(Non)local logistic equations with Neumann conditions
aa r X i v : . [ m a t h . A P ] J a n (NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS SERENA DIPIERRO, EDOARDO PROIETTI LIPPI, AND ENRICO VALDINOCI
Abstract.
We consider here a problem of population dynamics modeled on a logistic equationwith both classical and nonlocal diffusion, possibly in combination with a pollination term.The environment considered is a niche with zero-flux, according to a new type of Neumanncondition.We discuss the situations that are more favorable for the survival of the species, in terms ofthe first positive eigenvalue.Quite surprisingly, the eigenvalue analysis for the one dimensional case is structurally differ-ent than the higher dimensional setting, and it sensibly depends on the nonlocal character ofthe dispersal.The mathematical framework of this problem takes into consideration the equation − α ∆ u + β ( − ∆) s u = ( m − µu ) u + τ J ⋆ u in Ω , where m can change sign.This equation is endowed with a set of Neumann condition that combines the classical normalderivative prescription and the nonlocal condition introduced in [S. Dipierro, X. Ros-Oton,E. Valdinoci, Rev. Mat. Iberoam. (2017)].We will establish the existence of a minimal solution for this problem and provide a through-out discussion on whether it is possible to obtain non-trivial solutions (corresponding to thesurvival of the population).The investigation will rely on a quantitative analysis of the first eigenvalue of the associatedproblem and on precise asymptotics for large lower and upper bounds of the resource.In this, we also analyze the role played by the optimization strategy in the distribution ofthe resources, showing concrete examples that are unfavorable for survival, in spite of the largeresources that are available in the environment. Contents
1. Introduction 22. Functional analysis setting 103. Existence results and proofs of Theorems 1.1 and 1.2 124. Analysis of the eigenvalue problem in (1.13) and proof of Theorem 1.4 165. Optimization on m and proofs of Theorems 1.5, 1.7, 1.8, 1.9 and 1.10 186. Badly displayed resources, hectic oscillations and proof of Theorem 1.11 43Appendix A. Proofs of Theorems 1.7 and 1.8 when n = 2 Mathematics Subject Classification.
Key words and phrases.
Logistic equation, Fisher-KPP equation, long-range interactions, zero-flux condition.
Serena Dipierro : Department of Mathematics and Statistics, University of Western Australia, 35 StirlingHwy, Crawley WA 6009, Australia. [email protected]
Edoardo Proietti Lippi : Department of Mathematics and Computer Science, University of Florence, VialeMorgagni 67/A, 50134 Firenze, Italy. [email protected]
Enrico Valdinoci : Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Hwy,Crawley WA 6009, Australia. [email protected]
The authors are members of INdAM. The first and third authors are members of AustMS and are supportedby the Australian Research Council Discovery Project DP170104880 NEW “Nonlocal Equations at Work”. Thefirst author is supported by the Australian Research Council DECRA DE180100957 “PDEs, free boundariesand applications”. The third author is supported by the Australian Laureate Fellowship FL190100081 “Minimalsurfaces, free boundaries and partial differential equations”. Part of this work was carried out during a verypleasant and fruitful visit of the second author to the University of Western Australia, which we thank for thewarm hospitality.
Appendix B. Probabilistic motivations for the superposition of elliptic operators withdifferent orders 51References 531.
Introduction
We consider here a biological population with density u which is self-competing for theresources in a given environment Ω .These resources are described by a function m , which is allowed to change sign: the positivevalues of m correspond to areas of the environment favorable for life and produce a positive birthrate, whereas the negative values model a hostile environment whose byproduct is a positivedeath rate of linear type.The competition for the resource is encoded by a nonnegative function µ . Resources andcompetitions are combined into a standard logistic equation. In addition, the population isassumed to present a combination of classical and nonlocal diffusion (the cases of purely classicaland purely nonlocal diffusions are also included in our setting, and the results obtained are newalso for these cases). The population is also endowed with an additional birth rate possiblyprovided by pollination and modeled by a convolution operator (the case of no pollination isalso included in our setting, and the results obtained are new also for this case).The environment Ω describes an ecological niche and is endowed by a zero-flux conditionof Neumann type. Given the possible presence of both classical and nonlocal dispersal, thisNeumann condition appears to be new in the literature: when the diffusion is of purely classicaltype this new prescription reduces to the standard normal derivative condition along ∂ Ω , andwhen the diffusion is of purely nonlocal type it coincides with the nonlocal Neumann conditionset in R n \ Ω that has been recently introduced in [DROV17] – but in case the population issubject to both the classical and the nonlocal dispersion processes the Neumann condition thatwe introduce here takes into account the combination of both the classical and the nonlocalprescriptions (interestingly, without producing an overdetermined, or ill-posed, problem).The main question addressed in this paper is whether or not the environmental niche issuited for the survival of the population (notice that life is not always promoted by the ambientresource, since m can attain negative values). We will investigate this question by using spectralanalysis and providing a detailed quantification of favorable and unfavorable scenarios in termsof the first eigenvalue compared with the resource and pollination parameters.More precisely, the mathematical framework in which we work goes as follows. We consider abounded open set Ω ⊂ R n with boundary of class C : that is, we suppose that there exist R > and p , . . . , p K ∈ ∂ Ω such that ∂ Ω ⊂ B R ( p ) ∪ · · · ∪ B R ( p K ) , and, for each i ∈ { , . . . , K } ,(1.1) the set Ω ∩ B R ( p i ) is C -diffeomorphic to B +1 := { ( x , . . . , x n ) ∈ B s.t. x n > } .Given s ∈ (0 , , α , β ∈ [0 , + ∞ ) , with α + β > , m : Ω → R , µ : Ω → (cid:2) µ, + ∞ (cid:1) , with µ > , τ ∈ [0 , + ∞ ) and J ∈ L ( R n , [0 , + ∞ )) with(1.2) J ( x ) = J ( − x ) and(1.3) Z R n J ( x ) dx = 1 , While we use the name of pollination throughout this paper, we observe that the pollination analysisperformed is not limited to vegetable species: indeed, for animal species the convolution term that we studycan be seen as a birth rate of nonlocal type produced, for instance, by a mating call which attracts partnersfrom surrounding neighbors.
NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 3 we consider the mixed order logistic equation(1.4) − α ∆ u + β ( − ∆) s u = ( m − µu ) u + τ J ⋆ u in Ω , where J ⋆ u ( x ) := Z Ω J ( x − y ) u ( y ) dy. When β = 0 , we take the additional hypothesis that(1.5) Ω is connected.We observe that the operator in (1.4) is of mixed local and nonlocal type, and also of mixedfractional and integer order type. Interestingly, the nonlocal character of the operator is encodedboth in the fractional Laplacian ( − ∆) s u ( x ) := 12 Z R n u ( x ) − u ( x + ζ ) − u ( x − ζ ) | ζ | n +2 s dζ and in the convolution operator given by J .The use of the convolution operator in biological models to comprise the interaction ofthe population with the resource at a certain range has a very consolidated tradition, seee.g. [ABVV10, CDM13, ACR13, Cov15, BCV16, BCL17, CDV17] and the references therein.As for the nonlocal diffusive operator, for the sake of concreteness we stick here to theprototypical case of the fractional Laplacian, but the arguments that we develop are in factusable in more general contexts including various interaction kernels of singular type.Given the presence of both the Laplacian and the fractional Laplacian, the operator in (1.4)falls within the diffusive processes of mixed orders, which have been widely addressed by sev-eral methodologies and arose from a number of different motivations, see for instance vari-ous viscosity solutions approaches [JK05, JK06, BI08, BJK10, BCCI12, Cio12, BCCI14, dTEJ17,AdTEJ19], the Aubry-Mather theory for pseudo differential equations [dlLV09], Cahn-Hilliardand Allen-Cahn-type equations [CV13, CS16], probability and Harnack inequalities [BK05a,BK05b, CKSV10, CKSV12], decay for parabolic equations [DVV19, AV19b], friction and dissi-pation effects [DP18], smooth approximation with suitable solutions [CS10], Bernstein-type reg-ularity results [CDV], variational methods [BDVV20], nonlinear operators [AC20] and plasmaphysics [BdCN13].We endow the problem in (1.4) with a set of Neumann boundary conditions that correspondto a “zero-flux” condition according to the stochastic process producing the diffusive operatorin (1.4). This Neumann condition appears to be new in the literature and depends on thedifferent ranges of α and β according to the following setting. If α = 0 , we consider thenonlocal Neumann condition introduced in [DROV17], thus prescribing that(1.6) N s u ( x ) := Z Ω u ( x ) − u ( y ) | x − y | n +2 s dy = 0 for every x ∈ R n \ Ω . If instead β = 0 , we prescribe the classical Neumann condition(1.7) ∂u∂ν = 0 on ∂ Ω . Finally, if α = 0 and β = 0 , we prescribe both the classical and the nonlocal Neumannconditions, by requiring that(1.8) N s u = 0 in R n \ Ω ,∂u∂ν = 0 on ∂ Ω . We remark that the prescription in (1.8) is not an “overdetermined” condition (as it will beconfirmed by the existence result in Theorem 1.1 below).
SERENA DIPIERRO, EDOARDO PROIETTI LIPPI, AND ENRICO VALDINOCI
The set of boundary/external Neumann conditions in (1.6), (1.7) and (1.8), in dependenceof the different ranges of α and β , will be denoted by “ ( α, β ) -Neumann conditions”, and, withthis notation and (1.4), the main question studied in this paper focuses on the problem(1.9) (cid:26) − α ∆ u + β ( − ∆) s u = ( m − µu ) u + τ J ⋆ u in Ω , with ( α, β ) -Neumann condition.In this setting, the ( α, β ) -Neumann conditions provide an “ecological niche” for the populationwith density u , making Ω a natural environment in which a given species can live and competefor a resource m , according to a competition function µ . In this setting, the parameter τ , asmodulated by the interaction kernel J , describes an additional birth rate due to further inter-communication than just with the closest neighbors, as it happens, for instance, in pollination.As a matter of fact, the role of the ( α, β ) -Neumann conditions is precisely to make theboundary and the exterior of the niche Ω “reflective”: namely when an individual exits theniche, it is forced to immediately come back into the niche itself, following the same diffusiveprocess, see Section 2 in [DROV17] (see also [Von19] for a thoroughgoing probabilistic discussionabout this process).As a technical remark, we also observe that our ( α, β ) -Neumann condition is structurallydifferent (even when α = 0 and s = 1 / ) from the case of bounded domains with reflectingbarriers presented in [MPV13, PV18], and the diffusive operator taken into account in (1.9)cannot be obtained by the spectral decomposition of the classical Laplacian in Ω (except forthe special case of periodic environments, see e.g. Section 2.3 and Appendix Q in [AV19a]).The possible presence in (1.9) of two different diffusion operators, one of classical and theother of fractional flavor, has a clear biological interpretation, namely the population withdensity u can possibly alternate both short and long-range random walks, and this could bemotivated, for instance, by a superposition between local exploration of the environment andhunting strategies (see e.g. [VAB +
96, DHMP98, CCL07, CHL08, CDM12, KLS12, CCLR12, SV17,MV17]). A detailed presentation of this superposition of stochastic processes will be presentedin Appendix B; see also [DV] for the detailed description of the local/nonlocal reflecting barrieralso in terms of the population dynamics model.The notion of solution of (1.9) is intended here in the weak sense, as it will be discussedprecisely in formula (2.5). See however [GM02, BDVV20] for a regularity theory for weaksolutions of the equations driven by the mixed order operators as in (1.9).Our first result in this setting is that the problem in (1.9) admits a minimal energy solution(under very natural and mild structural assumptions). To state it, it is convenient to define q := ∗ ∗ − if β = 0 and n > , ∗ s ∗ s − if β = 0 and n > s, if β = 0 and n , or if β = 0 and n s, = n if β = 0 and n > ,n s if β = 0 and n > s, if β = 0 and n , or if β = 0 and n s .(1.10)As customary, the exponent ∗ s denotes the fractional Sobolev critical exponent for n > s andit is equal to nn − s . Similarly, the exponent ∗ denotes the classical Sobolev critical exponentfor n > and it is equal to nn − .We remark that q > n/ , and we have: NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 5
Theorem 1.1.
Assume that m ∈ L q (Ω) , for some q ∈ (cid:0) q, + ∞ (cid:3) and ( m + τ ) µ − ∈ L (Ω) .Then, there exists a nonnegative solution of (1.9) which can be obtained as a minimum of anenergy functional. The precise definition of energy functional used in Theorem 1.1 will be presented in (3.1):roughly speaking, the energy associated to Theorem 1.1 will be the “natural” functional for thevariational methods, and its Euler-Lagrange equation will correspond to the notion of weaksolution.While the functional analysis part of the proof of Theorem 1.1 relies on standard directmethods in the calculus of variations, the more interesting part of the argument makes use of astructural property of the nonlocal Neumann condition that will be presented in Theorem 2.1(roughly speaking, the nonlocal Neumann condition in (1.6) will be instrumental to minimizethe Gagliardo seminorm, thus clarifying the energetic role of the nonlocal reflection introducedin [DROV17]).Though the result in Theorem 1.1 has an obvious interest in pure mathematics, our mainanalysis will focus on whether problem (1.9) does admit a nontrivial solution (notice indeedthat u ≡ is always a solution of (1.9)). In particular, in view of Theorem 1.1, a usefulmathematical tool to detect nontrivial solutions consists in proving that the minimal energyconfiguration is not attained by the trivial solution (hence, in this case, the solution producedby Theorem 1.1 is nontrivial). The question of the existence of nontrivial solutions has acentral importance for the mathematical model, since it corresponds to the possibility of apopulation to survive in the environmental condition provided by the niche. Interestingly, inour model, the survival of the population can be enhanced by the possibility of exploitingresources by long-range interactions. Indeed, we stress that the nonlocal resource m in (1.4) isnot necessarily positive (hence, the natural environment can be “hostile” for the population):in this configuration, we show that the survival of the species is still possible if the “pollination”birth rate τ is sufficiently large. The quantitative result that we have is the following: Theorem 1.2.
Assume that m ∈ L q (Ω) , for some q ∈ (cid:0) q, + ∞ (cid:3) and ( m + τ ) µ − ∈ L (Ω) .Then,(i) if m ≡ and τ = 0 , then the only solution of (1.9) is the one identically zero;(ii) if (1.11) Z Ω (cid:0) m ( x ) + τ J ⋆ x ) (cid:1) dx > and (1.12) µ ∈ L (Ω) , then (1.9) admits a nonnegative solution u . A particular case of Theorem 1.2 is when the resource m is nonnegative. In this situation,Theorem 1.2(i) gives that no survival is possible without resources and pollination, i.e. whenboth m and τ vanish identically (unless also µ vanishes identically, then reducing the problem tothat of mixed operator harmonic functions), whereas Theorem 1.2(ii) guarantees survival if atleast one between the environmental resource and the pollination is favorable to life. Precisely,one can immediately deduce from Theorem 1.2 the following result: SERENA DIPIERRO, EDOARDO PROIETTI LIPPI, AND ENRICO VALDINOCI
Corollary 1.3.
Assume that m ∈ L q (Ω) , for some q ∈ (cid:0) q, + ∞ (cid:3) , m is nonnegative,and ( m + τ ) µ − ∈ L (Ω) .Then,(i) if m ≡ and τ = 0 , then the only solution of (1.9) is the one identically zero;(ii) if either m > or τ ( J ⋆ > in a set of positive measure and µ ∈ L (Ω) , then (1.9) admits a nonnegative solution u . Problems related to Corollary 1.3 have been studied in [CDV17] under Dirichelet (ratherthan Neumann) boundary conditions.From the biological point of view, assumption (1.11) states that the environment is “inaverage” favorable for the survival of the species. It is therefore a natural question to investigatethe situation in which the environment is “mostly hostile to life”. To study this phenomenon,when m ∈ L q with q > n/ , with m + and Z Ω m ( x ) dx < , we denote by λ the first positive eigenvalue associated with the diffusive operator in (1.9).More precisely, we consider the weighted eigenvalue problem(1.13) (cid:26) − α ∆ u + β ( − ∆) s u = λmu in Ω , with ( α, β ) -Neumann condition.As it will be discussed in detail in Proposition 4.1 here and in [DPLV], problem (1.13) admitsthe existence of two unbounded sequences of eigenvalues, one positive and one negative. Inthis setting, the smallest strictly positive eigengevalue will be denoted by λ . When we wantto emphasize the dependence of λ on the resource m , we will write it as λ ( m ) .We also denote by e an eigenfunction corresponding to λ normalized such that Z Ω m ( x ) e ( x ) dx = 1 . The first eigenvalue will be an important threshold for the survival of the species, quantifyingthe role of the necessary pollination parameter τ in order to overcome the presence of an hostilebehavior in average. The precise result that we obtain is the following one: Theorem 1.4.
Assume that m ∈ L q (Ω) , for some q ∈ (cid:0) q, + ∞ (cid:3) , and ( m + τ ) µ − ∈ L (Ω) .Then,(i) if m − τ , then the only solution of (1.9) is the one identically zero;(ii) if m + , µ ∈ L (Ω) , (1.14) Z Ω m ( x ) dx < , and (1.15) λ − < τ Z Ω ( J ⋆ e ( x )) e ( x ) dx, then (1.9) admits a nonnegative solution u . As customary, we freely use in this paper the standard notation m + ( x ) := max { , m ( x ) } and m − ( x ) := max { , − m ( x ) } . NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 7
Once again, in Theorem 1.4, the case described in (i) is the one less favorable to life, sincethe combination of both the resources and the pollination is in average negative, while the casein (ii) gives a lower bound of the pollination parameter τ which is needed for the survival ofthe species, as quantified by (1.15).We recall that the link between the survival ability of a biological population and the analysisof the eigenvalues of a linearized problem is a classical topic in mathematical biology, seee.g. [Ske51, Bel97, BHR05a, BHR05b, KS07, KAS12, MPV13, Maz20a, Maz20b] (yet, we believethat this is the first place in which a detailed analysis of this type is carried over to the case ofmixed operators with our new type of Neumann conditions).In light of (1.15), a natural question consists in quantifying the size of the first eigenvalue.Roughly speaking, from (1.15), the smaller λ , the smaller is the threshold for the pollinationguaranteeing survival, hence configurations with small first eigenvalues correspond to the onesof better chances of life.To address this problem, since the eigenvalue λ = λ ( m ) depends on the resource m , it isconvenient to consider an optimization problem for λ in terms of three structural parametersof the resource m , namely its minimum, its maximum and its average, in order to detect underwhich conditions on these parameters the first eigenvalue can be made conveniently small. Moreprecisely, given m , m ∈ (0 , + ∞ ) and m ∈ ( − m, we consider the class of resources M = M ( m, m, m ) := n m ∈ L ∞ (Ω) s.t. inf Ω m > − m, sup Ω m m, Z Ω m ( x ) dx = m | Ω | and m + o . (1.16)We will also consider the smallest possible first eigenvalue among all the resources in M , namelywe set(1.17) λ := inf m ∈ M λ ( m ) . When we want to emphasize the dependence of λ on the structural quantities m , m and m that characterize M , we will adopt the explicit notation λ ( m, m, m ) .Our main objective will be to detect whether or not λ can be made arbitrarily small in anumber of different regimes: we stress that the smallness of λ corresponds to a choice of anoptimal distribution of resources that is particularly favorable for survival.The first result that we present in this direction is a general estimate controlling λ with O (cid:0) m (cid:1) ,provided that the maximal hostility of the environment does not prevail with respect to themaximal and average resources. In terms of survival of the species, this is a rather encouragingoutcome, since it allows the existence of nontrivial solutions provided that the maximal resourceis sufficiently large. The precise result that we have is the following: Theorem 1.5.
Let (1.18) m + m m + m > d for some d > . Then, λ ( m, m, m ) Cm for some C = C (Ω , d ) > . A direct consequence of Theorem 1.5 gives that when the upper and lower bounds of theresource are the same and get arbitrarily large, then λ gets arbitrarily small (hence, in viewof (1.15), there exists a resource distribution which is favorable to survival). More precisely: To avoid notational confusion, we reserved the name of m to the resource in (1.4) and we denoted by m a“free variable” dimensionally related to the resource. SERENA DIPIERRO, EDOARDO PROIETTI LIPPI, AND ENRICO VALDINOCI
Corollary 1.6.
We have lim m ր + ∞ λ ( m , m , m ) = 0 . We now investigate the behavior of λ for large upper and lower bounds on the resource(maintaining constant the other parameters). Interestingly, this behavior sensibly depends onthe dimension n . In this setting, we first consider the asymptotics in dimension n > : we showthat large upper and lower bounds are both favorable for life for a given m < , according tothe following two results: Theorem 1.7.
Let n > . Then, lim m ր + ∞ λ ( m , m, m ) = 0 . Theorem 1.8.
Let n > . Then, lim m ր + ∞ λ ( m, m , m ) = 0 . While Theorem 1.7 is somehow intuitive (large resources are favorable to survival), at afirst glance Theorem 1.8 may look unintuitive, since it seems to suggest that a largely hostileenvironment is also favorable to survival: but we remark that in Theorem 1.8, being m given,an optimal strategy for m may well correspond to a very harmful environment confined in asmall portion of the domain, with a positive resource allowing for the survival of the species.Quite surprisingly, the structural analysis developed in Theorems 1.7 and 1.8 is significantlydifferent in dimension . Indeed, for n = 1 , we have that λ does not become infinitesimalfor large upper and lower bounds on the resource, unless the diffusion is purely nonlocal withstrongly nonlocal fractional parameter. Namely, we have the following two results. Theorem 1.9.
Let n = 1 , α > and β > . Then, for any m > and m ∈ ( − m, , (1.19) λ ( m , m, m ) > C for every m > , for some C = C ( m, m , α, β, Ω) > , and (1.20) lim m ց λ ( m , m, m ) = + ∞ . Moreover, for any m > and m < , (1.21) λ ( m, m , m ) > C for every m > − m , for some C = C ( m, m , α, β, Ω) > . Theorem 1.10.
Let n = 1 , α = 0 and β > .If s ∈ (1 / , , then, for any m > and m ∈ ( − m, (1.22) λ ( m , m, m ) > C for every m > , for some C = C ( m, m , α, β, Ω) > , and (1.23) lim m ց λ ( m , m, m ) = + ∞ . Moreover, for any m > and m < (1.24) λ ( m, m , m ) > C for every m > − m , for some C = C ( m, m , α, β, Ω) > .If s ∈ (0 , / , then, (1.25) lim m ր + ∞ λ ( m , m, m ) = 0 , and (1.26) lim m ր + ∞ λ ( m, m , m ) = 0 . NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 9
An interesting feature of Corollary 1.6, Theorems 1.7 and 1.8, (1.25) and (1.26) in terms ofreal-world applications is that their proofs are based on the explicit constructions of suitableresources: though perhaps not optimal, these resources are sufficiently well located to ensurethe maximal chances of survival for the population, and their explicit representation allows oneto use them concretely and to build on this specific knowledge.We also think that the phenomenon detected in Theorems 1.9 and 1.10 reveals an importantrole played by the nonlocal dispersal of the species in dimension : indeed, in this situation,the only configurations favorable to survival are the ones in (1.25) and (1.26), that are inducedby purely nonlocal diffusion (that is α = 0 ) with a strongly nonlocal diffusion exponent (thatis s / , corresponding to very long flies in the underlying stochastic process).To better visualize the results in Theorems 1.7, 1.8, 1.9 and 1.10, we summarize them inTable 1. For typographical convenience, in Table 1 we used the “check-symbol” ✓ to denote thecases in which λ gets as small as we wish (cases favorable to life) and the “x-symbol” ✗ to markthe situations in which λ remains bounded away from zero (cases unfavorable to life whichrequire stronger pollination for survival). Large m Large mn > ✓ ✓ n = 1 and α > ✗ ✗ n = 1 , α = 0 and s > / ✗ ✗ n = 1 , α = 0 and s / ✓ ✓ Table 1.
Summarizing the results in Theorems 1.7, 1.8, 1.9 and 1.10.
We stress that the optimization of the resources plays a crucial role in the survival resultsprovided by Corollary 1.6, Theorems 1.7 and 1.8, and formulas (1.25) and (1.26): that is,given m < , very large but badly displayed resources may lead to non-negligible first eigen-values (differently from the case of optimal distribution of resources discussed in Corollary 1.6,Theorems 1.7 and 1.8, and formulas (1.25) and (1.26)).To state precisely this phenomenon, given m < and Λ > − m , we let(1.27) M ♯ Λ ,m := (cid:26) m ∈ M (2Λ , , m ) s.t. inf Ω m − Λ2 and sup Ω m > Λ2 (cid:27) . Roughly speaking, the resources m in M ♯ Λ ,m have a prescribed average equal to m and attainmaximal positive and negative values comparable with a large parameter Λ , and a naturalquestion in this case is whether large Λ ’s provide sufficient conditions for the survival of thespecies. The next result shows that this is not the case, namely the abundance of the resourcewithout an optimal distribution strategy is not sufficient for prosperity: Theorem 1.11.
Given m < and Λ > − m , we have that sup m ∈ M ♯ Λ ,m λ ( m ) = + ∞ . Interestingly, the proof of Theorem 1.11 will be “constructive”, namely we will provide anexplicit example of a sequence of badly displayed resources which make the first eigenvaluediverge: a telling feature of this sequence is that it is highly oscillatory, thus suggesting thata hectic and erratic alternation of highly positive resources with very harmful surroundings ispotentially lethal for the development of the species.We recall that the investigation of the roles of fragmentation and concentration for resourcesis a classical topic in mathematical biology, and, in this sense, our result in Theorem 1.11confirms the main paradigm according to which concentrated resources favor survival (see e.g. [BHR05a, BHR05b, LLNP16]) – however, there are several circumstances in which thisgeneral paradigm is violated and fragmentation is better than concentration, see e.g. thesmall diffusivity regime analyzed in [LLL20, MRB20, LNY20]. In any case, the analysis offragmentation and concentration for mixed operators with our Neumann condition is, to thebest of our knowledge, completely new.We also remark that the results presented here are new even in the simpler cases in which noclassical diffusion and no pollination term is present in (1.4), as well as in the cases in whichthe death rate and the pollination functions are constant.The rest of this paper is organized as follows. In Section 2 we will introduce the functionalframework in which we work and the notion of weak solutions, also providing a new result show-ing that the nonlocal Neumann condition naturally produces functions with minimal Gagliardoseminorm (this is a nonlocal phenomenon, which has no counterpart in the classical setting,and will play a pivotal role in the minimization process).Then, in Section 3 we prove the existence results in Theorems 1.1 and 1.2. In Section 4 westudy the eigenvalue problemin (1.13), and we give the proof of Theorem 1.4. Not to overburdenthis paper, some technical proofs related to the spectral theory of the problem are deferred tothe article [DPLV].In Section 5, we deal with the proofs of Theorem 1.5, Theorems 1.7 and 1.8 when n > , andTheorems 1.9 and 1.10.When n = 2 , the proofs of Theorems 1.7 and 1.8 require some technical modification oflogarithmic type, hence their proofs is deferred to Appendix A.The proof of Theorem 1.11 is contained in Section 6.Finally, Section B contains some probabilistic motivations related to the diffusive operatorsof mixed integer and fractional order.2. Functional analysis setting
In this section we define the functional space in which we work. First, we recall the space H s Ω introduced in [DROV17] and defined as(2.1) H s Ω := (cid:26) u : R n → R s.t. u ∈ L (Ω) and Z Z Q | u ( x ) − u ( y ) | | x − y | n +2 s dx dy < + ∞ (cid:27) , where Q := R n \ ( R n \ Ω) . As customary, by u ∈ L (Ω) in (2.1) we mean that the restriction of the function u to Ω belongsto L (Ω) (we stress that functions in H s Ω are defined in the whole of R n ). Also, all functionsconsidered will be implicitly assumed to be measurable.Furthermore, we define(2.2) X α,β = X α,β (Ω) := H (Ω) if β = 0 ,H s Ω if α = 0 ,H (Ω) ∩ H s Ω if αβ = 0 . In light of this definition, X α,β is a Hilbert space with respect to the scalar product ( u, v ) X α,β := Z Ω u ( x ) v ( x ) dx + α Z Ω ∇ u ( x ) · ∇ v ( x ) dx + β Z Z Q ( u ( x ) − u ( y ))( v ( x ) − v ( y )) | x − y | n +2 s dx dy, (2.3)for every u, v ∈ X α,β .We also define the seminorm(2.4) [ u ] X α,β := α Z Ω |∇ u ( x ) | dx + β Z Z Q | u ( x ) − u ( y ) | | x − y | n +2 s dx dy. NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 11
From the compact embeddings of the spaces H (Ω) and H s Ω (see e.g. Corollary 7.2 in [DNPV12]when α = 0 ), we deduce the compact embedding of X α,β into L p (Ω) , for every p ∈ [1 , ∗ ) if α = 0 , and for every p ∈ [1 , ∗ s ) if α = 0 .We say that u ∈ X α,β is a solution of (1.9) if α Z Ω ∇ u ( x ) · ∇ v ( x ) dx + β Z Z Q ( u ( x ) − u ( y ))( v ( x ) − v ( y )) | x − y | n +2 s dx dy = Z Ω (cid:16)(cid:0) m ( x ) − µ ( x ) u ( x ) (cid:1) u ( x ) + τ ( x ) J ⋆ u ( x ) (cid:17) v ( x ) dx (2.5)for all functions v ∈ X α,β .Now we show that among all the functions in H s Ω , the ones minimizing the Gagliardo semi-norm are those satisfying the nonlocal Neumann condition in (1.6). This is a useful result initself, which also clarifies the structural role of the Neumann condition introduced in [DROV17]: Theorem 2.1.
Let u : R n → R with u ∈ L (Ω) , and set, for all x ∈ R n \ Ω , E u ( x ) := Z Ω u ( z ) | x − z | n +2 s dz. Then, if we define (2.6) ˜ u ( x ) := u ( x ) if x ∈ Ω ,E u ( x ) E ( x ) if x ∈ R n \ Ω , we have (2.7) Z Z Q | ˜ u ( x ) − ˜ u ( y ) | | x − y | n +2 s dx dy Z Z Q | u ( x ) − u ( y ) | | x − y | n +2 s dx dy. Also, the equality in (2.7) holds if and only if u satisfies (1.6) .Proof. We remark that the notation E in (2.6) stands for E u when u ≡ . Moreover, withoutloss of generality, we can suppose that Z Z Q | u ( x ) − u ( y ) | | x − y | n +2 s dx dy < + ∞ , otherwise the claim in (2.7) is obviously true.In addition,(2.8) Z Ω Z Ω | ˜ u ( x ) − ˜ u ( y ) | | x − y | n +2 s dx dy = Z Ω Z Ω | u ( x ) − u ( y ) | | x − y | n +2 s dx dy, so we only need to consider the integral on ( R n \ Ω) × Ω (being the integral on Ω × ( R n \ Ω) the same).Setting ϕ ( x ) := u ( x ) − ˜ u ( x ) , for every y ∈ R n \ Ω we have Z Ω | u ( x ) − u ( y ) | | x − y | n +2 s dx = Z Ω | u ( x ) − ˜ u ( y ) − ϕ ( y ) | | x − y | n +2 s dx (2.9) = Z Ω | u ( x ) − ˜ u ( y ) | − ϕ ( y )( u ( x ) − ˜ u ( y )) + | ϕ ( y ) | | x − y | n +2 s dx. Now, we observe that, for every y ∈ R n \ Ω , Z Ω u ( x ) − ˜ u ( y ) | x − y | n +2 s dx = E u ( y ) − E u ( y ) E ( y ) E ( y ) = 0 . Accordingly, (2.9) becomes Z Ω | u ( x ) − u ( y ) | | x − y | n +2 s dx = Z Ω | ˜ u ( x ) − ˜ u ( y ) | + | ϕ ( y ) | | x − y | n +2 s dx > Z Ω | ˜ u ( x ) − ˜ u ( y ) | | x − y | n +2 s dx, for every y ∈ R n \ Ω , and the equality holds if and only if ϕ ( y ) = 0 . Integrating over R n \ Ω (or, equivalently, on R n \ Ω ), we get Z R n \ Ω Z Ω | u ( x ) − u ( y ) | | x − y | n +2 s dx dy > Z R n \ Ω Z Ω | ˜ u ( x ) − ˜ u ( y ) | | x − y | n +2 s dx dy, and the equality holds if and only if ϕ ≡ in R n \ Ω . From this observation and (2.8) weobtain (2.7), as desired. (cid:3) Existence results and proofs of Theorems 1.1 and 1.2
The proof of Theorem 1.1 is based on a minimization argument. More precisely, given thefunctional setting introduced in Section 2 (recall in particular (2.2)), in order to deal withproblem (1.9), we consider the energy functional E : X α,β → R defined as E ( u ) := α Z Ω |∇ u | dx + β Z Z Q | u ( x ) − u ( y ) | | x − y | n +2 s dx dy + Z Ω (cid:18) µ | u | − mu − τ u ( J ⋆ u )2 (cid:19) dx. (3.1)As a technical remark, we observe that our objective here is to distinguish between trivial andnontrivial solutions, to detect appropriate conditions for the survival of the solutions and wedo not indulge in the distinction nonnegative and nontrivial versus strictly positive solutions.For the reader interested in this point, we mention however that, under appropriate conditions,one could develop a regularity theory (see e.g. Theorems 3.1.11 and 3.1.12 in [GM02]) thatallows the use of a strong maximum principle for smooth solutions (see e.g. Theorem 3.1.4in [GM02]).Now, we prove that the functional in (3.1) is the one associated with (1.9): Lemma 3.1.
The Euler-Lagrange equation associated to the energy functional E introducedin (3.1) at a non-negative function u is (1.9) .Proof. We compute the first variation of E , and we focus on the convolution term in (3.1) (beingthe computation for the other terms standard, see in particular Proposition 3.7 in [DROV17] todeal with the term involving the Gagliardo seminorm, which is the one producing the nonlocalNeumann condition).For this, we set J ( u ) := τ Z Ω u ( x )( J ⋆ u ( x )) dx. For any φ ∈ X α,β and ε ∈ ( − , , we have J ( u + εφ )= τ Z Ω ( u + εφ )( x )( J ⋆ ( u + εφ ))( x ) dx = τ Z Ω u ( x )( J ⋆ u )( x ) + ε (cid:2) u ( x )( J ⋆ φ )( x ) + φ ( x )( J ⋆ u )( x ) (cid:3) + ε φ ( x )( J ⋆ φ )( x ) dx. Accordingly,(3.2) d J dε ( u + εφ ) (cid:12)(cid:12)(cid:12) ε =0 = τ Z Ω u ( x )( J ⋆ φ )( x ) + φ ( x )( J ⋆ u )( x ) dx. NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 13
Now, since J is even (recall (1.2)), we see that Z Ω u ( x )( J ⋆ φ )( x ) dx = Z Ω u ( x ) (cid:18)Z Ω J ( x − y ) φ ( y ) dy (cid:19) dx = Z Ω φ ( y ) (cid:18)Z Ω J ( y − x ) u ( x ) dx (cid:19) dy = Z Ω φ ( x )( J ⋆ u )( x ) dx. Using this in (3.2) we obtain that d J dε ( u + εφ ) (cid:12)(cid:12)(cid:12) ε =0 = τ Z Ω φ ( x )( J ⋆ u )( x ) dx, which concludes the proof. (cid:3) As a consequence of Lemma 3.1, to find solutions of (1.9), we will consider the minimizingproblem for the functional E in (3.1). First, we show the following useful inequality: Lemma 3.2.
Let v, w ∈ L (Ω) . Then (3.3) Z Ω | v ( x ) | (cid:12)(cid:12) ( J ⋆ w )( x ) (cid:12)(cid:12) dx k v k L (Ω) k w k L (Ω) . Proof.
By the Cauchy-Schwarz Inequality, we have(3.4) Z Ω | v ( x ) | (cid:12)(cid:12) ( J ⋆ w )( x ) (cid:12)(cid:12) dx k v k L (Ω) k J ⋆ w k L (Ω) . Now, using the Young Inequality for convolutions with exponents 1 and 2 (see e.g. Theorem 9.1in [WZ15]), we obtain k J ⋆ w k L (Ω) = k J ∗ ( wχ Ω ) k L ( R n ) k J k L ( R n ) k wχ Ω k L ( R n ) = k w k L (Ω) , where (1.3) has been also used. This and (3.4) give (3.3), as desired. (cid:3) We are now able to provide a minimization argument for the functional in (3.1):
Proposition 3.3.
Assume that m ∈ L q (Ω) , for some q ∈ (cid:0) q, + ∞ (cid:3) , where q has been introducedin (1.10) , and that (3.5) ( m + τ ) µ − ∈ L (Ω) . Let also p := 2 qq − . Then, the functional E in (3.1) attains its minimum in X α,β . The minimal value is the sameas the one occurring among the functions u ∈ L p (Ω) for which Z Z Q | u ( x ) − u ( y ) | | x − y | n +2 s dx dy < + ∞ and such that N s u = 0 a.e. outside Ω .Moreover, there exists a nonnegative minimizer u , and it is a solution of (1.9) .Proof. First, we notice that p ∈ h , qq − (cid:17) and(3.6) p + 1 q = 1 . By (3.3) we have that(3.7) Z Ω τ u ( J ⋆ u )2 dx τ k u k L (Ω) k u k L (Ω) = τ Z Ω | u ( x ) | dx. Moreover, we use the Young Inequality with exponents / and to obtain that ( m + τ ) u µ u · m + τ − µ µ | u | | m + τ | µ . From this and (3.7) we have that Z Ω µ | u | − mu − τ u ( J ⋆ u )2 dx > Z Ω µ | u | − mu − τ u dx > − Z Ω | m + τ | µ dx =: − κ. (3.8)We point out that the quantity κ is finite, thanks to (3.5), and it does not depend on u .Recalling (3.1), formula (3.8) implies that(3.9) E ( u ) > α Z Ω |∇ u | dx + β Z Z Q | u ( x ) − u ( y ) | | x − y | n +2 s dx dy + Z Ω µ | u | dx − κ. Now, we take a minimizing sequence u j , and we observe that, in light of Theorem 2.1, wecan assume that(3.10) N s u j = 0 in ∈ R n \ Ω , for every j ∈ N .We can also suppose that E (0) > E ( u j ) > α Z Ω |∇ u j | dx + β Z Z Q | u j ( x ) − u j ( y ) | | x − y | n +2 s dx dy + Z Ω µ | u j | dx − κ, where (3.9) has been also exploited. This implies that α Z Ω |∇ u j | dx + β Z Z Q | u j ( x ) − u j ( y ) | | x − y | n +2 s dx dy + Z Ω µ | u j | dx κ. As a consequence,(3.11) α Z Ω |∇ u j | dx + β Z Z Q | u j ( x ) − u j ( y ) | | x − y | n +2 s dx dy κ. Moreover, by the Hölder Inequality with exponents / and , k u j k L (Ω) (cid:18)Z Ω | u j | dx (cid:19) / | Ω | / (cid:18)Z Ω µ | u j | dx (cid:19) / / | Ω | / µ / (cid:18)Z Ω µ | u j | dx (cid:19) / / | Ω | / µ / / | Ω | / κµ / . From this and (3.11), and using compactness arguments, we can assume, up to a subsequence,that u j converges to some u ∈ L p (Ω) (for every p ∈ [1 , ∗ s ) if α = 0 , and for every p ∈ [1 , ∗ ) if α = 0 , see e.g. Corollary 7.2 in [DNPV12]) and a.e. in Ω , and also | u j | h for some h ∈ L p (Ω) for every j ∈ N (see e.g. Theorem IV.9 in [Bre83]).Hence, if x ∈ R n \ Ω , by the Dominated Convergence Theorem, Z Ω u j ( y ) | x − y | n +2 s dy −→ Z Ω u ( y ) | x − y | n +2 s dy, as j ր + ∞ . Accordingly, in light of (3.10), when x ∈ R n \ Ω , we have(3.12) u j ( x ) = Z Ω u j ( y ) | x − y | n +2 s dy Z Ω dy | x − y | n +2 s −→ Z Ω u ( y ) | x − y | n +2 s dy Z Ω dy | x − y | n +2 s =: u ( x ) , NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 15 as j ր + ∞ (we stress that till now u was only defined in Ω , hence the last step in (3.12) isinstrumental to define u also outside Ω ). As a consequence, we obtain that u j converges a.e.in R n .Now, recalling (3.6), we have that lim sup j ր + ∞ (cid:12)(cid:12)(cid:12)(cid:12)Z Ω m ( u j − u ) dx (cid:12)(cid:12)(cid:12)(cid:12) lim sup j ր + ∞ Z Ω | m ( u j − u ) | dx = lim sup j ր + ∞ Z Ω | m ( u j − u )( u j + u ) | dx lim sup j ր + ∞ k m k L q (Ω) k u j − u k L p (Ω) k u j + u k L p (Ω) = 0 , so that lim j ր + ∞ Z Ω m ( u j − u ) dx = 0 . Also,(3.13) Z Ω (cid:0) u j ( J ⋆ u j ) − u ( J ⋆ u ) (cid:1) dx = Z Ω ( u j − u )( J ⋆ u j ) dx + Z Ω ( J ⋆ u j − J ⋆ u ) u dx. Using (3.3) with v := u j − u and w := u , we obtain(3.14) lim sup j ր + ∞ Z Ω | u j − u | (cid:12)(cid:12) J ⋆ u j (cid:12)(cid:12) dx lim sup j ր + ∞ k u j − u k L (Ω) k u j k L (Ω) = 0 . Similarly, exploiting (3.3) with v := u and w := u j − u , we have lim sup j ր + ∞ Z Ω (cid:12)(cid:12) J ⋆ u j − J ⋆ u (cid:12)(cid:12) | u | dx = lim sup j ր + ∞ Z Ω (cid:12)(cid:12) J ⋆ ( u j − u ) (cid:12)(cid:12) | u | dx (3.15) lim sup j ր + ∞ k u j − u k L (Ω) k u k L (Ω) = 0 . From (3.13), (3.14) and (3.15) we conclude that lim j ր + ∞ Z Ω ( u j ( J ⋆ u j ) − u ( J ⋆ u )) dx = 0 . We also have, by the Fatou Lemma and the lower semicontinuity of the L -norm, lim inf j ր + ∞ Z Z Q | u j ( x ) − u j ( y ) | | x − y | n +2 s dx dy > Z Z Q | u ( x ) − u ( y ) | | x − y | n +2 s dx dy, lim inf j ր + ∞ Z Ω |∇ u j | dx > Z Ω |∇ u | dx and lim inf j ր + ∞ Z Ω µ | u j | dx > Z Ω µ | u | dx. Gathering together these observations, we conclude that lim inf j ր + ∞ E ( u j ) > E ( u ) , and therefore u is the desired minimum.Also, since E ( | u | ) E ( u ) , we can suppose that u is nonnegative. Finally, u is a solutionof (1.9) thanks to Lemma 3.1. (cid:3) The claim of Theorem 1.1 follows from Proposition 3.3.Now, we provide the proof of Theorem 1.2, relying also on the existence result in Theorem 1.1:
Proof of Theorem 1.2.
Thanks to Theorem 1.1, we know that there exists a nonnegative solu-tion to (1.9).We now prove the claim in (i). For this, we assume that m ≡ and τ = 0 , and we arguetowards a contradiction, supposing that there exists a nontrivial solution u of (1.9).We notice that, since u > and µ > µ > in Ω , Z Ω µu dx > . As a consequence, taking v := u in (2.5) we obtain that α Z Ω |∇ u | dx + β Z Z Q | u ( x ) − u ( y ) | | x − y | n +2 s dx dy = − Z Ω µu dx < , which is a contradiction, and therefore the claim in (i) is proved.Now we deal with the claim in (ii). From Theorem 1.1 we know that there exists a nonnegativesolution u to (1.9) which is obtained by the minimization of the functional E in (3.1) (recallProposition 3.3). We claim that(3.16) u does not vanish identically.To prove this, we show that(3.17) is not a minimizer for E .For this, we consider the constant function v ≡ and a small parameter ε > . Then E ( εv ) = − ε (cid:20)Z Ω m + τ ( J ⋆ dx (cid:21) + ε Z Ω µ dx − c ε + c ε , where c := 12 Z Ω m + τ ( J ⋆ dx and c := 13 k µ k L (Ω) . We remark that c > , thanks to (1.11), and c ∈ (0 , + ∞ ) , in light of (1.12). Then, for small ε we have E ( εv ) < E (0) . This implies (3.17), which in turn proves (3.16). (cid:3) Analysis of the eigenvalue problem in (1.13) and proof of Theorem 1.4
In this section we focus on the proof of Theorem 1.4. For this, we need to exploit the analysisof the eigenvalue problem in (1.13) (some technical details are deferred to the article [DPLV]for the reader’s convenience).The first result towards the proof of Theorem 1.4 concerns the existence of two unboundedsequences of eigenvalues, one positive and one negative:
Proposition 4.1.
Let (4.1) m ∈ L q (Ω) , for some q ∈ (cid:0) q, + ∞ (cid:3) ,where q is given in (1.10) . Suppose that m + , m − and that (4.2) Z Ω m ( x ) dx = 0 . Then, problem (1.13) admits two unbounded sequences of eigenvalues: · · · λ − λ − < λ = 0 < λ λ · · · . In particular, if Z Ω m ( x ) dx < , NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 17 then (4.3) λ = min u ∈ X α,β (cid:26) [ u ] X α,β s.t. Z Ω mu dx = 1 (cid:27) where we use the notation in (2.4) . If instead Z Ω m ( x ) dx > , then λ − = − min u ∈ X α,β (cid:26) [ u ] X α,β s.t. Z Ω mu dx = − (cid:27) . The proof of Proposition 4.1 is contained in [DPLV].The first positive eigenvalue λ , as given by Proposition 4.1, has the following properties: Proposition 4.2.
Let m ∈ L q (Ω) , for some q ∈ (cid:0) q, + ∞ (cid:3) , where q is given in (1.10) . Supposethat m + and Z Ω m dx < . Then, the first positive eigenvalue λ of (1.13) is simple, and the first eigenfunction e can betaken such that e > .A similar statement holds if m − and Z Ω m dx > . See [DPLV] for the proof of Proposition 4.2.With this, we are now ready to give the proof of Theorem 1.4:
Proof of Theorem 1.4.
Thanks to Theorem 1.1, we know that there exists a nonnegative solu-tion to (1.9).We first prove the claim in (i). For this, we assume that m − τ , and we suppose bycontradiction that there exists a nontrivial solution u of (1.9).We observe that, applying (3.3) with v := u and w := u ,(4.4) τ Z Ω u (cid:0) J ⋆ u (cid:1) dx τ k u k L (Ω) = τ Z Ω u dx. Hence, taking u as a test function in (2.5), using (4.4) and recalling that u > and µ > µ , weget α Z Ω |∇ u | dx + β Z Z Q | u ( x ) − u ( y ) | | x − y | n +2 s dx dy = Z Ω ( m − µu ) u dx + τ Z Ω ( J ⋆ u ) u dx − τ Z Ω u dx − µ Z Ω u dx + τ Z Ω u dx< . This is a contradiction, whence the first claim is proved.Now we show the claim in (ii). From Theorem 1.1 we know that there exists a nonnegativesolution u to (1.9) which is obtained by the minimization of the functional E in (3.1) (recallProposition 3.3). We claim that(4.5) u does not vanish identically.To prove this, we show that(4.6) is not a minimizer for E . For this, we take an eigenfunction e associated to the first positive eigenvalue λ , as given byProposition 4.2. Namely, we take e ∈ X α,β such that(4.7) α Z Ω ∇ e · ∇ v dx + β Z Z Q ( e ( x ) − e ( y ))( v ( x ) − v ( y )) | x − y | n +2 s dx dy = λ Z Ω mev dx, for every v ∈ X α,β .By taking v := e in (4.7), we obtain that(4.8) α Z Ω |∇ e | dx + β Z Z Q | e ( x ) − e ( y ) | | x − y | n +2 s dx dy = λ Z Ω me dx. We also remark that, thanks to (1.14), we can use the characterization of λ given in for-mula (4.3) of Proposition 4.1, and hence we can normalize e in such a way that(4.9) Z Ω me dx = 1 . By Corollary 1.4 in [DPLV], we know that(4.10) e is bounded.We also take ε > . Then, by (4.8) and (4.9), E ( εe ) = ε " α Z Ω |∇ e | dx + β Z Z Q | e ( x ) − e ( y ) | | x − y | n +2 s dx dy − Z Ω me dx − Z Ω τ ( J ⋆ e ) e dx + ε Z Ω µe dx = ε (cid:20) ( λ − Z Ω me dx − Z Ω τ ( J ⋆ e ) e dx (cid:21) + ε Z Ω µe dx = ε (cid:20) ( λ − − Z Ω τ ( J ⋆ e ) e dx (cid:21) + ε Z Ω µe dx = − c ε + c ε , where c := 1 − λ + τ Z Ω ( J ⋆ e ) e dx and c := Z Ω µe dx. We notice that c > , thanks to (1.15), and c ∈ R , in light of (4.10). As a consequence, forsmall ε we have that E ( εe ) < E (0) , which proves (4.6). In turn, this implies (4.5), thuscompleting the proof of (ii). (cid:3) Optimization on m and proofs of Theorems 1.5, 1.7, 1.8, 1.9 and 1.10 This section is devoted to the understanding of the optimal configuration of the resource m ,which is based on the analysis of the minimal eigenvalue problem given in (1.17).First of all, we will see that the optimal resource distribution attaining the minimal eigenvaluein (1.17) is of bang-bang type, namely concentrated on its minimal and maximal values m and m . This property is based on the so called “bathtub principle”, see Lemma 3.3 in [DGT10](or [LL97, LY06]), that we recall here for the convenience of the reader: Lemma 5.1.
Let f ∈ L (Ω) and M be as in (1.16) . Then, the maximization problem sup m ∈ M Z Ω f m dx is attained by a suitable m ∈ M given by m := mχ D − mχ Ω \ D , NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 19 for some subset D ⊂ Ω such that (5.1) | D | = m + m m + m | Ω | . We now show that, in the light of Lemma 5.1, to optimize the eigenvalue λ in (1.17), wehave to consider m ∈ M of bang-bang type. More precisely, we define ˜ M = ˜ M ( m, m, m ) := ( m ∈ M s.t. m := mχ D − mχ Ω \ D , for some subset D ⊂ Ω with | D | = m + m m + m | Ω | ) , (5.2)and we have the following result: Proposition 5.2.
We have that λ = inf m ∈ ˜ M λ ( m ) . Proof.
We define ˜ λ := inf m ∈ ˜ M λ ( m ) . and we claim that(5.3) λ = ˜ λ. To this end, we observe that, since ˜ M ⊂ M , we have that(5.4) λ ˜ λ. Moreover, by the definition of λ in (1.17), we have that for every ε > there exists m ε ∈ M such that λ + ε > λ ( m ε ) . Then, we denote by e ε the nonnegative eigenfunction associatedto λ ( m ε ) , and we conclude that(5.5) λ + ε > λ ( m ε ) = [ e ε ] X α,β Z Ω m ε e ε dx . We also observe that, in light of Lemma 5.1, Z Ω m ε e ε dx Z Ω (cid:0) mχ D ε − mχ Ω \ D ε (cid:1) e ε dx, for a suitable D ε ⊂ Ω satisfying (5.1). Plugging this information into (5.5), and letting m ⋆ε := mχ D ε − mχ Ω \ D ε , we obtain that λ + ε > [ e ε ] X α,β Z Ω ( mχ D ε − mχ Ω \ D ε ) e ε dx > λ ( m ⋆ε ) > ˜ λ. Hence, taking the limit as ε goes to , we get that λ > ˜ λ . This, combined with (5.4), estab-lishes (5.3), as desired. (cid:3) We recall that many biological models describe optimal resourcesof bang-bang type, seee.g. [CC98, CC03, LY06, LLNP16, NY18, MNP20].In light of Proposition 5.2, from now on, when optimizing the eigenvalue λ ( m ) as in (1.17),we will suppose that m belongs to the set ˜ M introduced in (5.2).Now we provide the proof of Theorem 1.5. Proof of Theorem 1.5.
We take a ball B ⊂ Ω such that(5.6) | B | d | Ω | . We can assume, up to a translation, that Ω ⊂ { x n > } , and, for every ξ > , we define the set Ω ξ := B ∪ ( { x n < ξ } ∩ Ω) . We observe that | Ω ξ | is nondecreasing with respect to ξ , and we define ξ ∗ := sup (cid:26) ξ > | Ω ξ | < m + m m + m | Ω | (cid:27) . We claim that, for every ξ > ,(5.7) lim ξ → ξ | Ω ξ | = | Ω ξ | . To this end, we first show that(5.8) lim ξ → ξ χ Ω ξ ( x ) = χ Ω ξ ( x ) for a.e. x ∈ Ω . For this, we consider several cases. If x = ( x ′ , x n ) ∈ Ω ξ , then either x ∈ B or x n < ξ . If x ∈ B ,then x ∈ Ω ξ for each ξ > , and accordingly χ Ω ξ ( x ) = 1 = χ Ω ξ ( x ) , which implies (5.8).If instead x n < ξ , then there exists ˜ ξ ∈ ( x n , ξ ) such that, for every ξ ∈ ( ˜ ξ, ξ ) , we havethat χ Ω ξ ( x ) = 1 = χ Ω ξ ( x ) , which proves (5.8) also in this case.On the other hand, if x Ω ξ , then x B and x n > ξ . We notice that the set { x n = ξ } has zero Lebesgue measure, and therefore, in order to prove (5.8), we can assume that x n > ξ .Then, there exists ˜ ξ ∈ ( ξ, x n ) such that, for every ξ ∈ ( ξ, ˜ ξ ) , we have that x Ω ξ , andso χ Ω ξ ( x ) = 0 = χ Ω ξ ( x ) . This completes the proof of (5.8).By (5.8) and the Dominated Convergence Theorem we obtain (5.7), as desired.We also notice that if ξ = 0 , then Ω ξ = B , and therefore, by (1.18) and (5.6), | Ω ξ | = | B | d | Ω | m + m (cid:0) m + m (cid:1) | Ω | . This and the continuity statement in (5.7) guarantee that ξ ∗ > .Moreover, the continuity in (5.7) implies that(5.9) | Ω ξ ∗ | = m + m m + m | Ω | . Now, we set D := Ω ξ ∗ , and we observe that D satisfies (5.1), thanks to (5.9). Also, wetake v ∈ C ∞ ( B ) , with v . Then, recalling that B ⊂ D , λ [ v ] X α,β Z Ω (cid:0) mχ D − mχ Ω \ D (cid:1) v dx = [ v ] X α,β m Z B v dx Cm , for some positive constant C depending on Ω and d . This completes the proof of Theorem 1.5. (cid:3) With the aid of Theorem 1.5 we now prove Corollary 1.6, by arguing as follows:
Proof of Corollary 1.6.
We notice that lim m ր + ∞ m + m m = 12 . By taking m = m = m , this implies that m + m m + m = m + m m > , NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 21 as long as m is large enough. This says that the assumption (1.18) in Theorem 1.5 is satisfiedwith d := , and therefore, Theorem 1.5 gives that λ (cid:0) m , m , m (cid:1) C m , for some C > depending only on Ω . As a consequence, lim m ր + ∞ λ (cid:0) m , m , m (cid:1) = 0 , as desired. (cid:3) The next goal of this section is to prove Theorem 1.7. For concreteness, we give here theproof for n > , and we defer the case n = 2 to Appendix A.Without loss of generality, we suppose that(5.10) B ⊂ Ω . For n > and for any ρ ∈ (0 , , we define the function ϕ : R n → R as(5.11) ϕ ( x ) := c ⋆ + 1 if x ∈ B ρ ,c ⋆ + ρ γ − ρ γ (cid:18) | x | γ − (cid:19) if x ∈ B \ B ρ ,c ⋆ if x ∈ R n \ B , where γ > and(5.12) c ⋆ := − m + m m . We observe that, since m ∈ ( − m, , we have that c ⋆ > .Also, we set(5.13) D := B ρ . The idea to prove Theorem 1.7 is to use the function ϕ in (5.11) and the resource m = mχ D − mχ Ω \ D , with D as in (5.13), as competitors for the minimization of λ in (1.17).In this setting, we notice that, since m ∈ ˜ M , recalling (5.1), ρ n | B | = | B ρ | = | D | = m + m m + m | Ω | . This says that(5.14) sending m ր + ∞ is equivalent to sending ρ ց ,being m , m and | Ω | fixed quantities in this argument.In light of these observations, the next lemmata will be devoted to estimate in terms of ρ thequantities involving ϕ that appear in the minimization of λ .We point out that, in dimension n = 2 , the argument to prove Theorem 1.7 will be similar, butwe will need to introduce a logarithmic-type function as in (A.1) instead of a polynomial-typefunction as in (5.11) (as it often happens when passing from dimension to higher dimensions).The first result that we have in this setting deals with the H -seminorm of ϕ : Lemma 5.3.
Let n > and ϕ be as in (5.11) . Then, lim ρ ց Z Ω |∇ ϕ | dx = 0 . Proof.
By the definition of ϕ in (5.11), we have that ∇ ϕ = 0 only if x ∈ B \ B ρ . Accordingly,using polar coordinates, Z Ω |∇ ϕ | dx = γ (cid:18) ρ γ − ρ γ (cid:19) Z B \ B ρ | x | γ +2 dx = γ (cid:18) ρ γ − ρ γ (cid:19) Z ρ r n − γ − dr. (5.15)Now, we point out that(5.16) Z ρ r n − γ − dr n − γ − if γ < n − , − log ρ if γ = n − , − ρ n − γ − n − γ − if γ > n − . This and (5.15) entail that, for every γ > , lim ρ ց Z Ω |∇ ϕ | dx = 0 , which conludes the proof. (cid:3) Now, we deal with the Gagliardo seminorm of ϕ . For this, we point out the following usefulinequality: Lemma 5.4.
Let x , y ∈ R n \ { } and γ > . Then, there exists C γ > such that (5.17) (cid:12)(cid:12)(cid:12)(cid:12) | x | γ − | y | γ (cid:12)(cid:12)(cid:12)(cid:12) C γ || x | − | y || min {| x | γ +1 , | y | γ +1 } . Proof.
We can assume that | x | > | y | , being the other case analogous. In this way, formula (5.17)boils down to(5.18) | y | γ − | x | γ C γ | x | − | y || y | γ +1 . To prove (5.18), we first claim that, for every t > ,(5.19) − t γ C γ ( t − , for a suitable C γ > . Indeed, we set f ( t ) := Ct + 1 t γ − ( C + 1) , for some positive constant C (to be chosen in what follows), and we observe that(5.20) f (1) = 0 , and f ′ ( t ) = C − γt γ +1 > C − γ, for any t > . As a result, taking C := γ + 1 , we obtain that f ′ ( t ) > . This and (5.20) givethat f ( t ) > for every t > , which implies (5.19).Taking t := | x || y | in (5.19), we obtain that − | y | γ | x | γ C γ (cid:18) | x || y | − (cid:19) . Multiplying this inequality by | y | γ we deduce (5.18), as desired. (cid:3) With this, we now estimate the Gagliardo seminorm of ϕ as follows: NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 23
Lemma 5.5.
Let n > and ϕ be as in (5.11) . Then, lim ρ ց Z Z Q | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy = 0 . Proof.
In what follows, we will assume that ρ / . By the definition of ϕ in (5.11), it plainlyfollows that(5.21) Z Z B ρ × B ρ | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy = 0 and(5.22) Z Z ( R n \ B ) × ( R n \ B ) | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy = 0 . Moreover, by the change of variable z := y − x , Z Z B ρ × ( R n \ B ) | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy = Z Z B ρ × ( R n \ B ) | x − y | n +2 s dx dy Z B ρ dx Z R n \ B | z | n +2 s dz C Z B ρ dx = Cρ n , for some C > . As a consequence,(5.23) lim ρ ց Z Z B ρ × ( R n \ B ) | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy = 0 . Now, if x ∈ B \ B ρ and y ∈ B ρ , from (5.11) we have that | ϕ ( x ) − ϕ ( y ) | = (cid:18) ρ γ − ρ γ (cid:19) (cid:18) | x | γ − ρ γ (cid:19) . Hence, utilizing also (5.17) (applied here with | y | := ρ ), Z Z ( B \ B ρ ) × B ρ | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy = (cid:18) ρ γ − ρ γ (cid:19) Z Z ( B \ B ρ ) × B ρ (cid:18) | x | γ − ρ γ (cid:19) | x − y | n +2 s dx dy Cρ γ +2 (cid:18) ρ γ − ρ γ (cid:19) Z Z ( B \ B ρ ) × B ρ ( | x | − ρ ) | x − y | n +2 s dx dy (5.24)We observe that, since x ∈ B \ B ρ and y ∈ B ρ , | x | − ρ | x | − | y | | x − y | , and therefore, plugging this information into (5.24), Z Z ( B \ B ρ ) × B ρ | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy Cρ γ +2 (cid:18) ρ γ − ρ γ (cid:19) Z Z ( B \ B ρ ) × B ρ | x − y | − n − s dx dy Cρ γ +2 (cid:18) ρ γ − ρ γ (cid:19) Z B ρ dy Z B | z | − s − n dz Cρ γ +2 (cid:18) ρ γ − ρ γ (cid:19) Z B ρ dy C ρ n − , up to renaming C > from line to line. As a result,(5.25) lim ρ ց Z Z ( B \ B ρ ) × B ρ | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy = 0 . In addition, since R n \ Ω ⊂ R n \ B (recall (5.10)), changing variable z := y − x and usingpolar coordinates, Z Z ( B \ B ρ ) × ( R n \ Ω) | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy = (cid:18) ρ γ − ρ γ (cid:19) Z Z ( B \ B ρ ) × ( R n \ Ω) (cid:18) | x | γ − (cid:19) | x − y | n +2 s dx dy (cid:18) ρ γ − ρ γ (cid:19) Z B \ B ρ (cid:18) | x | γ − (cid:19) dx Z R n \ B | z | n +2 s dz C (cid:18) ρ γ − ρ γ (cid:19) Z B \ B ρ (cid:18) | x | γ − (cid:19) dx = C (cid:18) ρ γ − ρ γ (cid:19) Z B \ B ρ (cid:18) | x | γ − | x | γ + 1 (cid:19) dx C (cid:18) ρ γ − ρ γ (cid:19) Z ρ (cid:0) r n − γ − + r n − (cid:1) dr C (cid:18) ρ γ − ρ γ (cid:19) (cid:20) Z ρ r n − γ − dr (cid:21) , (5.26)possibly changing C > from line to line. We also remark that Z ρ r n − γ − dr n − γ if γ < n , − log ρ if γ = n , − ρ n − γ n − γ if γ > n . This and (5.26) imply that(5.27) lim ρ ց Z Z ( B \ B ρ ) × ( R n \ Ω) | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy = 0 . Furthermore, recalling (5.11) and making use of (5.17), we have that
Z Z ( B \ B ρ ) × (Ω \ B ) | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy = (cid:18) ρ γ − ρ γ (cid:19) Z Z ( B \ B ρ ) × (Ω \ B ) (cid:18) | x | γ − (cid:19) | x − y | n +2 s dx dy C (cid:18) ρ γ − ρ γ (cid:19) Z Z ( B \ B ρ ) × (Ω \ B ) (1 − | x | ) | x | γ +2 | x − y | n +2 s dx dy. Hence, noticing that, for every x ∈ B \ B ρ and every y ∈ Ω \ B , − | x | | y | − | x | | x − y | , we conclude that Z Z ( B \ B ρ ) × (Ω \ B ) | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 25 C (cid:18) ρ γ − ρ γ (cid:19) Z Z ( B \ B ρ ) × (Ω \ B ) | x | γ +2 | x − y | n +2 s − dx dy C (cid:18) ρ γ − ρ γ (cid:19) Z B \ B ρ | x | γ +2 dx C (cid:18) ρ γ − ρ γ (cid:19) Z ρ r n − γ − dr. Accordingly, recalling (5.16), we conclude that(5.28) lim ρ ց Z Z ( B \ B ρ ) × (Ω \ B ) | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy = 0 . We now claim that(5.29) lim ρ ց Z Z ( B \ B ρ ) × ( B \ B ρ ) | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy = 0 . For this, we observe that by (5.11)
Z Z ( B \ B ρ ) × ( B \ B ρ ) | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy = (cid:18) ρ γ − ρ γ (cid:19) Z Z ( B \ B ρ ) × ( B \ B ρ ) (cid:18) | x | γ − | y | γ (cid:19) dx dy | x − y | n +2 s = 2 (cid:18) ρ γ − ρ γ (cid:19) Z Z ( B \ Bρ ) × ( B \ Bρ ) {| x | | y |} (cid:18) | x | γ − | y | γ (cid:19) dx dy | x − y | n +2 s . Hence, from (5.17) we get
Z Z ( B \ B ρ ) × ( B \ B ρ ) | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy C (cid:18) ρ γ − ρ γ (cid:19) Z Z ( B \ Bρ ) × ( B \ Bρ ) {| x | | y |} | x − y | − n − s | x | γ +2 dx dy, up to renaming C > .Since ρ ∈ (0 , , we can take an integer k such that(5.30) k +1 < ρ k . In this way, we have that
Z Z ( B \ B ρ ) × ( B \ B ρ ) | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy C (cid:18) ρ γ − ρ γ (cid:19) Z Z (cid:0) B \ B / k +1 (cid:1) × (cid:0) B \ B / k +1 (cid:1) {| x | | y |} | x − y | − n − s | x | γ +2 dx dy C ρ γ k X i,j =0 Z Z (cid:0) B / i \ B / i +1 (cid:1) × (cid:0) B / j \ B / j +1 (cid:1) {| x | | y |} | x − y | − n − s | x | γ +2 dx dy. (5.31)We also observe that when x B / i +1 , y ∈ B / j and | x | | y | , we have that i +1 | x | | y | j , and accordingly j i + 1 . This implies that Z Z ( B \ Bρ ) × ( B \ Bρ ) {| x | | y |} | x − y | − n − s | x | γ +2 dx dy k X i =0 i +1 X j =0 Z Z (cid:0) B / i \ B / i +1 (cid:1) × (cid:0) B / j \ B / j +1 (cid:1) {| x | > | y |} | x − y | − n − s | x | γ +2 dx dy k X i =0 i +1 X j =0 Z Z(cid:0) B / i \ B / i +1 (cid:1) × (cid:0) B / j \ B / j +1 (cid:1) (2 γ +2)( i +1) | x − y | − n − s dx dy k X i =0 i +1 X j =0 Z B / i \ B / i +1 (2 γ +2)( i +1) dx Z B i + 12 j | z | − n − s dz C k X i =0 i +1 X j =0 − ni +(2 γ +2) i +(2 s − j C k X i =0 (2 γ +2 − n ) i C if γ < n − ,C k if γ = n − ,C (2 γ +2 − n ) k if γ > n − , C if γ < n − ,C | log ρ | if γ = n − ,C ρ n − γ − if γ > n − , up to renaming C > , where we used (5.30).Plugging this information into (5.31), we obtain (5.29).Putting together (5.21), (5.22), (5.23), (5.25), (5.27), (5.28) and (5.29), we obtain the desiredresult. (cid:3) We now estimate the weighted L -norm of the auxiliary function ϕ : Lemma 5.6.
Let n > and ϕ be as in (5.11) . Then, lim ρ ց m Z D ϕ dx − m Z Ω \ D ϕ dx = − m ( m + m ) m | Ω | > . Proof.
Recalling (5.12), (5.13) and (5.1), we see that m Z D ϕ dx = m ( c ⋆ + 1) | D | = m (cid:18) − m + m m + 1 (cid:19) m + m m + m | Ω | = m m ( m + m ) m ( m + m ) | Ω | . (5.32)Moreover, we observe that(5.33) ρ γ − ρ γ | x | γ χ B \ B ρ − ρ γ , as long as ρ is small enough. As a consequence, recalling (5.11) and (5.13), and using theDominated Convergence Theorem, we find that lim ρ ց m Z Ω \ D ϕ dx = lim ρ ց m Z Ω \ B c ⋆ dx + m Z B \ B ρ (cid:20) c ⋆ + ρ γ − ρ γ (cid:18) | x | γ − (cid:19)(cid:21) dx NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 27 = m c ⋆ | Ω \ B | + m c ⋆ | B | = m c ⋆ | Ω | = m (cid:18) m + m m (cid:19) | Ω | . From this and (5.32), and recalling (5.14), we conclude that lim ρ ց m Z D ϕ dx − m Z Ω \ D ϕ dx = lim m ր + ∞ m m ( m + m ) m ( m + m ) | Ω | − m (cid:18) m + m m (cid:19) | Ω | = m ( m + m ) m | Ω | − m (cid:18) m + m m (cid:19) | Ω | = m ( m + m ) m [ m − ( m + m )] | Ω | = − m ( m + m ) m | Ω | , which is positive, since m ∈ ( − m, , as desired. (cid:3) We are now in the position to give the proof of Theorem 1.7 for n > . Proof of Theorem 1.7 when n > . The strategy of the proof is to use the auxiliary function ϕ as defined in (5.11) and the resource m := mχ D − mχ Ω \ D , with D as in (5.13), as a competitorin the minimization problem (1.17). Indeed, in this way we find that λ ( m, m, m ) α Z Ω |∇ ϕ | dx + β Z Z Q | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dym Z D | ϕ | dx − m Z Ω \ D | ϕ | dx . Moreover, Lemmata 5.3 and 5.5 give that lim ρ ց α Z Ω |∇ ϕ | dx + β Z Z Q | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy = 0 . This, combined with (5.14) and Lemma 5.6, gives the desired result. (cid:3)
Now we deal with the proof of Theorem 1.8. The main strategy is similar to that of theproof of Theorem 1.7, but in this setting we introduce a different auxiliary function (and thisof course impacts the technical computations needed to obtain the desired results). Namely,we define(5.34) ψ ( x ) := c ♯ − if x ∈ B ρ ,c ♯ − ρ γ − ρ γ (cid:18) | x | γ − (cid:19) if x ∈ B \ B ρ ,c ♯ if x ∈ R n \ B , where(5.35) c ♯ := m − mm . We point out that c ♯ > , since m < < m . We also set(5.36) D := Ω \ B ρ . We remark that, in this setting, since m ∈ ˜ M , recalling (5.1), | Ω | − | B ρ | = | Ω \ B ρ | = | D | = m + m m + m | Ω | . This says that(5.37) sending m ր + ∞ is equivalent to sending ρ ց ,being m , m and | Ω | fixed quantities in this argument. The reader may compare the settingin (5.13) and (5.14) with the one in (5.36) and (5.37) to appreciate the structural differencebetween the two frameworks. Now, we list some useful properties of the auxiliary function ψ . Noticing that the function ψ in (5.34) differs by a constant from the function − ϕ in (5.11), we obtain the following tworesults directly from Lemmata 5.3 and 5.5: Lemma 5.7.
Let n > and ψ be as in (5.34) . Then, lim ρ ց Z Ω |∇ ψ | dx = 0 . Lemma 5.8.
Let n > and ψ be as in (5.34) . Then lim ρ ց Z Z Q | ψ ( x ) − ψ ( y ) | | x − y | n +2 s dx dy = 0 . We now deal with the weighted L -norm of the auxiliary function ψ : Lemma 5.9.
Let n > and ψ be as in (5.34) . Then, lim ρ ց m Z D ψ dx − m Z Ω \ D ψ dx = − m ( m − m ) m | Ω | > . Proof.
Recalling (5.34) and (5.36), we find that m Z D ψ dx = m Z Ω \ B c ♯ dx + m Z B \ B ρ (cid:20) c ♯ − ρ γ − ρ γ (cid:18) | x | γ − (cid:19)(cid:21) dx. Hence, recalling (5.33) and using the Dominated Convergence Theorem and (5.35), we deducethat(5.38) lim ρ ց m Z D ψ dx = m c ♯ | Ω \ B | + m c ♯ | B | = m c ♯ | Ω | = m (cid:18) m − mm (cid:19) | Ω | . Moreover, recalling (5.36), (5.1) and (5.35), m Z Ω \ D ψ dx = m ( c ♯ − | Ω \ D | = m (cid:18) m − mm − (cid:19) m − m m + m | Ω | = m m ( m − m ) m ( m + m ) | Ω | . As a consequence of this and (5.38), and recalling (5.37), we have that lim ρ ց m Z D ψ dx − m Z Ω \ D ψ dx = m (cid:18) m − mm (cid:19) | Ω | − lim m ր + ∞ m m ( m − m ) m ( m + m ) | Ω | = m (cid:18) m − mm (cid:19) | Ω | − m ( m − m ) m | Ω | = m ( m − m ) m [( m − m ) − m ] | Ω | = − m ( m − m ) m | Ω | , which is positive, since m < < m . (cid:3) Now we are ready to give the proof of Theorem 1.8 for n > . Proof of Theorem 1.8.
The strategy of the proof is to use the auxiliary function ψ as definedin (5.34) and the resource m := mχ D − mχ Ω \ D , with D as in (5.36), as a competitor in theminimization problem (1.17). Indeed, in this way we find that λ ( m, m, m ) α Z Ω |∇ ψ | dx + β Z Z Q | ψ ( x ) − ψ ( y ) | | x − y | n +2 s dx dym Z D | ψ | dx − m Z Ω \ D | ψ | dx . Moreover, from Lemmata 5.7 and 5.8 we have that lim ρ ց α Z Ω |∇ ψ | dx + β Z Z Q | ψ ( x ) − ψ ( y ) | | x − y | n +2 s dx dy = 0 . NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 29
This, combined with (5.37) and Lemma 5.9 implies the desired result. (cid:3)
Having completed the cases n > and deferred the case n = 2 to Appendix A, we now focuson the case n = 1 , by providing the proofs of Theorems 1.9 and 1.10.For this, when n = 1 we first establish the following lower bound for λ (as defined in (1.17)): Lemma 5.10.
Let n = 1 and α > . Then (5.39) λ ( m, m, m ) > − C m ( m + m ) mm ( m − m ) (cid:0) m (2 m + m ) − mm (cid:1) , for some C = C ( α, β, Ω) > .Proof. Without loss of generality, we can set α = 2 . We take an arbitrary resource m in theset ˜ M defined in (5.2). Moreover, we denote by e an eigenfunction associated to the firsteigengenvalue of problem (1.13), that is(5.40) λ ( m ) = Z Ω | e ′ | dx + β Z Z Q | e ( x ) − e ( y ) | | x − y | n +2 s dx dym Z D e dx − m Z Ω \ D e dx . In light of Proposition 4.2 here and Corollary 1.4 in [DPLV], up to a sign change, we know that e is nonnegative and bounded, and therefore we set a := inf Ω e and b := sup Ω e. By construction, we have that a ∈ [0 , b ] , and we can also normalize e such that b = 1 ; in thisway(5.41) e ( x ) for each x ∈ Ω . We also take x k , y k ∈ Ω such that e ( x k ) → a and e ( y k ) → as k ր + ∞ .We observe thatif there exist ¯ x and ¯ y such that | e (¯ x ) − e (¯ y ) | > − a which belongto the same connected component of Ω , then (1 − a ) C Z Ω | e ′ | dx, for some C > .(5.42)Indeed, for ¯ x and ¯ y as in the assumption of (5.42) we have that (cid:0) e (¯ y ) − e (¯ x ) (cid:1) = (cid:18)Z ¯ y ¯ x e ′ ( t ) dt (cid:19) C Z Ω | e ′ ( t ) | dt, for some positive C . Accordingly, we obtain the desired result in (5.42).Now we claim that(5.43) (1 − a ) C (cid:18)Z Ω | e ′ | dx + β Z Z Q | e ( x ) − e ( y ) | | x − y | n +2 s dx dy (cid:19) , for a suitable C > .To prove this claim, we need to consider different possibilities according to the possible lackof connectedness of Ω . For this, we first remark that, with no loss of generality, we can supposethat(5.44) a < , otherwise (5.43) is obviously satisfied. Furthermore, being Ω a bounded set with C boundary, necessarily it can have at most afinite number of connected components (otherwise, there would be accumulating components,violating the assumption in (1.1)). Hence, if Ω is not connected, we can define d to be thesmallest distance between the different connected components of Ω . We also let d to be thediameter of Ω and d the smallest diameter of all the connected components of Ω (of course, d , d and d are structural constants, and the other constants are allowed to depend on them,but we will write d , d and d explicitly in the forthcoming computations whenever needed toemphasize their roles). To prove (5.43), we distinguish two cases: the first case is when Ω has one connected component,or it has more than one connected component, with sup x,y ∈ Ω | e ( x ) − e ( y ) | > − a | e ( x ) − e ( y ) || x − y | > d , (5.45)and the second case is when Ω has more than one connected component, with sup x,y ∈ Ω | e ( x ) − e ( y ) | > − a | e ( x ) − e ( y ) || x − y | < d . (5.46)Let us first discuss case (5.45). If Ω has one connected component, then we can exploit (5.42)with ¯ x := x k and ¯ y := y k with k sufficiently large, and the claim in (5.43) plainly follows.Thus, to complete the study of (5.45), we suppose that Ω is not connected and, in the settingof (5.45), we find ¯ x , ¯ y ∈ Ω with(5.47) | e (¯ x ) − e (¯ y ) | > − a and | e (¯ x ) − e (¯ y ) || ¯ x − ¯ y | > d . In this framework, we have that(5.48) ¯ x and ¯ y belong to the same connected component.Indeed, if not, we would have that | ¯ x − ¯ y | > d , and thus, by (5.41), | e (¯ x ) − e (¯ y ) || ¯ x − ¯ y | | e (¯ x ) | + | e (¯ y ) | d d , which is in contradiction with (5.47), thus proving (5.48).Then, by (5.48), we can exploit (5.42), from which one deduces (5.43) in this case.Having completed the analysis of case (5.45), we now focus on the setting provided bycase (5.46) and we define(5.49) r := 1100 (cid:18) d (cid:19) min (cid:8) − a, d (cid:9) . We observe that, r > , due to (5.44), and, if ϑ ∈ Ω with | ϑ − x k | r , then(5.50) | e ( ϑ ) − e ( x k ) | − a . Indeed, suppose not. Then, the assumption in (5.46) guarantees that d > | e ( ϑ ) − e ( x k ) || ϑ − x k | > | e ( ϑ ) − e ( x k ) | r , and therefore | e ( ϑ ) − e ( x k ) | rd d (cid:18) d (cid:19) (1 − a ) − a , NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 31 against the contradiction assumption.This proves (5.50) and similarly one can show that if τ ∈ Ω with | τ − y k | r , then(5.51) | e ( ϑ ) − e ( y k ) | − a . Combining (5.50) and (5.51), we find that, for all ϑ , τ ∈ Ω with | ϑ − x k | r and | τ − y k | r , | e ( ϑ ) − e ( τ ) | > | e ( x k ) − e ( y k ) | − | e ( ϑ ) − e ( x k ) | − | e ( τ ) − e ( y k ) | > − a − − a > − a . For this reason, letting S := B r ( x k ) ∩ Ω and S := B r ( y k ) ∩ Ω , we have that Z Z Q | e ( x ) − e ( y ) | | x − y | n +2 s dx dy > Z Z Ω × Ω | e ( x ) − e ( y ) | | x − y | n +2 s dx dy > Z Z S × S | e ( ϑ ) − e ( τ ) | | ϑ − τ | n +2 s dϑ dτ > Z Z S × S (1 − a ) d n +2 s dϑ dτ = (1 − a ) | S | | S | d n +2 s > (1 − a ) r d n +2 s . (5.52)We also recall that in case (5.46) we have that Ω is not connected and consequently β = 0 , dueto (1.5). From this and (5.52), up to renaming constants, we deduce that Z Ω | e ′ | dx + β Z Z Q | e ( x ) − e ( y ) | | x − y | n +2 s dx dy > (1 − a ) r C , which, together with (5.49), proves (5.43), as desired.We also remark that, testing the weak formulation of (1.13) against a constant function, onesees that Z Ω me dx = 0 , and therefore, recalling (5.41),(5.53) m | D | > m Z D e dx = m Z Ω \ D e dx > a m | Ω \ D | . Recalling (5.1), we see that(5.54) | Ω \ D | = m − m m + m | Ω | , and therefore m Z D e dx − m Z Ω \ D e dx m | D | − a m | Ω \ D | = m m + m m + m | Ω | − a m m − m m + m | Ω | = mm (1 − a ) + m ( m + a m ) m + m | Ω | < mmm + m (1 − a ) | Ω | , (5.55)since m < and m , m > . Using this inequality and (5.43) in (5.40), we conclude that(5.56) C λ ( m ) > m + mmm · (1 − a ) − a = m + mmm · (1 − a ) a , up to renaming C > .Furthermore, from (5.53) we know that a mm · m + m m − m . Consequently, since the map [0 , ∋ t := (1 − t ) t is decreasing, we discover that (1 − a ) a > (cid:18) − mm · m + m m − m (cid:19) mm · m + m m − m = − m ( m + m ) m ( m − m ) · mm + m ( m − m ) . Combining this information and (5.56), we deduce that
C λ ( m ) > − m + mmm · m ( m + m ) m ( m − m ) · mm + m ( m − m )= − m ( m + m ) mm ( m − m ) (cid:0) mm + m ( m − m ) (cid:1) . Taking the infimum of this expression, we find the desired result. (cid:3)
With this, we are in the position to give the proof of Theorem 1.9.
Proof of Theorem 1.9.
For any m > and any m ∈ ( − m, , we define the function g m,m :(0 , + ∞ ) → (0 , + ∞ ) as g m,m ( m ) := − m ( m + m ) m m ( m − m ) (cid:0) m (2 m + m ) − mm (cid:1) . We observe that lim m ր + ∞ g m,m ( m ) = − m m (2 m + m ) > , and(5.57) lim m ց g m,m ( m ) = + ∞ . In particular,(5.58) inf m ∈ (0 , + ∞ ) g m,m ( m ) > . Now, by Lemma 5.10, we know that(5.59) λ ( m , m, m ) > C g m,m ( m ) . As a result, (1.19) follows from (5.58) and (5.59). Moreover, from (5.57) and (5.59) we ob-tain (1.20).To prove (1.21), for any m > and m < , we define the function ˜ g m,m : ( − m , + ∞ ) → (0 , + ∞ ) as ˜ g m,m ( m ) := − m ( m + m ) m m ( m − m ) (cid:0) m (2 m + m ) − m m (cid:1) . We notice that lim m ց− m ˜ g m,m ( m ) = − m − m mm > and that lim m ր + ∞ ˜ g m,m ( m ) = − m m ( m − m ) (2 m − m ) > . Accordingly, inf m ∈ ( − m , + ∞ ) ˜ g m,m ( m ) > . Using this and the fact that, by Lemma 5.10, λ ( m, m , m ) > C ˜ g m,m ( m ) , we obtain the desired result in (1.21). (cid:3) NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 33
Having established Theorem 1.9, we now deal with the case in which α = 0 , namely whenonly the nonlocal dispersal is active. This case is considered in Theorem 1.10, according to twodifferent ranges of the fractional parameter s . For this, we divide the proof of Theorem 1.10into two parts. Proof of Theorem 1.10 when s ∈ (1 / , . We denote by e the eigenfunction associated to thefirst eigengenvalue of problem (1.13), normalized such that(5.60) a := inf Ω e > and sup Ω e = 1 . We recall (5.53), (5.54) and (5.55) to write that(5.61) a mm · m + m m − m and(5.62) m Z D e dx − m Z Ω \ D e dx mmm + m (1 − a ) | Ω | . We stress that, in view of (5.61),(5.63) δ := 1 − a > − mm · m + m m − m = − m ( m + m ) m ( m − m ) > . In particular, by (5.60), we can find ¯ x and ¯ y in Ω such that(5.64) e (¯ x ) a + δ and e (¯ y ) > − δ . Now we claim that(5.65)
Z Z Q ( e ( x ) − e ( y )) | x − y | s dx dy > c (1 − a ) s +22 s − , for some c ∈ (0 , depending only on s and Ω (in particular, this c is independent of m ).Indeed, if the left hand side of (5.65) is larger that , we are done, therefore we can suppose,without loss of generality, that Z Z Q ( e ( x ) − e ( y )) | x − y | s dx dy . As a result,
Z Z Ω × Ω ( e ( x ) − e ( y )) | x − y | s dx dy + k e k L (Ω) | Ω | , and consequently we can exploit Theorem 8.2 in [DNPV12] and conclude that k e k C s − (Ω) C ,for some C > depending only on s and Ω .We let d be the diameter of Ω and d be the smallest diameter of all the connected compo-nents of Ω . We also define(5.66) r := min ((cid:18) δ C (cid:19) s − , d ) , and we observe that, for each x ∈ Ω ∩ B r (¯ x ) , e ( x ) e (¯ x ) + | e ( x ) − e (¯ x ) | e (¯ x ) + C | x − ¯ x | s − a + δ
100 + C r s − a + δ , thanks to (5.64).Similarly, for each y ∈ Ω ∩ B r (¯ y ) , e ( y ) > − δ , and consequently Z Z (Ω ∩ B r (¯ x )) × (Ω ∩ B r (¯ y )) ( e ( x ) − e ( y )) | x − y | s dx dy > (cid:18) − a − δ (cid:19) Z Z (Ω ∩ B r (¯ x )) × (Ω ∩ B r (¯ y )) dx dy | x − y | s > (cid:18) − a − δ (cid:19) | Ω ∩ B r (¯ x ) | | Ω ∩ B r (¯ y ) | d s > (1 − a ) d s | Ω ∩ B r (¯ x ) | | Ω ∩ B r (¯ y ) | > (1 − a ) r d s . From this and (5.66), noticing that s +22 s − > , we obtain (5.65), as desired.Gathering (5.65) and (5.62) we find that(5.67) λ ( m ) = β Z Z Q | e ( x ) − e ( y ) | | x − y | n +2 s dx dym Z D e dx − m Z Ω \ D e dx > m + mmm · C (1 − a ) s +22 s − − a , for some C > .We also notice that, in view of (5.63), (1 − a ) s +22 s − − a = (1 − a ) s +22 s − (1 − a )(1 + a ) = (1 − a ) s +32 s − a >
12 (1 − a ) s +32 s − > (cid:18) − m ( m + m ) m ( m − m ) (cid:19) s +32 s − . By inserting this inequality into (5.67), we conclude that(5.68) λ ( m ) > C ( m + m ) mm · (cid:18) − m ( m + m ) m ( m − m ) (cid:19) s +32 s − , up to renaming C .Now, for any m < and any m > − m , we define the function ¯ g m,m : (0 , + ∞ ) → (0 , + ∞ ) given by ¯ g m,m ( m ) := m + mm m · (cid:18) − m ( m + m ) m ( m − m ) (cid:19) s +32 s − . We remark that lim m ց ¯ g m,m ( m ) = + ∞ and lim m ր + ∞ ¯ g m,m ( m ) = 1 m · (cid:18) − m m (cid:19) s +32 s − > , and consequently(5.69) inf m ∈ (0 , + ∞ ) ¯ g m,m ( m ) > . Also, by (5.68), and making use of (5.2) and Proposition 5.2, we find that λ ( m , m, m ) = inf m ∈ ˜ M ( m ,m,m ) λ ( m ) > C ¯ g m,m ( m ) . In particular, λ ( m , m, m ) > C inf m ∈ (0 , + ∞ ) ¯ g m,m ( m ) , which, combined with (5.69), proves (1.22).Similarly, we see that lim m ց λ ( m , m, m ) > C lim m ց ¯ g m,m ( m ) = + ∞ , NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 35 which establishes (1.23).In addition, given m < and m > , if we consider the auxiliary function g ⋆m,m :( − m , + ∞ ) → (0 , + ∞ ) defined by g ⋆m,m ( m ) := m + mm m · (cid:18) − m ( m + m ) m ( m − m ) (cid:19) s +32 s − , we see that lim m ց− m g ⋆m,m ( m ) = − − m + mmm > and lim m ր + ∞ g ⋆m,m ( m ) = 1 m · (cid:16) − m m (cid:17) s +32 s − > , and these observations allow us to conclude that(5.70) inf m ∈ ( − m , + ∞ ) g ⋆m,m ( m ) > . Moreover, we deduce from Proposition 5.2, (5.2) and (5.68) that λ ( m, m , m ) = inf m ∈ ˜ M ( m, m ,m ) λ ( m ) > C g ⋆m,m ( m ) . Therefore λ ( m, m , m ) > C inf m ∈ ( − m , + ∞ ) g ⋆m,m ( m ) , which, together with (5.70), proves (1.24). (cid:3) Now we prove Theorem 1.10 in the case s ∈ (0 , / . This case is somehow conceptuallyrelated to the case n > , since the problem boils down to a subcritical situation.We suppose without loss of generality that(5.71) ( − ,
2) = B ⊂ Ω , and we define the function(5.72) ϕ ( x ) := c ⋆ + 1 if x ∈ B ρ ,c ⋆ + log | x | log ρ if x ∈ B \ B ρ ,c ⋆ if x ∈ R \ B . Here, c ⋆ > is the constant introduced in (5.12), and we set(5.73) D := B ρ . For our purposes, we recall the following basic inequality:
Lemma 5.11.
For every x , y ∈ R n \ { } , we have that (5.74) (cid:12)(cid:12) log | x | − log | y | (cid:12)(cid:12) (cid:12)(cid:12) | x | − | y | (cid:12)(cid:12) min {| x | , | y |} . Proof.
Without loss of generality, we assume that | y | | x | . To check (5.74), we take t := | x || y | − ,and we see that (cid:12)(cid:12) log | x | − log | y | (cid:12)(cid:12) = log | x | − log | y | = log | x || y | = log(1 + t ) t = | x || y | − | x | − | y || y | , as desired. (cid:3) With this, we now list some properties of the auxiliary function ϕ in (5.72). Lemma 5.12.
Let n = 1 , s ∈ (0 , / and ϕ be as in (5.72) . Then, lim ρ ց Z Z Q | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 0 . Proof.
Without loss of generality, we suppose that ρ < / . We observe that(5.75) Z Z B ρ × B ρ | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 0 and(5.76) Z Z ( R n \ B ) × ( R n \ B ) | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 0 . Moreover, lim ρ ց Z Z B ρ × ( R n \ B ) | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = lim ρ ց Z Z B ρ × ( R n \ B ) dx dy | x − y | s lim ρ ց Z B ρ dx Z R n \ B | z | s dz lim ρ ց Cρ = 0 . (5.77)Now we observe that if x ∈ B \ B ρ and y ∈ B ρ , then | ϕ ( x ) − ϕ ( y ) | = 1(log ρ ) | log | x | − log ρ | . As a consequence, changing variables X := x/ρ and Y := y/ρ , and taking k ∈ N suchthat k − /ρ k , we see that Z Z ( B \ B ρ ) × B ρ | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 1(log ρ ) Z Z ( B \ B ρ ) × B ρ | log | x | − log ρ | | x − y | s dx dy = ρ − s (log ρ ) Z Z ( B /ρ \ B ) × B | log | X || | X − Y | s dX dY ρ − s (log ρ ) k X j =2 Z Z ( B j \ B j − ) × B | log(2 j ) | (2 j − − s dX dY C ρ − s (log ρ ) k X j =2 j j j (1+2 s ) C ρ − s (log ρ ) k X j =1 j sj C ρ − s (log ρ ) , (5.78)up to renaming C > .In addition, using (5.74) (with | y | := ρ ) and changing variable z := x − y , Z Z ( B ρ \ B ρ ) × B ρ | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 1(log ρ ) Z Z ( B ρ \ B ρ ) × B ρ | log | x | − log ρ | | x − y | s dx dy ρ ) Z Z ( B ρ \ B ρ ) × B ρ ( | x | − ρ ) ρ | x − y | s dx dy ρ ) Z Z ( B ρ \ B ρ ) × B ρ | x − y | − s ρ dx dy ρ ) Z Z B ρ × B ρ | z | − s ρ dz dy = C ρ − s (log ρ ) , NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 37 for some
C > .From this and (5.78), we deduce that(5.79) lim ρ ց Z Z ( B \ B ρ ) × B ρ | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 0 . Moreover, recalling (5.71),
Z Z ( B \ B ρ ) × ( R n \ Ω) | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 1(log ρ ) Z Z ( B \ B ρ ) × ( R n \ Ω) (cid:12)(cid:12) log | x | (cid:12)(cid:12) | x − y | s dx dy ρ ) Z B \ B ρ (cid:12)(cid:12) log | x | (cid:12)(cid:12) dx Z R n \ B dz | z | s C (log ρ ) , for some C > . This implies(5.80) lim ρ ց Z Z ( B \ B ρ ) × ( R n \ Ω) | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 0 . Now, we observe that
Z Z ( B / \ B ρ ) × (Ω \ B ) | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 1(log ρ ) Z Z ( B / \ B ρ ) × (Ω \ B ) (cid:12)(cid:12) log | x | (cid:12)(cid:12) | x − y | s dx dy ρ ) Z B / \ B ρ (cid:12)(cid:12) log | x | (cid:12)(cid:12) dx Z Ω \ B | z | s dz C (log ρ ) , (5.81)for a suitable C > .Furthermore, taking R > such that Ω ⊂ B R , Z Z ( B \ B / ) × (Ω \ B ) | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 1(log ρ ) Z Z ( B \ B / ) × (Ω \ B ) (cid:12)(cid:12) log | x | (cid:12)(cid:12) | x − y | s dx dy (log ρ ) Z Z R ( y − x ) − − s dx dy = 2(log 2) s (log ρ ) Z [(1 − x ) − s − ( R − x ) − s ] dx = 2(log 2) s (1 − s )(log ρ ) " ( R − − s + (cid:18) (cid:19) − s − (cid:18) R − (cid:19) − s . This and (5.81) give that(5.82) lim ρ ց Z Z ( B \ B ρ ) × (Ω \ B ) | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 0 . Now, we take k ∈ N such that(5.83) k +1 < ρ k , and we observe that Z Z ( B \ B ρ ) × ( B \ B ρ ) | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 1(log ρ ) Z Z ( B \ B ρ ) × ( B \ B ρ ) (cid:12)(cid:12) log | x | − log | y | (cid:12)(cid:12) | x − y | s dx dy = 2(log ρ ) Z Z ( B \ Bρ ) × ( B \ Bρ ) {| x | > | y |} (cid:12)(cid:12) log | x | − log | y | (cid:12)(cid:12) | x − y | s dx dy ρ ) Z Z (cid:18) B \ B / k +1 (cid:19) × (cid:18) B \ B / k +1 (cid:19) {| x | > | y |} (cid:12)(cid:12) log | x | − log | y | (cid:12)(cid:12) | x − y | s dx dy = 2(log ρ ) k X i,j =0 Z Z ( B / i \ B / i +1 ) × (cid:18) B / j \ B / j +1 (cid:19) {| x | > | y |} (cid:12)(cid:12) log | x | − log | y | (cid:12)(cid:12) | x − y | s dx dy. (5.84)Moreover, we remark that if x ∈ B / i , y B / j +1 and | x | > | y | , we have that i > | x | > | y | > j +1 , and accordingly i j + 1 .This observation and (5.84) yield that (log ρ ) Z Z ( B \ B ρ ) × ( B \ B ρ ) | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy k X j =0 j +1 X i =0 Z Z ( B / i \ B / i +1 ) × (cid:18) B / j \ B / j +1 (cid:19) {| x | > | y |} (cid:12)(cid:12) log | x | − log | y | (cid:12)(cid:12) | x − y | s dx dy I + I , (5.85)where I := k X j =0 X j − i j +1 Z Z ( B / i \ B / i +1 ) × (cid:18) B / j \ B / j +1 (cid:19) {| x | > | y |} (cid:12)(cid:12) log | x | − log | y | (cid:12)(cid:12) | x − y | s dx dy and I := k X j =0 X i j − Z Z ( B / i \ B / i +1 ) × (cid:18) B / j \ B / j +1 (cid:19) {| x | > | y |} (cid:12)(cid:12) log | x | − log | y | (cid:12)(cid:12) | x − y | s dx dy. We point out that, if x ∈ B / i and | y | | x | , then | x − y | | x | + | y | | x | i − . NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 39
In light of this fact and (5.74), we have that I k X j =0 X j − i j +1 Z Z ( B / i \ B / i +1 ) × (cid:18) B / j \ B / j +1 (cid:19) {| x | > | y |} | x − y | − s | y | dx dy k X j =0 X j − i j +1 Z Z ( B / i \ B / i +1 ) × ( B / j \ B / j +1 ) 2 (1 − i )(1 − s ) − j +1) dx dy = k X j =0 X j − i j +1 − i − j (1 − i )(1 − s ) − j +1) = 2 − s k X j =0 X j − i j +1 − i (1 − s ) j · − s k X j =0 − j − − s ) j = 3 · − s k X j =0 (2 s − j . (5.86)In addition, if x ∈ B / i and y B / j +1 and | x | > | y | , (cid:12)(cid:12) log | x | − log | y | (cid:12)(cid:12) = log | x | − log | y | log 12 i − log 12 j +1 = (cid:0) j − i + 1 (cid:1) log 2 . As a result, I log k X j =0 X i j − Z Z ( B / i \ B / i +1 ) × ( B / j \ B / j +1 ) ( j − i + 1) | x − y | s dx dy. (5.87)Furthermore, if x ∈ B / i \ B / i +1 and y ∈ B / j , with i j − , we see that | x − y | > | x | − | y | > i +1 − j = 12 i +1 (cid:18) − j − i − (cid:19) > i +2 > | x | . Then, we insert this information into (5.87) and we conclude that I s log k X j =0 X i j − Z Z ( B / i \ B / i +1 ) × ( B / j \ B / j +1 ) ( j − i + 1) | x | s dx dy s log k X j =0 X i j − Z Z ( B / i \ B / i +1 ) × ( B / j \ B / j +1 ) ( j − i + 1) − ( i +1)(1+2 s ) dx dy = 4 s log k X j =0 X i j − − i − j ( j − i + 1) − ( i +1)(1+2 s ) = 2 s ) log k X j =0 X i j − si − j ( j − i + 1) = 2 s )+1 log k X j =0 X i j − (2 s − i i − j − ( j − i + 1) . Hence, changing index of summation by posing ℓ := j − i + 1 , I s )+1 log k X i =0 X ℓ > (2 s − i − ℓ ℓ ¯ C k X i =0 (2 s − i , where ¯ C := 2 s )+1 log + ∞ X ℓ =0 − ℓ ℓ . We plug this information and (5.86) into (5.85) and we find that(5.88) (log ρ ) Z Z ( B \ B ρ ) × ( B \ B ρ ) | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy C ⋆ k X m =0 (2 s − m , where C ⋆ := 2( ¯ C + 3 · − s ) . We observe that k X m =0 (2 s − m = k if s = 12 , while k X m =0 (2 s − m + ∞ X m =0 (2 s − m =: C ♯ if s ∈ (cid:18) , (cid:19) , and consequently, by (5.88), (log ρ ) Z Z ( B \ B ρ ) × ( B \ B ρ ) | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy C ⋆ k if s = 12 ,C ⋆ C ♯ if s ∈ (cid:18) , (cid:19) . From this and (5.83), it follows that (log ρ ) Z Z ( B \ B ρ ) × ( B \ B ρ ) | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy C ⋆ | log ρ | log 2 if s = 12 ,C ⋆ C ♯ if s ∈ (cid:18) , (cid:19) . This implies that lim ρ ց Z Z ( B \ B ρ ) × ( B \ B ρ ) | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 0 . From this, (5.75), (5.76), (5.77), (5.79), (5.80) and (5.82) we obtain the desired result. (cid:3)
An additional useful property of the function ϕ defined in (5.72) is the following: Lemma 5.13.
Let n = 1 , s ∈ (0 , / and ϕ be as in (5.72) . Then, lim ρ ց m Z D ϕ dx − m Z Ω \ D ϕ dx = − m ( m + m ) m | Ω | > . Proof.
From (5.72) and (5.73), and exploiting (5.1) and (5.12), we see that m Z D ϕ dx = m ( c ⋆ + 1) | D | = m ( c ⋆ + 1) m + m m + m | Ω | = mm ( m + m ) m ( m + m ) | Ω | and that m Z Ω \ D ϕ dx = m Z B \ B ρ ϕ dx + m Z Ω \ B ϕ dx NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 41 = m Z B \ B ρ (cid:18) c ⋆ + log | x | log ρ (cid:19) dx + m Z Ω \ B c ⋆ dx. We also remark that(5.89) (cid:18) log | x | log ρ (cid:19) χ B \ B ρ . Therefore, by the Dominated Convergence Theorem, and recalling (5.14), lim ρ ց m Z D ϕ dx − m Z Ω \ D ϕ dx = lim m ր + ∞ mm ( m + m ) m ( m + m ) | Ω | − lim ρ ց m Z B \ B ρ (cid:18) c ⋆ + log | x | log ρ (cid:19) dx + m Z Ω \ B c ⋆ dx ! = m ( m + m ) m | Ω | − m c ⋆ | Ω | = m ( m + m ) m | Ω | − m (cid:18) m + m m (cid:19) | Ω | = m ( m + m ) m ( m − ( m + m )) | Ω | = − m ( m + m ) m | Ω | , which is positive, since m ∈ ( − m, , as desired. (cid:3) In the case s ∈ (cid:0) , (cid:3) , it is also convenient to introduce the function(5.90) ψ ( x ) := c ♯ − if x ∈ B ρ ,c ♯ − log | x | log ρ if x ∈ B \ B ρ ,c ♯ if x ∈ R \ B , where c ♯ is defined in (5.35). We also set(5.91) D := Ω \ B ρ , and we study the main properties of the auxiliary function ψ .Comparing (5.72) with (5.90), we observe that | ψ ( x ) − ψ ( y ) | = | ϕ ( x ) − ϕ ( y ) | , and therefore,from Lemma 5.12, we obtain that: Lemma 5.14.
Let n = 1 , s ∈ (0 , / and ψ be as in (5.90) . Then, lim ρ ց Z Z Q | ψ ( x ) − ψ ( y ) | | x − y | s dx dy = 0 . We also have that:
Lemma 5.15.
Let n = 1 , s ∈ (0 , / and ψ be as in (5.90) . Then, lim ρ ց Z D ψ dx − m Z Ω \ D ψ dx = − m ( m − m ) m | Ω | > . Proof.
In view of (5.90), (5.91), (5.35) and (5.1), m Z Ω \ D ψ dx = m ( c ♯ − | Ω \ D | = m (cid:18) m − mm − (cid:19) m − m m + m | Ω | = mm ( m − m ) m ( m + m ) | Ω | , and m Z D ψ dx = m Z B \ B ρ (cid:18) c ♯ − log | x | log ρ (cid:19) dx + m Z Ω \ B c ♯ dx. Hence, recalling (5.37) and (5.89), and using the Dominated Convergence Theorem, lim ρ ց m Z D ψ dx − m Z Ω \ D ψ dx = lim ρ ց m Z B \ B ρ (cid:18) c ♯ − log | x | log ρ (cid:19) dx + m Z Ω \ B c ♯ dx − lim m ր + ∞ mm ( m − m ) m ( m + m ) | Ω | = mc ♯ | Ω | − m ( m − m ) m | Ω | = m ( m − m ) m | Ω | − m ( m − m ) m | Ω | = m ( m − m ) m (cid:0) ( m − m ) − m (cid:1) | Ω | = − m ( m − m ) m | Ω | , which is positive since m < < m . (cid:3) We are now ready to complete the proof of Theorem 1.10 in the case s ∈ (0 , / . Proof of Theorem 1.10 when s ∈ (0 , / . To prove (1.25), we exploit the auxiliary function ϕ introduced in (5.72) and the choice of the resource m ∈ ˜ M , as defined in (5.2), with D asin (5.73).In this way, in light of (1.17), λ ( m, m, m ) β Z Z Q | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dym Z D ϕ dx − m Z Ω \ D ϕ dx . Hence, taking m := m and utilizing Lemmata 5.12 and 5.13, we find that lim m ր + ∞ λ ( m , m, m ) lim ρ ց β Z Z Q | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy m Z D ϕ dx − m Z Ω \ D ϕ dx = 0 . This proves (1.25), and we now focus on the proof of (1.26). To this end, we exploit theauxiliary function ψ introduced in (5.90) and the choice of the resource m ∈ ˜ M , as definedin (5.2), with D as in (5.91).In this framework, in light of (1.17), λ ( m, m, m ) β Z Z Q | ψ ( x ) − ψ ( y ) | | x − y | s dx dym Z D ψ dx − m Z Ω \ D ψ dx . As a result, taking m := m and utilizing Lemmata 5.14 and 5.15, we conclude that lim m ր + ∞ λ ( m, m , m ) lim ρ ց β Z Z Q | ψ ( x ) − ψ ( y ) | | x − y | s dx dym Z D ψ dx − m Z Ω \ D ψ dx = 0 , thus establishing (1.26). (cid:3) NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 43 Badly displayed resources, hectic oscillations and proof of Theorem 1.11
This section contains the proof of Theorem 1.11, relying on an explicit example of sequenceof highly oscillating resources which make the first eigenvalue diverge. The technical details goas follows.
Proof of Theorem 1.11.
We suppose that B ⊂ Ω and we consider η ∈ C ∞ ( B / , [0 , with η =1 in B and k η k C ( R n ) . We let m ω := m − Λ | Ω | Z Ω η ( x ) sin( ωx ) dx and m ( x ) := m ω + Λ η ( x ) sin( ωx ) , with ω > to be taken arbitrarily large in what follows.We remark that(6.1) | Ω | Z Ω m ( x ) dx = m ω + Λ | Ω | Z Ω η ( x ) sin( ωx ) dx = m . Moreover, integrating by parts, | m ω − m | = Λ | Ω | (cid:12)(cid:12)(cid:12)(cid:12)Z Ω η ( x ) sin( ωx ) dx (cid:12)(cid:12)(cid:12)(cid:12) = Λ | Ω | ω (cid:12)(cid:12)(cid:12)(cid:12)Z Ω η ( x ) ddx cos( ωx ) dx (cid:12)(cid:12)(cid:12)(cid:12) = Λ | Ω | ω (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ∂ η ( x ) cos( ωx ) dx (cid:12)(cid:12)(cid:12)(cid:12) ω which is arbitrarily small provided that ω is large enough: in particular, we can suppose that(6.2) m m ω m . Also, for every x ∈ Ω ,(6.3) | m ( x ) | | m ω | + Λ | m | + Λ . Furthermore, for large ω we have that p ± := (cid:16) ± π ω , , . . . , (cid:17) ∈ B ⊂ Ω ∩ { η = 1 } . Therefore sup Ω m > m ( p + ) = m ω + Λ > Λ − | m | > Λ2 and inf Ω m m ( p − ) = m ω − Λ − Λ . (6.4)In view of (1.27), (6.1), (6.3) and (6.4), we obtain that(6.5) m ∈ M ♯ Λ ,m . Now, we take into account a function ϕ ∈ X α,β such that Z Ω m ( x ) ϕ ( x ) dx = 1 . Then, integrating by parts, we see that Z Ω (cid:16) m ω + Λ η ( x ) sin( ωx ) (cid:17) ϕ ( x ) dx = m ω Z Ω ϕ ( x ) dx + Λ Z R n η ( x ) sin( ωx ) ϕ ( x ) dx = m ω Z Ω ϕ ( x ) dx − Λ ω Z R n η ( x ) ddx cos( ωx ) ϕ ( x ) dx = m ω Z Ω ϕ ( x ) dx + Λ ω Z R n ∂ η ( x ) cos( ωx ) ϕ ( x ) dx + 2Λ ω Z R n η ( x ) cos( ωx ) ϕ ( x ) ∂ ϕ ( x ) dx m ω Z Ω ϕ ( x ) dx + 8Λ ω Z R n ϕ ( x ) dx + 2Λ ω Z R n ϕ ( x ) |∇ ϕ ( x ) | dx m ω Z Ω ϕ ( x ) dx + 9Λ ω Z R n ϕ ( x ) dx + Λ ω Z R n |∇ ϕ ( x ) | dx. As a consequence, if ω − m ω (which is the case for ω large, in view of (6.2)), Λ ω Z R n |∇ ϕ ( x ) | dx, and accordingly(6.6) α Z Ω |∇ ϕ ( x ) | dx Z Ω m ( x ) ϕ ( x ) dx > αω . Let now ζ ∈ ( − , and E ′ ∈ R n − with | E ′ | | ζ | and E := ( ζ , E ′ ) ∈ R n . We use thetrigonometric identity cos( y ) cos ζ − cos( y + ζ )sin ζ = sin y, for all ζ ∈ R \ ( π Z ) and y ∈ R ,together with the notation Φ := ηϕ and the change of variable X := x + 1 ω ( ζ , E ′ ) = x + Eω , to write that Z Ω (cid:16) m ω + Λ η ( x ) sin( ωx ) (cid:17) ϕ ( x ) dx = m ω Z Ω ϕ ( x ) dx + Λ Z R n sin( ωx )Φ( x ) dx = m ω Z Ω ϕ ( x ) dx + Λsin ζ Z R n (cid:16) cos( ωx ) cos ζ − cos( ωx + ζ ) (cid:17) Φ( x ) dx = m ω Z Ω ϕ ( x ) dx + Λsin ζ (cid:20)Z R n cos( ωx ) cos ζ Φ( x ) dx − Z R n cos( ωX ) Φ (cid:18) X − Eω (cid:19) dX (cid:21) = m ω Z Ω ϕ ( x ) dx + Λsin ζ Z R n cos( ωx ) (cid:20) cos ζ Φ( x ) − Φ (cid:18) x − Eω (cid:19)(cid:21) dx = m ω Z Ω ϕ ( x ) dx + Λsin ζ Z R n cos( ωx ) (cos ζ −
1) Φ( x ) dx + Λsin ζ Z R n cos( ωx ) (cid:18) Φ( x ) − Φ (cid:18) x − Eω (cid:19)(cid:19) dx. Since (cid:12)(cid:12)(cid:12)(cid:12) Φ( x ) − Φ (cid:18) x − Eω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) η ( x ) − η (cid:18) x − Eω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + η (cid:18) x − Eω (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( x ) − ϕ (cid:18) x − Eω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 45 ϕ ( x ) | E | ω + (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( x ) − ϕ (cid:18) x − Eω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , we thereby discover that, if E ∈ B and ω > , (cid:18) m ω + 8Λ | E | ω | sin ζ | (cid:19) Z Ω ϕ ( x ) dx + Λ | sin ζ | Z Ω (1 − cos ζ ) ϕ ( x ) dx + Λ | sin ζ | Z B (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( x ) − ϕ (cid:18) x − Eω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dx (cid:18) m ω + C Λ ω (cid:19) Z Ω ϕ ( x ) dx + C Λ | ζ | Z Ω ϕ ( x ) dx + C Λ | ζ | Z B (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( x ) − ϕ (cid:18) x − Eω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dx, for some C > .In particular, recalling also (6.2), it follows that there exists r ∈ (0 , , possibly dependingon m , Λ and n , such that, if ζ ∈ ( − r , r ) and ω is sufficiently large,(6.7) m ω Z Ω ϕ ( x ) dx + C Λ | ζ | Z B (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( x ) − ϕ (cid:18) x − Eω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dx. We also observe that, given an additional parameter κ > , to be taken conveniently small inwhat follows, (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( x ) − ϕ (cid:18) x − Eω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = 2 (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( x ) + ϕ (cid:18) x − Eω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( x ) − ϕ (cid:18) x − Eω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) κ | ζ | n +2 s − (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( x ) + ϕ (cid:18) x − Eω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + κ − | ζ | − n − s (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( x ) − ϕ (cid:18) x − Eω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) κ | ζ | n +2 s − ϕ ( x ) + 4 κ | ζ | n +2 s − ϕ (cid:18) x − Eω (cid:19) + κ − | ζ | − n − s (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( x ) − ϕ (cid:18) x − Eω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . Then, we plug this information into (6.7) and we conclude that, if r is small enough and ω islarge enough, m ω Z Ω ϕ ( x ) dx + C Λ | ζ | " κ Z B | ζ | n +2 s − ϕ ( x ) dx + κ Z B | ζ | n +2 s − ϕ (cid:18) x − Eω (cid:19) dx + 1 κ Z B | ζ | − n − s (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( x ) − ϕ (cid:18) x − Eω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dx m ω Z Ω ϕ ( x ) dx + C Λ | ζ | " κ | ζ | n +2 s − Z B ϕ ( x ) dx + 1 κ Z B | ζ | − n − s (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( x ) − ϕ (cid:18) x − Eω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dx . (6.8)We also remark that, in our notation, E = ζ , and accordingly Z Z B × ( B r ∩{| E ′ | | E |} ) | ζ | n +2 s − ϕ ( x ) dx dE Z Z B × B r | E | n +2 s − ϕ ( x ) dx dE ˜ Cr n +2 s − Z B ϕ ( x ) dx, for some ˜ C > . For this reason, letting ι be the measure of the set { x = ( x , x ′ ) ∈ B s.t. | x ′ | | x |} , weobtain that m ω Z Z Ω × ( B r ∩{| E ′ | | E |} ) ϕ ( x ) dx dE + 2 Cκ Λ Z Z B × ( B r ∩{| E ′ | | E |} ) | ζ | n +2 s − ϕ ( x ) dx dE m ω ι r n Z Ω ϕ ( x ) dx + 2 C ˜ Cκ Λ r n +2 s − Z B ϕ ( x ) dx , by choosing κ := min (cid:26) , − m ω ιC ˜ C Λ r n +2 s − (cid:27) . Therefore, we can integrate (6.8) over E ∈ B r ∩ {| E ′ | | E |} and up to renaming constantswe find that r n C Λ Z Z B × ( B r ∩{| E ′ | | E |} ) | E | − n − s (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( x ) − ϕ (cid:18) x − Eω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dx dE C Λ Z Z B × ( B r ∩{| E ′ | | E |} ) | E | − n − s (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( x ) − ϕ (cid:18) x − Eω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dx dE = C Λ ω − s Z Z B × ( B r /ω ∩{| z ′ | | z |} ) | z | − n − s | ϕ ( x ) − ϕ ( x − z ) | dx dz C Λ ω − s Z Z B × R n | ϕ ( x ) − ϕ ( x − z ) | | z | n +2 s dx dz C Λ ω − s Z Z Q | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy, and consequently β Z Z Q | ϕ ( x ) − ϕ ( y ) | | x − y | n +2 s dx dy Z Ω m ( x ) ϕ ( x ) dx > βr n ω s C Λ , up to renaming C > .This and (6.6), recalling (6.5), give that λ ( m ) > αω
2Λ + βr n ω s C Λ , which, taking ω as large as we wish, yields the desired result. (cid:3) Appendix A. Proofs of Theorems 1.7 and 1.8 when n = 2 The main strategy followed in this part is similar to the case n > , but when n = 2 we haveto define different auxiliary functions. We start with the proof of Theorem 1.7. For this, werecall the setting in (5.10) and (5.12), and we define(A.1) ϕ ( x ) := c ⋆ + 1 if x ∈ B ρ ,c ⋆ + log | x | log ρ if x ∈ B \ B ρ ,c ⋆ if x ∈ R \ B . We set(A.2) D := B ρ and we list below some interesting properties of ϕ : NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 47
Lemma A.1.
Let n = 2 and ϕ be as in (A.1) . Then lim ρ ց Z Ω |∇ ϕ | = 0 . Proof.
We compute ρ ) Z B \ B ρ | x | dx = 2 π (log ρ ) Z ρ r dr = − π log ρ −→ as ρ ց . (cid:3) Lemma A.2.
Let n = 2 and ϕ be as in (A.1) . Then lim ρ ց Z Z Q | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 0 . Proof.
As for the proof of Lemma 5.5, we have to consider several integral contributions (giventhe different expressions of the competitors the technical computations here are different fromthose in Lemma 5.5). First of all, we have that
Z Z B ρ × B ρ | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 0 and Z Z ( R \ B ) × ( R \ B ) | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 0 . Moreover, assuming ρ < / , Z Z B ρ × ( R \ B ) | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = Z Z B ρ × ( R \ B ) | x − y | s dx dy Z B ρ dx Z R \ B | z | s dz C | B | ρ −→ as ρ ց .Furthermore, if ( x, y ) ∈ ( B \ B ρ ) × B ρ we have | ϕ ( x ) − ϕ ( y ) | = 1(log ρ ) | log | x | − log ρ | . Consequently, from (5.74) (used here with | y | = ρ ), Z Z ( B \ B ρ ) × B ρ | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 1(log ρ ) Z Z ( B \ B ρ ) × B ρ | log | x | − log ρ | | x − y | s dx dy ρ (log ρ ) Z Z ( B \ B ρ ) × B ρ ( | x | − ρ ) | x − y | s dx dy ρ (log ρ ) Z Z ( B \ B ρ ) × B ρ | x − y | − s dx dy ρ (log ρ ) Z B ρ dy Z B | z | − s dz C (log ρ ) −→ as ρ ց .Also, exploiting (5.10), we have Z Z ( B \ B ρ ) × ( R \ Ω) | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 1(log ρ ) Z Z ( B \ B ρ ) × ( R \ Ω) (log | x | ) | x − y | s dx dy ρ ) Z B \ B ρ (log | x | ) dx Z R \ B | z | s dz C (log ρ ) Z B \ B ρ (log | x | ) dx C (log ρ ) −→ as ρ ց .Now from (5.74), used here with | y | = 1 , we have Z Z ( B \ B ρ ) × (Ω \ B ) | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 1(log ρ ) Z Z ( B \ B ρ ) × (Ω \ B ) (log | x | ) | x − y | s dx dy ρ ) Z Z ( B \ B ρ ) × (Ω \ B ) (1 − | x | ) | x | | x − y | s dx dy ρ ) Z B \ B ρ | x | dx Z B R +1 | z | − s dz C (log ρ ) Z B \ B ρ | x | dx = − C log ρ −→ as ρ ց , where we took R > sufficiently large such that Ω ⊂ B R .In addition, utilizing again (5.74), we first notice that Z Z ( B \ B ρ ) × ( B \ B ρ ) | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 1(log ρ ) Z Z ( B \ B ρ ) × ( B \ B ρ ) (log | x | − log | y | ) | x − y | s dx dy = 2(log ρ ) Z Z ( B \ Bρ ) × ( B \ Bρ ) | x | | y | (log | x | − log | y | ) | x − y | s dx dy. ρ ) Z Z ( B \ Bρ ) × ( B \ Bρ ) | x | | y | | x − y | | x | | x − y | s dx dy ρ ) Z B \ B ρ | x | dx Z B | z | − s dz C (log ρ ) Z B \ B ρ | x | dx = − C log ρ −→ as ρ ց , which concludes the proof. (cid:3) Lemma A.3.
Let n = 2 and ϕ be as in (A.1) . Then lim ρ ց m Z D ϕ dx − m Z Ω \ D ϕ dx = − m ( m + m ) | Ω | m > . Proof.
By (A.1) and (A.2), and recalling (5.1), we have that m Z D ϕ dx = m ( c ⋆ + 1) | B ρ | = m ( c ⋆ + 1) m + m m + m | Ω | . Hence, in light of (5.12) and (5.14), lim ρ ց m Z D ϕ dx = lim m ր + ∞ m (cid:18) − m + m m + 1 (cid:19) m + m m + m | Ω | = (cid:18) − m + m m + 1 (cid:19) ( m + m ) | Ω | . Moreover, by (5.12) and the Dominated Convergence Theorem, m Z Ω \ D ϕ dx = m Z B \ B ρ (cid:18) c ⋆ + log | x | log ρ (cid:19) dx + m c ⋆ | Ω \ B | NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 49 −→ m Z B (cid:18) − m + m m (cid:19) dx + m (cid:18) − m + m m (cid:19) | Ω \ B | = m (cid:18) − m + m m (cid:19) | Ω | , as ρ ց .As a result, lim ρ ց m Z D ϕ dx − m Z Ω \ D ϕ dx = "(cid:18) − m + m m + 1 (cid:19) ( m + m ) − m (cid:18) − m + m m (cid:19) | Ω | = (cid:2) m ( m + m ) − ( m + m ) m (cid:3) | Ω | m = [ m − ( m + m )] m ( m + m ) | Ω | m = − m ( m + m ) | Ω | m , which is positive, since m ∈ ( − m, . (cid:3) With this preliminary work, we can complete the proof of Theorem 1.7 in dimension n = 2 ,by arguing as follows: Proof of Theorem 1.7 when n = 2 . We use the function ϕ in (A.1) and the resource m := mχ D − mχ Ω \ D , with D as in (A.2), as competitors in the minimization problem in (1.17).In this way, we find that(A.3) λ ( m, m, m ) α Z Ω |∇ ϕ | dx + β Z Z Q | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dym Z D ϕ dx − m Z Ω \ D ϕ dx . From Lemmata A.1 and A.2, we have that lim ρ ց α Z Ω |∇ ϕ | dx + β Z Z Q | ϕ ( x ) − ϕ ( y ) | | x − y | s dx dy = 0 . Combining this with Lemma A.3 and (5.14), we obtain the desired result in Theorem 1.7. (cid:3)
We now focus on the proof of Theorem 1.8 when n = 2 . For this, we introduce the function(A.4) ψ ( x ) := c ♯ − if x ∈ B ρ ,c ♯ − log | x | log ρ if x ∈ B \ B ρ ,c ♯ if x ∈ R \ B , where c ♯ is the constant introduced in (5.35). Moreover, we set(A.5) D := Ω \ B ρ . The proofs of the next two results follow directly from Lemmata A.1 and A.2, since, com-paring (A.1) and (A.4), |∇ ϕ | = |∇ ψ | and | ϕ ( x ) − ϕ ( y ) | = | ψ ( x ) − ψ ( y ) | . Lemma A.4.
Let n = 2 and ψ be as in (A.4) . Then lim ρ ց Z Ω |∇ ψ | = 0 . Lemma A.5.
Let n = 2 and ψ be as in (A.4) . Then lim ρ ց Z Z Q | ψ ( x ) − ψ ( y ) | | x − y | s dx dy = 0 . Lemma A.6.
Let n = 2 and ψ be as in (A.4) . Then lim ρ ց m Z D ψ dx − m Z Ω \ D ψ dx = m ( m − m ) m | Ω | > . Proof.
By (A.4), (A.5) and (5.35), we see that(A.6) m Z Ω \ D ψ dx = m ( c ♯ − | Ω \ D | = m (cid:18) m − mm − (cid:19) | Ω \ D | . Also, recalling (5.1), we see that | Ω \ D | = m − m m + m | Ω | . Plugging this information into (A.6), we conclude that m Z Ω \ D ψ dx = m (cid:18) m − mm − (cid:19) m − m m + m | Ω | = m m ( m − m ) m ( m + m ) | Ω | . (A.7)In addition, by the Dominated Convergence Theorem and (5.37), lim ρ ց Z D ψ dx = lim ρ ց Z B \ B ρ (cid:18) c ♯ − log | x | log ρ (cid:19) dx + Z Ω \ B c ♯ dx = c ♯ | B | + c ♯ | Ω \ B | = c ♯ | Ω | = ( m − m ) m | Ω | . This and (A.7) give that lim ρ ց m Z D ψ dx − m Z Ω \ D ψ dx = m ( m − m ) m | Ω | − lim m ր + ∞ m m ( m − m ) m ( m + m ) | Ω | = m ( m − m ) m | Ω | − m ( m − m ) m | Ω | = m ( m − m ) m (cid:0) ( m − m ) − m (cid:1) | Ω | = − m ( m − m ) m | Ω | , which is positive since m < < m . (cid:3) With this preliminary work, we are in the position of completing the proof of Theorem 1.8in the case n = 2 . Proof of Theorem 1.8 when n = 2 . We use the function ψ in (A.4) and the resource m := mχ D − mχ Ω \ D , with D as in (A.5), as a competitor in the minimization problem in (1.17),thus obtaining that(A.8) λ ( m, m, m ) α Z Ω |∇ ψ | dx + β Z Z Q | ψ ( x ) − ψ ( y ) | | x − y | s dx dym Z D ψ dx − m Z Ω \ D ψ dx . From Lemmata A.4 and A.5, and recalling (5.37), we have that lim m ր + ∞ α Z Ω |∇ ψ | dx + β Z Z Q | ψ ( x ) − ψ ( y ) | | x − y | s dx dy = 0 . This, together with Lemma A.6, gives the desired result. (cid:3)
NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 51
Appendix B. Probabilistic motivations for the superposition of ellipticoperators with different orders
The goal of this appendix is to provide a natural framework in which sums of local/nonlocaloperators naturally arise. Though the argument provided can be extended to more generalsuperpositions of operators, for the sake of concreteness we limit ourselves to the operatorin (1.9).For this, extending a presentation in [Val09], we consider a discrete stochastic process on thelattice h Z n , with time increment τ . The space scale h > and the time step τ > will beconveniently chosen to be infinitesimal in what follows.We denote by B := { e , . . . , e n } the standard Euclidean basis of R n , we let B := B ∪ ( − B ) = { e , . . . , e n , − e , . . . , − e n } , and we suppose that a particle moves on h Z n , and,given p ∈ [0 , , λ ∈ N , and s ∈ (0 , , its probability of jumping from a point hk to h ˜ k (with k , ˜ k ∈ Z n ) is given by(B.1) P ( k, ˜ k ) := pc | k − ˜ k | n +2 s + (1 − p ) U ( k − ˜ k )2 n , where c := X j ∈ Z n \{ } | j | n +2 s and U ( j ) := (cid:26) if j ∈ λ B = { λe , . . . , λe n , − λe , . . . , − λe n } , otherwise . We point out that P (˜ k, k ) = P ( k, ˜ k ) = P ( k − ˜ k,
0) = P (˜ k − k, , and that X j ∈ Z n \{ } P ( j,
0) = X j ∈ Z n \{ } (cid:18) pc | j | n +2 s + (1 − p ) U ( j )2 n (cid:19) = p + X j ∈ λ B (1 − p )2 n = p + (1 − p ) = 1 . (B.2)The heuristic interpretation of the probability described in (B.1) is that, at any time step, theparticle has a probability p of following a jump process, and a probability − p of followinga classical random walk. Indeed, with probability p , the particle experiences a jump governedby the power law c | j | n +2 s , while with probability − p it walks to one of the closest neighborsscaled by the additional parameter λ (being all closest neighbors equally probable, and beingthe probability for the particle of not moving at all equal to zero).Therefore, given x ∈ h Z n and t ∈ τ N , we define u ( x, t ) to be the probability density of theparticle to be at the point x at the time t , and we write that the probability of being somewhere,say at x , at the subsequent time step is equal to the superposition of the probabilities of beingat another point of the lattice, say x + hj , at the previous time step times the probability ofgoing from x + hj to x , namely, letting k := xh ∈ Z n , u ( x, t + τ ) = X j ∈ Z n \{ } u ( x + hj, t ) P ( k, k + j ) = X j ∈ Z n \{ } u ( x + hj, t ) P ( j, . As a result, in view of (B.2), u ( x, t + τ ) − u ( x, t )= X j ∈ Z n \{ } (cid:16) u ( x + hj, t ) − u ( x, t ) (cid:17) P ( j, X j ∈ Z n \{ } (cid:16) u ( x + hj, t ) − u ( x, t ) (cid:17) (cid:18) pc | j | n +2 s + (1 − p ) U ( j )2 n (cid:19) = pc X j ∈ Z n \{ } u ( x + hj, t ) − u ( x, t ) | j | n +2 s + 1 − p n X j ∈ λ B (cid:0) u ( x + hj, t ) − u ( x, t ) (cid:1) = p c X j ∈ Z n \{ } u ( x + hj, t ) + u ( x − hj, t ) − u ( x, t ) | j | n +2 s + 1 − p n X j ∈ λ B (cid:0) u ( x + hj, t ) + u ( x − hj, t ) − u ( x, t ) (cid:1) . (B.3)Now we consider two specific situations, namely the one in which(B.4) τ := h s , λ := h s − ∈ N and p is independent on the time step, and the one in which(B.5) τ = h , p := αh − s , and λ := 1 , for a given α > , independent on the time step.We observe that the case in (B.4) corresponds to having the closest neighborhood walk scaledby a suitably large factor (for small h ), while the case in (B.5) corresponds to having the usualnotion of closest neighborhood random walk, with the probability − p that the particle followsit being large (for small h ).In case (B.4), we consider N ∈ N and define h := N s − . In this way, taking N ր + ∞ , onehas that h ց , and thus we deduce from (B.3) that u ( x, t + τ ) − u ( x, t ) τ = ph n c X j ∈ Z n \{ } u ( x + hj, t ) + u ( x − hj, t ) − u ( x, t ) | hj | n +2 s + 1 − p n X j ∈ h s − B u ( x + hj, t ) + u ( x − hj, t ) − u ( x, t ) h s . (B.6)With a formal Taylor expansion, we observe that u ( x + hj, t ) + u ( x − hj, t ) − u ( x, t ) = h D x u ( x, t ) j · j + O ( h ) , therefore the latter sum in (B.6) can be written as X j ∈ h s − B h − s ) D x u ( x, t ) j · j + O ( h − s ) = X j ∈ N B D x u ( x, t ) jN · jN + O (cid:18) N − s − s (cid:19) = X i ∈ B D x u ( x, t ) i · i + o (1) = 2∆ u ( x, t ) + o (1) , as N ր + ∞ (i.e., as h ց ).Hence, recognizing a Riemann sum in the first term of the right hand side of (B.6), takingthe limit as h ց (that is τ ց ), we formally conclude that ∂ t u ( x, t ) = p c Z R n u ( x + y, t ) + u ( x − y, t ) − u ( x, t ) | y | n +2 s dy + 1 − p n ∆ u ( x, t ) , which is precisely the heat equation associated to the operator in (1.9) (up to defining correctlythe structural constants). NON)LOCAL LOGISTIC EQUATIONS WITH NEUMANN CONDITIONS 53
A similar argument can be carried out in case (B.5). Indeed, in this situation one deducesfrom (B.3) that u ( x, t + τ ) − u ( x, t ) τ = αh n c X j ∈ Z n \{ } u ( x + hj, t ) + u ( x − hj, t ) − u ( x, t ) | hj | n +2 s + 1 − αh − s n X j ∈ B u ( x + hj, t ) + u ( x − hj, t ) − u ( x, t ) h . Hence, since X j ∈ B u ( x + hj, t ) + u ( x − hj, t ) − u ( x, t ) h = X j ∈ B D x u ( x + hj, t ) j · j + O ( h )= 2∆ u ( x, t ) + o (1) as h ց , we conclude that in this case ∂ t u ( x, t ) = α c Z R n u ( x + y, t ) + u ( x − y, t ) − u ( x, t ) | y | n +2 s dy + 12 n ∆ u ( x, t ) , which, once again, constitutes the parabolic equation associated to the operator in (1.9) (up torenaming the structural constants). References [AC20] Nicola Abatangelo and Matteo Cozzi,
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