Food structure: a key role in nutrition?

• Foods are not random mixtures of compounds but fine materials with multi-scale organisations What is innovation ?

• Lever for the competitiveness, the profitability and the sustainability of companies • Real benefits are very complex to explain to consumers (who will give relevant advices?) • Possible effects on social divide (increase of inegality) regarding access to healthy foods (price + access to information) • Take into account the legislation (ex European rules for new foods (a brake for innovation but set up to avoid drifts in labels …) • Risk evaluation: who will make it?

Innovation for the food industry

• Don't forget traditional and/or less processed foods

•

  Food transition: more animal proteins, more fats, less crude plant products • Ageing: 22% > 65y in 2025 (F) • Environment: air, water, food polluants • Food contaminants • Sedentarity  How supply an optimal nutrition to consumers ? … A major stake for health public policies … a challenge and an opportunity for the industry  Prevalence of chronic non-contagious diseases Specific nutritionnal needs 2 2 2 C. Genot Food structure: a key role in nutrition and a driver for innovation Context : trends for Human health status in developped countries + Individual factors • Physiology • Genetics • Way of life

Bioavailability

  Oral challenge with native OVA C. Genot Food structure: a key role in nutrition and a driver for innovation Sensitization by small aggregates lead to lower allergy symptoms with native ovalbumin Small OVA aggregates exhibit lower capacity to sensitize the mice. acids, 2-monoacylglycerols, lyso-phospholipids, phospholipids, oxilipins, sterols, vitamins A, D, E, K… Amounts bioavailable for the target organ or function Lipid structures influence fatty acid bioavailability and metabolism Oxidation • Lipid molecular structures : sn-1,3 vs sn-2, fatty acid carying molecules (TAG, PL, …) • Physical state (solid vs liquid) • Dispersion state • Interface & matrix composition and structure Excretion Eating is not only nutrition • The diet is composed of different selected foods ingested in specified order at specific times (meals vs snacking) during the day• With food, we also eat Food structure: a key role in nutrition and a driver for innovationSustainabilityFood structures influence food acceptance, release of nutrients, their bioavailability and metabolism 8

  • A new offer to answer the evolution of the demand • Added value for the companies and for the consumers • A link beween future and present • … 2005 Innovation = the implementation of a new or significantly improved product (good or service), or process, a new marketing method, or a new organisational method in business practices, workplace organisation or external relations.

Opportunities•

  Demand for new products in link with new modes of consumption • Technological evolution (process) Threats /constraints• Regulatory requirements • Economic tensions Major challenges : • sustainable development • merging food and health --> functional food, new physical processes, new products with high demand C. Genot Food structure: a key role in nutrition and a driver for innovation Food structures and innovation: 3 levels • For research: new experimental facilities • For industry: new products ; new or revisited processes ; tools to predict and control • For consumers: new structure-designed foods with proven health benefitsFood structures -health and innovation Conclusive remarks • The effects on health of a food /a diet: a combination of the effects of its nutrients, as affected by the molecular and supramolecular structures, and determined by the preparation methods. • Take into account all aspects of what make/define food and food quality: Include the oral phase, sensorial perception, and consumer behavior (acceptability, choice's drivers) • Define the target(s) ; general population or specific targets ? (ill persons, elder, infants, preterms, …) • Not one consumer but different consumers depending on socioeconomical, cultural, physiological aspects (age, health state) • Based on proven facts (scientific publications) and not on opinions, self-designed experts, industry lobbies, … C. Genot Food structure: a key role in nutrition and a driver for innovation

  Food structure: a key role in nutrition and a driver for innovation

C. Genot Food structure: a key role in nutrition and a driver for innovation micro meso macro

Objective : model mechanical behavior and destructuration of foods during mastication, formation of the food bolus and its fate during digestion

BIA-Team MC2

Innovation for research and industry:

Food structures and the digital revolution…

Multicriteria optimisation of food structures and nutrient fate

In the future

Multiscale modeling

Innovate in molecular/supramolecular food structures by using new ressources


Maksym Loginov

Psm -Stlo

PSM -STLO recent works on filtration separation

  recent works on filtration-separation" • method we use • things we do • originality of our approach In 2016-2019 • analytical centrifugal consolidation • analytical centrifugal ultrafiltration • cross-flow filtration • filtration-consolidation theory


Tatiana Malanina Swelling

Ditions Stlopendays

published or not. The documents may come

Swelling of dry heated protein microparticles in different physicochemical conditions

Swelling of dry heated protein microparticles in different physicochemical conditions

Tatiana MALANINA-Research engineer Interactions-Structures-Functionalities -Marie-Hélène FAMELART

Socio-economic context

• Innovation and new product development.

• Using a 100% dairy ingredient as a high-viscosity additive in food matrices.

Patent [1].

Scientific context

These protein microparticles are able to swell when placed in an aqueous phase: at pH 6.5, they entrap a huge amount of water and lead to high-viscosity suspensions

What are the effects of different physicochemical conditions on the ability of the PMI to swell, entrap water and give high-viscosity values?

• Find the best physicochemical conditions to express a high-functionality of PMI amongst:  pH (7.0-4.5)  Addition of sodium or calcium chloride (0-1 M)  Addition of caseins (10-60 g/kg)

Expected Results :

• Prepare dairy model food matrices such as yoghurt, cheese, dessert, sauce… with these PMI with the aim of replacing food additives and understanding their texture and structure.

POSTER NUMBER 20

Tatiana MALANINA (mv.tatiana@hotmail.com)


Olivier Harle

Olivier Harle Optimizing

Stlopendays

Optimizing lactic fermentation of soy juice

teaching and research institutions in France or abroad, or from public or private research centers.

Proteins

How to optimize lactic fermentation of soy juice?

• Understand the interactions between lactic acid bacteria, their impact on fermentation and on product composition Genomic

  based proteins = 40 % of our diet in developed countries

Figure 1 .

 1 Figure 1. Present and projected environmental pressures. (Springmann et al. 2018)


Louise Chevalier

François Jacques Labonne

Xavier Dechaume-Moncharmont

François Xavier

A dynamic game theoretical model predicts variance in choosiness when mate availability fluctuates

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

A dynamic game theoretical model predicts variance in choosiness when mate availability fluctuates

Louise Chevalier(1), Jacques Labonne(2), (1) UMR 1224 Evolution of mate choosiness is a central topic in the Darwinian framework and has stimulated many experimental approaches in sexual selection. The optimal choosiness may depend on the quality of both the chooser and the potential partner, but it should also account for the complexity of intra-sexual competition for mate, mutual choice and fluctuation of partner availability. Most existing models, however, have focused on single-sex discrimination, treating mate choice as an optimization problem for either males or females alone. Johnstone (1997) thus developed a game theoretical model of mutual mate choice to calculate the Evolutionary Stable Strategy (ESS) of choosiness flexibility when individuals pair only once per breeding season under constant and even operational sex ratio (OSR). Yet, OSR is rarely even in natural populations. On the contrary, it fluctuates and depends on adult sex ratio (ASR) and the duration of the refractory period (i.e. time to become available for another reproduction). The results of Johnstone's seminal work have yet to be generalized to this larger picture that encompasses different options to multiple mating for both sexes. In the present work, we develop his model by integrating biased ASR and variable refractory period, while still considering the dynamic mutual mate choice framework under scramble competition and the variation in quality for each sex.

Our results evidence for a predominant role of refractory period (via multiple mating) over ASR, controlling the choosiness of both sexes, different combinations of periods between sexes producing different outcomes in ESS of choosiness. When males and females mate more than once, variance in choosiness is substantial and is correlated to variance in quality. Additionally, patterns of positive homogamy demonstrated by Johnstone hold in many cases, although with decreasing strength with the biased ASR. This theoretical investigation confirms that evolution of optimal flexible choosiness leads to positive assortative mating and explains maintenance of variance in choosiness within populations. Keyword: Game theory, Dynamic Programming, Mate choice, Mutual Choice, Choosiness, Evolutionary Stable Strategy.

Type of presentation: Oral communication Presenting author : Louise Chevalier

	ECOBIOP, INRA -UNIV PAU & PAYS ADOUR, 64310 Saint-Pée sur Nivelle,
	France; (2) UMR 1224 ECOBIOP, INRA -UNIV PAU & PAYS ADOUR, 64310 Saint-Pée sur
	Nivelle, France; (3) UMR CNRS 6282 Biogéosciences, University Bourgogne Franche-Comté,
	21000 Dijon, France


Claude Genot

Xiaochen

come

Steady-state and time-resolved fluorescence to evaluate protein oxidation in oxidizing model food emulsion


Claude Genot

Xiaochen Han

Soumia Charaf

Valérie Beaumal

Lucie Ribourg

Michèle Dalgalarrondo

Fate of adsorbed and non-adsorbed proteins in oxidizing emulsions studied by steady-state and time-resolved fluorescence

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

Rapeseed oil

  at 37 °C (h) Fluorescence (F/F t0 )Oxygen consumption (mmol 02/kg oil) Conjugated dienes (mmol eq. Hpx/kg oil)Effect of ageing of the emulsion at 37°C on mean lifetimeEffect protein location l exc/em = 290 nm / 330 nm Sample t1 (ns) a 1 t2 (ns) a 2 <T> (ns) n BLG 2.5 g/L 1.10 ± 0.14 73.9 ± 12.6 2.41 ± 0.39 25.6 ± 12.1 1.79 ± 0.12 3Aqueous phase 1.34 ± 0.11 91.4 ± 3.4 3.89 ± 0.73 8.6 ± 3


Florence Gondret

Bénédicte Lebret

Marie-Hélène Perruchot

Dolores Batonon

Yves Mercier

"Flash" dietary methionine

14h15

  Swine production in Spain: outlook and evolutions. Estelle ANTOINE, Studies engineer, Economy department, IFIP. 15h00 Quality and prices of grains and meals supplies in the European Union during fi rst half of 2017. Alexandre BOY, Analyst, AGRITEL. 15h45 Reducing the use of antibiotics in swine production in Europe : Which reality beyond the fi gures? Focus on the French strategy. Anne HEMONIC, Doctor of Veterinary Medicine (DVM), IFIP. 16h30 Coffee break. 16h50 Quality of digestion in swine nutrition: Reconsider the use of NSPases in swine feed. Pierre COZANNET, Enzymes Research Manager, Adisseo. 17h35 Flash dietary methionine supply as an innovative way to improve pork quality traits. Florence GONDRET, Scientist, UMR Pegase, Team "Physiology and metabolisms of growth", INRA. 18h20 Conclusion. Quality and prices of grains and meals supplies in the European Union during fi rst half of 2017. Reducing the use of antibiotics in swine production in Europe : Which reality beyond the fi gures? FROM FEED TO MEAT PRODUCTS: LOOKING FOR QUALITY P le a s e c o n fi r m b e f o r e Ja nu ary 9 th at d e lp h in e .d u m as @a disseo.c om A d i s s e o F r a n c e S. A.S. -10, pl ac e d u G é n é ra l d e Ga ulle -921 6 0 A n t o n y -F r a n c e SWINE CONFERENCE Monday January 30 th 2017


The "Réseau hydrique" network Sites on a water deficit gradient

Thank you for your aeen;on !

Topics••Fuel

  Photo : Pierre Girard, DSF.


domain: sdv shs

text:
Jean-Marc Touzard

Jean-Marc Marc Barbier

Laure Hossard

Opening the TATA-BOX to raise new questions on agroecological transition

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

Introduction

The urgency of ecological challenges not only calls for changing the ways we produce, exchange and consume our food, but also the ways we are doing research and supporting the processes of change in agriculture. The TATA-BOX project is in line with these perspectives by proposing, experimenting and evaluating a participatory approach to support local actors towards agroecological transition (AET). This project has mobilised a multidisciplinary research team, including agronomists, economists, sociologists, geographers, etc. In this book the team reports its common Thanks to Roy Hammond for his help in revising English of this paper.

J.-M. Touzard (*) • J.-M. Barbier • L. Hossard Innovation, INRA, CIRAD, Montpellier SupAgro, Univ Montpellier, Montpellier, France e-mail: jean-marc.touzard@inra.fr; jean-marc.barbier@inra.fr; laure.hossard@inra.fr analytical frameworks, its results and its evaluations, going as far as integrating external critical analysis. This is the aim of our paper, written by researchers at UMR Innovation (Montpellier) who study innovations in agricultural and agro-food systems [START_REF] Faure | Innovation and development in agriculture and food systems, Quae[END_REF] and have discovered the TATA-BOX project by reading the other chapters of this book. Firstly, we will highlight the contributions of the book and then develop three questions that echo our own work on innovations and AET: How to associate agroecological issues with the diversity of practices and projects in a given area? What are the conditions for disseminating or outscaling the TATA-BOX approach? How to better integrate the political dimensions of agroecological transition?

Strengthening the Capacities of Farmers and Researchers

First of all, reading this book generates much interest. The contributions cover different stages of a participatory research project, by crossing several disciplinary viewpoints. Few studies in this area have taken the time to report and analyse their approach with such precision and reflexivity: an updated survey of the issues and principles of AET; a conceptual framework to analyse this transition by combining three systems, usually mobilised separately (farming systems, Socio Ecological Systems, Socio Technical Systems); the proposal of a five-step approach to support the stakeholders of a territory; the operationalization of this approach in two local areas with feedback on the participatory process; a focus on several tools used during the project (surveys, network analysis, graphical tools, etc.); and an evaluation of both local participatory processes and research management.

These contributions refer to recent theoretical and methodological knowledge that fuel the increasing number of participatory research projects dedicated to transitions and innovations in agriculture, particularly in the SAD department of INRA (Barbier and Elzen 2012;Meynard et al. 2012;Prost et al. 2017): the reaffirmation of a systemic framework; the taking into account of the knowledge, the judgments and the interactions of the stakeholders; the articulation of several scales and fields of action; the attention given to pathways more than to the states of the systems; the application of design theory to agriculture; the perspective of "adaptive management", etc. The book is thus a resource to better understand the current research on AETs.

Admittedly, each contribution is not necessarily original and the project remains focused on the co-design phase of an action plan, without going so far as to analyse the actions implemented afterwards. But the combination of these contributions, their confrontations and their reflective dimension offer a precious source of knowledge for all the categories of actors potentially involved in AET: researchers and research managers, farmers, policy makers, local development organizations, etc. The TATA-BOX project thus appears as a real laboratory for the development of participatory research responding to agroecological issues at a local scale. The uncompromising evaluation of the experience (Chap. 11) shows that the project has clearly strengthened the capacity of the stakeholders participating to the local participatory processes. By providing tools, references and ideas, the book may also strengthen the capacity of its readers, and thus contribute to scaling up local initiatives of AET [START_REF] Tourdonnet | Agro-ecological innovation: how to mobilize ecological processes into agricultural systems[END_REF]. But more than that, the interest of the book is also to raise questions that are sometimes evacuated by other researches on AET.

Which Agroecological Transition?

The first question is about the way in which the vision of a sustainable agriculture in a territory can be designed as well as the AET that can lead to it. The TATA-BOX project assumes the objective of favoring a "strong ecological modernization" based on "radical redesign of agricultural systems". This option gives a normative dimension to the participatory process promoted by the researchers, even if it relies on maieutic principles. The objective is indeed to help stakeholders co-design -or "discover"-through their interactions with researchers (i) a shared diagnosis of the current situation, (ii) a common vision of a territorial agroecological system, and (iii) a pathway for the transition. The implementation of such an approach then calls for several remarks:

The approach is applied in two local areas where agriculture still plays an important role and is already well engaged in AET. Indeed some actors and organizations carry individual and collective initiatives that refer to strong agroecology, for instance the development of organic agriculture. They also seem to have a consistent role in the two territories and in the workshops of the project. But what about territories where agriculture is less present or dominated by conventional agriculture or weak agroecology? Should we then support activist groups, though they be marginalized or in conflict in the territory? Should we instead propose work oriented more towards a weak form of agroecology? Should the approach be integrated in a wider territorial approach that takes more into account the evolution of nonagricultural dynamics, whether economic or ecological? In any case, the TATA-BOX approach deserves to be tested in more contrasted territories in terms of their commitment to agroecological transition.

The proposed approach has the merit of taking into account the plurality of views on agroecology (cf. Chap. 3), the uncertainties (cf. Chap. 6) and the diversity of current practices and projects in local agriculture and food systems. Studies in the Tarn French département (FADN NUTS III, http://ec.europa.eu/agriculture/rica/) show that many farmers hybridise strong and weak agroecology, or integrate strong agroecology into their practices, networks and conventional agriculture organizations (cf. Chaps. 7 and 8). But the challenge of TATA-BOX is ultimately to obtain convergence on a desirable and shared future, and then to propose an action plan. This future certainly includes the coexistence of a diversity of projects and prac-tices, more or less close to strong agroecology, but it remains linked to the search for a compromise that can limit the range of proposals. This methodological choice is shared with many participatory research projects and facilitates reflections on "the" transition. But we can also follow a prospective approach that holds several scenarios and several possible paths [START_REF] Delmotte | From stakeholders narratives to modelling plausible future agricultural systems. Integrated assessment of scenarios for Camargue, southern France[END_REF]. It can maintain comparability of different options, not necessarily dominated by strong agroecology.

In fact, there is a possible contradiction between the need for a strong Ecological Modernization of Agriculture and the ambition to develop participatory approaches. The TATA-BOX approach begins with a diagnosis of the territory and its agriculture, but participation could well result in more than just the capture of knowledge and data from the stakeholders. What happens if they want to deal with "their" problems and propose solutions that are not referring to "strong ecology" or even "weak ecology"? What happens if there is no possible convergence towards a "Territorial Agroecological System"? We agree that AET cannot be promoted without participation, even with more punitive and coercive public policies, but participation does not necessarily create an AET.

Can TATA-BOX Be Used Without Researchers?

Promoting the development of "strong agroecology" through a participatory approach raises a second question: what is the reproducibility of the approach proposed by TATA-BOX and what are the conditions for its adoption by other groups of local stakeholders?

The characteristics of local agriculture, its territory and its more or less "receptive" and innovating actors have already been mentioned. But we must emphasise another condition for the success of TATA-BOX: the very strong involvement of research. More than 40 researchers have been mobilised around the project and their role has been multiple, as Chaps. 10 and 11 show. Researchers are indeed at the initiative of the process, have set up groups and workshops, have strongly contributed to the diagnosis, have supported stakeholders towards the search of consensus, have ensured the follow-up, the evaluation and capitalizing on the project. The researchers who implemented the TATA-BOX project were themselves accompanied by other researchers who supported them in their reflexivity and analysed the impacts on stakeholders. These circumstances are exceptional, and even a luxury for research. But the implementation of such a device may be expensive, timeconsuming, and thus difficult to reproduce. How to ensure that the TATA-BOX approach can be "decoupled" from the networks of actors who created and applied it, and in the first place from the researchers? This question can be addressed by analysing three major contributions of researchers to the project.

An initial contribution of researchers has been the production of references, methods and knowledge on local agriculture, ecosystems and food-chains. This cognitive function contributed to the initial diagnosis and the analysis of the diversity of strategies. It also played a role to legitimise the implementation of the approach and to provide references used by stakeholders during the workshops including during the discussions on action plans. This function could be extended in the longer term through the organization of an "action research project" (Faure et al. 2014). What is the possibility of launching the TATA-BOX approach in other territories without this contribution of scientific knowledge? The introduction of "selfdiagnosis" methods and the use of collaboration through student training are often proposed solutions.

A second contribution of researchers is the design and implementation of a combination of tools, methods and actions constituting the global TATA-BOX approach.

In fact, these tools and methods already exist separately. The stakeholders themselves considered them "unoriginal" at the end of the three workshops (Chap. 11). But at the same time, they recognized as "very stimulating" the TATA-BOX experience as a whole, because it stimulated collective learning processes. Researchers then played a role of "animation for local development". This function must be implemented autonomously by other actors to ensure the dissemination of the approach. A guide has been produced for this purpose [START_REF] Audouin | Petit guide de l'accompagnement à la conception collective d' une transition agroécologique à l' échelle du territoire[END_REF]. But what is the capacity or willingness of these animators and their organizations to mobilise the TATA-BOX process and, if necessary, readjust the combination of tools according to new local conditions? This question applies to the duplication of design approaches in other territories, but also for the implementation of action plans. Because once conceived, it is not the researchers who will support all the local changes.

The third main contribution from the researchers refers to the evaluation of the approach. TATA-BOX has been monitored and evaluated by researchers, with in particular a survey of a sample of participants. This external evaluation cannot be done each time and could be largely integrated into the participatory process. Indeed the empowerment of the approach calls for proposing a participatory evaluation where the stakeholders could define the objectives, criteria and evaluation methods for the future agroecological systems, the actions they propose and their collective process of design. Here, we can draw attention to the risk that the assessment could be limited to the impacts on local agriculture and ecosystems, whereas all the dimensions of sustainability must be considered, as well as the possible impacts outside the territory [START_REF] Andrieu | Co-conception of technical and organisational changes into agricultural systems[END_REF].

The diffusion of the TATA-BOX approach without the participation of researchers refers to the principle of autonomy, put forward by agroecology (cf. Chap. 4). AET should not hide a new form of "dependence" on research! The importance of links with research and universities has already been shown in several emblematic examples of agroecology, including to ensure their economic viability, for example by hosting trainees and projects or by providing training (Morel 2016). As suggested by the authors of this book, we must consider the TATA-BOX approach neither as a reproducible "turnkey" solution, nor as a pure product of research not reproducible as it stands, but as a living lab, which allows methods and tools to be tested, provides references and shows clearly that the essential thing is to create spaces of debate and exchange in the territories where agroecology is ongoing.

To Highlight the Political Dimensions of Ecological Transition

Finally, the TATA-BOX project and the book that reports this experience lead us to question and better understand the political dimensions of AET.

First of all it is clear that TATA-BOX results from a political project. By announcing a "preferential option" for strong agroecology and bottom-up/participatory development, the researchers' perspective is to go beyond the mere understanding of the world, to engage in a transformation of the world. In France, in scientific practices, including in social sciences, this political dimension is often overshadowed, for fear that political commitment could call into question the scientific nature of an approach. But as soon as the political purpose is clarified and contextualized, and that the social categories that can benefit from the project are debated, there is room for "research capable of engaging and disengaging" [START_REF] Callon | Ni intellectuel engagé, ni intellectuel dégagé: la double stratégie de l'attachement et du détachement[END_REF]. In this sense, the research presented in this book goes beyond the vision of an agroecology "detached from political action" that INRA puts forward (i.e. defining it as the study of relations between ecosystems and agriculture). At the same time, this research does not enclose itself in a vision that is undoubtedly too "committed" that [START_REF] Altieri | Agroecology: the science of sustainable agriculture[END_REF] advocates, even if the defense of family farming makes sense through the TATA-BOX project. At the same time it does not just focus on the defense of organic farming [START_REF] Lamine | Sustainability and resilience in agrifood systems: reconnecting agriculture, food and the environment[END_REF]. The political project of TATA-BOX tends towards "territorial agroecology", which opens spaces of dialogue and negotiation and leads to the elaboration of an action plan. But to fully assume the political dimensions of this "territorial agroecology", several points undoubtedly deserve to be developed.

Firstly, the inscription of the TATA-BOX process and workshops in the local political system can be discussed. The project has implemented ad hoc groups, bringing together a diversity of stakeholders, but not necessarily having a strong political legitimacy. Another option would have been to work more directly with an existing political body, such as a steering group of mayors or representatives dealing with the agricultural and ecological issues within their different municipalities. This is the choice that has been made by the UMR Innovation in support of the agroecological and food policy of the Montpellier metropolis (Soulard et al. 2017). The political impact is stronger, but probably with less debate; less agronomic knowledge and a lower degree of detail on agricultural specificities. This could compromise the final implementation. It's about finding a tradeoff between political impact and political debate.

Moreover, as noted by the authors of the book, the process of political design calls for continuing the experiment, following the evolution of local political action.

In this respect, one can think that the "weak" ties of TATA-BOX with local political authorities can limit the commitment of concrete actions. One could imagine that the project could play a more diffuse role within the local community, promoting the emergence of solutions, and strengthening local capacity to maintain pressure on policy makers. In any case, there is room for complementary work in political science aimed at analysing and supporting the future evolution of political actions and tools for the AET of these territories. Highlighting the notions of "local resource governance" and "socio-technical systems" provides a favorable framework for inviting political science in the wake of this project.

Lastly, the TATA-BOX approach could be considered as a political object that can be mobilised by researchers and stakeholders for action at different scales. The experience can thus lead to the writing of "policy briefs" for national or European policy makers. The presentation of the lessons of the approach in conferences or professional media is to be continued, in agricultural as well as scientific circles. In fact scientists have also to integrate these approaches to participate in AET. The challenge is then to amplify debates, feedback and questioning in the political field. This is also what can be done at the level of other countries through the promotion of this book which is a reference for all those who want to build an "international AET". The images or other third party material in this chapter are included in the chapter's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

  Meynard JM, Dedieu B, Bos AP (2012) Re-design and co-design of farming systems: an overview of methods and practices. In: Darnhofer I, Gibbon D, Dedieu B (eds) Farming systems research into the 21st century: the new dynamic. Springer, Dordrecht, pp 407-431 Morel K (2016) Viabilité des microfermes maraîchères biologiques. Une étude inductive combinant méthodes qualitatives et modélisation. Université Paris-Saclay Prost L, Berthet ETA, Cerf M et al (2017) Innovative design for agriculture in the move towards sustainability: scientific challenges. Res Eng Des 28:119-129. https://doi.org/10.1007/ s00163-016-0233-4 Soulard CT, Perrin C, Valette E (eds) (2017) Toward sustainable relations between agriculture and the city. Springer International Publishing AG, Cham Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence and indicate if changes were made.


J Ér

Ôme Coville

A NOTE ON LIOUVILLE TYPE RESULTS FOR A FRACTIONAL OBSTACLE PROBLEM

This note is a synthesis of my reflexions on some questions that have emerged during the MATRIX event "Recent Trends on Nonlinear PDEs of Elliptic and Parabolic Type" concerning the qualitative properties of solutions to some non local reaction-diffusion equations of the form

where K ⊂ R N is a bounded smooth compact "obstacle", L is non local operator and f is a bistable nonlinearity. When K is convex and the nonlocal operator L is a continuous operator of convolution type then some Liouville-type results for solutions satisfying some asymptotic limiting conditions at infinity have been recently established by Brasseur, Coville, Hamel and Valdinoci [4]. Here, we show that for a bounded smooth convex obstacle K, similar Liouville type results hold true when the operator L is the regional s-fractional Laplacian.

Introduction

A classical topic in applied analysis consists in the study of "diffusive processes" in complex media e.g. media containing obstacles. Roughly speaking, this corresponds to study dispersal processes that follows a random motion in an environment that possess an inaccessible region. At the macroscopic level, this problem can be translated into a reaction -diffusion equation that is defined outside a set K, which acts as an impenetrable obstacle.

One of the cornerstones in the study of these processes lies in suitable rigidity results of Liouvilletype, which allow the classification of stationary solutions, at least under some geometric assumption on the obstacle K.

In this note, we investigate further non local version of a diffusion equation and provide new Liouville-type result (whose precise statements will be given in Section 2).

Concretely, we will suppose that the random motion is modelled by a Lévy flight which at the macroscopic level, leads to consider an integral operator with a singular positive kernel. For such type of processes we will show that the solutions of the stationary equation with a prescribed behaviour at infinity are necessarily constant, at least when the obstacle is convex.

We now provide the detailed mathematical description of the problem that we take into account.

1.1. A fractional obstacle problem. Throughout this note, K denotes a smooth compact set of R n with n 2, |•| denotes the Euclidean norm in R n and L denotes the regional fractional nonlocal operator ( [START_REF] Guan | Boundary problems for fractional laplacians[END_REF][START_REF]Reflected symmetric α-stable processes and regional fractional laplacian[END_REF]) defined for s ∈ (0, 1) by (1.1) L[u](x) := C n,s lim ε→0 ˆ|x-y|>ε,y∈R n \K u(y) -u(x) |x -y| n+2s dy .

We are interested in qualitative properties of smooth bounded solutions to the following non local semilinear equation

L[u](x) + f (u(x)) = 0 for all x ∈ R n \ K, (1.2)
where f is a C 1 "bistable" non-linearity and when necessary with the Neumann boundary condition below ∇u(x) • ν(x) = 0 for all x ∈ ∂K, (1.3) where ν(x) denotes the outer normal derivative of the set K. The precise assumptions on K, u, f and L will be given later on. Typically, this homogeneous Neumann boundary condition is required to define properly the regional fractional Laplacian L on the boundary of the obstacle when s ∈ 1 2 , 1 . This problem may be thought of as a fractional version of the following problem

(1.4) ∆u + f (u) = 0 in R n \ K, ∇u • ν = 0 on ∂K.
For problem (1.4) with the local diffusion operator ∆u, it was shown in [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF] that there exist a timeglobal classical solution u(t, x) to the parabolic problem

(1.5)    ∂u ∂t = ∆u + f (u) in R × R n \ K, ∇u • ν = 0 on R × ∂K satisfying 0 < u(t, x) < 1 for all (t, x) ∈ R × R n \ K, and a classical solution u ∞ (x) to the elliptic problem          ∆u ∞ + f (u ∞ ) = 0 in R n \ K, ∇u ∞ • ν = 0 on ∂K, 0 u ∞ 1 in R n \ K, u ∞ (x) → 1 as |x| → +∞. (1.6)
The function u ∞ is a stationary solution of (1.5) and it is actually obtained as the large time limit of u(t, x), in the sense that u(t, x) → u ∞ (x) as t → +∞ locally uniformly in x ∈ R n \ K. Under some geometric conditions on K (e.g. if K is starshaped or directionally convex, see [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF] for precise assumptions) it is shown in [3, Theorems 6.1 and 6.4] that solutions to (1.6) are actually identically equal to 1 in the whole set R n \ K. This Liouville property shows that the solutions u(t, x) of (1.5) constructed in [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF] then satisfy

u(t, x) -→ t→+∞ 1 locally uniformly in x ∈ R n \ K. (1.7)
From an ecological point of view, such a results can be interpreted as follows. Let us consider that u(t, x) represents the density of a population that moves according to a Brownian motion in a environment consisting of the whole space R n with a compact obstacle K and that the demography of this population can be described by the nonlinear function f . Then, the equation (1.5) can be understood as the evolution of this population int the region R n \ K. In this context, (1.7) means that, at large time, the population tends to occupy the whole space.

Assuming now that the random movement of the individuals follows, say, a reflected symmetric α-stable Lévy process, then the resulting reaction-diffusion equation will be ∂u ∂t

(t, x) = L[u](t, x) + f (u(t, x)) for all t > 0, x ∈ R n \ K (1.8)
+ A Neumann type boundary condition when necessary whereL is the regional fractional Laplacian defined above.

The numerical simulations below (see Figure 1) obtained for a non singular version of the fractional operator L namely when we replace the singular measure δ+|z| n+2s with δ = 0.01, the bistable non-lineartity f is a cubic non-linearity f (s) = s(s -0.1)(1 -s) and the initial condition is of Heaviside type. We can see that the density u(t, x) tends to 1 on the all space and the influence of the obstacle on the shape of the transition In this note, we deal with qualitative properties of the stationary solutions of equation (1.8), together with some asymptotic limiting conditions at infinity similar to those appearing in (1.6). Namely, we will be mainly concerned with solutions of

     L[u] + f (u) = 0 in R n \ K, 0 u 1 in R n \ K, u(x) → 1 as |x| → +∞.
(1.9) with, when necessary, the additional homogeneous Neumann boundary condition

(1.10) ∇u • ν = 0 on ∂K.
1.2. General assumptions. Let us now state precisely the assumptions we will use. Along this note, we suppose that the domain K is a smooth (at least C 0,1 ) bounded compact domain of R n , and that f is a smooth bistable non-linearity, that is f will always satisfies

(1.11) f ∈ C 1 ([0, 1]), f (0) = 0 = f (1) 0, f (1) < 0, (1.12)      ∃ θ ∈ (0, 1), f (0) = f (θ) = f (1) = 0, f < 0 in (0, θ), f > 0 in (θ, 1), ˆ1 0 f > 0, f (0) < 0, f (θ) > 0.
Observe that the assumption on f implies that the associated potential is unbalanced which is a necessary condition to observe the propagation of a front with a positive speed [START_REF] Achleitner | Traveling waves for a bistable equation with nonlocal diffusion[END_REF][START_REF] Gui | Traveling wave solutions of allen-cahn equation with a fractional laplacian[END_REF]. Thus it seems reasonable to assume such condition in our setting since we expect that the solution u of (1.9) reflect the outcome of the invasion of the population in the environment R n \ K.

Main results

For the local problem (1.6), the Liouville property obtained in [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF] says that u = 1 in R n \ K under some geometric conditions on K,in particular when K is convex. Similar Liouville type property was recently obtained for continuous solution of (1.9) when the singular kernel 1 |z| n+2s is replaced by a non negative integrable kernel J, i.e. J ∈ L 1 (R), see [START_REF] Brasseur | Liouville type results for a non-local equation in a presence of an obstacle[END_REF]. More precisely, if J is assume to satisfy the assumptions below (2.13) J ∈ L 1 (R n ) is a non-negative, radially symmetric kernel with unit mass, there are 0 r 1 < r 2 such that J(x) > 0 for a.e. x with r 1 < |x| < r 2 , and there exists a function φ ∈ C(R) satisfying

(2.14) J 1 * φ -φ + f (φ) 0 in R, φ is increasing in R, φ(-∞) = 0, φ(+∞) = 1,
where J 1 ∈ L 1 (R) is the non-negative even function with unit mass given for a.e. x ∈ R by

J 1 (x) := ˆRn-1 J(x, y 2 , • • • , y n ) dy 2 • • • dy n .
then in [START_REF] Brasseur | Liouville type results for a non-local equation in a presence of an obstacle[END_REF] the authors prove the following Theorem 2.1 (Brasseur, Coville, Hamel,Valdinoci [START_REF] Brasseur | Liouville type results for a non-local equation in a presence of an obstacle[END_REF]). Let K ⊂ R n be a compact convex set. Assume that f satisfies (1.11) and (1.12) and J satisfies (2.13) and (2.14) and let

u ∈ C(R n \ K, [0, 1]) (2.15) be a function satisfying (2.16)      ˆRn \K J(x -y)(u(y) -u(x)) dy + f (u(x)) 0 for x ∈ R n \ K, u(x) → 1 as |x| → +∞. Then, u = 1 in R n \ K.
Observe that the problems (2.16) and (1.9) only differ in their formulation by the singularity of the kernels used. In particular, the problem (2.16) can be reformulated in to the framework of problem (1.9) since for all J ∈ L 1 (R n ) and for all

x ∈ R n \ K and u ∈ L ∞ (R n ) lim ε→ ˆRn \K,|x-y|>ε J(x -y)(u(y) -u(x)) dy = ˆRn \K J(x -y)(u(y) -u(x)) dy.
Therefore, it is expected that (2.16) and (1.9) share some common properties. On of the goals of the present note is to extend the results known for (2.16) to the solutions of (1.9) and when possible to highlight the role of the singularity of the kernel in this context.

Our main results show that under the right regularity assumptions we can transposed the results of theorem 2.1 to solutions to (1.9). Namely, we first prove that Theorem 2.2. Let K ⊂ R n be a compact smooth convex set (C 0,1 ). Assume (1.11), (1.12) and s ∈ (0, 1 2

). Let u ∈ C 0,β (R n \ K, [0, 1]) with β > 2s be a function satisfying L[u] + f (u) 0 in R n \ K, u(x) → 1 as |x| → +∞. (2.17) Then, u = 1 in R n \ K.
Our second result complete the picture, namely we show that Theorem 2.3. Let K ⊂ R n be a compact smooth convex set (C 0,1 ). Assume (1.11), (1.12) and s

∈ 1 2 , 1 . Let u ∈ C 1,β (R n \ K, [0, 1]) with β > 2s -1 be a function satisfying      L[u] + f (u) 0 in R n \ K, ∇u • ν = 0 on ∂K, u(x) → 1 as |x| → +∞.
(2.18)

Then, u = 1 in R n \ K.
We can already see clearly the effect of the singularity of the kernel. Indeed, unlike the non local operators with integrable kernel the s-fractional Laplacian is well defined in R n \ K only for regular function, i.e. u should be at least C 0,β . In this singular setting, requiring that the super-solution u is solely continuous is not enough.

These results complete our knowledge on the validity of such type of Liouville property for a broad class of reaction diffusion equation. They show some universality of such type of property and prove that such rigidity type result can be viewed as an intrinsic property of the problem which can be related to a generic property of the equation rather than a special property of the diffusion process considered.

2.1. Further comments and strategy of proofs. Prior to proving these results, let us make some comments on our hypotheses and highlight some of the differences that arise when the singular measure 1 |z| n+2s is replaced by an integrable kernel J. First, let us observe that thanks to the regularising property of the regional fractional Laplacian L, see [START_REF] Caffarelli | Regularity theory for fully nonlinear integro-differential equations[END_REF][START_REF] Guan | Boundary problems for fractional laplacians[END_REF][START_REF]Reflected symmetric α-stable processes and regional fractional laplacian[END_REF] the continuity assumption made on u can be easily weakened when u is assumed to be a solution to (1.9) instead of a super-solution. Thus, in this situation, the result of Theorems 2.2 and 2.3 hold as well for bounded solution u that satisfies the equation (2.17) respectively (2.18) in the sense of viscosity solutions. Note that contrary to the regional fractional Laplacian L, the nonlocal operator M[u] := ´Rn \K J(x -y)(u(y) -u(x)) dy has no regularising properties and as a consequence weakening the regularity assumption on the solution u is a hard task which, for the moment, can only been achieved by imposing further restrictions on the data f and J. Nevertheless, in this non regularising context, the regularity of the obstacle K is no more an issue and K can be any arbitrary convex domain. The regularisation effect on the solutions induced by the singularity is in fact the only main distinction between the problem (2.16) and the singular problems (2.17) and (2.18).

This distinction appears also clearly in the set of assumption needed for the existence of monotone travelling front with a positive speed, which is an essentiel key element of the proof of Theorem 2.1. In particular, as already mentioned at Section 1.2, assumptions (1.11) and (1.12) are actually necessary and sufficient for the existence of a travelling wave solution with positite speed c to the one dimentional fractional equation, i.e. a monotone solution to

c ∂ z ϕ = ∂ s z ϕ + f (ϕ)
, with a positive speed c > 0. Such assumptions are not any more sufficient in the context of (2.16), where there exists data f and J that satisfy (1.11)-(1.12) for which only discontinuous null speed fronts exist.

Let us also note that, similar condition are also necessary and suffisent for the existence of one dimentional travelling wave with positive speed for the local problem, namely solution of c ϕ = ϕ + f (ϕ) with positive speed (see e.g. [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF]). This fact, then suggest a strong connexion between the regularity of the front and the minimal set of assumptions that are required to produced a front of positive speed.

Let us emphasize that the motivation behind these assumptions are that, by analogy with the local problem (1.5), we expect a solution to (1.9) to be the large time limit of an entire solution to the evolution problem (1.8) which behaves like ϕ(x 1 + ct) when t → -∞. For this interpretation to even make sense it is necessary to work in a setting where the function ϕ exists.

Let us now say a word on our strategy of proofs. The proofs are a rather straightforward adaptation of the arguments developed in [START_REF] Brasseur | Liouville type results for a non-local equation in a presence of an obstacle[END_REF] for the non local obstacle problem (2.16). The main idea is to compare by means of adequate sliding method, a family of planar function of the type ϕ(x • e -r), where e ∈ ∂B 1 , r ∈ R and ϕ ∈ C 1,1 (R) a given monotone function with a given super-solution u. To adapt such technique to our situation, we need first to verify that, as proved for (2.16) similar comparison principles in half-spaces hold true as well for the fractional equations (2.17) and (2.18).

The outline of this note will be as follow. In Section 3 we provide several comparison principles and recall some known results on the 1d travelling fronts for fractional bistable equation. Then we prove in Section 4, following the arguments developed in [START_REF] Brasseur | Liouville type results for a non-local equation in a presence of an obstacle[END_REF], we prove the Liouville property described in Theorems 2.2 and 2.3.

Some mathematical background

In this section, we start by collecting some comparison principles that fit for our purposes and to shortened the presentation we only fully state the necessary comparison principle for regional fractional Laplacian with s ∈ 1 2 , 1 . Throughout this section, K is any compact subset of R n , f is any C 1 (R) function.

We start with a weak maximum principle Lemma 3.1 (Weak maximum principle). Assume that s ∈ 1 2 , 1 and

(3.19) f -c 1 in [1 -c 0 , +∞), for some c 0 > 0, c 1 > 0. Let H ⊂ R n be an open affine half-space such that K ⊂⊂ H c = R n \ H. Let u, v ∈ L ∞ (R n \ K) ∩ C 1,β H for some β > 1 -2s be such that (3.20) L[u] + f (u) 0 in H, L[v] + f (v) 0 in H.
Assume also that

(3.21) u 1 -c 0 in H, that (3.22) lim sup |x|→+∞ v(x) -u(x) 0
and that

(3.23) v u a.e. in H c \ K.
Then, v u a.e. in R n \ K.

The next lemma is concerned with a strong maximum principle.

Lemma 3.2 (Strong maximum principle). Assume that s ∈ 1 2 , 1 and let H ⊂ R n be an open affine half-space such that K ⊂⊂ H c . Let u, v ∈ L ∞ (R n \ K) ∩ C 1,
β H for some β > 1 -2s be such that (3.20) holds true. Assume also that

(3.24) v u a.e. in R n \ K
and that there exists x ∈ H such that v(x) = u(x). Then, v = u a.e. in H.

These comparison principles are in essence identical to the one derived in [START_REF] Brasseur | Liouville type results for a non-local equation in a presence of an obstacle[END_REF] and as such we point the interested reader to [START_REF] Brasseur | Liouville type results for a non-local equation in a presence of an obstacle[END_REF] for a detailed proof of these results.

Remark 3.3. The above comparison principles have only been stated for regional fractional operators with exponent s ∈ 1 2 , 1 . Identical weak and strong maximum principles can be formulated for the regional fractional Laplacian with exponent s ∈ 0, 1 2 as soon as we impose the adequate regularity to the functions u and v in order to properly define the regional fractional Laplacian of u and v. In such case, the above statement will holds true if instead of having u

, v ∈ L ∞ (R n \ K) ∩ C 1,β H we assume that u, v ∈ L ∞ (R n \ K) ∩ C 0,β H with β > 2s.
Lastly, we recall some known result on the existence and properties of travelling fronts ϕ(x • e + ct), solution of the fractional evolution equation

∂ t u(t, x) = ∆ s u(t, x) + f (u(t, x)) for t ∈ R, x ∈ R n
that is, solution of the following

-c∂ z ϕ(z) + ∂ s z ϕ(z) + f (ϕ(z)) = 0 for z ∈ R (3.25) lim z→+∞ ϕ(z) = 1, lim z→-∞ ϕ(z) = 0 (3.26)
where ∂ s z ϕ denotes the one dimentional s-fractional Laplacian. The existence, uniqueness and some asymptotic properties of such solution ϕ have been obtained in several context [START_REF] Achleitner | Traveling waves for a bistable equation with nonlocal diffusion[END_REF][START_REF] Cabré | Nonlinear equations for fractional laplacians ii: existence, uniqueness, and qualitative properties of solutions[END_REF][START_REF] Gui | Traveling wave solutions of allen-cahn equation with a fractional laplacian[END_REF][START_REF] Palatucci | Local and global minimizers for a variational energy involving a fractional norm[END_REF]. The next statement is a summary of these results. Theorem 3.4 (Fractional Travelling wave [START_REF] Achleitner | Traveling waves for a bistable equation with nonlocal diffusion[END_REF][START_REF] Cabré | Nonlinear equations for fractional laplacians ii: existence, uniqueness, and qualitative properties of solutions[END_REF][START_REF] Gui | Traveling wave solutions of allen-cahn equation with a fractional laplacian[END_REF][START_REF] Palatucci | Local and global minimizers for a variational energy involving a fractional norm[END_REF]). Assume f is a bistable function that satisfies (1.11) and (1.12) and let s ∈ (0, 1). Then there exists a unique c ∈ R and a monotone smooth (at least C 1,1 ) increasing function ϕ such that (c, ϕ) is a solution to (3.25)-(3.26). Moreover, if f is unbalanced with ´1 0 f (s) ds > 0, then c > 0.

As a trivial consequence of the existence of a smooth front of positive speed, for any separating open affine half-space H ⊂ R n such that K ⊂⊂ H c , we can derive a family of function which will be a "sub-solution to the problem (1.9)" for all x ∈ H. More precisely, let H be an affine subspace of R n such that K ⊂ H c . By definition of the affine space, there exists a unit vector e ∈ ∂B 1 and x 0 ∈ R n such that H = x 0 + H e with H e an open-halfspace of direction e, i.e.

H e := {x ∈ R n |x • e 0}.
For this direction e and for all real r ∈ R, we can define the family of functions φ e,r (x) := ϕ(x•e-r) where ϕ is the smooth increasing profile obtained in Theorem 3.4. By construction, since ϕ is monotone increasing we have

(3.27) ∀ x ∈ H, ∀ y ∈ K φ r,e (y) -φ r,e (x) 0.
In addition, we can check that for all x ∈ H, we have Hence, for all x ∈ H, we get

L[φ r,e ](x) = lim ε→0 ˆRn \K,|x-y|>ε φ r,e (
(3.28) L[φ e,r ](x) + f (φ e,r ) ∆ s φ e,r (x) + f (φ e,r ) = ∂ s ϕ(x.e -r) + f (ϕ(x.e -r)) = c∂ z ϕ(x • e -r) > 0.

The case of convex obstacles: proofs of the main Theorem

In this section, based on the arguments introduced in [START_REF] Brasseur | Liouville type results for a non-local equation in a presence of an obstacle[END_REF] we sketch the proof our main results (Theorems 2.2 and 2.3). The proofs of the Theorems 2.3 and 2.2 being identical, we only sketch the proof of Theorem 2.3.

But before we start our argumentation,let us first start with the following simple observation.

Lemma 4.1. Let s ∈ 1 2 , 1 , K ⊂ R n be a smooth compact convex set and assume (1.11) and (1.12). Let u ∈ C 1,β (R n \ K), [0, 1]) with β > 1 -2s be such that Then there exists γ ∈ (0, 1] such that γ u 1 in R n \ K.

L[u] + f (u) 0 in R n \ K,
The proof of this Lemma being an elementary adaptation of the argument used in [START_REF] Brasseur | Liouville type results for a non-local equation in a presence of an obstacle[END_REF], we will refer to [START_REF] Brasseur | Liouville type results for a non-local equation in a presence of an obstacle[END_REF] for its proof.

We now turn to the proof of Theorem 2.3 .

Proof of Theorem 2.3. Let us fix s ∈ 1 2 , 1 and let K, f , and u be as in Theorem 2.3. Let us now follow the argument developed in [START_REF] Brasseur | Liouville type results for a non-local equation in a presence of an obstacle[END_REF]. Firstly, without loss of generality, one can assume by (1.11) that f is extended to a C 1 (R) function satisfying (3.19). Secondly, by (2.17) and the boundedness of K, there exists R 0 > 0 large enough so that K ⊂ B R 0 and u 1 -c 0 in R n \ B R 0 , where c 0 > 0 is given in (3.19).

We now proceed by contradiction, and suppose that inf

R n \K u < 1. (4.32)
From (2.18) and (4.32), together with the continuity of u, there exists then

x 0 ∈ R n \ K such that u(x 0 ) = min R n \K u ∈ [0, 1).
We observe that, by Lemma 4.1, one has u(x 0 ) > 0. In addition, since K is convex, there exists e ∈ ∂B 1 such that K ⊂ H c e , where H e is the open affine half-space defined by H e := x 0 + x ∈ R n ; x • e > 0 .

As in section 3, let us define for all r ∈ R, the family of functions

φ r (x) := φ r,e (x) = ϕ(x • e -r), x ∈ R n ,
where ϕ is a smooth monotone increasing function given by Theorem 3.4. Note that by construction, since K ⊂ H c e , we can check (as in the section 3) that for any r ∈ R, φ r satisfies (4.33)

L[φ r ](x) + f (φ r (x)) > 0 for x ∈ H e .
First, we claim that Claim 4.2. There exists r 0 ∈ R such that φ r 0 u in R n \ K.

Again the proof of this claim is an elementary adaptation of a proof done in [START_REF] Brasseur | Liouville type results for a non-local equation in a presence of an obstacle[END_REF] that for the sake of clarity we give the details.

Proof. First let us define H := x 1 + H e with x 1 to be chosen such that B R 0 ⊂ H c . Let us fix x 1 such that H ⊂⊂ H e . By construction the function ϕ is monotone increasing and satisfies lim z→-∞ ϕ(z) = 0. So we can find r 0 >> 1 such that φ r 0 (x) = ϕ(x • e -r 0 ) u(x 0 ) u(x) for all x ∈ H c . Now thanks to our choice of x 1 we have H ⊂⊂ H e and from (4.33) we deduce

       L[u](x) + f (u(x)) 0 for x ∈ H, L[φ r 0 ](x) + f (φ r 0 (x)) > 0 for x ∈ H, u(x) φ r 0 (x) for x ∈ H c \ K,
We then get the desired results by applying the weak-maximum principle (Lemma 3.1).

Equipped with the claim 4.2, we can now define the following quantity

r * := inf r ∈ R ; φ r u in R n \ K .
All the game now is to show that r * = -∞. So, we claim that Claim 4.3. r * = -∞.

Assume for the claim is true, then the proof of Theorem 2.3 is thereby complete. Indeed, from this claim we infer that φ r u in R n \ K for any r ∈ R. In particular, recalling that ϕ(+∞) = 1, we get that

1 > u(x 0 ) lim r→-∞ φ r (x 0 ) = lim r→-∞ ϕ(x 0 • e -r) = 1,
a contradiction. Therefore, (4.32) can not hold. In other words, inf

R n \K u = 1, i.e. u = 1 in R n \ K proving thereby Theorem 2.3.
Let us now conclude our proof by establishing the Claim 4.3. Again, the proof of this last claim is done by a very elementary adaption of the arguments used to prove Theorem 2.1. As a consequence we will only highlights the main differences.

Proof of the Claim 4.3. The proof of (4.3) is by contradiction. We assume that r * ∈ R. Then, there exists a sequence (ε j ) j∈N of positive real numbers such that φ r * +ε j (x) = ϕ(x • e -r * -ε j ) u(x) for all x ∈ R n \ K and ε j → 0 as j → +∞. Thus, passing to the limit as j → +∞, we obtain that φ r * (x) u(x) for all x ∈ R n \ K.

Let us denote H the open affine half-space

H = x ∈ R n ; x • e > R 0 .
Notice that H ∩ K = ∅ and that u is well defined and continuous in H. We also observe that, by construction, sup H c φ r * < 1. (4.34) Two cases may occur. Case 1: inf H c \K (u -φ r * ) > 0. In this situation, the argument is identical as in for (2.16), and we point the reader to [START_REF] Brasseur | Liouville type results for a non-local equation in a presence of an obstacle[END_REF] for the details.

Case 2: inf H c \K (u -φ r * ) = 0. In this situation, by (4.31) and (4.34), and by continuity of u and φ r * , there exists a point x ∈ H c \ K such that u(x) = φ r * (x). Note that x ∈ H e , since otherwise x ∈ R n \ H e , namely x • e < x 0 • e, and the chain of inequalities u(x) = φ r * (x) < φ r * (x 0 ) u(x 0 ) = min This last contradiction then rules out also this situation and therefore rules out Case 2 too.

Remark 4.4. The proof of Theorem 2.2 is identical to the one given in [START_REF] Brasseur | Liouville type results for a non-local equation in a presence of an obstacle[END_REF]. This is due to the fact that the s-fractional operator is well defined and continuous up to the boundary of ∂K when s ∈ 0, 1 2 and as such the strong maximum principle (Lemma 3.2) holds also true for any half space H such that K ⊂ H c . In this situation, the case x ∈ ∂K does not need to be analysed separately from the other cases.

Figure 1 .

 1 Figure 1. Simulation of the singular non local evolution problem (1.8) where the singular Levy kernel has been replaced by a non singular measure1

  y) -φ r,e (x) |x -y| n+2s dy = lim ε→0 ˆRn ,|x-y|>ε φ r,e (y) -φ r,e (x) |x -y| n+2s dy -ˆK,|x-y|>ε φ r,e (y) -φ r,e (x) |x -y| n+2s dy = ∆ s φ r,e (x) -ˆK φ r,e (y) -φ r,e (x) |x -y| n+2s dy. which combined with (3.27) enforces L[φ r,e ](x) ∆ s φ r,e (x) for x ∈ H.

  (4.29) ∇u • ν = 0 in ∂K, (4.30) u(x) → 1 as |x| → +∞. (4.31)

  R n \K u leads to a contradiction. Therefore, we have φ r * u in R n \ K with equality at a point x ∈R n \ K ∩ H e .Again, two situations can occur either x ∈ R n \ K or x ∈ ∂K. Assume for the moment that the latter situation occurs. Then thanks to convexity of K, ν(x) the outward normal to ∂K at x is then the vector e, i.e. ν(x) = e and thanks to (2.18) ∇u • ν(x) = 0 we deduce 0 ∇(u -φ r * ) • ν(x) -∇(φ r * ) • ν(x) = -ϕ (x • e -r * ) < 0.This contradiction then rules out this situation. Lastly assume that x ∈ R n \ K, then in this situation K ⊂⊂ H c e and ϕ r * and u satisfy respectivelyL[u] + f (u) 0 in H e , L[φ r * ] + f (φ r * ) > 0 in H e (by (4.33)), .In particular, it follows from the strong maximum principle (Lemma 3.2) that φ r * = u in H e . Thus, for any e ⊥ ∈ ∂B 1 such that e ⊥ • e = 0, one infers from (4.31) and the definition of φ r * that 1 = lim t→+∞ u(x 0 + t e ⊥ ) = lim t→+∞ φ r * (x 0 + t e ⊥ ) = φ r * (x 0 ) < 1.

  1 |z| n+2s by 1 δ+|z| n+2s for δ << 1, suggest that the long time behaviour of solution of positive equation (1.8) should be identical as those observed in the classical reaction-diffusion equation.

Acknowledgements. The author would like to thank Professor Changfeng Gui for raising to my attention this topic during the MATRIX event "Recent Trends on Nonlinear PDEs of Elliptic and Parabolic Type"

The author has been supported by the ANR DEFI project NONLOCAL (ANR-14-CE25-0013). The author want to thank professor Changfeng Gui for bringing to my attention this question during the Matrix event "