ∇ • (A∇ψ) + 2λẽ • A∇ψ + q • ∇ψ +[λ 2 ẽAẽ + λ∇ • (Aẽ) + λq • ẽ + ζ]ψ (1.19)
acting on the set

E λ = ψ ∈ C 2 (Ω), ψ is L-periodic with respect to x and ν • A∇ψ = -λ(ν • Aẽ)ψ on ∂Ω} .
In our last result, we prove that formula (1.17) implies formula (1.18) in the "KPP" case, but under some additional assumptions on the advection and the diffusion coefficients. This result gives an alternate proof of the formula (1.18). Theorem 1.12 Let e be a fixed unit vector in R d and let ẽ = (e, 0, . . . , 0) ∈ R N . Assume that f is a "KPP" nonlinearity and that Ω, A and q satisfy (1.2), (1.3) and (1.4) respectively. Assume, furthermore, that ν • Aẽ = 0 on ∂Ω (in the case where ∂Ω = ∅). Then, formula (1.17) implies formula (1.18).

Main tools: change of variables and maximum principles

In this section, we introduce some tools that will be used in different places of this paper in order to prove the main results. Throughout this paper, ẽ will denote the vector in R N defined by ẽ = (e, 0,

• • • , 0) = (e 1 , • • • , e d , 0, • • • , 0),
where e 1 , • • • , e d are the components of the vector e.

Our study is concerned with the model (1.14). Having a "combustion", a "ZFK", or a "KPP" nonlinearity, together with the assumptions (1.3) and (1.4), problem (1.14) has at least a classical solution (c, u) such that c > 0 and u t > 0 (see Theorems 1.5 and 1.6). The function u is globally C 1,µ (R × Ω) and C 2,µ with respect to (x, y) variables (for every µ ∈ [0, 1)). It follows that ∇ x,y .(A∇u) ∈ C(R × Ω). Having a unit direction e ∈ R d , and having a bounded classical solution (c, u) of (1.14) with c = c(e) (combustion case) or c ≥ c * (e) (ZFK or KPP case), we make the same change of variables as Xin [START_REF] Xin | Existence of planar flame fronts in connective-diffusive periodic media[END_REF]. Namely, let φ = φ(s, x, y) be the function defined by φ(s, x, y) = u s -x • e c , x, y for all s ∈ R and (x, y) ∈ Ω.

(2.1)

Then, for all (s, x, y

) ∈ R × Ω, [∇ x,y • (A∇ x,y φ) + (ẽAẽ)φ ss + ∇ x,y • (Aẽφ s ) + ∂ s (ẽA∇ x,y φ) ] (s, x, y) = ∇ x,y • (A∇u)(t, x, y),
where s = x • e + ct. Consequently,

F [φ](s, x, y) = ∇ x,y • (A∇ x,y φ) + (ẽAẽ)φ ss + ∇ x,y • (Aẽφ s ) + ∂ s (ẽA∇ x,y φ) is defined at each point (s, x, y) ∈ R × Ω and the map (s, x, y) → F [φ](s, x, y) belongs to C(R × Ω).
In all this paper, L = L c will denote the operator acting on the set E (given in Theorem 1.8) and which is defined by

Lϕ = ∇ x,y • (A∇ x,y ϕ) + (ẽAẽ)ϕ ss + ∇ x,y • (Aẽϕ s ) + ∂ s (ẽA∇ x,y ϕ) + q • ∇ x,y ϕ + (q • ẽ -c)ϕ s in C(R × Ω) = F [ϕ] + q • ∇ x,y ϕ + (q • ẽ -c)ϕ s in C(R × Ω), (2.2)
for all ϕ ∈ E.

It follows from above that if φ = φ(s, x, y) is a function that is given by a pulsating travelling (c, u) solving (1.14) (under the change of variables (2.1)), then

F [φ] ∈ C(R × Ω), φ is globally bounded in C 1,µ (R × Ω) (for every µ ∈ [0, 1)
) and it satisfies the following degenerate elliptic equation

Lφ(s, x, y) + f (x, y, φ) = F [φ](s, x, y) + q • ∇ x,y φ(s, x, y) + (q • ẽ -c)φ s (s, x, y) + f (x, y, φ) = 0 (2.3)
in R × Ω, together with the boundary and periodicity conditions

φ is L-periodic with respect to x, ν • A(∇ x,y φ + ẽφ s ) = 0 on R × Ω. ( 2
f in (1.14) is C 1,δ (Ω × [0, 1]
), one can then conclude that (see the proof of Proposition 6.3 in [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF]) any solution u of (1.14) satisfies:

∀(t, x, y) ∈ R × Ω, |∂ tt u(t, x, y)| ≤ M ∂ t u(t, x, y)
for some constant M independent of (t, x, y). In other words, the function

φ(s, x, y) = u((s -x • e)/c, x , y)
(where c = c(e) in the "combustion" case, and c = c * (e) in the "ZFK" or the "KPP" case) satisfies

∀(s, x, y) ∈ R × Ω, |∂ ss φ(s, x, y)| ≤ (M/c) ∂ s φ(s, x, y).
Let E be the functional subset of E defined by

E = ϕ ∈ E, ∃C > 0, ∀ (s, x, y) ∈ R × Ω, |∂ ss ϕ(s, x, y)| ≤ C ∂ s ϕ(s, x, y) .
The previous facts together with the discussions at the beginning of this section imply that the functions φ and φ * of Theorems 1.8 and 1.9 are elements of E ⊂ E. These theorems also yield that the max-min and the min-max formulae can also hold over the subset E of E.

Namely, in the case of a "combustion" nonlinearity c(e) = min (2.7)

Moreover, the min and the max are attained at, and only at, the function φ(s, x, y) and its shifts φ(s + τ, x, y) for any τ ∈ R.

On the other hand, only a min-max formula holds in the case of "ZFK" or "KPP" nonlinearities. That is c * (e) = min ϕ∈E sup (s,x,y)∈R×Ω R ϕ(s, x, y).

(2.8)

Moreover, the min is attained at the function φ * (s, x, y) and its shifts φ * (s + τ, x, y) for any τ ∈ R.

In the proofs of the variational formulae which were given in Theorem 1.8 and Theorem 1.9, we will use two versions of the maximum principle in unbounded domains for some problems related to (2.2-2.4) and (2.5). Such generalized maximum principles were proved in Berestycki, Hamel [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF] in a slightly more general framework:

Lemma 2.3 ([1]
) Let e be a fixed unit vector in R d . Let g(x, y, u) be a globally bounded and globally Lipschitz-continuous function defined in Ω × R and assume that g is non-increasing with respect to u in Ω×(-∞, δ] for some δ > 0. Let h ∈ R and Σ - h := (-∞, h) × Ω. Let c = 0 and φ 1 (s, x, y), φ 2 (s, x, y) be two bounded and globally C 1,µ Σ - h functions (for some µ > 0) such that

             L φ 1 + g(x, y, φ 1 ) ≥ 0 in D (Σ - h ), L φ 2 + g(x, y, φ 2 ) ≤ 0 in D (Σ - h ), ν • A [ẽ(φ 1 s -φ 2 s ) + ∇ x,y (φ 1 -φ 2 )] ≤ 0 on (-∞, h] × ∂Ω, lim s 0 →-∞ sup {s≤s 0 , (x,y)∈Ω} [φ 1 (s, x, y) -φ 2 (s, x, y)] ≤ 0, (2.9)
where L φ := ∇ x,y • (A∇ x,y φ) + (ẽAẽ)φ ss + ∇ x,y • (Aẽφ s ) + ∂ s (ẽA∇ x,y φ)

+ q • ∇ x,y φ + (q • ẽ -c)φ s , (2.10)
and ẽ denotes the vector (e, 0,

• • • , 0) ∈ R N .
If φ 1 ≤ δ in Σ - h and φ 1 (h, x, y) ≤ φ 2 (h, x, y) for all (x, y) ∈ Ω, then

φ 1 ≤ φ 2 in Σ - h .
Remark 2.4 Note here that φ 1 , φ 2 , q, A and g are not assumed to be L-periodic in x and that q is not assumed to satisfy (1.4).

Changing φ 1 (s, x, y), φ 2 (s, x, y) and g(x, y, s) into 1 -φ 1 (-s, x, y), 1φ 2 (-s, x, y) and -g(x, y, 1 -s) respectively in Lemma 2.3 leads to the following Lemma 2.5 ( [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF]) Let e be a fixed unit vector in R d . Let g(x, y, u) be a globally bounded and globally Lipschitz-continuous function defined in Ω × R and assume that g is non-increasing with respect to u in Ω × [1 -δ, +∞) for some δ > 0. Let h ∈ R and Σ + h := (h, +∞) × Ω. Let c = 0 and φ 1 (s, x, y), φ 2 (s, x, y) be two bounded and globally C 1,µ Σ + h functions (for some µ > 0) such that

             L φ 1 + g(x, y, φ 1 ) ≥ 0 in D (Σ + h ), L φ 2 + g(x, y, φ 2 ) ≤ 0 in D (Σ + h ), ν • A [ẽ(φ 1 s -φ 2 s ) + ∇ x,y (φ 1 -φ 2 )] ≤ 0 on [h, +∞) × ∂Ω, lim s 0 →+∞ sup {s≥s 0 , (x,y)∈Ω} [φ 1 (s, x, y) -φ 2 (s, x, y)] ≤ 0, (2.11)
where L is the same operator as in Lemma 2.3.

If φ 2 ≥ 1 -δ in Σ + h and φ 1 (h, x, y) ≤ φ 2 (h, x, y) for all (x, y) ∈ Ω, then

φ 1 ≤ φ 2 in Σ + h .

Case of a "combustion" nonlinearity

This section is devoted to prove Theorem 1.8, where the nonlinearity f satisfies the assumptions (1.5-1.6) and (1.7).

Proof of formula (1.15)

Having a prefixed unit direction e ∈ R d , and since the coefficients A and q of problem (1.14) satisfy the assumptions (1.3) and (1.4), it follows, from Theorem 1.5, that there exists a unique pulsating travelling front (c(e), u) (u is unique up to a translation in the time variable) which solves problem (1.14). Moreover, ∂ t u > 0 in R × Ω. We will complete the proof of (1.15) via two steps.

Step 1. After the discussions done in the section 2, the existence of a classical solution (c(e), u), satisfying (1.14), implies the existence of a globally φ(-∞, ., .) = 0, and φ(+∞, ., .) = 1 uniformly in (x, y) ∈ Ω, (3.1) where L is the operator defined in (2.2) for c = c(e). We also recall that the two functions u and φ satisfy the relation u(t, x, y) = φ(x • e + c(e)t, x, y), (t, x, y) ∈ R × Ω. One has ∂ s φ > 0 in R × Ω and this is equivalent to say that the function u = u(t, x, y) is increasing in t, since c(e) > 0.

C 1 (R × Ω) function φ(s, x, y) satisfying 0 ≤ φ ≤ 1 in R × Ω, with        φ is L-periodic with respect to x, Lφ(s, x, y) + f (x, y, φ) = 0 in D (R × Ω), ν • A(∇ x,y φ + ẽφ s ) = 0 in R × ∂Ω,
Together with the facts in section 2.1, one gets that the function φ ∈ E. Furthermore, (3.1) yields that

∀ s ∈ R, ∀(x, y) ∈ Ω, c(e) = R φ(s, x, y), ( 3

Rϕ(s, x, y).

Then, there exists a function ψ = ψ(s, x, y) ∈ E such that c(e) > sup (s,x,y)∈R×Ω Rψ (s, x, y).

Since the function ψ ∈ E, one then has ψ s (s, x, y) > 0 for all (s, x, y) ∈ R×Ω.

This yields that

Lψ(s, x, y) + f (x, y, ψ) < 0 in R × Ω, (3.4) where L is the operator defined in (2.2) for c = c(e).

Notice that the later holds for each function of the type

ψ τ (s, x, y) := ψ(s + τ, x, y)
because of the invariance of (3.4) with respect to s and because the advection field q and the diffusion matrix A depend on the variables (x, y) only. That is

Lψ τ (s, x, y) + f (x, y, ψ τ ) < 0 in R × Ω. (3.5)
Step 2. In order to draw a contradiction, we are going to slide the function ψ with respect to φ. From the limiting conditions satisfied by these two functions, there exists a real number B > 0 such that φ(s, x, y) ≤ θ for all s ≤ -B, (x, y) ∈ Ω, ψ(s, x, y) ≥ 1 -ρ for all s ≥ B, (x, y) ∈ Ω, and φ(B, x, y) ≥ 1 -ρ for all (x, y) ∈ Ω, (3.6) where θ and ρ are the values that appear in the conditions (1.7) satisfied by the "combustion" nonlinearity f. Taking τ ≥ 2B, and since ψ is increasing with respect to s, one gets that φ(-B, x, y) ≤ ψ τ (-B, x, y) for all (x, y) ∈ Ω and

ψ τ ≥ 1 -ρ in Σ + -B . It follows from Lemma 2.3 (take δ = θ, h = -B, φ 1 = φ, and φ 2 = ψ τ ) that φ ≤ ψ τ in Σ - -B . Moreover, Lemma 2.5 (take δ = ρ, h = -B, φ 1 = φ, and φ 2 = ψ τ ) implies that φ ≤ ψ τ in Σ + -B . Consequently, φ ≤ ψ τ in R × Ω for all τ ≥ 2B.
Let us now decrease τ and set

τ * = inf{τ ∈ R, φ ≤ ψ τ in R × Ω }.
First one notes that τ * ≤ 2B. On the other hand, the limiting conditions ψ(-∞, ., .) = 0 and φ(+∞, ., .) = 1 imply that τ * is finite. By continuity, φ ≤ ψ τ * in R × Ω. Two cases may occur according to the value of sup Owing to (3.6) and to the above inequality, it follows that

[-B,B]×Ω φ -ψ τ * .
ψ τ (B, x, y) ≥ 1 -ρ in Ω.
Moreover, since the function ψ is increasing in s, one gets that ψ τ ≥ 1 -ρ in Σ + B . Lemma 2.5, applied to φ and ψ τ in Σ + B , yields that φ(s, x, y) ≤ ψ τ (s, x, y) for all s ≥ B, (x, y) ∈ Ω.

As a consequence, one obtains φ ≤ ψ τ in R × Ω, and that contradicts the minimality of τ * . Therefore, case 1 is ruled out. case 2: suppose that sup

[-B,B]×Ω φ -ψ τ * = 0.
Then, there exists a sequence of points (

s n , x n , y n ) in [-B, B] × Ω such that φ(s n , x n , y n ) -ψ τ (s n , x n , y n ) → 0 as n → +∞.
Due to the L-periodicity of the functions φ and ψ, one can assume that (x n , y n ) ∈ C. Consequently, one can assume, up to extraction of a subsequence, that (s n , x n , y n ) → (s, x, ȳ) ∈ [-B, B] × C as n → +∞. By continuity, one gets φ(s, x, ȳ) = ψ τ * (s, x, ȳ).

We return now to the variables (t, x, y). Let

z(t, x, y) = φ(x • e + c(e) t, x, y) -ψ(x • e + c(e) t + τ * , x, y) = u(t, x, y) -ψ(x • e + c(e) t + τ * , x, y) for all (t, x, y) ∈ R × Ω.
Since the functions φ and ψ are in E, it follows that the function z is globally C 1 (R × Ω) and it satisfies

∀ (t, x, y) ∈ R × Ω, ∇ x,y • (A∇z)(t, x, y) = F [φ](s, x, y) -F [ψ τ * ](s, x, y),
where s = x•e+c(e)t. Thus, ∇ x,y •(A∇z) ∈ C(R×Ω). Moreover, the function z is non positive and it vanishes at the point ((s -x • e)/c(e), x, ȳ). It satisfies the boundary condition ν • (A∇z) = 0 on R × ∂Ω. Furthermore, it follows, from (3.2) and (3.4), that

∂ t z -∇ x,y • (A∇z) + q(x, y) • ∇ x,y z ≤ f (x, y, φ) -f (x, y, ψ τ * ).
However, the function f is globally Lipschitz-continuous in Ω × R; hence, there exists a bounded function b(t, x, y) such that

∂ t z -∇ x,y • (A∇z) + q(x, y) • ∇ x,y z + b(t, x, y) z ≤ 0 in R × Ω, with z(t, x, y) ≤ 0 for all (t, x, y) ∈ R × Ω.
Applying the strong parabolic maximum principle and Hopf lemma, one gets that z(t, x, y) = 0 for all t ≤ (s -x • e)/c(e) and for all (x, y) ∈ Ω. On the other hand, it follows from the definition of z and from the L-periodicity of the functions φ and ψ that z(t, x, y) = 0 for all (t, x, y) ∈ R×Ω. Consequently, φ(s, x, y) = ψ τ * (s, x, y) = ψ(s + τ * , x, y) for all (s, x, y) ∈ R × Ω.

Referring to the equations (3.3) and (3.5), one gets a contradiction. Thus, case 2 is ruled out too, and that completes the proof of the formula (1.15). Remark 3.1 (The uniqueness, up to a shift, of the minimizer in (1.15)) If ψ ∈ E is a minimizer in (1.15). The above arguments imply that case 2 necessarily occurs, and that ψ is equal to a shift of φ. In other words, the minimum in (1.15) is realized by and only by the shifts of φ.

3.2 Proof of formula (1.16) In this subsection, we are going to prove the "max-min" formula of the speed of propagation c(e) whenever the nonlinearity f is of the "combustion" type. The tools and techniques which one uses here are similar to those used in the previous subsection. However, we are going to sketch the proof of formula (1.16) for the sake of completeness.

As it was justified in the previous subsection, one easily gets that

c(e) ≤ sup ϕ∈E inf (s,x,y)∈R×Ω Rϕ(s, x, y) and ∀ (s, x, y) ∈ R × Ω, c (e) = Rφ(s, x, y), where φ(s, x, y)
= u ((s -x • e)/c(e), x, y) , for all (s, x, y) ∈ R × Ω, and u = u(t, x, y) is the unique (up to a translation in t) pulsating travelling front solving problem (1.14) and propagating in the speed c(e). We recall that the function φ ∈ E (see section 2). It follows that the function φ satisfies the following

           φ is L-periodic with respect to x, φ is increasing in s ∈ R, Lφ(s, x, y) + f (x, y, φ) = 0 in R × Ω, ν • A(∇ x,y φ + ẽφ s ) = 0 in R × ∂Ω, φ(-∞, .
, .) = 0, and φ(+∞, ., .) = 1 uniformly in (x, y) ∈ Ω,

where L is the operator defined in (2.2) for c = c(e).

Notice that the later holds also for each function of the type

φ τ (s, x, y) := φ(s + τ, x, y)
because of the invariance of (3.8) with respect to s and because the advection field q and the diffusion matrix A depend on the variables (x, y) only.

To complete the proof of formula (1.16), we assume that c(e) < sup ϕ∈E inf (s,x,y)∈R×Ω Rϕ(s, x, y).

Hence, there exists ψ ∈ E such that c(e) < Rψ (s, x, y), for all (s, x, y) ∈ R × Ω.

Since the function ψ ∈ E, one then has ψ s (s, x, y) > 0 for all (s, x, y) ∈ R×Ω.

This yields that

Lψ(s, x, y)

+ f (x, y, ψ) > 0 in R × Ω. (3.8)
To get a contradiction, we are going to slide the function φ with respect to ψ. In fact, the limiting conditions satisfied by ψ and φ, which are elements of E, yield that there exists a real positive number B such that ψ(s, x, y) ≤ θ for all s ≤ -B, (x, y) ∈ Ω, φ(s, x, y) ≥ 1 -ρ for all s ≥ B, (x, y) ∈ Ω, and ψ(B, x, y) ≥ 1 -ρ for all (x, y) ∈ Ω, (

where θ and ρ are the values appearing in the conditions (1.7) satisfied by the nonlinearity f. Having τ ≥ 2B, one applies Lemma 2.3 (taking δ = θ, h = -B, φ 1 = ψ, and φ 2 = φ τ ) and Lemma 2.5 (taking δ = ρ, h = -B, φ 1 = ψ, and φ 2 = φ τ ) to the functions φ τ and ψ, over the domains Σ - -B and Σ + -B respectively, to get that ψ ≤ φ τ in Σ - -B and ψ ≤ φ τ in Σ + -B . Consequently, one can conclude that

∀ τ ≥ 2B, ψ ≤ φ τ in R × Ω.
Let us now decrease τ and set

τ * = inf{τ ∈ R, ψ ≤ φ τ in R × Ω }.
It follows, from the limiting conditions ψ(+∞, ., .) = 1 and φ(-∞, ., .) = 0, that τ * is finite. By continuity, we have ψ ≤ φ τ * . In this situation, two cases may occur. Namely, case A: sup

[-B,B]×Ω ψ -φ τ * < 0, or case B: sup [-B,B]×Ω ψ -φ τ * = 0.
Imitating the ideas and the skills used in case 1 and case 2 during the proof of formula (1.15) 4 Case of "ZFK" or "KPP" nonlinearities: proof of formula (1.17) This section is devoted to the proof of Theorem 1.9. We assume that the nonlinear source f is of "ZFK" type. Remember that this case includes the class of "KPP" nonlinearities. Namely, f = f (x, y, u) is a nonlinearity satisfying (1.5-1.6) and (1.8). We will divide the proof of formula (1.17) into 3 steps:

Step 1. Under the assumptions (1.2), (1.3), and (1.4) on the domain Ω, the diffusion matrix A, and the advection field q respectively, and having a nonlinearity f satisfying the above assumptions, Theorem 1.6 yields that for c = c * Ω,A,q,f (e), there exists a solution u * = u * (t, x, y) of (1.14) such that u * t (t, x, y) > 0 for all (t, x, y) ∈ R × Ω. In other words, the function φ * defined by φ * (s, x, y) = u * s -x • e c * (e) , x, y , (s, x, y) ∈ R × Ω is increasing in s ∈ R. Owing to section 2, φ * satisfies

F [φ * ] + q • ∇ x,y φ * + (q • ẽ -c * (e))φ * s , +f (x, y, φ * ) = 0 in R × Ω (4.
1) together with boundary and periodicity conditions

φ * is L-periodic with respect to x, ν • A(∇ x,y φ * + ẽφ * s ) = 0 on R × Ω. (4.2)
Moreover, (4.1) implies that F [ϕ](s, x, y) + q • ∇ x,y ϕ + f (x, y, ϕ) ∂ s φ(s, x, y) + q(x, y) • ẽ.

∀ (s, x, y) ∈ R × Ω, c * (e) = F [φ * ](s, x, y) + q • ∇ x,y φ * (s, x, y) + f (x, y, φ * ) ∂ s φ * (s, x, y) + q(x, y) • ẽ = Rφ * (s, x, y),
In order to prove equality, we argue by contradiction. Assuming that the above inequality is strict, one can find δ > 0 such that

c * (e) -δ > inf ϕ∈E sup (s,x,y)∈R×Ω F [ϕ](s, x, y) + q • ∇ x,y ϕ + f (x, y, ϕ)
∂ s ϕ(s, x, y) + q(x, y) • ẽ.

(4.4) To draw a contradiction, we are going to approach the "ZFK" nonlinearity f by a sequence of "combustion" nonlinearities (f θ ) θ and the minimal speed of propagation by the sequence of speeds (c θ ) θ corresponding to the functions (f θ ) θ . The details will appear in the next step.

Step 2. Let χ be a C 1 (R) function such that 0 ≤ χ ≤ 1 in R, χ(u) = 0 for all u ≤ 1, 0 < χ(u) < 1 for all u ∈ (1, 2) and χ(u) = 1 for all u ≥ 2. Assume moreover that χ is non-decreasing in R. For all θ ∈ (0, 1/2), let χ θ be the function defined by

∀ u ∈ R, χ θ (u) = χ(u/θ).
The function χ θ is such that 0 ≤ χ θ ≤ 1, 0 < χ θ < 1 in (-∞, θ], 0 < χ θ < 1 in (θ, 2θ) and χ θ = 1 in [2θ, +∞). Furthermore, the functions χ θ are nonincreasing with respect to θ, namely,

χ θ 1 ≥ χ θ 2 if 0 < θ 1 ≤ θ 2 < 1/2.
We set

f θ (x, y, u) = f (x, y, u) χ θ (u) for all (x, y, u) ∈ Ω × R.
In other words, we cut off the source term f near u = 0. For each θ ∈ (0, 1/2), the function f θ is a nonlinearity of "combustion" type that satisfies (1.5-1.6) and (1.7) with the ignition temperature θ. Therefore, Theorem 1.5 yields that the existence of a classical solution (c θ , u θ ) of (1.14) with the nonlinearity f θ . Furthermore, the function u θ is increasing in t and unique up to translation in t and the speed c θ is unique and positive.

It was proved, through Lemma 6.1 and Lemma 6.2 in Berestycki, Hamel [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF], that the speeds c θ are non-increasing with respect to θ and c θ c * (e) as θ 0.

Consider a sequence θ n 0. Then, there exists n 0 ∈ N such that c θn ≥ c * (e) -δ for all n ≥ n 0 (or equivalently θ n ≤ θ n 0 ).

In what follows, we fix θ such that θ < θ n 0 . One consequently gets c θ ≥ c * (e) -δ. On the other hand, it follows, from the construction of f θ , that f ≥ f θ in Ω × R. Together with (4.4), one obtains

c θ > inf ϕ∈E sup (s,x,y)∈R×Ω F [ϕ](s, x, y) + q • ∇ x,y ϕ + f θ (x, y, ϕ) ∂ s ϕ(s, x, y) + q • ẽ. (4.5)
Thus, there exists a function ψ ∈ E such that

c θ > F [ψ](s, x, y) + q • ∇ x,y ψ(s, x, y) + f θ (x, y, ψ) ∂ s ψ(s, x, y) + q(x, y) • ẽ. (4.6)
However, ψ s (s, x, y) > 0 for all (s, x, y) ∈ R × Ω. Thus, the inequality (4.6) can be rewritten as

Lψ(s, x, y) + f θ (x, y, ψ) < 0 in R × Ω, (4.7)
with ψ ∈ E and L is the operator defined in (2.2) for c = c θ .

For each τ ∈ R, we define the function ψ τ by ψ τ (s, x, y) = ψ(s + τ, x, y) for all (s, x, y) ∈ R × Ω.

Since the coefficients of L are independent of s, the later inequality also holds for all functions ψ τ with τ ∈ R. That is,

Lψ τ (s, x, y) + f θ (x, y, ψ τ ) < 0 in R × Ω. ( 4

.8)

Step 3. For the fixed θ (in step 2), the function f θ is a "combustion" nonlinearity whose ignition temperature is θ. There corresponds a solution (c θ , u θ ) of (1.14) within the nonlinear source f θ . We define φ θ by φ θ (s, x, y) = u θ s -x • e c θ , x, y , for all (s, x, y) ∈ R × Ω.

Referring to section 2, one knows that φ θ ∈ E and thus it satisfies the following equation

Lφ θ (s, x, y) + f θ (x, y, φ θ ) = 0 in R × Ω. (4.9)
Now, the situation is exactly the same as that in step 2 of the proof of formula (1.15) because the nonlinearity f θ is of "combustion" type. The little difference is that f (in step 2 of the proof of formula (1.15)) is replaced here by f θ , and the function φ of equation (3.3) is replaced by the function φ θ of (4.9). Thus, following the arguments of subsection 3.1 and using the same tools of "step 2" as in the proof of formula (1.15), one gets that the (4.4) is impossible and that completes the proof of formula (1.17).

Remark 4.1 We found that one can use another argument (details are below) different from the sliding method in order to prove the min -max formulae for the speeds of propagation whenever f is a homogenous (i.e f = f (u)) nonlinearity of "combustion" or "ZFK"type and Ω = R N . Meanwhile, the sliding method, that we used in the proofs of formulae (1.15) and (1.17), is a unified argument that works in the general heterogenous periodic framework.

Another proof of formulae (1.15) and (1.17) in a particular framework: Here, we assume that f = f (u), and Ω = R N . Following the same procedure of "step 1" in the previous proof, one gets the inequality c * (e) ≥ inf 1 The case where f is of "combustion" type follows in a similar way.

Let 0 ≤ u 0 (x, y) ≤ 1 be a function in C(R N ) such that u 0 (x, y) → 0 as x•e → -∞, and u 0 (x, y) → 1 as x•e → +∞, uniformly in y and all directions of R d which are orthogonal to e. Let u be a pulsating front propagating in the direction of -e with the speed c θ and solving the initial data problem      u t = ∇ • (A(x, y)∇u) + q(x, y) • ∇u + f θ (u), t > 0, (x, y) ∈ Ω, u(0, x, y) = u 0 (x, y), ν • A ∇u(t, x, y) = 0, t ∈ R, (x, y) ∈ ∂Ω. Having f θ (u) as a "combustion" nonlinearity, it follows from J. Xin [START_REF] Xin | Front propagation in heterogeneous media[END_REF] (Theorem 3.5) and Weinberger [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF], that ∀r > 0, lim This means that the speed of propagation c θ corresponding to (1.14) is equal to the spreading speed in the direction of -e when the nonlinearity is of "combustion" type and the initial data u 0 satisfies the above conditions.

For all (t, x, y) ∈ [0, +∞) × Ω, let w(t, x, y) = ũ(t, x, y) -u(t, x, y). It follows, from (4.11) and (4.12), that      w t -∇ • (A(x, y)∇w) -q(x, y) • ∇w + bw > 0, t > 0, (x, y) ∈ Ω, ∀(x, y) ∈ Ω, w(0, x, y) ≥ 0, ν • A ∇w(t, x, y) = 0, t ∈ R, (x, y) ∈ ∂Ω, (4.14) for some b = b(t, x, y) ∈ C(R × Ω). The parabolic maximum principle implies that w ≥ 0 in [0, +∞) × Ω. In other words, ∀(t, x, y) ∈ [0, +∞) × Ω, u(t, x, y) ≤ ũ(t, x, y).

However, for all c > d, Referring to (4.13), one concludes that d ≥ c θ which is impossible (d < c θ ). Therefore, our assumption that c * (e) > inf ϕ∈E sup (s,x,y)∈R×Ω Rϕ(s, x, y) is false and that completes the proof of formula (1.17) in the case where f = f (u) and Ω = R N .

1 Introduction and main results 2 1 . 1 A 7 2

 117 description of the periodic framework . . . . . . . . . . . . . 2 1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . Main tools: change of variables and maximum principles 10

  14) is given by c(e) = min ϕ∈E sup (s,x,y)∈R×Ω R ϕ(s, x, y), (1.15) c(e) = max ϕ∈E inf (s,x,y)∈R×Ω R ϕ(s, x, y). (1.16)

. 4 )Remark 2 . 1 Remark 2 . 2 (

 42122 Moreover, since u(t, x, y) → 0 as x•e → -∞ and u(t, x, y) → 1 as x•e → +∞ locally in t and uniformly in y and in the directions of R d which are orthogonal to e, and since φ is L-periodic with respect to x, the change of variables s = x • e + ct guarantees that φ(-∞, ., .) = 0 and φ(+∞, ., .) = 1 uniformly in (x, y) ∈ Ω.(2.5)Therefore, one can conclude that φ ∈ E. It is now clear that a max-min formula of the type (1.16) can not hold for the minimal speed c * (e) > 0 in the "ZFK" or the "KPP" case. Indeed, for each speed c ≥ c * (e), there is a solution (c, u) of (1.14) such that u t > 0, which gives birth to a function φ = φ(s, x, y) under the change of variables (2.1). Owing to the above discussions the function φ ∈ E and it satisfies c = Rφ(s, x, y) for all (s, x, y) ∈ R × Ω. Therefore sup ϕ∈E inf (s,x,y)∈R×Ω R ϕ(s, x, y) ≥ c. Since one can choose any c ≥ c * (e), one concludes that sup ϕ∈E inf (s,x,y)∈R×Ω R ϕ(s, x, y) = +∞ in the "ZFK" or the "KPP" case. The same formulae, but over a subset of E) If the restriction of the nonlinear source

case 1 :

 1 suppose that sup [-B,B]×Ω φ -ψ τ * < 0.Since the functions ψ and φ are globally C 1 (R × Ω) there exists η > 0 such that the above inequality holds for all τ ∈ [τ * -η, τ * ]. Choosing any τ in the interval [τ * -η, τ * ], and applying Lemma 2.3 to the functions ψ τ and φ, one gets that φ(s, x, y) ≤ ψ τ (s, x, y) for all s ≤ -B, (x, y) ∈ Ω, together with the inequality φ(s, x, y) < ψ τ (s, x, y) for all s ∈ [-B, B], and for all (x, y) ∈ Ω.

  ϕ∈E sup (s,x,y)∈R×Ω Rϕ(s, x, y). Now, to prove the other sense of inequality, we assume that c * (e) > inf ϕ∈E sup (s,x,y)∈R×Ω Rϕ(s, x, y), and we assume that f is of "ZFK" type 1 . Then, as it was explained in "step 2" of the previous proof, one can find ψ ∈ E, δ > 0, θ > 0, and d > 0 such that c * (e) -δ < d < c θ < c * (e) where ∀(s, x, y) ∈ R × Ω, d > c * (e) -δ > Rψ(s, x, y), and f θ (u) = f (u) χ θ (u) ≤ f (u) for all u ∈ R is of "combustion" type (c θ is the speed of propagation, in the direction of -e, of pulsating travelling fronts solving (1.14) with the nonlinearity f θ and the domain Ω = R N ).Hence, for all (t, x, y) ∈ R × R N , d > F [ψ](s, x, y) + q • ∇ x,y ψ(s, x, y) + f θ (ψ) ∂ s ψ(s, x, y) + q(x, y) • ẽ.(4.10)Let ũ(t, x, y) = ψ(x•e+dt, x, y). As it was explained in section 2• (A(x, y)∇ũ) -q(x, y) • ∇ũ -f θ (ũ) > 0, t ∈ R, (x, y) ∈ Ω, ν • A ∇ũ(t, x, y) = 0, t ∈ R, (x, y) ∈ ∂Ω, ∀ k ∈ d i=1L i Z, ∀ (t, x, y) ∈ R × Ω, ũ(t + k • e d , x, y) = ũ(t, x + k, y),0 ≤ ũ ≤ 1.

  t→+∞ sup |x|≤r u(t, x -cte, y) = 0 uniformly in y, for every c > c θ , and lim t→+∞ inf |x|≤r u(t, x -cte, y) = 1 uniformly in y, for every c < c θ . (4.13)

  lim t→+∞ ũ(t, x -cte, y) = lim t→+∞ ψ(x • e + (d -c)t, x -cte, y) = 0 locally in x and uniformly in y (since ψ ∈ E). Consequently, ∀c > d, ∀r > 0, lim t→+∞ sup |x|≤r u(t, x -cte, y) = 0 uniformly in y.

  Rφ is the function defined in Theorem 1.8. In other words, the L-periodic (with respect to x) function Rφ is constant over R × Ω and it is equal to c(e).

		.2)
	and
	Lφ(s, x, y) + f (x, y, φ) = 0,	(3.3)
	where It follows, from (3.2) and from the above explanations, that
	c(e) ≥ inf

ϕ∈E sup (s,x,y)∈R×Ω

Rϕ(s, x, y).

To complete the proof of formula (1.15), we assume that c(e) > inf ϕ∈E sup (s,x,y)∈R×Ω

  Similar to what we have already mentioned in Remark 3.1, if ψ ∈ E is a maximizer in (1.16), then the above arguments yield that case B necessarily occurs, and that ψ is equal to a shift of φ. One then concludes that the maximum in(1.16) is realized by, and only by, the shifts of φ.

	Therefore, the assumption that
	c(e) < sup	inf	Rϕ(s, x, y)
	ϕ∈E	(s,x,y)∈R×Ω
	is false, and that completes the proof of formula (1.16).
	Remark 3.2 (The uniqueness, up to a shift, of the maximizer in (1.16))
	, one gets that case A (owing to minimality of τ * ) and case
	B (owing to (3.7) and (3.8)) are ruled out.

Acknowledgements

I am deeply grateful to Professor François Hamel for his valuable directions and advices during the preparation of this paper. I would like also to thank Professor Andrej Zlatoš for his important comments, given while reading a preprint of this work, which allowed me to consider a wider family of "ZFK" nonlinearities in Theorem 1.9 and thus get a more general result.