domain: math info

text:
Julien Brasseur

Analysis of some nonlocal models in population dynamics

Keywords: nonlocal reaction-diffusion equations, rigidity results, Besov spaces, calculus of variations, perforated domains, function space theory

This thesis is mainly devoted to the mathematical analysis of some nonlocal models arising in population dynamics. In general, the study of these models meets with numerous difficulties owing to the lack of compactness and of regularizing effects. In this respect, their analysis requires new tools, both theoretical and qualitative. We present several results in this direction.

In the first part, we develop a functional analytic toolbox which allows one to handle some quantities arising in the study of these models. In the first place, we extend the characterization of Sobolev spaces due to Bourgain, Brezis and Mironescu to low regularity function spaces of Besov type. This results in a new theoretical framework that is more adapted to the study of some nonlocal equations of Fisher-KPP type. In the second place, we study the regularity of the restrictions of these functions to hyperplanes. We prove that, for a large class of Besov spaces, a surprising loss of regularity occurs. Moreover, we obtain an optimal characterization of the regularity of these restrictions in terms of spaces of so-called "generalized smoothness".

In the second part, we study qualitative properties of solutions to some nonlocal reaction-diffusion equations set in (possibly) heterogeneous domains. In collaboration with J. Coville, F. Hamel and E. Valdinoci, we consider the case of a perforated domain which consists of the Euclidean space to which a compact set, called an "obstacle", is removed. When the latter is convex (or close to being convex), we prove that the solutions are necessarily constant. In a joint work with J. Coville, we study in greater detail the influence of the geometry of the obstacle on the classification of the solutions. Using tools of the type of those developed in the first part of this thesis, we construct a family of counterexamples when the obstacle is no longer convex. Lastly, in a work in collaboration with S. Dipierro, we study qualitative properties of solutions to nonlinear elliptic systems in variational form. We establish various monotonicity results in a fairly general setting that covers both local and fractional operators.

Résumé

Cette thèse est consacrée principalement à l'analyse mathématique de modèles nonlocaux issus de la dynamique des populations. En général, l'étude de ces modèles se heurte à de nombreuses difficultés dues à l'absence de compacité et d'effets régularisants. A ce titre, leur analyse requiert de nouveaux outils tant théoriques que qualitatifs. Nous présentons des résultats recouvrant ces deux aspects.

Dans une première partie, nous développons une "boîte à outils" destinée à traiter certaines quantités récurrentes dans l'étude de ces modèles. En premier lieu, nous étendons la caractérisation des espaces de Sobolev due à Bourgain, Brezis et Mironescu à des espaces de fonctions moins réguliers de type Besov, offrant ainsi un cadre théorique plus adapté à l'étude de certaines équations du type Fisher-KPP. En second lieu, nous étudions la régularité de ces fonctions par restriction sur des hyperplans. Nous montrons que, pour une large classe d'espaces de Besov, une surprenante perte de régularité a lieu. En outre, nous obtenons une caractérisation optimale de la régularité de ces restrictions via des espaces dits à "régularité généralisée".

Dans une seconde partie, nous nous intéressons aux propriétés qualitatives des solutions d'équations de réaction-diffusion non-locales posées dans des domaines possiblement hétérogènes. En collaboration avec J. Coville, F. Hamel et E. Valdinoci, nous considérons le cas d'un domaine perforé consistant en l'espace euclidien privé d'un ensemble compact appelé "obstacle". Lorsque ce dernier est convexe (ou presque convexe), nous montrons que les solutions sont nécessairement constantes. Dans un travail conjoint avec J. Coville, nous étudions plus en détail l'influence de la géométrie de l'obstacle sur la classification des solutions. En utilisant des outils du type de ceux développés dans la première partie de cette thèse, nous construisons une famille de contre-exemples lorsque l'obstacle n'est plus convexe. Enfin, dans un travail en collaboration avec S. Dipierro, nous étudions les propriétés qualitatives des solutions de systèmes d'équations elliptiques non-linéaires sous forme variationnelle. Nous y démontrons plusieurs résultats de monotonicité dans un cadre très général qui couvre à la fois le cas des opérateurs locaux et fractionnaires.

Mots-clés : équations de réaction-diffusion non-locales, résultats de rigidité, espaces de Besov, calcul des variations, domaines perforés, théorie des fonctions.

Introduction 1 Foreword

Over past decades, it has become clear that some phenomena in population dynamics cannot be described by local equations. Inter alia, the experimental data have shown that the dispersal of biological populations often presents "nonlocal features". This has led mathematicians and biologists to consider nonlocal models. But if they share some resemblances with the classical models, their treatment is often a delicate matter. In general, these models enjoy neither good compactness properties nor good regularizing effects. To make matters worse, there may not even be such thing as (fractional) Sobolev spaces to cope with situations where they would be the natural tool to use. The usual techniques of proof are therefore more complex to implement. One thus expects new approaches to be helpful to handle these issues.

This thesis is intended as a contribution in this direction and, more generally, to the field of nonlocal equations. Of course, this is a very wide topic and we will broach only a small part of it. We will be primarily interested in equations involving a "nonlocal diffusion term" and a "nonlinear reaction term".

This manuscript consists of two parts which have different styles. In Part I, we address functional analytic issues and we provide new results regarding smoothness and function spaces in connection with nonlocal equations; whereas, in Part II, we study qualitative properties of solutions to some nonlocal reaction-diffusion problems, with special emphasis on rigidity results.

The motivation of this thesis and the outline of our main contributions are explained in greater details in the next two sections.

State of art 2.1 A brief history of population dynamics

Before going to the heart of this thesis, it is worth taking a small walk through history. As a matter of fact, population dynamics is a much older topic than usually thought. The very first to show preoccupations of this kind was Plato (ca. 427-347 BC) in his Laws (Nóµoι in Greek). In the fifth book of this monumental essay, he wondered what should be the optimal organization of a city-state in order for it to remain stable over the course of time. Although this might seem quite remote from population dynamics, he raised there important questions on the relationship of the population to the environment. Based on both ecological and governance grounds, he argued that the total number of citizens should be kept constant equal to 5,040 because it can be divided into many lesser parts which should, in his view, be very convenient to optimize both the organization of the polis and the demographic needs. 1 To maintain this number constant he also pointed out the need for a demographic legislation, foreshadowing Malthusian ideas on demography. 2It is striking to see that Plato, 2,400 years ago, already foresaw the importance of population dynamics and the possibility of a mathematical approach (although rather fancy3 ) to address the problem! More than 1,500 years later, Fibonacci (1175-1250) in his 1202 book Liber Abaci introduced the famous sequence named after him. What is however less known, is that it was introduced to solve a problem which one would nowadays call a typical problem in population dynamics. Here is an English translation of the original statement (in Latin) of the problem:

A certain man had one pair of rabbits together in a certain enclosed place. One wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair and in the second month those born to bear also.

Interestingly enough, the solution to this problem, the Fibonacci sequence (F n ) n 0 , grows like ϕ n / √ 5 as n → ∞, where ϕ = (1 + √ 5)/2 is the golden ratio (see e.g. [START_REF] Vorobiev | Fibonacci numbers[END_REF]Theorem 12,p.27]). In other words, it involves an exponential growth of the population. In a certain sense -that will be made clear in the next page -this feature somehow instates the Fibonacci sequence as a forerunner of some discoveries which would play an important role in the establishment of population dynamics.

But modern population dynamics really start to expand from the seminal work of Malthus (1766-1834). In his celebrated Essay on the Principle of Population (1798), he developed what is now regarded as the "first principle" of population dynamics. In substance, it says that 1. all life forms have a propensity to exponential population growth when the resources are abundant;4 2. the population growth is limited by the availability of the resources. 5 This amounts to say that, for a population with unlimited resources, the size u(t) of the population at time t is given by u(t) = u 0 e λt , where u 0 is the initial population size and λ > 0 is the growth rate of the population. In other words, u obeys u (t) = λu(t) for t > 0, u(0) = u 0 .

(2.1) This is known as the Malthusian growth model. However, this model is generally unrealistic: it does not take into account the competition between individuals for the resource, which thereby yields exponential growth of the population. A couple of decades later, Verhulst [START_REF] Verhulst | Recherches mathématiques sur la loi d'accroissement de la population[END_REF] (1804-1849), inspired by the work of Malthus, proposed a new model that adjusted (2.1) to take into account the resource limitation phenomenon. This model, known as the logistic growth model, takes the form

u (t) = λu(t) 1 - u(t) κ , (2.2)
where λ > 0 is the growth rate of the population and κ > 0 is a constant corresponding to the maximum population size of the species that the environment can sustain indefinitely given the availability of the resource, called the carrying capacity. Of course, if κ → ∞, then this boils down to the Malthusian growth model. Equation (2.2) gives rise to an unstable equilibrium at 0 and a stable equilibrium at κ. Indeed, this is because u (t) = 0 when u(t) ∈ {0, κ} while u (t) > 0 in {0 < u < κ} and u (t) < 0 in {u > κ}. Therefore, κ attracts any function starting from u(0) ∈ (0, κ). That is, u(t) → κ as t → ∞. For this reason, the nonlinear function of u on the right-hand side of (2.2) is sometimes referred to as monostable.

But the mathematical theory of population dynamics really took flight in the 1930's. Until then, investigations were based on the Malthusian principle (exponential growth) and on the Verhulst model (logistic growth). Two fundamental discoveries would change this old paradigm.

The first one is due to Allee . According to the Verhulst model, the per capita growth rate of the population is all the greater when the population is small. That is, the less individuals there are, the less mortality there is, as the available resource per individual is higher. In his 1931 book Animal Aggregations: A Study in General Sociology, Allee discovered that this might not always be the case. Another phenomenon must be taken into account: cooperation between individuals. He pointed out that there is a positive correlation between population size and the individual fitness (or per capita growth rate) of the population. This is called the "Allee effect", but some authors refer to it as the "second principle" of population dynamics. It manifests through different mechanisms such as mate limitation or cooperative feeding. Loosely speaking, the higher the population size is, the more likely the individuals are to cooperate, which then results in a demographic increase. Contrariwise, the lower the population size is, the less likely the individuals are to cooperate, which then results in a demographic decrease. From the mathematical point of view, this can be modelled by introducing a critical threshold above which the population tends to increase and below which it tends to decrease. It is usually translated into models of the type

u (t) = λu(t) 1 - u(t) κ (u(t) -θ) , (2.3)
where λ and κ have the same meaning as in (2.2) and θ > 0 is the critical size of the population.

By contrast with the Verhulst model, there are now two stable equilibria at 0 and κ, respectively, and an unstable equilibrium at θ. For this reason, the nonlinear function of u on the right-hand side of (2.3) is called bistable. This is indeed consistent with Allee's observations: if it holds that u(0) ∈ (0, θ), then, after some large time, the population will decline to extinction, namely u(t) → 0 as t → ∞. Conversely, if u(0) ∈ (θ, κ), then the population will tend to reach the maximal amount of individuals that the environment is able to sustain, namely u(t) → κ as t → ∞.

The second milestone in the history of population dynamics arose with the discovery, in 1937, by Fisher [START_REF] Fisher | The wave of advance of advantageous genes[END_REF] and, independently, by Kolmogorov, Petrovskii and Piskunov [START_REF] Kolmogorov | A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem[END_REF] of the so-called Fisher-KPP equation:

∂u ∂t = ρ∆u + λu 1 - u κ , (2.4)
where ρ > 0 is a diffusion coefficient analogous to that used in physics. The principal novelty of this model is that it takes into account the spatial interactions between individuals. All the models considered hitherto were not spatialized, meaning that, in each of the populations considered, each individual was assumed to interact with all the other individuals. However, in most situations, the heterogeneity of the spatial distribution cannot be neglected. The population size u(t) is thus not sufficiently informative and one must look for its density u(t, x) instead. This difficulty is overcome by considering that the movement of individuals is approximated by a random motion which then results in a reaction-diffusion equation (see the next section for more explanations). Since these seminal works, much attention has been paid to models taking both into account the diffusion of the species and general demographic variations, such as

∂u ∂t = ∆u + f (u), (2.5)
where f is some nonlinear reaction term, see e.g. [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF][START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF][START_REF] Levermore | Multidimensional stability of traveling waves in a bistable reaction-diffusion equation[END_REF][START_REF] Weinberger | Long-time behavior of a class of biological models[END_REF][START_REF]On spreading speed and travelling waves for growth and migration models in a periodic habitat[END_REF].

It is worth mentioning that, over the past century, equations of the type of (2.5) have emerged from a number of seemingly remote areas, most notably in the study of combustion phenomena and phase transitions in liquid crystals, superconductivity and material sciences in general. In the latter case, one typically has f (X) = X(1 -X 2 ), and (2.5) is then called the Allen-Cahn or Ginzburg-Landau equation (see e.g. [START_REF] Bethuel | Ginzburg-landau vortices[END_REF][START_REF]Mathematical problems related to liquid crystals, superconductors and superfluids[END_REF]). As a matter of fact, mechanistic models in ecology often intertwine with other branches of natural sciences. Although the mathematical theories of the abovementioned fields are rather disjoint, the tools and methods developed for either one of them often turn out to find applications in the other. These equations are customary gathered together under the umbrella term "reaction-diffusion problems".

We refer the interested reader to [START_REF] Bacaër | A short history of mathematical population dynamics[END_REF][START_REF] Roques | Modèles de réaction-diffusion pour l'écologie spatiale[END_REF][START_REF] Volpert | Reaction-diffusion waves in biology[END_REF] (and references therein) for further historical remarks and a comprehensive study of the above models.

From local to nonlocal models: towards a new paradigm

Since the seminal work of Fisher [START_REF] Fisher | The wave of advance of advantageous genes[END_REF], Kolmogorov, Petrovskii and Piskunov [START_REF] Kolmogorov | A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem[END_REF], the study of spatial distribution of individuals has become a fundamental cornerstone of population dynamics. In fact, the patterns formed by the individuals at the microscopic level can be shown to determine the diffusion at the macroscopic level. Therefore a better knowledge of the former is expected to improve our understanding of the latter. From our perspective, this means that one can recover the Fisher-KPP equation starting from microscopic considerations only. Indeed, interpreting the trajectories of the individuals as realizations of a Lévy process (X t ) t 0 , we can then consider its associated Feller semi-group (P t ) t 0 . In turn, this semi-group is generated by an operator L given by Lu(x) = lim

t→0 + P t u(x) -u(x) t .
This operator, called the infinitesimal generator of (P t ) t 0 , describes the diffusion of the species at the macroscopic level. Since diffusion is accompanied by demographic variations, we then obtain -at least formally -an equation of the type

∂u ∂t = Lu + f (u), (2.6)
where f is some nonlinear reaction term. If (X t ) t 0 is a Brownian motion, then the operator L boils down to the Laplacian, namely

Lu = 1 2 ∆u,
(see e.g. [START_REF] Applebaum | Lévy processes and stochastic calculus[END_REF]Example 3.3.4,p.141]) and we recover essentially (2.4).

The recent development of GPS technologies has allowed the detailed tracking of individuals within a given species. This has been the source of a new revolution in the development of population dynamics and has shed new lights on how to describe diffusion phenomena. If the collected sets of data highly support the assumption that the trajectories follow Lévy processes (see e.g. [START_REF] Bartumeus | Lévy processes in animal movement: an evolutionary hypothesis[END_REF][START_REF]Behavioral intermittence, Lévy patterns, and randomness in animal movement[END_REF][START_REF] Lomholt | Lévy strategies in intermittent search processes are advantageous[END_REF][START_REF] Reynolds | Scale-free animal movement patterns: Lévy walks outperform fractional Brownian motions and fractional Lévy motions in random search scenarios[END_REF][START_REF] Seuront | Anomalous diffusion and multifractality enhance mating encounters in the ocean[END_REF]), it has been observed that the Brownian motion does not always account for the patterns formed by individuals. In particular, they may exhibit long-range jumps which are not accounted by Brownian motions. In this case, dispersion can occur over large distances and may exhibit "nonlocal features".

From the PDE standpoint, this means that, depending on the specific behavior one wishes to describe, different operators from the Laplacian may arise.

In the case of marine predators, the Brownian motion is relevant where prey is abundant, but a Lévy type behavior occurs when prey is sparsely distributed [START_REF] Hallatschek | Acceleration of evolutionary spread by long-range dispersal[END_REF][START_REF] Humphries | Environmental context explains Lévy and Brownian movement patterns of marine predators[END_REF][START_REF] Reynolds | The Lévy flight paradigm: random search patterns and mechanisms[END_REF][START_REF] Sims | Lévy flight search patterns of marine predators not questioned: a reply to Edwards et al[END_REF]. In this situation, random patterns of individuals are best described by processes with long-range jumps. Typical examples of such processes are α-stable Lévy processes with 0 < α < 2. Although they are somehow related to the Brownian motion (which may be seen as a 2-stable Lévy process) their infinitesimal generator happen to be of a completely different nature. It is given by the singular integral Lu(x) = -(-∆) α/2 u(x) := C(N, α) p.v.

ˆRN

u(y) -u(x) |x -y| N +α dy, (2.7) where C(N, α) > 0 is a normalization constant and "p.v." stands for the Cauchy principal value (see e.g. [START_REF] Garbaczewski | Fractional laplacians and Lévy flights in bounded domains[END_REF][START_REF] Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF][START_REF] Valdinoci | From the long jump random walk to the fractional Laplacian[END_REF]). This operator is called the fractional Laplacian.

Another example of processes which arise in this situation are compound Poisson processes. Their infinitesimal generator is, again, of a completely different nature.

Precisely, one has

Lu(x) = ˆRN J(x -y)(u(y) -u(x))dy, (2.8)
where J ∈ L 1 (R N ) is a nonnegative kernel with J(x -y) encoding the probability to jump from a location x to a location y (see e.g. [START_REF] Applebaum | Lévy processes and stochastic calculus[END_REF]Example 3.3.7,p.141]).

What is here striking is that both (2.7) and (2.8) are nonlocal integral operators. 6This has strong consequences and induces strange and surprising phenomena.

To name only a few, Dipierro, Savin and Valdinoci [START_REF]All functions are locally s-harmonic up to a small error[END_REF] have shown that every reasonably smooth function can be locally approximated by α 2 -harmonic functions, i.e. such that (-∆) α/2 u = 0. This highly contrasts with the rigidity of classical harmonic functions and is a purely nonlocal feature. In the same vein, Cafferelli, Dipierro and Valdinoci [START_REF] Cafferelli | A logistic equation with nonlocal interactions[END_REF] proved that a similar "density result" holds with nonlocal Fisher-KPP type equations of the form

∂u ∂t = -(-∆) α/2 u + u(σ -u),
where σ is a function to be thought of as a resource producing a birth rate proportional to it (see also [START_REF]Local approximation of arbitrary functions by solutions of nonlocal equations[END_REF]). This means that equations of the type of (2.6) may enjoy very different properties depending on the type of diffusion considered. The fractional Laplacian shares common features with the classical Laplacian. In particular, compactness properties are preserved and the function spaces naturally associated to it, although of a nonlocal nature, have similar properties. A result due to Caffarelli and Silvestre [START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF] shows that it may even be localized by adding a new variable.

By contrast with the fractional Laplacian, convolution operators of the type of (2.8) lack of strong compactness properties which makes their analytical treatment much more involved. In particular, there are no a priori regularity results in general and no powerful functional analytic framework similar to that provided by the (fractional) Sobolev spaces.

But the nonlocal setting presents a further difficulty. As a matter of fact, boundary value problems, for both (2.7) and (2.8), cannot be handled in the same way as in the local case because of the contribution of the diffusion coming from outside the domain. Moreover, if various results are known for general equations of the type of (2.6) in the whole space R N (see e.g. [START_REF] Bates | Travelling waves in a convolution model for phase transitions[END_REF][START_REF] Chen | Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations[END_REF][START_REF] Coville | Travelling fronts in asymmetric nonlocal reaction diffusion equations: the bistable and ignition cases[END_REF][START_REF] Fife | An integrodifferential analog of semilinear parabolic PDEs[END_REF][START_REF] Yagisita | Existence of traveling wave solutions for a nonlocal bistable equation: an abstract approach[END_REF]), numerous questions remain open when it comes to heterogeneous domains. Such problems are a major issue in population dynamics. In this regard, let us quote Turchin [START_REF] Turchin | Quantitative analysis of movement: measuring and modeling population redistribution in animals and plants[END_REF] who, in his 1998 monograph Quantitative Analysis of Movement, wrote that "the spatial dimension and, in particular, the interplay between environmental heterogeneity and individual movement, is an extremely important aspect of ecological dynamics."

The purpose of this thesis is to study some of these nonlocal features, especially those arising when considering operators of convolution type as in (2.8).

Contributions

The topics addressed in this thesis are quite different from a chapter to another, with the notable exception of Chapter 4 which can be seen as a follow-up to Chapter 3. A significant part of this thesis is dedicated to the study of models of the type ∂u ∂t (t, x) = ˆΩ J(x -y)(u(t, y) -u(t, x))dy + f (u(t, x)), set in some domain Ω ⊂ R N , albeit we sometimes emphasize more on the functional analytic aspects behind it than on the equation itself. Some of our contributions are thus of interest per se even outside the realm of population dynamics.

Our results sometimes overlap with other areas such as phase transitions in physics and material sciences (in particular Chapter 5 and, to some extent, Chapter 2). But, as pointed out at the end of Section 2.1 above, these topics are closely related to population dynamics.

This thesis is divided into five (mutually independent) chapters, split into two parts. The first part deals with regularity issues arising in nonlocal reaction-diffusion problems. The second part is mainly concerned with qualitative properties of solutions to some nonlocal problems with special emphasis on rigidity results. Possible extensions of our results are briefly discussed at the end of the thesis.

We present below the content of each chapter and our main contributions.

A functional analytic toolbox

Part I contains two chapters which are adapted from the following papers: * J. Brasseur: A Bourgain-Brezis-Mironescu characterization of higher order Besov-Nikol'skii spaces, Ann. Inst. Fourier (2017), (forthcoming).

A Bourgain-Brezis-Mironescu characterization of Besov-Nikol'skii spaces

In Chapter 1, we study a problem in functional analysis that emerges in the study of some nonlocal Fisher-KPP type equations. Precisely, we consider the equation

1 ε m (J ε * u(x) -u(x)) + u(x)(a(x) -u(x)) = 0 in R N , (3.1)
where ε > 0, m ∈ [0, 2], u is the density of a given population, J ε (z) := 1 ε N J z ε , with J ∈ C ∩ L 1 (R N ), is a symmetric positive dispersal kernel with unit mass having finite m-th order moment, and a ∈ C 2 (R N ) is a function satisfying lim sup |x|→∞ a(x) < 0.

The parameter ε is a measure of the spread of dispersal of the species and 1/ε m is a rate of dispersal which arises when considering a "cost function" (see [START_REF]Persistence criteria for populations with non-local dispersion[END_REF]).

As an example, one may think of a population of trees producing and dispersing seeds. Several dispersal strategies are then possible: either it disperses few seeds but over large distances (ε 1) or it disperses many of them but over smaller distances (ε 1). The parameter m measures the influence of the cost function on the different possible strategies.

Studying persistence of the population amounts to seeking for a positive solution to (3.1). Of particular interest is the asymptotic behavior of solutions, which allows to determine whether one of the extreme strategies (ε 1 or ε 1) yield persistence or extinction. But if we have a clear picture when ε → ∞ (see [START_REF]Persistence criteria for populations with non-local dispersion[END_REF]Theorems 1.3 and 1.4]) it is only poorly understood when ε → 0 + and 0 < m < 2.

The best result in this direction is due to Berestycki, Coville and Vo [START_REF]Persistence criteria for populations with non-local dispersion[END_REF]. They have proved that if J behaves sufficiently well, for example if it is compactly supported with J(0) > 0, and if a is such that max{a, 0} ≡ 0, then, when ε → 0 + , u ε converges pointwise almost everywhere to some non-negative bounded function v satisfying v(x)(a(x) -v(x)) = 0 in R N .

(3.2)

Unfortunately, this equation admits infinitely many solutions, so one cannot directly infer a persistence strategy for that case. However, it is known that solutions to (3.1) satisfy ˆRN ˆRN ρ ε (x -y) |u ε (x) -u ε (y)| 2 |x -y| m dxdy C for all ε > 0, (3.3) where ρ ε (z) = ε -m |z| m J ε (z), see [START_REF]Persistence criteria for populations with non-local dispersion[END_REF]Lemma 5.1(ii)]. This inequality is in fact the key tool which allows to handle the case m = 2. Indeed, (3.3) together with the recent characterization of Sobolev spaces derived by Bourgain, Brezis, Mironescu [START_REF] Bourgain | Another look at Sobolev spaces[END_REF] and Ponce [START_REF] Ponce | An estimate in the spirit of Poincaré's inequality[END_REF] implies that (u ε ) ε>0 is relatively compact in L 2 loc (R N ) and that it converges along a subsequence to some function v satisfying lim ε→0 + ˆRN ˆRN ρ ε (x -y) |v(x) -v(y)| 2 |x -y| 2 dxdy C, which, by the result of Bourgain et al. [START_REF] Bourgain | Another look at Sobolev spaces[END_REF]Theorem 2], is equivalent to saying that v belongs to the Sobolev space H 1 (R N ). Then, relying on standard elliptic theory, it can be shown that v is the unique nontrivial solution of

β ∆v(x) + v(x)(a(x) -v(x)) = 0,
where β > 0 is some constant depending on N and J.

Unfortunately, there is no such characterization when 0 < m < 2, which prevents from using this strategy. Whence, a detailed study of functionals of the type of (3.3) is needed. This is the main purpose of Chapter 1.

More precisely, given s ∈ (0, 1] and p ∈ [1, ∞), we study the properties of func-

tions f ∈ L p (R N ) satisfying ˆRN ˆRN ρ ε (x -y) |f (x) -f (y)| p |x -y| sp dxdy C as ε → 0 + , (3.4)
where (ρ ε ) ε>0 ⊂ L 1 (R N ) is a standard sequence of mollifiers, i.e. such that

            
ρ ε 0 a.e. in R N for any ε > 0, ˆRN ρ ε (z)dz = 1 for any ε > 0, lim ε→0 + ˆ|z| δ ρ ε (z)dz = 0 for all δ > 0.

(3.5)

In their paper, Berestycki et al. [START_REF]Persistence criteria for populations with non-local dispersion[END_REF] were expecting (3.4) with 0 < s < 1 to provide a description of fractional Sobolev spaces. However, the underlying space happens to be of a different nature. Precisely, we show that it coincides with the so-called Besov-Nikol'skii space B s p,∞ (R N ) (see Definition 3.3) provided that (ρ ε ) ε>0 satisfies

ρ ε (z) = 1 ε N ρ z ε for some ρ ∈ L 1 (R N ). (3.6)
That is, we prove the following Theorem -Let s ∈ (0, 1) and p ∈ [1, ∞). Let (ρ ε ) ε>0 ⊂ L 1 (R N ) be a sequence of radial functions satisfying (3.5) and (3.6). Then, f ∈ L p (R N ) satisfies (3.4) if, and only if, f ∈ B s p,∞ (R N ). Moreover,

[f ] p B s p,∞ (R N ) ∼ sup ε>0 ˆRN ˆRN ρ ε (x -y) |f (x) -f (y)| p |x -y| sp dxdy.
The main difficulty in establishing this result is that the techniques of proof used in [START_REF] Bourgain | Another look at Sobolev spaces[END_REF][START_REF] Brezis | How to recognize constant functions. connections with Sobolev spaces[END_REF][START_REF] Ponce | An estimate in the spirit of Poincaré's inequality[END_REF] to handle the case of Sobolev spaces do not adapt. There are at least two reasons for this: on the one hand, many properties of the growth rate which arises in (3.4) when s = 1 are lost in the fractional case s ∈ (0, 1) and, on the other hand, smooth functions are not dense in B s p,∞ (R N ). To cope with this, we develop a new strategy relying essentially on elementary arguments.

Further, we prove that an important property does no longer hold: compactness. When s = 1, it is known that any bounded sequence (f ε ) ε>0 ⊂ L p (R N ) satisfying ˆRN ˆRN ρ ε (x -y) |f ε (x) -f ε (y)| p |x -y| p dxdy C as ε → 0 + , must be relatively compact in L p loc (R N ) provided that (ρ ε ) ε>0 satisfies some mild symmetry properties (see e.g. [START_REF] Bourgain | Another look at Sobolev spaces[END_REF][START_REF] Ponce | An estimate in the spirit of Poincaré's inequality[END_REF]).

We show that, surprisingly, this does not extend to the fractional setting.

Theorem -Let s ∈ (0, 1) and p ∈ [1, ∞). Let (ρ ε ) ε>0 be a sequence of mollifiers of the form (1.8)

with ρ ∈ L 1 (R N ) satisfying ˆRN ρ(z)|z| p(1-s) dz < ∞.
Then, there exists a bounded sequence (f ε ) ε>0 ⊂ L p (R N ) satisfying ˆRN ˆRN ρ ε (x -y) |f ε (x) -f ε (y)| p |x -y| sp dxdy C as ε → 0 + , but which is not relatively compact in L p loc (R N ). Chapter 1 is in fact concerned with more general functionals than (3.4). Beyond theoretical interest, this extra degree of generality allows one to handle both the (delicate) case p = ∞ and the higher order case s 1 (see Theorem 2.3). A remarkable consequence of this is that it enables to derive new characterizations for both Lipschitz and Zygmund-Hölder spaces (see Theorem 2.5). So that, in the end, we obtain a unified approach for representing as diverse scales as Sobolev, BV , Lipschitz, Besov-Nikol'skii and Zygmund-Hölder spaces.

On another note, we exhibit several consequences of our results which allow to clarify the relationship between the usual Besov spaces B s p,q (R N ) with q < ∞ and the Nikol'skii spaces B s p,∞ (R N ), see in particular Examples 2.10, 2.11 and 2.19. Chapter 1 contains several other results with different flavors. In particular, we extend the by-now celebrated limiting embedding:

q -1/q ∇f L p (R N ) ∼ lim r→1 -(1 -r) 1/q f B r p,q (R N ) for 1 < p, q < ∞,
to the Lipschitz case p = ∞ (see Theorem 2.12). In the same vein, we investigate the analogue of this for Besov-Nikol'skii spaces. Namely, we ask wether it holds that sup 0<r<s (s -r) 1/q f B r p,q (R N ) , yields a equivalent semi-norm on B s p,∞ (R N ). It turns out that this is not true (see Theorem 2.14) which suggests that, differently from the integer order case, the restriction to (3.6) in Theorem 3.1 cannot be removed.

On restrictions of Besov functions

Chapter 2 is certainly our most theoretical contribution. There, we study the "restriction property" in the context of Besov spaces B s p,q (R N ). Function spaces X(R N ) having this property are those which admit that

f (•, y) ∈ X(R d ) for a.e. y ∈ R N -d ,
whenever f ∈ X(R N ). This property, which holds true on the vast majority of functions spaces, plays a fundamental role in lifting theory and, by extension, in some reaction-diffusion problems of Ginzburg-Landau type, see [START_REF] Bourgain | Another look at Sobolev spaces[END_REF][START_REF] Mironescu | Lifting in Besov spaces[END_REF].

With moderate work, it can be shown to hold in B s p,q (R N ) whenever q p in the whole range of relevant parameters, that is: 0 < q p ∞ and s > σ p where

σ p := N 1 p -1 + .
We prove that this is no longer the case when p < q. Namely, we obtain the following

Theorem -Let N 2, 1 d < N , 0 < p < q ∞ and let s > σ p . Then, there exists a function f ∈ B s p,q (R N ) such that f (•, y) / ∈ B s p,q (R d ) for a.e. y ∈ R N -d .
This result is doubly surprising since it has no comparable antecedents and because Besov spaces are mere microscopic modifications of fractional Sobolev spaces which are known to satisfy this property.

Our construction is nontrivial and requires advanced decomposition techniques, most notably subatomic decompositions.

One of the major property of these spaces (and what makes them so useful) is that they embed in nice spaces. This is particularly relevant for applications as it allows to grasp some additional regularity. For example, the standard embedding theorem (which holds independently of the value of q) says that

B s p,q (R N ) → A s,p (R N ), (3.7)
where A s,p (R N ) stands for

C s-N p (R N ), BMO(R N ) and L N p N -sp ,∞ (R N ),
when respectively sp > N , sp = N and sp < N (the precise definitions will be given in Chapter 2, see Definition 2.19). In particular, since B s p,q (R N ) enjoys the restriction property when q p, it is easily seen that

f (•, y) ∈ A s,p (R d ) for a.e. y ∈ R N -d . (3.8) whenever f ∈ B s p,q (R N ).
It is quite natural to ask wether (3.8) still holds when p < q. But we prove that even this weaker property fails.

Theorem -Let N 2, 1 d < N , 0 < p < q ∞ and let s > σ p . Then, there exists a function f ∈ B s p,q (R N ) such that f (•, y) / ∈ A s,p (R d ) for a.e. y ∈ R N -d .
Furthermore, we establish a positive result about the regularity of these restrictions. Precisely, we exhibit a function space X(R N ) that is intermediate between B s p,q (R N ) and B s p,q (R N ) for all 0 < s < s and such that

∀f ∈ B s p,q (R N ), f (•, y) ∈ X(R d ) for a.e. y ∈ R N -d
. This space X is a so-called Besov space of generalized smoothness, usually denoted by B (s,Ψ) p,q (we refer to Definition 2.11 for the definition of these spaces). In this setting, s remains the dominant smoothness parameter and Ψ is a positive function of log-type, called admissible, which allows encoding more general types of smoothness.

Of course, this depends on the interplay between Ψ and the parameters p and q. We prove that restrictions of Besov functions to almost every hyperplanes belong to the space B (s,Ψ) p,q (R d ), whenever

j 0 Ψ(2 -j ) χ < ∞, (3.9)
where χ = qp q-p (resp.

χ = p if q = ∞). Theorem -Let N 2, 1 d < N , 0 < p < q ∞,
s > σ p and let Ψ be an admissible function satisfying (3.9). Suppose that f ∈ B s p,q (R N ). Then,

f (•, y) ∈ B (s,Ψ) p,q (R d ) for a.e. y ∈ R N -d .
We also prove that the condition (3.9) is optimal, at least when q = ∞. When q < ∞, it still can be shown to be sharp but under the additional requirement that

χ < 1 c ∞ where c ∞ := sup 0<t 1 log 2 Ψ(t) Ψ(t 2 ) , (3.10)
and χ is as in (3.9). In other words, we arrive at a sharp characterization of the aforementioned loss of regularity.

Theorem -Let N 2, 1 d < N , 0 < p < q ∞, s > σ p and let Ψ be an admissible function that does not satisfy (3.9). If q < ∞ and Ψ is increasing suppose, in addition, that (3.10) holds true. Then, there is a function f ∈ B s p,q (R N ) such that f (•, y) / ∈ B (s,Ψ) p,q (R d ) for a.e. y ∈ R N -d . In the course of Chapter 2 we also establish the analogue of the above results in the setting of Besov spaces of generalized smoothness. This is of independent interest given their relevance in stochastic calculus and in the theory of pseudo-differential operators (where they appear in a natural way), see e.g. [START_REF] Almeida | Real interpolation of generalized Besov-Hardy spaces and applications[END_REF][START_REF] Jacob | Pseudo differential operators & Markov processes[END_REF][START_REF] Leopold | On Besov spaces of variable order of differentiation[END_REF][START_REF] Leopold | Trace theorems for Sobolev spaces of variable order of differentiation[END_REF][START_REF] Schneider | Wavelet solution of variable order pseudodifferential equations[END_REF].

These results may find applications in lifting theory and in the study of some turbulence phenomena in fluid mechanics, where the need for this type of property recently appeared, see [START_REF] Jaffard | Théorèmes de trace et "dimensions négatives[END_REF] and references therein. Also, they suggest an implicit link with other results related to fractal geometry and may lead to further developments in this direction, see [START_REF] Aubry | Local behavior of traces of Besov functions: prevalent results[END_REF][START_REF] Edmunds | Spectral theory for isotropic fractal drums[END_REF][START_REF]Eigenfrequencies of isotropic fractal drums[END_REF].

Rigidity results

Part II contains three chapters which are adapted from the following papers:

Liouville type results for a nonlocal obstacle problem

Chapter 3 is a work in collaboration with J. Coville, F. Hamel and E. Valdinoci. To a certain extent, it may be seen as a contribution towards the following problem: how does the geometry of the environment affects the evolution of a population with nonlocal dispersal?

(3.11)

The focus here is on perforated domains, that is when the environment possesses some inaccessible regions. To be more specific, Chapter 3 is concerned with qualitative properties of solutions to nonlocal reaction-diffusion equations of the form

Lu + f (u) = 0 in R N \ K, (3.12)
where K ⊂ R N is a compact "obstacle", L is the nonlocal operator given by

Lu(x) := ˆRN \K J(x -y)(u(y) -u(x))dy, (3.13)
and f ∈ C 1 ([0, 1]) is a bistable nonlinearity. To clarify the ideas, suppose that

f (u) = λu(1 -u)(u -θ)
for some θ ∈ (0, 1) and λ > 0.

Before we go any further, some comments are in order. This problem may be thought of as a nonlocal version of

∆u + f (u) = 0 in R N \ K, ∇u • ν = 0 on ∂K, (3.14)
where ν is the outward unit vector normal to K (assuming that K is smooth enough).

The local problem (3.14) was first studied by Berestycki, Hamel and Matano in [START_REF] Berestycki | Bistable travelling waves around an obstacle[END_REF]. There, it is shown that there exist an entire solution u(t, x) to the parabolic problem

   ∂u ∂t = ∆u + f (u) in R × R N \ K, ∇u • ν = 0 on R × ∂K, (3.15)
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| → +∞. (3.16)
This latter solution is 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.
Moreover, this result is independent of the geometry of K. Of course, this does not mean that the geometry does not play a role: its influence is encoded in the stationary solution u ∞ . Berestycki et al. proved that if the obstacle K satisfies some "good" geometrical properties, for example if K is starshaped, then the solution u ∞ to (3.16) must be identically equal to 1 in the whole set R N \ K (see [START_REF] Berestycki | Bistable travelling waves around an obstacle[END_REF]Theorem 6.1]). Per contra, if K is no longer starshaped but merely simply connected, they show that this Liouville type property may fail.

If we interpret u as the density of a population with bistable growth this means that, after some large time, whether the population tends to occupy the whole space depends on the geometry of K.

In Chapter 3, we deal with qualitative properties of solutions to equation (3.12), together with some asymptotic limiting conditions at infinity similar to those appearing in (3.16). Namely, we will be concerned with solutions to

     Lu + f (u) = 0 in R N \ K, 0 u 1 in R N \ K, u(x) → 1 as |x| → +∞. (3.17)
By analogy, it is natural to expect the influence of the environment in the nonlocal setting to be encoded by (3.17). Therefore, a first step towards answering (3.11) lies in the study of solutions to (3.17).

The goal of Chapter 3 is to find geometrical conditions on K ensuring that the solutions to (3.17) are identically equal to 1 in the whole set R N \ K.

We show that this holds true whenever K is convex.

Theorem -Let K ⊂ R N be a compact convex set and let u ∈ C(R N \ K, [0, 1]) be a function satisfying Lu + f (u) 0 in R N \ K, u(x) → 1 as |x| → +∞. (3.18)
Then, u = 1 in R N \ K.

Whereas the solutions to (3.16) are automatically classical C 2 solutions (by standard elliptic theory), there is, in general, no smoothing effect for (3.18) and the solutions may even not be continuous at all. This is the reason why we require some a priori continuity.

Nonetheless, if we ask for a solution instead of a super-solution, we prove that it is possible to get rid of this a priori assumption provided that the nonlinearity does not vary "too much". Precisely,

Theorem -Let K ⊂ R N be a compact convex set. Suppose that max [0,1] f < 1 2 . (3.19) Let u : R N \ K → [0, 1] be a measurable function satisfying Lu + f (u) = 0 a.e. in R N \ K, u(x) → 1 as |x| → +∞.
Then, u = 1 a.e. in R N \ K.

We prove several improvement of these results. In particular, we prove that when J is compactly supported and square integrable, then the requirement on the asymptotic behavior of u can be weakened to sup

R N \K u = 1,
which is even stronger than what is known in the local case (see Theorem 2.4).

Chapter 3 also contains a "robustness" result. Namely, we prove that if a compact set K is sufficiently close to a compact convex set (in the C 0,α topology), then the Liouville property still holds (see Theorem 2.6). In fact, this result is a sort of dichotomy: if the Liouville property can be shown to hold for a given set K, then it still holds for all compact sets sufficiently close it (in the C 0,α topology).

Our results are mainly based on sub-and super-solutions techniques and, to this end, we prove various comparison principles which are of independent interest (see Lemmata 4.1, 4.2 and 4.3).

A counterexample to the Liouville property of some nonlocal problems

Chapter 4 is a work in collaboration with J. Coville. It is meant as a follow-up to Chapter 3, although it may be read independently. The purpose here is to better pinpoint the geometrical conditions under which the Liouville property established at the previous chapter remains valid. An angle of this question consists in finding an obstacle K for which it does not hold. Chapter 4 provides an answer to this question. Precisely, we establish the following Theorem -There are (non-starshaped) simply connected compact obstacles K and data f and J for which problem

Lu + f (u) = 0 in R N \ K, u(x) → 1 as |x| → +∞, has a solution u ∈ C(R N \ K, [0, 1]) which is not identically equal to 1 in R N \ K.
The main difficulty in our construction lies in the lack of compactness. We show how to circumvent this difficulty relying on the Bourgain-Brezis-Mironescu characterization of Sobolev spaces. This enables us to obtain a priori estimates which allow the use of variational methods.

Remarkably, our construction is flexible enough to handle broader classes of nonlocal operators where the dispersal process need not be isotropic but instead depends on the geodesic distance between points in R N \ K. For example, we prove that the same result holds when L is replaced by

L g u(x) := ˆRN \K J(d g (x, y))(u(y) -u(x))dy, (3.20)
where

d g (•, •) is the geodesic distance on R N \ K and J ∈ L 1 loc (0, ∞) is a locally integrable kernel such that sup x∈R N \K ˆRN \K J(d g (x, y))dy < ∞.
Operators of the type of (3.20) are actually of interest in their own right. They give an alternative way to describe the evolution of individuals in a heterogeneous medium which, in some situations, may be regarded as more realistic. Their most interesting feature is that the individuals can no longer travel through the obstacle: they are compelled to bypass it as if it were a material obstacle.

Some monotonicity results for general systems of nonlinear elliptic PDEs

Chapter 5 is a work in collaboration with S. Dipierro. It is concerned with symmetry and monotonicity properties of general systems of nonlinear elliptic partial differential equations. Although the topic is rather old, recent years have seen a renewed interest in these questions. A typical example which has raised a lot of attention is:

     ∆u = uv 2 , ∆v = vu 2 , u, v > 0. (3.21)
This system emerges in the study of phase separation phenomena for Bose-Einstein condensates with multiple states. A natural question to ask is: under which conditions do u and v enjoy monotonicity properties? (3.22) Such rigidity property plays an important role in the study of (3.21), inter alia, it allows the classification of solutions. In this perspective, it is also of interest to ask for De Giorgi type results. For example:

under which conditions are u and v one-dimensional? (3.23) Some answers to (3.22) and (3.23) are known, see for example [START_REF] Berestycki | On phase-separation models: asymptotics and qualitative properties[END_REF][START_REF] Berestycki | On entire solutions of an elliptic system modeling phase separations[END_REF][START_REF] Farina | Some symmetry results for entire solutions of an elliptic system arising in phase separation[END_REF][START_REF] Wang | On the uniqueness of solutions of a nonlocal elliptic system[END_REF]. However, if (3.21) has a relatively "simple" form, things can get significantly tougher when considering more general nonlinearities and (possibly anomalous) diffusion processes. For example, one may want to address similar questions for systems of the form:

(-∆) s 1 u = F 1 (u, v), (-∆) s 2 v = F 2 (u, v), (3.24)
where s 1 , s 2 ∈ (0, 1] and F 1 , F 2 are the derivatives with respect to the first and the second variable, respectively, of some function F ∈ C 1,1 (R 2 ). Systems of the type of (3.24) have recently been investigated (see e.g. [START_REF] Terracini | Uniform Hölder regularity with small exponent in competition-fractional diffusion systems[END_REF][START_REF]Uniform Hölder bounds for strongly competing systems involving the square root of the Laplacian[END_REF][START_REF] Verzini | Strong competition versus fractional diffusion: the case of Lotka-Volterra interaction[END_REF][START_REF] Wang | On the uniqueness of solutions of a nonlocal elliptic system[END_REF]) and some symmetry results have been obtained under various assumptions (see e.g. [START_REF] Dipierro | A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian[END_REF][START_REF] Fazly | Symmetry results for fractional elliptic systems and related problems[END_REF]). The purpose of Chapter 5 is to find a method providing an answer to (3.22) and (3.23) in a general setting (including (3.21) and (3.24) as particular cases).

Precisely, the goal of Chapter 5 is to show that minima and stable solutions of general energy functionals of the form

E (u, v) = ˆΩ F (∇u, ∇v, u, v, x)dx,
enjoy some monotonicity properties, under an assumption on the growth at infinity of the energy, which we call the "stability inequality" (see (3.26)

below).

This setting allows not only to handle systems with general nonlinearities, but also with quite general diffusion operators, possibly of degenerate type. The most distinguished examples are the p-Laplacian and the mean curvature operator. By the Caffarelli-Silvestre extension theorem [START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF], this covers also fractional (and, hence, nonlocal) operators such as the fractional Laplacian.

The usual approach to obtain such rigidity results is to apply some stability inequality to a cut-off function. But this approach is generally difficult to implement because it requires to work with the precise form of the energy (which can be quite complicated).

Our strategy, inspired by [START_REF] Savin | Regularity of nonlocal minimal cones in dimension 2[END_REF][START_REF]Some monotonicity results for minimizers in the calculus of variations[END_REF], relies on a completely different argument. Loosely speaking, it consists in comparing the energies of (u, v) and a perturbed translation of itself. Then, using the stability inequality and a contradiction argument, we can prove that the solutions indeed enjoy some rigidity properties. In doing so, we do not need to work with the precise form of the potential F and we can therefore deal with general energy functionals.

Our working hypotheses are the following. First, we assume that both the domain Ω and the potential F are invariant under translation in the e N -direction, namely

Ω = V × R for some V ⊆ R N -1 ,
and F does not depend on the x N -coordinate. We also suppose that

F = F (p 1 , p 2 , z 1 , z 2 , x ) ∈ C(R 2N × R 2 × V ), (3.25)
and that F is C 2 and convex with respect to the first two variables. Also, we assume that there exists a constant C > 0 such that, for any

p 1 , p 2 , q 1 , q 2 ∈ R N with |q 1 | |p 1 • e N |/4 and |q 2 | |p 2 • e N |/4, it holds that |F p 1 p 1 (p 1 + q 1 , p 2 , z 1 , z 2 , x )| C|F p 1 p 1 (p 1 , p 2 , z 1 , z 2 , x )|, |F p 2 p 2 (p 1 , p 2 + q 2 , z 1 , z 2 , x )| C|F p 2 p 2 (p 1 , p 2 , z 1 , z 2 , x )|, |F p 1 p 2 (p 1 + q 1 , p 2 + q 2 , z 1 , z 2 , x )| C|F p 1 p 2 (p 1 , p 2 , z 1 , z 2 , x )|.
Notice that these assumptions are fairly general and apply, in particular, to (3.21) and (3.24).

The first result of Chapter 5 deals with minimizers. Namely, we obtain the following

Theorem -Let u, v ∈ C 1 (Ω) be such that (u, v) is a minimizer of E and that the growth condition ˆΩ∩B R |F p 1 p 1 ||∇u| 2 + |F p 2 p 2 ||∇v| 2 + |F p 1 p 2 ||∇u||∇v| dx R 2 , ( 3.26)
is satisfied for large enough R > 0, where the derivatives of F are evaluated at (∇u, ∇v, u, v, x ). Then, u and v are monotone on each line in the e N -direction, i.e., for any x ∈ Ω, either u N (x+te N ) 0 or u N (x+te N ) 0, and either v N (x+te N ) 0 or v N (x + te N ) 0, for any t ∈ R. In particular, u and v are one-dimensional.

Moreover, we show that a similar result holds for stable solutions up to a slight additional regularity requirement on F and (u, v).

Theorem -Suppose that F ∈ C 3,α (R 2N × R 2 × V ). Let (u, v) be such that either u, v ∈ C 0,1 (Ω) are convex or u, v ∈ C 1,1 (Ω)
. Moreover, suppose that (u, v) is a stable solution of E , and that the growth condition (3.26) holds true. Then, u and v are monotone in the e N -direction, i.e. either u N 0 or u N 0 and either v N 0 or v N 0 in Ω.

The results contained in Chapter 5 are, in fact, slightly more general. They still hold when replacing the notions of stable and minimal solutions by weaker ones. Indeed, we prove that it suffices to consider minimizers and stable solutions among the class of functions which are obtained by piecewise Lipschitz deformations in the e N -direction (see Theorems 1.5 and 1.10).

Several applications of our results are presented at the end of Chapter 5.

Part I A functional analytic toolbox

Chapter 1

A Bourgain-Brezis-Mironescu characterization of higher order Besov-Nikol'skii spaces

            ρ ε 0 a.e. in R N for any ε > 0, ˆRN ρ ε (z)dz = 1 for any ε > 0, lim ε→0 + ˆ|z| δ ρ ε (z)dz = 0 for all δ > 0. (1.1) Let M ∈ N * , 1 p < ∞ and s ∈ (0, M ]. We are interested in the properties of functions f ∈ L p (R N ) satisfying ˆRN ρ ε (h) ω ˆRN |∆ M h f (x)| p |h| sp dx dh C as ε → 0 + , (1.2)
where ω : R + → R + is an increasing, concave function and ∆ M h f (x) stands for the usual M -th order forward difference of f given by

∆ M h f (x) := M j=0 (-1) M -j M j f (x + hj), x, h ∈ R N . (1.
3)

The assumptions on ω will be made precise later on. Functionals of the type of (1.2) were initially introduced by Bourgain, Brezis and Mironescu [START_REF] Bourgain | Another look at Sobolev spaces[END_REF][START_REF] Brezis | How to recognize constant functions. connections with Sobolev spaces[END_REF] to obtain a new characterization of the Sobolev space W 1,p (R N ). Namely, for M = s = 1 and ω(t

) = t, (1.2) reads ˆRN ˆRN ρ ε (h) |f (x + h) -f (x)| p |h| p dxdh C as ε → 0 + , (1.4)
and the result of Bourgain, Brezis and Mironescu states that, any

f ∈ L p (R N ) satisfying (1.4) belongs to the Sobolev space W 1,p (R N ) if 1 < p < ∞, or to BV (R N ) if p = 1, provided (ρ ε ) ε>0 is radial.
More precisely, they have shown that

lim ε→0 + ˆRN ˆRN ρ ε (h) |f (x + h) -f (x)| p |h| p dxdh = K p,N ∇f p L p (R N ) , where K p,N := ˆSN-1 |σ • e| p dH N -1 (σ), e ∈ S N -1 .
As a result, they were able to establish the following limiting embedding

lim r→1 -(1 -r)p f p W r,p (R N ) = K p,N ∇f p L p (R N ) . (1.5)
Since this work, numerous new characterizations of the Sobolev spaces W k,p (R N ) or BV (R N ) have been obtained [START_REF] Bojarski | How to recognize polynomials in higher order Sobolev spaces[END_REF][START_REF] Borghol | Some properties of Sobolev spaces[END_REF][START_REF] Dávila | On an open question about functions of bounded variations[END_REF][START_REF] Ferreira | Characterization of polynomials and higher-order Sobolev spaces in terms of functionals involving difference quotients[END_REF][START_REF] Ponce | An estimate in the spirit of Poincaré's inequality[END_REF][START_REF]A new approach to Sobolev spaces and connections to Gammaconvergence[END_REF][START_REF] Spector | Characterization of Sobolev and BV spaces[END_REF] and various asymptotic formulae characterizing the Sobolev norms in terms of fractional norms have been derived [START_REF] Karadzhov | Limits of higher-order Besov spaces and sharp reiteration theorems[END_REF][START_REF] Kolyada | On limiting embeddings of Besov spaces[END_REF][START_REF] Maz | On the Bourgain, Brezis and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces[END_REF][START_REF]Limits of Besov norms[END_REF]. For instance, Maz'ya and Shaposhnikova [START_REF] Maz | On the Bourgain, Brezis and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces[END_REF] obtained the counterpart of (1.5) in the critical case r → 0 + , that is

lim r→0 + rp f p W r,p (R N ) = 2σ N f p L p (R N ) ,
whenever f ∈ 0<r<1 W r,p (R N ) and where σ N stands for the superficial measure of the unit sphere S N -1 . Also, let us mention the work of Ponce [START_REF]A new approach to Sobolev spaces and connections to Gammaconvergence[END_REF] who was the first to obtain a characterization of the space

BV (R N ) in terms of a class of functions in L 1 (R N ) satisfying ˆRN ˆRN ρ ε (h) Ω |f (x + h) -f (x)| |h| dxdh C as ε → 0 + , (1.6)
under suitable growth assumptions on Ω ∈ C(R + , R + ).

More recently, such type of characterizations were extended by Borghol [START_REF] Borghol | Some properties of Sobolev spaces[END_REF], Bojarski, Ihnatsyeva, Kinnunen [START_REF] Bojarski | How to recognize polynomials in higher order Sobolev spaces[END_REF] and Ferreira, Kreisbeck and Ribeiro [START_REF] Ferreira | Characterization of polynomials and higher-order Sobolev spaces in terms of functionals involving difference quotients[END_REF], who considered the cases 1 < p < ∞ in higher order Sobolev spaces. Typically, in [START_REF] Ferreira | Characterization of polynomials and higher-order Sobolev spaces in terms of functionals involving difference quotients[END_REF] it is shown that the spaces W k,p (R N ), with p ∈ (1, ∞) and k ∈ N * , can be characterized by quantities of the type

ˆRN ˆRN ρ ε (h) Ω |∆ k h f (x)| |h| k dxdh, (1.7)
where Ω : R + → R + is an increasing, convex function such that

m 1 t p Ω(t) m 2 t p ,
for all t 0 and some positive constants 0 < m 1 < m 2 .

To our knowledge, very few is known in the case 0 < s < M . Nonetheless, recent works of Lamy and Mironescu [START_REF] Lamy | Characterization of function spaces via low regularity mollifiers[END_REF] suggest a connection between expressions of the type of (1.2) and Besov spaces. In [START_REF] Lamy | Characterization of function spaces via low regularity mollifiers[END_REF], the authors prove the following Theorem 1.1 (Lamy,Mironescu,[START_REF] Lamy | Characterization of function spaces via low regularity mollifiers[END_REF]). -Let s > 0, p, q ∈ [1, ∞] and let

(ρ ε ) ε>0 ⊂ L 1 (R N ) satisfying (1.1) and such that ρ ε (h) = 1 ε N ρ h ε for some ρ ∈ L 1 (R N ). (1.8)
Then,

f B s p,q (R N ) f L p (R N ) + 1 ε s f * ρ ε -f L p (R N )
L q ((0,1), dε ε )

.

(1.9)

The converse of this holds under some additional moment condition on ρ (see [START_REF] Lamy | Characterization of function spaces via low regularity mollifiers[END_REF] for further details). In fact, the case q = ∞ is not properly stated nor explicitly proven in [START_REF] Lamy | Characterization of function spaces via low regularity mollifiers[END_REF]. To fill this gap, we shall give some additional details at the end of the chapter. A consequence of this, which has not been noticed in [START_REF] Lamy | Characterization of function spaces via low regularity mollifiers[END_REF], is the following

Proposition 1.2. -Let s ∈ (0, 1), p ∈ [1, ∞) and (ρ ε ) ε>0 ⊂ L 1 (R N ) satisfying (1.1)
and (1.8). Then, the following statements are equivalent:

(i) f ∈ B s p,∞ (R N ), (ii) f ∈ L p (R N ) satisfies ˆRN ˆRN ρ ε (h) |f (x + h) -f (x)| p |h| sp dxdh C as ε → 0 + . (1.10) Moreover, f p B s p,∞ (R N ) ∼ f p L p (R N ) + sup ε∈(0,1) ˆRN ˆRN ρ ε (h) |f (x + h) -f (x)| p |h| sp dxdh. (1.11)
It is worth noticing that, by contrast with the representation of B s p,∞ (R N ) obtained in [START_REF] Lamy | Characterization of function spaces via low regularity mollifiers[END_REF], no moment condition on ρ ε is needed. Moreover, since ρ ε does not need to be radial, some directions may be privileged, yet with no impact on the resulting norm. This is in clear contrast with the case s = 1 (see also [START_REF] Chiron | On the definitions of Sobolev and BV spaces into singular spaces and the trace problem[END_REF]Remark 10] or [START_REF]A new approach to Sobolev spaces and connections to Gammaconvergence[END_REF]Corollary 3,p.232]).

This sheds new lights on how to describe smoothness and could be of potential interest in some problems of the calculus of variations and in the study of some integro-differential equations (see e.g. [START_REF] Andreu-Vaillo | Nonlocal diffusion problems[END_REF][START_REF] Aubert | Can the nonlocal characterization of Sobolev spaces by Bourgain et al. be useful to solve variational problems?[END_REF][START_REF]Persistence criteria for populations with non-local dispersion[END_REF][START_REF] Gilboa | Nonlocal linear image regularization and supervised segmentation[END_REF][START_REF]Nonlocal operators with applications to image processing[END_REF][START_REF] Trageser | Local and nonlocal models in thin-plate and bridge dynamics[END_REF]).

Also, in view of Proposition 1.2, it is natural to ask for corresponding assertions of (1.6) and (1.7) in the framework of the fractional Besov-Nikol'skii spaces B s p,∞ (R N ). For example: what can be said about the limiting behavior of (1.10) when ε → 0 + ? Can one describe higher order Besov-Nikol'skii spaces via expressions of the type (1.2)? It is the main concern of this chapter to deal with these issues.

Main Motivation

This work originates in a problem raised in [START_REF]Persistence criteria for populations with non-local dispersion[END_REF]. Consider the heterogeneous Fisher-KPP equation:

1 ε m J ε * u(x) -u(x) + f (x, u) = 0, u = u ε , x ∈ R N , ε > 0, (1.12)
where m ∈ [0, 2], u is the density of a given population, J ε (z) := 1 ε N J z ε , with J ∈ C ∩ L 1 (R N ) a symmetric positive dispersal kernel with unit mass and having finite m-th order moment, and f ∈ C 1,α (R N +1 ) is a heterogeneous KPP type nonlinearity, that is:

     f (•, 0) = 0, for all x ∈ R N , f (x, s)/s is decreasing with respect to s ∈ (0, ∞), there exists S(x) ∈ C(R N ) ∩ L ∞ (R N ) such that f (•, S(•)) 0.
For the sake of simplicity, we restrict our attention to non-linearities of the form

f (x, s) = s(a(x) -s), with lim sup |x|→∞ a(x) < 0.
Roughly speaking, f models the growth rate of the population and J the probability to jump from one location to another. The parameter ε is a measure of the spread of dispersal of the species. The scaling term 1 ε m can be interpreted as the rate of dispersal of the species. It arises when considering a cost function (see [START_REF]Persistence criteria for populations with non-local dispersion[END_REF]Section 2] for a more detailed explanation on the matter). Consider for instance a tree producing and dispersing seeds. Then, ε 1 represents a strategy where the dispersal rate is large but the seeds are spread over smaller distances, and ε 1 represents the opposite strategy (i.e. smaller dispersal rate but the seeds are spread over larger distances). As for the parameter m, it measures the influence of the cost function on the different strategies.

Existence of positive solutions to (1.12) is naturally expected to provide a persistence criteria for the population under consideration. Nonetheless, if the asymptotic behavior of solutions to (1.12) is quite well understood when ε → ∞ (see [START_REF]Persistence criteria for populations with non-local dispersion[END_REF]), it is not the case when ε → 0 + and 0 < m < 2. Berestycki et al. [START_REF]Persistence criteria for populations with non-local dispersion[END_REF] were able to prove the following result.

Theorem 1.3 (Berestycki, Coville, Vo, [START_REF]Persistence criteria for populations with non-local dispersion[END_REF]). -Assume that J is compactly supported with J(0) > 0, that m ∈ (0, 2), that max{a, 0} ≡ 0 and that a ∈ C 2 (R N ). Then, when ε → 0 + , the solution u ε of (1.12) converges almost everywhere to some non-negative bounded function v satisfying

v(x)(a(x) -v(x)) = 0 in R N .
(1.13)

Unfortunately, equation (1.13) admits infinitely many solutions, so it may happen that v ≡ 0 (extinction) or that v = a + 1 K for some compact K ⊂ supp(a + ) (persistence in a given area of the ecological niche). Whence, one cannot directly infer a persistence strategy for that case.

However, it is known that solutions to (1.12), when they exist, satisfy

ˆRN ˆRN ρ ε (x -y) |u ε (x) -u ε (y)| 2 |x -y| m dxdy C for all ε > 0, (1.14)
with ρ ε (z) = ε -m |z| m J ε (z) a smooth mollifier satisfying (1.1) (see [START_REF]Persistence criteria for populations with non-local dispersion[END_REF]Lemma 5.1(ii)] for a proof).

To quote Berestycki et al.: "If for the case m = 2 we could rely on elliptic regularity and the new description of Sobolev spaces developed in Bourgain et al. [START_REF] Bourgain | Another look at Sobolev spaces[END_REF], Brezis [START_REF] Brezis | How to recognize constant functions. connections with Sobolev spaces[END_REF], Ponce [START_REF] Ponce | An estimate in the spirit of Poincaré's inequality[END_REF][START_REF]A new approach to Sobolev spaces and connections to Gammaconvergence[END_REF] to get some compactness, this characterization does not allow us to treat the case m < 2. We believe that a new characterization of fractional Sobolev spaces in the spirit of the work of Bourgain, Brezis and Mironescu [START_REF] Bourgain | Another look at Sobolev spaces[END_REF][START_REF] Brezis | How to recognize constant functions. connections with Sobolev spaces[END_REF] will be helpful to resolve this issue."

This motivates the study of general classes of functions of the type of (1.2), in particular the forthcoming Theorems 2.3 and 2.15.

Comments

If (1.10) is very similar to (1.4), the underlying spaces, W 1,p (R N ) and B s p,∞ (R N ), are very different in nature and one has to cope with some technicalities. Among others, it is not clear anymore whether the limit of (1.10) as ε → 0 + exists nor, even if it does, whether it provides an equivalent semi-norm. In the integer order case, things are not too controversial in the sense that

∇f L p (R N ) ∼ lim sup |h|→0 ∆ 1 h f L p (R N ) |h| = sup h =0 ∆ 1 h f L p (R N ) |h| . (1.15)
(see e.g. [START_REF]Limits of Besov norms[END_REF] or Lemma 7.1), while the counterpart of (1.15) in the fractional case s ∈ (0, 1) is not true in general. Indeed, every nontrivial function

f ∈ C ∞ c (R N ) satisfies lim |h|→0 ∆ 1 h f L p (R N ) |h| s = 0 < sup h =0 ∆ 1 h f L p (R N ) |h| s = [f ] B s p,∞ (R N ) , (1.16)
whenever s ∈ (0, 1), p ∈ [1, ∞]. Finiteness of either or both the two first terms in the left-hand side of (1.16) equally describes B s p,∞ (R N ) in the sense that they define the same set of functions. But the respective (semi-)norms induced by these quantities are not equivalent (see Section 5). For these reasons, at some places, it will be more convenient to state our results in terms of suprema as in (1.11) instead of limits.

On the other hand, smooth functions are not dense in B s p,∞ (R N ), so that the arguments used in the integer case do not simply adapt. We show how to do this in a way that allows, not only to give a meaning, but also to handle the tricky case p = ∞ in both the integer and the fractional case, using only elementary arguments. Also, in the particular case where ρ is radially symmetric, we improve (1.11) to a seminorm equivalence at all orders s > 0. More general quantities are also investigated as well as compactness in the case of a sequence (f ε ) ε>0 ⊂ L p (R N ).

At the end, this yields a common nonlocal description for the Besov-Nikol'skii spaces B s p,∞ (R N ), the Hölder-Zygmund spaces C s (R N ), the BV (R N ) space, the Sobolev spaces W k,p (R N ) and the Lipschitz space C 0,1 (R N ). As a by-product, we obtain new characterizations for these spaces and a new limiting embedding between Lipschitz and Besov spaces which extends the previous known results.

Main results

A new characterization of Besov-Nikol'skii spaces

To state our results, we shall introduce some notations and terminology. Definition 2.1. -A function ω : R + → R + is said to be roughly subadditive if there exists a constant A > 0 such that, ω(t 1 + t 2 ) A {ω(t 1 ) + ω(t 2 )} , for every t 1 , t 2 ∈ R + . If A = 1, then we say that ω is subadditive.

To shorten our statements, it will be more convenient to call C inc the set of all continuous, increasing functions ω : [0, ∞) → [0, ∞) satisfying ω(0) = 0 and lim t→∞ ω(t) = ∞. Also, we set [START_REF] Bruckner | Some function classes related to the class of convex functions[END_REF]Theorem 5]). As indicated by the example of t α with α > 1, C + inc contains also some convex functions as long as they do not increase too fast. Indeed, a direct computation

C + inc := ω ∈ C inc such that ω is roughly subadditive . Remark 2.2. -Observe that if ω 1 , ω 2 ∈ C + inc , then ω 1 • ω 2 ∈ C + inc . Typical examples of functions in C + inc are: (i) ω 1 (t) = t α with α > 0, (ii) ω 2 (t) = ln(1 + t), (iii) ω 3 (t) = t tanh(t), (iv) ω 4 (t) = arsinh(t), ... More generally, if ω ∈ C inc is concave, then ω ∈ C + inc (see e.g.
shows that if ω : [0, ∞) → [0, ∞) is a continuous, convex function with ω(0) = 0 and if ω(2t)
κ ω(t), for all t 0 and some constant κ > 0 (independent of t), then ω ∈ C + inc . Our first result reads as follows

Theorem 2.3. -Let M ∈ N * , s ∈ (0, M ) and p ∈ [1, ∞]. Let ω ∈ C +
inc and (ρ ε ) ε>0 ⊂ L 1 (R N ) be a sequence of radial functions satisfying (1.1) and (1.8). Then, the following statements are equivalent:

(i) f ∈ B s p,∞ (R N ), (ii) f ∈ L p (R N ) is such that ˆRN ρ ε (h) ω ∆ M h f L p (R N ) |h| s dh C as ε → 0 + . (2.1) Moreover, ω [f ] B s p,∞ (R N ) ∼ sup ε>0 ˆRN ρ ε (h) ω ∆ M h f L p (R N ) |h| s dh.
Remark 2.4. -It is noteworthy that the assumptions of Theorem 2.3 are somehow self-improving. For example, if ω ∈ C inc is such that

α 1 ω ω α 2 ω a.e. in R N ,
for some ω, ω ∈ C + inc and α 1 , α 2 > 0, then ω still characterizes B s p,∞ (R N ). Note also that the Jensen inequality allows to extend this result to convex ω ∈ C inc .

Moreover, the conclusion of Theorem 2.3 still holds under the slightly weaker assumption that (ρ ε ) ε>0 ⊂ L 1 (R N ) satisfies (1.1) and (1.8) with ρ ∈ L 1 (R N ) such that there exists a number δ > 0 and a nonnegative radial function ϕ with ρ ϕ a.e. in B δ and ´Bδ ϕ > 0.

Also, when 1 p < ∞, the fact that ω ∈ C + inc allows one to replace (2.1) by

ˆRN ρ ε (h) ω ˆRN Ω |∆ M h f (x)| |h| s dx dh, (2.2)
for any continuously increasing Ω : R + → R + with Ω(0) = 0 and

m 1 t p Ω(t) m 2 t p , (2.3)
for all t 0 and some 0 < m 1 m 2 .

By the same token, we obtain the following counterpart for the Lipschitz space.

Theorem 2.5. -Let ω ∈ C + inc and (ρ ε ) ε>0 ⊂ L 1 (R N ) be a sequence of radial functions satisfying (1.1) and (1.8). Then, the following statements are equivalent:

(i) f ∈ C 0,1 (R N ), (ii) f ∈ L ∞ (R N ) is such that ˆRN ρ ε (h) ω f (• + h) -f L ∞ (R N ) |h| dh C as ε → 0 + . Moreover, ω [f ] C 0,1 (R N ) ∼ lim sup ε→0 + ˆRN ρ ε (h) ω f (• + h) -f L ∞ (R N ) |h| dh.
In fact, our proof also allows to cover first order Sobolev spaces. For example, in view of (2.2), we have the

Theorem 2.6. -Let 1 p < ∞, (ω, Ω) ∈ C + inc × C inc with Ω satisfying (2.
3) and (ρ ε ) ε>0 ⊂ L 1 (R N ) be a sequence of radial functions satisfying (1.1) and (1.8). Then, the following statements are equivalent:

(i) f ∈ W 1,p (R N ) (resp. f ∈ BV (R N ) if p = 1), (ii) f ∈ L p (R N ) is such that ˆRN ρ ε (h) ω ˆRN Ω |f (x + h) -f (x)| |h| dx dh C as ε → 0 + .
Moreover,

ω ∇f p L p (R N ) ∼ lim sup ε→0 + ˆRN ρ ε (h) ω ˆRN Ω |f (x + h) -f (x)| |h| dx dh. (2.4)
Note that the limit superior in the right-hand side of (2.4) may not necessarily coincide with the limit inferior, depending on the choices of ω and Ω.

Remark 2.7. -If ω(t) = t and Ω is convex, then the corresponding assertion still holds in higher order Sobolev spaces, see [START_REF] Ferreira | Characterization of polynomials and higher-order Sobolev spaces in terms of functionals involving difference quotients[END_REF] for a proof.

Here are some straightforward consequences of Theorem 2.3.

Example 2.8. -Let M ∈ N * , s ∈ (0, M ) and J ∈ L 1 (R N ) be a radial function such that J := ˆRN J(z)|z| sq dh < ∞ for some 1 q < ∞.

Then, choosing

ρ ε (h) = 1 J |h| sq ε sq J ε (h), and ω(t) = t q we obtain [f ] B s p,∞ (R N ) ∼ sup ε>0 1 ε sq ˆRN J ε (h) ∆ M h f q L p (R N ) dh 1/q . (2.5)
Remark 2.9. -Notice that the quantity (1.14) appearing in the study of the nonlocal Fisher-KPP equation (1.12) can be seen as a particular case of (2.5).

Other choices of ρ ε highlight interesting links with the more classical Besov spaces B s p,q (R N ) with 1 q < ∞ (see Definition 3.3 on Section 3 for the definition of these spaces).

Example 2.10. -Given 1 q < ∞, the choice ω(t) = t q and

ρ ε (h) = 1 C|h| N 1 (ε,2ε) (|h|), (2.6)
where

C = σ N ln(2), yields [f ] B s p,∞ (R N ) ∼ sup ε>0 ˆε<|h|<2ε ∆ M h f q L p (R N ) |h| N +sq dh 1/q . Example 2.11. -Given 1 q < ∞, the choice ω(t) = t q and ρ ε (h) = 1 σ N ε (s-r)q (s -r)q |h| N -(s-r)q 1 (0,ε) (|h|),
for some r ∈ (0, s), gives

q -1/q [f ] B s p,∞ (R N ) ∼ sup ε>0 (s -r) 1/q ε s-r ˆ|h|<ε ∆ M h f q L p (R N ) |h| N +rq dh 1/q .

Limits of Besov norms

Following the original result of Bourgain, Brezis and Mironescu in [START_REF] Bourgain | Another look at Sobolev spaces[END_REF]; Karadzhov, Milman, Xiao [START_REF] Karadzhov | Limits of higher-order Besov spaces and sharp reiteration theorems[END_REF] and Triebel [START_REF]Limits of Besov norms[END_REF] proved the following limiting embedding

q -1/q ∇f L p (R N ) ∼ lim r→1 -(1 -r) 1/q [f ] B r p,q (R N )
for 1 < p, q < ∞.

(2.7) See e.g. [START_REF]Limits of Besov norms[END_REF] where higher order derivatives are also studied. The counterpart of Example 2.11 for the Lipschitz space leads one to ask wether (2.7) still holds in the critical case p = ∞ (recall W 1,∞ (R N ) is the same as C 0,1 (R N )). However, because of the restriction to (1.8) in Theorem 2.5 one cannot directly infer that this is the case. In addition, spaces of the type W 1,∞ (R N ) or B s ∞,q (R N ) do not admit spaces such as C ∞ c (R N ) as dense subset (they are not even separable) and they inherit from the "bad" properties of L ∞ (R N ). This makes the validity of (2.7) in the case p = ∞ rather unclear.

We prove that a weaker version of (2.7) still holds when p = ∞.

Theorem 2.12.

-Let q ∈ [1, ∞) and assume f ∈ L ∞ (R N ) is such that lim sup r→1 - (1 -r) 1/q f B r ∞,q (R N ) < ∞. (2.8)
Then, f ∈ C 0,1 (R N ). Moreover,

q -1/q [f ] C 0,1 (R N ) ∼ lim sup r→1 - (1 -r) 1/q f B r ∞,q (R N ) . (2.9)
Remark 2.13. -Due to the lack of continuity of the translations in L ∞ (R N ) it is not clear wether the lim sup in (2.8) (resp. in (2.9)) can be replaced by a lim inf.

The proof can be carried out using subadditivity and monotonicity arguments via an improvement of the Chebychev inequality due to Bourgain, Brezis and Mironescu [START_REF] Bourgain | Another look at Sobolev spaces[END_REF] together with Theorem 2.3.

However, in the fractional case, one lose the aforementioned monotonicity and the arguments fail. In view of Example 2.11 and

C s (R N ) = B s ∞,∞ (R N
) it is natural to ask wether or not the counterpart holds for B s p,∞ (R N ). Using subatomic decompositions we were able to show that this is not the case.

Theorem 2.14. -Let s > 0, p ∈ [1, ∞) and q ∈ [1, ∞). Then, there exists a function f ∈ L p (R N ) \ B s p,∞ (R N ) such that sup 0<r<s (s -r) 1/q f B r p,q (R N ) < ∞.
Here, " f B r p,q (R N ) " stands for the B r p,q (R N )-norm of f in the sense of subatomic decomposition theory (see Definition 4.2 below).

In particular, this suggests that the restriction to (1.8) in Theorems 1.2 and 2.3 (and actually also in (1.9) when q = ∞) is not far from being optimal.

A non-compactness result

In the integer case s = 1, it is known that any bounded sequence (f

ε ) ε>0 ⊂ L p (R N ) satisfying ˆRN ˆRN ρ ε (h) |f ε (x + h) -f ε (x)| p |h| p dxdh C as ε → 0 + ,
must be relatively compact in L p loc (R N ) provided (ρ ε ) ε>0 is a suitable sequence of mollifiers (e.g. nonincreasing if N = 1 [START_REF] Bourgain | Another look at Sobolev spaces[END_REF] or radially symmetric if N 2 [START_REF] Ponce | An estimate in the spirit of Poincaré's inequality[END_REF]).

Per contra, we show that this phenomenon does not extend to s ∈ R + \ N, at least if ρ ε exhibits a reasonable decay at infinity.

Theorem 2.15. -Let M ∈ N * , s ∈ (0, M ) and p ∈ [1, ∞). Let (ρ ε ) ε>0 be a sequence of mollifiers of the form (1.8) with ρ ∈ L 1 (R N ) satisfying the moment condition ˆRN ρ(z)|z| p(M -s) dz < ∞.
Then, there exists a bounded sequence

(f ε ) ε>0 ⊂ L p (R N ) satisfying ˆRN ˆRN ρ ε (h) |∆ M h f ε (x)| p |h| sp dxdh C as ε → 0 + ,
but which is not relatively compact in L p loc (R N ). Remark 2.16. -In some particular cases it is possible to get rid of assumption (1.8). For instance, if the ρ ε 's are non-increasing and supported in some ball of the form B rε for all ε > 0 and some r > 0, then the result still holds. Notice also that the conclusion of Theorem 2.15 still holds for slightly more general functionals in the spirit of (2.1) with, say, ω = |•| q/p , Ω = |•| p , for any q 1.

In the same vein, we obtain the following Theorem 2.17.

-Let s > 0, p ∈ [1, ∞) and q ∈ [1, ∞). Then, there exists a bounded sequence (f ε ) ε>0 ⊂ L p (R N ) satisfying lim sup ε→0 + ε f ε q B s-ε p,q (R N ) * < ∞, (2.10)
but which is not relatively compact in L p loc (R N ). The subscript " * " in (2.10) means that the B s-ε p,q -norm of f ε is calculated using ( s + 1)-th order finite differences (according to Definition 3.3). This is no longer true if, instead, we use smaller order differences. For example, if (f ε ) ε>0 is bounded in L p (R N ), then lim sup

ε→0 + ε f ε p W 1-ε,p (R N ) < ∞, implies that (f ε ) ε>0 is relatively compact in L p loc (R N ), while lim sup ε→0 + ε f ε p B 1-ε p,p (R N ) * < ∞,
does not. Evidently, this restriction is immaterial if 0 < s / ∈ N.

An approximation criteria

It is well-known that neither

C ∞ c (R N ) nor S (R N ) are dense in B s p,∞ (R N ).
If the question of how to approximate a given f ∈ B s p,q (R N ) in a "suitable manner" has already been well-studied (see e.g. [START_REF] Kowalski | A method of approximation of Besov spaces[END_REF][START_REF] Nikol'skii | Approximation of functions of several variables and embedding theorems[END_REF][START_REF] Stasyuk | Approximation of functions from Nikol'skii-Besov type[END_REF]), to the author's knowledge it seems, however, that no criterion to recognize a function f ∈ B s p,∞ (R N ) which can be approximated by smooth functions in its natural (strong) topology is available in the literature.

An interesting consequence of (the proof of) Theorem 2.3 is that it gives such a criterion.

Corollary 2.18. -Let M ∈ N * , s ∈ (0, M ), p ∈ [1, ∞). Let (ρ ε ) ε>0 ⊂ L 1 (R N )
be a sequence of radial functions satisfying (1.1) and (1.8), and let ω ∈ C + inc . Then, the following statements are equivalent:

(i) f ∈ L p (R N ) is such that lim ε→0 + ˆRN ρ ε (h) ω ∆ M h f L p (R N ) |h| s dh = 0, (ii) f ∈ B s p,∞ (R N ) and there exists (f n ) n 0 ⊂ C ∞ c (R N ) such that f -f n B s p,∞ (R N ) → 0 as n → ∞.
A noteworthy consequence of Corollary 2.18 is the following Example 2.19. -Let s ∈ (0, 1) and p ∈ [1, ∞). Then, with the choice (2.6) and ω(t) = t p we find that condition (ii) above is equivalent to

lim ε→0 + ˆRN ˆε<|x-y|<2ε |f (x) -f (y)| p |x -y| N +sp dxdy = 0,
or, more generally, to

lim ε→0 + ˆε<|h|<2ε ∆ M h f q L p (R N )
|h| N +sq dh = 0, in the higher order case.

In Sections 3 and 4 we detail all our notations and useful definitions. In Section 5, we show some preliminary estimates which aims to simultaneously open the way to Corollary 2.18 and to explain why it is more convenient to represent B s p,∞ (R N ) in terms of the supremum of (2.1) rather than in terms of its limits. Section 6 is devoted to the proof of Theorem 2.3 and Section 7 to that of Theorems 2.5, 2.6, and 2.12. In Section 8 we establish Theorem 2.14. In Section 9, we prove Theorems 2.15 and 2.17. Finally, in the Appendix, we discuss Proposition 1.2.

Notations and definitions

Throughout this chapter we will make use of the following notations.

S N -1 : is the unit sphere of R N ; H N -1 : is the (N -1)-dimensional Hausdorff measure;

|K| : is the Lebesgue measure of the set K; 1 K : is the characteristic function of the set K; B R : is the ball of radius R > 0 centered at the origin; B R (x) : is the ball of radius R > 0 centered at x ∈ R N ; τ h : is the translation operator τ h f (x) = f (x + h), x, h ∈ R N ; f * g : is the convolution of f and g;

: is the "approximatively-less-than" symbol: a b ⇔ a Cb; ∼ : is the equivalence symbol: a ∼ b ⇔ a b and b a; ffl A : is the integral mean symbol:

ffl A f = 1
|A| ´A f . We denote by L p (R N ) the Lebesgue space of (equivalence classes of) functions for which the p-th power of the absolute value is Lebesgue integrable (resp. essentially bounded functions when p = ∞); by C ∞ c (R N ) the space of smooth compactly supported functions; by S (R N ) the Schwartz space of rapidly decaying functions; and, by S (R N ), its dual, the space of tempered distributions. The Lipschitz space C 0,1 (R N ) is the space of functions f ∈ L ∞ (R N ) for which the semi-norm

[f ] C 0,1 (R N ) := sup h =0 τ h f -f L ∞ (R N ) |h| , (3.1)
is finite. The space C 0,1 (R N ) is a Banach space for the norm

f C 0,1 (R N ) := f L ∞ (R N ) + [f ] C 0,1 (R N ) .
The number (3.1) is called the Lipschitz constant of f . For the sake of clarity, we recall some further definitions.

Definition 3.1. -Let p ∈ [1, ∞) and k ∈ N * . The k-th order Sobolev space W k,p (R N ) is defined as the closure of C ∞ c (R N ) under the norm f W k,p (R N ) := f L p (R N ) + 1 |α| k D α f p L p (R N ) 1/p . Definition 3.2.
-The space of functions of bounded variation, denoted by

BV (R N ), is the space of all f ∈ L 1 (R N ) such that [f ] BV (R N ) := sup ˆRN f (x) div φ(x) dx : φ ∈ C 1 c (R N ), φ L ∞ (R N ) 1 < ∞,
naturally endowed with the norm

f BV (R N ) := f L 1 (R N ) + [f ] BV (R N ) . Definition 3.3. -Let M ∈ N * , s ∈ (0, M ) and p, q ∈ [1, ∞]. The Besov space B s p,q (R N ) consists of all functions f ∈ L p (R N ) such that [f ] B s p,q (R N ) := ˆRN ∆ M h f q L p (R N ) dh |h| N +sq 1 q < ∞, (3.2)
which, in the case q = ∞, is to be understood as

[f ] B s p,∞ (R N ) := sup h∈R N \{0} ∆ M h f L p (R N ) |h| s < ∞,
where ∆ M h f is given by (1.3). The space B s p,q (R N ) is naturally endowed with the norm

f B s p,q (R N ) := f L p (R N ) + [f ] B s p,q (R N ) . Remark 3.4. -Of course, if one denotes the semi-norm (3.2) by [f ] (M ) B s p,q (R N ) , then for M 1 , M 2 ∈ N * with M 1 < M 2 and s ∈ (0, M 1 ) it holds [f ] (M 1 ) B s p,q (R N ) ∼ [f ] (M 2 ) B s p,q (R N ) ,
(similarly when q = ∞), so that the definition above is indeed consistent. We refer to [START_REF] Triebel | Theory of function spaces[END_REF] (e.g. estimate (45) on p.99) or Lemma 6.3 for further details.

Remark 3.5. -The integral in (3.2) can be indifferently replaced by an integral over {|h| δ} for any δ > 0, or on the whole R N since the singular part in h in the integral arises when h is close to zero, while the integral on {|h| > δ} can always be dominated by the L p -norm of f . Of special interest are the cases q = p, p = ∞ and/or q = ∞. The fractional Sobolev spaces W s,p (R N ) (sometimes also called Slobodeckij, Gagliardo, or Aronszajn spaces) is defined by W s,p (R N ) = B s p,p (R N ) for s / ∈ N. In this context, the semi-norm (3.2) when s ∈ (0, 1) is often referred to as the Gagliardo semi-norm.

When q = ∞, the space B s p,∞ (R N ) is called the Nikol'skii space. This scale gives another interesting way to measure the convergence rate of the translate of a given function to itself. It is well-known that, for any p, q ∈ [1, ∞) and s > 0,

B s p,q (R N ) → B s p,∞ (R N )
, where " →" stands for the continuous imbedding symbol. We refer to [START_REF] Simon | Sobolev, Besov and Nikolskii fractional spaces: imbeddings and comparisons for vector valued spaces on an interval[END_REF][START_REF]Interpolation theory, function spaces, differential operators[END_REF] for a proof of this fact. When p = q = ∞, then the space B s ∞,∞ (R N ) coincides with the Hölder-Zygmund space C s (R N ).

Moreover, by contrast with W s,p (R N ) (see e.g. [START_REF] Fiscella | Density properties for fractional Sobolev spaces[END_REF] for a simple proof of this fact) or, more generally, with the spaces

B s p,q (R N ) with p, q ∈ (1, ∞), neither C ∞ c (R N ) nor S (R N ) are dense in B s p,∞ (R N ),

Subatomic decompositions

There exists many ways to decompose a function f ∈ B s p,q (R N ) into "building blocks". The theory of subatomic (or quarkonial) decompositions developed by Triebel in [START_REF]Fractals and spectra related to Fourier analysis and function spaces[END_REF]139] is one of them of particular interest because, unlike related decompositions of atomic or, say, Littlewood-Paley type, it yields a decomposition of any function f ∈ B s p,q (R N ) on a suitable system of functions which is independent of f and the resulting coefficients are linearly dependent on f . In such a framework, the search for a function amounts, roughly speaking, to seeking for a discrete sequence of numbers.

For the convenience of the reader we recall some basic definitions.

Definition 4.1. -Let ν 0, m ∈ Z N and ψ ∈ C ∞ c (R N
) be a non-negative function with supp(ψ) ⊂ B 2 r for some r 0 and

k∈Z N ψ(x -k) = 1, if x ∈ R N .
Let Q ν,m be the cube of sides parallel to the coordinate axis with side-length 2 -ν and centered at 2 -ν m. Let s ∈ R, 1 p ∞, β ∈ N N and

ψ β (x) = x β ψ(x) := x β 1 1 ... x β N N ψ(x).
Then,

(βqu) ν,m (x) := 2 -ν(s-N p ) ψ β (2 ν x -m), x ∈ R N , is called an (s, p)-β-quark relative to Q ν,m . Definition 4.2.
-Let s > 0, 1 p, q ∞ and (βqu) ν,m be (s, p)-β-quarks according to Definition 2.4. Let > r where r has the same meaning as in Definition 2.4. For all

λ = {λ β ν,m ∈ C : (ν, m, β) ∈ N × Z N × N N } we set λ ,p,q := sup β∈N N 2 |β| ν 0 m∈Z N |λ β ν,m | p q/p 1/q
, with obvious modification if p = ∞ and/or q = ∞. We call B s p,q (R N ) the collection of all f ∈ S (R N ) which can be represented as

f (x) = β∈N N ∞ ν=0 m∈Z N λ β ν,m (βqu) ν,m (x), (4.1)
endowed with the norm

f B s p,q (R N ) := inf λ ,p,q , (4.2)
where the infinimum is taken over all admissible representations (4.1).

The standard fact of subatomic decompositions states as follows Theorem 4.3. -Let s > 0 and 1 p, q ∞. Then, (4.2) does not depend upon the choice of nor on ψ, and B s p,q (R N ) is a Banach space which coincides with the space B s p,q (R N ) introduced in Definition 3.3. Moreover,

f B s p,q (R N ) ∼ f B s p,q (R N ) .
We refer to [139] and references therein for a proof of this. In fact, there are optimal subatomic coefficients, i.e. coefficients λ β ν,m (f ) realizing the infinimum in (4.2) and which can be obtained as a dual pairing of the form f, Ψ β, ν,m S ,S where (Ψ β, ν,m ) ⊂ S (R N ) is an appropriate sequence of functions. We refer to [139] for further details.

5 The space N s,p (R N )

The aim of this section is twofold. On the one hand, we point out that, even though the spaces B s p,∞ (R N ) can be characterized as limits superior (see Proposition 5.2 below), it does not yield an equivalent norm (as it does for the Sobolev spaces W 1,p (R N ) with p > 1, see e.g. Lemma 7.1). As will become clear in the next section, this is the reason why B s p,∞ (R N ) is more conveniently described as the supremum of (2.1) rather than as its limit superior. On the other hand, we provide some preliminary results towards Corollary 2.18. For simplicity, we consider only first order differences ∆ 1 h f = τ h f -f but all the results of this section also hold for higher order differences.

For the sake of convenience, we define a "new" function space which, in fact, is merely another way to look at the Nikol'skii space B s p,∞ (R N ) as shown hereafter.

Definition 5.1. -Let s ∈ (0, 1) and p ∈ [1, ∞]. Then, the space N s,p (R N ) consists of all functions f ∈ L p (R N ) such that [f ] N s,p (R N ) := lim sup |h|→0 τ h f -f L p (R N ) |h| s < ∞.
It is endowed with the following norm:

f N s,p (R N ) := f L p (R N ) + [f ] N s,p (R N ) .
In addition, we also define

N s,p 0 (R N ) := f ∈ N s,p (R N ) : [f ] N s,p (R N ) = 0 .
As expected, we have the

Proposition 5.2. -Let s ∈ (0, 1) and p ∈ [1, ∞]. Then, B s p,∞ (R N ) = N s,p (R N ).
Remark 5.3. -The equality here holds in the sense of sets: the topology of both are not precisely equivalent as shown below. In fact, "[•] N s,p (R N ) " is a quite crude way to characterize the Nikol'skii space. For these reasons (and in order not to mix with both topologies) we shall write

B s p,∞ (R N ) = (B s p,∞ (R N ), • B s p,∞ (R N ) ) and N s,p (R N ) = (B s p,∞ (R N ), • N s,p (R N ) ). Proof. -Let f ∈ B s p,∞ (R N ).
Then, for all δ > 0, we have

[f ] B s p,∞ (R N ) := sup h∈R N \{0} τ h f -f L p (R N ) |h| s sup 0<|h|<δ τ h f -f L p (R N ) |h| s . Letting δ → 0 + , we get [f ] B s p,∞ (R N ) lim sup |h|→0 τ h f -f L p (R N ) |h| s =: [f ] N s,p (R N ) , (5.1)
and so f ∈ N s,p (R N ). Conversely, let f ∈ N s,p (R N ). Then, for all η > 0 there is a δ 0 > 0 such that for all δ ∈ (0, δ 0 ) we have

sup 0<|h|<δ τ h f -f L p (R N ) |h| s -[f ] N s,p (R N ) < η.
Now fix such η and δ. By the triangle inequality we obtain

sup 0<|h|<δ τ h f -f L p (R N ) |h| s < η + [f ] N s,p (R N ) < ∞.
On the other hand, sup

δ |h| τ h f -f L p (R N ) |h| s 2 δ s f L p (R N ) < ∞.
Therefore, f ∈ B s p,∞ (R N ).

Proposition 5.4. -Let s ∈ (0, 1) and p ∈ (1, ∞]. Then,

W 1,p (R N ) ⊂ N s,p 0 (R N ) and BV (R N ) ⊂ N s,1 0 (R N ). Proof. -First, let f ∈ W 1,p (R N ) (resp. f ∈ BV (R N ) if p = 1). Then, τ h f -f L p (R N ) |h| s |h| 1-s ∇f L p (R N ) , ∀h ∈ R N .
Taking the limit superior as |h| → 0 gives f ∈ N s,p 0 (R N ).

Proposition 5.5. -Let s ∈ (0, 1), p ∈ [1, ∞) and N s,p (R N ) denote the closure of C ∞ c (R N ) in N s,p (R N ). Then, N s,p (R N ) = N s,p 0 (R N ).
In particular, N s,p 0 (R N ) is a closed subspace of N s,p (R N ).

Proof. -"⊂": By definition,

C ∞ c (R N ) is dense in N s,p (R N ), whence the inclu- sion N s,p (R N ) ⊂ N s,p 0 (R N ) is straightforward. "⊃": Let f ∈ N s,p 0 (R N ) and let (f n ) n 0 ⊂ C ∞ c (R N ) be such that f -f n L p (R N ) → 0 as n → ∞.
Then, clearly,

f -f n N s,p (R N ) := f -f n L p (R N ) + [f -f n ] N s,p (R N ) f -f n L p (R N ) + [f ] N s,p (R N ) + [f n ] N s,p (R N ) = f -f n L p (R N ) → 0 as n → ∞. Whence, f ∈ N s,p (R N ). Moreover, the map Θ : f ∈ N s,p (R N ) → [f ] N s,p (R N ) is continuous. Therefore N s,p 0 (R N ) = Θ -1 ({0}) is closed in N s,p (R N ). Proposition 5.6. -Let s ∈ (0, 1), p ∈ [1, ∞) and Bs p,∞ (R N ) (resp. N s,p (R N )) denote the closure of C ∞ c (R N ) in B s p,∞ (R N ) (resp. N s,p (R N )). Then, f ∈ N s,p (R N ) if, and only if, f ∈ Bs p,∞ (R N ). Proof. -Let f ∈ N s,p (R N ) and (f n ) n 0 ⊂ C ∞ c (R N ) be such that f n → f in N s,p (R N ) as n → ∞.
Thus, for all η > 0 there exists n 0 = n 0 (η) 0 and δ 0 = δ 0 (η) > 0 such that

n n 0 , δ ∈ (0, δ 0 ) =⇒ sup |h|<δ ∆ 1 h (f -f n ) L p (R N ) |h| s < η.
Now, fix such η, δ and n 0 . On the other hand, for all η > 0 and all δ > 0 there is a

n 1 = n 1 (η, δ) 0 such that n n 1 =⇒ sup |h| δ ∆ 1 h (f -f n ) L p (R N ) |h| s < η.
Indeed, this is because

sup |h| δ ∆ 1 h (f -f n ) L p (R N ) |h| s 2 δ s f -f n L p (R N ) → 0 as n → ∞.
Therefore, for all n max{n 0 , n 1 },

sup h =0 ∆ 1 h (f -f n ) L p (R N ) |h| s < η.
Summing up, we find that, for all η > 0, there exists M 0 such that

n M =⇒ [f -f n ] B s p,∞ (R N ) < η. Thus, f ∈ Bs p,∞ (R N ). Conversely, let f ∈ Bs p,∞ (R N ) and (f n ) n 0 ⊂ C ∞ c (R N ) be such that f n → f in B s p,∞ (R N ). Using (5.1) we find [f ] N s,p (R N ) [f -f n ] N s,p (R N ) + [f n ] N s,p (R N ) = [f -f n ] N s,p (R N ) [f -f n ] B s p,∞ (R N ) → 0 as n → ∞. Thus f ∈ N s,p (R N ).
6 Characterization of Besov-Nikol'skii spaces

Preliminary

For the sake of clarity we shall introduce the following short notation

D ω (ρ ε , f ) := ˆRN ρ ε (h) ω ∆ M h f L p (R N ) |h| s dh.
First, an easy observation.

Proposition 6.1. -Let M ∈ N * , s > 0, p ∈ [1, ∞] and (ρ ε ) ε>0 be a sequence of mollifiers. Assume ω ∈ C + inc . Then, lim sup ε→0 + D ω (ρ ε , f ) ω lim sup |h|→0 ∆ M h f L p (R N ) |h| s ,
and

sup ε>0 D ω (ρ ε , f ) ω sup h =0 ∆ M h f L p (R N ) |h| s .
Proof. -Let η > 0 be any fixed number. Then, we have

D ω (ρ ε , f ) = ˆ0 |h| η + ˆ|h|>η ρ ε (h) ω ∆ M h f L p (R N ) |h| s dh.
On the one hand, ˆ0

|h| η ρ ε (h) ω ∆ M h f L p (R N ) |h| s dh sup 0 |h| η ω ∆ M h f L p (R N ) |h| s ˆ0 |h| η ρ ε (h)dh sup 0 |h| η ω ∆ M h f L p (R N ) |h| s .
On the other hand, since ω is non-decreasing

ˆ|h|>η ρ ε (h) ω ∆ M h f L p (R N ) |h| s dh ω 2 M f L p (R N ) η s ˆ|h|>η ρ ε (h)dh -→ 0 as ε → 0 + . Therefore, lim sup ε→0 + ˆRN ρ ε (h) ω ∆ M h f L p (R N ) |h| s dh sup 0 |h| η ω ∆ M h f L p (R N ) |h| s .
Taking now the limit as η → 0 + and using ω ∈ C + inc we obtain lim sup

ε→0 + D ω (ρ ε , f ) ω lim sup |h|→0 ∆ M h f L p (R N ) |h| s .
The remaining inequality follows by a direct application of Hölder's inequality.

Here is another estimate we shall need.

Lemma 6.2. -Let p ∈ [1, ∞], M ∈ N * , h 1 , h 2 ∈ R N and h = h 1 + h 2 . Then, ∆ 2M h f L p (R N ) ∆ M h 1 f L p (R N ) + ∆ M h 2 f L p (R N ) , for all f ∈ L p (R N ).
This is essentially covered by [START_REF] Triebel | Theory of function spaces[END_REF]Estimate (16), p.112] but, for the sake of completeness, we choose to provide the details.

Proof. -Let f ∈ S (R N ). Since translations τ h f have Fourier transform e ih•ξ f , the Fourier transform of ∆ M h f writes F [∆ M h f ](ξ) = f (ξ) M j=0 M j (-1) M -j (e ih•ξ ) j .
And so, by applying the binomial formula and taking the inverse Fourier transform of the result one gets

∆ M h f = F -1 [(e ih•ξ -1) M f ]. Now let h 1 , h 2 ∈ R N and h = h 1 + h 2 . Notice that we have e ih•ξ -1 = e ih 1 •ξ (e ih 2 •ξ -1) + e ih 1 •ξ -1. Let P ∈ C[X, Y ] be the polynomial defined by P (X, Y ) = (X(Y -1) + (X -1)) 2M .
By the binomial formula one may find

Q 1 , Q 2 ∈ C[X, Y ] such that P (X, Y ) = (X -1) M Q 1 (X, Y ) + X M (Y -1) M Q 2 (X, Y ).
This holds for any X, Y ∈ C. In particular

(e ih•ξ -1) 2M = (e ih 1 •ξ -1) M Q 1 (e ih 1 •ξ , e ih 2 •ξ ) + e iM h 1 •ξ (e ih 2 •ξ -1) M Q 2 (e ih 1 •ξ , e ih 2 •ξ ).
Multiplying this equality by f (ξ) and taking the inverse Fourier transform of the result, we obtain:

∆ 2M h f = F -1 M k, =0 α k, (e ih 1 •ξ -1) M F f (• + kh 1 + h 2 ) + F -1 M k, =0 β k, e iM h 1 •ξ (e ih 2 •ξ -1) M F f (• + kh 1 + h 2 ) ,
where α k, and β k, are the respective coefficients of

Q 1 and Q 2 . Otherwise said, ∆ 2M h f = M k, =0 α k, ∆ M h 1 f • +kh 1 + h 2 + M k, =0 β k, ∆ M h 2 f • +(k + M )h 1 + h 2 .
We therefore obtain that, for each

f ∈ S (R N ) ∆ 2M h f L p (R N ) C ∆ M h 1 f L p (R N ) + ∆ M h 2 f L p (R N ) , for some constant C > 0 depending only on M , Q 1 and Q 2 . Since S (R N ) is dense in L p (R N )
for p < ∞ the result follows for every f ∈ L p (R N ). When p = ∞, the above still holds in the S sense and, thus, extends to L ∞ (R N ) as well. Also, we recall the following well-known fact.

Lemma 6.3. -Let M ∈ N * , s ∈ (0, M ) and f ∈ L p (R N ). Then, sup h =0 ∆ M h f L p (R N ) |h| s C(s, M ) sup h =0 ∆ 2M h f L p (R N ) |h| s , (6.1)
for some constant C(s, M ) > 0 depending only on s and M . Similarly,

lim sup |h|→0 ∆ M h f L p (R N ) |h| s C(s, M ) lim sup |h|→0 ∆ 2M h f L p (R N ) |h| s . (6.2)
This is a consequence of [START_REF] Triebel | Theory of function spaces[END_REF]Estimate (45), p.99], but the proof being very short we chose to provide all the details.

Proof. -Let f ∈ L p (R N ) and P ∈ C[X] be the unique polynomial such that P (X)(X -1) = 1 - X + 1 2 M . (6.3)
Note that P exists because X -1 is a divisor of the right-hand side of (6.3). In particular, we have that

(X -1) M = 1 2 M (X 2 -1) M + (X -1) M +1 P (X).
Hence, for every h, ξ ∈ R N we have

(e ih•ξ -1) M = 1 2 M (e i2h•ξ -1) M + (e ih•ξ -1) M +1 P (e ih•ξ ).
Whence, reasoning as in Lemma 6.2, we obtain

∆ M h f (x) = 1 2 M ∆ M 2h f (x) + ∆ M +1 h ∈L a f (x + h ) ,
for some finite set of indices L ⊂ N and coefficients a depending on P . Thus, for every s

∈ (0, M ), h = 0 and f ∈ L p (R N ) it holds, ∆ M h f L p (R N ) |h| s 1 2 M -s ∆ M 2h f L p (R N ) |2h| s + C ∆ M +1 h f L p (R N ) |h| s .
We obtain that

1 - 1 2 M -s sup h =0 ∆ M h f L p (R N ) |h| s C sup h =0 ∆ M +1 h f L p (R N ) |h| s .
Therefore (6.1) follows by induction. The proof of (6.2) is similar.

Proof of Theorem 2.3

Let M ∈ N * , s > 0, p ∈ [1, ∞], ω ∈ C + inc and (ρ ε ) ε>0 be as in Theorem 2.3. Here again, we will make use of the short notation

D ω (ρ ε , f ) := ˆRN ρ ε (h) ω ∆ M h f L p (R N ) |h| s dh.
In addition, we call M(R N ) the set of mollifiers (ρ ε ) ε>0 ⊂ L 1 (R N ) satisfying (1.8) for some ρ ∈ L 1 (R N ) such that there exists a number δ > 0 and a nonnegative, nondecreasing, radial function

Ψ ∈ C(R N ) with ρ(h) Ψ(h) for a.e. h ∈ B δ and ˆBδ/4 Ψ > 0. (6.4)
Note that, by Proposition 6.1, we only need to establish a one-sided inequality.

We begin with a few preliminary facts (Claim A and Claim B) showing that the proof of Theorem 2.3 reduces to the case where

(ρ ε ) ε>0 ∈ M(R N ). Claim A. -It is enough to establish Theorem 2.3 for radial ρ's such that ess inf A ρ > 0, (6.5)
for some annulus A ⊂ R N centered at zero.

Proof of Claim A. -Let ρ ∈ L 1 (R N ) be a nonnegative radial function with unit mass. Then, there is a nonnegative function ρ ∈ L 1 loc (R + ) with ρ(z) = ρ(|z|). In particular, we may find some 0 < c 1 < c 2 such that ˆc2

c 1 ρ(θ)dθ > 0.
Let 0 < θ 0 < 1 be such that c 1 > c 2 θ 0 and let ρ * be the radial function given by

ρ * (z) := C 1 θ 0 ρ(θz)dθ = C 1 θ 0 ρ(θ|z|)dθ for z ∈ R N ,
where C > 0 is given by

C := (1 -θ 0 ) ˆ1 θ 0 dθ θ N -1
.

Notice that, by the Fubini theorem, ρ * ∈ L 1 (R N ) and ρ * has unit mass. Indeed, this is because

ρ * L 1 (R N ) = C 1 -θ 0 ˆ1 θ 0 ˆRN ρ(θz)dzdθ = C 1 -θ 0 ˆ1 θ 0 dθ θ N = 1.
Furthermore, one easily checks that ρ * satisfies (6.5). Indeed, we have ess inf

|z|∈ c 2 , c 1 θ 0 ρ * (z) = C 1 -θ 0 ess inf |z|∈ c 2 , c 1 θ 0 ˆ|z| θ 0 |z| ρ(θ) dθ |z| Cθ 0 c 1 (1 -θ 0 ) ˆc2 c 1 ρ(θ)dθ > 0.
On the other hand, we have

ρ ε (θ •) = θ -N ρ ε/θ θ -N 0 ρ ε/θ for any θ ∈ [θ 0 , 1]. Hence, 1 C D ω (ρ * ε , f ) = 1 θ 0 D ω (ρ ε (θ •), f )dθ θ -N 0 sup θ 0 θ 1 D ω (ρ ε/θ , f ). (6.6)
Assuming that Theorem 2.3 holds for mollifiers with ρ satisfying (6.5), we finally obtain

ω [f ] B s p,∞ (R N ) sup ε>0 D ω (ρ ε , f ).
Thus, the claim follows.

Claim B. -It is enough to establish Theorem 2.3 for (ρ ε ) ε>0 ∈ M(R N ). Proof of Claim B. -Let ρ ∈ L 1 (R N
) be a nonnegative radial function with unit mass. Then, there is a nonnegative function ρ ∈ L 1 loc (R + ) with ρ(z) = ρ(|z|). On account of Claim A, we may assume that there are some 0 < r 1 < r 2 and some α > 0 with

ρ(t) α 1 (r 1 ,r 2 ) (t) =: Ψ(t) for a.e. t 0. If r 1 < r 2 4 , then (ρ ε ) ε>0 ∈ M(R N
) and the claim is trivial. Hence, we may assume that r 1 r 2 4 . To show that the latter case reduces to the former, we simply clip together rescaled copies of Ψ as follows. For any j 0, define

θ j := r 1 r 2 j and Ψ θ j (t) := θ -N j Ψ t θ j .
Observe that, by construction, θ j → 0 as j → ∞ and

0 < • • • < θ j+1 r 1 < θ j+1 r 2 = θ j r 1 < θ j r 2 < • • • < θ 1 r 2 = r 1 < r 2 .
Thus, the supports of the Ψ θ j 's are mutually disjoint and they form a countable partition of [0, r 2 ]. Now take an integer k ∈ N such that

k > ln 1 5 ln r 1 r 2 .
By construction, this guarantees that θ k < 1 5 and, in turn, that

supp(Ψ θ k ) ⊂ 0, r 2 5 .
In particular, we have

r 2 5 , r 2 ⊂ k j=0 supp(Ψ θ j ). Fix such a k ∈ N and set J = [[0, k]].
Then, the function

η * (t) := j∈J Ψ θ j (t), for t 0, is bounded and r 2 5 , r 2 ⊂ supp(η * ) ⊂ [0, r 2 ].
Moreover, η * satisfies the following monotonicity property:

η * (t 1 ) η * (t 2 ) α whenever r 2 5 < t 1 < t 2 < r 2 .
Thus, there is a nondecreasing function Φ * ∈ C(R + ) with η * Φ * a.e. in [0, r 2 ] and

ˆr2 /4 0 Φ * (t) t N -1 dt > 0. (6.7)
Indeed, it suffices to take e.g.

Φ * (t) := 5α 4r 2 t - r 2 5 1 ( r 2 5 ,∞) (t).
See Whence, (η ε ) ε>0 ∈ M(R N ). On the other hand, Hence, one obtains

c D ω (η ε , f ) = j∈J D ω Ψ θ j ε (|•|), f j∈J D ω (ρ θ j ε , f ). ( 6
sup ε>0 D ω (η ε , f ) #J c sup ε>0 D ω (ρ ε , f ).
Assuming that Theorem 2.3 holds for mollifiers belonging to M(R N ), we finally obtain

ω [f ] B s p,∞ (R N ) sup ε>0 D ω (ρ ε , f ).
Thus, the claim follows.

Remark 6.4. -By (6.6) and (6.8) we also have that

ω lim sup |h|→0 ∆ M h f L p (R N ) |h| s lim sup ε→0 + D ω (ρ ε , f ),
holds for any radial (ρ ε ) ε>0 satisfying (1.1) and (1.8) whenever it holds for any (ρ ε ) ε>0 belonging to M(R N ).

We may now complete the proof of Theorem 2.3.

Step 1: case M = 1 and s ∈ (0, 1).

Let p ∈ [1, ∞], (ρ ε ) ε>0 ∈ M(R N ), ω ∈ C + inc and f ∈ L p (R N ) satisfying (2.

1)

. Let h ∈ R N (to be fixed later) and let z ∈ R N . Then, we have

τ z f -f = τ h f -f + τ h (τ z-h f -f ). (6.9)
This implies

τ h f -f L p (R N ) τ z f -f L p (R N ) + τ z-h f -f L p (R N ) . (6.10) Now, choose z ∈ B |h| (h). Then, since z and z -h belong to B 2|h| , it comes 1 2 s τ h f -f L p (R N ) |h| s τ z f -f L p (R N ) |z| s + τ z-h f -f L p (R N ) |z -h| s . (6.11)
Since ω is roughly subadditive, there exists a constant A ω > 0 depending only on ω and such that, for every x, y ∈ R + , ω(x + y) A ω {ω(x) + ω(y)} . (6.12) Remark 6.5. -Note that (6.12) implies ω(2x) 2A ω ω(x) and, since ω

∈ C + inc , it is increasing, thus ω(2 s x) 2A ω ω(x) for s 1. Similarly, when s M ∈ N * , one has ω(2 s x) (2A ω ) M ω(x).
From (1.8), Remark 6.5 and since s 1, using the short notation ∆

1 h f = τ h f -f we have ω ∆ 1 h f L p (R N ) |h| s ω 2 s ∆ 1 z f L p (R N ) |z| s + 2 s ∆ 1 z-h f L p (R N ) |z -h| s A ω ω 2 s ∆ 1 z f L p (R N ) |z| s + ω 2 s ∆ 1 z-h f L p (R N ) |z -h| s 2A 2 ω ω ∆ 1 z f L p (R N ) |z| s + ω ∆ 1 z-h f L p (R N ) |z -h| s . (6.13)
Using (ρ ε ) ε>0 ∈ M(R N ) we know there exist a radially nondecreasing Ψ ∈ C(R N ) and a number δ > 0 such that ρ ε (z) Ψ ε (|z|) for a.e. z ∈ B εδ and all ε > 0. (

As seen in Figure 1.2, we clearly have

Ψ ε (|z -h|) Ψ ε (|z|) for all h ∈ B εδ/2 , z ∈ B |h|/2 (h) and ε > 0. ( 6

.15)

Let h ∈ B εδ/2 . Multiplying (6.13) by Ψ ε (|z -h|) and using (6.14)-(6.15) we obtain

Ψ ε (|z -h|) ω ∆ 1 h f L p (R N ) |h| s 2A 2 ω ρ ε (z) ω ∆ 1 z f L p (R N ) |z| s + ρ ε (z -h) ω ∆ 1 z-h f L p (R N ) |z -h| s ,
C(Ψ, δ) ω ∆ 1 h f L p (R N ) |h| s 2A 2 ω ˆB|h|/2 (h) ρ ε (z) ω ∆ 1 z f L p (R N ) |z| s dz + 2A 2 ω ˆB|h|/2 (h) ρ ε (z -h) ω ∆ 1 z-h f L p (R N ) |z -h| s dz 4A 2 ω ˆRN ρ ε (z) ω ∆ 1 z f L p (R N ) |z| s dz,
where

C(Ψ, δ) := ˆB|h|/2 (h) Ψ ε (|z -h|)dz = ˆBδ/4 Ψ(|z|)dz > 0. Whence, ω ∆ 1 h f L p (R N ) |h| s 4A 2 ω C(Ψ, δ) ˆRN ρ ε (z) ω ∆ 1 z f L p (R N ) |z| s dz. (6.16)
Passing to the limit superior as |h| → 0 in (6.16) it follows

ω [f ] N s,p (R N ) 4A 2 ω C(Ψ, δ) lim sup ε→0 + ˆRN ρ ε (z) ω ∆ 1 z f L p (R N ) |z| s dz,
where we used the continuity of ω. This, together with Proposition 6.1 yields

ω [f ] N s,p (R N ) ∼ lim sup ε→0 + ˆRN ρ ε (z) ω ∆ 1 z f L p (R N ) |z| s dz. (6.17)
Similarly, taking the supremum over h = 0 in (6.16), we obtain

ω [f ] B s p,∞ (R N ) ∼ sup ε>0 ˆRN ρ ε (h) ω ∆ 1 h f L p (R N ) |h| s dh.
Remark 6.6. -Estimate (6.17) together with Proposition 5.6 and Remark 6.4 prove Corollary 2.18 for 1 p < ∞ and s ∈ (0, 1) (recall we have assumed ω(0) = 0).

Step 2: case M 2 and s ∈ (0, M ).

The assumption s ∈ (0, 1) being artificial (by Remark 6.5) the above still holds for general s > 0. In particular, replacing (6.10) by the estimate of Lemma 6.2, for

f ∈ L p (R N ), one obtains ω ∆ 2M h f L p (R N ) |h| s C(M, ρ, ω) ˆRN ρ ε (z) ω ∆ M z f L p (R N ) |z| sp dz, (6.18)
for any s ∈ (0, M ). Taking the supremum over ε > 0 (i.e. over |h| > 0) and recalling that ω is a continuous, non-decreasing function, we find that

ω [f ] B s p,∞ (R N ) sup ε>0 ˆRN ρ ε (z) ω ∆ M z f L p (R N ) |z| s dz. This is because the space B s p,∞ (R N ) with s ∈ (0, M ) is characterized by finite differ- ences of order M , i.e. [f ] B s p,∞ (R N ) ∼ sup |h| =0 ∆ 2M h f L p (R N ) |h| s , ∀s ∈ (0, M ), Indeed, recall Lemma 6.3 and ∆ 2M h f L p (R N ) C(M ) ∆ M h f L p (R N ) . Remark 6.7.
-As above, we still have

ω lim sup |h|→0 ∆ M h f L p (R N ) |h| s ∼ lim sup ε→0 + D ω (ρ ε , f ).
So that Corollary 2.18 follows in that case too.

Remark 6.8. -Note that, when (ρ ε ) ε>0 ∈ M(R N ) (with corresponding Ψ and δ), we have actually proved a stronger estimate than needed. Indeed, we have shown that for any h

∈ R N \ {0}, s ∈ (0, 1], p ∈ [1, ∞] and (ρ ε ) ε>0 ∈ M(R N ) it holds ω ∆ 1 h f L p (R N ) |h| s C(Ψ, δ, A ω ) ˆRN ρ ε(|h|) (z) ω ∆ 1 z f L p (R N ) |z| s dz,
where ε(t) = 2t δ and A ω is as in Definition 2.1.

Step 3: proof of Remark 2.4.

Let 1 p < ∞, ω ∈ C + inc and Ω ∈ C inc satisfying (2.
3). Then, we have

ω ˆRN Ω |∆ M h f (x)| |h| s dx ω m 1 ∆ M h f p L p (R N ) |h| sp K 1 (m 1 , A ω ) ω ∆ M h f p L p (R N ) |h| sp ,
where

K 1 (m 1 , A ω ) > 0 and A ω > 0 is a number such that ω satisfies the condition of Definition 2.1 with A = A ω . Similarly, for some K 2 (m 2 , A ω ) > 0, ω ˆRN Ω |∆ M h f (x)| |h| s dx K 2 (m 2 , A ω ) ω ∆ M h f p L p (R N ) |h| sp . Now, since ω = ω • |•| p lies in C + inc (by Remark 2.
2) we obtain the desired claim, i.e. that

ω [f ] p B s p,∞ (R N ) ∼ sup ε>0 ˆRN ρ ε (h) ω ˆRN Ω |∆ M h f (x)| |h| s dx dh.
The remaining claims of Remark 2.4 follow by a similar argument of comparison.

Characterization of Sobolev and BV spaces

We begin with a preliminary result.

Lemma 7.1. -Let p ∈ [1, ∞], f ∈ L p (R N ) and let A := sup h =0 ∆ 1 h f L p (R N ) |h| .
If A is finite, then,

A = lim sup |h|→0 ∆ 1 h f L p (R N ) |h| .
Although our argument is much simpler, a proof of a similar result (involving moduli of continuity) may be found in [START_REF]Limits of Besov norms[END_REF]. However, the argument in [START_REF]Limits of Besov norms[END_REF] heavily relies on the continuity of ∆ 1 h f L p (R N ) and, thus, does not cover the case p = ∞. We show that, in fact, it is enough to ask only for some kind of subadditivity.

Proof. -Let f ∈ L p (R N ), 1 p ∞. For any t ∈ R, define F (t) := sup σ∈S N -1 ∆ 1 σt f L p (R N ) .
Clearly,

F is measurable. Now, let t 1 , t 2 ∈ R. Specializing (6.10) in h = (t 1 + t 2 )σ and z = t 1 σ, for some σ ∈ S N -1 , yields ∆ 1 σ(t 1 +t 2 ) f L p (R N ) ∆ 1 σt 1 f L p (R N ) + ∆ 1 -σt 2 f L p (R N ) F (t 1 ) + F (t 2 ).
Consequently,

F (t 1 + t 2 ) F (t 1 ) + F (t 2 ) for all t 1 , t 2 ∈ R. Whence, F : R → [0, ∞) is a measurable, subadditive function. Now suppose that A := sup t>0 F (t) t < ∞.
Then, by the limit theorem of subadditive functions [96, Theorem 16.3.3, p.467],

A = lim t→0 + F (t) t = lim t→0 + sup σ∈S N -1 ∆ 1 σt f L p (R N ) t .
This proves the lemma.

Proofs of Theorems 2.5 and 2.6

Proof. -The proof follows from a straightforward adaptation of the proof of Theorem 2.3 in the case M = 1 and s ∈ (0, 1) with Lemma 7.1 and the fact that, using for example [27, Theorem 2, Theorem 3'],

∇f L p (R N ) ∼ lim sup |h|→0 f (• + h) -f L p (R N ) |h| , (7.1)
for all 1 p < ∞ (when p = 1 the left-hand side of (7.1) is to be understood in the BV -sense, i.e. as the total mass of the Radon measure ∇f ).

The case p = ∞ follows from the arguments above and the definition of the Lipschitz semi-norm.

A limiting embedding between Lipschitz and Besov spaces

This subsection is devoted to the proof of Theorem 2.12. To this end, we recall the following improvement of the Chebychev inequality due to Bourgain, Brezis and Mironescu [START_REF] Bourgain | Another look at Sobolev spaces[END_REF].

Lemma 7.2 (Bourgain,Brezis,Mironescu,[START_REF] Bourgain | Another look at Sobolev spaces[END_REF]). -Let g, h : (0, δ) → R + . Assume that g(t) g(t/2) for all t ∈ (0, δ) and that h is non-increasing. Then, for some constant

C = C(N ) > 0, δ -N ˆδ 0 t N -1 g(t)dt ˆδ 0 t N -1 h(t)dt C ˆδ 0 t N -1 g(t)h(t)dt.
We are now ready to prove Theorem 2.12.

Proof of Theorem 2.12. -Let q ∈ [1, ∞) and let (ρ ε ) ε∈(0,1] be defined by

ρ ε (t) := 1 |B 1 | ε 1-ε t N -ε 1 (0,ε) (t) for all ε ∈ (0, 1] and all t 0. Note that ˆ∞ 0 ρ ε (t) t N -1 dt = 1 |B 1 | for all ε ∈ (0, 1]. (7.2)
In addition, we also set

η ε (h) := ε -N C 2 |h| ε 1 Bε (h) for all ε ∈ (0, 1] and all h ∈ R N ,
where C 2 > 0 is a constant such that η ε has unit mass for each ε. Notice that (η ε ) ε>0 ⊂ L 1 (R N ) is a sequence of radial functions satisfying (1.1) and (1.8). In particular, by Theorem 2.5, we know that

[f ] C 0,1 (R N ) lim sup ε→0 + ˆBε η ε (h) ∆ 1 h f L ∞ (R N ) |h| dh. (7.3)
Next, for every t > 0, define

F (t) := ˆSN-1 ∆ 1 σt f L ∞ (R N ) dH N -1 (σ).
By the triangle inequality we have F (2t) 2F (t) so that if we let

g(t) := F (t) t ,
we have g(t) g(t/2). In these notations, we have the identity:

ˆBε ρ ε (|h|) ∆ 1 h f L ∞ (R N ) |h| dh = ˆε 0 t N -1 ρ ε (t)g(t)dt.
Invoking Lemma 7.2 and (7.2) we deduce that, for every ε

∈ (0, 1], ˆBε ρ ε (|h|) ∆ 1 h f L ∞ (R N ) |h| dh ε -N ˆε 0 t N -1 ρ ε (t)dt ˆε 0 t N -1 g(t)dt = Bε ∆ 1 h f L ∞ (R N ) |h| dh ε -1 Bε ∆ 1 h f L ∞ (R N ) dh ˆBε η ε (h) ∆ 1 h f L ∞ (R N ) |h| dh.
Whence, using (7.3), we come up with

[f ] C 0,1 (R N ) lim sup ε→0 + ˆBε ρ ε (|h|) ∆ 1 h f L ∞ (R N ) |h| dh.
Now, use the Jensen inequality to deduce that

[f ] q C 0,1 (R N ) lim sup ε→0 + ˆBε ρ ε (|h|) ∆ 1 h f q L ∞ (R N ) |h| q dh lim sup ε→0 + ε -ε ε ˆBε ∆ 1 h f q L ∞ (R N ) |h| N +q-ε dh lim sup ε→0 + ε ˆRN ∆ 1 h f q L ∞ (R N ) |h| N +q-ε dh. Define σ ∈ (1 -1 q , 1) by the relation ε = q(1 -σ). Then, [f ] q C 0,1 (R N ) lim sup σ→1 - q(1 -σ)[f ] q B σ ∞,q (R N ) .
The converse of this is covered by Proposition 6.1.

A non-limiting embedding theorem

This section is devoted to the proof of Theorem 2.14. The idea of the proof is very similar to that of Theorem 4.4 (ii) on p.36 in [START_REF] Triebel | Theory of function spaces[END_REF] (see in particular pp.39-40 there).

Nevertheless, we choose to give more details in order to make the dependence of the constants involved on s, p and q as explicit as possible. We will need some preliminary estimates.

Lemma 8.1. -Let (u j ) j 0 be the sequence defined by [START_REF] Andreu-Vaillo | Nonlocal diffusion problems[END_REF] .

u j := k if j = 2 k for some k ∈ N 0 else Then, (u j ) j 0 / ∈ ∞ (N) and sup ε>0 ε j 0 2 -jε u j 2 e ln
Proof. -Let ε > 0 and set

A ε := j 0 ε2 -jε u j = k 0 ε2 -2 k ε k.
Using the (trivial) estimate e -x 1/(ex), we have 2 -x 1/(ex ln( 2)). Thus,

A ε 1 e ln(2) k 0 k 2 -k .
Recalling the well-known identity k x k = x/(1 -x) 2 (for 0 x < 1), we finally obtain A ε 2 e ln [START_REF] Andreu-Vaillo | Nonlocal diffusion problems[END_REF] .

Since this holds for every ε > 0, we obtain the desired claim.

Lemma 8.2. -Let M ∈ N * and (u k ) k 1 be a sequence of non-negative numbers. Let ψ ∈ C ∞ c (R)
be such that ψ is not a polynomial of degree less than or equal to M -1, and such that supp(ψ) ⊂ [-η, η] for some η 1, and set

f (x 1 , ... , x N ) = k 1 u k ψ x 1 -2(M + η)k 2 -k ... ψ x N -2(M + η)k 2 -k .
Then, for any fixed j 1, we have

sup 1 2 j+1 |h| 1 2 j ∆ M h f L p (R N ) c u j 2 -j N p ,
for some constant c > 0 depending only on N , p, M and ψ.

Proof. -We begin with the case N = 1. Fix any j 1 and let |h| 2 -j . Let us set

x j := 2(M + η)j and R j := 2 -j (M + η).

Since supp(ψ) ⊂ [-η, η] and |h| 2 -j , for any ∈ [[0, M ]], we have

x ∈ supp ψ • + h -x j 2 -j ⇔ x + h ∈ x j -η2 -j , x j + η2 -j ⇔ x ∈ x j -h -η2 -j , x j -h + η2 -j =: B ,j .
And, clearly

supp ∆ M h ψ • -x j 2 -j ⊂ ∈[[0,M ]] B ,j . Thus, supp ∆ M h ψ • -x j 2 -j ⊂ [x j -R j , x j + R j ] =: B j .
Furthermore,

R j+1 + R j = 2 -j (M + η) 1 + 1 2 < 2(M + η) = x j+1 -x j ,
and so, the B j 's are mutually disjoint. Therefore, given any fixed j 1 and ε > 0 a small parameter less than R j , we have

∆ M h f p L p (R) = k 1 u k ∆ M h ψ • -x k 2 -k p L p (R) ˆxj +ε x j -ε k 1 u k ∆ M h ψ x -x k 2 -k p dx = u p j ˆxj +ε x j -ε ∆ M h ψ x -x j 2 -j p dx = u p j 2 -j ˆε/2 -j -ε/2 -j |∆ M h/2 -j ψ(x)| p dx. Whence, writing K j := B 2 -j \ B 2 -(j+1) for j 0 we have sup h∈K j ∆ M h f p L p (R) u p j 2 -j sup h∈K j ˆε -ε |∆ M h/2 -j ψ(x)| p dx = c p ε u p j 2 -j .
where

c ε = c ε (M, p, ψ) := sup 1 2 |h| 1 ˆε -ε |∆ M h ψ(x)| p dx 1/p .
Since ε > 0 is an arbitrary small parameter and ψ is not a polynomial of degree less than or equal to M -1, we may find a number ε 0 > 0 such that c ε 0 > 0.

The proof when N 2 follows by a straightforward adaptation of the case N = 1 using the product structure ψ(x 1 ) ... ψ(x N ) and Fubini's theorem which gives the result with c = c N ε 0 . We are now ready to prove Theorem 2.14.

Proof of Theorem 2.14. -Let M ∈ N * be such that s ∈ (0, M ) and let u j be the sequence of Lemma 8.1. Also, we choose

ψ ∈ C ∞ c (R N ) such that supp(ψ) ⊂ B 2 and m∈Z N ψ(x -m) = 1 for any x ∈ R N .
In addition, we suppose that ψ has the product structure

ψ(x) = Ψ(x 1 ) ... Ψ(x N ),
for some Ψ ∈ C ∞ c (R) different from a polynomial of degree less than or equal to M -1. Then, we set

m j := 2(M + 2)j 1 ∈ Z N with 1 := (1, ... , 1) ∈ Z N ,
and we define

f (x) := j 1 u 1/q j 2 -j(s-N p ) ψ(2 j (x -m j )) = j 1 u 1/q j 2 -jε 2 -j(s-ε-N p ) ψ(2 j (x -m j )).
where x ∈ R N . It follows from Definition 2.4 that

2 -j(s-ε-N p ) ψ(2 j (x -m j ))
can be interpreted as (s -ε, p)-0-quarks. Accordingly, by Definition 4.2, we have that

ε f q B s-ε p,q (R N ) ε j 1 2 -jε u 1/q j q .
Using Lemma 8.1 we obtain that

ε f q B s-ε p,q (R N ) 2q -1 e ln(2) < ∞, ∀ε ∈ (0, s).
In particular, recalling Theorem 4.3, f ∈ L p (R N ). Also, for all j 1, we write

K j := {2 -(j+1) |h| 2 -j }.
Recall that

f B s p,∞ (R N ) ∼ f L p (R N ) + sup j 1 2 js sup h∈K j ∆ M h f L p (R N ) ,
is an equivalent norm on B s p,∞ (R N ) (this is a discretized version of Theorem 2.5.12 on p.110 in [START_REF] Triebel | Theory of function spaces[END_REF]). Using this together with Lemma 8.2 we get

f B s p,∞ (R N ) c sup j 1 u 1/q j = ∞. Here c = c(N, p, M, ψ) > 0. Thus f / ∈ B s p,∞ (R N
). This completes the proof.

Non-compactness results

This section is devoted to the proofs of Theorem 2.15 and Theorem 2.17. We begin with the former one.

Proof of Theorem 2.15. -For simplicity, we replace ε > 0 by 1/n with n 1 and write ρ n instead of ρ 1/n . We write

x = (x 1 , ... , x N ) ∈ R N , y = (x 1 , ... , x N -1 ) ∈ R N -1 ,
and, for all n 1, we let

f n (x) := n M -s M p Φ(n M -s M x N ) ϕ(y), for some arbitrary Φ ∈ C ∞ c (R) and ϕ ∈ C ∞ c (R N -1 ) (if N = 1, replace ϕ by 1) with max Φ W M,p (R) , ϕ W M,p (R N -1 ) C 0 . (9.1)
Note that f n → 0 a.e. in R N . Further, from Fubini's theorem we infer that

ˆRN |f n (x)| p dx = ˆRN-1 |ϕ(y)| p dy n M -s M ˆR |Φ(n M -s M x N )| p dx N = ϕ p L p (R N -1 ) Φ p L p (R) = C 1 .
On the one hand, we observe that (9.1) gives

D α f n p L p (R N ) 1 for each α = (α 1 , ... , α N -1 , 0) ∈ N N with |α| M.
While, on the other hand, for all j ∈ [[1, M ]],

∂ j f n ∂x j N p L p (R N ) = n (M -s) M jp ϕ p L p (R N -1 ) n M -s M ˆR |Φ (j) (n M -s M x N )| p dx N = n (M -s) M jp ϕ p L p (R N -1 ) Φ (j) p L p (R) n (M -s)p ϕ p L p (R N -1 ) max j∈[[1,M ]] Φ (j) p L p (R) .
Whence, using the product structure of f n we get

sup |α| M D α f n p L p (R N ) C 2 n (M -s)p for all n 1.
Moreover, for all h = 0, it holds

∆ M h f n p L p (R N ) |h| M p sup |α| M D α f n p L p (R N ) C 2 |h| M p n (M -s)p . Then, ˆRN ˆRN ρ n (h) |∆ M h f n (x)| p |h| sp dxdh ˆRN ρ n (h)|h| p(M -s) n p(M -s) dh = ˆRN ρ(h)|h| p(M -s) dh.
We thus conclude that

ˆRN ˆRN ρ n (h) |∆ M h f n (x)| p |h| sp dxdh C 3 for any n 1. Yet, (f n ) n 1 is not relatively compact in L p loc (R N
). The proof of Theorem 2.17 is as follows.

Proof of Theorem 2.17. -The proof in this case is very similar to that of Theorem 2.15. We let M ∈ N * , s ∈ (0, M ) and pick a slightly different sequence of functions, for example

f n (x) := n γ M p Φ(n γ M x N ) ϕ(y), where 0 γ 1 q , Φ ∈ C ∞ c (R) and ϕ ∈ C ∞ c (R N -1
). Also, we set

ρ n (h) := 1 nσ N |h| N -1/n 1 (0,1) (|h|).
As above, one has

f n L p (R N ) = Φ L p (R) ϕ L p (R N -1 ) and sup |α| M D α f n L p (R N ) n γ . Whence, we arrive at ˆB1 ρ n (h) ∆ M h f n q L p (R N ) |h| sq dh ˆB1 ρ n (h)|h| (M -s)q n γq dh = n γq-1 ˆ1 0 dr r 1-(q(M -s)+1/n) = n γq 1 + (M -s)qn 1 n 1-γq . Since 0 γ 1 q , we obtain ˆB1 ρ n (h) ∆ M h f n q L p (R N ) |h| sq dh C for all n 1. However, (f n ) n 1 is not relatively compact in L p loc (R N ).

Appendix

In [START_REF] Lamy | Characterization of function spaces via low regularity mollifiers[END_REF], Lamy and Mironescu proved the Theorem 9.1 (Lamy, Mironescu, [START_REF] Lamy | Characterization of function spaces via low regularity mollifiers[END_REF]). -Let s > 0, p ∈ [1, ∞) and (ρ ε ) ε>0 satisfying (1.1) and (1.8). Then,

f B s p∞ (R N ) f L p (R N ) + sup ε∈(0,1) ρ ε * f -f L p (R N ) ε s . (9.2)
Since Theorem 9.1 is not properly stated in [START_REF] Lamy | Characterization of function spaces via low regularity mollifiers[END_REF] nor its proof, we shall give a brief sketch of the proof in order to justify that their result indeed applies to the scale

B s p,∞ (R N ). Sketch of the proof. -It is well-known that each tempered distribution f ∈ S (R N ) can be decomposed as f = j 0 f j , (9.3)
where

f 0 = f * ζ, f j = f * ϕ 2 1-j , j 1, and ζ, ϕ ∈ S (R N ) are functions satisfying (i) supp( ζ) ⊂ B 2 and ζ ≡ 1 in a neighborhood of B1 , (ii) ϕ := ζ 1/2 -ζ with ϕ = ζ(•/2) -ζ and supp( ϕ) ⊂ B 4 \ B1 .
where the subscript ϕ k means k -N ϕ(•/k) and ϕ stands for the Fourier transform of ϕ (similarly for ζ). Formula (9.3) is called the Littlewood-Paley decomposition of f . Furthermore, it is known that each function in B s p,∞ (R N ) is a tempered distribution, so that this decomposition makes sense here and may even serve to formulate an equivalent norm on this space via the formula

f B s p,∞ (R N ) ∼ sup j 0 2 js f j L p (R N ) .
To see that Theorem 9.1 holds it suffices to discretize the last term on the right-hand side of (9.2) as sup

ε∈(1/2,1) sup j 0 2 js f -f * ρ 2 -j ε L p (R N ) .
At this stage, all the estimates obtained in [START_REF] Lamy | Characterization of function spaces via low regularity mollifiers[END_REF] directly apply because it is the terms f j L p (R N ) which are estimated there (and not their sum nor their integral) in terms of the quantity

f -f * ρ 2 -j ε L p (R N ) .
Using this result, Proposition 1.2 can be proved by arguing as follows.

Proof of Proposition 1.2. -Suppose without loss of generality that the ρ ε 's are compactly supported and that supp(ρ) ⊂ B 1 . Also, up to replace ρ ε by ρε(h)+ρε(-h) 2 , we can always assume that each ρ ε is even. Then, by the Jensen inequality,

ρ ε * f -f p L p (R N ) ε sp = 1 ε sp ˆRN ˆRN ρ ε (h)[f (x -h) -f (x)]dh p dx ˆRN ˆBε ρ ε (-h) |f (x + h) -f (x)| p ε sp dhdx ˆRN ˆBε ρ ε (h) |f (x + h) -f (x)| p |h| sp dhdx.
Whence, sup ε∈(0,1)

ρ ε * f -f p L p (R N ) ε sp sup ε∈(0,1) ˆRN ˆRN ρ ε (h) |f (x + h) -f (x)| p |h| sp dhdx.
And so, by Theorem 9.1, f ∈ B s p,∞ (R N ). The proof when ρ is not compactly supported follows by a simple comparison argument: cutting off ρ as ρ := ρ1 B R for some R > 0 with |B R ∩ supp(ρ)| > 0, we clearly have ρ ρ and (1.10) implies that the same property holds for ρ instead of ρ (up to some multiplicative factor ρ L 1 (B R ) to make ρ ε a sequence of mollifiers), i.e. that f ∈ B s p,∞ (R N ).

Chapter 2

On restrictions of Besov functions

This chapter is inspired by the paper [START_REF]On restrictions of Besov functions[END_REF], published in Nonlinear Analysis.

Introduction

In this chapter, we address the following question: given a function

f ∈ B s p,q (R N ),
what can be said about the smoothness of f (•, y) for a.e. y ∈ R N -d ?

In order to formulate this as a meaningful question, one is naturally led to restrict oneself to 1 d < N , 0 < p, q ∞ and s > σ p , where

σ p = N 1 p -1 + , (1.1)
since otherwise f ∈ B s p,q (R N ) need not be a regular distribution. Let us begin with a simple observation. If f ∈ L p (R N ) for some 0 < p ∞, then

f (•, y) ∈ L p (R d ) for a.e. y ∈ R N -d .
This is a straightforward consequence of Fubini's theorem. Using similar Fubini-type arguments, one can show that, if f ∈ W s,p (R N ) for some 0 < p ∞ and σ p < s / ∈ N, then we have f (•, y) ∈ W s,p (R d ) for a.e. y ∈ R N -d . We say that these spaces have the restriction property.

Unlike their cousins, the Triebel-Lizorkin spaces F s p,q (R N ), Besov spaces do not enjoy the Fubini property unless p = q, that is

N j=1 f (x 1 , ... , x j-1 , •, x j+1 , ... , x N ) B s p,q (R) L p (R N -1 )
, is an equivalent quasi-norm on B s p,q (R N ) if, and only if, p = q; while the counterpart for F s p,q (R N ) holds for any given values of p and q where it makes sense (see [139, Theorem 4.4, p.36] for a proof). In particular, B s p,p (R N ) and F s p,q (R N ) have the restriction property. It is natural to ask wether or not this feature holds in B s p,q (R N ) for an arbitrary q = p.

Let us recall some known facts.

Fact 1.1. -Let N 2, 1 d < N , 0 < q p ∞, s > σ p and f ∈ B s p,q (R N ). Then, it holds that f (•, y) ∈ B s p,q (R d ) for a.e. y ∈ R N -d .
(A proof of a slightly more general result will be given in the sequel, see Proposition 5.1.)

In fact, there is a weaker version of Fact 1.1, which shows that this stays "almost" true when p < q. This can be stated as follows

Fact 1.2. -Let N 2, 1 d < N , 0 < p < q ∞, s > σ p and f ∈ B s p,q (R N ). Then, it holds that f (•, y) ∈ s <s B s p,q (R d ) for a.e. y ∈ R N -d .
See e.g. [86, Theorem 1] or [9, Theorem 1.1].

Mironescu [START_REF] Mironescu | Personal communication[END_REF] suggested that it might be possible to construct a counterexample to Fact 1.1 when p < q. We prove that this is indeed the case. This is quite remarkable since, to our knowledge, the list of properties of the spaces B s p,q where q plays a crucial role is rather short.

Our first result is the following

Theorem 1.3. -Let N 2, 1 d < N , 0 < p < q ∞ and let s > σ p . Then, there exists a function f ∈ B s p,q (R N ) such that f (•, y) / ∈ B s p,∞ (R d ) for a.e. y ∈ R N -d . Remark 1.4.
-Note that this is actually stronger than what we initially asked for, since B s p,q → B s p,∞ . Remark 1.5. -We were informed that, concomitant to our work, a version of Theorem 1.3 for N = 2 and p 1 was proved by Mironescu, Russ and Sire in [START_REF] Mironescu | Lifting in Besov spaces[END_REF]. We present another proof independent of it with different techniques. In fact, we will even prove a generalized version of Theorem 1.3 that incorporates other related function spaces (see Theorem 6.1) which is of independent interest. Despite the negative conclusion of Theorem 1.3, one may ask if something weaker than Fact 1.1 still holds when p < q. For example, by standard embeddings, we know that B s p,q (R N ) → A s,p (R N ) for any 0 < q < ∞, where A s,p (R N ) stands for

C s-N p (R N ), BMO(R N ) and L N p N -sp ,∞ (R N ), (1.2)
when respectively sp > N , sp = N and sp < N (see Subsection 2.3). In particular, we may infer from Fact 1.1 that if q p, then for every f ∈ B s p,q (R N ) it holds

f (•, y) ∈ A s,p (R d ) for a.e. y ∈ R N -d .
It is tempting to ask wether the same is true when p < q. But, as it turns out, even this fails to hold. This is the content of our next result.

Theorem 1.6. -Let N 2, 1 d < N , 0 < p < q ∞ and let s > σ p . Then, there exists a function f ∈ B s p,q (R N ) such that f (•, y) / ∈ A s,p (R d ) for a.e. y ∈ R N -d .
It is nonetheless possible to refine the conclusions of Fact 1.2 and Theorem 1.3. We find that a natural way to characterize such restrictions is to look at a more general scale of functions known as Besov spaces of generalized smoothness, denoted by B (s,Ψ) p,q (R N ) (see Definition 2.11). This type of spaces was first introduced by the Russian school in the mid-seventies (see e.g. [START_REF] Kalyabin | Spaces of functions of generalized smoothness[END_REF][START_REF] Kudryavtsev | Spaces of differentiable functions of several variables and imbedding theorem[END_REF][START_REF] Lizorkin | Spaces of generalized smoothness[END_REF]) and was shown to be useful in various problems ranging from Black-Scholes equations [START_REF] Schneider | Wavelet solution of variable order pseudodifferential equations[END_REF] to the study of pseudo-differential operators [START_REF] Almeida | Real interpolation of generalized Besov-Hardy spaces and applications[END_REF][START_REF] Jacob | Pseudo differential operators & Markov processes[END_REF][START_REF] Leopold | On Besov spaces of variable order of differentiation[END_REF][START_REF] Leopold | Trace theorems for Sobolev spaces of variable order of differentiation[END_REF]. A comprehensive state of art covering both old and recent material can be found in [START_REF] Farkas | Characterization of function spaces of generalized smoothness[END_REF]. Several versions of these spaces were studied in the literature, from different points of view and different degrees of generality. We choose to follow the point of view initiated by Edmunds and Triebel in [START_REF] Edmunds | Spectral theory for isotropic fractal drums[END_REF] (see also [START_REF] Cobos | Abstract and concrete logarithmic interpolation spaces[END_REF][START_REF]Eigenfrequencies of isotropic fractal drums[END_REF][START_REF]Embeddings and entropy numbers in Besov spaces of generalized smoothness[END_REF][START_REF] Moura | Function spaces of generalized smoothness[END_REF]139]), which seems better suited to our purposes. Here, s remains the dominant smoothness parameter and Ψ is a positive function of log-type called admissible (see Definition 2.9). That admissible function is a finer tuning that allows encoding more general types of smoothness. The simplest example is the function Ψ ≡ 1 for which one has B (s,Ψ) p,q (R N ) = B s p,q (R N ). More generally, the spaces B (s,Ψ) p,q (R N ) are intercalated scales between B s-ε p,q (R N ) and B s+ε p,q (R N ). For example: if Ψ is increasing, then we have

B s p,q (R N ) → B (s,Ψ) p,q (R N ) → B s p,q (R N ) for every s < s, see [111, Proposition 1.9(vi)].
We prove that restrictions of Besov functions to almost every hyperplanes belong to B (s,Ψ) p,q (R d ), whenever Ψ satisfies the following growth condition

j 0 Ψ(2 -j ) χ < ∞, (1.3)
with χ = qp q-p (resp. χ = p if q = ∞). More precisely, we prove the following Theorem 1.7. -Let N 2, 1 d < N , 0 < p < q ∞, s > σ p and let Ψ be an admissible function satisfying (1.3). Suppose that f ∈ B s p,q (R N ). Then,

f (•, y) ∈ B (s,Ψ) p,q (R d ) for a.e. y ∈ R N -d .
It turns out that the condition (1.3) on Ψ in Theorem 1.7 is optimal, at least when q = ∞. In other words, we obtain a sharp characterization of the aforementioned loss of regularity.

Theorem 1.8. -Let N 2, 1 d < N , 0 < p < q ∞,
s > σ p and let Ψ be an admissible function that does not satisfy (1.3). If q < ∞ and Ψ is increasing suppose, in addition, that

qp q -p < 1 c ∞ where c ∞ := sup 0<t 1 log 2 Ψ(t) Ψ(t 2 ) . (1.4)
Then, there is a function f ∈ B s p,q (R N ) such that

f (•, y) / ∈ B (s,Ψ) p,q (R d ) for a.e. y ∈ R N -d .
Remark 1.9. -Notice that condition (1.4) is sufficient and also not far from being necessary to ensure that (1.3) does not hold, as it happens that for some particular choices of Ψ, (1.3) is equivalent to qp q-p > 1 c∞ . A fine consequence of Theorem 1.7 is that it provides a substitute for A s,p (R d ) when p < q (in Theorem 1.6), which could be of interest in some applications (see e.g. [START_REF] Bourgain | Another look at Sobolev spaces[END_REF][START_REF] Mironescu | Lifting in Besov spaces[END_REF]). For example, if sp > d, p < q and (1.3) is satisfied, then by Theorem 1.7 and [40, Proposition 3.4] we have

∀f ∈ B s p,q (R N ), f (•, y) ∈ C (s-d p ,Ψ) (R d ) for a.e. y ∈ R N -d ,
where

C (α,Ψ) (R d ) is the generalized Hölder space B (α,Ψ) ∞,∞ (R d ) (see Remark 2.

below).

Remark 1.10. -It is actually possible to formulate Theorems 1.7 and 1.8 in terms of the space B w(•) p,q (R d ) introduced by Ansorena and Blasco in [START_REF] Ansorena | Atomic decomposition of weighted Besov spaces[END_REF][START_REF]Characterization of weighted Besov spaces[END_REF], even though their results do not allow to handle higher orders s 1 and neither the case 0 < p < 1 nor 0 < q < 1. Nevertheless, this is merely another side of the same coin and we wish to avoid unnecessary complications. Beyond technical matters, our approach is motivated by the relevance of the scale B (s,Ψ) p,q (R d ) in physical problems and in fractal geometry (see e.g. [START_REF] Edmunds | Spectral theory for isotropic fractal drums[END_REF][START_REF]Eigenfrequencies of isotropic fractal drums[END_REF][START_REF] Moura | Function spaces of generalized smoothness[END_REF][START_REF]Fractals and spectra related to Fourier analysis and function spaces[END_REF]139]).

In the course of the chapter we will also address the corresponding problem with f ∈ B (s,Ψ) p,q (R N ) instead of f ∈ B s p,q (R N ) which is of independent interest. In fact, as we will show, our techniques allow to extend Theorems 1.3, 1.7 and 1.8 to this generalized setting with almost no modifications, see Theorems 6.1, 7.1, 7.2 and Remark 7.3.

Chapter 2 is organized as follows. In the forthcoming Section 2 we recall some useful definitions and results related to Besov spaces. In Section 3, we give some preliminary results on sequences which will be needed for our purposes. In Section 4, we establish some general estimates within the framework of subatomic decompositions and, in Section 5, we use these estimates to prove a generalization of Fact 1.1 which will be used to prove Theorem 1.7. In Section 6, we prove at a stroke Theorems 1.3 and 1.6 using the results collected at Section 3. Finally, in Section 7, we prove Theorems 1.7 and 1.8.

Notations and definitions

For the convenience of the reader, we specify below some notations used all along this chapter.

As usual, R denotes the set of all real numbers, C the set of all complex numbers and Z the collection of all integers. The set of all nonnegative integers {0, 1, 2, ...} will be denoted by N, and the set of all positive integers {1, 2, ...} will be denoted by N * .

The N -dimensional real Euclidean space will be denoted by R N . Similarly, N N (resp. Z N ) stands for the lattice of all points m = (m 1 , ... , m

N ) ∈ R N with m j ∈ N (resp. m j ∈ Z).
Given a real number x ∈ R we denote by x its integral part and by x + its positive part max{0, x}. By analogy, we write R + := {x + : x ∈ R}.

The cardinal of a discrete set E ⊂ Z will be denoted by #E. Given two integers a, b ∈ Z with a < b we denote by [[a, b]] the set of all integers belonging to the segment line

[a, b], namely [[a, b]] := [a, b] ∩ Z.
We will sometimes make use of the approximatively-less-than symbol " ", that is we write a b for a C b where C > 0 is a constant independent of a and b.

Similarly, a b means that b a. Also, we write a ∼ b whenever a b and b a.

We will denote by L N the N -dimensional Lebesgue measure and by B R the Ndimensional ball of radius R > 0 centered at zero.

The characteristic function of a set E ⊂ R N will be denoted by 1 E . We recall that a quasi-norm is similar to a norm in that it satisfies the norm axioms, except that the triangle inequality is replaced by x + y K( x + y ) for some K > 0.

Given two quasi-normed spaces (A, • A ) and (B, • B ), we say that A → B when A ⊂ B with continuous embedding, i.e. when

f B f A for all f ∈ A.
Further, we denote by p (N), 0 < p < ∞, the space of sequences u = (u j ) j 0 such that

u p (N) := j 0 |u j | p 1/p < ∞,
and by ∞ (N) the space of bounded sequences.

As usual, S (R N ) denotes the (Schwartz) space of rapidly decaying functions and S (R N ) its dual, the space of tempered distributions.

Given 0 < p ∞, we denote by L p (R N ) the space of measurable functions f in R N for which the p-th power of the absolute value is Lebesgue integrable (resp. f is essentially bounded when p = ∞), endowed with the quasi-norm

f L p (R N ) := ˆRN |f (x)| p dx 1/p
, (resp. the essential sup-norm when p = ∞).

We collect below the different representations of Besov spaces which will be in use in this chapter.

Classical Besov spaces

Perhaps the simplest (and the most intuitive) way to define Besov spaces is through finite differences. This can be done as follows.

Let

f be a function in R N . Given M ∈ N * and h ∈ R N , let ∆ M h f (x) = M j=0 (-1) M -j M j f (x + hj),
be the iterated difference operator. Within these notations, Besov spaces can be defined as follows.

Definition 2.1. -Let M ∈ N * , 0 < p, q ∞ and s ∈ (0, M ) with s > σ p where σ p is given by (1.1). The Besov space B s p,q (R N ) consists of all functions f ∈ L p (R N ) such that

[f ] B s p,q (R N ) := ˆ|h| 1 ∆ M h f q L p (R N ) dh |h| N +sq 1/q < ∞,
which, in the case q = ∞, is to be understood as

[f ] B s p,∞ (R N ) := sup |h| 1 ∆ M h f L p (R N ) |h| s < ∞.
The space B s p,q (R N ) is naturally endowed with the quasi-norm

f B s p,q (R N ) := f L p (R N ) + [f ] B s p,q (R N ) . (2.1) Remark 2.2. -Different choices of M in (2.1) yield equivalent quasi-norms. Remark 2.3. -If p, q 1, then • B s p,q (R N
) is a norm. However, if either 0 < p < 1 or 0 < q < 1, then the triangle inequality is no longer satisfied and it is only a quasinorm. Nevertheless, we have the following useful inequality

f + g B s p,q (R N ) f η B s p,q (R N ) + g η B s p,q (R N ) 1/η
, where η := min{1, p, q}, which compensates the absence of a triangle inequality.

For our purposes, we shall require a more abstract apparatus which will be provided by the so-called subatomic (or quarkonial) decompositions. This provides a way to decompose any f ∈ B s p,q (R N ) along elementary building blocks (essentially made up of a single function independent of f ) and to, somehow, reduce it to a sequence of numbers (depending linearly on f ). This type of decomposition first appeared in the monograph [START_REF]Fractals and spectra related to Fourier analysis and function spaces[END_REF] of Triebel and was further developed in [139] (see also [START_REF] Izuki | Weighted Besov-Morrey spaces and Triebel-Lizorkin spaces[END_REF][START_REF] Knopova | Spaces of generalized smoothness on h-sets and related Dirichlet forms[END_REF][START_REF] Nakamura | Generalized Morrey spaces and trace operator[END_REF][START_REF]Theory of function spaces III[END_REF]). We outline below the basics of the theory.

Given ν ∈ N and m ∈ Z N , we denote by Q ν,m ⊂ R N the cube with sides parallel to the coordinate axis, centered at 2 -ν m and with side-length 2 -ν .

Definition 2.4. -Let ψ ∈ C ∞ (R N ) be a nonnegative function with supp(ψ) ⊂ y ∈ R N : |y| < 2 r ,
for some r 0 and

k∈Z N ψ(x -k) = 1 for any x ∈ R N . Let s > 0, 0 < p ∞, β ∈ N N and ψ β (x) = x β ψ(x). Then, for ν ∈ N and m ∈ Z N , the function (βqu) ν,m (x) := 2 -ν(s-N p ) ψ β (2 ν x -m) for x ∈ R N , (2.2
)

is called an (s, p)-β-quark relative to the cube Q ν,m . Remark 2.5. -When p = ∞, (2.2) means (βqu) ν,m (x) := 2 -νs ψ β (2 ν x -m).
Definition 2.6. -Given 0 < p, q ∞, we define b p,q as the space of sequences λ = (λ ν,m ) ν 0,m∈Z N such that

λ bp,q := ∞ ν=0 m∈Z N |λ ν,m | p q/p 1/q < ∞.
For the sake of convenience we will make use of the following notations

λ = λ β : β ∈ N N with λ β = λ β ν,m ∈ C : (ν, m) ∈ N × Z N . Then, we have the Theorem 2.7. -Let 0 < p, q
∞, s > σ p and (βqu) ν,m be (s, p)-β-quarks according to Definition 2.4. Let > r (where r has the same meaning as in Definition 2.4). Then, B s p,q (R N ) coincides with the collection of all f ∈ S (R N ) which can be represented as

f (x) = β∈N N ∞ ν=0 m∈Z N λ β ν,m (βqu) ν,m (x), (2.3)
where λ β ∈ b p,q is a sequence such that λ bp,q, := sup

β∈N N 2 |β| λ β bp,q < ∞.
Moreover,

f B s p,q (R N ) ∼ inf (2.3) λ bp,q, , (2.4)
where the infimum is taken over all admissible representations (2.3). In addition, the right hand side of (2.4) is independent of the choice of ψ and > r.

An elegant proof of this result may be found in [137, Section 14.15, pp.101-104] (see also [139, Theorem 2.9, p.15]).

Remark 2.8. -It is known that, given f ∈ B s p,q (R N ) and a fixed > r, there is a decomposition λ β ν,m (depending on the choice of (βqu) ν,m and ) realizing the infimum in (2.4) and which is said to be an optimal subatomic decomposition of f . We refer to [139] (especially Corollary 2.12 on p.23) for further details.

Besov spaces of generalized smoothness

Before we define what we mean by "Besov space of generalized smoothness", we first introduce some necessary definitions. Definition 2.9. -A real function Ψ on the interval (0, 1] is called admissible if it is positive and monotone on (0, 1], and if

Ψ(2 -j ) ∼ Ψ(2 -2j ) for any j ∈ N. Example 2.10. -Let 0 < c < 1 and b ∈ R. Then, Ψ(x) := | log 2 (cx)| b for x ∈ (0, 1],
is an example of admissible function. Another example is

Ψ(x) := (log 2 | log 2 (cx)|) b for x ∈ (0, 1].
Roughly speaking, admissible functions are functions having at most logarithmic growth or decay near zero. They may be seen as particular cases of a class of functions introduced by Karamata in the mid-thirties [START_REF]Über einen Konvergenzsatz des Herrn Knopp[END_REF][START_REF] Karamata | Théorèmes sur la sommabilité exponentielle et d'autres sommabilités s'y rattachant[END_REF] known as slowly varying functions (see e.g. [23, Definition 1.2.1, p.6] and also [START_REF] Bricchi | Complements and results on h-sets, Function Spaces, Differential Operators and Nonlinear Analysis[END_REF]Section 3,p.226] where the reader may find a discussion on how these two notions relate to each other as well as various examples and references).

We refer the interested reader to [START_REF] Moura | Function spaces of generalized smoothness[END_REF]139] for a detailed review of the properties of admissible functions. Definition 2.11. -Let M ∈ N * , 0 < p, q ∞, s ∈ (0, M ) with s > σ p and let Ψ be an admissible function. The Besov space of generalized smoothness B (s,Ψ)

p,q (R N ) consists of all functions f ∈ L p (R N ) such that [f ] B (s,Ψ) p,q (R N ) := ˆ1 0 sup |h| t ∆ M h f q L p (R N ) Ψ(t) q t 1+sq dt 1/q < ∞,
which, in the case q = ∞, is to be understood as

[f ] B (s,Ψ) p,∞ (R N ) := sup 0<t 1 t -s Ψ(t) sup |h| t ∆ M h f L p (R N ) < ∞.
The space B (s,Ψ) p,q (R N ) is naturally endowed with the quasi-norm

f B (s,Ψ) p,q (R N ) := f L p (R N ) + [f ] B (s,Ψ) p,q (R N ) . (2.5)
Remark 2.12. -Different choices of M in (2.5) yield equivalent quasi-norms.

Remark 2.13. -Observe that, by taking Ψ ≡ 1, we recover the usual Besov spaces, that is we have f B (s,1) p,q (R N ) ∼ f B s p,q (R N ) , see [136, Theorem 2.5.12, p.110] for a proof of this.

Remark 2.14. -As already mentioned in the introduction, these spaces were introduced by Triebel and Edmunds in [START_REF] Edmunds | Spectral theory for isotropic fractal drums[END_REF][START_REF]Eigenfrequencies of isotropic fractal drums[END_REF] to study some fractal pseudo-differential operators, but the first comprehensive studies go back to Moura [START_REF] Moura | Function spaces of generalized smoothness[END_REF] (see also [START_REF]Tailored Besov spaces and h-sets[END_REF][START_REF]Local growth envelopes of spaces of generalized smoothness: the subcritical case[END_REF][START_REF] Caetano | Local growth envelopes of spaces of generalized smoothness: the critical case[END_REF][START_REF] Haroske | Continuity envelopes of spaces of generalised smoothness, entropy and approximation numbers[END_REF][START_REF] Knopova | Spaces of generalized smoothness on h-sets and related Dirichlet forms[END_REF][START_REF]Theory of function spaces III[END_REF][START_REF]Bases in function spaces, sampling, discrepancy, numerical integration[END_REF]). In the literature these spaces are usually defined from the Fourier-analytical point of view (e.g. in [START_REF] Moura | Function spaces of generalized smoothness[END_REF]139]) but, as shown in [81, Theorem 2.5, p.161], the two approaches are equivalent.

Remark 2.15. -Notice that, here as well, the triangle inequality fails to hold when either 0 < p < 1 or 0 < q < 1, but, in virtue of the Aoki-Rolewicz lemma, we have the same kind of compensation as in the classical case, see [87, Lemma 1.1, p.3]. That is, there exists η ∈ (0, 1] and an equivalent quasi-norm

• B (s,Ψ) p,q (R N ), * with f + g B (s,Ψ) p,q (R N ), * f η B (s,Ψ) p,q (R N ), * + g η B (s,Ψ) p,q (R N ), * 1/η .
A fine property of these spaces is that they admit subatomic decompositions. In fact, it suffices to modify the definition of (s, p)-β-quarks to this generalized setting in the following way. Definition 2.16. -Let r, ψ and ψ β with β ∈ N N be as in Definition 2.4. Let s > 0 and 0 < p ∞. Let Ψ be an admissible function. Then, in generalization of

(2.2), (βqu) ν,m (x) := 2 -ν(s-N p ) Ψ(2 -ν ) -1 ψ β (2 ν x -m) for x ∈ R N , is called an (s, p, Ψ)-β-quark.
Then, we have the following Theorem 2.17. -Let 0 < p, q ∞, s > σ p and Ψ be an admissible function. Let (βqu) ν,m be (s, p, Ψ)-β-quarks according to Definition 2.16. Let > r (where r has the same meaning as in Definition 2.16). Then, B (s,Ψ) p,q (R N ) coincides with the collection of all f ∈ S (R N ) which can be represented as

f (x) = β∈N N ∞ ν=0 m∈Z N λ β ν,m (βqu) ν,m (x), (2.6)
where λ β ∈ b p,q is a sequence such that λ bp,q, := sup

β∈N N 2 |β| λ β bp,q < ∞.
Moreover,

f B s p,q (R N ) ∼ inf (2.6) λ bp,q, , (2.7)
where the infimum is taken over all admissible representations (2.6). In addition, the right hand side of (2.7) is independent of the choice of ψ and > r.

This result can be found in [111,

Related spaces and embeddings

Let us now say a brief word about embeddings. Given a locally integrable function f in R N and a set B ⊂ R N having finite nonzero Lebesgue measure, we let

f B := B f (y) dy = 1 L N (B)
ˆB f (y) dy, be the average of f on B. Moreover, we denote by f * : R + → R + the decreasing rearrangement of f , given by f * (t) := inf λ 0 : µ f (λ) t , for all t 0, where

µ f (λ) := L N x ∈ R N : |f (x)| > λ , is the so-called distribution function of f . Definition 2.19. -Let s > 0 and 0 < p < ∞. (i) The Zygmund-Hölder space C s (R N ) is the Besov space B s ∞,∞ (R N ). (ii)
The space of functions of bounded mean oscillation, denoted by BMO(R N ), consists of all locally integrable functions f such that

f BMO(R N ) := sup B B |f (x) -f B | dx < ∞, (2.8)
where the supremum in (2.8) is taken over all balls B ⊂ R N . (iii) The weak L p -space, denoted by L p,∞ (R N ), consists of all measurable functions f such that

f L p,∞ (R N ) := sup t>0 t 1/p f * (t) < ∞,
where f * is the decreasing rearrangement of f .

Let us now state the following Theorem 2.20 (Sobolev embedding theorem for B s p,q ). -Let 0 < p, q < ∞ and let s > σ p . Then, the following hold true:

(i) if sp > N , then B s p,q (R N ) → C s-N p (R N ); (ii) if sp = N , then B s p,q (R N ) → BMO(R N ); (iii) if sp < N , then B s p,q (R N ) → L N p N -sp ,∞ (R N ).
In particular,

B s p,q (R N ) → A s,p (R N ) where A s,p (R N ) is as in (1.2).
Proof. -The cases (i), (ii) and (iii) are respectively covered by [START_REF] Triebel | Theory of function spaces[END_REF]Formula (12) p,q (R N ). As already mentioned in the introduction, the space B (s,Ψ) p,q (R N ) is embedded in a generalized version of the Hölder space when sp > N . When sp < N , it is shown in [START_REF]Local growth envelopes of spaces of generalized smoothness: the subcritical case[END_REF]

that B (s,Ψ) p,q (R N ) embeds in a weighted version of L N p N -sp ,∞ (R N ).
Yet, when sp = N , the corresponding substitute for BMO does not seem to have been clearly identified nor considered in the literature, see however [START_REF] Caetano | Local growth envelopes of spaces of generalized smoothness: the critical case[END_REF][START_REF] Gurka | Sharp embeddings of Besov spaces with logarithmic smoothness[END_REF][START_REF]Sharp embeddings of Besov-type spaces[END_REF][START_REF] Moura | Spaces of generalized smoothness in the critical case: optimal embeddings, continuity envelopes and approximation numbers[END_REF] where some partial results are given.

Preliminaries

In this section, we study the properties of some discrete sequences which will play an important role in the sequel. More precisely, we will be interested in the convergence of series of the type j 0 2 j |λ j, 2 j x | for x > 0, where λ = (λ j,k ) j,k 0 is an element of some Besov sequence space, say, b 1,q with q > 1.

Some technical lemmata

Let us start with a famous result due to Cauchy. Theorem 3.1 (Cauchy's condensation test). -Let λ ∈ 1 (N) be a nonnegative, nonincreasing sequence. Then,

j 1 λ j j 0 2 j λ 2 j 2 j 1 λ j . Remark 3.2.
-The monotonicity assumption on λ is central here. Indeed, there exist nonnegative sequences λ ∈ 1 (N) which are not nonincreasing and such that j 0 2 j λ 2 j = ∞. Take for example:

λ j = 1/k 2 if j = 2 k and k = 0, 2 -j else.
Then, clearly, λ ∈ 1 (N). However, 2 j λ 2 j = 2 j j 2 when j 1, so that (2 j λ 2 j ) j 0 / ∈ 1 (N).

A simple consequence of Cauchy's condensation test is the following

Lemma 3.3. -Let λ ∈ 1 (N) be a nonnegative, nonincreasing sequence. Then, j 0 2 j λ 2 j x φ(x) j 1 λ j for any x > 0,
where φ(x)

:= 4 |x| 1 [1,∞) (x) + (1 -log 2 |x|)1 (0,1) (x) .
Proof. -Let k ∈ N and 2 k x 2 k+1 . Then, by Cauchy's condensation test

j 0 2 j λ 2 j x j 0 2 j λ 2 k+j = 2 -k j k 2 j λ 2 j 2 -k j 0 2 j λ 2 j 4 x j 1 λ j .
In like manner, for 2 -(k+1) x 2 -k , we have j 0

2 j λ 2 j x j 0 2 j λ 2 j-k-1 = 2 k+1 j -k-1 2 j λ 2 j 2 k+1 (k + 1) j 0 2 j λ 2 j .
Finally, invoking again Cauchy's condensation test, we have j 0

2 j λ 2 j x 2 k+2 (k + 1) j 1 λ j 4 x 1 -log 2 (x) j 1 λ j .
This completes the proof.

In some sense, this "functional version" of Cauchy's condensation test may be generalized to sequences which are not necessarily nonincreasing.

Indeed, one can show that

L 1 x ∈ R + : j 0 2 j |λ 2 j x | = +∞ = 0, whenever λ ∈ 1 (N)
. This is due to the fact that p -spaces can be seen as "amalgams" of L p (1, 2) and a weighted version of p . More precisely, we have Lemma 3.4. -Let 0 < p < ∞ and let λ ∈ p (N). Then,

j 1 |λ j | p 1/p = ˆ[1,2] j 0 2 j |λ 2 j x | p dx 1/p .
Proof. -It suffices to assume p = 1 and that λ is nonnegative. Then,

1 2 k 2 k j<2 k+1 λ j = [2 k ,2 k+1 ] λ x dx = 1 2 k ˆ[1,2] λ 2 k y 2 k dy = ˆ[1,2] λ 2 k y dy, which yields j∈N * λ j = k∈N 2 k j<2 k+1 λ j = k∈N 2 k ˆ[1,2] λ 2 k x dx = ˆ[1,2] k∈N λ 2 k x 2 k dx.
The proof is complete.

We now establish a technical inequality which will be needed in the sequel.

Lemma 3.5. -Let N 1 and 0 < p < ∞. Let λ = (λ β j,k ) (j,β,k)∈N×N N ×N N be a sequence such that the partial sequences (λ β j,k ) k∈N N belong to p (N N ) for all (j, β) ∈ N × N N . Then, for any positive (α j ) j 0 ∈ 1 (N) there exists C = C(λ, α, N, d) > 0 such that for any (j, β) ∈ N × N N , 2 jN |λ β j, 2 j x | p C max{1, |β| N +1 } α j k∈N N |λ β j,k | p ,
holds for a.e. x = (x 1 , ... , x N ) ∈ [1, 2] N where

2 j x = ( 2 j x 1 , ... , 2 j x N ) ∈ N N .
Proof. -For the sake of convenience, we use the following notations

U j,β (x) := 2 jN |λ β j, 2 j x | p and U j,β := k∈N N |λ β j,k | p .
We have to prove that

U j,β (x) C max{1, |β| N +1 } α j U j,β ,
for a.e. x ∈ [1, 2] N and any (j, β) ∈ N × N N . By iterated applications of Lemma 3.4, we have

ˆ[1,2] N U j,β (x) dx U j,β . (3.1)
Now, define

Γ j,β := x ∈ [1, 2] N : U j,β (x) max{1, |β| N +1 } α j U j,β .
Then, applying Markov's inequality and using (3.1), we have

L N (Γ j,β ) α j max{1, |β| N +1 } for any (j, β) ∈ N × N N .
In turn, this gives

β∈N N j 0 L N (Γ j,β ) < ∞.
Therefore, we can apply the Borel-Cantelli lemma and deduce that there exists j 0 , β 0 0 such that

U j,β (x) max{1, |β| N +1 } α j U j,β ,
for any j > j 0 and/or |β| > β 0 and a.e. x ∈ [1, 2] N . On the other hand, for any j j 0 and |β| β 0 we have

U j,β (x) 2 j 0 N max{1, |β| N +1 } max 0 j j 0 α j α j U j,β .
This completes the proof.

Some useful sequences

We now construct some key sequences which will be at the heart of the proofs of Theorems 1.3, 1.6 and 1.8.

Lemma 3.6. -There exists a sequence

(ζ k ) k 0 ⊂ R + satisfying sup j 0 1 2 j 2 j k<2 j+1 ζ k 1, (3.2)
and such that Let (λ k ) k 0 be a sequence such that λ 0 = λ 1 = 0 and such that, for any j 1, the 2 j j first terms of the sequence (λ k ) k 0 on the discrete interval [[2 j , 2 j+1 -1]] have value j and the remaining terms are all equal to zero. Then, for any j 1, we have

sup j 0 ζ 2 j x = ∞ for all x ∈ [1, 2). ( 3
1 2 j 2 j k<2 j+1 λ k = j + • • • + j + 0 + • • • + 0 2 j = j 2 j j 2 j 1.
For the sake of convenience, we set

T j := [[2 j , 2 j+1 -1]
] for any j 0.

We will construct a sequence (ζ k ) k 0 satisfying both (3.2) and (3.3) by rearranging the terms of (λ k ) k 0 . To this end, we follow the following procedure. For k ∈ [[0, 2 3 -1]] we impose ζ k = λ k . For j = 3, we shift the values of (λ k ) k 0 on T 3 in such a way that the smallest x ∈ [1, 2) such that ζ 2 3 x is nonzero coincides with the limit superior of the set of all z ∈ [1, 2) such that ζ 2 2 z is nonzero. For j = 4, we shift the values of (λ k ) k 0 on T 4 in such a way that the smallest x ∈ [1, 2] such that ζ 2 4 x is nonzero coincides with the limit superior of the set of all z ∈ [1, 2) such that ζ 2 3 z is nonzero, and so on. When the range of nonzero terms has reached the last term on T j for some j 1, we start again from T j+1 and set ζ k = λ k on T j+1 , and we repeat the above procedure. See Figure 2.1 for a visual illustration.

If, for some j 0, it happens that the above shifting of the λ k 's on T j exceeds T j , then we shift the λ k 's on T j in such a way that the limit superior of the set of all x ∈ [1, 2] for which ζ 2 j x is nonzero coincides with x = 2.

Note that this procedure is well-defined because the proportion of nonzero terms on each T j is 2 -j 2 j j which has a divergent series thus allowing us to fill as much "space" as needed.

Then, by construction, for any x ∈ [1, 2) there are infinitely many values of j 0 such that ζ 2 j x = j. Consequently, (3.3) holds. Moreover, (3.2) is trivially satisfied.

This completes the proof.

As an immediate corollary, we have

Corollary 3.7. -Let 0 < p < q ∞.
Then, there exists a sequence

(λ j,k ) j,k 0 ⊂ R + satisfying j 0 k 0 λ p j,k q/p 1/q < ∞,
(with the usual modification if q = ∞) and such that sup j 0 2 j/p λ j, 2 j x = ∞ for all x ∈ [1, 2).

Remark 3.8. -By "usual modification if q = ∞" we mean that, when q = ∞, the quasi-norm

• q (N) = ( j 0 |•| q ) 1/q is replaced by • ∞ (N) = sup j 0 |•|. Proof. -When q = ∞, it suffices to set λ j,k = 2 -j/p ζ 1/p k if 2 j k < 2 j+1 , 0 otherwise, (3.4)
where (ζ k ) k 0 is the sequence constructed at Lemma 3.6. When q < ∞, we simply replace (ζ k ) k 0 in (3.4) by (ξ k ) k 0 where

ξ k = j - √ p q ζ k for any k ∈ [[2 j , 2 j+1 -1]] with j 1,
and ξ 0 = ξ 1 = 0. Then, we obtain

j 1 1 2 j 2 j k<2 j+1 ξ k q/p = j 1 j - √ p q • 1 2 j 2 j k<2 j+1 ζ k q/p j 1 j - √ q p < ∞.
Moreover, by construction of (ζ k ) k 0 , for any x ∈ [1, 2), there is a countably infinite set J x ⊂ N such that ζ 2 j x = j for any j ∈ J x . In particular, ξ 2 j x = j α for any j ∈ J x and x ∈ [1, 2), where α = 1 -p/q > 0. Thus, sup j 0

2 j λ p j, 2 j x = sup j 0 ξ 2 j x sup j∈Jx j α = ∞ for any x ∈ [1, 2),
which is what we had to show.

We conclude this section by a weighted version of Corollary 3.7.

Lemma 3.9. -Let 0 < p < q ∞. Let Ψ be an admissible function that does not satisfy (1.3). If q < ∞ and Ψ is increasing assume, in addition, that

χ = qp q -p < 1 c ∞ ,
where c ∞ is as in Theorem 1.8. Then, there exists a sequence (λ j,k ) j,k 0 ⊂ R + such that

j 0 k 0 λ p j,k q/p 1/q < ∞, (3.5)
(with the usual modification if q = ∞) and j 0

2 j q p λ q j, 2 j x Ψ(2 -j ) q 1/q = ∞ for all x ∈ [1, 2), (3.6)
(with the usual modification if q = ∞).

Proof. -The proof is essentially the same as in the unweighted case with minor changes that we shall now detail.

Let us begin with the case q = ∞. Let β j := Ψ(2 -j ) p . Since β j > 0 and (β j ) j 0 / ∈ 1 (N) we may find another positive sequence (γ j ) j 0 which has a divergent series and such that

β j γ j → ∞ as j → ∞,
i.e. (γ j ) j 0 diverges slower than (β j ) j 0 . Take, for example

γ j = β j j k=0 β k
, see e.g. [START_REF] Ash | Neither a worst convergent nor a best divergent series exists[END_REF]. Note that 0 < γ j 1 for all j 0. Let ( k ) k 0 be a sequence such that 0 = 1 = 0 and such that, for any j 1, the 2 j γ j first terms of the sequence ( k ) k 0 on the discrete interval T j := [[2 j , 2 j+1 -1]] have value 1 γ j and the remaining terms are all equal to zero. Then, for any j 1, we have

1 2 j 2 j k<2 j+1 k = 1 γ j + • • • + 1 γ j + 0 + • • • + 0 2 j = 1 γ j 2 j γ j 2 j 1.
Now, since the proportion of nonzero terms on each T j is 2 -j 2 j γ j which has a divergent series, we may apply to ( j ) j 0 the same rearrangement as in the proof of Lemma 3.6. That is, we can construct a sequence ( * j ) j 0 such that 1 2 j

2 j k<2 j+1 * k
1 for all j 0, and for any x ∈ [1, 2) there is a countably infinite set J x ⊂ N such that

β j * 2 j x = β j γ j = j k=0
β k for all j ∈ J x , i.e. we have sup j 0

β j * 2 j x lim j→∞ j∈Jx j k=0 β k = ∞.
Therefore, letting

λ j,k = 2 -j/p ( * k ) 1/p if 2 j k < 2 j+1 , 0 otherwise,
we obtain a sequence satisfying both (3.5) and (3.6).

Let us now prove the lemma when q < ∞. Notice that if Ψ is either constant or decreasing there is nothing to prove since the result is a consequence of Corollary 3.7. Hence, we may assume that Ψ is increasing. By our assumptions, we have

1 p log 2 β j β 2j c ∞ < 1 χ ,
which implies that

β j 2 c∞p β 2j • • • 2 kc∞p β 2 k j for any k ∈ N. (3.7)
By Cauchy's condensation test, we have 2

j 0 β q q-p j j 0 2 j β q q-p 2 j β q q-p 1 j 0 2 j(1-c∞χ) = ∞.
Thus, we may infer as above that the following positive sequence has divergent series:

γ j = β q q-p j j k=0 β q q-p k .
Notice that γ j 1/(j+1). Now define (τ j ) j 0 by τ j := γ j /β j . Since 2 -c∞p β 1 j c∞p β j for any j 1 (by (3.7) and the monotonicity of β j ), our assumptions on χ and c ∞ then imply

j 0 τ q/p j τ q/p 0 + j 1 β -q/p j (j + 1) q/p τ q/p 0 + j 1 2 c∞q β -q/p 1 j q(1/p-c∞) < ∞.
The conclusion now follows by letting λ j,k := τ 1/p j λ j,k where λ j,k is the sequence constructed above with γ j instead of γ j . Indeed, we have

j 0 k 0 λ p j,k q/p j 0 τ q/p j < ∞,
and, for each x ∈ [1, 2), there is a countably infinite set J x ⊂ N such that

2 j/p λ j, 2 j x β 1/p j = 2 j/p τ 1/p j β j γ j 1/p 2 -j/p = 1 for any j ∈ J x .
Therefore, (2 j/p λ j, 2 j x β 1/p j ) j 0 / ∈ q (N). This completes the proof.

General estimates

Throughout this section we will write

x ∈ R N as x = (x 1 , ... , x N ) = (x , x ) with x ∈ R d , x ∈ R N -d and, similarly, m = (m , m ) ∈ Z N and β = (β , β ) ∈ N N . Also, we set D := {0, 1} N -d . Let ψ ∈ C ∞ 0 (R N , [0, 1]) be such that supp(ψ) ⊂ B 1 and that 2 -ν(s-N p ) ψ β (2 ν x -m),
are (s, p)-β-quarks. Also, we assume that ψ has the product structure

ψ(x 1 , ... , x N ) = ψ(x 1 ) ... ψ(x N ). (4.1)
Let > 0 and f ∈ B s p,q (R N ). Then, by Theorem 2.7, there are coefficients λ β ν,m such that

f (x) = β∈N N ∞ ν=0 m∈Z N λ β ν,m 2 -ν(s-N p ) ψ β (2 ν x -m). (4.2)
We can further assume that

f B s p,q (R N ) ∼ sup β∈N N 2 |β| ν 0 m∈Z N |λ β ν,m | p q/p 1/q , ( 4.3)
i.e. that λ β ν,m = λ β ν,m (f ) is an optimal subatomic decomposition of f . Note, however, that the optimality of the decomposition λ β ν,m (f ) depends on the choice of > 0 (this can be seen from [139, Corollary 2.12, p.23]). Of course, by Theorem 2.7, we still have

f B s p,q (R N ) sup β∈N N 2 |β| ν 0 m∈Z N |λ β ν,m | p q/p 1/q
, for any positive = . Using (4.1) and (4.3), we can decompose f (•, x ) as

f (x , x ) = ν 0 β ∈N d m ∈Z d b β ν,m (λ, x )2 -ν(s-d p ) ψ β (2 ν x -m ),
where we have set

b β ν,m (λ, x ) := 2 ν N -d p β ∈N N -d m ∈Z N -d λ β ν,m ψ β (2 ν x -m ). (4.4)
Then, defining

J p,q (λ, x ) := sup β ∈N d 2 |β | ν 0 m ∈Z d |b β ν,m (λ, x )| p q/p 1/q , ( 4.5)
we obtain

f (•, x ) B s p,q (R d ) J p,q (λ, x ).
In fact, we also have

f (•, x ) B s p,q (R d ) J p,q (λ, x ), (4.6)
for any > 0. For the sake of convenience, we introduce some further notations. Given any δ ∈ D, we set

b β ,δ ν,m (λ, x ) := 2 ν N -d p β ∈N N -d |λ β ν,m , 2 ν x d+1 +δ d+1 ,..., 2 ν x N +δ N |, (4.7)
J ,δ p,q (λ, x ) := sup

β ∈N d 2 |β | ν 0 m ∈Z d |b β ,δ ν,m (λ, x )| p q/p 1/q . ( 4.8)
Notice that since supp(ψ β ) ⊂ B 1 , we have

ψ β (2 ν x -m ) = 0 =⇒ m i ∈ { 2 ν x i , 2 ν x i + 1} for all i ∈ [[d + 1, N ]].
And so, using (4.4) and (4.5), we can derive the following bounds

b β ν,m (λ, x ) δ∈D b β ,δ ν,m (λ, x ),
and

J p,q (λ, x ) c δ∈D J ,δ p,q (λ, x ), (4.9)
for some c > 0 depending only on #D, p and q. As a consequence of (4.6) and (4.9), to estimate f (•, x ) B s p,q (R d ) from above one only needs to estimate the terms (4.8) from above, for each δ ∈ D.

Within these notations, we have the following Lemma 4.1. -Let N 2, 0 < p, q ∞, δ ∈ D and 0 < < 0 . Then, with the notations above

J ,δ p,q (λ, x ) sup β∈N N 2 0 |β| ν 0 m ∈Z d |λ β ν,m , 2 ν x +δ | p 2 ν(N -d)
q/p 1/q , for a.e. x ∈ R N -d (with the usual modification if p = ∞ and/or q = ∞), where

2 ν x + δ = ( 2 ν x d+1 + δ d+1 , ... , 2 ν x N + δ N ) ∈ Z N -d .
Proof. -Suppose first that p, q < ∞. For simplicity, we will write m ν,δ := 2 ν x + δ.

Using (4.7) and (4.8) we get J ,δ p,q (λ, x ) sup

β ∈N d 2 |β | ν 0 m ∈Z d β ∈N N -d |λ β ν,m ,m ν,δ | p 2 ν(N -d) q/p 1/q . Write λ β ν,m = 2 -0 |β| Λ β ν,m and let a := 0 -. Then, 2 |β | |λ β ν,m | = 2 |β |-0 |β| |Λ β ν,m | 2 -a|β| |Λ β ν,m | 2 -a|β | |Λ β ν,m |.
Hence, by Hölder's inequality we have J ,δ p,q (λ, x ) sup

β ∈N d ν 0 m ∈Z d β ∈N N -d 2 -a|β | |Λ β ν,m ,m ν,δ | p 2 ν(N -d) q/p 1/q K a/2 sup β ∈N d ν 0 m ∈Z d sup β ∈N N -d 2 -p a 2 |β | |Λ β ν,m ,m ν,δ | p 2 ν(N -d) q/p 1/q
, where we have used the notation

K α = β ∈N N -d 2 -α|β | for α > 0.
Since the p spaces are increasing with p, by successive applications of the Hölder inequality, we have

J ,δ p,q (λ, x ) K a/2 sup β ∈N d ν 0 β ∈N N -d 2 -p a 2 |β | m ∈Z d |Λ β ν,m ,m ν,δ | p 2 ν(N -d) q/p 1/q K a/2 K 1/p p a 4 sup β ∈N d ν 0 sup β ∈N N -d 2 -p a 4 |β | m ∈Z d |Λ β ν,m ,m ν,δ | p 2 ν(N -d) q/p 1/q = K a/2 K 1/p p a 4 sup β ∈N d ν 0 sup β ∈N N -d 2 -q a 4 |β | m ∈Z d |Λ β ν,m ,m ν,δ | p 2 ν(N -d) q/p 1/q K a/2 K 1/p p a 4 sup β ∈N d ν 0 β ∈N N -d 2 -q a 4 |β | m ∈Z d |Λ β ν,m ,m ν,δ | p 2 ν(N -d) q/p 1/q = K a/2 K 1/p p a 4 sup β ∈N d β ∈N N -d 2 -q a 4 |β | ν 0 m ∈Z d |Λ β ν,m ,m ν,δ | p 2 ν(N -d) q/p 1/q K a/2 K 1/p p a 4 K 1/q q a 4 sup β∈N N ν 0 m ∈Z d |Λ β ν,m ,m ν,δ | p 2 ν(N -d) q/p 1/q . Letting K a,p,q := K a/2 K 1/p p a 4 K 1/q q a 4 and recalling λ β ν,m = 2 -0 |β| Λ β ν,m we get J ,δ p,q (λ, x ) K a,p,q sup β∈N N 2 0 |β| ν 0 m ∈Z d |λ β ν,m ,m ν,δ | p 2 ν(N -d) q/p 1/q
, which is the desired estimate. The proof when p = ∞ and/or q = ∞ is similar but technically simpler.

Remark 4.2. -Of course, when p = ∞, the term "2 ν(N -d) " disappears (recall Remark 2.5) so that, in this case, Fact 1.1 follows directly from the above lemma.

f (•, x ) B (s,Ψ) p,q (R d ) δ∈D J 0 ,δ p,q (λ, x ), with J 0 ,δ p,q (λ, x ) := sup β∈N N 2 0 |β| ν 0 Ψ(2 -ν ) p m ∈Z d |λ β ν,m , 2 ν x +δ | p 2 ν(N -d) q/p 1/q .
Similarly, given a function f ∈ B (s,Ψ)

p,q (R N ), we can estimate the B (s,Ψ)

p,q (R d )-quasinorm of its restrictions f (•, x ) in the same spirit. This is done up to a slight modification in the discussion above. It suffices to multiply the (s, p)-β-quarks considered above by a factor of Ψ(2 -ν ) -1 and to take η β ν,m , the optimal subatomic decomposition of f ∈ B (s,Ψ) p,q (R N ) with respect to these new quarks. Then, the B (s,Ψ) p,q (R N )-quasinorm of f and the B (s,Ψ) p,q (R d )-quasi-norm of its restrictions f (•, x ) satisfy the same relations as when Ψ ≡ 1 with η β ν,m instead of λ β ν,m . That is, we still have

f B (s,Ψ) p,q (R N ) ∼ sup β∈N N 2 |β| ν 0 m∈Z N |η β ν,m | p q/p 1/q , and f (•, x ) B (s,Ψ) p,q (R d ) δ∈D J 0 ,δ p,q (η, x ),
where , 0 > 0 and J 0 ,δ p,q (η, x ) is as in (4.8).

The case q p

This section is concerned with Fact 1.1 (Fact 1.2 being only a consequence of Theorem 1.7). We will use subatomic decompositions together with the estimate given at Lemma 4.1 to get the following generalization of Fact 1.1.

Proposition 5.1. -Let N 2, 1 d < N , 0 < q p ∞ and s > σ p . Let Ψ be an admissible function. Let K ⊂ R N -d be a compact set and let f ∈ B (s,Ψ)

p,q (R N ). Then, it holds that ˆK f (•, x ) q B (s,Ψ) p,q (R d ) dx 1/q C f B (s,Ψ) p,q (R N ) ,
for some constant C = C(K, N, d, p, q) > 0 (with the usual modification if q = ∞).

Proof. -Without loss of generality, we may consider the case K = [1, 2] N -d only (the general case follows from standard scaling arguments). Also, we can suppose that p < ∞ since otherwise, when p = ∞, the desired result is a simple consequence of Lemma 4.1 (recall Remark 4.2). Let us first prove Lemma 5.1 for Ψ ≡ 1 (it will be clear at the end why this is enough to deduce the general case).

Let f ∈ B s p,q (R N ). Given the (s, p)-β-quarks (βqu) ν,m and > r defined at Section 4 we let λ β ν,m = λ β ν,m (f ) be the corresponding optimal subatomic decomposition. In particular

f (x) = β∈N N ∞ ν=0 m∈Z N λ β ν,m (βqu) ν,m (x), with f B s p,q (R N ) ∼ sup β∈N N 2 |β| ν 0 m∈Z N |λ β ν,m | p q/p 1/q
.

By the discussion in Section 4, we have that

f (•, x ) B s p,q (R d ) δ∈D J ,δ p,q (λ, x ), (5.1)
for all ∈ (0, ), where J ,δ p,q (λ, x ) is given by (4.8). Define

Λ β ν,m := m ∈Z d |λ β ν,m ,m | p 1/p .
In particular,

f B s p,q (R N ) ∼ sup β∈N N 2 |β| ν 0 m ∈Z N -d |Λ β ν,m | p q/p 1/q .
Then, the conclusion of Lemma 4.1 rewrites

J ,δ p,q (λ, x ) q sup β∈N N 2 0 q|β| ν 0 2 νq N -d p |Λ β ν, 2 ν x +δ | q for all δ ∈ D,
and some 0 ∈ ( , ). Integration over [1, 2] N -d yields

I δ := ˆ[1,2] N -d J ,δ p,q (λ, x ) q dx ˆ[1,2] N -d sup β∈N N 2 0 q|β| ν 0 2 νq N -d p |Λ β ν, 2 ν x +δ | q dx β∈N N 2 0 q|β| ν 0 ˆ[1,2] N -d 2 νq N -d p |Λ β ν, 2 ν x +δ | q dx .
Now, we observe that

2 νq N -d p |Λ β ν, 2 ν x +δ | q k∈N N -d |Λ β ν, 2 k d+1 x d+1 +δ d+1 ,••• , 2 k N x N +δ N | p 2 k d+1 +•••+k N q/p .
Hence, using the fact that q p and applying N -d times Lemma 3.4, we get

I δ β∈N N 2 0 q|β| ν 0 k∈N N -d |Λ β ν,k+δ | p q/p β∈N N 2 ( 0 -)q|β| sup β∈N N 2 q|β| ν 0 k∈N N -d |Λ β ν,k+δ | p q/p = K ,N,q sup β∈N N 2 q|β| ν 0 k∈N N -d |Λ β ν,k+δ | p q/p K ,N,q sup β∈N N
2 q|β| λ β q bp,q .

Thus, recalling (5.1), we arrive at

ˆ[1,2] N -d f (•, x ) q B s p,q (R d ) dx 1/q f B s p,q (R N ) .
Now, having in mind Remark 4.3, we can reproduce exactly the same proof when Ψ ≡ 1 with almost no modifications. This completes the proof.

The case p < q

In this section we prove, at a stroke, Theorem 1.3 and Theorem 1.6. As will become clear, the proof of Theorem 1.6 will easily follow from that of Theorem 1.3. Let us begin with the following more general result: Theorem 6.1. -Let N 2, 1 d < N , 0 < p < q ∞ and s > σ p . Let Ψ be an admissible function. Then, there exists a function f ∈ B (s,Ψ)

p,q (R N ) such that f (•, x ) / ∈ B (s,Ψ) p,∞ (R d ) for a.e. x ∈ R N -d .
Proof. -We will essentially follow two steps.

Step 1: case d = N -1.

We will construct a function satisfying the requirements of Theorem 6.1 (and hence of Theorem 1.3) via its subatomic coefficients.

Let Ψ be an admissible function. Let 0 < p < q ∞, s > σ p , M = s + 1 and let (λ j,k ) j,k 0 ∈ b p,q be the sequence constructed at Corollary 3.7.

Also, we let

ψ ∈ C ∞ c (R N ) be a function such that supp(ψ) ⊂ [-2, 2] N , inf z∈[0,1] N ψ(z) > 0 and m∈Z N ψ(• -m) ≡ 1. (6.1)
In addition, we will suppose that ψ has the product structure

ψ(x) = ψ(x 1 ) ... ψ(x N ). (6.2)
Notice that such a ψ always exists. 1 Then, we define

f (x) = j 0 k 0 λ j,k 2 -j(s-N p ) Ψ(2 -j ) -1 × ψ(2 j (x 1 -C M j)) ... ψ(2 j (x N -1 -C M j))ψ(2 j x N -k), (6.3)
where C M = 2(M + 2). It follows from Definition 2.16 that

Ψ(2 -j ) -1 2 -j(s-N p ) ψ(2 j x -m) for x ∈ R N , with m = (C M 2 j j, ... , C M 2 j j, k) ∈ Z N ,
can be interpreted as (s, p, Ψ)-0-quarks relative to the cube Q j,m . Consequently, by Theorem 2.17 and Corollary 3.7, we have

f B (s,Ψ) p,q (R N ) c j 0 k 0 λ p j,k q/p 1/q < ∞, (modification if q = ∞). Therefore, f ∈ B (s,Ψ)
p,q (R N ). In particular, the sum in the right-hand side of (6.3) converges in L p (R N ) and is unconditionally convergent for 1 Here is an example. Let u(t) := e -1/t 2 1 (0,∞) (t) (extended by 0 in (-∞, 0]) and let v(t) = u(1 + t)u(1 -t). Then,

ψ(x) := N j=1 1 2 ψ 0 x j 2 where ψ 0 (t) = v(t) v(t -1) + v(t) + v(t + 1)
, is a smooth function satisfying (6.1) and (6.2).

a.e. x ∈ R N (notice the terms involved are all nonnegative) and, by Fubini, f (•, x N ) also converges in L p (R N -1 ) for a.e. x N ∈ R. Thus, letting

Λ j (x N ) := k 0 λ j,k 2 j/p ψ(2 j x N -k),
we may rewrite (6.3) as

f (x , x N ) = j 0 Λ j (x N )2 -j(s-N -1 p ) Ψ(2 -j ) -1 ψ(2 j (x 1 -C M j)) ... ψ(2 j (x N -1 -C M j)).
Notice that assumption (6.1) implies that there is a c 0 > 0 such that

ψ(2 j x N -2 j x N ) c 0 > 0 for all x N ∈ [1, 2
] and j 0.

In particular, we have

Λ j (x N ) c 0 λ j, 2 j x N 2 j/p . (6.4)
Now, for all j 0, we write

K j := h ∈ R N -1 : 2 -(j+1) |h| 2 -j . (6.5)
By [START_REF] Brasseur | A Bourgain-Brezis-Mironescu characterization of higher order Besov-Nikol'skii spaces[END_REF]Lemma 8.2] (in fact in [START_REF] Brasseur | A Bourgain-Brezis-Mironescu characterization of higher order Besov-Nikol'skii spaces[END_REF] it is implicitly supposed that 1 p < ∞ but the proof still works when 0 < p < 1) and (6.4), we have sup

h∈K j ∆ M h f (•, x N ) L p (R N -1 ) c 2 -js Ψ(2 -j ) -1 Λ j (x N ) c 2 -js Ψ(2 -j ) -1 2 j/p λ j, 2 j x N , (6.6)
for any j 0 and some c > 0 independent of j. Recall that

g B (s,Ψ) p,∞ (R N -1 ) ∼ g L p (R N -1 ) + sup j 1 2 js Ψ(2 -j ) sup h∈K j ∆ M h g L p (R N -1 ) , is an equivalent quasi-norm on B (s,Ψ) p,∞ (R N -1
) (this is a discretized version of Definition 2.11). This together with (6.6) and Corollary 3.7 gives

f (•, x N ) B (s,Ψ) p,∞ (R N -1 ) sup j 0 2 j/p λ j, 2 j x N = ∞ for a.e. x N ∈ [1, 2]. Therefore, f (•, x N ) / ∈ B (s,Ψ) p,∞ (R N -1 ) for a.e. x N ∈ [1, 2].
We will show that one can construct a function satisfying the requirements of Theorem 6.1 by considering a weighted sum of translates of the function f constructed above. To this end, we let

f l (x , x N ) := f (x , x N + l) for l ∈ Z,
and we define

g := l∈Z 2 -|l| f l .
Then, by the triangle inequality for Besov quasi-norms, we have

g η B (s,Ψ) p,q (R N ) l∈Z 2 -η|l| f l η B (s,Ψ) p,q (R N ) c η f η B (s,Ψ) p,q (R N ) < ∞,
for some 0 < η 1. Hence, g ∈ B (s,Ψ)

p,q (R N ). To complete the proof we need to show that

g(•, x N ) / ∈ B (s,Ψ) p,∞ (R N -1 ) for a.e. x N ∈ R. (6.7)
Let m ∈ Z. Then, by the triangle inequality for Besov quasi-norms we have

2 -η|m| f m (•, x N ) η B (s,Ψ) p,∞ (R N -1 ) g(•, x N ) η B (s,Ψ) p,∞ (R N -1 ) + l =m 2 -η|l| f l (•, x N ) η B (s,Ψ) p,∞ (R N -1 ) g(•, x N ) η B (s,Ψ) p,∞ (R N -1 ) + c η sup l =m f l (•, x N ) η B (s,Ψ) p,∞ (R N -1 )
. (6.8)

Clearly, the left-hand side of (6.8) is infinite for a.e.

x N ∈ [1 -m, 2 -m]
. Thus, to prove (6.7), one only needs to make sure that the last term on the right-hand side of (6.8) is finite for a.e.

x N ∈ [1 -m, 2 -m].
For it, we notice that, by construction, it is necessary to have j 1 and 2 j k < 2 1+j , (6.9) for λ j,k = 0 to hold. In particular, Λ 0 ≡ 0 and Λ j (x N ) consists only in finitely many terms for a.e. x N ∈ R. In addition, by our assumptions on the support of ψ, we have

ψ(2 j x N -k) = 0 provided x N - k 2 j 2 1-j . (6.10)
By (6.9) and (6.10), we deduce that if

x N ∈ R \ [1, 2]
, then there are only finitely many values of j 1 such that Λ j (x N ) ≡ 0. In particular,

f (•, x N + l) ∈ B (s,Ψ) p,∞ (R N -1 ) for a.e. x N ∈ [1, 2] and all l ∈ Z \ {0}. (6.11)
Moreover, a consequence of (6.9) and (6.10) is that

j 1 and x N ∈ supp(Λ j ) =⇒ 1 -2 1-j x N < 2 + 2 1-j .
In turn, this implies that the support of Λ j is included in [0, 3]. Therefore, f (•, x N + l) ≡ 0 for a.e. x N ∈ [1, 2] and all l ∈ Z with |l| 2. (6.12)

Hence, by (6.11) and (6.12), we infer that max

l =0 f l (•, x N ) B (s,Ψ) p,∞ (R N -1 ) < ∞ for a.e. x N ∈ [1, 2].
In like manner, for every m ∈ Z, we have max

l =m f l (•, x N ) B (s,Ψ) p,∞ (R N -1 ) < ∞ for a.e. x N ∈ [1 -m, 2 -m].
This proves the theorem for d = N -1.

Step 2: case

1 d < N -1.
By the above, we know that Theorem 1.3 holds for any N 2 and d = N -1. In particular, there exists a function f ∈ B (s,Ψ)

p,q (R d+1 ) such that f (•, x d+1 ) / ∈ B (s,Ψ) p,∞ (R d ) for a.e. x d+1 ∈ R. Now, pick a function w ∈ S (R N -d-1 ) with w > 0 on R N -d-1 and set g(x) = g(x 1 , ... , x N ) = f (x 1 , ... , x d , x d+1 )w(x d+2 , ... , x N ).
It is standard that g ∈ L p (R N ) where p := max{1, p}. Then, letting M = s + 1 and using [START_REF] Triebel | Theory of function spaces[END_REF]Formula (16), p.112], we have that

sup |h| t ∆ 2M h g L p (R N ) f L p (R d+1 ) sup |h | t ∆ M h w L p (R N -d-1 ) + w L p (R N -d+1 ) sup |h | t ∆ M h f L p (R d+1 ) ,
for any h = (h , h ) ∈ R N \ {0} with h = (h 1 , ... , h d+1 ) and h = (h d+2 , ... , h N ). In particular, recalling Remark 2.12, we see that this implies

g B (s,Ψ) p,q (R N ) f L p (R d+1 ) w B (s,Ψ) p,q (R N -d-1 ) + w L p (R N -d-1 ) f B (s,Ψ) p,q (R d+1 ) .
Hence, g ∈ B (s,Ψ) p,q (R N ). Moreover, it is easily seen that

g(•, x d+1 , ... , x N ) = f (•, x d+1 )w(x d+2 , ... , x N ) / ∈ B (s,Ψ) p,∞ (R d ),
for a.e. (x d+1 , ... , x N ) ∈ R N -d . This completes the proof.

The function we have constructed above (in the proof of Theorem 6.1) turns out to verify the conclusion of Theorem 1.6.

Proof of Theorem 1.6. -For simplicity, we outline the proof for N = 2 and d = 1 only (the general case follows from the same arguments as above). Let f be the function constructed in the proof of Theorem 6.1 with Ψ ≡ 1, namely

f (x 1 , x 2 ) := j 0 Λ j (x 2 )2 -j(s-1 p ) ψ(2 j (x 1 -C M j)), with Λ j (x 2 ) := k 0 λ j,k 2 j/p ψ(2 j x 2 -k),
where ψ, C M and (λ j,k ) j,k 0 are as in the proof of Theorem 6.1. Clearly,

f B s p,q (R 2 ) c j 0 k 0 λ p j,k q/p 1/q < ∞.
Hence, f ∈ B s p,q (R 2 ). We now distinguish the cases sp > 1, sp = 1 and sp < 1.

Step 1: case sp > 1.

This case works as in Theorem 6.1. Indeed, by the supports of the functions involved, we have for a.e.

x 2 ∈ [1, 2], f (•, x 2 ) C s-1 p (R) ∼ sup j 0 2 j(s-1 p ) sup h∈K j ∆ M h f (•, x 2 ) L ∞ (R) sup j 0 2 j/p λ j, 2 j x 2 = ∞,
where K j is given by (6.5). We may now conclude as in the proof of Theorem 6.1.

Step 2: case sp = 1.

It suffices to notice that, for any k 0, we have

f (•, x 2 ) BMO(R) C M k+2 -k C M k-2 -k C M k+2 -k C M k-2 -k f (x, x 2 ) -f (z, x 2 ) dz dx = C M k+2 -k C M k-2 -k j 0 Λ j (x 2 ) C M k+2 -k C M k-2 -k ψ(2 j (x -C M j)) -ψ(2 j (z -C M j)) dz dx.
Hence, by the support of the functions involved we deduce that

f (•, x 2 ) BMO(R) Λ k (x 2 ) C M k+2 -k C M k-2 -k C M k+2 -k C M k-2 -k ψ(2 k (x -C M k)) -ψ(2 k (z -C M k)) dz dx = Λ k (x 2 ) 1 -1 1 -1 ψ(x) -ψ(z) dz dx c Λ k (x 2 ).
Therefore, we have

f (•, x 2 ) BMO(R) sup j 0 Λ j (x 2 ) = sup j 0 2 j/p λ j, 2 j x 2 = ∞ for a.e. x 2 ∈ [1, 2].
Thus, we may again conclude as in the proof of Theorem 6.1.

Step 3: case sp < 1.

Define r := p 1-sp and rewrite f as

f (x 1 , x 2 ) := j 0 c j (x 2 ) 2 j/r f j (x 1 ),
where we have set f j (x 1 ) := ψ(2 j (x 1 -C M j)) and

c j (x 2 ) := 2 -j(s-2 p ) 2 -j/r k 0 λ j,k ψ(2 j x 2 -k).
Since the f j 's have mutually disjoint support we find that f (•, x 2 ) * (t) c j (x 2 ) 2 j/r f * j (t) for any t 0 and j 0.

Moreover, it is easy to see that f * j (t) = ψ * (2 j t). In turn, this implies that

f (•, x 2 ) L r,∞ (R) c j (x 2 ) 2 j/r sup t>0 t 1/r ψ * (2 j t) = c j (x 2 ) ψ L r,∞ (R) 2 j/p λ j, 2 j x 2 .
Hence, for a.e.

x 2 ∈ [1, 2], f (•, x 2 ) L r,∞ (R) sup j 0 2 j/p λ j, 2 j x 2 = ∞.
This completes the proof.

Characterization of restrictions of Besov functions

In this section, we prove that Besov spaces of generalized smoothness are the natural scale in which to look for restrictions of Besov functions. More precisely, we will prove Theorems 1.7 and 1.8. We present several results, with different assumptions and different controls on the norm of f (•, x ). Let us begin with the following Theorem 7.1. -Let N 2, 1 d < N , 0 < p < q ∞ and let s > σ p . Let K ⊂ R N -d be a compact set. Let Φ and Ψ be two admissible functions such that

j 0 Φ(2 -j ) -p Ψ(2 -j ) p < ∞. (7.1) Let f ∈ B (s,Φ)
p,q (R N ). Then, there exists a constant C > 0 such that

f (•, x ) B (s,Ψ) p,q (R d ) C f B (s,Φ) p,q (R N )
for a.e. x ∈ K.

Moreover, the constant C is independent of x but may depend on f , K, N , d, p, q, Φ and Ψ.

Proof. -Let us first suppose that q < ∞ and that Φ ≡ 1. Without loss of generality we may take K = [1, 2] N -d , the general case being only a matter of scaling. Here again, we use the following short notation

2 ν x = ( 2 ν x d+1 , ... , 2 ν x N ) ∈ N N -d .
Let f ∈ B s p,q (R N ) and write its subatomic decomposition as

f (x) = β∈N N ν 0 m∈Z N λ β ν,m (βqu) ν,m (x),
where the (s, p)-β-quarks (βqu) ν,m are as in Section 4 and λ β ν,m = λ β ν,m (f ) is the optimal subatomic decomposition of f , i.e. such that

f B s p,q (R N ) ∼ sup β∈N N 2 |β| λ β bp,q . (7.2)
Let ε > 0 be small. Rewriting f as in the discussion at Section 4 and using Lemma 4.1 together with Remark 4.3 we have

f (•, x ) B (s,Ψ) p,q (R d ) δ∈D J -ε,δ p,q (λ, x ), (7.3)
where

J -ε,δ p,q (λ, x ) := sup β∈N N 2 ( -ε)|β| ν 0 2 ν(N -d) Ψ(2 -ν ) p m ∈Z d |λ β ν,m , 2 ν x +δ | p q/p 1/q
. By Lemma 3.5, we know that for any positive sequence (α ν

) ν 0 ∈ 1 (N) there is a constant C = C(λ, α, N, d) > 0 such that 2 ν(N -d) m ∈Z d |λ β ν,m , 2 ν x +δ | p C max{1, |β| N -d+1 } α ν m∈Z N |λ β ν,m | p ,
for a.e. x ∈ [1, 2] N -d and any (ν, β) ∈ N × N N . In particular, we have

J -ε,δ p,q (λ, x ) sup β∈N N 2 ( -ε)|β| max 1, |β| N -d+1 p ν 0 Ψ(2 -ν ) p α ν m∈Z N |λ β ν,m | p q/p 1/q
. Now, by assumption (7.1), we can choose α ν = Ψ(2 -ν ) p . Therefore, recalling (7.3), we have

f (•, x ) B (s,Ψ) p,q (R d ) sup β∈N N 2 ( -ε)|β| max 1, |β| N -d+1 p ν 0 m∈Z N |λ β ν,m | p q/p 1/q sup β∈N N 2 |β| ν 0 m∈Z N |λ β ν,m | p q/p 1/q .
Finally, recalling (7.2), we have

f (•, x ) B (s,Ψ) p,q (R d ) f B s p,q (R N ) for a.e. x ∈ [1, 2] N -d .
The proof when q = ∞ and/or Φ ≡ 1 is similar (recall Remark 4.3). It this latter case, one only have to adjust the (s, p)-β-quarks by a factor of Φ(2 -ν ) -1 and to replace λ β ν,m by η β ν,m , the optimal decomposition of f ∈ B (s,Φ)

p,q (R N ) along these (s, p, Φ)-βquarks. Then, as in Remark 4.3, it suffices to replace Ψ(2 -ν )λ β ν,m in the estimate

(7.3) of f (•, x ) B (s,Ψ) p,q (R d ) by Ψ(2 -ν )/Φ(2 -ν ) η β ν,m
and the same proof yields the desired conclusion.

We carry on with the following generalization of Theorem 1.7.

Theorem 7.2. -Let N 3, 1 d < N , 0 < r p < q ∞, s > σ p and let χ = qr q-r (resp. χ = r if q = ∞). Let Φ and Ψ be two admissible functions such that

j 0 Φ(2 -j ) -χ Ψ(2 -j ) χ < ∞. (7.4) Let K ⊂ R N -d be a compact set and suppose that f ∈ B (s,Φ) p,q (R N ). Then, ˆK f (•, x ) r B (s,Ψ) p,r (R d ) dx 1/r C f B (s,Φ) p,q (R N ) ,
for some constant C = C(K, N, d, p, q, Φ, Ψ) > 0.

Proof. -By Hölder's inequality, the definition of the norms involved and our assumptions on s, p, q, r, χ, we see that (7.4)

implies that B (s,Φ) p,q (R N ) ⊂ B (s,Ψ) p,r (R N ) continuously, i.e. f B (s,Ψ) p,r (R N ) j 0 Φ(2 -j ) -χ Ψ(2 -j ) χ 1/χ f B (s,Φ) p,q (R N ) .
Using now Proposition 5.1, we obtain the desired conclusion.

We now prove Theorem 1.8.

Proof of Theorem 1.8. -The proof works exactly as in Theorem 1.3 and, here again, it suffices to prove the result for N 2, d = N -1. We prove the case N = 2 only but the general case N 2 is similar. Let (λ j,k ) j,k 0 be the sequence constructed at Lemma 3.9. Let M ∈ N * with s < M . We consider the following function

f (x 1 , x 2 ) = j 0 k 0 λ j,k 2 -j(s-2 p ) ψ(2 j (x 1 -C M j))ψ(2 j x 2 -k),
where ψ and C M are as in the proof of Theorem 1.3. There, we have shown that

f B s p,q (R 2 )
c λ bp,q , (thus implying f ∈ B s p,q (R 2 )) and sup

h∈K j ∆ M h f (•, x 2 ) L p (R) c 2 -js 2 j/p λ j, 2 j x 2 ,
for any j 0 and a.e.

x 2 ∈ [1, 2]
, where K j is as in (6.5). From this it follows that

2 js Ψ(2 -j ) sup h∈K j ∆ M h f (•, x 2 ) L p (R) c Ψ(2 -j )2 j/p λ j, 2 j x 2 .
Now, by Lemma 3.9 and since

g B (s,Ψ) p,q (R) ∼ g L p (R) + j 0 2 jsq Ψ(2 -j ) q sup h∈K j ∆ M h g q L p (R) 1/q
, is an equivalent quasi-norm on B (s,Ψ) p,q (R), we have f ∈ B s p,q (R 2 ) and f (•, x 2 ) / ∈ B (s,Ψ) p,q (R) for a.e. x 2 ∈ [1, 2]. Then, arguing exactly as in Theorem 1.3, we obtain a function satisfying the requirements of Theorem 1.8. This completes the proof.

Remark 7.3. -This can be generalized in the spirit of Theorem 7.1. Indeed, repeating the arguments of the proof of Theorem 6.1, we can prove that if (7.4) is violated, then there is a function

f ∈ B (s,Φ) p,q (R N ) such that f (•, x ) / ∈ B (s,Ψ) p,q (R d ) for a.e. x ∈ R N -d .

Part II

Rigidity results

Chapter 3 Liouville type results for a nonlocal obstacle problem 1 Introduction

A classical topic in applied analysis consists in the study of diffusive processes in media with an obstacle: roughly speaking, a dispersal follows a Brownian motion in an environment that possess an inaccessible region. At the level of partial differential equations, this translates into a reaction-diffusion equation that is defined outside a set K, which acts as an impenetrable obstacle and along which Neumann conditions are prescribed.

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

In this chapter, we will study a nonlocal version of a diffusion equation and provide a series of Liouville-type results (whose precise statements will be given in Section 2). Not only the results obtained have a theoretical interest in the development of the theory of nonlocal equations, but they also possess several potential applications (especially in mathematical biology, where the dispersal of biological populations often presents nonlocal features, see e.g. formula (1) in [START_REF] Cosner | Evolutionary stability of ideal free nonlocal dispersal[END_REF], or in [START_REF] Berestycki | A non-local bistable reaction-diffusion equation with a gap[END_REF]).

Concretely, we will suppose that the diffusion operator arises by convolution with an integrable kernel and we will show that solutions of bistable stationary equations with fixed behavior at infinity are necessarily constant, at least when the obstacle is convex or "close to being convex" (we also observe that similar rigidity results do not hold in general for nonconvex obstacles).

Interestingly, in the nonlocal case, the boundary conditions along the obstacle do not need to be prescribed a priori (differently from the classical case).

In addition, the nonlocal operator that we consider here is not "regularizing", so some care is needed in our case to deal with a possible lack of regularity of the solutions.

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

A nonlocal obstacle problem

Throughout this chapter, K denotes a compact set of R N with N 2, and |•| denotes the Euclidean norm in R N . We are interested in qualitative properties of bounded solutions to the following nonlocal semilinear equation

Lu + f (u) = 0 in R N \ K, (1.1)
where L is the nonlocal diffusion operator given by

Lu(x) := ˆRN \K J(x -y)(u(y) -u(x))dy. (1.2)
The kernel J ∈ L 1 (R N ) is a radially symmetric non-negative function with unit mass and f is a C 1 "bistable" nonlinearity (precise assumptions on f and J will be given later on). This problem may be thought of (see the next page for more explanations) as a nonlocal version of the following problem

∆u + f (u) = 0 in R N \ K, ∇u • ν = 0 on ∂K, (1.3)
where ν is the outward unit vector normal to K, assuming for (1.4) that K is smooth enough. For problem (1.4) with the local diffusion operator ∆u, it was shown in [START_REF] Berestycki | Bistable travelling waves around an obstacle[END_REF] that there exist a time-global classical solution u(t, x) to the parabolic problem

   ∂u ∂t = ∆u + f (u) in R × R N \ K, ∇u • ν = 0 on R × ∂K, (1.4)
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| → +∞. (1.5)
The function u ∞ is a stationary solution of (1.4) 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 travelling waves around an obstacle[END_REF] for precise assumptions) it is shown in [17, Theorems 6.1 and 6.4] that solutions to (1.5) are actually identically equal to 1 in the whole set R N \ K. This Liouville property shows that the solutions u(t, x) of (1.4) constructed in [START_REF] Berestycki | Bistable travelling waves around an obstacle[END_REF] then satisfy

u(t, x) -→ t→+∞ 1 locally uniformly in x ∈ R N \ K. (1.6)
To some extent, this result can be given an ecological interpretation. Consider a population with trajectories describing a Brownian motion in an environment consisting of the whole space R N with a compact obstacle K, and suppose that f represents the demographic rate of the population. Then, the solution u(t, x) to (1.4) can be understood as the density of the population at time t and location x. In this context, (1.6) means that, at large time, the population tends to occupy the whole space.

Assuming now that the trajectories follow, say, a compound Poisson process, then the diffusion phenomena are better described by a convolution-type operator such as (1.2). The reaction-diffusion equation ∂u ∂t = ∆u + f (u) is then replaced by the equation

∂u ∂t = Lu + f (u),
with the nonlocal dispersion operator L, see [START_REF] Fife | An integrodifferential analog of semilinear parabolic PDEs[END_REF][START_REF] Hutson | The evolution of dispersal[END_REF]. In this chapter, we deal with qualitative properties of the stationary solutions of equation (1.1), together with some asymptotic limiting conditions at infinity similar to those appearing in (1.5).

Namely, we will be mainly concerned with solutions of

     Lu + f (u) = 0 in R N \ K, 0 u 1 in R N \ K, u(x) → 1 as |x| → +∞. (1.7)
It is expected that (1.5) and (1.7) share some common properties. One of the goals of the present chapter is, as for (1.5), to find some geometric conditions on K which guarantee that the solutions u to (1.7) are identically equal to 1. Moreover, as in [START_REF] Bouhours | Robustness for a Liouville type theorem in exterior domains[END_REF] for (1.5), we will also show the robustness of the Liouville type results for (1.7).

We notice however that, whereas the solutions of (1.5) are automatically classical C 2 solutions in R N \ K if the boundary ∂K is smooth enough (by standard interior and boundary elliptic estimates), there is in general no smoothing effect for the nonlocal problems (1.1) or (1.7). The solutions u may even not be continuous in general. Yet some regularity results (uniform or Hölder continuity) will be shown here under additional assumptions on the data J and f . Actually, one of the difficulties and novelties of this chapter, as compared to [START_REF] Berestycki | Bistable travelling waves around an obstacle[END_REF], is to deal with this a priori lack of regularity in general.

We observe also that, in (1.1) or (1.7), we do not ask for any additional boundary condition on ∂K. To understand why this is so, let us give some heuristic motivation. First of all, the most intuitive nonlocal counterpart of (1.5) would be to replace ∆u in (1.5) by L ε u with ε > 0 small, where

L ε u(x) := 1 βε 2 ˆRN \K J ε (x -y)(u(y) -u(x))dy,
and J ε (z) = ε -N J(ε -1 z), J being a radially symmetric kernel with

β = (2N ) -1 ˆRN J(z)|z| 2 dz.
In other words, the nonlocal dispersion operator Lu in (1.1) or (1.7) would be replaced by L ε u and the kernel J would be given by (βε 2 ) -1 J ε . Furthermore, using for example [START_REF] Bourgain | Another look at Sobolev spaces[END_REF], the associated energy of (1.7), with L ε in place of L, can be thought of as an approximation of that of (1.5) (see [START_REF] Andreu-Vaillo | Nonlocal diffusion problems[END_REF] and also [START_REF]Persistence criteria for populations with non-local dispersion[END_REF][START_REF] Berestycki | A non-local bistable reaction-diffusion equation with a gap[END_REF][START_REF] Hutson | The evolution of dispersal[END_REF] where similar quantities are considered in a biological framework). Now, to see how the Neumann boundary condition in (1.5) can be recovered from (1.1) or (1.7) with L ε as ε → 0 + , let us consider for simplicity the case where ∂K is of class C 1 with unit normal ν and the bounded function u is of class C 1 (R N \ K) and is extended as a C 1 (R N ) function still denoted by u. Formula (1.1) then also holds by continuity in R N \ K and, for every x, y ∈ R N \ K, there exists a point c x,y ∈ [x, y] such that u(y) -u(x) = ∇u(c x,y ) • (y -x). It follows that, for every

x ∈ R N \ K, -f (u(x)) = 1 βε ˆRN \K J ε (x -y) ∇u(c x,y ) • y -x |y -x| dy,
where J ε (z) = ε -N J(ε -1 z) and J(z) = J(z)|z|. Then, for all x ∈ ∂K, a formal computation leads to

γ ∇u(x) • ν = lim ε→0 + ˆRN \K J ε (x -y) ∇u(c x,y ) • y -x |y -x| dy = lim ε→0 + -εβf (u(x)) = 0,
where γ = (1/2) ´RN J(z)|z 1 |dz > 0. Hence, ∇u • ν = 0 on ∂K and (1.7) is then a reasonable nonlocal counterpart for (1.5). The above calculation justifies, at least formally, why no additional boundary condition on ∂K is required in (1.1) or (1.7).

General assumptions, notations and definitions

Let us now specify the detailed assumptions made throughout the chapter. As already mentioned above, we suppose that f is of "bistable" type and J is a radially symmetric kernel. More precisely, we will assume that

f ∈ C 1 ([0, 1]), f (0) 0, f (1) < 0, (1.8)
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 , (1.9) and there exists a function φ ∈ C(R) satisfying

J 1 * φ -φ + f (φ) 0 in R, φ is increasing in R, φ(-∞) = 0, φ(+∞) = 1, (1.10)
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 .
We notice that, in addition to the first property in (1.9), the second one is immediately fulfilled if J is assumed to be continuous. Moreover, we notice that condition (1.10) implies immediately that 0 < φ < 1 in R. As is well-known (see e.g. [START_REF] Bates | Travelling waves in a convolution model for phase transitions[END_REF][START_REF] Coville | Travelling fronts in asymmetric nonlocal reaction diffusion equations: the bistable and ignition cases[END_REF]), condition (1.10) is satisfied if, in addition to (1.8) and (1.9), the following assumptions are made on f and J:

           ∃ 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, f (1) < 0, f < 1 in [0, 1], ˆR J 1 (x)|x|dx < +∞ and J ∈ W 1,1 (R N ).
(1.11)

Let us also list in this subsection a few notations and definitions that will be used all along the chapter:

|E| : is the Lebesgue measure of the measurable set E; 1 E : is the characteristic function of the set E; B R : is the open Euclidean ball of radius R > 0 centered at the origin;

B R (x) : is the open Euclidean ball of radius R > 0 centered at x ∈ R N ; A (R 1 , R 2 ) : is the open annulus B R 2 \ B R 1 for 0 R 1 < R 2 , by setting B 0 = {0}; A (x, R 1 , R 2 ) : is the open annulus x + A (R 1 , R 2 );
g * h : is the convolution of g and h; g + : is the positive part of g, i.e. g + := max{0, g}.

123

Given Ω ⊂ R N and p ∈ [1, ∞], we denote by L p (Ω) the Lebesgue space of (equivalence classes of) measurable functions g for which the p-th power of the absolute value is Lebesgue integrable when p < ∞ (resp. essentially bounded when p = ∞). When the context is clear, we will write g p instead of g L p (Ω) . Given α ∈ (0, 1] and p ∈ [1, ∞], B α p,∞ (R N ) stands for the Nikol'skii space consisting in all measurable functions g ∈ L p (R N ) such that

[g] B α p,∞ (R N ) := sup h =0 g(• + h) -g L p (R N ) |h| α < +∞.
We note that, when p = ∞, the space B α ∞,∞ (R N ) coincides with the classical Hölder space C 0,α (R N ). For a set E ⊂ R N and g : E → R, we set

[g] C 0,α (E) = sup x∈E, y∈E, x =y |g(x) -g(y)| |x -y| α .
Let us finally recall some useful notions of regularity of a compact set K.

Definition 1.1. -Let α ∈ (0, 1]. We say that a compact set K ⊂ R N has C 0,α boundary if there exist r > 0, p ∈ N, p rotations (R i ) 1 i p of R N , p points (z i ) 1 i p of ∂K and p functions (ψ i ) 1 i p defined in the (N -1)-dimensional ball B N -1 r = x ∈ R N -1 ; |x | < r of class C 0,α (B N -1 r
) and such that

∂K = 1 i p R i x N = ψ i (x ); x ∈ B N -1 r , (1.12)
and

• K ∩ B r (z i ) = R i x N > ψ i (x ); x ∈ B N -1 r ∩ B r (z i ), (1.13)
for every 1 i p.

Definition 1.2. -Let α ∈ (0, 1], let K ⊂ R N be a compact convex set with non-empty interior (∂K is then automatically of class C 0,α ) and let (K ε ) 0<ε 1 ⊂ R N be a family of compact, simply connected sets having C 0,α boundary. We say that (K ε ) 0<ε 1 is a family of C 0,α deformations of K if the following conditions are fulfilled:

(i) K ⊂ K ε 1 ⊂ K ε 2 for all 0 < ε 1 ε 2 1; (ii) K ε → K as ε ↓ 0 in C 0,α , in the sense that there exist r > 0, p ∈ N, p rotations (R i ) 1 i p of R N , p points (z i ) 1 i p of ∂K, p functions (ψ i ) 1 i p and p families of functions (ψ i,ε ) 1 i p, 0<ε 1 of class C 0,α (B N -1 r
) describing ∂K and ∂K ε as in (1.12) and (1.13) above, and such that

ψ i -ψ i,ε C 0,α (B N -1 r
) → 0 as ε ↓ 0, for every 1 i p.

Main results

The Liouville property for the local problem (1.5) says that u = 1 in R N \ K under some geometric conditions on K, in particular when K is convex, see [START_REF] Berestycki | Bistable travelling waves around an obstacle[END_REF]. When the obstacle K is convex, we prove that this Liouville property still holds for (1.7) with the nonlocal operator L. We will actually prove several results, which correspond to various assumptions on the solutions u and the data f and J. We will also show the robustness of the Liouville type property with respect to small deformations of the obstacle K. The assumptions (1.8), (1.9) and (1.10) will be common assumptions of almost all results. In some statements, assumption (1.10) is replaced by the stronger assumption (1.11).

A first rough Liouville type result

Under rather mild additional assumptions on K, we first state a "rough" Liouville type property for the solutions of (1.7), if f is assumed to be non-negative on the range of u.

Proposition 2.1. -Let K ⊂ R N be a compact set such that R N \ K is con- nected. Assume that f ∈ C 1 ([0, 1]) and J satisfies (1.9). Let θ ∈ [0, 1) and assume that f 0 in [θ, 1]. Let u : R N \ K → [θ, 1] be a continuous solution of Lu + f (u) = 0 in R N \ K, u(x) → 1 as |x| → +∞.
(2.1)

Then, u = 1 in R N \ K.
One of the main goals of the chapter is to understand under which conditions on K this Liouville type property still holds or does not hold when u ranges in the whole interval [0, 1].

Liouville type properties for convex obstacles

Our first main theorem is the following result dealing with continuous super-solutions to

L(u) + f (u) = 0 ranging in [0, 1]. Theorem 2.2. -Let K ⊂ R N be a compact convex set. Assume (1.8), (1.9), (1.10) and let u ∈ C(R N \ K, [0, 1]), (2.2)
be a function satisfying

Lu + f (u) 0 in R N \ K, u(x) → 1 as |x| → +∞. (2.3) Then, u = 1 in R N \ K.
If we ask for a solution of (1.7) instead of a super-solution, it turns out that the regularity or limiting conditions required on u to obtain a Liouville type result can be considerably weakened, by strengthening the assumptions made on f and/or J.

Firstly, the continuity assumption (2.2) can be relaxed provided the nonlinearity does not vary too much. More precisely, we will prove the following result.

Theorem 2.3. -Let K ⊂ R N be a compact convex set. Assume (1.8), (1.9), (1.10) and suppose that max

[0,1] f < 1 2 . ( 2

.4)

Let u : R N \ K → [0, 1] be a measurable function satisfying

Lu + f (u) = 0 a.e. in R N \ K, u(x) → 1 as |x| → +∞.
Then, u = 1 a.e. in R N \ K.

Secondly, assuming that f and J satisfy (1.11) instead of (1.10), that J ∈ L 2 (R N ) and is compactly supported, and that f does not vary too much or u is a priori uniformly continuous, then the assumptions on the asymptotic behaviour of u at infinity can be noticeably weakened. More precisely, the following result holds.

Theorem 2.4. -Let K ⊂ R N be a compact convex set and assume that f and J satisfy (1.8), (1.9) and (1.11). Assume further that J is compactly supported and

J ∈ L 2 (R N ). If u : R N \ K → [0, 1] is uniformly continuous in R N \ K and obeys    Lu + f (u) = 0 in R N \ K, sup R N \K u = 1, (2.5
)

then u = 1 in R N \ K. Similarly, if (2.4) holds and if u : R N \ K → [0, 1] is a measurable function satisfying    Lu + f (u) = 0 a.e. in R N \ K, ess sup R N \K u = 1, (2.6)
then u = 1 a.e. in R N \ K.

Remark 2.5. -Condition (2.4) ensures that u actually has a uniformly continuous representative in R N \ K (see Lemma 2.2 and Remark 3.3). However, if u is already known to be uniformly continuous, then Theorem 2.4 provides the same conclusion without assumption (2.4) (see Lemma 7.2 for further details).

Robustness of the Liouville property for nearly convex obstacles K

Under some flatness assumptions on f , and following a line of ideas in [START_REF] Bouhours | Robustness for a Liouville type theorem in exterior domains[END_REF], it turns out that the Liouville property is still available under small Hölder perturbations of a given convex obstacle K. Namely, the following result holds.

Theorem 2.6. -Let α ∈ (0, 1], let K ⊂ R N be a compact convex set with non-empty interior and let (K ε ) 0<ε 1 be a family of C 0,α deformations of K. Assume (1.8), (1.9), (1.11) and suppose that J ∈ B α 1,∞ (R N ) and max

[0,1] f < inf 0<ε 1 inf x∈R N \Kε J(x -•) L 1 (R N \Kε) .
For 0 < ε 1, let L ε be the operator given by, for every v ∈ L ∞ (R N \ K ε ),

L ε v(x) := ˆRN \Kε J(x -y)(v(y) -v(x))dy.
Then there exists ε 0 ∈ (0, 1] such that for all ε ∈ (0, ε 0 ] the unique measurable solution

u ε of      L ε u ε + f (u ε ) = 0 a.e. in R N \ K ε , 0 u ε 1 a.e. in R N \ K ε , u ε (x) → 1 as |x| → +∞, (2.7
)

is u ε = 1 a.e. in R N \ K ε .
Remark 2.7. -It should be noted that the monotonicity assumption (i) in Definition 1.2 has been made for simplicity and is not necessary for our purposes. Moreover, the conclusion of Theorem 2.6 remains true whenever (K ε ) 0<ε 1 is a family of C 0,α deformations of any compact set K for which the conclusion of Theorem 2.3 is valid. Since there exist some smooth, compact, non-convex and simply connected sets which are C 0,α close to a smooth, compact and convex set, Theorem 2.6 implies that the Liouville property holds for some smooth, compact and non-convex obstacles, and then also for their C 0,α perturbations. Finally, we conjecture that the Liouville property of Theorem 2.3 holds for any starshaped compact obstacle as well.

However, as in the local case (see [17, Theorem 6.5]), the above Liouville type properties cannot be expected for general obstacles. For example, one can easily find counterexamples if K is no longer simply connected. Take for instance K = A (1, 2) = B 2 \ B 1 and suppose that J is supported in B 1/2 . Then, the function u defined by

u(x) = 1 if x ∈ R N \ B 2 , 0 if x ∈ B 1 ,
is a continuous solution of (1.1). Yet, u is not identically 1 in the whole set R N \ K.

Outline of the chapter. The following first sections are concerned with general results on the solutions to problems (1.1) or (1.7). Namely, in Section 3, we show that the solutions are uniformly continuous, more precisely they have a uniformly continuous representative, if rather mild assumptions are made on f . In Section 4, we give several comparison principles that fit our purposes. We then use these comparison principles in Section 5 to construct a radially symmetric lower bound for the solutions. In Section 6, we study an auxiliary problem which will enable us to pave the way towards the proof of Theorem 2.4. The remaining part of the chapter is devoted to the proofs of our main results. In Section 7, we prove, at a stroke, Theorems 2.2 and 2.3, and with more work we show how to relax the assumptions on u when the kernel J is compactly supported, that is we prove Theorem 2.4. In Section 8, as a preliminary result we prove the rough Liouville-type result Proposition 2.1 and then we establish our robustness result Theorem 2.6.

Some auxiliary regularity results

Throughout this section, K is any compact subset of R N , f is any C 1 (R) function, and J is any L 1 (R N ) non-negative and radially symmetric kernel with unit mass. For x ∈ R N , we write

J (x) := ˆRN \K J(x -y)dy.
Notice that J is a uniformly continuous function in R N . In the sequel, for any δ > 0, we will denote K δ the closed thickening of K with width δ, defined by

K δ := K + B δ .
We now prove that, when J is compactly supported and f is not too large in [0, 1], then the measurable solutions u to (1.7) are continuous far away from the obstacle.

Lemma 3.1. -Suppose that K ⊂ R N is a compact set and that J is supported in the ball B δ for some δ > 0. Suppose that

max [0,1] f < 1. (3.1) Let u ∈ L ∞ (R N \ K, [0, 1]) be a solution of Lu + f (u) = 0 a.e. in R N \ K. Then u is uniformly continuous in R N \ K δ , in the sense that u has a representative in its class of equivalence that is uniformly continuous in R N \ K δ . If, in addition, J ∈ B α 1,∞ (R N ) for some α ∈ (0, 1], then u ∈ C 0,α (R N \ K δ ) and 1 -max [0,1] f [u] C 0,α (R N \K δ ) [J] B α 1,∞ (R N ) .
Proof. -For every x and y in R N \ K δ , we have

Lu(x) -Lu(y) = ˆRN \K u(z) J(x -z) -J(y -z) dz -u(x) ˆRN \K J(x -z) dz + u(y) ˆRN \K J(y -z)dz = -J (x)u(x) +J (y)u(y) + ˆRN \K u(z) J(x -z) -J(y -z) dz.
Since J has unit mass and is supported in B δ , we get that J (x) = J (y) = 1. Therefore,

Lu(x) -Lu(y) + u(x) -u(y) = ˆRN \K u(z) J(x -z) -J(y -z) dz.
Now, remember that u is a solution to Lu + f (u) = 0 a.e. in R N \ K. In particular, there exists a measurable negligible set E such that Lu(z) + f (u(z)) = 0 (and u(z) ∈ [0, 1]) for all z ∈ R N \ (K ∪ E). Hence, letting g(t) := t -f (t), for t ∈ [0, 1], we obtain that g(u(x)) -g(u(y)) = ˆRN \K u(z) J(x-z)-J(y-z) dz =: h(x, y),

for all x, y ∈ R N \ (K δ ∪ E). Notice that, since J ∈ L 1 (R N ) and u ∈ L ∞ (R N \ K), the function h defined by the right-hand side of the previous equation can actually be defined in R N ×R N and it is uniformly continuous in R N ×R N . Furthermore, by (3.1), the function

g ∈ C 1 ([0, 1]) is such that g > 0 in [0, 1]. It is then a C 1 diffeomorphism from [0, 1] to [g(0), g(1)] = [-f (0), 1 -f (1)]. Let us denote g -1 : [g(0), g(1)] → [0, 1] its reciprocal. Fix y 0 ∈ R N \ (K δ ∪ E). For every x ∈ R N \ (K δ ∪ E), (3.2) yields g(u(y 0 )) + h(x, y 0 ) = g(u(x)) ∈ [g(0), g (1)
].

Since the function x → g(u(y 0 )) + h(x, y 0 ) is continuous (in the whole R N ) and since E is negligible, it follows that g(u(y 0 )) + h(x, y 0 ) ∈ [g(0), g(1)] for all x ∈ R N \ K δ (since any point of the open set R N \ K δ is the limit of a sequence of points in

R N \ (K δ ∪ E)). Define now u(x) = g -1 g(u(y 0 )) + h(x, y 0 ) for x ∈ R N \ K δ . By (3.2), one has u = u in R N \ (K δ ∪ E). Furthermore, u is uniformly continuous in R N \ K δ owing to its definition, since h is uniformly continuous in R N × R N and g -1 is C 1 hence Lipschitz continuous in [g(0), g(1)
].

Even if it means redefining u by u in R N \ K δ , it follows that u is uniformly continuous in R N \ K δ and that (3.2) holds, by continuity, for all x, y ∈ R N \ K δ . In particular, since 0 u 1 in R N \ K, we get that

∀ x, y ∈ R N \ K δ , |g(u(x)) -g(u(y))| J(• + x -y) -J L 1 (R N ) . (3.3) Finally, if J ∈ B α 1,∞ (R N ) with α ∈ (0, 1], then (3.3) yields g(u) ∈ C 0,α (R N \ K δ ) and [g(u)] C 0,α (R N \K δ ) sup h =0 J(• + h) -J L 1 (R N ) |h| α = [J] B α 1,∞ (R N ) . Since max [g(0),g(1)] |(g -1 ) | (1 -max [0,1] f ) -1 and 0 u 1, one concludes that u ∈ C 0,α (R N \ K δ ) and 1 -max [0,1] f [u] C 0,α (R N \K δ ) [J] B α 1,∞ (R N ) .
We now establish a regularity result for u in the whole set R N \ K for flatter nonlinearities.

Lemma 3.2. -Suppose that K ⊂ R N is a compact set and that max [0,1] f < inf R N \K J . (3.4) Let u ∈ L ∞ (R N \K, [0, 1]
) be a solution of Lu+f (u) = 0 a.e. in R N \K. Then, u can be redefined up to a negligible set and extended as a uniformly continuous function in

R N \ K. If, in addition, J ∈ B α 1,∞ (R N ) for some α ∈ (0, 1], then u ∈ C 0,α (R N \ K) and inf R N \K J -max [0,1] f [u] C 0,α (R N \K) 2 [J] B α 1,∞ (R N ) .
Remark 3.3. -In the case of a compact convex obstacle K, then the conclusion of Lemma 2.2 still holds if (2.5) is replaced by max

[0,1] f < 1 2 .
Indeed, J 1/2 in R N \ K when K is convex (remember also that J is always assumed to be a non-negative radially symmetric kernel with unit mass). The bound 1/2 is somehow optimal, since K can be as large as desired, still in the class of compact convex sets. However, this bound deteriorates considerably if K is only starshaped, as the infimum of J in R N \ K can become arbitrarily small. Roughly speaking, the less convex the obstacle K, the flatter the nonlinearity f needs to be to insure (2.5) and the interior continuity of the solution u.

Proof of Lemma 2.2. -Reasoning exactly as in the proof of Lemma 3.1, there exists a measurable negligible set E such that

∀ x, y ∈ R N \ (K ∪ E), G(x, u(x)) -G(y, u(y)) = h(x, y), (3.5)
where h(x, y) is defined in (3.2) (remember also that h is uniformly continuous in

R N × R N ) and G(x, s) = J (x) s -f (s) for (x, s) ∈ R N × [0, 1].
By (2.5) and the continuity of J , the function )] . Even if it means redefining u by u in R N \ K and extending it by u in R N \ K, it follows that u is continuous in R N \ K and that (3.5) holds, by continuity, for all x, y in the open set R N \ K and then in R N \ K. In particular, since 0 u 1 in R N \ K, we get that

G is such that ∂ s G(x, s) > 0 for all (x, s) ∈ R N \ K × [0, 1]. For every x ∈ R N \ K, the function G(x, •) is then a C 1 dif- feomorphism from [0, 1] to [G(x, 0), G(x, 1)]. Let us denote H x : [G(x, 0), G(x, 1)] → [0, 1] its reciprocal, that is, H x (G(x, t)) = t for all x ∈ R N \ K and t ∈ [0, 1]. Fix y 0 ∈ R N \ (K ∪ E). For every x ∈ R N \ (K ∪ E), (3.5) yields G(y 0 , u(y 0 )) + h(x, y 0 ) = G(x, u(x)) ∈ [G(x, 0), G(x, 1)]. Since the function x → G(y 0 , u(y 0 )) + h(x, y 0 ) is continuous (in the whole space R N ), since G is itself continuous in R N × [0, 1] and since E is negligible, it follows that G(y 0 , u(y 0 )) + h(x, y 0 ) ∈ [G(x, 0), G(x, 1)] for all x in the open set R N \ K. Define now u(x) = H x G(y 0 , u(y 0 )) + h(x, y 0 ) for x ∈ R N \ K. By (3.5), one has u = u in R N \ (K ∪ E). Furthermore, u is continuous in R N \ K owing to its definition, since h is continuous in R N × R N and (x, s) → H x (s) is continuous in the set (x, s) ∈ R N \ K × R; s ∈ [G(x, 0), G(x, 1
∀ x, y ∈ R N \ K, |G(x, u(x))) -G(y, u(y))| J(• + x -y) -J L 1 (R N ) . (3.6)
Finally, define

β := inf R N \K J -max [0,1] f > 0,
the positivity of β resulting from (2.5). From (3.6) together with the definition of G and the inequalities 0 u 1 in R N \ K, one infers that, for all x, y ∈ R N \ K,

J (x) (u(x) -u(y)) -(f (u(x)) -f (u(y))) J(• + x -y) -J L 1 (R N ) + u(y) (J (x) -J (y)) 2 J(• + x -y) -J L 1 (R N ) .
It follows from the mean value theorem and the above definition of β that

β |u(x) -u(y)| 2 J(• + x -y) -J L 1 (R N ) ,
for all x, y ∈ R N \ K. In particular, the function u is uniformly continuous in

R N \ K. Furthermore, if J ∈ B α 1,∞ (R N ) for some α ∈ (0, 1], then u ∈ C 0,α (R N \ K) and β [u] C 0,α (R N \K) 2 [J] B α 1,∞ (R N )
. The proof of Lemma 2.2 is thereby complete.

Comparison principles

In this section, we collect some comparison principles that fit for our purposes. Throughout this section, K is any compact subset of R N , f is any C 1 (R) function, and J is any L 1 (R N ) non-negative and radially symmetric kernel with unit mass.

We start with a weak maximum principle.

Lemma 4.1 (Weak maximum principle). -Assume that f -c 1 in [1 -c 0 , +∞), for some c 0 > 0, c 1 > 0. (4.1) 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) be such that u, v ∈ C H , (4.2)
and

Lu + f (u) 0 in H, Lv + f (v) 0 in H. (4.3) Assume also that u 1 -c 0 in H, (4.4
)

that lim sup |x|→+∞ v(x) -u(x) 0, (4.5)
and that v u a.e. in H c \ K. (4.6)

Then, v u a.e. in R N \ K.

Proof. -We let w := v -u. We want to prove that w 0 a.e. in R N \ K. From (4.6), we only have to show that w 0 in H (remember that from (4.2) both functions u and v are assumed to be continuous in H). We argue by contradiction and we suppose that sup H w > 0. Then, thanks to (4.6), one has sup R N \K w = sup H w > 0 and there exists a sequence (x j ) j∈N in H such that lim j→+∞ w(x j ) = sup

R N \K w > 0.
It follows then from (4.5) that the sequence (x j ) j∈N is bounded. Thus, up to extraction of a subsequence, there exists a point x ∈ H such that x j → x as j → +∞, hence w(x) = lim j→+∞ w(x j ) > 0 by (4.2). As a consequence, (4.3) yields

Lw(x) = Lv(x) -Lu(x) -f (v(x)) + f (u(x)) = -w(x) ˆ1 0 f (tv(x) + (1 -t)u(x))dt. (4.7)
Moreover, combining (4.4) and w(x) > 0, we obtain that v(x) = w(x) + u(x) > u(x) 1 -c 0 , and so tv(x) + (1 -t)u(x) 1 -c 0 for all t ∈ [0, 1]. From this and (4.1), we conclude that f tv(x) + (1 -t)u(x) -c 1 < 0 for all t ∈ [0, 1]. This inequality, together with w(x) > 0, yields

-w(x) ˆ1 0 f (tv(x) + (1 -t)u(x))dt > 0.
By inserting this information into (4.7), we get Lw(x) > 0. That is, recalling (1.2) and the nonnegativity of J,

0 < Lw(x) = ˆRN \K J(x -y)(w(y) -w(x))dy = ˆRN \K J(x -y) w(y) -sup R N \K w dy 0.
This is a contradiction, and so the desired result is established.

The next lemma is concerned with a strong maximum principle.

Lemma 4.2 (Strong maximum principle). -Assume that J satisfies (1.9), with 0

r 1 < r 2 . Let H ⊂ R N be an open affine half-space such that K ⊂ H c . Let u, v ∈ L ∞ (R N \ K) satisfy (4.2) and (4.3). Assume also that v u a.e. in R N \ K, (4.8)
and that there exists x ∈ H such that v(x) = u(x). Then, v = u a.e. in (H + B r 2 ) \ K.

Remark 4.4. -The previous arguments immediately show that, when r 1 = 0 in (1.9), the sweeping principle holds in any compact connected set F . Namely, if J satisfies (1.9) with r 1 = 0, if u ∈ C(F ) satisfies (4.9) with F instead of A (s 1 , s 4 ) and the strict inequality somewhere in

F , if (w τ ) τ ∈[a,b] is a continuous family in C(F ) satisfying (4.11) with F instead of A (s 1 , s 4 ) and if w τ 0 u in F for some τ 0 ∈ [a, b], then w τ u in F for every τ ∈ [a, b].

Construction of radially symmetric lower bounds

In this section, we derive a first lower bound on continuous non-negative supersolutions u of (2.3) that we constantly use along this chapter. Throughout this section, K is any compact subset of R N , f is any C 1 (R) function, and J is any L 1 (R N ) non-negative and radially symmetric kernel with unit mass. We recall that J 1 is the non-negative even L 1 (R) kernel with unit mass defined for a.e. y 1 ∈ R by

J 1 (y 1 ) := ˆRN-1 J(y 1 , y 2 , • • • , y N ) dy 2 • • • dy N ,
and that assumption (1.10) means the existence of a continuous increasing function φ : R → (0, 1) such that

   ˆR J 1 (τ -σ)(φ(σ) -φ(τ ))dσ + f (φ(τ )) 0 for all τ ∈ R, φ(-∞) = 0, φ(+∞) = 1.
(5.1)

Then, for such φ, we establish the following lemma:

Lemma 5.1. -Assume that f and J satisfy (4.1) and (1.10), let γ ∈ (0, 1] and let u ∈ C R N \ K, [γ, 1] be a function satisfying (2.3). Then, there exists r 0 > 0 such that φ(|x| -r 0 ) u(x) for all x ∈ R N \ K.

Proof. -Since u(x) → 1 as |x| → +∞, there exists R 0 > 0 so large that K ⊂ B R 0 and u 1 -c 0 in R N \ B R 0 , where c 0 > 0 is given in (4.1). By (5.1), there exists A > 0 such that φ γ in (-∞, -A]. Define

r 0 = R 0 + A > 0,
and let us check that the conclusion of Lemma 5.1 holds with this real number r 0 .

Let e be any unit vector of R N , that is, e ∈ ∂B 1 = S N -1 . For r ∈ R, let φ r,e be the function defined by φ r,e (x) := φ(e • x -r) for x ∈ R N , where e • x stands for the standard inner product in R N . Let (e 1 , • • • , e N ) be the canonical basis of R N and let R be a rotation such that e = Re 1 . Set now e i := Re i for all i ∈ {2, • • • , N } and, for x, y ∈ R N and r ∈ R, let us define x * = x -re, y * = y -re and

X = (X 1 , • • • , X N ) = (x * • e, x * • e 2 , • • • , x * • e N ) = R -1 y * , Y = (Y 1 , • • • , Y N ) = (y * • e, y * • e 2 , • • • , y * • e N ) = R -1 y * .
Since J is rotationally invariant, we deduce from (5.1) that, for all x ∈ R N and r ∈ R, Accordingly, by (5.2) and the definition of H e , for any r r 0 and x ∈ H e , Lφ r,e (x) = L R N φ r,e (x) -ˆK J(x -y)(φ r,e (y) -φ r,e (x))dy -f (φ r,e (x)).

L R N φ r,e (x) := ˆRN J(x -y)(φ r,e (y) -φ r,e (x))dy = ˆR J 1 (X 1 -Y 1 )(φ(Y 1 ) -φ(X 1 ))dY 1 -f (φ(X 1 )) = -f (φ(x • e -r)) = -f (φ r,e (x)). (5.2) Set H e := {x ∈ R N ; x • e > R 0 } (notice that H e ∩ K = ∅). We remark that, if r r 0 and x ∈ H c e \ K, then φ r,e (x) = φ(x • e -r) φ(R 0 -r 0 ) = φ(-A) γ u(x).
(5.3)

Consequently, we can exploit the weak comparison principle of Lemma 4.1 (used here with H = H e ⊂ R N \ K and v = φ r 0 ,e ) and deduce that

φ(x • e -r 0 ) = φ r 0 ,e (x) u(x),
for every x ∈ R N \ K and also for every x ∈ R N \ K by continuity. This inequality holds for every e ∈ ∂B 1 , while r 0 > 0 does not depend on e. In particular, taking into account the possible choice of e = x/|x| if x = 0 (and any e ∈ ∂B 1 if x = 0), we conclude that φ(|x| -r 0 ) u(x) for all x ∈ R N \ K.

This proves Lemma 5.1.

Remark 5.2. -If R N \ K is connected, if f (0)
0 and if J satisfies (1.9) with r 1 = 0 (for instance, if J is continuous at the origin with J(0) > 0), then Lemma 5.1 holds with γ = 0. Indeed, these additional assumptions imply that inf R N \K u > 0. If not, then by continuity of u and the limiting conditions in (2.3), there exists x 0 ∈ R N \ K such that u(x 0 ) = 0. Thus, by (2.3) and f (0) 0, 0 Lu(x 0 ) = ˆRN \K J(x 0 -y)(u(y) -u(x 0 ))dy, and u(y) = u(x 0 ) = 0 for all y ∈ B r 2 (x 0 ) ∩ R N \ K. Therefore, u(y) = 0 for all y ∈ R N \ K by repeating this argument and by connectedness of R N \ K. This contradicts the limit u(y) → 1 as |y| → +∞. Finally, inf R N \K u > 0 and the conclusion of Lemma 5.1 holds.

Construction of solutions in large balls

We recall that B R (x) denotes the open Euclidean ball of R N centered at x ∈ R N and of radius R > 0, and that B R = B R (0). Throughout this section we suppose that f and J satisfy (1.8), (1.9) and (1.11). Here, for any R > 0 large enough and any x 0 ∈ R N , we will construct and study the properties of positive continuous solutions of the following auxiliary problem

L B R (x 0 ) [v](x) -v(x) + f (v(x)) = 0 for x ∈ B R (x 0 ), (6.1)
where

L B R (x 0 ) [v](x) := ˆBR (x 0 ) J(x -y)v(y)dy. (6.2)
Besides the own interest of (6.1), the properties of some particular solutions v of (6.1) are essential in the proof of Theorem 2.4, as they will provide key estimates ensuring to derive the asymptotic behaviour of the solutions u of (2.5) or (2.6). So in Sections 6.1 and 6.2, our main concern will be to establish, for any x 0 ∈ R N and R > 0 large enough, the existence of a positive maximal solution v x 0 ,R to (6.1), such that v x 0 ,R → 1 locally uniformly in R N as R → +∞. Based on the construction of these solutions in closed balls B R (x 0 ), we will next show in Section 6.3 the existence of continuous and compactly supported sub-solutions in R N .

Existence of a positive solution in

B R (x 0 )
This section is devoted to the proof of the existence of a positive continuous solution of (6.1) in B R (x 0 ), for any R > 0 large enough and any x 0 ∈ R N .

Lemma 6.1. -Assume that f and J satisfy (1.8), (1.9) and (1.11). Assume further that J is compactly supported and J ∈ L 2 (R N ). Then there exists d 0 = d 0 (f, J) > 0 such that for every x 0 ∈ R N and R d 0 , problem (6.1) admits a positive continuous solution v : B R (x 0 ) → (0, 1) such that max B R (x 0 ) v > θ, where θ ∈ (0, 1) is defined in (1.11).

Proof. -Let x 0 ∈ R N be fixed, let also R J > 0 be fixed (independently of x 0 ) such that supp(J) ⊂ B R J , and pick any R > R J . To construct a solution, we adapt the strategy used in [START_REF] Clément | Existence and multiplicity results for a semilinear elliptic eigenvalue problem[END_REF] for the construction of a solution of a local reaction-diffusion equation. The proof is divided into three main steps.

Step 1: definition and elementary properties of an energy functional E

In the proof of Lemma 6.1, let us extend f by f (1)(s -1) for s 1 and by -f (-s) for s 0 and denote f this extension. Now, define

F (t) := ˆt 0 f (s)ds for t ∈ R, c(x) := 1 - ˆBR (x 0 ) J(x -y)dy ∈ [0, 1] for x ∈ R N ,
and consider the following energy functional

E (u) := 1 4 ˆBR (x 0 ) ˆBR (x 0 ) J(x -y)(u(y) -u(x)) 2 dxdy + 1 2 ˆBR (x 0 ) c(x)u 2 (x)dx - ˆBR (x 0 ) F (u(x))dx, (6.3) defined for u ∈ L 2 (B R (x 0 )). Since J ∈ L 1 (R N ), E is well defined in L 2 (B R (x 0 )).
Moreover, using the definition of F and the oddness of f , we have

ˆBR (x 0 ) F (u(x))dx = ˆBR (x 0 ) F (|u(x)|)dx, (6.4)
for any u ∈ L 2 (B R (x 0 )), while elementary computations yield

E (u) = - 1 2 ˆBR (x 0 ) ˆBR (x 0 ) J(x -y)u(x)u(y)dxdy + 1 2 ˆBR (x 0 ) u 2 (x)dx - ˆBR (x 0 ) F (u(x))dx. (6.5)
From the last two formulas, one infers that, for any u ∈ L 2 (B R (x 0 )),

E (|u|) = - 1 2 ˆBR (x 0 ) ˆBR (x 0 ) J(x -y)|u(y)||u(x)|dxdy + 1 2 ˆBR (x 0 ) |u(x)| 2 dx - ˆBR (x 0 ) F (|u(x)|)dx E (u).
To complete Step 1, let us check that the functional E is bounded from below in L 2 (B R (x 0 )). From (1.11) and (6.4), the definition of F and f , and since f (s) 0 for s 1, we see that, for any u ∈ L 2 (B R (x 0 )),

ˆBR (x 0 ) F (u(x))dx ˆBR (x 0 ) ˆmin{1,|u(x)|} 0 f (s)ds R N |B 1 | ˆ1 0 f (s)ds,
where |B 1 | denotes the Lebesgue measure of the unit ball. Setting

C 0 := |B 1 | ˆ1 0 f (s)ds > 0,
we thus get that

E (u) -C 0 R N for any u ∈ L 2 (B R (x 0 )). (6.6)
Hence, the quantity γ := inf

u∈L 2 (B R (x 0 )) E (u), (6.7)
is finite.

Step 2: the infimum of E in L 2 (B R (x 0 )) is achieved

We shall now see that γ is achieved for some v ∈ L 2 (B R (x 0 )). So, let (u n ) n∈N be a minimising sequence. From the inequality E (|u|) E (u), we may assume without loss of generality that the functions u n are all non-negative.

Let us first check that the sequence (u n ) n∈N is bounded in L 2 (B R (x 0 )). To do so, we recall the definition (6.2) of L B R (x 0 ) and we notice that the principal eigenvalue λ p of the operator L B R (x 0 ) -Id is negative (see for example [START_REF] Andreu-Vaillo | Nonlocal diffusion problems[END_REF][START_REF] Berestycki | On the definition and the properties of the principal eigenvalue of some nonlocal operators[END_REF][START_REF]On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators[END_REF][START_REF] García-Melián | On the principal eigenvalue of some nonlocal diffusions problem[END_REF] for a precise definition of λ p and some of its properties) and satisfies

-λ p = inf ϕ L 2 (B R (x 0 )) =1 E (ϕ),
where

E (ϕ) := 1 2 ˆBR (x 0 ) ˆBR (x 0 ) J(x -y)(ϕ(y) -ϕ(x)) 2 dxdy + ˆBR (x 0 ) c(x)ϕ 2 (x)dx.
As a consequence, from (6.6), we get

E (u n ) - λ p 2 ˆBR (x 0 ) u 2 n (x)dx -C 0 R N ,
for all n ∈ N. Therefore the sequence (u n ) n∈N is bounded in L 2 (B R (x 0 )) since it is a minimising sequence and since λ p < 0. Up to extraction of a subsequence, the sequence (u n ) n∈N converges weakly in L 2 (B R (x 0 )) to a non-negative function v ∈ L 2 (B R (x 0 )). We actually claim that

E (v) = γ. (6.8)
Due to the lack of compactness in this non-local minimisation problem, we cannot expect to get a strong convergence in L 2 (B R (x 0 )) for the minimising subsequence and therefore passing to the limit in the energy (6.3) is not immediate. To overcome this difficulty, let us observe that by introducing the function

G(t) := ˆt 0 (s -f (s))ds = t 2 2 -F (t),
we get from (6.5) that, for any n ∈ N,

E (u n ) = - 1 2 ˆBR (x 0 ) ˆBR (x 0 ) J(x -y)u n (x)u n (y)dxdy + ˆBR (x 0 ) G(u n (x))dx,
and therefore

E (u n ) -E (v) = - 1 2 ˆBR (x 0 ) ˆBR (x 0 ) J(x -y)[u n (x)u n (y) -v(x)v(y)]dxdy + ˆBR (x 0 ) [G(u n (x)) -G(v(x))]dx.
(6.9)

Observe that the double integral in the right-hand side of (6.9) can be rewritten as

ˆBR (x 0 ) ˆBR (x 0 ) J(x -y)u n (x)u n (y)dxdy = ˆBR (x 0 ) u n (x) ˆBR (x 0 ) J(x -y)[u n (y) -v(y)]dy dx + ˆBR (x 0 ) v(y) ˆBR (x 0 ) J(x -y)u n (x)dx dy.
Using Lebesgue's dominated convergence theorem, together with the assumption J ∈ L 2 (R N ) and the L 2 (B R (x 0 )) weak convergence of the sequence (u n ) n∈N , it is easy to see that lim n→+∞ ˆBR (x 0 ) ˆBR (x 0 ) J(x -y)u n (x)u n (y)dxdy = ˆBR (x 0 ) ˆBR (x 0 ) J(x -y)v(x)v(y)dxdy, and therefore

lim n→+∞ - 1 2 ˆBR (x 0 ) ˆBR (x 0 ) J(x -y)[u n (x)u n (y) -v(x)v(y)]dxdy = 0. (6.10)
On the other hand, since f (s) < 1 for all s ∈ R (by assumption), the function G is convex and, for all n ∈ N, we get

ˆBR (x 0 ) [G(u n (x)) -G(v)(x)]dx ˆBR (x 0 ) G (v(x))[u n (x) -v(x)]dx.
From the definition of G and f , together with the fact that v

∈ L 2 (B R (x 0 )), we infer that G (v) ∈ L 2 (B R (x 0 )). Using the L 2 (B R (x 0 )) weak convergence of (u n ) n∈N to v, it follows that lim inf n→+∞ ˆBR (x 0 ) [G(u n (x)) -G(v(x))]dx 0. (6.11)
Thus passing to the limit in (6.9), and using (6.10) and (6.11), we obtain γ-E (v) 0. Together with the definition (6.7) of γ, this shows that v is a minimiser of the energy E , that is, (6.8) holds.

Step 3: v is a continuous positive solution u of (6.1)

We first show in this step that v is a solution to (6.1) with f instead of f . From (6.8), v is a critical point of E and in particular, it follows from the formulation (6.5) of E that v is a non-negative weak solution of

L B R (x 0 ) [v] -v + f (v) = 0 in B R (x 0 ). Since all functions L B R (x 0 ) [v], v and f (v) belong to L 2 (B R (x 0 )), the function v satisfies L B R (x 0 ) [v](x) -v(x) + f (v(x)) = 0 for a.e. x ∈ B R (x 0 ). Further- more, since J ∈ L 2 (R N ), the Cauchy-Schwarz inequality implies that L B R (x 0 ) [v] ∈ L ∞ (B R (x 0 )). Therefore, since by (1.11) the function s → s -f (s) is a C 1 diffeomor- phism from R + onto R + and since v is non-negative, it follows from the equation L B R (x 0 ) [v] -v + f (v) = 0 a.e. in B R (x 0 ) that v ∈ L ∞ (B R (x 0 )).
Thus by reproducing the arguments of the proof of Lemma 3.1 we deduce that v has a representative, still denoted by v, which is continuous in B R (x 0 ) and satisfies

L B R (x 0 ) [v](x) -v(x) + f (v(x)) = 0 for all x ∈ B R (x 0 ).
(6.12)

Remember now that, from (1.9), J > 0 a.e. in A (r 1 , r 2 ) with 0 r 1 < r 2 , and that R R J r 2 > r 1 , with supp(J) ⊂ B R J . As a consequence, if there exists a point x ∈ B R (x 0 ) such that v(x) = 0, then, arguing as in the proof of the strong maximum principle (Lemma 4.2) or as in the proof of the sweeping principle (Lemma

4.3), it follows that v = 0 in A (x, r 1 , r 2 ) ∩ B R (x 0 ), hence v = 0 in A (y, r 1 , r 2 ) ∩ B R (x 0 ) for all y ∈ A (x, r 1 , r 2 ) ∩ B R (x 0 ) and finally v = 0 in B r (x) ∩ B R (x 0 ) for some r > 0.
Therefore, the non-empty set x ∈ B R (x 0 ); v(x) = 0 is both (obviously) closed and open relatively to B R (x 0 ) and is thus equal to B R (x 0 ). As a consequence, either

v ≡ 0 in B R (x 0 ) or v > 0 in B R (x 0 ).
In this paragraph, we prove that the solution v constructed is a solution of (6.1), namely we just need to show that v 1 in B R (x 0 ). To do so, define M = max B R (x 0 ) v 0 and let x ∈ B R (x 0 ) be such that v(x) = M . Assume by contradiction that M > 1. By evaluating (6.12) at x and using the definition of f , we get that

ˆBR (x 0 ) J(x -y)v(y)dy = L B R (x 0 ) [v](x) = M -f (M ) > M.
Since v M in B R (x 0 ), this leads to a contradiction. Hence M 1 and thus v is a non-negative continuous solution of (6.1) in B R (x 0 ). Furthermore, as for the positivity of v, one gets that either

v ≡ 1 in B R (x 0 ) or v < 1 in B R (x 0 ). The former case is impossible since L B R (x 0 ) [v] ≡ 1 in B R (x 0 ) (indeed, ´BR (x 0 ) J(x -y)dy < 1 for all x ∈ ∂B R (x 0 )). Thus, 0 v < 1 in B R (x 0 ).
Finally, let us verify that the solution v constructed is not the trivial solution. To do so, it is enough to show that E (v) = E (0) = 0. We claim that, for R > R J large enough, E (v) < 0. Indeed, let us consider the test function

ϕ := 1 B R (x 0 ) ∈ L 2 (B R (x 0 )).
We have

E (ϕ) = 1 4 ˆBR (x 0 ) ˆBR (x 0 ) J(x -y) ϕ(y) -ϕ(x) 2 dxdy + 1 2 ˆBR (x 0 ) c(x)ϕ 2 (x)dx - ˆBR (x 0 ) F (ϕ(x))dx = 1 2 ˆBR (x 0 ) c(x)dx -R N |B 1 | ˆ1 0 f (s)ds = 1 2 ˆBR (x 0 ) ˆRN \B R (x 0 ) J(x -y)dxdy -R N |B 1 | ˆ1 0 f (s)ds.
Since supp(J) ⊂ B R J , the above equality yields

E (ϕ) = 1 2 ˆBR (x 0 )\B R-R J (x 0 ) ˆRN \B R (x 0 ) J(x -y)dxdy -R N |B 1 | ˆ1 0 f (s)ds, 1 2 |B 1 | R N -(R -R J ) N -R N |B 1 | ˆ1 0 f (s)ds.
Thus, since ´1 0 f (s)ds > 0, there exists d 0 = d 0 (J, f ) ∈ (R J , +∞), independent of x 0 , such that, for every R d 0 , the right-hand side of the above inequality is negative First, let us observe that, from the assumptions on J, the linear operator L B R (x 0 ) is a continuous operator in C(B R (x 0 )). Next let us choose a real number k > 0 large enough such that the function s → -ks -f (s) is decreasing in R. We can increase further k if necessary to ensure that k + 1 ∈ ρ(L B R (x 0 ) ), where ρ(L B R (x 0 ) ) denotes the resolvent of the operator L B R (x 0 ) . We note that, by this choice of k, the operator [START_REF]On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators[END_REF][START_REF] Coville | On eigenvalue problems arising from nonlocal diffusion models[END_REF]). Now, set v 0 = 1 and let v 1 ∈ C(B R (x 0 )) be the solution of the following linear problem

L B R (x 0 ) -(k + 1) satisfies a comparison principle, in the sense that if w ∈ C(B R (x 0 )) satisfies L B R (x 0 ) [w] -(k + 1)w 0 in B R (x 0 ) then w 0 in B R (x 0 ) (see
L B R (x 0 ) [v 1 ](x) -(k + 1)v 1 (x) = -kv 0 (x) -f (v 0 (x)) for x ∈ B R (x 0 ). (6.13)
The function v 1 is well defined, since by construction the continuous operator

L B R (x 0 ) - (k + 1) is invertible. We claim that v v 1 v 0 in B R (x 0 )
. Indeed, since v ( 1) and v 0 = 1 are respectively a solution and a super-solution of (6.1), we have, for

x ∈ B R (x 0 ), L B R (x 0 ) [v 1 -v 0 ](x) -(k + 1)(v 1 (x) -v 0 (x)) = -L B R (x 0 ) [1](x) + 1, L B R (x 0 ) [v 1 -v](x) -(k + 1)(v 1 (x) -v(x)) = -kv 0 (x) -f (v 0 (x)) + kv(x) + f (v(x)).
Thus implying that

L B R (x 0 ) [v 1 -v 0 ](x) -(k + 1)(v 1 (x) -v 0 (x)) 0, L B R (x 0 ) [v 1 -v](x) -(k + 1)(v 1 (x) -v(x)) 0.
So, the inequality v v 1 v 0 in B R (x 0 ) follows from the comparison principle satisfied by the operator L B R (x 0 ) -(k + 1). In particular, 0 < v 1 1 in B R (x 0 ). Now let v 2 ∈ C(B R (x 0 )) be the solution of (6.13) with v 2 instead of v 1 in the lefthand side and v 1 instead of v 0 in the right-hand side. From the monotonicity of s → -ks -f (s) and from the comparison principle, we have v

v 2 v 1 v 0 in B R (x 0 )
. By induction, we can construct a non-increasing sequence of functions

(v n ) n∈N in C(B R (x 0 )) satisfying v v n+1 v n v 0 in B R (x 0 ) and L B R (x 0 ) [v n+1 ](x) -(k + 1)v n+1 (x) = -kv n (x) -f (v n (x)) for x ∈ B R (x 0 ). (6.14)
Since the sequence is non-increasing and bounded from below, the quantity

v x 0 ,R (x) := inf n∈N v n (x) = lim n→+∞ v n (x) ∈ [v(x), 1] (⊂ (0, 1])
is well defined for every x ∈ B R (x 0 ). Moreover, by passing to the limit in the equation (6.14) and using Lebesgue's dominated convergence theorem, it follows that v x 0 ,R is a solution of (6.1). As in the proof of Lemma 3.1, we infer that v x 0 ,R is continuous in B R (x 0 ) and, as in the proof of Lemma 6.1, we get that v x 0 ,R < 1 in B R (x 0 ). To sum up, v x 0 ,R is a solution of (6.1) belonging to C(B R (x 0 ), (0, 1)).

We finally claim that v x 0 ,R is a maximal solution to (6.1). Indeed, let w ∈ C(B R (x 0 ), [0, 1]) be any solution to (6.1). By replacing v with w in the arguments of the previous paragraph and using the fact that the sequence (v n ) n∈N is defined with the same initial value v 0 = 1, we get that w v n in B R (x 0 ) for every n ∈ N, hence w v x 0 ,R in B R (x 0 ). The proof of Lemma 6.3 is thereby complete.

The maximal solutions v x 0 ,R possess some important properties, in particular they are monotone non-decreasing with respect to the domains. Lemma 6.4. -Let us assume that f and J satisfy (1.8), (1.9) and (1.11). Assume further that J is compactly supported and J ∈ L 2 (R N ). Let d 0 = d 0 (f, J) > 0 be given as in Lemmata 6.1 and 6.3. The following properties hold:

(i) for every

x 1 , x 2 ∈ R N and d 0 R 1 R 2 such that B R 1 (x 1 ) ⊂ B R 2 (x 2 ), then v x 1 ,R 1 (x) v x 2 ,R 2 (x) for all x ∈ B R 1 (x 1 ); (ii) for every x 0 ∈ R N and R d 0 , the function v 0,R (• -x 0 ) defined in B R (x 0 ) satisfies v 0,R (• -x 0 ) = v x 0 ,R in B R (x 0 ); (iii) for every x 0 ∈ R N and R d 0 , min B R (x 0 ) v x 0 ,4R max B R (x 0 ) v x 0 ,2R .
Proof. -The proof of (i) is straightforward. Indeed, let

x 1 , x 2 ∈ R N and d 0 R 1 R 2 be such that B R 1 (x 1 ) ⊂ B R 2 (x 2 ). Recall from the proof of Lemma 6.3 that v x 2 ,R 2 ∈ C(B R 2 (x 2 ), (0, 1)) can be defined as v x 2 ,R 2 = lim n→+∞ v n , where (v n ) n∈N is the sequence of positive functions in C(B R 2 (x 2 ), (0, 1]) defined by induction by v 0 = 1 in B R 2 (x 2 ) and, for n ∈ N, L B R 2 (x 2 ) [v n+1 ](x) -(k + 1)v n+1 (x) = -kv n (x) -f (v n (x)) for x ∈ B R 2 (x 2 ).
Here

k > 0 is such that k + 1 ∈ ρ(L B R 2 (x 2 )
) and the function s → -ks -f (s) is decreasing. By increasing k if necessary we may assume that k

+ 1 ∈ ρ(L B R 2 (x 2 ) ) ∩ ρ(L B R 1 (x 1 )
). Now observe that, for any n ∈ N, v n satisfies (6.15) that is, the function v n+1 is a super-solution to problem (6.14) in B R 1 (x 1 ). We claim that, for every n ∈ N,

L B R 1 (x 1 ) [v n+1 ](x) -(k + 1)v n+1 (x) -kv n (x) -f (v n (x)) for x ∈ B R 1 (x 1 ),
v x 1 ,R 1 (x) v n (x) for all x ∈ B R 1 (x 1
). and 6.3 (we recall that d 0 > R J , where supp(J) ⊂ B R J ). Thanks to part (i) of Lemma 6.4, the sequence (v x 0 ,Rn ) n∈N is non-decreasing, in the sense that v x 0 ,Rn v x 0 ,Rp in B Rn (x 0 ) for all n p. Moreover, 0 < v x 0 ,Rn < 1 in B Rn (x 0 ) for each n ∈ N. As a consequence, the sequence (v x 0 ,Rn ) n∈N converges pointwise in R N to a function 0 < v 1 which, thanks to Lebesgue's dominated convergence theorem, satisfies

J * v(x) -v(x) + f (v(x)) = 0 for all x ∈ R N . (6.16)
As in the proof of Lemma 3.1, the function v can be viewed as a uniformly continuous function and therefore the limit v x 0 ,Rn → v holds locally uniformly in R N . Consider now any x 1 ∈ R N and any δ ∈ [d 0 , +∞). We can then extract a subsequence of (R n ) n∈N that we still denote (R n ) n∈N such that, for all n ∈ N, B δ (x 1 ) ⊂ B Rn (x 0 ) and R n+1 4R n . By Lemma 6.1 and parts (i) and (iii) of Lemma 6.4, we get that

1 > min B δ (x 1 ) v x 0 ,R n+1 min B Rn (x 0 ) v x 0 ,R n+1 min B Rn (x 0 ) v x 0 ,4Rn max B Rn (x 0 ) v x 0 ,2Rn • • • • • • max B Rn (x 0 ) v x 0 ,Rn max B δ (x 1 ) v x 0 ,Rn max B δ (x 1 ) v x 1 ,δ > θ. (6.17)
Taking the limit as n → +∞ in the inequality min

B δ (x 1 ) v x 0 ,R n+1 max B δ (x 1 ) v x 0 ,Rn ,
we obtain that min

B δ (x 1 )
v max

B δ (x 1 ) v.
Hence, v is equal to a constant C x 1 ,δ in B δ (x 1 ) and, thanks to (6.17), there holds

θ < C x 1 ,δ 1. Furthermore, since x 1 ∈ R N is arbitrary, it follows that v is equal to a constant C ∈ (θ, 1] in R N .
Lastly, (6.16) yields f (C) = 0. Since f satisfies (1.11) and θ < C 1, we infer that C = 1. Therefore, v = 1 in R N and thus the sequence (v x 0 ,Rn ) n∈N converges to 1 locally uniformly in R N as n → +∞. Since the non-decreasing sequence (R n ) n∈N converging to +∞ is arbitrary, and so is δ ∈ [d 0 , +∞), it follows that v x 0 ,R converges to 1 locally uniformly in R N as R → +∞. The proof of Lemma 6.5 is thereby complete.

Compactly supported continuous sub-solutions in R N

In this section, we construct compactly supported continuous sub-solutions from R N to [0, 1] of problems of the type (6.1). Such continuous sub-solutions will then serve as a building block of some lower bounds in the proof of Theorem 2.4.

Let us first introduce some useful notations. For x 0 ∈ R N , R > 0 and x ∈ R N , let P x 0 ,R (x) be the projection of x on the closed convex set B R (x 0 ), that is,

P x 0 ,R (x) ∈ B R (x 0 ) and |x -P x 0 ,R (x)| = dist(x, B R (x 0 )) = min y∈B R (x 0 )
|x -y|.

Lemma 6.6. -Assume that f and J satisfy (1.8), (1.9) and (1.11). Assume further that J is compactly supported and J ∈ L 2 (R N ). Let d 0 = d 0 (f, J) > 0 be given as in Lemmata 6.1 and 6.3 and, for any x 0 ∈ R N and R d 0 , let v x 0 ,R ∈ C(B R (x 0 ), (0, 1)) be the maximal solution of (6.1). Then there exists δ 0 > 0 such that, for any x 0 ∈ R N , R d 0 and δ ∈ (0, δ 0 ], the continuous function w x 0 ,R,δ : R N → [0, 1) defined by

w x 0 ,R,δ (x) = max v x 0 ,R (P x 0 ,R (x)) -δ -1 |x -P x 0 ,R (x)|, 0 , (6.18)
satisfies ˆBR (x 0 ) J(x -y)w x 0 ,R,δ (y)dy

=L B R (x 0 ) [w x 0 ,R,δ ](x) -w x 0 ,R,δ (x) + f (w x 0 ,R,δ (x)) 0, (6.19)
for all x ∈ R N and all R R + δ.

Proof. -In view of (6.18), we see that, for every x 0 ∈ R N , R d 0 and δ > 0, the function

w x 0 ,R,δ is continuous R N , that 0 w x 0 ,R,δ < 1 in R N , that w x 0 ,R,δ = v x 0 ,R in B R (x 0 ) and that w x 0 ,R,δ = 0 in R N \ B R+δ (x 0 ).
We set g(s) := s -f (s) for s ∈ [0, 1]. From (1.11), we see that

γ := min [0,1] g > 0. (6.20)
We recall that, by (1.11), J is assumed to belong to W 1,1 (R N ), and set

δ 0 := γ • ˆRN |∇J(z)| dz -1 > 0. ( 6.21)
Let us now fix any x 0 ∈ R N , R d 0 , δ ∈ (0, δ 0 ] and let us check that (6.19) holds for any R R + δ. Since both w x 0 ,R,δ and J are non-negative and since w x 0 ,R,δ = 0 in R N \B R+δ (x 0 ), recalling also that f (0) = 0 due to (1.11), we see that it is sufficient to show (6.19) for x ∈ B R+δ (x 0 ). Furthermore, by monotonicity of the integral with respect to R , it is enough to show (6.19) for R = R + δ.

For any x ∈ B R+δ (x 0 ), there holds ˆBR+δ (x 0 ) J(x -y)w x 0 ,R,δ (y)dy = ˆBR+δ (x 0 )\B R (x 0 ) J(x -y)w x 0 ,R,δ (y)dy

Proof. -We proceed by contradiction. Suppose that the conclusion does not hold. Then, by continuity of u and (7.2), there exists a point x 0 ∈ R N \ K, such that u(x 0 ) = 0. Arguing as in the proof of the strong maximum principle (Lemma 4.2) or in the proof of the sweeping principle (Lemma 4.3), we get that u = 0 in A (x 0 , r 1 , r 2 ) ∩ R N \ K, where 0 r 1 < r 2 are given in (1.9), and then u = 0 in

A (x 1 , r 1 , r 2 ) ∩ R N \ K for all x 1 ∈ A (x 0 , r 1 , r 2 ) ∩ R N \ K. Since K is convex, it follows in particular that u = 0 in B r (x 0 ) ∩ R N \ K for some r > 0. Finally, the non- empty set x ∈ R N \ K; u(x) = 0 is both (obviously) closed and open relatively to the connected set R N \ K. Hence u = 0 in R N \ K, contradicting (7.2).
We now turn to the proof of Theorem 2.2.

Proof of Theorem 2.2. -Let K, f , J and u be as in Theorem 2.2. Firstly, without loss of generality, one can assume by (1.8) that f is extended to a C 1 (R) function satisfying (4.1). Secondly, by (2.3) 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 (4.1).

We proceed the proof by contradiction, and suppose that inf

R N \K u < 1. (7.3)
From (2.3) and (7.3), 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 7.1, one has u(x 0 ) > 0. Now, 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 .
In light of assumption (1.10), there exists an increasing function φ ∈ C(R) such that

J 1 * φ -φ + f (φ) 0 in R, φ(-∞) = 0, φ(+∞) = 1.
Let us also define the function

ϕ r (x) := φ r,e (x) = φ(x • e -r), x ∈ R N ,
and the following quantity

r * := inf r ∈ R ; ϕ r u in R N \ K .
From Lemmata 5.1 and 7.1, we know that r * ∈ [-∞, r 0 ], where r 0 > 0 is given in Lemma 5.1. We claim that in fact r * = -∞.

The proof of (7.4) 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. (7.5)
Two cases may occur.

Case 1: inf H c \K (u -ϕ r * ) > 0. In this situation, thanks to the uniform continuity of φ, there exists ε > 0 such that inf

H c \K (u -ϕ r * -ε ) > 0.
Now, we observe that u and ϕ r * -ε satisfy

     Lu + f (u) 0 in H, Lϕ r * -ε + f (ϕ r * -ε ) 0 in H (by (5.3)), u ϕ r * -ε in H c \ K, together with u 1 -c 0 in R N \B R 0 ⊃ H and lim |x|→+∞ u(x) = 1, while ϕ r * -ε 1 in R N .
Thus, by the weak maximum principle (Lemma 4.1) and the continuity of u and ϕ r * -ε in R N \ K, we get that u ϕ r * -ε in R N \ K. This contradicts the minimality of r * and therefore Case 1 is ruled out. Case 2: inf H c \K (u -ϕ r * ) = 0. In this situation, by (7.2) and (7.5), 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 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 . Since K ⊂ H c
e and ϕ r * and u satisfy respectively

Lu + f (u) 0 in H e , Lϕ r * + f (ϕ r * ) 0 in H e (by (5.3)),
it follows in particular from the strong maximum principle (Lemma 4.2) that ϕ r * = u in H e . Thus, for any e ⊥ ∈ ∂B 1 such that e ⊥ • e = 0, one infers from (7.2) and the definition of ϕ r * that

1 = lim t→+∞ u(x 0 + te ⊥ ) = lim t→+∞ ϕ r * (x 0 + te ⊥ ) = ϕ r * (x 0 ) < 1.
This contradiction rules out Case 2 too. Hence (7.4) holds true and as a consequence we have 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, (7.3) can not hold. In other words, inf

R N \K u = 1, i.e. u = 1 in R N \ K. The proof of Theorem 2.2 is thereby complete.
We observe that, by the same token, we obtain Theorem 2.3.

Proof of Theorem 2.3. -By Lemma 2.2, Remark 3.3 and our assumptions on f , we know that u has a (uniformly) continuous representative u * ∈ C(R N \ K) in its class of equivalence and we can identify u with u * . The desired result now follows as a consequence of Theorem 2.2.

Compactly supported kernels: proof of Theorem 2.4

In this subsection we prove Theorem 2.4. That is, provided some additional assumptions on f and J are satisfied, we show that the Liouville result obtained in Theorem 2.2 holds true when the uniform limit of u as |x| → +∞, namely condition (7.2), is replaced by the following weaker condition ess sup

R N \K u = 1, (7.6)
where u :

R N \ K → [0, 1] is a measurable solution of Lu + f (u) = 0 a.e. in R N \ K.
The condition (7.6) can be rewritten as sup

R N \K u = 1, (7.7)
if u is already assumed to be uniformly continuous in R N \ K. Note that the extra assumptions (2.4) made on f (namely f < 1/2 in [0, 1]) actually guarantees that u has a uniformly continuous representative in its class of equivalence, as follows from Lemma 2.2 and Remark 3.3. As a consequence, in the proof of Theorem 2.4 we can assume without loss of generality that u : R N \ K → [0, 1] is uniformly continuous and satisfies (7.7). Notice immediately that the same arguments as in the proof of Lemma 7.1 imply that u > 0 in R N \ K.

Otherwise u would be identically equal to 0 in R N \ K, contradicting the assumption (7.7).

The key-point in the proof of Theorem 2.4 is the following lemma.

Lemma 7.2. -Let K ⊂ R N be a compact set and assume that f and J satisfy (1.8), (1.9) and (1.11). Assume further that J is compactly supported and J ∈ L 2 (R N ). Let u : R N \ K → [0, 1] be a uniformly continuous solution of (2.5). Then, u(x) → 1 as |x| → +∞.

The proof of Lemma 7.2 is postponed in Section 7.3. In this section, we complete the proof of Theorem 2.4.

Proof of Theorem 2.4. -From the previous paragraphs, the function u can be assumed to be uniformly continuous in R N \ K without loss of generality. Then, since the condition (1.11), together with (1.8) and (1.9), implies the condition (1.10), the assumptions of Theorem 2.2 are all fulfilled, thanks to Lemma 7.2. Therefore u = 1 in R N \ K, completing the proof of Theorem 2.4.

Proof of Lemma 7.2

This section is devoted to the proof of Lemma 7.2. It is divided into four main steps. To prove Lemma 7.2, it suffices to show that for any ε > 0 small enough there exists

R(ε) > 0 such that u 1 -ε in R N \ B R(ε)
. To obtain such a lower bound, our strategy relies on the existence of continuous families of continuous sub-solutions w τ which satisfy w τ 1 -ε in B 1 (x τ ) for some x τ ∈ R N (these sub-solutions are drawn from Section 6.3). Then, we use the sweeping principle to propagate the lower bound satisfied by the w τ 's to a lower bound for u.

Step 1: the solution u is close to 1 in some large balls In this step, we show that, for any ε > 0, > 0, and R > 0, there exists a point

x * ∈ R N \ K such that |x * | > , B R (x * ) ⊂ R N \ K, and u 1 -ε in B R (x * ). (7.9)
To do so, notice first that, from (2.5) and the continuity of u in R N \ K, two situations may occur: namely, either there exists a sequence (

x n ) n∈N ⊂ R N \ K such that lim n→+∞ |x n | = +∞ and lim n→+∞ u(x n ) = 1, (7.10)
or there exists a point x ∈ R N \ K such that u(x) = 1. In the latter case, since f (u(x)) = f (1) = 0, we get, as in the proof of Lemma 7.1, that u = 1 in R N \ K: the claim (7.9) is therefore trivial in this case. Thus, it suffices to treat the former case (7.10) only. Consider the functions u n defined in

R N \ K -x n by u n (x) = u(x + x n ).
Since u is uniformly continuous in R N \ K and since K is compact and lim n→+∞ |x n | = +∞, it follows that, for every r > 0, the functions u n 's, ranging in [0, 1], are defined in B r for all n large enough and are uniformly equicontinuous in B r . From Arzela-Ascoli theorem and the diagonal extraction process, there exists a continuous function u ∞ : R N → [0, 1] such that, up to extraction of a subsequence, u n → u ∞ locally uniformly in R N as n → +∞. Furthermore, u ∞ (0) = 1 by (7.10). On the other hand, the functions u n 's satisfy

ˆ(R N \K)-xn J(x -y)u n (y)dy - ˆ(R N \K)-xn J(x -y)dy u n (x) + f (u n (x)) = 0, for all x ∈ R N \ K -x n . Lebesgue's dominated convergence theorem implies that J * u ∞ -u ∞ + f (u ∞ ) = 0 in R N .
Since f (u ∞ (0)) = f (1) = 0 and u ∞ 1 in R N , we get as in the proof of Lemma 7.1 that u ∞ = 1 in R N . In particular, for any fixed ε > 0, > 0, and R > 0, it follows that, for every n ∈ N large enough, there holds

|x n | > , B R (x n ) ⊂ R N \ K and u n 1 -ε in B R , that is, u 1 -ε in B R (x n ).
In other words, the claim (7.9) holds with x * = x n and n large enough.

Step 2: a sub-solution in a ball Fix ε > 0 small enough so that f < 0 in [1 -ε, 1], and let us now establish a lower bound for u in a ball far away from K, by using a sub-solution drawn from Section 6.3. We recall here that R J > 0 is such that supp(J) ⊂ B R J .

We first claim that there exist

x * ∈ R N , 0 < R J R K R and a function w ∈ C(R N , [0, 1)) such that B R+1 (x * ) ⊂ R N \B R K ⊂ R N \K, u 1 -ε in B R+1 (x * ), L B R+1 (x * ) [w]-w+f (w) 0 in R N , w 1-ε in B 1 (x * ), w = 0 in R N \B R+1 (x * ). (7.11) To show this claim, let R K max{1, R J } be such that K ⊂ B R K . Then choose R max{R K , d 0 } 1 (d 0 > 0 is
given as in Lemmata 6.1 and 6.3) such that the maximal solution v 0,R ∈ C(B R , (0, 1)) to problem (6.1) in B R satisfies

v 0,R 1 -ε in B 1 . (7.12)
Note that such a real number R exists according to Lemmata 6.3 and 6.5. On the one hand, as far as the first line in (7.11) is concerned, formula (7.9), applied here

with = R + 1 + R K > 0 and R + 1 > 0 in place of R, yields the existence of x * ∈ R N such that |x * | > R + 1 + R K , (7.13)
(hence, B R+1 (x * ) ⊂ R N \B R K ⊂ R N \K) and u 1 -ε in B R+1 (x * ). (7.14)
Thanks to (7.12) and part (ii) of Lemma 6.4, the maximal solution

v x * ,R ∈ C(B R (x * ), (0, 1)), to problem (6.1
) in B R (x * ) satisfies v x * ,R 1 -ε in B 1 (x * ).
On the other hand, as far as the second line in (7.11) is concerned, Lemma 6.6 provides the existence of a function w ∈ C(R N , [0, 1)) such that

L B R+1 (x * ) [w] -w + f (w) 0 in R N , w = v x * ,R in B R (x * ) ⊃ B 1 (x * ),
and w = 0 in R N \ B R+1 (x * ). As a consequence, x * , R K , R and w fulfill (7.11).

We then claim that w u in R N \ K.

Since w = 0 in R N \ B R+1 (x * ) and u 0 in R N \ K, we only need to show that

w u in B R+1 (x * ) (⊂ R N \ K). Denote z := w -u, in B R+1 (x * ) and assume that max B R+1 (x * ) z = z(x) > 0,
for some x ∈ B R+1 (x * ). Since B R+1 (x * ) ⊂ R N \ K and u and J are non-negative with J having a unit mass in L 1 (R N ), it follows from the equation

Lu + f (u) = 0 satisfied by u in R N \ K that L B R+1 (x * ) [u](x) -u(x) + f (u(x)) 0.
Together with the first inequality of the second line of (7.11) applied at x, we get that

L B R+1 (x * ) [z](x) -z(x) + f (w(x)) -f (u(x)) 0. (7.16) Since z z(x) in B R+1 (x * ), one has L B R+1 (x * ) [z](x) -z(x)
0. Furthermore, remembering (7.14) and the choice of ε, there holds 1 -ε u(x) = w(x) -z(x) < w(x) < 1 and f < 0 in [1-ε, 1], hence f (w(x))-f (u(x)) < 0. This contradicts (7.16). Therefore, max B R+1 (x * ) z 0, that is, w u in B R+1 (x * ) and then in R N \ K.

Step 3: a lower bound in annuli with large inner radii Let us now construct some families of sub-solutions and exploit the sweeping principle (Lemma 4.3) to get a lower bound of u in some annuli. To do so, let

x * ∈ R N , 0 < R J R K R and w ∈ C(R N , [0, 1
)) be as in (7.11). Consider any orthonormal basis (e 1 , • • • , e N ) of R N and, for τ ∈ [0, 2π], let R τ be the rotation of angle τ in the plane spanned by (e 1 , e 2 ) (that is, R τ e 1 = (cos τ )e 1 + (sin τ )e 2 and R τ e 2 = -(sin τ )e 1 + (cos τ )e 2 ) and leaving invariant the vectors e 3 , • • • , e N . We set

A := A (|x * | -R -1, |x * | + R + 1) = B |x * |+R+1 \ B |x * |-R-1 . From (7.11), note that A ⊂ R N \B R K ⊂ R N \K (hence, A ∩ K = ∅). Now for each τ ∈ [0, 2π] and x ∈ R N , we set w τ (x) := w(R τ x).
Thanks to the rotational invariance of J and A, and since B R+1 (x * ) ⊂ A and both J and w are non-negative, it follows from (7.11) that each function w τ satisfies

L A [w τ ] -w τ + f (w τ ) 0 in R N .
On the other hand, it follows from (2.5) that the function u obeys

L A [u](x)-u(x) + f (u(x)) = - ˆRN \(K∪A) J(x -y)u(y)dy -u(x) 1 - ˆRN \K J(x -y)dy 0,
for all x ∈ R N \ K and therefore for all x ∈ A. In addition, thanks to positivity of u in R N \ K (remember (7.8)) and the fact that J > 0 a.e. in A (r 1 , r 2 ) with 0 r 1 < r 2 R J R K R (remember (1.9) and supp(J) ⊂ B R J ), one infers that ˆRN \(K∪A) J(x-y)u(y)dy > 0 for all

x ∈ A := A (|x * |+R+1-r 2 , |x * |+R+1) (⊂ A), hence L A [u](x) -u(x) + f (u(x)) < 0 for all x ∈ A .
Since w u in A (⊂ R N \ K) by (7.15) and r 2 R K |x * |-R-1 by (7.13), it follows from the sweeping principle (Lemma 4.3) applied to u, to the family (w τ ) τ ∈[0,2π] and to

(s 1 , s 2 , s 3 , s 4 ) = (|x * | -R -1, |x * | + R + 1 -r 2 , |x * | + R + 1, |x * | + R + 1), that w τ u in A for every τ ∈ [0, 2π]. (7.17)
Notice also (even if the following inequalities will not explicitly be used in the next step) that, since w 1 -ε in B 1 (x * ) by (7.11), the family of estimates in (7.17) implies in particular that u

1 -ε in τ ∈[0,2π] B 1 (R -1 τ x * ).
Since the previous arguments are independent of the choice of the orthonormal basis (e 1 , . . . , e N ), we also get that u

1 -ε in A (|x * | -1, |x * | + 1).

Step 4: conclusion

Let us now finish our argument. To complete the proof of Lemma 7.2, we will again construct an adequate family of sub-solutions and use the sweeping principle to push further the estimates obtained in the previous step. To do so, pick some ρ > 0 and consider the domain

A ρ := A (|x * | -R -1, |x * | + R + 1 + ρ),
where R > 0 is defined in Steps 2 and 3. From (7.11), we note that

A ρ ⊂ R N \ B R K ⊂ R N \ K (hence, A ρ ∩ K = ∅). Next, consider any rotation R of R N , let e := x * /|x * | ∈ ∂B 1 and, for each σ ∈ [0, ρ] and x ∈ R N , denote W σ (x) := w(R x -σ e).
As in the previous step, from the rotational invariance of J and A ρ , and since B R+1 (x * + σe) ⊂ A ρ for every σ ∈ [0, ρ] and both J and w are non-negative, it follows from (7.11) that each function W σ satisfies

L Aρ [W σ ] -W σ + f (W σ ) 0 in R N .
Similarly, it follows from (2.5) that the function u obeys

L Aρ [u] -u + f (u) 0 in R N \ K,
(and therefore in A ρ ), while

L Aρ [u] -u + f (u) < 0 in A (|x * | + R + 1 + ρ -r 2 , |x * | + R + 1 + ρ) (⊂ A ρ ).
From the inequality (7.17) of the previous step (which holds for every τ ∈ [0, 2π] and for every orthonormal basis (e 1 , • • • , e N )), we have W 0 u in A and then in

R N \ K (since W 0 = 0 in R N \ A and u > 0 in R N \ K. As a consequence, W 0 u in A ρ .
Finally, it follows from the sweeping principle (Lemma 4.3) applied to u, to the family (W σ ) σ∈[0,ρ] and to

(s 1 , s 2 , s 3 , s 4 ) = (|x * | -R -1, |x * | + R + 1 + ρ -r 2 , |x * | + R + 1 + ρ, |x * | + R + 1 + ρ), that W σ u in A ρ for every σ ∈ [0, ρ]. Since w 1 -ε in B 1 (x *
) by (7.11), we obtain in particular that

u 1 -ε in σ∈[0,ρ] B 1 (R -1 (x * + σe)).
The previous arguments being independent of the choice of ρ > 0 and the rotation R of R N , we conclude that

u(x) 1 -ε for all |x| |x * | -1.
Since ε > 0 can be arbitrarily small, the proof of Lemma 7.2 is thereby complete.

The case of small perturbations of convex obstacles

In this section, we explore further the validity of the Liouville Theorem 2.2 and we prove Theorem 2.6, a kind of stability result for the Liouville property. In the spirit of the results of Bouhours [START_REF] Bouhours | Robustness for a Liouville type theorem in exterior domains[END_REF], we show that the Liouville property obtained in Theorem 2.2 still holds true for small perturbations of convex obstacles, provided some additional assumptions are made on f and J. To do so, we adapt to our problem the arguments developed in [START_REF] Bouhours | Robustness for a Liouville type theorem in exterior domains[END_REF] and, in particular, we will rely on the following Lemma 8.1. -Assume all hypotheses of Theorem 2.6. Then, for every δ ∈ (0, 1), there exists a real number R δ > 0 such that, for any ε ∈ (0, 1] and any measurable solution

u ε : R N \ K ε → [0, 1] of (2.7), there holds u ε (x) 1 -δ for a.e. |x| R δ .
Before proving Lemma 8.1, let us first establish a preliminary "rough" Liouvilletype result, namely Proposition 2.1.

Proof of Proposition

2.1. -We recall that f ∈ C 1 ([0, 1]), that J is assumed to satisfy (1.9), that K is a compact set such that R N \ K is connected, and that u : R N \ K → [θ, 1] is a continuous solution of (1.1) such that f 0 on [θ, 1]. Let us set m = inf R\K u ∈ [θ, 1].
Suppose, by contradiction, that m < 1. Let (x n ) n∈N ⊂ R N \ K be a sequence such that u(x n ) → m as n → +∞. Since u(x) → 1 as |x| → +∞, the sequence (x n ) n∈N is bounded and, up to extraction of a subsequence, we may assume that it converges to some x ∈ R N \ K. Evaluating the equation satisfied by u at x n , we obtain

ˆRN \K J(x n -y)(u(y) -u(x n ))dy + f (u(x n )) = 0.
By assumption, f (u(x)) 0 for all x ∈ R N \ K and therefore ˆRN \K J(x n -y)(u(y) -u(x n ))dy 0.

Since J ∈ L 1 (R N ) passing to the limit in the above inequality results in 0 ˆRN \K J(x -y)(u(y) -m)dy 0.

Thus, arguing as in Section 4 and using (1.9) and the connectedness of R N \ K, we obtain that u = m (< 1) in R N \ K. Since u(x) → 1 as |x| → +∞, we get a contradiction. The proof of Proposition 2.1 is thereby complete.

Let us now turn our attention to the proof of Lemma 8.1.

Proof of Lemma 8.1.

-First of all, in virtue of Lemma 2.2, we know that, for every ε ∈ (0, 1], every measurable solution u ε : R N \ K ε → [0, 1] of (2.7) possesses a Hölder continuous representative u * ε ∈ C 0,α (R N \ K ε ). Consequently, we are allowed to identify u ε with u * ε . For simplicity, we omit the superscript * and write simply u ε instead of u * ε . Let us then continuously extend f by f (0)s for s 0 and by f (1)(s -1) for s 1 and still denote f this extension. We also observe that, since (K ε ) 0<ε 1 is a family of (at least) C 0,α deformations of K in the sense of Definition 1.2, there exists a real number R 0 > 0 such that

K ε ⊂ B R 0 for all 0 < ε 1. ( 8.1)
Notice now that it is sufficient to show the conclusion of Lemma 8.1 for δ > 0 small enough. For any δ > 0 small enough, we are going to consider an auxiliary problem whose solutions will provide an appropriate lower bound for u ε , allowing us to prove the desired uniform convergence as |x| → +∞. To this end, for δ ∈ (0, 1), denote

f δ (s) := f (s) -f (1 -δ/2) for s ∈ R, and s δ := f (1 -δ/2) f (0) .
It is immediate to check that there exists δ 1 ∈ (0, 1) such that, for every δ ∈ (0, δ 1 ), one has s δ < 0 < 1 -δ/2 < 1 and

     f δ f in R, f δ = f < 1/2 in R, f δ (s δ ) = 0, f δ (s δ ) < 0, f δ (1 -δ/2) = 0, f δ (1 -δ/2) < 0, ´1-δ/2 s δ f δ (r)dr > 0, f δ vanishes only once in (s δ , 1 -δ/2).
Using the results obtained in [START_REF] Bates | Travelling waves in a convolution model for phase transitions[END_REF][START_REF] Chen | Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations[END_REF][START_REF] Coville | Travelling fronts in asymmetric nonlocal reaction diffusion equations: the bistable and ignition cases[END_REF][START_REF] Yagisita | Existence of traveling wave solutions for a nonlocal bistable equation: an abstract approach[END_REF], we know that, for every δ ∈ (0, δ 1 ), there exists a continuous function

φ δ : R → (s δ , 1 -δ/2) satisfying      L R φ δ + f δ (φ δ ) = J 1 * φ δ -φ δ + f δ (φ δ ) 0 in R, φ δ is increasing in R, φ δ (-∞) = s δ , φ δ (0) = 0, φ δ (+∞) = 1 -δ/2.
Fix in the sequel any δ ∈ (0, δ 1 ), any ε ∈ (0, 1] and any (Hölder-continuous) function

u ε : R N \ K ε → [0, 1] solving (2.7). For A > 0, we let Φ δ,A be the function defined in R N by Φ δ,A (x) := φ δ (|x| -A).
We observe that, by construction, we have

Φ δ,R 0 (x) 0 u ε for all x ∈ B R 0 \ K ε . (8.2)
Our aim is to extend the above relation to all

x ∈ R N \ B R 0 . Since u ε (x) → 1 as |x| → +∞, there exists R ε > R 0 such that u ε (x) max(1 -c 0 , 1 -δ/2) for all |x| R ε where c 0 > 0 is such that f < 0 in [1 -c 0 , +∞).
Then, reasoning as in Lemma 5.1 (or using directly that Φ δ,A → s δ < 0 as A → +∞ locally uniformly in R N and Φ δ,A < 1 -δ/2 < 1 in R N ), we obtain that, for some

A ε > 0, Φ δ,Aε u ε in R N \ K ε .
Consequently, it makes sense to define

A * := inf A ∈ R ; Φ δ,A u ε in R N \ K ε A ε . We claim that A * R 0 , (8.3)
We argue by contradiction and assume that A * > R 0 . From the definition of A * and the continuity of φ δ , we have

Φ δ,A * u ε in R N \ K ε . (8.4)
If min B Rε \Kε (u ε -Φ δ,A * ) > 0, then from the uniform continuity of φ δ , there exists

τ > 0 small enough such that Φ δ,A * -τ u ε in B Rε \ K ε . On the other hand, Φ δ,A * -τ < 1 -δ/2 u ε in R N \ B Rε . Hence, Φ δ,A * -τ u ε in R N \ K ε , a contradiction with the definition of A * . Therefore, min B Rε \Kε (u ε -Φ δ,A * ) = 0. Since u ε and Φ δ,A * are continuous, there exists x 0 ∈ B Rε \ K ε such that Φ δ,A * (x 0 ) = u ε (x 0 ).
Since A * > R 0 by assumption, it follows from (8.2) and the strict monotonicity of Φ δ,A with respect to A that x 0 ∈ B Rε \ B R 0 . Let us set e 0 = x 0 /|x 0 | and define the open affine half-space

H := x ∈ R N ; x • e 0 > R 0 (⊂ R N \ K ε ).
From (8.4) and the definition of Φ δ,A * , we have

u ε (x) ϕ(x) := φ δ (x • e 0 -A * ) for all x ∈ R N \ K ε .
Reasoning as in Lemma 5.1 and recalling the assumptions on f δ , we have that

         L ε u ε + f (u ε ) = 0 in H, L ε ϕ + f (ϕ) 0 in H (as in (5.3)), u ε ϕ in R N \ K ε , u ε (x 0 ) = ϕ(x 0 ) with x 0 ∈ H.
Applying the strong maximum principle (Lemma 4.2) we obtain in particular that u ε = ϕ in H. This is impossible since u ε (x) → 1 as |x| → +∞, while ϕ < 1 -δ/2 < 1 in R N . As a consequence, the claim (8.3) holds true.

From (8.3) and the monotonicity of Φ δ,A with respect to A, we then deduce that

Φ δ,R 0 Φ δ,A * u ε in R N \ K ε .
Since ε ∈ (0, 1] and u ε : R N \ K ε → [0, 1] solving (2.7) were arbitrary, since R 0 > 0 verifying (8.1) was independent of ε, and since φ δ (+∞) = 1-δ/2 > 1-δ, the desired conclusion follows.

We are now ready to prove Theorem 2.6.

Proof of Theorem 2.6. -First of all, as in the proof of Lemma 8.1, it follows from Lemma 2.2 that, for every ε ∈ (0, 1], every measurable solution u ε : R N \ K ε → [0, 1] of (2.7) can be identified with its Hölder continuous C 0,α (R N \ K ε ) representative. Furthermore, Lemma 2.2 yields

[u ε ] C 0,α (R N \Kε) A := 2[J] B α 1,∞ (R N ) inf 0<η 1 inf x∈R N \Kη J(x -•) L 1 (R N \Kη) -max [0,1]
f .

Note that A is independent of ε. In particular, for every ε * ∈ (0, 1] and every R R 0 , where R 0 > 0 is chosen as in (8.1), the family (u ε ) 0<ε ε * is uniformly bounded in C 0,α (B R \ K ε * ). Recalling that K ε → K as ε → 0 + in the C 0,α sense, there exists a sequence (ε j ) j∈N ∈ (0, 1] converging to 0 + and a function u 0 ∈ C 0,α (R N \ K) such that, for all R R 0 and β ∈ (0, α),

u ε j -u 0 C 0,β (B R \Kε j ) → 0 as j → +∞. ( 8.5)
Notice that 0 u 0 1 in R N \ K. By Lemma 8.1 we know that u ε (x) → 1 uniformly in ε > 0 as |x| → +∞. Consequently,

u 0 (x) → 1 as |x| → +∞. (8.6)
Now, we claim that

Lu 0 (x) + f (u 0 (x)) = 0 in R N \ K, (8.7)
where L is given by (1.2). This can be seen as follows. First, fix x in the open set R N \ K and an integer j 0 large enough such that x ∈ R N \ K ε j for all j j 0 . Notice that f (u ε j (x)) → f (u 0 (x)) as j → +∞ since f is continuous. Next, for all j j 0 we have

L ε j u ε j (x) -Lu 0 (x) = ˆRN \Kε j J(x -y) (u ε j -u 0 )(y) -(u ε j -u 0 )(x) dy - ˆKε j \K J(x -y)(u 0 (y) -u 0 (x))dy.
For every R R 0 and j j 0 , there holds

|L ε j u ε j (x) -Lu 0 (x)| 2 ˆKε j \K J(x -y) dy + 2 ˆRN \B R J(x -y) dy + u ε j -u 0 L ∞ (B R \Kε j ) + |u ε j (x) -u 0 (x)|.
Since K ε j → K in the C 0,α sense and J ∈ L 1 (R N ), we have in particular that the first term in the right-hand side converges to 0 as j → +∞. Recalling (8.5) and letting first j → +∞ and then R → +∞, we find that

L ε j u ε j (x) -Lu 0 (x) → 0 as j → +∞.
Therefore, (8.7) holds for all x ∈ R N \ K and finally for all x ∈ R N \ K by continuity and boundedness of u 0 in R N \ K. Remember now that u 0 ∈ C(R N \ K, [0, 1]). By (8.6), (8.7) and Theorem 2.2, we infer that u 0 = 1 in R N \ K. This also shows that the limit of the functions u ε j is unique and, hence, u ε → 1 as ε → 0 + in the sense of (8.5), not only along a subsequence.

We conclude by contradiction. Suppose then that there exists countably infinitely many numbers in (0, 1], which we label in decreasing order as (ε j ) j∈N , such that ε j → 0 + as j → +∞ and

∀ j ∈ N, ∃ x j ∈ R N \ K ε j , u ε j (x j ) = min R N \Kε j u ε j < 1. (8.8)
Note that this makes sense since, without loss of generality, we have identified the functions u ε j with their continuous representatives in R N \ K ε j . We observe that (1.11), (8.8) and Proposition 2.1 yield that u ε j (x j ) < θ for all j ∈ N.

Now, since the functions u ε j converge uniformly to 1 as |x| → +∞ (by Lemma 8.1), the sequence (x j ) j∈N is bounded. Hence, up to extraction of a subsequence, we may assume that x j → x as j → +∞, for some x ∈ R N \ K. Furthermore, since the functions u ε j converge to u 0 ≡ 1 as j → +∞ in the sense of (8.5), we obtain that

1 > θ > u ε j (x j ) -→ j→+∞ u 0 (x) = 1.
This is a contradiction. Therefore, there exists an ε 0 ∈ (0, 1] such that u ε = 1 in R N \ K ε for every ε ∈ (0, ε 0 ) and for every measurable solution u ε : R N \ K ε → [0, 1] of (2.7) (after identification with its continuous representative). The proof of Theorem 2.6 is thereby complete.

comes from population dynamics. In this setting, the movement of the individuals is modelled by a stochastic process that is defined in a domain that possesses several inaccessible regions (reflecting the heterogeneity of the environment). At the macroscopic level, the corresponding density of population u(t, x) satisfies a reactiondiffusion equation that is defined outside a set K, which acts as an obstacle. When the individuals follow isotropic Poisson jump processes, this reaction-diffusion equation is given by

∂u ∂t = Lu + f (u) in R × R N \ K, (1.3)
and the solutions to (1.1) are particular stationary solutions to (1.3).

In recent years, much attention has been paid to the case of Brownian diffusion. In this situation, the reaction-diffusion equation takes the form

   ∂u ∂t = ∆u + f (u) in R × R N \ K, ∇u • ν = 0 on R × ∂K. (1.4)
This problem was first studied by Berestycki, Hamel and Matano in [START_REF] Berestycki | Bistable travelling waves around an obstacle[END_REF]. There, it is shown that there exists a solution to (1.4) that satisfies 0 < u(t, x) < 1 for all (t, x) ∈ R × R N \ K, as well as a classical solution, u ∞ , to

         ∆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| → +∞. (1.5)
This latter solution is actually obtained as the large time limit of u(t, x); more precisely:

u(t, x) → u ∞ (x) as t → ∞, locally uniformly in x ∈ R N \ K.
In addition, they were able to classify the solutions u ∞ to (1.5) under some geometric assumptions on K. When the obstacle K is either starshaped of directionally convex (see [START_REF] Berestycki | Bistable travelling waves around an obstacle[END_REF]Definition 1.2]), they prove that the solutions to (1.5) are actually identically equal to 1 in the whole set R N \ K. This was further extended to more complex obstacles by Bouhours who showed a sort of "stability" of this Liouville type property with respect to small regular perturbations of the obstacle, see [START_REF] Bouhours | Robustness for a Liouville type theorem in exterior domains[END_REF]. From the biological standpoint, this means that, after some large time, the population tends to occupy the whole space.

Yet, when the domain is no longer starshaped nor directionally convex but merely simply connected, it is shown in [START_REF] Berestycki | Bistable travelling waves around an obstacle[END_REF] that this Liouville type property may fail.

In other words, the geometry of the domain may force the population to diffuse heterogeneously in space, even after some large time.

It is expected that (1.1) and (1.5) share some common properties. In particular, some of the results obtained for (1.5) should, to some extent, hold true as well for (1.1).

Recently, Brasseur et al. [START_REF] Brasseur | Liouville type results for a nonlocal obstacle problem[END_REF] have shown that (1.1) enjoys a similar Liouville type property when K is convex (or close to being convex) and when the data f and J satisfy some rather mild assumptions. That is, any solution u to (1.1) is identically equal to 1 in the whole set R N \ K. They also point out that this cannot be expected for general obstacles since one can easily find counterexamples when K is no longer simply connected. Indeed, take for instance K = A (1, 2) = B 2 \ B 1 and suppose that J is supported in B 1/2 . Then, the function u defined by

u(x) = 1 if x ∈ R N \ B 2 , 0 if x ∈ B 1 ,
is a continuous solution to (1.1); yet, u is not identically 1 in the whole set R N \ K.

In view of this, it is natural to ask:

what are the optimal geometric assumptions on K ensuring that (1.1) enjoys such a Liouville property?

So far, this question remains open. In this chapter, our main concern is to find out whether it is possible to construct a nontrivial simply connected obstacle K, as well as data f and J, for which (1.1) has a continuous solution u which is not identically equal to 1.

Note that this is actually a quite reasonable question. Indeed, since the Liouville property does not hold true on annuli it is quite natural to expect counterexamples on simply connected obstacles which are "ε-close" to an annulus. We will see that this is indeed the case. Precisely, we will construct a family of simply connected compact sets K ε and data f ε and J ε for which the solution to (1.1) need not be identically equal to 1.

Main results

Before we state our main results, let us first specify the assumptions made all along this chapter. We will assume that J is such that

       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 , M 1 (J) := ˆRN J(x)|x|dx < +∞ and J ∈ W 1,1 (R N ), (1.6)
and that f ∈ C 1 ([0, 1]) is a "bistable" nonlinearity, namely The obstacles constructed in Theorem 1.1 are almost of the same nature as those given in [START_REF] Berestycki | Bistable travelling waves around an obstacle[END_REF] for the local case. Namely, we consider an annulus A into which a small channel is pierced, see Figure 4.1 below for a visual illustration.

   ∃ θ ∈ (0, 1), f (0) = f (θ) = f (1) = 0, f < 0 in (0, θ), f > 0 in (θ, 1), ˆ1 0 f (s)ds > 0, f (0) < 0, f (θ) > 0, f (1) < 0, f < 1 in [0, 1]. ( 1
By contrast with the classical reaction-diffusion, the operator L does not enjoy strong compactness properties and has no regularising effects. So our construction is not a simple adaptation of the techniques of proof used for the local problem (1.5). One of the novelties of this chapter is that we show how to circumvent these issues. As we shall explain in the sequel, our argument is in fact general enough to recover the local problem as a limit case (see our remarks below).

Let us briefly describe our approach. Our strategy relies essentially on two ingredients. First, we take advantage of the fact that the kernel J and the nonlinearity f may be chosen at our convenience. That is, instead of considering the problem (1.1), we can consider a rescaled version of (1.1) given an appropriate choice of J. In our setting, J will be such that supp(J) = B r for some r > 0, and J ∈ L 2 (R N ) is radially non-increasing. (1.8) Then, given a small parameter ε, we look for a nonconstant positive solution u ε to

ˆRN \K J ε (x -y)(u ε (y) -u ε (x))dy + f ε (u ε (x)) = 0 in R N \ K, (1.9)
that further satisfies 0 u ε 1 in R N \ K and u ε (x) → 1 as |x| → +∞, where

f ε (s) := ε 2 f (s) and J ε (z) = 1 ε N J z ε .
In order to prove Theorem 1.1, we only need to show that, for some ε > 0, there is some obstacle K ε such that (1.9) admits a positive nonconstant solution u ε . Second, we consider a well-chosen family of smooth simply connected obstacles (K ε ) 0<ε<1 that look like an annulus with a tiny channel of diameter of the order of ε N/(N -1) pierced in it (see the Figure 4.1). Given such a family, we prove that, for ε small enough, (1.9) indeed admits a positive nonconstant continuous solution. More precisely, we prove the following Theorem 1.2. -Let N 2. Let J and f be such that (1.6), (1.7) and (1.8) hold. Then, there exist ε * > 0 and a family of smooth simply connected obstacles (K ε ) 0<ε<1 ⊂ R N such that, for all 0 < ε < ε * , there is a positive nonconstant solution

u ε ∈ C(R N \ K ε , [0, 1]) to (1.9).
Due to the lack of a strong regularising property of (1.9), the construction of u ε relies essentially on elementary arguments. In particular, we obtain a solution u ε to (1.9) using an adequate monotone iterative scheme and elementary estimates. The main difficulty in our proof lies in the construction of an adequate pair of ordered continuous sub-and super-solution in a context where the equation (1.1) does not allow the use of traditional schemes based on compactness arguments. To cope with this major difficulty, we make a detailed construction of the obstacle K ε and design it in such a way that we still can obtain standard L 2 -estimates by elementary means. This requires a detailed analysis of all the parameters involved at each steps of our construction, especially when we construct our super-solution. To construct our super-solution we rely on the fact that a solution u ε to (1.9) satisfies in particular

1 ε 2 ˆRN \Kε J ε (x -y)(u ε (y) -u ε (x))dx + f (u ε (x)) = 0, (1.10)
and, from there, relying essentially on the Bourgain-Brezis-Mironescu characterisation of Sobolev spaces (see e.g. [START_REF] Bourgain | Another look at Sobolev spaces[END_REF][START_REF] Ponce | An estimate in the spirit of Poincaré's inequality[END_REF]), we can interpret the first term on the left-hand side as a nonlocal approximation of ∆u in the sense that its energy approximates the L 2 -variation of u. This, in turn, with a pertinent choice of K ε and a well-chosen auxiliary problem, allows one to derive a priori bounds to construct a super-solution by means of variational methods.

A striking consequence of our construction is that it adapts almost straightforwardly to other situations. For example, it applies to the standard reaction-diffusion equation (1.5) providing so an alternative proof of the existence of a counterexample. But it also extends to broader classes of nonlocal operators where the dispersal process need not be isotropic but instead depends on the geodesic distance between points in R N \ K. Indeed, our proof also adapts (with almost no changes) to operators of the form

L g u(x) := ˆRN \K J(d g (x, y))(u(y) -u(x))dy, (1.11)
where

d g (•, •) is the geodesic distance on R N \ K and J ∈ L 1 loc (0, ∞) is such that sup x∈R N \K ˆRN \K J(d g (x, y))dy < ∞, (1.12)
and z → J(|z|) satisfies (1.6). More precisely, we have

Theorem 1.3. -Let N 2.
Then, there are smooth (non-starshaped) simply connected compact obstacles K and data f and J satisfying (1.7) and (1.12) for which the problem

L g u + f (u) = 0 in R N \ K, u(x) → 1 as |x| → +∞, (1.13)
has a solution u ∈ C(R N \ K, [0, 1]) which is not identically equal to 1 in R N \ K.

The obstacle K and the data f and J (= J(|•|)) constructed at Theorem 1.3 are exactly the same as in Theorem 1.1.

Problem (1.13) is of interest in its own right. It gives an alternative way to describe the evolution of particles within a perforated domain which, in some situations, may be regarded as more realistic. The point here is that particles cannot travel through K (as is it the case for problem (1.1)). Instead, they are compelled to "bypass" K as if it was a material obstacle. This particularity may be helpful to study the dynamics of some species (such as worms or spores) for which this behavior is well-suited.

When needed we will state in side remarks the necessary changes to make to the proofs in order to handle this type of dispersal processes.

Remark 1.4. -It turns out that the techniques of proof used in [START_REF] Brasseur | Liouville type results for a nonlocal obstacle problem[END_REF] to establish the Liouville property of (1.1) for convex domains also apply to this modified setting (at least when J is non-increasing), but we leave this to a subsequent paper.

The chapter is organized as follows. After describing our notations, we recall some results from the literature in Section 2. In Section 3, given a pair (J, f ) we construct an adequate family of obstacles. Then, in Section 4, we construct some particular super-solutions to the problem (1.9). Finally, in Section 5, we use the super-solution constructed at Section 4 to prove Theorem 1.2.

Notations

Let us list a few notations that will be used throughout the chapter.

As usual, S N -1 denotes the unit sphere of R N and B R (x) the open Euclidean ball of radius R > 0 centred at x ∈ R N (when x = 0, we simply write B R ). We denote by

A (R 1 , R 2 ) the open annulus B R 2 \ B R 1 .
For a compact set Ω ⊂ R N , we denote by diam(Ω) its diameter, given by diam(Ω) := sup

x,y∈Ω |x -y|.

The N -dimensional Hausdorff measure will be denoted by H N . For a measurable set E ⊂ R N , we denote by |E| its Lebesgue measure and by 1 E its characteristic

Construction of a family of obstacles

This section is devoted to the construction of an appropriate family of obstacles (K ε ) 0<ε<1 . Our construction will depend on the interplay with the datum (J, f ). As mentioned in the introduction, we will assume that J satisfies (1.6) and (1.8) and that f satisfies (1.7). However, before constructing (K ε ) 0<ε<1 , we need to define some important quantities depending on f and J. We will call C 0 > 0 and M 2 (J) > 0 the constants respectively defined by

     C 0 := max s∈[0,1] f (s), (3.1)
M 2 (J) := ˆRN J(z)|z| 2 dz. (3.2)
Note that the assumptions (1.6) and (1.7) guarantee that these two numbers are well-defined. Furthermore, we introduce two quantities, C N,J and R * 0 , respectively defined by

       C N,J := π 2 M 2 (J) 32N , (3.3)
R * 0 (J, f ) := θC N,J 5C 0 . (3.4)
Let us now start the construction of the obstacle. Fix some R 1 > 2 and let 0 < R 0 < R * 0 (J, f ) (where R * 0 (J, f ) is as in (3.4)). Let 0 < ε < 1 be a small Thus, passing to polar coordinates, the mass carried by J ε (x -•) in F ε can be written as ˆ

Fε J ε (x -y)dy = ˆΛε(x) J ε (y)dy = ˆSN-1 ˆε/2 0 1 Λε(x) (σt) J ε (t)t N -1 dt dH N -1 (σ).
Notice that Λ ε (x) is a convex set and that 0 ∈ Λ ε (x). In particular, both t → 1 Λε(x) (σt) and t → J ε (t) are non-increasing functions. Hence, using Chebyshev's integral inequality (see e.g. [109, Theorem 2.5.10, p.40]), we have

ˆ Fε J ε (x-y)dy N (ε/2) N ˆSN-1 ˆε/2 0 1 Λε(x) (σt)t N -1 dt ˆε/2 0 J ε (t)t N -1 dt dH N -1 (σ).
Since J ε has unit mass and supp(J ε ) = B ε/2 , one has ˆε/2

0 J ε (t)t N -1 dt = σ -1 N = N |B 1 | -1 ,
where σ N = H N -1 (S N -1 ). Ergo,

ˆ Fε J ε (x -y)dy 1 |B ε/2 | ˆSN-1 ˆε/2 0 1 Λε(x) (σt)t N -1 dt dH N -1 (σ) = 1 |B ε/2 | ˆBε/2
1 Λε(x) (y)dy.

Since

F ε ⊂ R N \ K ε and Λ ε (x) = B ε/2 ∩ ( F ε -x), we get ˆRN \Kε J ε (x -y)dy |B ε/2 (x) ∩ F ε | |B ε/2 | , for any x ∈ F ε . (3.7)
Let us now estimate the quantity |B ε/2 (x) ∩ F ε |. Observe that for ε small enough, say when 0 < ε < ε 2 := 4 -(N -1) , one has ε/2 > 2ε γ . In particular, this implies that B ε/2 (x) ∩ F ε always contains an hyper-rectangle of the form T (0, ε/4) × (0, 2ε γ ) × • • • × (0, 2ε γ ) for some translation T of R N , so that

|B ε/2 (x) ∩ F ε | (ε/4) × (2ε γ ) N -1 = 2 N -3 ε N +1 .
Therefore, recalling (3.7), we obtain that, for all 0 < ε < ε 2 and all x ∈ F ε , it holds

ˆRN \Kε J ε (x -y)dy C 1 ε, (3.8)
for some C 1 > 0 depending on N only.

Step 2: Lower bound in

R N \ (K ε ∪ F ε )
Let us now consider the case where x ∈ R N \ (K ε ∪ F ε ). For it, we first note that, since 0 < ε γ < ε < R 0 /2 (remember 0 < ε < ε 1 ), the point

x 0 := (R 0 , ε γ , 0, • • • , 0) ∈ ∂ F ε satisfies |x 0 | 2 = R 2 0 + ε 2γ < R 2 0 + R 0 ε γ /2 < (R 0 + ε γ /4) 2 , which implies that F ε ∩ B R 0 +ε γ /4 = ∅. On the other hand, it is clear from the definition of F ε that F ε \ B R 1 -ε γ /4 = ∅. A consequence of this is that F ε ∩A (R 0 +ε γ /4, R 1 -ε γ /4) = A (R 0 +ε γ /4, R 1 -ε γ /4)∩ z ∈ R N ; z 1 > 0, |z | < ε γ .
Whence, recalling properties (i) and (iv) in the definition of K ε , we deduce that

A (R 0 + ε/4, R 1 -ε/4) ⊂ A (R 0 + ε γ /4, R 1 -ε γ /4) ⊂ K ε ∪ F ε ,
where, in the left-hand side, we have used the fact that ε γ < ε. In turn, this implies that

x ∈ R N \ A (R 0 + ε/4, R 1 -ε/4).
In particular, since 0 < ε < R 0 /2 < R 0 , we may find a point z ∈ R N such that

|x -z| = 3ε 8 and B ε/8 (z) ⊂ B ε/2 (x) \ A ⊂ R N \ K ε . (3.9)
Indeed, when x ∈ R N \ B R 1 -ε/4 , this follows from the convexity of B R 1 ; and, when x ∈ B R 0 +ε/4 , the constraint 0 < ε < R 0 /2 allows one to choose z on the diagonal of B R 0 +ε/4 containing x. On account of this, we may write ˆRN \Kε J ε (x -y)dy ˆBε/8 (z) J ε (x -y)dy = ˆBε/8 (z-x) J ε (y)dy = ˆB1/8( z-x ε )

J(y)dy. Now, by (3.9), we have (z -x)/ε ∈ ∂B 3/8 . Thus, ˆRN \Kε J ε (x -y)dy ˆB1/8 (ex) J(y)dy =: M J (e x ) for some e x ∈ ∂B 3/8 .

Notice that B 1/8 (e x ) ⊂ B 1/2 = supp(J) (because e x ∈ ∂B 3/8 ) which implies M J (e x ) > 0. Moreover, since J is radially symmetric, the quantity M J (e x ) does not depend on the choice of e x ∈ ∂B 3/8 , namely M J (e x ) = M J (e) ≡ M J > 0, for every e ∈ ∂B 3/8 , and some constant M J depending on J only. Therefore, for any 0 < ε < ε 1 and x ∈ R N \ (K ε ∪ F ε ), it holds ˆRN \Kε J ε (x -y)dy M J > 0.

(3.10)

Step 3:

Conclusion Since R N \ K ε = F ε ∪ (R N \ ( F ε ∪ K ε ))
, by (3.8) and (3.10), we obtain

inf x∈R N \Kε ˆRN \Kε J ε (x -y)dy min {M J , C 1 } ε,
for any 0 < ε < ε 3 := min{ε 1 , ε 2 }. Whence, letting

ε 0 := min ε 3 , min {M J , C 1 } max [0,1] f ,
and recalling that f ε (s) = ε 2 f (s), we obtain max

[0,1] f ε < inf x∈R N \Kε ˆRN \Kε J ε (x -y)dx for any ε ∈ (0, ε 0 ),
which is the desired inequality.

Remark 3.2. -Since J ε is radially non-increasing and satisfies (1.6), there is some non-increasing J ε ∈ L 1 loc (0, ∞) satisfying J ε (z) = J ε (|z|). In particular, since d g (x, y) |x -y|, it holds that

sup x∈R N \Kε ˆRN \Kε J ε (d g (x, y))dy sup x∈R N \Kε ˆRN \Kε J ε (x -y) dy = 1,
thus implying that J ε satisfies (1.12). Moreover, Proposition 3.1 still holds when J ε (x -y) is replaced by J ε (d g (x, y)), i.e. we still have max

[0,1] f ε < inf x∈R N \Kε ˆRN \Kε J ε (d g (x, y))dy. (3.11)
Indeed, this is because our proof reduces to estimate the mass carried by J ε (x -•) on convex sub-domains of R N \ K ε and, in this case, the geodesic distance coincides with the Euclidean distance, namely it holds that J ε (x -y) = J ε (d g (x, y)).

Construction of a global super-solution

In this section we construct a global super-solution to (1.9). Precisely, given a pair (J, f ) satisfying (1.7) and (1.8) and given the family of obstacles (K ε ) 0<ε<1 associated to (J, f ) (as defined in Section 3), we construct a global super-solution ūε to

ˆRN \Kε J ε (x -y)(ū ε (y) -ūε (x))dy + f ε (ū ε (x)) 0 for x ∈ R N \ K ε , (4.1)
that further satisfies

ūε ≡ 1 for x ∈ R N \ B R , (4.2)
for some large R > 0, where f ε and J ε are as in (3.5). More precisely, we prove the following Lemma 4.1. -Let N 2 and let (J, f ) be a pair satisfying (1.7) and (1.8). Let (K ε ) 0<ε<1 be the family of obstacles associated to the pair (J, f ) (as defined in Section 3). Let f ε and J ε be as in (3.5). Then, there exists R * > 0 and ε * > 0 such that, for all 0 < ε < ε * and all R R * , there is a continuous positive nonconstant function ūε satisfying (4.1) and (4.2).

The proof of Lemma 4.1 follows essentially two steps. In the first step, we construct a positive solution to a suitable auxiliary problem defined in B R \ K ε for some large R. Then, in a second step, we regularise this solution to obtain a super-solution that satisfies both (4.1) and (4.2). To simplify the presentation each step of the proof corresponds to a subsection.

An auxiliary problem in

B R \ K ε
Let us first construct an adequate auxiliary problem. To do so, we define a new nonlinearity, f , satisfying

f (s) :=          -κs for s 3θ 4 , f 0 (s) for 3θ 4 < s < θ, f (s) for θ s 1, f (1)(s -1) for s > 1, (4.3)
where θ ∈ (0, 1) is as in (1.7), κ > 0 is a small number and f 0 is a smooth function such that f ∈ C 1 (R). From (1.7), we can choose κ > 0 and f 0 such that

f f in [0, 1], max [0,1] f (s) = max [0,1] f and sup R f sup [0,1] f . (4.4)
Now, for R > R 1 + 2, we let L R,ε be the operator given by L R,ε w(x) := ˆBR \Kε J ε (x -y)(w(y) -w(x))dy, (

and we consider the following problem

L R,ε u ε,R (x) + c ε (x)(1 -u ε,R (x)) + f ε (u ε,R (x)) = 0 for all x ∈ B R \ K ε , (4.6)
where

f ε (s) = ε 2 f (s) for s ∈ R and c ε (x) := ˆRN \B R J ε (x -y)dy for x ∈ B R \ K ε . (4.7)
Our goal in this step is to show that, for each ε ∈ (0, 1) small enough, there exists a continuous function u ε,R : B R \ K ε → (0, 1) satisfying (4.6).

Remark 4.2. -Observe that, by construction (remember (4.4)), the function

u ε,R := u ε,R in B R \ K, 1 in R N \ B R ,
provides a discontinuous super-solution to (4.1) satisfying (4.2). We are thus on the right track to construct the required super-solution.

For it, we observe that, by setting v ε,R := 1 -u ε,R , (4.6) rewrites

L R,ε v ε,R (x) -c ε (x)v ε,R (x) + g ε (v ε,R (x)) = 0 for x ∈ B R \ K ε , (4.8)
with g ε (s) := -ε 2 f (1 -s). Therefore, to construct u ε,R it suffices to construct a positive solution v ε,R : B R \ K → (0, 1) to (4.8). As in [START_REF] Berestycki | Bistable travelling waves around an obstacle[END_REF], this will be done using a variational argument. To do so, we define

g(s) := -f (1 -s), G(t) := ˆt 0 g(s)ds and G ε (t) := ε 2 G(t),
for all s, t ∈ R and ε ∈ (0, 1). Now, for any ε ∈ (0, 1) and any domain Ω ⊂ B R \ K ε , we consider the following energy functional

E ε,Ω (w) := 1 4 ˆΩ ˆΩ J ε (x-y)(w(x)-w(y)) 2 dxdy+ 1 2 ˆΩ c ε (x)w 2 (x)dx-ˆΩ G ε (w(x))dx, ( 4
.9) for w ∈ L 2 (Ω). Observe that for any ε > 0 and any domain Ω ⊂ B R \ K ε , the null function w ≡ 0 is a global minimiser of E ε,Ω . Therefore, we have to construct a local minimiser. However, unlike its local analogue, the energy functional E ε,Ω does not posses strong compactness properties, rendering this type of approach very delicate to implement.

With this in mind, we will show that, for the family K ε constructed in Section 3 and ε small enough, the above energy has indeed a nontrivial local minimiser when Ω = B R \ K ε .

Following the scheme of construction introduced in [START_REF] Berestycki | Bistable travelling waves around an obstacle[END_REF], we first show that the function w 0 := 1 B R 0 is a strict minimiser of the functional E ε,B R 0 when ε ∈ (0, 1) is small enough.

More precisely,

Proposition 4.3. -Let N 2, 0 < R 0 < R * 0 (J, f ) (where R * 0 (J, f
) is given by (3.4)) and let w 0 := 1 B R 0 . Then, there exists κ 0 > 0, 0 < ε 1 (J, N, R 0 ) < 1 and

0 < δ 0 (R 0 ) < |B R 0 | 1/2 such that, for each 0 < ε < ε 1 , it holds that E ε,B R 0 (w) -E ε,B R 0 (w 0 ) κ 0 ε 2 w -w 0 2 L 2 (B R 0 ) , for all w ∈ L 2 (B R 0 ) such that w -w 0 L 2 (B R 0 ) δ 0 .
Proof. -Let us begin with some preliminary observations. First, we notice that since g is linear around 1 (because f is linear around 0), the function G ε is smooth in a neighborhood of 1. In particular, there exists τ 0 ( f ) > 0 such that

G ε (t) = G ε (1) + G ε (1)(t -1) + 1 2 G ε (1)(t -1) 2 for any |t -1| < τ 0 . But since G ε (1) = ε 2 G (1) = ε 2 g(1) = 0 and G ε (1) = ε 2 G (1) = ε 2 g (1) = -ε 2 f (0) = -ε 2 κ
, this expansion can be rewritten as

G ε (t) = ε 2 G(1) - κε 2 2 (t -1) 2 for any |t -1| < τ 0 . (4.10)
Using the number τ 0 , we define

δ 0 := min θ 4 , C 0 κ , τ 0 2 |B R 0 | 1/2 , (4.11)
where θ, C 0 and κ are as in (1.7), (3.1) and (4.3); and we let w ∈ L 2 (B R 0 ) be any function such that

w -w 0 L 2 (B R 0 ) δ 0 . (4.12)
Second, denoting by w 0 := span L 2 (B R 0 ) (w 0 ) the vector space spanned by w 0 and letting w 0 ⊥ be its orthogonal with respect to the standard scalar product of L 2 (B R 0 ), we can write the space L 2 (B R ) as the direct sum L 2 (B R 0 ) = w 0 ⊕ w 0 ⊥ . This means that we may always find a constant α ∈ R and a function h ∈ w 0 ⊥ such that w decomposes as w = αw 0 + h. In particular, the orthogonality of h with respect to w 0 implies that ˆBR 0 h(x)dx = 0 and w -w 0 2

L 2 (B R 0 ) = (1 -α) 2 w 0 2 L 2 (B R 0 ) + h 2 L 2 (B R 0 ) . (4.13)
In view of this, assumption (4.12) gives

- δ 0 |B R 0 | 1/2 (1 -α) δ 0 |B R 0 | 1/2 and h L 2 (B R 0 ) δ 0 . (4.14)
This fact will be abundantly used in the sequel. This being said, we are now in position to prove Proposition 4.3. For it, we observe that, since w 0 ≡ 1 in B R 0 , we have that

E ε,B R 0 (w 0 ) = - ˆBR 0 G ε (w 0 (x))dx = -G ε (1)|B R 0 | = -ε 2 G(1)|B R 0 |. Furthermore, thanks to R > R 0 + 2 and supp(J ε ) ⊂ B ε 2 , we have that c ε (x) ≡ 0 in B R 0 , for any 0 < ε < 1. Consequently, E ε,B R 0 (w) rewrites E ε,B R 0 (w) = 1 4 ˆBR 0 ˆBR 0 J ε (x -y)(w(x) -w(y)) 2 dxdy - ˆBR 0 G ε (w(x))dx .

II I

Let us first estimate II . In view of the Bourgain-Brezis-Mironescu representation of H 1 (B R 0 ) (see [START_REF] Bourgain | Another look at Sobolev spaces[END_REF]), one can interpret II as a nonlocal approximation of ∇w 2 L 2 (B R 0 ) . The crux of our strategy is that, as shown by Ponce [START_REF] Ponce | An estimate in the spirit of Poincaré's inequality[END_REF]Theorem 1.1], this nonlocal approximation enjoys a Poincaré-type inequality. Let us now proceed. Let (ρ ε ) 0<ε<1 be the family of radially symmetric mollifiers defined by

ρ ε (z) := M 2 (J) -1 J ε (z)|z| 2 ε -2 for ε ∈ (0, 1),
where M 2 (J) is given by (3.2). Notice that, by construction, it satisfies ρ ε 0 a.e. in R N , ˆRN ρ ε (z)dz = 1 and lim ε→0 + ˆ|z| τ ρ ε (z)dz = 0, for each 0 < ε < 1 and each τ > 0. Moreover, II can be rewritten as

II = ε 2 M 2 (J) 4 ˆBR 0 ˆBR 0 ρ ε (x -y) |w(x) -w(y)| 2 |x -y| 2 dxdy.
Now, by [118, Theorem 1.1], we know that there exists some ε 1 = ε 1 (J, N, R 0 ) > 0 such that the following Poincaré-type inequality

w - B R 0 w 2 L 2 (B R 0 ) 2A 0 K 2,N ˆBR 0 ˆBR 0 ρ ε (x -y) |w(x) -w(y)| 2 |x -y| 2 dxdy = 8A 0 ε -2 K 2,N M 2 (J) × II ,
holds for all ε ∈ (0, ε 1 ) and all w ∈ L 2 (B R 0 ). Here,

K 2,N := ˆSN-1 (σ • e 1 ) 2 dH N -1 (σ) = 1 N ,
and A 0 > 0 is the smallest constant such that the standard Poincaré-Wirtinger inequality holds. That is, A 0 is the smallest positive constant such that w -

B R 0 w 2 L 2 (B R 0 ) A 0 ∇w 2 L 2 (B R 0 ) ,
holds for any w ∈ H 1 (B R 0 ). In our case, A 0 satisfies the upper bound:

A 0 diam(B R 0 ) 2 π 2 = 4R 2 0 π 2 ,
see [14, Theorem 3.2] (see also [START_REF] Payne | An optimal Poincaré inequality for convex domains[END_REF]). In particular, this gives

ε 2 π 2 M 2 (J) 32N R 2 0 w - B R 0 w 2 L 2 (B R 0 ) II .
Now, since w = αw 0 + h, since w 0 ≡ 1 on B R 0 and since h is integral free (by (4.13)) we have

II ε 2 C N,J R 2 0 h 2 L 2 (B R 0 ) , (4.15)
where C N,J is given by (3.3). We are now left to estimate I. For it, we rewrite I as follows

I = E ε,B R 0 (w 0 ) - ˆBR 0 G ε (w(x)) -G ε (w 0 (x)) dx. ( 4

.16)

To estimate the last integral, we split it into two parts, I 1 and I 2 , where

I 1 := - ˆBR 0 G ε (w 0 +(α-1)w 0 +h)-G ε (w 0 +(α-1)w 0 ) , I 2 := - ˆBR 0 G ε (w 0 +(α-1)w 0 )-G ε (w 0 ) .
Let us first estimate I 2 . Using (4.11), (4.12) and (4.14) we have in particular that |1 -α| < τ 0 . This, together with (4.10), gives

I 2 = - ˆBR 0 G ε (w 0 +(α-1)w 0 )-G ε (w 0 ) = κ 2 ε 2 |B R 0 |(α -1) 2 .
Therefore, recalling (4.16), we get

I = E ε,B R 0 (w 0 ) + κ 2 ε 2 |B R 0 |(α -1) 2 + I 1 . (4.17)
Let us now estimate I 1 . On account of (4.14), we may write

α = 1 -η for some |η| δ 0 |B R 0 | 1/2 . (4.18)
Then, a standard change of variables yields

I 1 = ε 2 ˆBR 0 ˆ1-η+h(x) 1-η f (1 -τ )dτ dx = -ε 2 ˆBR 0 ˆ-h(x) 0 f (τ + η)dτ dx. Now, we set Σ := x ∈ B R 0 ; -h(x) > θ 2 ,
and we decompose I 1 as

I 1 = -ε 2 ˆΣ ˆ-h(x) 0 f (τ + η)dτ dx + ˆBR 0 \Σ ˆ-h(x) 0 f (τ + η)dτ dx . (4.19)
We will estimate these two integrals separately. In view of (4.11) and (4.18), we have that |η| θ/4. In turn, this implies that

-h(x) + |η| 3θ 4 for any x ∈ B R 0 \ Σ.
Since, by construction, f is linear in (-∞, 3θ/4], we get

ˆBR 0 \Σ ˆ-h(x) 0 f (τ + η)dτ dx = -κ ˆBR 0 \Σ ˆη-h(x) 0 τ dτ - ˆη 0 τ dτ dx = - κ 2 ˆBR 0 \Σ h 2 (x)dx + κη ˆBR 0 \Σ h(x)dx = - κ 2 ˆBR 0 \Σ h 2 (x)dx -κη ˆΣ h(x)dx,
where, in the last equality, we have used the fact that h is integral free, that is:

ˆBR 0 \Σ h(x)dx + ˆΣ h(x)dx = 0.
Using now the Cauchy-Schwarz inequality, we get

ˆBR 0 \Σ ˆ-h(x) 0 f (τ + η)dτ dx - κ 2 ˆBR 0 \Σ h 2 (x)dx + κ|η| |Σ| h L 2 (Σ) .
By the Bienaymé-Chebyshev inequality, we have

|Σ| 2 θ 2 h 2 L 2 (Σ) , (4.20)
and thus

ˆBR 0 \Σ ˆ-h(x) 0 f (τ + η)dτ dx - κ 2 ˆBR 0 \Σ h 2 (x)dx + 2κ|η| θ h 2 L 2 (Σ) . (4.21)
Thanks to (4.11) and (4.18), (4.21) reduces to

ˆBR 0 \Σ ˆ-h(x) 0 f (τ + η)dτ dx - κ 2 ˆBR 0 \Σ h 2 (x)dx + 2C 0 θ h 2 L 2 (Σ) . (4.22)
Let us now estimate the first integral on the right-hand side of (4.19). For it, we observe that

τ + η -|η| -δ 0 |B R 0 | 1/2 - C 0 κ
for any x ∈ Σ and any τ ∈ (0, -h(x)).

Recalling (4.4), we then obtain

sup x∈Σ sup τ ∈(0,-h(x)) f (τ + η) sup s - C 0 κ f (s) = max s∈[0,1] f (s) = C 0 .
This, together with the Cauchy-Schwarz inequality, gives

ˆΣ ˆ-h(x) 0 f (τ + η)dτ dx C 0 ˆΣ |h(x)|dx C 0 |Σ| h L 2 (Σ) ,
Using the Bienaymé-Chebyshev inequality (4.20), we finally get ˆΣ ˆ-h(x)

0 f (τ + η)dτ dx 2C 0 θ h 2 L 2 (Σ) . ( 4
E ε,B R 0 (w) -E ε,B R 0 (w 0 ) ε 2 κ 2 |B R 0 |(α -1) 2 + κ 2 h 2 L 2 (B R 0 \Σ) - 4C 0 θ h 2 L 2 (Σ) + C N,J R 2 0 h 2 L 2 (B R 0 ) , for all 0 < ε < ε 1 and all w ∈ L 2 (B R 0 ) with w -w 0 L 2 (B R 0 ) δ 0 .
Recalling that 0 < R 0 R * 0 (J, f ) and using (3.4), we have C N,J /R 2 0 5C 0 /θ. This, together with the above inequality, yields

E ε,B R 0 (w) -E ε,B R 0 (w 0 ) ε 2 κ 2 |B R 0 |(α -1) 2 + C 0 θ h 2 L 2 (B R 0 ) .
Therefore, letting

κ 0 := inf κ 2 , C 0 θ ,
and recalling (4.13), we obtain

E ε,B R 0 (w) -E ε,B R 0 (w 0 ) ε 2 κ 0 w -w 0 2 L 2 (B R 0 )
, for all 0 < ε < ε 1 and all w ∈ L 2 (B R 0 ) with w -w 0 L 2 (B R 0 ) δ 0 .

Remark 4.4. -Note that the proof of Proposition 4.3 relies only on L 2 -estimates and on a Poincaré-type inequality. Remarkably, this allows one to adapt straightforwardly our arguments to the local analogue of E ε,B R 0 . Using Proposition 4.3, we now prove the following Proposition 4.5. -Let N 2, and let E ε,R be the energy functional defined by (4.9) with Ω = B R \K ε . Then, there exists

C * > 0, 0 < δ 0 < |B R 0 | 1/2 and 0 < ε δ 0 < 1 such that, for any 0 < ε < ε δ 0 and any w ∈ L 2 (B R \ K ε ) with w -w 0 L 2 (B R \Kε) = δ 0 , it holds that E ε,R (w) -E ε,R (w 0 ) > C * ε 2 .
Proof. -Let us first notice that our assumptions on f imply that there is some κ 1 > 0 such that -G(t) κ 1 t 2 for every t ∈ R.

(4.24)

Let us now compute the energy of w 0 . Since supp(J ε ) = B ε/2 and R 1 -R 0 > ε, a straightforward calculation yields

E ε,R (w 0 ) = E ε,B R 0 (w 0 )+ 1 2 ˆFε ˆBR 0 J ε (x -y)dxdy.
In addition, elementary computations yield

1 2 ˆFε ˆBR 0 J ε (x -y)dxdy = 1 2 ˆFε∩B R 0 + ε 2 ˆBR 0 J ε (x -y)dx dy |F ε ∩ B R 0 + ε 2 | 2 Cε N +1 , for come constant C = C(N ) > 0.
As a consequence, we obtain

E ε,R (w 0 ) E ε,B R 0 (w 0 ) + Cε N +1 . (4.25) Next, developing E ε,R (w), we get E ε,R (w) = E ε,B R 0 (w) + E ε,Fε (w) + E ε,B R \B R 1 (w) + 1 2 ˆFε ˆBR 0 + ˆBR \B R 1 J ε (x -y)(w(x) -w(y)) 2 dxdy.
Using (4.24) we obtain that E ε,Ω (w)

κ 1 ε 2 w 2 L 2 (Ω) for any domain Ω ⊂ B R \ K ε . In particular, since w 0 = 0 in F ε ∪ B R \ B R 1 we have E ε,B R \Kε (w) E ε,B R 0 (w) + κ 1 ε 2 w -w 0 2 L 2 (Fε∪B R \B R 1 ) . (4.26)
Gluing together (4.25) and (4.26), we obtain

E ε,R (w) -E ε,R (w 0 ) E ε,B R 0 (w) -E ε,B R 0 (w 0 ) + κ 1 ε 2 w -w 0 2 L 2 (Fε∪B R \B R 1 ) -Cε N +1 . (4.27)
Now, by Proposition 4.3, there exists

κ 0 > 0, 0 < δ 0 < |B R 0 | 1/2 and ε 1 > 0 such that, for any 0 < ε < ε 1 and any w ∈ L 2 (B R 0 ) with w -w 0 L 2 (B R 0 ) δ 0 , we have E ε,B R 0 (w) -E ε,B R 0 (w 0 ) κ 0 ε 2 w -w 0 2 L 2 (B R 0 ) . ( 4

.28)

Letting κ := min{κ 1 , κ 0 } and combining (4.28) and (4.27), we obtain

E ε,R (w) -E ε,R (w 0 ) ε 2 κ w -w 0 2 L 2 (B R \Kε) -Cε N +1 = ε 2 κδ 2 0 -Cε N -1 , (4.29)
for all 0 < ε < ε 1 and all w ∈ L2 (B R \ K ε ) with w -w 0 L 2 (B R \Kε) = δ 0 . The conclusion now follows from (4.29) and the choice

C * = κδ 2 0 2 and ε δ 0 := min ε 1 , κδ 2 0 2C 1 N -1 .
The proof is thereby complete.

We are now in position to construct a positive solution to (4.8).

Proposition 4.6. -Let N 2 and let (J, f ) be a pair satisfying (1.7) and (1.8). Let (K ε ) 0<ε<1 be the family of obstacles associated to the pair (J, f ) (as defined in Section 3). Let f be the extension of f given by (4.3) and let f ε and J ε be respectively given by (4.7) and (3.5). Then, there exists ε > 0 such that, for all

0 < ε < ε, there is a function v ε,R ∈ C(B R \ K ε ) satisfying (4.8) and 0 < v ε,R < 1 in B R \ K ε . Proof. -Let w 0 := 1 B R 0 and let 0 < δ 0 < |B R 0 | 1/2 and 0 < ε δ 0 < 1 be quantities constructed in the proof of Proposition 4.5, namely such that E ε,R (w) -E ε,R (w 0 ) > C * ε 2 ,
holds for some constant C * > 0 and for any 0 < ε < ε δ 0 and any w ∈ L 2 (B R \ K ε ) with w -w 0 L 2 (B R \Kε) = δ 0 . Let us fix 0 < ε < ε := min{ε 0 , ε δ 0 } where ε 0 is as in Proposition 3.1. Further, we denote by B δ 0 (w 0 ) the following set:

B δ 0 (w 0 ) := w ∈ L 2 (B R \ K ε ) ; w -w 0 L 2 (B R \Kε) δ 0 ,
and we define m := inf

w∈B δ 0 (w 0 ) E ε,R (w). Note that m is well-defined since E ε,R is a non-negative continuous functional in L 2 (B R \ K ε ).
Using Lemma 4.5, we will show that there is a local minimum v ε,R of the energy E ε,R in the ball B δ 0 (w 0 ) which is also a solution to (4.8). However, it must be noted that E ε,R lacks of strong compactness properties and passing to the limit along a subsequence is not straightforward. So let us first show that m is achieved in B δ 0 (w 0 ). Take a minimising sequence (v j ) j∈N ⊂ B δ 0 (w 0 ). Notice that |w| ∈ B δ 0 (w 0 ) for all w ∈ B δ 0 (w 0 ). Moreover, a straightforward computation shows that E ε,R (|v j |) E ε,R (v j ) for all j 0. Thus, we may assume that the v j 's are a.e. non-negative for every j 0. By (4.24), we have -G ε (t) κ 1 ε 2 t 2 for all t ∈ R. In particular, E ε,R (v j ) κ 1 ε 2 v j K ε ). Whence, up to extract a subsequence, we obtain that v j converges weakly in

L 2 (B R \ K ε ) to some v ε,R ∈ B δ 0 (w 0 ) (notice that B δ 0 (w 0 ) is closed in L 2 (B R \ K ε )).
Let us check that v ε,R is indeed a minimiser of E ε,R in B δ 0 (w 0 ). To this end, we shall introduce the following notations

J ε (x) := ˆRN \Kε J ε (x -y)dy and H ε (x, s) := ˆs 0 J ε (x)τ -g ε (τ ) dτ.
Since 0 < ε < ε 0 , by Proposition 3.1, we have max

[0,1] f ε < inf R N \Kε J ε .
Therefore, from the construction of g ε (remember (4.4)), we have

g ε (s) = ε 2 f (1 -s) max R f ε max [0,1] f ε < inf R N \Kε J ε for any s ∈ R. (4.30)
Whence, H ε (x, •) is convex for each fixed x. Developing the terms involved in the definition of E ε,R we arrive at

E ε,R (w) = - 1 2 ˆBR \Kε ˆBR \Kε J ε (x -y)w(x)w(y)dxdy + ˆBR \Kε H ε (x, w(x))dx.
Using the weak convergence of (v j ) j∈N towards v ε,R and the dominated convergence theorem, we can pass to the limit in the double integral and get that

lim j→+∞ ˆBR \Kε ˆBR \Kε J ε (x -y)v j (x)v j (y)dxdy = ˆBR \Kε ˆBR \Kε J ε (x -y)v ε,R (x)v ε,R (y)dxdy. Moreover, since H ε (x, •) is convex, we have ˆBR \Kε H ε (x, v j (x))-H ε (x, v ε,R (x)) dx ˆBR \Kε ∂ s H ε (x, v ε,R (x))(v j (x)-v ε,R (x))dx.
From the definition of H ε , g ε and from (4.30) a quick computation shows that

|∂ s H ε (x, s)| = |J ε (x)s -g ε (s)| A|s| for all s ∈ R and some constant A > 0. Since v ε,R ∈ L 2 (B R \ K ε ), it follows that ∂ s H ε (•, v ε,R (•)) ∈ L 2 (B R \ K ε ).
Therefore, using the previous two displayed formulas and the weak convergence of v j towards v ε,R , we obtain lim j→∞

[E ε,R (v j ) -E ε,R (v ε,R )] 0. Since, on the other hand, lim j→∞ E ε,R (v j ) = m E ε,R (v ε,R ), we finally obtain E ε,R (v ε,R ) = m = inf w∈B δ 0 (w 0 ) E ε,R (w) E ε,R (w 0 ).
(4.2). Let us briefly explain our strategy. Since, by construction, v ε,R satisfies (4.8), the function u ε,R = 1 -v ε,R verifies (4.6) and, as already noted above, extending the function u ε,R by 1 outside B R , we obtain a (discontinuous) super-solution to (4.1) that satisfies (4.2). The aim of this section is to find the right extension of u ε,R that provides the desired super-solution.

To do so, we first introduce some useful notations. Given R > 0 and x ∈ R N , we let P R (x) be the projection of x to the ball B R , that is

P R (x) ∈ B R and |x -P R (x)| = dist(x, B R ) = min y∈B R |x -y|. For σ > 0, we let u ε,σ ∈ C(R N \ K ε ) be the following function u ε,σ (x) := min u ε,R (P R (x)) + σ -1 |x -P R (x)|, 1 . (4.31)
We shall see that, for well-chosen σ, the function u ε,σ will satisfy

L ε u ε,σ (x) + f ε (u ε,σ (x)) 0 for all x ∈ R N \ K ε , (4.32)
where L ε is the nonlocal operator given by

L ε w(x) := ˆRN \Kε J ε (x -y)(w(y) -w(x))dy. (4.33)
Namely, we claim Claim 4.7. -There exists σ ε > 0 such that u ε,σ satisfies (4.32) for all σ < σ ε .

Observe that by proving Claim 4.7, we end the proof of Lemma 4.1. Indeed, by construction, we have f f so that u ε,σ trivially satisfies (4.1). As for condition (4.2) it is also satisfied (by construction of u ε,σ ) provided that R is taken sufficiently large.

Proof. -Define A R := R N \ B R . As in the previous section, we set

J ε (x) = ˆRN \Kε J(x -y)dy and c ε (x) = ˆRN \B R J ε (x -y)dy.
Then, in view of (4.31), we have

L ε u ε,σ (x) + f ε (u ε,σ (x)) ˆBR \Kε J ε (x -y)(u ε,R (y) -u ε,σ (x))dy + c ε (x)(1 -u ε,σ (x)) + f (u ε,σ (x)). ( 4

.34)

Since u ε,σ (x) = u ε,R (x) for all x ∈ B R \ K ε , using (4.6) we easily get that

L ε u ε,σ (x) + f ε (u ε,σ (x)) 0 for x ∈ B R \ K ε . (4.35)
To complete the proof, it remains to show that u ε,σ satisfies (4.32) in the set A R . We shall consider two sub-domains, Π + and Π -, defined as follows

Π -:= A R ∩ u ε,σ < 1 , Π + := A R ∩ u ε,σ = 1 .
Note that since u ε,σ (x) = 1 for all x ∈ Π + , it follows directly from (4.34) that

L ε u ε,σ (x) + f ε (u ε,σ (x)) = ˆRN \Kε J ε (x -y)(u ε,R (y) -1)dy 0, (4.36)
for any x ∈ Π + . Thus, to conclude the proof we need only to check that (4.36) still holds in Π -. To this end, for any x ∈ Π -and any s ∈ [0, 1], we set

g R (x, s) := J ε (P R (x))s -f ε (s). (4.37)
Now, since 0 < ε < ε 0 , it follows from Proposition 3.1 that there exists a γ > 0 such that inf

z∈R N \Kε min s∈[0,1] ∂ s g R (z, s) > γ. (4.38)
Next, since J ∈ W 1,1 (R N ) (by (1.6)) we may set

σ ε := εγ • ˆRN |∇J(z)|dz -1 > 0. ( 4

.39)

Let us also set

s(x) := u ε,R (P R (x)) and τ (x) := dist(x, B R ) = |x -P R (x)| > 0. Then, u ε,σ rewrites u ε,σ (x) = s(x) + σ -1 τ (x) and 0 < s(x) + σ -1 τ (x) < 1 for any x ∈ Π -. (4.40)
Now, using (4.31) and the definition of L ε , we can rewrite L ε u ε,σ (x) as

L ε u ε,σ (x) = L ε u ε,σ (P R (x)) + ˆRN \Kε [J ε (x-y)-J ε (P R (x)-y)](u ε,σ (y)-u ε,σ (x))dy - τ (x) σ ˆRN \Kε J ε (P R (x) -y)dy. Since P R (x) ∈ B R \ K ε , since J ∈ W 1,1 (R N
) and since J 0 a.e. in R N , by (4.35) we obtain

L ε u ε,σ (x) - τ (x) σ J ε (P R (x)) -f ε (s(x)) + ˆRN \Kε |J ε (x -y) -J ε (P R (x) -y)| dy. - τ (x) σ J ε (P R (x)) -f ε (s(x)) + ˆRN |J ε (x -y) -J ε (P R (x) -y)| dy. - τ (x) σ J ε (P R (x)) -f ε (s(x)) + τ (x) ε ˆRN |∇J(z)|dz.
Therefore, we get

L ε u ε,σ (x) + f ε (s(x) + σ -1 τ (x)) f ε (s(x) + σ -1 τ (x)) -f ε (s(x)) - τ (x) σ J ε (P R (x)) + τ (x) ε ˆRN |∇J(z)|dz.
By adding and subtracting s(x)J ε (P R (x)) on the right hand side of the above inequality and recalling (4.37), we obtain

L ε u ε,σ (x) + f ε (u ε,σ (x)) g R (x, s(x)) -g R x, s(x) + σ -1 τ (x) + γ σ -1 ε τ (x).
where we have used (4.39). By (4.38), (4.40) and the mean value theorem, we deduce that there exists some

ξ ∈ s(x), s(x) + σ -1 τ (x) ⊂ [0, 1], such that g R (x, s(x)) -g R x, s(x) + σ -1 τ (x) = -∂ s g R (x, ξ)σ -1 τ (x) -γ σ -1 τ (x).
Therefore, for every 0 < σ < σ ε , we obtain that

L ε u ε,σ (x) + f ε (u ε,σ (x)) γ τ (x) 1 σ ε - 1 σ < 0 for any x ∈ Π -.
The proof of Claim 4.7 is thereby complete.

Remark 4.8. -An analogue version of Lemma 4.1 holds when J ε (x-y) is replaced by J ε (d g (x, y)) where J ε is a locally integrable function such that J ε (|z|) = J ε (z) and d g (x, y) is the geodesic distance on R N \ K ε . Indeed, the only places where the structure of the radial kernel J ε came into place is when we used the Poincarétype inequality [START_REF] Ponce | An estimate in the spirit of Poincaré's inequality[END_REF]Theorem 1.1] in Proposition 4.5, when we asserted that the solutions to (4.8) satisfying max [0,1] f ε < inf B R \Kε J ε are continuous and when we Remark 5.4. -The same arguments also apply when the operator L is replaced by L g provided that J = J(|•|) satisfies (1.8), since it still holds that if w(x) → ∈ R as |x| → ∞, then L g w(x) → 0 as |x| → ∞. Moreover, the continuity of w still implies the continuity of L g w but the proof is less obvious since one can no longer rely on the continuity of translations in L 1 (R N ). For the sake of completeness, we state a last lemma below which justifies why this is true.

Lemma 5.5. -Let K ⊂ R N be a compact set and assume that J satisfies (1.12) and that J is supported in the segment line [0, r] for some r > 0.

Let w ∈ C(R N \K). Then, L g w ∈ C(R N \ K).
Proof. -Let x 1 , x 2 ∈ R N \ K with x 1 fixed and x 2 arbitrarily close to x 1 . For w ∈ C(R N \ K), the analogue of (5.10) is here:

|L g w(x 1 ) -L g w(x 2 )| 2 w ∞ ˆRN [ J(d g (x 1 , y)) -J(d g (x 2 , y))]dy + J ∞ |w(x 1 ) -w(x 2 )|,
where J is as in (2.7). Since w ∈ C(R N \ K), the delicate part is to show that the first term on the right-hand side vanishes as x 2 → x 1 . This can be done as follows.

Let δ > 0 be small enough so that

x 2 ∈ B δ/2 (x 1 ) ⊂ B δ (x 1 ) ⊂ R N \ K. Then, we may write ˆRN \K [ J(d g (x 1 , y)) -J(d g (x 2 , y))]dy ˆRN \(B δ (x 1 )∪K) | J(d g (x 1 , y)) -J(d g (x 2 , y))|dy + ˆBδ(x 1 ) | J(d g (x 1 , y)) -J(d g (x 2 , y))|dy =: I 1 (x 1 , x 2 ) + I 2 (x 1 , x 2 ).
Since d g (x i , y) = |x i -y| for any i ∈ {1, 2} and y ∈ B δ (x 1 ), we have

I 2 (x 1 , x 2 ) J(• + x 1 -x 2 ) -J L 1 (R N ) -→ x 2 →x 1 0.
On the other hand, since J is radially symmetric, supp(J) = B r and J ∈ W 1,1 (B r ), by [71, Theorems 1.1 and 2.3], we have that J ∈ W 1,1 ((0, r), t N -1 ), J is almost everywhere equal to a continuous function, J exists almost everywhere and

ˆr 0 | J (t)|t N -1 dt C 1 ˆBr |∇J(z)|dz. (5.19)
Therefore, using the fact that d g (x i , y) |x i -y| δ/2 for any y ∈ R N \(B δ (x 1 )∪K), we have

I 1 (x 1 , x 2 ) ˆRN \(B δ (x 1 )∪K) ˆdg(x1,y) dg(x 2 ,y) | J (t)|dtdy 2 δ N -1 ˆRN \(B δ (x 1 )∪K) ˆdg(x1,y) dg(x 2 ,y) | J (t)|t N -1 dtdy. (5.20) Now, since x 1 , x 2 ∈ B δ/2 (x 1 ) ⊂ R N \ K and d g (•, •) is a distance, we have |d g (x 1 , y) -d g (x 2 , y)| d g (x 1 , x 2 ) = |x 1 -x 2 | -→ x 2 →x 1 0.
Therefore, using (5.19), (5.20) and the dominated convergence theorem, we obtain that

I 1 (x 1 , x 2 ) → 0 as x 2 → x 1 .
This completes the proof.

In particular, in [START_REF] Berestycki | On phase-separation models: asymptotics and qualitative properties[END_REF], the authors establish the existence, symmetry and nondegeneracy of solutions to (1.2) in one dimension. They show that entire solutions are reflectionally symmetric, i.e. there exists x 0 ∈ R such that u(x -x 0 ) = v(x 0 -x) for any x ∈ R. They also establish a result which may be seen as the analogue of a celebrated conjecture of De Giorgi for problem (1.2) in dimension 2. More precisely, they show that monotone solutions to (1.2) in dimension 2 have one-dimensional symmetry under the growth condition

u(x) + v(x) C(1 + |x|),
for some C > 0. On the other hand, the linear growth is the lowest possible for positive solutions to (1.2). Namely, if there exists α ∈ (0, 1) such that

u(x) + v(x) C(1 + |x|) α ,
then both u and v are constants, and at least one of them is 0, see [START_REF] Noris | Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition[END_REF].

In dimension 2, Farina [START_REF] Farina | Some symmetry results for entire solutions of an elliptic system arising in phase separation[END_REF] improves the result in [START_REF] Berestycki | On phase-separation models: asymptotics and qualitative properties[END_REF] showing that if (u, v) is a monotone solution to (1.2) and has at most algebraic growth at infinity, then it must be one-dimensional. It turns out that the monotonicity condition can be weakened in order to get the one-dimensional symmetry for solutions to (1.2). Indeed, it has been proved in [START_REF] Berestycki | On entire solutions of an elliptic system modeling phase separations[END_REF] that stable solutions to (1.2) are one-dimensional in R 2 .

Let us also mention [START_REF] Fazly | De Giorgi type results for elliptic systems[END_REF] where symmetry results for systems in dimension 2 involving more general nonlinearities have been obtained. Quasilinear (possibly degenerate) elliptic systems in R 2 have also been considered in [START_REF] Dipierro | Geometric inequalities and symmetry results for elliptic systems[END_REF]. As for dimensions higher than 2, in [START_REF] Farina | Monotonicity and 1-dimensional symmetry for solutions of an elliptic system arising in Bose-Einstein condensation[END_REF], the authors prove that if (u, v) has algebraic growth and lim

x N →±∞ u(x , x N ) -v(x , x N ) = ±∞ uniformly in x ∈ R N -1 ,
then (u, v) depends on x N only. Also, in [START_REF] Wang | On the De Giorgi type conjecture for an elliptic system modeling phase separation[END_REF][START_REF]Harmonic approximation and improvement of flatness in a singular perturbation problem[END_REF], it has been proved that if (u, v) has linear growth, then it is one-dimensional.

Recently, the nonlocal counterpart of (1.2) has also been investigated, see e.g. [START_REF] Terracini | Uniform Hölder regularity with small exponent in competition-fractional diffusion systems[END_REF][START_REF]Uniform Hölder bounds for strongly competing systems involving the square root of the Laplacian[END_REF][START_REF] Verzini | Strong competition versus fractional diffusion: the case of Lotka-Volterra interaction[END_REF][START_REF] Wang | On the uniqueness of solutions of a nonlocal elliptic system[END_REF], and some symmetry results have been obtained in [START_REF] Dipierro | A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian[END_REF] for a quite general system of nonlocal PDEs of the form

(-∆) s 1 u = F 1 (u, v), (-∆) s 2 v = F 2 (u, v),
where F 1 and F 2 denote the derivatives of a function F ∈ C 1,1 loc (R 2 ) with respect to the first and the second variable, respectively; s 1 , s 2 ∈ (0, 1) and, given s ∈ (0, 1), (-∆) s stands for the fractional Laplace operator (-∆) s u(x) := p.v.

ˆRN

u(x) -u(y) |x -y| N +2s dy, where p.v. denotes the Cauchy principal value (see [START_REF] Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF] for the definition and further details). See also [START_REF] Fazly | Symmetry results for fractional elliptic systems and related problems[END_REF] for symmetry results for nonlocal systems of equations.

Our aim in this chapter is to provide monotonicity results for minima and stable solutions of energy functionals of the form (1.1) (thus embracing all the systems considered above). We will consider the case in which both the domain Ω and the functional F are invariant under translations in the e N -direction, and we will deal with symmetry properties of minima or stable solutions in the class of functions which are obtained by piecewise Lipschitz domain deformations in the e N -direction (see Definitions 1.1 and 1.2). We will also require some rather mild assumptions on F and a growth condition (see (1.5)).

The key tool of our proofs relies on a technique introduced in [START_REF] Savin | Regularity of nonlocal minimal cones in dimension 2[END_REF] to study the regularity of fractional minimal surfaces in dimension 2 (see [START_REF] Caffarelli | Nonlocal minimal surfaces[END_REF] where these object were introduced), and developed in [START_REF]Some monotonicity results for minimizers in the calculus of variations[END_REF] to work in a more general setting. See also [START_REF] Caffarelli | Minimization of a fractional perimeter-Dirichlet integral functional[END_REF][START_REF] Dipierro | A nonlocal free boundary problem[END_REF] where these techniques have been used in the context of free boundary problems.

Here we adapt the new strategy of [START_REF]Some monotonicity results for minimizers in the calculus of variations[END_REF] to the case of general systems of equations. The idea is to look at the stability inequality without dealing with its precise form (which, in some cases, can be very hard to handle), and to simply compare the energy of the couple (u, v) to that of its translations. To this end, one also needs to modify the "translated" couple at infinity to make it a compact perturbation of (u, v). Here the growth condition comes into play and ensures that the energy of the perturbed couple can be made arbitrarily close to the energy of (u, v). Then, if u and v are not monotone in the e N -direction, one can modify locally the perturbed couple in order to get lower energy, but this is in contradiction with the minimality of (u, v). This strategy allows to deal with quite general form of energy functionals.

We now introduce the setting in which we will work and give precise definitions and statements of our results.

The mathematical setting

We consider a domain Ω ⊂ R N and study the symmetry properties of minima for functionals of the type

E (u, v) := ˆΩ F (∇u, ∇v, u, v, x)dx.
We assume that the domain Ω and the functional F are invariant under translations in the e N -direction, namely

Ω = V × R, V ⊆ R N -1 ,
and F does not depend on the x N -coordinate. We will denote a point in Ω by x = (x , x N ) ∈ V × R. We also suppose that F is convex with respect to the first two variables. Precisely, we assume that

F = F (p 1 , p 2 , z 1 , z 2 , x ) ∈ C(R 2N × R 2 × V ), (1.3)
and F is C 2 and uniformly convex in p 1 and p 2 at all p 1 , p 2 with p 1 • e N = 0 and p 2 • e N = 0, for all (z 1 , z 2 , x ) ∈ R 2 × V .

Finally, we assume that there exists a constant C > 0 such that

|F p 1 p 1 (p 1 + q 1 , p 2 , z 1 , z 2 , x )| C|F p 1 p 1 (p 1 , p 2 , z 1 , z 2 , x )|, |F p 2 p 2 (p 1 , p 2 + q 2 , z 1 , z 2 , x )| C|F p 2 p 2 (p 1 , p 2 , z 1 , z 2 , x )|, (1.4) |F p 1 p 2 (p 1 + q 1 , p 2 + q 2 , z 1 , z 2 , x )| C|F p 1 p 2 (p 1 , p 2 , z 1 , z 2 , x )|, for any p 1 , p 2 , q 1 , q 2 ∈ R N with |q 1 | |p 1 • e N |/4 and |q 2 | |p 2 • e N |/
4. We will make these assumptions throughout the chapter.

Given R > 0, we consider the following energy functional

E R (u, v) := ˆΩ∩B R F (∇u(x), ∇v(x), u(x), v(x), x )dx,
where, as usual, B R denotes the ball of radius R centered at the origin. Following the line in [START_REF]Some monotonicity results for minimizers in the calculus of variations[END_REF], we want to study the symmetry properties of minimal or stable solutions of the above functional among suitable perturbations, obtained by domain deformations in the direction given by e N . To do this, we recall the following definitions introduced in [START_REF]Some monotonicity results for minimizers in the calculus of variations[END_REF].

where ϕ ∈ C 0,1 (R N ) is a function supported in B R with ∂ N ϕ L ∞ (R N ) < 1. Definition 1.2. -Given a function w ∈ C 0,1 (Ω), a piecewise e N -Lipschitz deformation of w in B R is a function w ∈ C 0,1 (Ω) defined by w(x) = w (i) (x) for some i (depending on x ∈ Ω),
where w (1) , . . . , w (m) constitute a finite number of e N -Lipschitz deformations of w in B R . In this case, we write w ∈ D R (w).

Also, if all w (i) satisfy

w (i) (x) = w(x + ϕ (i) (x)e N ) with ϕ (i) C 0,1 (Ω)
δ for some δ, we write w ∈ D δ R (w).

A monotonicity result for stable solutions

The second result that we state concerns stable critical points of the energy instead of e N -minimizers. We give a definition of the stability condition that involves the second variation of the functional E for deformations of the solution both in the e Ndirection and in the vertical e N +1 -direction. Precisely, we give the following:

Definition 1.7. -Given a function w, a piecewise Lipschitz deformation of w in the {e N , e N +1 }-directions is a function w defined as

w = w + φ,
where w ∈ D δ R (w) and φ is a Lipschitz function with compact support in Ω ∩ B R , and φ C 0,1 (Ω) δ. In this case, we write w ∈ D δ R (w).

Definition 1.8. -We say that (u, v) is an {e N , e N +1 }-stable solution of E if for every R > 0 and ε > 0 there exists δ > 0 depending on R, ε, u and v such that, for any t ∈ (0, δ), we have that E R (u, v) is finite and

E R ( u, v) -E R (u, v) -εt 2 , (1.6)
for any u ∈ D t R (u) and any v ∈ D t R (v). Remark 1.9. -As we shall see later on (see Lemma 2.1), this notion of {e N , e N +1 }stable solution shares intimate links with the classical notion of stability. In fact, this follows from the same arguments as in the case of a single equation (see [START_REF]Some monotonicity results for minimizers in the calculus of variations[END_REF]). Actually, we can infer directly from the definition that (i) if (u, v) is a classical minimizer of E , then (u, v) is {e N , e N +1 }-stable for E ; (ii) if (u, v) is {e N , e N +1 }-stable for E , then (u, v) is a critical point for E . This is because we allow perturbation in the e N +1 -direction.

Next, we state our monotonicity result for {e N , e N +1 }-stable solutions.

Theorem 1.10. -Let F ∈ C 3,α (R 2N × R 2 × V ) satisfy (1.4) and let (u, v) be such that either u, v ∈ C 0,1 (Ω) are convex or u, v ∈ C 1,1 (Ω). Moreover, suppose that (u, v) is an {e N , e N +1 }-stable solution of E , and that the growth condition (1.5) holds true. Then, u and v are monotone in the e N -direction, i.e. either u N 0 or u N 0 and either v N 0 or v N 0 in Ω. By Theorem 1.10 and Remark 1.6 one obtains that an {e N , e N +1 }-stable solution of E is one-dimensional, in the sense that there exist u, v : R → R and ω u , ω v ∈ S N -1 such that (u(x), v(x)) = u(ω u • x), v(ω v • x) . We remark that, in general, it is not possible to conclude that u and v have the same direction of monotonicity. It is not true, for instance, for an uncoupled system of PDEs, such as ∆u = 0, ∆v = 0.

See also [START_REF] Dipierro | Geometric inequalities and symmetry results for elliptic systems[END_REF]Remark 1.4] for a discussion on this fact.

Nonetheless, there are cases in which it is possible to obtain that u and v have the same direction of monotonicity, even for quite general systems, see e.g. [START_REF] Dipierro | Geometric inequalities and symmetry results for elliptic systems[END_REF] and [START_REF] Dipierro | A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian[END_REF].

Organization of the chapter

In Section 2 we show that, under suitable assumptions, the notion of {e N , e N +1 }stability is equivalent to the notion of classical stability. In Section 3 we perform a local analysis and, in Section 4, we estimate the energy of the perturbation at infinity, collecting the basic facts in order to prove Theorems 1.5 and 1.10 in Sections 5 and 6, respectively. Finally, in Section 7, we investigate some applications of our results in some concrete cases.

Stability property

The following lemma justifies Remark 1.9. Proof. -Suppose first that (u, v), with u, v ∈ C 2 (R N ), is a classical stable solution. That is, by definition, for any ϕ 1 , ϕ 2 ∈ C 0,1 0 (B R ), we have

lim inf t→0 E R (u + tϕ 1 , v + tϕ 2 ) -E R (u, v) t 2 0. ( 2
Given u ∈ D t R (u) and v ∈ D t R (v) we may choose ϕ 1 := u-u t and ϕ 2 := v-v t . By construction, the Lipschitz norm of ϕ 1 and ϕ 2 is bounded by a quantity that does not depend on t. Therefore (2.5) implies (1.6), and so it follows that (u, v) is {e N , e N +1 }stable.

Reciprocally, let (u, v) be an {e N , e N +1 }-stable solution of E (and so, a critical point). In this case, we choose u := u + tϕ 1 and v := v + tϕ 2 in (1.6). Then, taking ε arbitrarily small, we prove (2.1). Therefore (u, v) is a stable solution of E . This completes the proof.

Local analysis

In this section, we consider local perturbations of (u, v) in the e N -direction. More precisely, we will show that we can perturb the couples max{u(x), u(x) + te N }, v(x) and u(x), max{v(x), v(x) + te N } in such a way that the energy of the "perturbed couples" decreases.

We first show that if one of the two elements of a couple (u, v) is a maximum of two functions that form an angle at an intersection point, then it cannot be an e N -minimizer for E . Now, we set f 0 := max {u 0 , u 1 0 } and we consider the couple g 0 := (f 0 , v 0 ). The fact that F is strictly convex in the variable p 1 implies that g 0 is not a minimizer for F 0 .

(3.4)

To see this, given the function β R (x ) := c δ max {0, |x | -R} , with R large and c, δ small, we define

h 0 := 1 + α u 0 + (1 -α)u 1 0 -β R (x ), (3.5)
where α ∈ (0, 1) is small. Then, we have that the couple max {f 0 , h 0 } , v 0 agrees with g 0 outside the ball B R+R 1 .

(3.6)

On the other hand, in B R , considering h 0 , we cut the graphs of two transversal linear functions by a single one. Therefore, if we take R sufficiently large, the couple max {f 0 , h 0 } , v 0 has lower energy for F 0 than g 0 .

To see this, we observe that, by definition, ∇f 0 coincides with either ∇u 0 or ∇u 1 0 , and so from (3.2) we have that F 0 (∇u 0 , ∇v 0 ) = F 0 (∇u 1 0 , ∇v 0 ) = F 0 (∇f 0 , ∇v 0 ).

Therefore, by the strict convexity of F 0 with respect to the first variable, we have that, fixed α and δ, F 0 (∇h 0 , ∇v 0 ) = F 0 (α∇u 0 + (1 -α)∇u 1 0 , ∇v 0 αF 0 (∇u 0 , ∇v 0 ) + (1 -α)F 0 (∇u 1 0 , ∇v 0 ) -η = F 0 (∇f 0 , ∇v 0 ) -η, (3.8) in B R ∩ {h 0 > f 0 }, for some η > 0. On the other hand, the set {h 0 > f 0 } is contained in a strip in the e N -direction. This and (3.6) imply that, when we integrate (3.8) in B R , then the energy of (max {f 0 , h 0 } , v 0 ) is less than the energy of g 0 minus a term of order CηR N -1 , whereas, when we integrate in B R+R 1 , the energy of (max {f 0 , h 0 } , v 0 ) is less than the energy of g 0 minus a term of order CηR N -2 . All in all, if R is large enough we obtain (3.7). In turn, this implies (3.4). Now, we set f r := max u r , u 1 r .

(3.9)

Since the convergence in (3.3) is uniform, we have that h r := max {f r , h 0 } , v r , has lower energy for F r than g r as well. Hence, we can scale back and obtain that h * (x) := rh r (x/r) has lower energy for F in B r(R+R 1 ) ⊆ B η than g 1 .

In order to get a contradiction, it remains to prove that h * is an allowed perturbation of g 1 (according to Definition 1.2). Notice that this is equivalent to check that h r is an allowed perturbation of g r . That is, recalling (3.9), we have to prove that max {f r , h 0 } is a piecewise Lipschitz domain deformation of f r with Lipschitz norm bounded by δ.

(3.10)

To do this, we recall the uniform convergence of u r and u 1 r to u 0 and u 1 0 respectively (as given by (3.3)) and the definition of h 0 given in (3.5) to obtain that h 0 (x) = 1 + α u r (x) + (1 -α)u 1 r (x) -β R (x ) + ω r (x), (3.11) where ω r → 0 as r → 0 + locally uniformly, together with its derivatives. Now we notice that our hypothesis in (3.1) gives that ∇u 1 0 • e N < 0 < ∇u 0 • e N . This, together with the uniform convergence in (3.3), implies that we can apply the Implicit Function Theorem, and we have that the part of the graph of max {f r , h 0 } where h 0 > f r can be obtained from u r by a Lipschitz domain deformation with Lipschitz norm less than δ, provided that we take α sufficiently small. Indeed, fixed x (and so looking at the 1-dimensional problem in the last variable only) and recalling (3.11), we obtain that u r (x , x N + ϕ(x)) = h 0 (x) = 1 + α u r (x) + (1 -α)u 1 r (x) -β R (x ) + ω r (x), thanks to the Implicit Function Theorem in 1-dimension. Furthermore, if α and r are sufficiently small, the perturbation function ϕ has norm bounded by δ/2. This shows (3.10) and finishes the proof of Lemma 3.1. Now we deal with perturbations of the couples max{u(x), u(x) + te N }, v(x) and u(x), max{v(x), v(x) + te N } with lower energy.

Lemma 3.2. -Let u, v ∈ C 2 (Ω) be such that (u, v) is a critical point for the functional E in a neighborhood of the origin, and let F be C 2 in a neighborhood of (∇u(0), ∇v(0), u(0), v(0), 0).

Suppose that

u N (0) = 0, ∇u N (0) = 0, and set w 1 (x) := max {u(x), u(x + te N )} .

(3.12)

Then, for every η > 0, there exists a function ψ 1 which is Lipschitz and has compact support in B η such that, for any small t,

E η (w 1 + tψ 1 , v) -E η (w 1 , v) -c t 2 ,
where c > 0 is a small constant that depends on F , η and u.

Remark 3.3. -We point out that we have an analogous result if we consider the function v instead of u. Precisely, if we assume that v N (0) = 0, ∇v N (0) = 0, and we consider w 2 (x) := max {v(x), v(x + te N )} , then, for every η > 0 there exists a function ψ 2 which is Lipschitz and has compact support in B η such that, for any small t,

E η (u, w 2 + tψ 2 ) -E η (u, w 2 ) -ct 2 ,
where c > 0 is a small constant depending on F, η and v.

Proof of Lemma 3. We consider a Lipschitz function g 1 and, using the fact that F ∈ C 2 in the variables p 1 and z 1 , we compute F (∇u + t∇g 1 , ∇v, u + tg 1 , v, x ) = F (∇u, ∇v, u, v, x ) + t (F p 1 ∇g 1 + F z 1 g 1 )

+ t 2 (∇g 1 ) T F p 1 p 1 ∇g 1 + F z 1 z 1 g 2 1 + 2g 1 F p 1 z 1 • ∇g 1 + o(t 2 ),
where the derivatives of F are evaluated at (∇u, ∇v, u, v, x ) and the constant in the error term o(t 2 ) depends on u, F and g 1 C 0,1 (Bη) . Hence, we obtain E η (u + tg 1 , v) = E η (u, v) + t L(g 1 ) + t 2 Q(g 1 ) + o(t 2 ), Now, we notice that u N , 0 and G satisfy the hypotheses of Remark 4.3 in [START_REF]Some monotonicity results for minimizers in the calculus of variations[END_REF], and therefore (u N ) + is not a minimizer of Q. So we can choose the function ψ 1 in such a way that Q((u N ) + + ψ 1 ) Q((u N ) + ) -c for some small c, which may depend on u, F and η. This together with (3.19) gives that, for small t, E η (w 1 + tψ 1 , v) -E η (w 1 , v) -ct 2 and this concludes the proof. Now we show that, under an additional regularity hypothesis on F , the nondegeneracy condition ∇u N = 0 in Lemma 3.2 is satisfied. For this we use the Hopf Lemma, by adapting the proof of Lemma 4.6 in [START_REF]Some monotonicity results for minimizers in the calculus of variations[END_REF] to the slightly more delicate case of a system of equations. Furthermore, assume that (u, v) is a critical point for E and F ∈ C 3,α in a neighborhood of (∇u(0), ∇v(0), u(0), v(0), 0). Then, u and v are of class C 3,α in a neighborhood of 0. If, in addition, u N (0) = 0 (resp. v N (0) = 0) and u N (resp. v N ) does not vanish identically in a neighborhood V 0 of 0, then there exists a point x 0 ∈ V 0 (resp. x 1 ∈ V 0 ) such that u N (x 0 ) = 0, ∇u N (x 0 ) = 0 and v N (x 1 ) = 0, ∇v N (x 1 ) = 0.

Proof. -Since (u, v) is a critical point for E it satisfies the elliptic system of equation G 1 (∇ 2 u, ∇ 2 v, ∇u, ∇v, u, v, x ) := div x F p 1 (M ) -F z 1 (M ) = 0, G 2 (∇ 2 u, ∇ 2 v, ∇u, ∇v, u, v, x ) := div x F p 2 (M ) -F z 2 (M ) = 0, where M := (∇u, ∇v, u, v, x ). Consider the first equation of the above system, with v fixed. If v ∈ C 0,1 (Ω) is convex, then ∇v ∈ L ∞ (Ω) N and ∇ 2 v ∈ L ∞ (Ω) N ×N exist almost everywhere (this follows from Rademacher's theorem and Alexandrov's theorem). On the other hand, if v ∈ C 1,1 (Ω), then ∇ 2 v ∈ L ∞ (Ω) N ×N also exists almost everywhere (this follows from Rademacher's theorem). Thus, at fixed v, the first equation (in the variable u = u v ) is satisfied in the classical sense and the corresponding solution u v belongs to C 3,α (Ω) (this follows from Theorem 2.1 in [START_REF] Trudinger | Hölder gradient estimates for fully nonlinear elliptic equations[END_REF], the Schauder estimates and the fact that F ∈ C 3,α ). In like manner, for any fixed u, the solution v u of the second1 equation belongs to C 3,α (Ω). Therefore, (u, v) ∈ C 3,α (Ω) × C 3,α (Ω). This shows the first claim of Lemma 3.4. Now, we focus on the second claim of Lemma 3.4. To prove it, we observe that G 1 = G 1 (q 1 , q 2 , p 1 , p 2 , z 1 , z 2 , x ) ∈ C 1,α .

(3.20)

Hence, differentiating in the e N -direction we see that w := u N ∈ C2,α (Ω) satisfies the linearized equation (in the viscosity sense)

G 1 (q 1 ) ij w ij + G 1 (p 1 ) i w i + G 1 z 1 w = -G 1 (q 2 ) ij (v N ) ij -G 1 (p 2 ) i (v N ) i -G 1 z 2 v N . (3.21)
Here, the derivatives of G 1 are evaluated at (∇ 2 u, ∇ 2 v, ∇u, ∇v, u, v, x ). For the convenience of the reader, we rewrite (3.21) as

L 1 w = f 1 ,
where

L 1 w := G 1 (q 1 ) ij w ij + G 1 (p 1 ) i w i + G 1 z 1 w, f 1 := -G 1 (q 2 ) ij (v N ) ij -G 1 (p 2 ) i (v N ) i -G 1 z 2 v N .
Moreover, we set P := {x ∈ V 0 : f 1 (x) > 0}, E := {x ∈ V 0 : f 1 (x) = 0}, N := {x ∈ V 0 : f 1 (x) < 0}.

In addition, denote by (P j ) j∈J 1 (resp. (E j ) j∈J 2 ) the connected components of P (resp. E). This makes sense since f 1 ∈ C 0,α (V 0 ) (recall (3.20) and the fact that v is C 3,α in V 0 ).

Note that if P = N = ∅ then f 1 ≡ 0, and so the conclusion follows as in Lemma 4.6 in [START_REF]Some monotonicity results for minimizers in the calculus of variations[END_REF]. Thus, we may suppose now that either P or N are nonempty. Moreover, we can assume that P is nonempty, otherwise it is enough to replace w by -w.

Also, we observe that if there exists a compact set Σ V 0 such that w |Σ ≡ 0, then, necessarily, Σ ⊂ E.

(3.22)

Now we take 2 x 0 ∈ {w = 0} ∩ P . Then, there exists j 1 ∈ J 1 such that x 0 ∈ P j 1 . It follows from (3.22) and the continuity of f 1 that w cannot vanish identically in P j 1 .

If we consider u - R,t and v - R,t , we can write the same inequality with A replaced by -A. Therefore, we obtain We conclude this section recalling the following integral formulas: We recall the notation introduced at the beginning of Section 4, and we observe that, since (u, v) is an e N -minimizer,

E R (u + R,t , v + R,t ) + E R (u - R,t , v - R,t ) -2E R (u, v) C ˆΩ∩B R |F p 1 p 1 | |∇u|
E R max u - R,t , u , v + E R min u - R,t , u , v = E R (u - R,t , v) + E R (u, v), E R u, max v - R,t , v + E R u, min v - R,t , v = E R (u, v - R,t ) + E R (u, v).
E R (u, v) E R (u + R,t , v).
From this and (4.6) we obtain that lim

R→+∞ E R (u - R,t , v) -E R (u, v) = 0, (5.1)
at t fixed. Using again the minimality of (u, v), we have that

E R (u, v) E R min u - R,t , u , v ,
which, together with the first relation in (4.8), implies that

E R max u - R,t , u , v -E R (u - R,t , v) = E R (u, v) -E R min u - R,t , u , v 0. (5.2)
Putting together (5.1) and (5.2) we obtain lim

R→+∞ E R max u - R,t , u , v -E R (u, v) = 0. (5.3)
We set f R,t := max u - R,t , u , (

and we observe that f R,t = max {u(x), u(x + te N )} , and f R,t ∈ D t R (u). Now, we argue by contradiction, assuming that u ∈ C 1 (Ω) is not monotone on a line in the direction e N . This implies that we can take t > 0 in such a way that u(x) and u(x + te N ) satisfy the hypotheses of Lemma 3.1, say at some point x 0 ∈ Ω (see Remark 4.2 in [START_REF]Some monotonicity results for minimizers in the calculus of variations[END_REF]). Therefore, we have that g R,t := (f R,t , v) is not an e N -minimizer for E . Hence, in a neighborhood of x 0 , we can perturb g R,t into g R,t in such a way that E R ( g R,t ) E R (g R,t ) -c, for some c > 0 which depends only on (u, v). From the last inequality and (5.3) we reach a contradiction with the minimality of (u, v) as R → +∞. If we assume that v ∈ C 1 (Ω) is not monotone on a line in the e N -direction, we get again a contradiction with the fact that (u, v) is an e N -minimizer. Indeed we can repeat the same argument as above, using (4.7) and the second integral formula in (4.8). This concludes the proof of Theorem 1.5.

Proof of Theorem 1.10

We recall the notation introduced at the beginning of Section 4. From (4.6) we deduce that, for ε > 0, we can take R large such that

E R (u + R,t , v) + E R (u - R,t , v) -2E R (u, v) εt 2 . (6.1)
Moreover, we know that (u, v) is {e N , e N +1 }-stable, and so E R (w 1 , v) -E R (u, v) -εt 2 for any w 1 ∈ D t R (u), (

for every t sufficiently small (see Definition 1.8). Now, recalling the definition of f R,t in (5.4), we use the first integral formula in (4.8) to obtain that

E R (f R,t , v) -E R (u, v) = E R (u - R,t , v) -E R min u - R,t , u , v = E R (u - R,t , v) + E R (u + R,t , v) -2E R (u, v) + E R (u, v) -E R min u - R,t , u , v + E R (u, v) -E R (u + R,t , v) 3εt 2 ,
where we have used (6.1) and (6.2).

Suppose, by contradiction, that u N changes sign in Ω. Then, by Lemma 3.4, there exists a point x 0 ∈ Ω such that the hypotheses of Lemma 3.2 are satisfied in a neighborhood of x 0 . Hence, we have that we can perturb f R,t into f R,t near x 0 in such a way that

E R ( f R,t , v) E R (f R,t , v) -ct 2 ,
where f R,t ∈ D Ct R (u), for some c, C > 0 which depend only on u. Then, we obtain E R ( f R,t , v) E R (u, v) + (3ε -c)t 2 , which gives a contradiction with the stability inequality (1.6) if we take ε c. If we assume that v N changes sign in Ω, we reason in a similar way, using (4.7), the second integral formula in (4.8) and Remark 3.3 to reach the same contradiction. Therefore, either u N 0 or u N 0 and either v N 0 or v N 0 in Ω. This completes the proof of Theorem 1.10.

7 Some applications As already mentioned in the Introduction, the above system arises in the analysis of phase separation phenomena in binary mixtures of Bose-Einstein condensates with multiple states. The energy associated to (7.1) is the following:

E (u, v) := 1 2 ˆRN |∇u| 2 + |∇v| 2 + u 2 v 2 dx.
Under a suitable growth condition, we show here that stable solutions of (7.1) are one-dimensional, i.e. there exist u, v : R → R and ω u , ω v ∈ S N -1 such that (u(x), v(x)) = u(ω u • x), v(ω v • x) .

Precisely:

Proposition 7.1. -Let u, v ∈ C 2 (R N ). Suppose that (u, v) is a stable solution to (7.1), and that the following growth condition holds true: there exists a constant C > 0 such that, for R sufficiently large, ˆBR |∇u| 2 + |∇v| 2 dx CR 2 . (

Then, (u, v) possesses one-dimensional symmetry.

Proof. -We define

F (p 1 , p 2 , z 1 , z 2 , x ) := 1 2 |p 1 | 2 + |p 2 | 2 + z 2 1 z 2 2 .
Notice that F is convex in p 1 and p 2 and satisfies (1.4). Moreover F p 1 p 1 = 1 = F p 2 p 2 and F p 1 p 2 = 0, therefore the growth condition (1.5) is ensured by (7.2). Then, we can apply Theorem 1.10 (recall also Lemma 2.1) and we obtain that (u, v) is onedimensional.

As a corollary, we obtain the one-dimensional symmetry in dimension 2 for stable solutions to (7.1) which have linear growth at infinity (see [START_REF] Berestycki | On entire solutions of an elliptic system modeling phase separations[END_REF]). Then, (u, v) possesses one-dimensional symmetry.

Proof. -We observe that, by Theorem 5.1 in [START_REF] Wang | On the De Giorgi type conjecture for an elliptic system modeling phase separation[END_REF], if (u, v) satisfies (7.3) then |∇u| and |∇v| are bounded in R 2 . Therefore, condition (7.2) is trivially satisfied, and so we get the desired result thanks to Proposition 7.1. for some C M > 0. As a consequence, we have It remains to check the growth condition (1.5). For this, notice that (7.6) and formula (4.13) in [START_REF] Farina | Bernstein and De Giorgi type problems: new results via a geometric approach[END_REF] give that As paradigmatic examples in Proposition 7.3 one can take the p-Laplacian, with p ∈ (1, +∞) if {∇u = 0} = ∅ and p ∈ [2, +∞) if {∇u = 0} = ∅ (in this case a(t) = t p-2 ), or the mean curvature operator (in this case a(t) = (1 + t 2 ) -1/2 ). We stress on the fact that one can also take different operators a and b satisfying the hypotheses (for instance, one can take a to be the p-Laplacian and b an operator of mean curvature type).

|F
|F p 1 p 1 (p 1 , p 2 ,z 1 , z 2 , x )| |p 1 | 2 + |F p 2 p 2 (p 1 , p 2 , z 1 , z 2 , x )| |p 2 | 2 C M λ 2 (|p 1 |)|p 1 | 2 + γ 2 (|p 2 |)|p 2 | 2 = C M a(|p 1 |)|p 1 | 2 + b(|p 2 |)|p 2 | 2 C M Λ 2 (|p 1 |) + Γ 2 (|p 2 |) .
Furthermore, we observe that condition (7.5) is satisfied, for instance, when N = 2 and ∇u and ∇v are bounded (thanks to the hypotheses on a and b, see page 3474 in [START_REF] Dipierro | Geometric inequalities and symmetry results for elliptic systems[END_REF]), This means that we recover Theorem 7.1 of [START_REF] Dipierro | Geometric inequalities and symmetry results for elliptic systems[END_REF] for stable solutions, without requiring conditions on the sign of F 12 .

Systems involving the fractional Laplacian

The general setting of our results allows us to treat also the case of nonlocal systems of equations, i.e.

(-∆) s 1 u = F 1 (u, v),

(-∆) s 2 v = F 2 (u, v), (7.7)
where s 1 , s 2 ∈ (0, 1), F is a C 1,1 loc function on R 2 and F 1 and F 2 denote the derivatives of F with respect to the first and the second variable respectively.

As a matter of fact, as in [125, Remark 2.12], we observe that one can generalize the functional considered in (1.1) to the following functional ˆΩ F (∇u, ∇v, u, v, x ) dx + ˆ∂Ω G(u, v, x ) dH N -1 , (

where G satisfies the same regularity assumptions as F . Furthermore, the growth condition in (1.5) can be weakened in the following way. We define 0 (R) := R and instead of (1.5). Notice that if k = 0 then (7.9) corresponds to (1.5). See Remark 2.13 and Section 9 in [START_REF]Some monotonicity results for minimizers in the calculus of variations[END_REF] for the proof and further discussion on this fact. We also recall that, using the extension result in [START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF], one can localize problem (7.7) by adding one variable, see e.g. formula (1.7) in [START_REF] Dipierro | A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian[END_REF]. We will denote by U and V the extension functions of u and v, respectively (see e.g. formulae (2.3) and (2.4) in [START_REF] Dipierro | A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian[END_REF]).

So one has to deal with an energy functional like (7. With this, we can prove the following:

Proposition 7.4. -Let u, v ∈ C 2 (R N ). Suppose that (u, v) is a stable solution to (7.7). Also, assume that the following growth condition holds true in the extension: there exist a constant C > 0 and k ∈ N such that, for R sufficiently large, ˆBR x 1-2s

1 1 |∇U | 2 + x 1-2s 2 1 |∇V | 2 dx 1 • • • dx N +1 CRπ k (R). (7.11)
Then, (u, v) possesses one-dimensional symmetry.

Proof. -With the notation introduced in (7.10), we observe that the thesis simply follows from (7.11) (that ensures the growth condition (7.9)) and Theorem 1.10.

We remark that (7.11) is a reasonable energy growth condition, since it is satisfied for instance in the case of a single equation when N = 2 for any s ∈ (0, 1) and when N = 3 for s ∈ (1/2, 1), see [START_REF] Cabré | Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian[END_REF][START_REF]Sharp energy estimates for nonlinear fractional diffusion equations[END_REF]. Moreover, if s 1 = s 2 = 1/2, it can be checked as in [START_REF] Cabré | Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian[END_REF] with suitable modifications under an additional assumption on the bound of ∇U and ∇V (see in particular formula (1.16) and Section 4 in [START_REF] Cabré | Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian[END_REF]). We finally remark that, differently from [START_REF] Dipierro | A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian[END_REF], we do not need here any sign assumption on F .

Open problem -Let K ⊂ R N be a compact convex set. Suppose that J is compactly supported. If u : R N \ K → [0, 1] is uniformly continuous in R N \ K and obeys

   Lu + f (u) = 0 in R N \ K, sup R N \K u = 1,
then, is it true that u = 1 in the whole set R N \ K?

It is also tempting to ask for a characterization of the set of all obstacles K for which the Liouville property holds (however, this is not known even for the local case which is a priori simpler).

And, of course, it remains the question of the existence and large-time convergence of solutions to the evolution problem

∂u ∂t = Lu + f (u) in R × R N \ K.

On Chapter 5

If the results of Chapter 5 are fairly general, they still require a variational structure. One therefore cannot handle "prey-predator" type systems such as (-∆) s u = u(α -β v), (-∆) s v = -v(γ -δ u).

It would be interesting to find a general method that could handle at least some systems in non-variational form.

Figure 1 .

 1 1 for a visual evidence. Now, set Φ(x) := 1 c Φ * (|x|) and η(x) := 1 c η * (|x|) where c := ˆBr 2 η * (|x|)dx. By construction, η ∈ L 1 (R N ) and η has unit mass. Moreover, by (6.7), we have η Φ a.e. in B r 2 and ˆBr 2 /4 Φ > 0.

. 8 )Figure 1 . 1 .

 811 Figure 1.1. Construction of η * and Φ * .

Figure 1 . 2 .

 12 Figure 1.2. Spatial conditions on h and z.

. 3 )

 3 Proof. -Let us first construct an auxiliary sequence satisfying (3.2).

Figure 2 . 1 .

 21 Figure 2.1. Construction of the first terms of (ζ k ) k 0 . The hatched zone corresponds to the values of x for which ζ 2 j x takes nonzero values.

Remark 4 . 3 .

 43 -The same kind of estimate holds in the setting of Besov spaces of generalized smoothness. That is, given a function f ∈ B s p,q (R N ) decomposed as above by (4.2) with (4.1) and (4.3), we can estimate the B (s,Ψ) p,q (R d )-quasi-norm of its restrictions to almost every hyperplanes f (•, x ) exactly in the same fashion. It suffices to replace the (s, p)-β-quarks (βqu) ν,m in the decomposition of f (•, x ) by Ψ(2 -ν ) -1 (βqu) ν,m in order to get (s, p, Ψ)-β-quarks. From here, we can reproduce the same reasoning as in Lemma 4.1 with Ψ(2 -ν )λ β ν,m instead of λ β ν,m and we obtain

Furthermore, if r r 0

 0 , y ∈ K and x ∈ H e , then y • e -r |y| -r R 0 -r and φ r,e (x) = φ(x • e -r) φ(R 0 -r) φ(y • e -r) = φ r,e (y).

. 7 )

 7 Our first result reads as follows Theorem 1.1. -Let N 2. Then, there are smooth (non-starshaped) simply connected compact obstacles K and data f and J satisfying (1.6) and (1.7) for which problem (1.1) has a positive nonconstant solution u ∈ C(R N \ K, [0, 1]).

Figure 4 . 1 .

 41 Figure 4.1. Illustration of K ε in dimension 2.

Definition 1 . 1 .

 11 -Given a function w, an e N -Lipschitz deformation of w in B R is a function w defined by w(x) = w(x + ϕ(x)e N ) for any x ∈ Ω,

Lemma 2 . 1 .

 21 -If Ω = R N , F ∈ C 2 and u, v ∈ C 2 (R N ), then (u, v) is a classical stable solution for the energy functional E if, and only if, (u, v) is an {e N , e N +1 }stable solution of E according to Definition 1.8.

2 . -We define u 1

 21 (x) := u(x + te N ) -u(x) t (3.13)and we observe thatu 1 -u N C 0,1 (Bη) = o(1) as t → 0. (3.14)

Lemma 3 . 4 .

 34 -Assume that either u, v ∈ C 0,1 (Ω) are convex, or u, v ∈ C 1,1 (Ω).

(4. 8 ) 5

 85 Proof of Theorem 1.5

7. 1

 1 Two-stated mixture of Bose-Einstein condensate Here, we consider the following system      ∆u = uv 2 , ∆v = vu 2 , u, v > 0.

Corollary 7 . 2 .

 72 -Let N = 2 and let u, v ∈ C 2 (R 2 ). Suppose that (u, v)is a stable solution to (7.1), and that the following growth condition holds true: there exists a constant C > 0 such that |u(x)| + |v(x)| C(1 + |x|). (7.3)

  from above and below by (λ1 (|p 1 |) + λ 2 (|p 1 |)) and (γ 1 (|p 2 |) + γ 2 (|p 2 |)), respectively, up to multiplicative constants. Furthermore, since conditions (B1) and (B2) in[START_REF] Farina | Bernstein and De Giorgi type problems: new results via a geometric approach[END_REF] are satisfied for a, b, A and B, one can use Lemma 4.2 there to see that, if|p 1 |, |p 2 | M , |q 1 | |p 1 |/2 and |q 2 | |p 2 |/2, then λ 1 (|p 1 |) C M λ 2 (|p 1 |), λ 2 (|p 1 + q 1 |) C M λ 2 (|p 1 |), γ 1 (|p 2 |) C M γ 2 (|p 2 |), γ 2 (|p 2 + q 2 |) C M γ 2 (|p 2 |),

  This and (7.5) imply thatˆBR |F p 1 p 1 (∇u, ∇v, u, v, x )| |∇u| 2 + |F p 2 p 2 (∇u, ∇v, u, v, x )| |∇v| 2 dx C M ˆBR (Λ 2 (|∇u|) + Γ 2 (|∇v|)) dx C M R 2 ,up to rename C M . This shows (1.5) holds true. Hence, we can apply Theorem 1.10, thus obtaining the desired monotonicity property.

k 2 +

 2 (R) := log k-1 (R) = log • . . . • log k times R for any k ∈ N * . can require that, for some k ∈ N, ˆΩ∩B R |F p 1 p 1 (∇u, ∇v, u, v, x )||∇u| 2 + |F p 2 p 2 (∇u, ∇v, u, v, x )||∇v| |F p 1 p 2 (∇u, ∇v, u, v, x )||∇u||∇v| dx CRπ k (R),(7.9)

1 |p 1 | 2 + x 1 -2s 2 1 |p 2

 112112 (0, +∞) × R N , V := (0, +∞) × R N -1 , F (p 1 , p 2 , z 1 , z 2 , x ) := x 1-2s 1 | 2 , G(z 1 , z 2 , x ) := F (z 1 , z 2 ),(7.10) (notice that in this application one has to replace N by N +1 to apply Theorem 1.10).

  Theorem 1.23, (see also[START_REF] Knopova | Spaces of generalized smoothness on h-sets and related Dirichlet forms[END_REF] Theorem 10, p.284]).

	Remark 2.18. -The counterpart of Remark 2.8 for B p,q (R N ) remains valid, (s,Ψ)
	see [111, Remark 1.26, p.48].

  , p.131], [108, Lemma 6.5] and [117, Théorème 8.1, p.301].

	Remark 2.21. -Let us briefly mention that a corresponding result holds for the
	spaces B	(s,Ψ)

  ,ϕ 2 F := F (∇u+t∇ϕ 1 , ∇v+t∇ϕ 2 , u+tϕ 1 , v+tϕ 2 , x )-F (∇u, ∇v, u, v, x ), (2.2) , ϕ 2 ) := F (p 1 ) i ϕ 1 i + F z 1 ϕ 1 + F (p 2 ) i ϕ 2 i + F z 2 ϕ 2 , L(ϕ 1 , ϕ 2 ) := F (p 1 ) i (p 1 ) j ϕ 1 i ϕ 1 j + F z 1 z 1 (ϕ 1 ) 2 + 2F (p 1 ) i z 1 ϕ 1 ϕ 1 i + F (p 2 ) i (p 2 ) j ϕ 2 i ϕ 2 j + F z 2 z 2 (ϕ 2 ) 2 + 2F (p 2 ) i z 2 ϕ 2 ϕ 2 i + F (p 1 ) i (p 2 ) j ϕ 1 i ϕ 2 j + 2F z 2 z 1 ϕ 1 ϕ 2 + 2F (p 2 ) i z 1 ϕ 2 ϕ 2 i + F (p 2 ) i (p 1 ) j ϕ 2 i ϕ 1 j + 2F (p 1 ) i z 2 ϕ 2 ϕ 1 i, and the derivatives of F are evaluated at (∇u, ∇v, u, v, x ). Now, we observe that, since (u, v) is a critical point of E , it holds that ˆBR H(ϕ 1 , ϕ 2 ) = 0.So integrating (2.3) and recalling (2.2), we obtain thatE R (u + tϕ 1 , v + tϕ 2 ) -E R (u, v) = t 2 2 ˆBR L(ϕ 1 , ϕ 2 )dx + o(t 2 ). (2.4)Dividing by t 2 and recalling (2.1) we find that ˆBR L(ϕ 1 , ϕ 2 )dx 0.

	In addition, setting ∆ 1 ϕ 1 we have where H(ϕ 1 Thus, (2.4) yields ∆ 1 ϕ 1 ,ϕ 2 F = tH(ϕ 1 , ϕ 2 ) + E	t 2 2	L(ϕ 1 , ϕ 2 ) + o(t 2 ),	.1) (2.3)

R (u + tϕ 1 , v + tϕ 2 ) -E R (u, v) o(t 2 ).

  2 |A| 2 + |F p 2 p 2 | |∇v| 2 |A| 2 + 2 |F p 1 p 2 | |∇u||∇v||A| 2 dx |F p 1 p 1 ||∇u| 2 + |F p 2 p 2 ||∇v| 2 + |F p 1 p 2 ||∇u||∇v| dx |x| 2 , (4.4) where the derivatives of F are evaluated at (∇u, ∇v, u, v, x ). Now we set h(r) := ˆΩ∩Br |F p 1 p 1 ||∇u| 2 + |F p 2 p 2 ||∇v| 2 + |F p 1 p 2 ||∇u||∇v| dx. Recalling (1.5), we have that h(r) Cr 2 . Also, we see that Remark 4.1. -We point out that in a similar way, by perturbing only one of the element of the couple (u, v), one can obtain lim sup

	Ct 2 (log R) 2	ˆΩ∩(B R \B √	R )
			ˆR √ R	h (r) r 2 dr	h(R) R 2 + 2 ˆR √ R	h(r) r 3 dr C log R.	(4.5)
	Passing to polar coordinates in the last integral in (4.4) and using (4.5) we get
	lim sup R→+∞	sup t∈(0, √ R/4)	E R (u + R,t , v + R,t ) + E R (u -R,t , v -R,t ) -2E R (u, v) t 2	0.
		R→+∞	sup t∈(0, √ R/4)	E R (u + R,t , v) + E R (u -R,t , v) -2E R (u, v) t 2	0,	(4.6)
	and
		lim sup R→+∞	sup t∈(0, √ R/4)	E R (u, v + R,t ) + E R (u, v -R,t ) -2E R (u, v) t 2	0.	(4.7)

  p 1 p 1 (p 1 , p 2 , z 1 , z 2 , x )| C M λ 2 (|p 1 |), |F p 2 p 2 (p 1 , p 2 , z 1 , z 2 , x )| C M γ 2 (|p 2 |),(7.6)up to rename C M . Therefore, recalling also formulae (1.4)-(1.7) in[START_REF] Dipierro | Geometric inequalities and symmetry results for elliptic systems[END_REF], we get|F p 1 p 1 (p 1 + q 1 ,p 2 , z 1 , z 2 , x )| C M λ 2 (|p 1 + q 1 |) C M λ 2 (|p 1 |) C M |F p 1 p 1 (p 1 , p 2 , z 1 , z 2 , x )|,and|F p 2 p 2 (p 1 , p 2 +q 2 , z 1 , z 2 , x )| C M γ 2 (|p 2 + q 2 |) C M γ 2 (|p 2 |) C M |F p 2 p 2 (p 1 , p 2 , z 1 , z 2 , x )|,for any 2|q 1 | |p 1 | M and 2|q 2 | |p 2 | M . This means that (1.4) is satisfied.

"We must fix at the right total the number of citizens; next, we must agree about the distribution of them, into how many sections, and each of what size they are to be divided; and among these sections we must distribute, as equally as we can, both the land and the houses. [...] Of land we need as much as is capable of supporting so many inhabitants of temperate habits. [...] The number 5,040 is here chosen because, for a number of moderate size, it has the greatest possible number of divisors[START_REF]Local approximation of arbitrary functions by solutions of nonlocal equations[END_REF], including all the digits from 1 to 10. [...] And in order that these things may remain in this state forever, these further rules must be observed: the number of hearths, as now appointed by us, must remain unchanged, and must never become either more or less." Plato, Laws, Book V.

"The magistrates [...] shall consider how to deal with the excess or deficiency in families, and contrive means as best they can to secure that the 5,040 households shall remain unaltered. There are many contrivances possible: where the fertility is great, there are methods of inhibition, and contrariwise there are methods of encouraging and stimulating the birth-rate, by means of honours and dishonours." Plato, op. cit.

It is also amusing to notice that 7! = 1 × 2 × 3 ×

×

×

×

= 5,040 which suggests an influence of Pythagoreanism.

"Population, when unchecked, increases in a geometrical ratio. [...] Taking the population of the world at any number, a thousand millions, for instance, the human species would increase in the ratio of1, 2, 4, 8, 16, 32, 64, 128, 256,

512, etc." Malthus, Essay on the Principle of Population. 5 "The increase of population is necessarily limited by the means of subsistence", Malthus, op. cit.

It is worth pointing out that the equation that was actually derived by Kolmogorov, Petrovskii and Piskunov in[START_REF] Kolmogorov | A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem[END_REF] was a nonlocal equation of the type of (2.6) with L of the form (2.8) (see[START_REF] Kolmogorov | A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem[END_REF] Formula (7.32), p.174]). In fact, the local equation (2.4) was obtained as a sort of approximation using a formal Taylor expansion, which suggests that nonlocal equations are natural in this context.

This chapter is inspired by the paper[START_REF] Brasseur | Liouville type results for a nonlocal obstacle problem[END_REF] written in collaboration with J. Coville, F. Hamel and E. Valdinoci.

L 2 (B R \Kε) for all j 0. Therefore (v j ) j∈N is bounded in L 2 (B R \

We may actually consider indifferently the first or the second equation, as only one equation is needed.

We use here the short notation {w = 0} = {x ∈ V 0 : w(x) = 0}, and similarly for {w < 0} and {w > 0}.

Remerciements

Proof. -We let w := v-u. Notice that w(x) = 0. As a consequence, using (4.3), we can write Lw(x) = Lv(x) -Lu(x) -f (v(x)) + f (u(x)) = 0.

On the other hand, w(y) 0 = w(x) for a.e. y ∈ R N \ K, thanks to (4.8), and therefore Lw(x) = ˆRN \K J(x -y)(w(y) -w(x))dy 0.

Hence, Lw(x) = 0 and 0 = ˆRN \K J(x -y)(w(y) -w(x))dy = ˆRN \K J(x -y)w(y)dy.

From our assumptions, we have w 0 a.e. in R N \ K. Accordingly, since J is such that J > 0 a.e. in the annulus A (r 1 , r 2 ) from the general assumption (1.9), it follows that w(x) = 0 i.e. v(x) = u(x) for a.e. x ∈ A (x, r 1 , r 2 ) ∩ R N \ K.

In particular, since u and v are continuous in H and H ⊂ R N \ K, we get that v(x) = u(x) for all x ∈ A (x, r 1 , r 2 ) ∩ H =: Ω 1 (x).

Applying the same arguments as above to the new set of contact points Ω 1 (x) we obtain that v(x) = u(x) for all x ∈ A (x 1 , r 1 , r 2 ) ∩ H and for all x 1 ∈ Ω 1 (x). As a consequence, v(x) = u(x) for all x ∈ B µ (x) ∩ H with µ := r 2 -r 1 . Iterating this procedure over again implies that v(x) = u(x) for each x in B 2µ (x) ∩ H and so on in B kµ (x) ∩ H for any k ∈ N. Hence, v = u in H. Therefore, as in the beginning of the proof, it follows that Lw(x) = 0 for all x ∈ H and v = u a.e. in H + A (r 1 , r 2 ) ∩ R N \ K = (H + B r 2 ) \ K.

The proof of Lemma 4.2 is thereby complete.

Finally, we derive a sweeping-type result in the spirit of Serrin's sweeping theorem [START_REF] Serrin | A symmetry problem in potential theory[END_REF] (see also [START_REF] Mcnabb | Strong comparison theorems for elliptic equations of second order[END_REF], and page 29 in [START_REF] Pucci | The maximum principle[END_REF] for a very clear explanation of the method). Lemma 4.3 (Sweeping principle). -Assume that J satisfies (1.9), with 0 r 1 < r 2 . Let g : R → R be a continuous function, let a, b, s 1 , s 2 , s 3 , s 4 be some real numbers such that a b and r 2 s 1 s 2 < s 3 s 4 . Let u ∈ C(A (s 1 , s 4 )) satisfy ˆA (s 1 ,s 4 ) J(x -y)u(y)dy -u(x) + g(u(x)) 0 for all x ∈ A (s 1 , s 4 ), (4.9) and ˆA (s 1 ,s 4 ) J(x -y)u(y)dy -u(x) + g(u(x)) < 0 for all x ∈ A (s 2 , s 3 ). (4.10)

Let (w τ ) τ ∈[a,b] be a continuous family in C(A (s 1 , s 4 )) such that ˆA (s 1 ,s 4 ) J(x -y)w τ (y)dy -w τ (x) + g(w τ (x)) 0 for all x ∈ A (s 1 , s 4 ). (4.11) Assume further that there exists τ 0 ∈ [a, b] such that w τ 0 u in A (s 1 , s 4 ). Then w τ u in A (s 1 , s 4 ) for every τ ∈ [a, b].

Proof. -Let us define Σ ⊂ [a, b] to be the following set:

To prove the theorem, we will show that Σ is a non-empty open and closed set relatively to [a, b]. It will then follow that Σ = [a, b] and the theorem will be proved. First of all, by definition, Σ is a closed subset of [a, b] and τ 0 ∈ Σ. To finish our proof, it remains to show that Σ is an open set relatively to [a, b]. So let us pick τ ∈ Σ. We have w τ u in A (s 1 , s 4 ). By continuity of u and w τ in the compact set A (s 1 , s 4 ), either max A (s 1 ,s 4 ) (w τ -u) < 0 or there exists z ∈ A (s 1 , s 4 ) such that w τ (z) = u(z).

In the latter case, using w τ u in A (s 1 , s 4 ) together with (4.9) and (4.11) at the point z, we get that 0 ˆA (s 1 ,s 4 ) J(z -y)(u(y) -w τ (y))dy 0.

Using the continuity of both u and w τ and the fact that J > 0 a.e. in A (r 1 , r 2 ) for some 0 r 1 < r 2 by (1.9), it follows that w τ = u in A (z, r 1 , r 2 ) ∩ A (s 1 , s 4 ) (which is nonempty since r 2 s 1 ) and then w τ = u in A (z , r 1 , r 2 ) ∩ A (s 1 , s 4 ) for all z ∈ A (z, r 1 , r 2 ) ∩ A (s 1 , s 4 ). In particular, it is easy to see that there exists r > 0 such that w τ = u in B r (z) ∩ A (s 1 , s 4 ). As a consequence, the non-empty set

x ∈ A (s 1 , s 4 ); w τ (x) = u(x) is both (obviously) closed and open relatively to the (connected) set A (s 1 , s 4 ) and it is thus equal to A (s 1 , s 4 ). In other words, w τ = u in A (s 1 , s 4 ), hence ˆA (s 1 ,s 4 ) J(x -y)u(y)dy -u(x) + g(u(x)) = 0 for all x ∈ A (s 1 , s 4 ), contradicting (4.10) in A (s 2 , s 3 ). Therefore, we must have max A (s 1 ,s 4 ) (w τ -u) < 0. Since w τ is continuous with respect to τ in the uniform norm, there exists δ > 0 such that w τ u in A (s 1 , s 4 ) for all τ ∈ (τ -δ,

and thus E (v) E (ϕ) < 0, which proves our claim. Furthermore, since 0

(remember that v was either positive or identically equal to 0 in B R (x 0 )). As a conclusion, for every R such that R d 0 , there exists a solution v ∈ C(B R (x 0 ), (0, 1)) to (6.1) with max B R (x 0 ) v > θ. The point x 0 ∈ R N being arbitrary and the constant d 0 being independent of x 0 , the proof of Lemma 6.1 is thereby complete.

Existence and properties of the maximal solution in

Let us now look more closely at the properties of positive solutions of (6.1) and in particular at the maximal solution, if any. To this end, let us in this subsection extend continuously f by f (1)(s -1) for s 1 and by 0 for s 0. To simplify our presentation let us still denote f this extension.

Let us first recall the notion of maximal solution for problem (6.1).

The following lemma provides the existence and uniqueness of a maximal solution to the problem (6.1) when R > 0 is large enough. Lemma 6.3. -Assume that f and J satisfy (1.8), (1.9) and (1.11). Assume further that J is compactly supported and J ∈ L 2 (R N ). Then there exists d 0 = d 0 (f, J) > 0, given as in Lemma 6.1, such that for every x 0 ∈ R N and R d 0 , problem (6.1) admits a unique maximal solution

Proof. -Let x 0 ∈ R N be fixed and let R be fixed such that R d 0 , where d 0 = d 0 (f, J) > 0 is given in Lemma 6.1. We will check that the conclusion holds with this quantity d 0 . First of all, the uniqueness of the maximal solution in B R (x 0 ), if any, is a trivial consequence of its definition.

Let us then focus on the construction of a maximal solution. From Lemma 6.1, there exists a solution v ∈ C(B R (x 0 ), (0, 1)) to (6.1). Now, remember that 1 is a strict super-solution to (6.1). Therefore, since f is Lipschitz continuous, it follows that we can construct a maximal solution v x 0 ,R ∈ C(B R (x 0 ), (0, 1)) to (6.1) such that 0 < v v x 0 ,R < 1 in B R (x 0 ), by using standard monotone iterative scheme as in [START_REF]On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators[END_REF]Theorem A.1]. For the sake of completeness, let us describe this scheme in the next paragraph.

To do so, we proceed by induction. By construction of v x 1 ,R 1 and the definition of v 0 , we know that v (6.15), since the function s → -ks -f (s) is decreasing and v x 1 ,R 1 (x) v n (x) for all x ∈ B R 1 (x 1 ), we see that w satisfies

(ii) follows from the following observations. For any

To prove (iii), we simply observe that, for any x 1 ∈ B 2R (x 0 ), one has B 2R (x 1 ) ⊂ B 4R (x 0 ) and, by (i),

Now, since for every x, y ∈ B R (x 0 ) there exists (a unique) x 1 ∈ B 2R (x 0 ) such that y = x -(x 1 -x 0 ) and x ∈ B R (x 1 ) ⊂ B 2R (x 1 ), the latter inequality implies that

for all x, y ∈ B R (x 0 ), which completes the proof.

We can now state our last property about the maximal solution. Lemma 6.5. -Let us assume that f and J satisfy (1.8), (1.9) and (1.11). Assume further that J is compactly supported and J ∈ L 2 (R N ). Then, for every

Proof. -Let x 0 ∈ R N be fixed. Consider any non-decreasing sequence (R n ) n∈N in [d 0 , +∞) and converging to +∞, where d 0 = d 0 (f, J) > 0 is given in Lemmata 6.1

Therefore, it follows from the above equality and the definitions of v x 0 ,R and w x 0 ,R,δ that, for x ∈ B R (x 0 ), ˆBR+δ (x 0 ) J(x -y)w x 0 ,R,δ (y)dy-w x 0 ,R,δ (x) + f (w x 0 ,R,δ (x))

= ˆBR+δ (x 0 )\B R (x 0 ) J(x -y)w x 0 ,R,δ (y)dy 0.

To complete our proof, we have to show that the above inequality holds also for x ∈ B R+δ (x 0 ) \ B R (x 0 ). To this end, let us consider x ∈ B R+δ (x 0 ) \ B R (x 0 ) and set

that is, w x 0 ,R,δ (x) = max{s(x)-δ -1 τ (x), 0}. From the nonnegativity of J and w x 0 ,R,δ and the fact that

J(x-y)v x 0 ,R (y)dy-max{s(x)-δ -1 τ (x), 0}+f (max{s(x)-δ -1 τ (x), 0}).

(6.22) Now, two situations may occur: either s(x) δ -1 τ (x) (that is, w x 0 ,R,δ (x) = 0), or s(x) > δ -1 τ (x) (that is, w x 0 ,R,δ (x) > 0). In the first situation, we easily conclude that ˆBR+δ (x 0 ) J(x -y)w x 0 ,R,δ (y)dy-w x 0 ,R,δ (x) + f (w x 0 ,R,δ (x)) ˆBR (x 0 ) J(x -y)v x 0 ,R (y)dy 0.

So let us now assume that s(x) > δ -1 τ (x), that is,

Let us rewrite the first integral in the right-hand side of (6.22) as

Since v x 0 ,R solves (6.1) in B R (x 0 ), since

and since J ∈ W 1,1 (R N ), the above equality yields

Combining now the above inequality with (6.22) and s(x) -δ -1 τ (x) > 0, and using (4.38), (4.39) and (4.40), we get ˆBR+δ (x 0 ) J(x -y)w x 0 ,R,δ (y)dy -w x 0 ,R,δ (x) + f (w x 0 ,R,δ (x))

. This is the desired inequality and the proof of Lemma 6.6 is thereby complete.

The case of convex obstacles: proofs of the main Liouville type results

In this section, we prove our main results. We first consider in Section 7.1 the case where J is a general kernel satisfying (1.9), namely we prove Theorems 2.2 and 2.3. Once this is done, we consider in Section 7.2 kernels having compact support and we prove Theorem 2.4. Section 7.3 is devoted to the proof of a lemma used in the proof of Theorem 2.4. Throughout Section 7, we always assume that K is a compact convex set and that f and J satisfy the conditions (1.8), (1.9) and (1.10).

General kernels: proofs of Theorems 2.2 and 2.3

Let us start our proof of Theorem 2.2 with the following simple observation.

Lemma 7.1. -Let K ⊂ R N be a compact convex set and assume (1.8) and (1.9).

Then there exists γ ∈ (0, 1] such that γ u 1 in R N \ K.

Chapter 4 A counterexample to the Liouville property of some nonlocal problems

This chapter is inspired by the paper [START_REF] Brasseur | A counterexample to the Liouville property of some nonlocal problems[END_REF] written in collaboration with J. Coville.

Introduction 1.A nonlocal problem in heterogeneous media

Let K be a compact set of R N with N 2, and let |•| be the Euclidean norm in R N . We are interested in the qualitative properties of positive solutions u to the following problem

where f is a bistable nonlinearity with f (0) = f (1) = 0 and L is the nonlocal operator

with J ∈ L 1 (R N ) a non-negative kernel with unit mass. The precise assumptions on f and J will be given later on. This type of model naturally arises in the study of the behavior of particles evolving in a heterogeneous medium. The typical kind of problem we have in mind function. If 0 < |E| < ∞ and if g : R N → R is locally integrable, we denote by

the average of g in the set E. Also, we denote by L p (E), 1 p ∞, the Lebesgue space of (equivalence classes of) measurable functions g for which the p-th power of the absolute value is Lebesgue integrable when p < ∞ (resp. essentially bounded when p = ∞).

Preliminaries

In this section, we recall some known results that will be used throughout the chapter.

In most cases, we will omit their proofs and point the interested reader to the relevant references.

We first state a general existence result.

Lemma 2.1. -Assume that f and J satisfy (1.6) and

Assume, in addition, that

and that

Although the proof of Lemma 2.1 relies on rather standard arguments it is not straightforward. For this reason, we will give a detailed proof (which is postponed to the Appendix at the end of the chapter).

Next, we recall a regularity result for nonlocal equations of the form

where

(2.5)

) be a solution to (2.3) a.e. in Ω\K. Then, u can be redefined up to a negligible set and extended as a uniformly continuous function in Ω \ K.

For a detailed proof, we refer to [31, Lemma 3.2] (see also [START_REF] Bates | Travelling waves in a convolution model for phase transitions[END_REF][START_REF] Berestycki | A non-local bistable reaction-diffusion equation with a gap[END_REF]).

Remark 2.3. -Note that Ω need not be bounded. In particular, Lemma 2.2 holds when Ω = R N .

Finally, we recall the following result Lemma 2.4. -Let K ⊂ R N is a compact set and suppose that f and J satisfy (1.6) and (1.7). Assume further that J is compactly supported and that

Then, u(x) → 1 as |x| → ∞.

The proof may be found in [START_REF] Brasseur | Liouville type results for a nonlocal obstacle problem[END_REF]Lemma 7.2].

Remark 2.5. -The above results still hold when J(x-y) is replaced by J(d g (x, y)). For the validity of Lemma 2.1 in this case, we refer to Remark 5.4 in the Appendix. On the other hand, a careful inspection of the proof of [START_REF] Brasseur | Liouville type results for a nonlocal obstacle problem[END_REF]Lemma 3.2] shows that the condition (2.6) with J replaced by

still implies the continuity of solutions to

in Ω \ K. Similarly, Lemma 2.4 holds as well with L g (as given by (1.11)) instead of L since its proof requires only estimates on convex regions on which it trivially holds that d g (x, y) = |x -y|.

parameter and set γ := N N -1 . We call A the annulus A := A (R 0 , R 1 ) and we consider a smooth compact simply connected set K ε ⊂ A satisfying the following properties:

where x = (x 1 , x ) and x = (x 2 , . . . , x N ) (see Figure 4.1). Furthermore, we define the following open set:

We will refer to (K ε ) 0<ε<1 as the family of obstacles associated to the pair (J, f ).

Let us also list in this section a preparatory lemma.

Proposition 3.1. -Let N 2 and let (J, f ) be a pair satisfying (1.7) and (1.8). Let (K ε ) 0<ε<1 be the family of obstacles associated to the pair (J, f ). Let

Then, there exists some ε 0 > 0 depending only on N , R 0 , J and f , such that

Proposition 3.1 will play an important role in the sequel. Inter alia, it guarantees that the solutions of some nonlocal equations defined in the sequel are continuous.

Proof. -By assumption (1.8), up to rescale J, we may assume without loss of generality that supp(

We will estimate from below the integral in the right-hand side of (3.6). For it, we will treat separately the case where x ∈ F ε and the case where

Step 1: Lower bound in F ε Let x ∈ F ε . Since J ε is radially non-increasing, non-negative and supported in B ε/2 , there is some

Now, thanks to Proposition 4.5, we deduce that v ε,R ∈ B δ 0 (w 0 ) is a local minimiser and, as such, v ε,R solves (4.8) almost everywhere in B R \ K ε .

Let us now check that v ε,R is a continuous solution to (4.8) in the whole set

To complete the proof it remains to show that 0

Then, evaluating (4.8) at x 0 , one obtains

which is a contradiction. Therefore v ε,R < 1. Since, by construction, we have that v ε,R 0, it remains to check that v ε,R cannot cancel. Assume, by contradiction, that this is the case, namely that there exists a point x 0 ∈ B R \ K ε such that v ε,R (x 0 ) = 0. Then, by (4.8), we have that

and, as above, this implies that v ε,R ≡ 0. However, since v ε,R ∈ B δ 0 (w 0 ) and

, which is a contradiction. The proof of Proposition 4.6 is thereby complete.

From now on (and until the end of Section 4), ε will be fixed and taken so small that 0 < ε < ε, where ε is as defined in Proposition 4.6.

An extension procedure

Let us now complete the proof of Lemma 4.1. We will modify the function v ε,R constructed above in order to get a continuous super-solution to (4.1) satisfying made our extension procedure. But the Poincaré inequality was only needed in the ball B R 0 and, by convexity, it trivially holds that J ε (x -y) = J ε (d g (x, y)) for any (x, y) ∈ B R 0 × B R 0 . Similarly, the extension procedure required only to evaluate the new function on the annulus B R+σ \ B R but, since R -R 1 > 0 is large and ε is small, it still holds that J ε (x -y) = J ε (d g (x, y)) for any x ∈ B R+σ \ B R and any y ∈ R N \ K ε . Moreover, as already noted in Remark 2.5, condition (3.11) still implies the continuity of solutions to the corresponding auxiliary problem:

where, by analogy, we have set

In fact, the only place where some care should be taken is when justifying that if

where x ∈ B R \ K ε is a point at which v ε,R reaches an extremum, then it holds that v ε,R (y) ≡ v ε,R (x) for any y ∈ B R \ K ε (which is needed to establish the analogue of Proposition 4.6). But, fortunately, the geometry of K ε is simple enough to ensure that this is still the case. Indeed, (4.41) implies that v ε,R (y) ≡ v ε,R (x) for any

. By iteration, one finds that v ε,R (y) ≡ v ε,R (x) for any y ∈ Π j (x) and any j 1, where Π j (x) is given by

Then, one can show that, for some j 0 1 (independent of x), it holds that

Whence, iterating the same reasoning over again, one gets that v ε,R (y) ≡ v ε,R (x) for any y ∈ B kε/4 (x) ∩ B R \ K ε and any k ∈ N; which then gives the desired result.

Construction of continuous global solutions

In this final section we construct a positive nonconstant solution to (1.9). Our goal will be to find an ordered pair of global continuous sub-and super-solution. That is, given 0 < ε < ε * (where ε * has the same meaning as in Lemma 4.1), we aim to construct two functions, u ε and u ε , such that

(where L ε is as in (4.33)) and which further satisfy lim

Here,

Then, by Lemmata 2.1 and 2.2 we automatically obtain the existence of a continuous solution u ε to

satisfying 0

This, together with (5.1), yields a continuous solution to (5.2) satisfying 0 < u ε < 1 and u ε (x) → 1 as x 1 → ∞. In particular, we have sup x∈R N \Kε u ε (x) = 1. Since (1.6), (1.7) are satisfied, u ε is continuous, J ε is compactly supported and J ε ∈ L 2 (R N ) (by (1.8)), we may apply Lemma 2.4 and we obtain that lim |x|→+∞ u ε (x) = 1, which proves that u ε satisfies the requirements of Theorem 1.2 and thus Theorem 1.1 is proved.

Therefore, to complete the proof of Theorem 1.2, we need only to prove the following lemma.

Lemma 5.1. -Let (J, f ) be a pair satisfying (1.7) and (1.8). Let (K ε ) 0<ε<1 be the family of obstacles associated to the pair (J, f ) (as defined in Section 3). Let (J ε , f ε ) be as in (3.5) and let ε * > 0 be as in Lemma 4.1. Then, there exists r 0 > 0 such that, for all 0 < ε < ε * , there is (i) a continuous global sub-solution u ε to (5.2)

Proof. -By Lemma 4.1, we know that there exists some R * > 0 and some 0 < ε * < 1 such that, for all 0 < ε < ε * , there is a nonconstant super-solution

So, we are left to prove that there exists a sub-solution u ε to (5.2) satisfying (i) and such that u ε u ε .

To do so, let us extend f outside [0, 1] by f (0)s when s 0 and f (1)(s -1) for s 1. For simplicity, we still denote by f this extension. Now, we take δ ∈ (0, 1) and we let f δ be a C 1 function defined in R such that

always exists provided that δ is taken sufficiently small, say if 0 < δ < δ 1 for some small δ 1 > 0.

Let f ε,δ (s) := ε 2 f δ (s) and let L R N be the operator given by

Since J ε is radially symmetric (because J is), using the results obtained in [START_REF] Bates | Travelling waves in a convolution model for phase transitions[END_REF][START_REF] Chen | Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations[END_REF][START_REF] Coville | Travelling fronts in asymmetric nonlocal reaction diffusion equations: the bistable and ignition cases[END_REF][START_REF] Yagisita | Existence of traveling wave solutions for a nonlocal bistable equation: an abstract approach[END_REF], we know that, for any 0 < ε < 1, there exists an increasing function φ ε,δ ∈ C 1 (R) and a number c ε,δ > 0 such that the function ϕ ε,δ (x) := φ ε,δ (x • e 1 ) satisfies

where H e 1 is the hyperplane H e 1 := {x 1 = 0}. Now, for any r 0 > 0, we let ϕ ε,δ,r 0 be the function defined by ϕ ε,δ,r 0 (x) := ϕ ε,δ (x -r 0 ).

By construction, for every r 0 > 0, we have

Now, we set

Note that, for all 0 < ε < ε * , it holds that K ε ⊂ R N \ H * provided that r 0 is chosen sufficiently large. Let us now prove that, for r 0 large enough, u ε is a sub-solution to (5.2). First, if x ∈ R N \ (K ∪ H * ), then u ε (x) = 0 and

Next, if x ∈ H * , then, since J ε is compactly supported, we have

provided that r 0 is chosen sufficiently large. From this and (5.5), we deduce that

Together with (5.6), we obtain that u ε is a global sub-solution to (5.2) which, by (5.4), satisfies u ε (x) → 1 as x 1 → ∞ and u ε (x) = 0 if x 1 r 0 . By increasing r 0 to R * (if necessary) we then achieve u ε < u ε when 0 < ε < ε * . The proof of Lemma 5.1 is thereby complete.

Remark 5.2. -Observe that, on account of Remarks 2.5, 3.2 and 4.8, the same proof as above yields an analogous result with L g in place of L. To see this, it suffices to notice that our arguments are essentially focused on what is happening far away from K and, since the kernel we consider is compactly supported, the operator L g will then coincide with L (possibly up to take R sufficiently large). In like manner, as already mentioned in Remark 2.5, the fact that "sup R N \Kε u = 1" implies that "lim |x|→∞ u(x) = 1" still holds with L g in place of L since, here as well, the proof relies only on estimates of the behaviour of u far away from K ε .

Appendix

In this appendix, we prove Lemma 2.1. Our strategy closely follows [START_REF] Brasseur | Liouville type results for a nonlocal obstacle problem[END_REF][START_REF] Coville | On a non-local reaction diffusion equation arising in population dynamics[END_REF] and relies on the well-known monotone iterative method. Before doing so, we first state a preliminary lemma.

Lemma 5.3. -Let K ⊂ R N be a compact set and assume that J satisfies (1.6). Let k > 0 and let w ∈ C(R N \ K) be such that

and that lim sup |x|→∞ w(x) 0.

(5.8)

Proof. -Suppose, by contradiction, that sup R N \K w > 0. Then, by assumption (5.8), there exists a number r > 0 with K ⊂ B r and a sequence (x j ) j 0 ⊂ B r \ K such that lim

(5.9)

Since (x j ) j 0 is bounded, up to extraction of a subsequence, there exists a point x ∈ B r \ K such that x j → x as j → ∞. Moreover, since w is continuous and (5.7) is satisfied everywhere in R N \K, it makes sense to evaluate (5.7) at x j for any j 0.

That is, we have ˆRN \K J(x j -y)(w(y) -w(x j ))dy kw(x j ) for any j 0.

But, since k > 0, using (5.9) and the dominated convergence theorem, we obtain

which is a contradiction. The proof is thereby complete.

We are now in position to prove Lemma 2.1.

Proof of Lemma 2.1. -Let us first observe that, from the assumptions made on J, the operator L is linear and continuous on (C 0 (R N \ K), • ∞ ), where

Indeed, this is because, given any w ∈ C 0 (R N \ K), we have

where J is as in (2.4), and, by the dominated convergence theorem, we have that Lw(x) → 0 as |x| → ∞. The continuity of Lw is a mere consequence of the continuity of translations in L 1 (R N ) and of the continuity of w, as is easily seen from the (trivial) inequality (5.10) which holds for any

Moreover, the continuity of the operator L follows from the fact that

Next, we let k > 0 be a number large enough so that the map s → -ks -f (s) is decreasing in [0, 1] and that k ∈ ρ(L) where ρ(L) denotes the resolvent of the operator L.

Let u and u be continuous global sub-and super-solutions to

satisfying (2.1) and (2.2). We will construct a solution u to (5.11) satisfying u u u using a monotone iterative scheme. That is, we will construct u as the limit of an appropriate sequence of functions. The main tool behind our construction is the comparison principle Lemma 5.3. To this end, we have to make sure that the sequence we construct has the right asymptotic behavior as |x| → ∞ (as required by Lemma 5.3). With this aim in mind, we first construct an appropriate sequence of auxiliary functions. Namely, we define v 0 ≡ 0 and, for x ∈ R N \ K and j 0, we let

(5.12)

Let us check that the v j 's are well-defined elements of

and

Similarly, if, for some j 0, it holds that v j ∈ C 0 (R N \ K), then, given that k ∈ ρ(L) and that Lu ∈ C 0 (R N \ K), v j+1 is a well-defined element of C 0 (R N \ K) as soon as

which trivially holds since f is continuous, f (1) = 0 and u(x) → 1, v j (x) → 0 as |x| → ∞. Whence, by induction, we infer that the v j 's are, indeed, well-defined elements of C 0 (R N \ K).

Let us now define a sequence (u j ) j 0 ⊂ C(R N \ K) by setting u j := u + v j . Then, by construction, for any x ∈ R N \ K and j 0, we have Lu j+1 (x) -ku j+1 (x) = -ku j (x) -f (u j (x)), (5.13) and the u j 's satisfy the limit condition lim |x|→∞ u j (x) = 1 for any j 0.

(5.14) 200

We will show that the desired solution to (5.11) can be obtained as the pointwise limit of (u j ) j 0 . Let us proceed step by step. First, when j = 0, we have

We claim that u u 1 u 0 = u in R N \ K. Indeed, we have

Since u 0 = u is a super-solution to (5.11), u u and s → -ks -f (s) is decreasing, we obtain that

By construction of u 1 (remember (2.1) and (5.14)), we have

This, together with Lemma 5.3, then gives that u u 1 u 0 = u in R N \ K. Similarly, by (5.13), the function u 2 ∈ C(R N \ K) solves (5.15) with u 2 in place of u 1 and u 1 in place of u 0 . Thus, from (2.1), (5.14) and the monotonicity of s → -ks -f (s), we deduce that (5.16) and (5.17) still hold with u 2 instead of u 1 and u 1 instead of u 0 . We may then apply the comparison principle Lemma 5.3 and we deduce that u u 2 u 1 u 0 = u in R N \ K. By induction, we infer that the u j 's satisfy the monotonicity relation

Since (u j ) j 0 is non-increasing and bounded from below by u, the function

is well-defined for any x ∈ R N \ K. In particular, since 0 u u 1, it follows from (5.18) that u ∈ L ∞ (R N \ K). It remains only to check that the function u is a solution to (5.11). For it, it suffices to let j → ∞ in (5.13) (using the dominated convergence theorem), which then gives

The proof is thereby complete.

Chapter 5 Some monotonicity results for general systems of nonlinear elliptic PDEs

This chapter is inspired by the paper [START_REF] Brasseur | Some monotonicity results for general systems of nonlinear elliptic PDEs[END_REF] written in collaboration with S. Dipierro, published in the Journal of Differential Equations.

Introduction 1.Monotonicity and 1-dimensional symmetry for systems

In this chapter, we study monotonicity properties of minima and stable solutions of general energy functionals of the type ˆΩ F (∇u, ∇v, u, v, x) dx, (

where Ω ⊂ R N . Recent years have seen numerous ongoing research activities in investigating symmetry properties of systems of PDEs. A typical example is

which arise in phase separation for Bose-Einstein condensates with multiple states.

The interested reader may refer to [START_REF] Berestycki | On phase-separation models: asymptotics and qualitative properties[END_REF][START_REF] Berestycki | On entire solutions of an elliptic system modeling phase separations[END_REF] (and references therein) for a derivation of this phase separation model.

We recall also some elementary properties of piecewise e N -Lipschitz deformations, which follow easily from the definitions above.

Proposition 1.3. -The following properties hold true:

In the sequel, we will also assume a growth condition on the functional E . Namely, we suppose that there exists a constant C > 0 such that, for R sufficiently large,

(1.5)

A monotonicity result for minimizers of the energy functional

The first result here deals with e N -minimizers of E . To state it, we give the following:

-We say that (u, v), with u, v ∈ C 0,1 (Ω), is an e N -minimizer for E if, for any R > 0, we have that E R (u, v) is finite and

We are now in the position to state our first monotonicity result.

Theorem 1.5. -Let u, v ∈ C 1 (Ω) and let F satisfy (1.3) and (1.4). Suppose that (u, v) is an e N -minimizer for the energy E and that the growth condition (1.5) is satisfied. Then, u and v are monotone on each line in the e N -direction, i.e., for any x ∈ Ω, either u N (x + te N ) 0 or u N (x + te N ) 0, and either v N (x + te N ) 0 or v N (x + te N ) 0, for any t ∈ R.

Remark 1.6. -We notice that if a continuous function w is monotone on each line in R N , then it is one-dimensional, that is there exist a function w 0 : R → R and a unit direction ω ∈ S N -1 such that w(x) = w 0 (ω • x) (see [START_REF]Some monotonicity results for minimizers in the calculus of variations[END_REF]Section 9] for a proof of this fact). Lemma 3.1. -Suppose that 0 ∈ Ω, and that u, u 1 , v, v 1 are C 1 functions which satisfy u(0) = u 1 (0), u 1 N (0) < 0 < u N (0), (

and v(0) = v 1 (0), v N (0) < 0 < v 1 N (0). Then, the couples

are not e N -minimizers for E in any ball B η with η > 0.

Proof. -We prove the lemma for the couple g 1 , the proof for g 2 being similar with obvious modifications.

We argue by contradiction and we assume that g 1 is an e N -minimizer in a ball B η , for some η > 0. We define F 0 (p 1 , p 2 ) := F (p 1 , p 2 , 0, 0, 0). Moreover, we observe that if we subtract a linear functional from F then this does not affect the minimality. Indeed, if we consider

and if E is the associated energy functional, then we have

This means that the difference between E R (u, v) and E R (u, v) is a term which depends only on the boundary value of u. Therefore, if (u, v) is an e N -minimizer for E , then it is an e N -minimizer for E as well. In view of this, we may assume that F 0 (∇u(0), ∇v(0)) = F 0 (∇u 1 (0), ∇v(0)).

Moreover, we can suppose that u(0) = u 1 (0) = 0 (up to translate F in the variable z 1 ). Now, for small r > 0, we define the rescaled functions u r (x) := r -1 u(rx), u 1 r (x) := r -1 u 1 (rx) and v r (x) := r -1 v(rx). We also set g r := (max {u r , u

is the rescaled functional, we have that g r is an e N -minimizer for F r in B η/r . Sending r → 0 + , we obtain the following limits, which are uniform on compact sets:

and

Now, we take a Lipschitz function ψ 1 with compact support in B η , and we use the fact that (u, v) is a critical point for E to obtain

Using (3.12) and (3.13), we can write

Now, we claim that, for η sufficiently small,

as t → 0. We focus on the first equality in (3.17), since the second is similar. To prove it, fixed µ > 0, we define

Notice that (3.14) implies that

As for the contribution coming from B 1 µ , we observe that, since ∇u N (0) = 0, the measure of B 1 µ is at most of the order of µ. This, together with (3.18), gives that

for some C > 0. Since µ can be taken arbitrarily small, this implies (3.17). Formula (3.17) means that, for η sufficiently small, we can write (u N ) + instead of u 1 + in the right hand side of (3.15). Hence, recalling also (3.16), we have that

Thus, we can apply the Hopf Lemma to w at the point x 0 , which admits a tangent ball from either {w < 0} ∩ P j 1 or {w > 0} ∩ P j 2 , and the conclusion follows. If x 0 ∈ {w = 0} ∩ N , we replace w with -w and we reason as above. If x 0 ∈ {w = 0} ∩ E, we let E j 2 , for some j 2 ∈ J 2 , be the connected component of E such that x 0 ∈ E j 2 . By the continuity of f 1 , there exists j 1 ∈ J 1 such that K := E j 2 ∪ P j 1 is connected (up to exchanging w with -w). Again, (3.22) implies that w cannot vanish identically in K, and so we can apply the Hopf Lemma to w at x 0 , which admits a tangent ball from either {w < 0} ∩ K or {w > 0} ∩ K. The same arguments apply to v N . This completes the proof of Lemma 3.4.

Remark 3.5. -The above result states that the non-degeneracy hypothesis ∇u N = 0 of Lemma 3.2 is always satisfied under a slightly stronger hypothesis on the potential F . On the other hand, notice that we obtain a C 3,α regularity for u and v near 0, even though we asked only for, say, a C 1,1 regularity. Thus, Lemma 3.2 is consistent.

Perturbations at infinity

In this section we modify the couples max{u(x), u(x) + te N }, v(x) and u(x), max{v(x), v(x) + te N } at infinity in such a way that they become compact perturbations of the couple (u, v). For this, for any R > 1 we define the function ϕ R : R → R, which is Lipschitz, even and with compact support, as

We have that

Also, for any t ∈ (0, √ R/4], we consider the following bi-Lipschitz change of coordinates:

x → y(x

In these new coordinates, we define the functions u + R,t and v + R,t as u + R,t (y) := u(x) and v + R,t (y) := v(x).

We observe that u + R,t (x) and v + R,t coincide with u(x -te N ) and v(x -te N ) respectively in B √ R/2 and with u(x) and v(x) respectively outside B R . We also define u - R,t and v - R,t by replacing t with -t in (4.3). Now, we want to obtain an estimate of

For this, we notice that

.

Furthermore, the following relations hold:

x and dy = 1 + tr(A) dx.

Hence, we have

General systems with p-Laplacian type operators

Theorem 1.10 actually applies to a broader class of operators and nonlinearities. Indeed, with the notation introduced in [START_REF] Dipierro | Geometric inequalities and symmetry results for elliptic systems[END_REF] (see in particular pages 3474-3475 there), one can consider div a(|∇u|

where F is a C 1,1 loc function on R 2 (it corresponds to the function F introduced in [START_REF] Dipierro | Geometric inequalities and symmetry results for elliptic systems[END_REF], here denoted as F to avoid confusion), and F 1 , F 2 denote the derivatives of F with respect to the first and the second variable respectively.

Then, we have the following:

) is a stable solution to (7.4), and that conditions (B1) and (B2) in [START_REF] Farina | Bernstein and De Giorgi type problems: new results via a geometric approach[END_REF] are satisfied for a, b, A and B. Also, assume that the following growth condition holds true: there exists a constant C > 0 such that, for R sufficiently large,

where Λ 2 and Γ 2 are as in [START_REF] Dipierro | Geometric inequalities and symmetry results for elliptic systems[END_REF]. Then, (u, v) possesses one-dimensional symmetry.

Proof

Some open problems

In this last chapter, we list a few possible extensions of our results.

On Chapter 1

A natural question is whether Theorem 2.3 extends to the case of Besov-Nikol'skii spaces on domains, at least under some regularity assumptions.

Open problem -Let Ω ⊂ R N be a smooth domain (possibly unbounded). Let s ∈ (0, 1), p ∈ [1, ∞) and let (ρ ε ) ε>0 ⊂ L 1 (R N ) be a family of mollifiers of the form ρ ε (z) = ε -N ρ(ε -1 z). Then, is it true that Our guess is that this holds true for every domain Ω ⊂ R N whose complement satisfies the measure density condition: ∃c > 0, |B r (x) ∩ c Ω| cr N for any (r, x) ∈ (0, 1] × c Ω.

Although this condition is quite weak it seems however sufficiently strong to avoid "cusp" like singularities (such as a disk with a slit).

A related question would be to investigate whether functionals of the form ˆΩ ˆΩ ρ ε (d g (x, y)) |f (x) -f (y)| p d g (x, y) sp dxdy, where d g (•, •) is the geodesic distance on Ω, can provide a description of B s p,∞ (Ω) and, if so, under which conditions on Ω. This could pave the way towards a new description of Besov-Nikol'skii spaces in general metric spaces and may be useful in problems of the type of those addressed at Chapters 3 and 4.

Another interesting perspective of research would be to figure out whether there are some reasonable conditions ensuring that a bounded sequence (f ε ) ε>0 ⊂ L p (R N ) satisfying ˆRN ˆRN ρ ε (x -y) |f ε (x) -f ε (y)| p |x -y| sp dxdy C as ε ↓ 0, is relatively compact in L p loc (R N ).

On Chapter 2

In Chapter 2, we have shown that, for every f ∈ B s p,q (R N ) with p < q, it holds that f (•, y) ∈ B (s,Ψ) p,q (R d ) for a.e. y ∈ R N -d , provided that Ψ satisfies the growth condition

If we could show that this latter condition is optimal when q = ∞, our results are less satisfying when q < ∞ since we required the further assumption that qp q -p < 1 c ∞ .

Is it possible to get rid of this extra condition?

Open problem -Let N 2, 1 d < N , 0 < p < q < ∞, s > σ p and let Ψ be an admissible function that does not satisfy (7.12). Then, does there exist a function f ∈ B s p,q (R N ) such that f (•, y) / ∈ B (s,Ψ) p,q (R d ) for a.e. y ∈ R N -d ?

On Chapters 3 and 4

In Chapter 3, we have shown that the requirement on the asymptotic behavior of u can be replaced by a condition on the supremum of u provided that J is square integrable. However, this assumption is quite unnatural and it would be interesting to find out whether it is possible to get rid of it. The reason why we had to make this hypothesis is because we had to cope with the lack of compactness which forced us to make some additional assumptions on J. In some sense, it is a bit counterintuitive because it says that J cannot be too singular, while, in the case of the fractional Laplacian, it is precisely the singularity which yields compactness.