On the definition and the properties of the principal eigenvalue of some nonlocal operators
aa r X i v : . [ m a t h . A P ] J un On the definition and the properties of the principal eigenvalue ofsome nonlocal operators
Henri Berestycki ∗ , J´erˆome Coville † , Hoang-Hung Vo ‡ September 24, 2018
Abstract
In this article we study some spectral properties of the linear operator L Ω + a defined on thespace C ( ¯Ω) by : L Ω [ ϕ ] + aϕ := Z Ω K ( x, y ) ϕ ( y ) dy + a ( x ) ϕ ( x )where Ω ⊂ R N is a domain, possibly unbounded, a is a continuous bounded function and K is a continuous, non negative kernel satisfying an integrability condition.We focus our analysis on the properties of the generalized principal eigenvalue λ p ( L Ω + a )defined by λ p ( L Ω + a ) := sup { λ ∈ R | ∃ ϕ ∈ C ( ¯Ω) , ϕ > , such that L Ω [ ϕ ] + aϕ + λϕ ≤ } . We establish some new properties of this generalized principal eigenvalue λ p . Namely, we provethe equivalence of different definitions of the principal eigenvalue. We also study the behaviourof λ p ( L Ω + a ) with respect to some scaling of K .For kernels K of the type, K ( x, y ) = J ( x − y ) with J a compactly supported probabilitydensity, we also establish some asymptotic properties of λ p (cid:0) L σ,m, Ω − σ m + a (cid:1) where L σ,m, Ω isdefined by L σ, , Ω [ ϕ ] := 1 σ N Z Ω J (cid:18) x − yσ (cid:19) ϕ ( y ) dy . In particular, we prove thatlim σ → λ p (cid:18) L σ, , Ω − σ + a (cid:19) = λ (cid:18) D ( J )2 N ∆ + a (cid:19) , where D ( J ) := R R N J ( z ) | z | dz and λ denotes the Dirichlet principal eigenvalue of the ellipticoperator. In addition, we obtain some convergence results for the corresponding eigenfunction ϕ p,σ . ∗ CAMS - ´Ecole des Hautes ´Etudes en Sciences Sociales, 190-198 avenue de France, 75013, Paris, France, email: [email protected] † UR 546 Biostatistique et Processus Spatiaux, INRA, Domaine St Paul Site Agroparc, F-84000 Avignon, France, email: [email protected] ‡ Institute for Mathematical Sciences, Renmin University of China, 59 Zhongguancun, Haidian, Beijing, 100872,China. present email: [email protected] ontents λ p , λ ′ p , λ ′′ p and λ v λ p (cid:18) L σ,m, Ω − σ m + a (cid:19) . . . . . . . . . . . . . . . . . . . . . . . . 284.2.1 The case 0 < m <
2: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2.2 The case m = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2.3 The case m = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ϕ p,σ The principal eigenvalue of an operator is a fundamental notion in modern analysis. In particular, thisnotion is widely used in PDE’s literature and is at the source of many profound results especially in thestudy of elliptic semi linear problems. For example, the principal eigenvalue is used to characterise thestability of equilibrium of a reaction-diffusion equation enabling the definition of persistence criteria[18, 19, 20, 5, 33, 44, 53]. It is also an important tool in the characterisation of maximum principleproperties satisfies by elliptic operators [12, 8] and to describe continuous semi-groups that preservean order [1, 32, 46]. It is further used in obtaining Liouville type results for elliptic semi-linearequations [10, 6].In this article we are interested in such notion for linear operators L Ω + a defined on the space ofcontinuous functions C ( ¯Ω) by : L Ω [ ϕ ] + aϕ := Z Ω K ( x, y ) ϕ ( y ) dy + a ( x ) ϕ ( x )where Ω ⊂ R N is a domain, possibly unbounded, a is a continuous bounded function and K is a nonnegative kernel satisfying an integrability condition. The precise assumptions on Ω , K and a will begiven later on.To our knowledge, for most of positive operators, the principal eigenvalue is a notion related tothe existence of an eigen-pair, namely an eigenvalue associated with a positive eigen-element. For2he operator L Ω + a , when the function a is not constant, for any real λ , neither L Ω + a + λ nor itsinverse are compact operators. Moreover, as noticed in [25, 30, 41, 60], the operator L Ω + a may nothave any eigenvalues in the space L p (Ω) or C ( ¯Ω). For such operator, the existence of an eigenvalueassociated with a positive eigenvector is then not guaranteed. Studying quantities that can be usedas surrogates of a principal eigenvalue and establishing their most important properties are thereforeof great interest for such operators.In this perspective, we are interested in the properties of the following quantity: λ p ( L Ω + a ) := sup { λ ∈ R | ∃ ϕ ∈ C ( ¯Ω) , ϕ > , such that L Ω [ ϕ ] + a ( x ) ϕ + λϕ ≤ } , (1.1)which can be expressed equivalently by the sup inf formula: λ p ( L Ω + a ) = sup ϕ ∈ C (¯Ω) ϕ> inf x ∈ Ω (cid:18) − L Ω [ ϕ ]( x ) + a ( x ) ϕ ( x ) ϕ ( x ) (cid:19) . (1.2)This number was originally introduced in the Perron-Frobenius Theory to characterise the eigen-values of an irreducible positive matrix [21, 63]. Namely, for a positive irreducible matrix A, theeigenvalue λ ( A ) associated with a positive eigenvector can be characterised as follows: λ p ( A ) := sup x ∈ R N x> inf i ∈{ ,...,N } (cid:18) − ( Ax ) i x i (cid:19) = λ ( A ) = inf x ∈ R N x ≥ ,x =0 sup i ∈{ ,...,N } (cid:18) − ( Ax ) i x i (cid:19) =: λ ′ p ( A ) , (1.3)also known as the Collatz-Wieldandt characterisation.Numerous generalisation of these types of characterisation exist in the literature. Generalisationsof the characterisation of the principal eigenvalue by variants of the Collatz-Wielandt characterisation(i.e. (1.3)) were first obtained for positive compact operators in L p (Ω) [42, 43, 57] and later for generalpositive operators that posses an eigen-pair [35].In parallel with the generalisation of the Perron-Frobenius Theory, several inf sup formulas havebeen developed to characterise the spectral properties of elliptic operators satisfying a maximumprinciple, see the fundamental works of Donsker and Varadhan [32], Nussbaum, Pinchover [50],Berestycki, Nirenberg, Varadhan [8] and Pinsky [51, 52]. In particular, for an elliptic operator definedin a bounded domain Ω ⊂ R N and with bounded continuous coefficients, E := a ij ( x ) ∂ ij + b i ( x ) ∂ i + c ( x ),several notions of principal eigenvalue have been introduced. On one hand, Donsker and Varadhan[32] have introduced a quantity λ V ( E ), called principal eigenvalue of E , that satisfies λ V ( E ) := inf ϕ ∈ dom ( E ) ϕ> sup x ∈ Ω (cid:18) − E [ ϕ ]( x ) ϕ ( x ) (cid:19) = inf { λ ∈ R | ∃ ϕ ∈ dom ( E ) , ϕ > E [ ϕ ]( x ) + λϕ ( x ) ≥ } , where dom ( E ) ⊂ C ( ¯Ω) denotes the domain of definition of E . On the other hand Berestycki, Nirenbergand Varadhan [8] have introduced λ ( E ) defined by: λ ( E ) := sup { λ ∈ R | ∃ ϕ ∈ W ,N (Ω) , ϕ > E [ ϕ ]( x ) + λϕ ( x ) ≤ } , = sup ϕ ∈ W ,N (Ω) ϕ> inf x ∈ Ω (cid:18) − E [ ϕ ]( x ) ϕ ( x ) (cid:19)
3s another possible definition for the principal eigenvalue of E . When Ω is a smooth boundeddomain and E has smooth coefficients, both notions coincide (i.e. λ V ( E ) = λ ( E )). The equivalenceof this two notions has been recently extended for more general elliptic operators, in particular theequivalence holds true in any bounded domains Ω and in any domains when E is an elliptic self-adjointoperator with bounded coefficients [12]. It is worth mentioning that the quantity λ V ( E ) was originallyintroduced by Donsker and Varadhan [32] to obtain the following variational characterisation of λ ( E )in a bounded domain: λ ( E ) = sup dµ ∈ P (Ω) inf ϕ ∈ dom ( E ) ϕ> Z Ω (cid:18) − E [ ϕ ]( x ) ϕ ( x ) (cid:19) dµ ( x ) , where P (Ω) is the set of all probability measure on Ω. Such characterisation is still valid when Ω isunbounded, see Nussbaum and Pinchover [50].Lately, the search of Liouville type results for semilinear elliptic equations in unbounded domains[10, 56] and the characterisation of spreading speed [7, 48] have stimulated the studies of the propertiesof λ ( E ) and several other notions of principal eigenvalue have emerged. For instance, several newnotions of principal eigenvalue have been introduced for general elliptic operators defined on (limitor almost) periodic media [6, 10, 49, 56]. For the interested reader, we refer to [12], for a reviewand a comparison of the different notions of principal eigenvalue for an elliptic operator defined in aunbounded domain.For the operator L Ω + a , much less is known and only partial results have been obtained when Ωis bounded [25, 28, 32, 37, 40, 41] or in a periodic media [29, 30, 60, 59]. More precisely, λ p ( L Ω + a )has been compared to one of the following definitions : λ ′ p ( L Ω + a ) := inf { λ ∈ R | ∃ ϕ ∈ C (Ω) ∩ L ∞ (Ω) , ϕ ≥ , s. t. L Ω [ ϕ ]( x ) + ( a ( x ) + λ ) ϕ ( x ) ≥ } or when L Ω + a is a self-adjoint operator: λ v ( L Ω + a ) := inf ϕ ∈ L (Ω) ,ϕ − hL Ω [ ϕ ] + aϕ, ϕ ik ϕ k L (Ω) , = inf ϕ ∈ L (Ω) ,ϕ RR Ω × Ω K ( x, y )[ ϕ ( x ) − ϕ ( y )] dxdy − R Ω (cid:2) a ( x ) + R Ω K ( x, y ) dy (cid:3) ϕ ( x ) dx k ϕ k L (Ω) , where h , i denotes the scalar product of L (Ω). For Ω ⊂ R N a bounded domain and for particularkernels K , an equality similar to λ V ( E ) = λ ( E ) has been obtained in [25], provided that K ∈ C ( ¯Ω × ¯Ω)satisfies some non-degeneracy conditions. The author shows that λ p ( L Ω + a ) = λ ′ p ( L Ω + a ) . (1.4)In a periodic media, an extension of this equality was obtained in [29, 30] for kernels K of theform K ( x, y ) := J ( x − y ) with J a symmetric positive continuous density of probability. In suchcase, they prove that λ p ( L R N + a ) = λ ′ p ( L R N + a ) = λ v ( L R N + a ) . (1.5)In this paper, we pursue the works begun in [25, 30, 28] by one of the present authors and weinvestigate more closely the properties of λ p ( L Ω + a ). Namely, we first look whether λ p ( L Ω + a )4an be characterised by other notions of principal eigenvalue and under which conditions on Ω , K and a the equality (1.4) or (1.5) holds true. In particular, we introduce a new notion of principaleigenvalue, λ ′′ p ( L Ω + a ), defined by : λ ′′ p ( L Ω + a ) := inf { λ ∈ R | ∃ ϕ ∈ C c (Ω) , ϕ ≥ , such that L Ω [ ϕ ]( x ) + ( a ( x ) + λ ) ϕ ( x ) ≥ } and we compare this new quantity with λ p , λ ′ p and λ v .Another natural question is to obtain a clear picture on the dependence of λ p with respect to all theparameters involved. If the behaviour of λ p ( L Ω + a ) with respect to a or Ω can be exhibited directlyfrom the definition, the impact of scalings of the kernel is usually unknown and has been largelyignored in the literature except in some specific situations involving particular nonlocal dispersaloperators defined in a bounded domain [2, 24, 41, 58].For a particular type of K and a , we establish the asymptotic properties of λ p with respect tosome scaling parameter. More precisely, let K ( x, y ) = J ( x − y ) and let us denote J σ ( z ) := σ N J (cid:0) zσ (cid:1) .When J is a non negative function of unit mass, we study the properties of the principal eigenvalueof the operator L σ,m, Ω − σ m + a , where the operator L σ,m, Ω is defined by: L σ,m, Ω [ ϕ ] := 1 σ m Z Ω J σ ( x − y ) ϕ ( y ) dy. In this situation, the operator L σ,m, Ω − σ m refers to a nonlocal version of the standard diffusionoperator with a homogeneous Dirichlet boundary condition. Such type of operators has appearedrecently in the literature to model a population that have a constrained dispersal [4, 34, 39, 41, 58].In this context, the pre-factor σ m is interpreted as a frequency at which the events of dispersal occur.For m ∈ [0 ,
2] and a large class of J , we obtain the asymptotic limits of λ p (cid:18) L σ,m, Ω − σ m + a (cid:19) as σ → σ → + ∞ . Our interest in studying the properties of λ p ( L Ω + a ) stems from the recent studies of populationshaving a long range dispersal strategy [25, 28, 4, 41, 60]. For such a population, a commonly usedmodel that integrates such long range dispersal is the following nonlocal reaction diffusion equation ([34, 38, 39, 47, 62]): ∂ t u ( t, x ) = Z Ω J ( x − y ) u ( t, y ) dy − u ( t, x ) Z Ω J ( y − x ) dy + f ( x, u ( t, x )) in R + × Ω . (1.6)In this context, u ( t, x ) is the density of the considered population, J is a dispersal kernel and f ( x, s )is a KPP type non-linearity describing the growth rate of the population. When Ω is a boundeddomain [3, 25, 28, 36, 41, 59], an optimal persistence criteria has been obtained using the sign of λ p ( M Ω + ∂ u f ( x, M Ω stands for the operator: M Ω [ ϕ ] := Z Ω J ( x − y ) ϕ ( y ) dy − j ( x ) ϕ ( x ) , j ( x ) := R Ω J ( y − x ) dy .In such model, a population will persists if and only if λ p ( M Ω + ∂ u f ( x, <
0. We can easily checkthat λ p ( M Ω + ∂ u f ( x, λ p ( L Ω − j ( x ) + ∂ u f ( x, R N and in periodic media, adapted versions of λ p have been recently used to definean optimal persitence criteria [29, 30, 60, 58]. The extension of such type of persistence criteria formore general environments is currently investigated by ourself [4] by means of our findings on theproperties of λ p .The understanding of the effect of a dispersal process conditioned by a dispersal budget is anotherimportant question. The idea introduced by Hutson, Martinez, Mischaikow and Vickers [39], is simpleand consists in introducing a cost function related to the amount of energy an individual has to useto produce offspring, that jumps on a long range. When a long range of dispersal is privileged, theenergy consumed to disperse an individual is large and so very few offsprings are dispersed. On thecontrary, when the population chooses to disperse on a short range, few energy is used and a largeamount of the offsprings is dispersed. In R N , to understand the impact of a dispersal budget on therange of dispersal, we are led to consider the family of dispersal operator : M σ,m [ ϕ ]( x ) := 1 σ m ( J σ ⋆ ϕ ( x ) − ϕ ( x )) , where J σ ( z ) := σ N J (cid:0) zσ (cid:1) is the standard scaling of the probability density J . For such family, thestudy of the dependence of λ p ( M σ,m + a ) with respect to σ and m is a first step to analyse the impactof the range of the dispersal σ on the persistence of the population. In particular the asymptoticlimits σ → + ∞ and σ → Let us now state the precise assumptions we are making on the domain Ω, the kernel K and thefunction a . Here, throughout the paper, Ω ⊂ R N is a domain (open connected set of R N ) and for a and K we assume the following: a ∈ C ( ¯Ω) ∩ L ∞ (Ω) , (1.7)and K is a non-negative Caratheodory function, that is K ≥ ∀ x ∈ Ω K ( x, · ) is measurable, K ( · , y ) is uniformly continuous for almost every y ∈ Ω . (1.8)For our analysis, we also require that K satisfies the following non-degeneracy condition: There exist positive constants r ≥ r > , C ≥ c > such that K satisfies: C Ω ∩ Br x ) ( y ) ≥ K ( x, y ) ≥ c Ω ∩ Br x ) ( y ) for all x, y ∈ Ω , (1.9) where A denotes the characteristic function of the set A ⊂ R N and B r ( x ) is the ball centred at x of radius r . These conditions are satisfied for example for kernels like K ( x, y ) = J (cid:16) x − yg ( y ) h ( x ) (cid:17) with h and g positive and bounded in Ω and J ∈ C ( R N ) , J ≥
0, a compactly supported function such that J (0) >
0. Note that when Ω is bounded, any kernel K ∈ C ( ¯Ω × ¯Ω) which is positive on the diagonal,6atisfies all theses assumptions. Under this assumptions, we can check that the operator L Ω + a iscontinuous in C ( ¯Ω),[45].Let us now state our main results. We start by investigating the case of a bounded domain Ω.In this situation, we prove that λ p , λ ′ p and λ ′′ p represent the same quantity. Namely, we show thefollowing Theorem 1.1.
Let Ω ⊂ R N be a bounded domain and assume that K and a satisfy (1.7) – (1.9) .Then, the following equality holds : λ p ( L Ω + a ) = λ ′ p ( L Ω + a ) = λ ′′ p ( L Ω + a ) . In addition, if K is symmetric, then λ p ( L Ω + a ) = λ v ( L Ω + a ) . When Ω is an unbounded domain, the equivalence of λ p , λ ′ p and λ ′′ p is not clear for generalkernels. Namely, let consider Ω = R , K ( x, y ) = J ( x − y ) with J a density of probability with acompact support and such that R R J ( z ) z dz >
0. For the operator L R , which corresponds to thestandard convolution by J , by using e λx and constants as test functions, we can easily check that λ ′ p ( L R ) ≤ − < − min λ> Z R J ( z ) e − λz dz ≤ λ p ( L R ). However some inequalities remain true in generaland the equivalence of the three notions holds for self-adjoint operators. More precisely, we provehere the following Theorem 1.2.
Let Ω ⊂ R N be an unbounded domain and assume that K and a satisfy (1.7) – (1.9) .Then the following inequalities hold λ ′ p ( L Ω + a ) ≤ λ ′′ p ( L Ω + a ) ≤ λ p ( L Ω + a ) . When K is symmetric and such that p ( x ) := R Ω K ( x, y ) dy ∈ L ∞ (Ω) then the following equality holds: λ v ( L Ω + a ) = λ ′ p ( L Ω + a ) = λ ′′ p ( L Ω + a ) = λ p ( L Ω + a ) . Another striking property of λ p refers to the invariance of λ p under a particular scaling of thekernel K . More precisely, we show Proposition 1.3.
Let Ω ⊂ R N be a domain and assume that a and K satisfy (1.7) – (1.9) . For all σ > , let Ω σ := σ Ω , a σ ( x ) := a (cid:0) xσ (cid:1) and L σ, Ω σ [ ϕ ]( x ) := 1 σ N Z Ω σ K (cid:16) xσ , yσ (cid:17) ϕ ( y ) dy. Then for all σ > , one has λ p ( L Ω + a ) = λ p ( L σ, Ω σ + a σ ) . λ p is still validfor Ω = R N . In this case, since R N is invariant under the scaling, we get λ p ( L R N + a ) = λ p ( L σ, R N + a σ ) . Next, for particular type of kernel K , we investigate the behaviour of λ p with respect of somescaling parameter. More precisely, let K ( x, y ) = J ( x − y ) and let J σ ( z ) := σ N J (cid:0) zσ (cid:1) . We considerthe following operator L σ,m, Ω [ ϕ ] := 1 σ m Z Ω J σ ( x − y ) ϕ ( y ) dy. For J is a non negative function of unit mass, we study the asymptotic properties of the principaleigenvalue of the operator L σ,m, Ω − σ m + a when σ → σ → + ∞ .To simplify the presentation of our results, let us introduce the following notation. We denote by M σ,m, Ω , the following operator: M σ,m, Ω [ ϕ ]( x ) := 1 σ m (cid:18) σ N Z Ω J (cid:18) x − yσ (cid:19) ϕ ( y ) dy − ϕ ( x ) (cid:19) . (1.10)For any domains Ω, we obtain the limits of λ p ( M σ,m, Ω + a ) when σ tends either to zero or to + ∞ .Let us denote the second moment of J by D ( J ) := Z R N J ( z ) | z | dz, the following statement describes the limiting behaviour of λ p ( M σ,m, Ω + a ): Theorem 1.4.
Let Ω be a domain and assume that J and a satisfy (1.7) – (1.9) . Assume furtherthat J is even and of unit mass. Then, we have the following asymptotic behaviour: • When < m ≤ , lim σ → + ∞ λ p ( M σ,m, Ω + a ) = − sup Ω a • When m = 0 , lim σ → + ∞ λ p ( M σ, , Ω + a ) = 1 − sup Ω a .In addition, when Ω = R N and if a is symmetric ( a ( x ) = a ( − x ) for all x ) and the map t → a ( tx ) is non increasing for all x, t > then λ p ( M σ, , R N + a ) is monotone non decreasingwith respect to σ . • When ≤ m < , lim σ → λ p ( M σ,m, Ω + a ) = − sup Ω a • When m = 2 and a ∈ C ,α (Ω) for some α > , then lim σ → λ p ( M σ, , Ω + a ) = λ (cid:18) D ( J )2 N ∆ + a (cid:19) and λ (cid:18) D ( J )2 N ∆ + a (cid:19) := inf ϕ ∈ H (Ω) ,ϕ D ( J )2 N R Ω |∇ ϕ | ( x ) dx k ϕ k − R Ω a ( x ) ϕ ( x ) dx k ϕ k . R N .Having established the asymptotic limits of the principal eigenvalue λ p ( M σ, , Ω + a ), it is naturalto ask whether similar results hold for the corresponding eigenfunction ϕ σ,p when it exists. In thisdirection, we prove that for m = 2, such convergence does occur : Theorem 1.5.
Let Ω be any domain and assume that J and a satisfy (1.7) – (1.9) . Assume furtherthat J is even and of unit mass. Then there exists σ such that for all σ ≤ σ , there exists a positiveprincipal eigenfunction ϕ p,σ associated to λ p ( M σ, , Ω + a ) . In addition, when ϕ p,σ ∈ L (Ω) for all σ ≤ σ , we have ϕ p,σ → ϕ in L loc (Ω) , where ϕ ∈ H (Ω) is a positive principle eigenfunction associated to λ (cid:16) D ( J )2 N ∆ + a (cid:17) . Remark 1.
When Ω is bounded, then the condition ϕ p,σ ∈ L is always satisfied. Moreover, in thissituation, the above limits ϕ p,σ → ϕ as σ → L (Ω) instead of L loc (Ω). First, we can notice that the quantity λ V defined by Donsker and Varadhan [32] for elliptic operatorscan also be defined for the operator L Ω + a and is equivalent to the quantity λ ′ p . The equality (1.4)can then be seen as the nonlocal version of the equality λ = λ V where λ is the notion introducedby Berestycki-Nirenberg-Varadhan [8].Next, we would like to emphasize, that unlike the classical elliptic operators, due to the lack ofa regularising effect of the operator L Ω + a , the quantity λ p ( L Ω + a ) may not be an eigenvalue, i.e.the spectral problem L Ω [ ϕ ]( x ) + a ( x ) ϕ ( x ) + λϕ ( x ) = 0 in Ω , may not have a solution in spaces of functions like L p (Ω) , C (Ω)[29, 27, 32, 41]. As a consequence,even in bounded domains, the relations between λ p , λ ′ p , λ ′′ p and λ v are quite delicate to obtain.Another difficulty inherent to the study of nonlocal operators in unbounded domains concerns thelack of natural a priori estimates for the positive eigenfunction thus making standard approximationschemes difficult to use in most case.Lastly, we make some additional comments on the assumptions we have used on the dispersalkernel K . The non-degeneracy assumption (1.9) we are using, is related to the existence of LocalUniform Estimates [22, 23] (Harnack type estimates) for a positive solution of a nonlocal equation: L Ω [ ϕ ] + b ( x ) ϕ = 0 in Ω . (1.11)Such type of estimates is a key tool in our analysis, in particular in unbounded domains, where weuse it to obtain fundamental properties of the principal eigenvalue λ p ( L Ω + a ), such as the limit: λ p ( L Ω + a ) = lim n →∞ λ p ( L Ω n + a ) , where Ω n is a sequence of set converging to Ω. As observed in [26], some local uniform estimates canalso be obtained for some particular kernels K which does not satisfies the non-degeneracy condition91.9). For example, for kernels of the form K ( x, y ) = g N ( y ) J (cid:16) x − yg ( y ) (cid:17) with J satisfying (1.8) and (1.9)and g ≥ { x | g ( x ) = 0 } is a bounded set and with Lebesgue measurezero, some local uniform estimates can be derived for positive solutions of (1.11). As a consequence,the Theorems 1.1 and 1.2 hold true for such kernels. We have also observed that the condition (1.9)can be slightly be relaxed and the Theorems 1.1 and 1.2 hold true for kernels K such that, for somepositive integer p , the kernel K p defined recursively by : K ( x, y ) := K ( x, y ) ,K n +1 ( x, y ) := Z Ω K n ( x, z ) K ( z, y ) dz for n ≥ , satisfies the non-degeneracy condition (1.9).For a convolution operator, i.e. K ( x, y ) := J ( x − y ), this last condition is optimal. It is relatedto a geometric property of the convex hull of { y ∈ R N | J ( y ) > } : K p satisfies (1.9) for some p ∈ N if and only if the convex hull of { y ∈ R N | J ( y ) > } contains . Note that if a relaxed assumption on the lower bound of the non-degeneracy condition satisfiedby K appears simple to find, the condition on the support of K seems quite tricky to relax. To tacklethis problem, it is tempting to investigate the spectrum of linear operators involving the FractionalLaplacian, ∆ α :∆ α ϕ := C N,α
P.V. (cid:18)Z Ω ϕ ( y ) − ϕ ( x ) | x − y | N +2 α dy (cid:19) , ϕ ≡ R N \ ΩThat is, to look for the properties of the principal eigenvalue of the spectral problem:∆ α ϕ + ( a + λ ) ϕ = 0 in Ω . (1.12)As for elliptic operators and L Ω + a , analogues of λ , λ ′ and λ can be defined for ∆ α + a and therelations between all possible definitions can be investigated. When Ω is bounded or a is periodic, thedifferent definitions are equivalent [9]. However, in the situations considered in [9] the operator ∆ α + a has a compact resolvent enabling the use of the Krein Rutmann Theory. Thus, the corresponding λ p is associated with a positive eigenfunction, rendering the relations much more simpler to obtain.Moreover, in this analysis, the regularity of the principal eigenfunction and a Harnack type inequality[16, 17, 61] for some non negative solution of (1.12) are again the key ingredients in the proofs yieldingto the inequality λ ′ p (∆ α + a, Ω) ≤ λ p (∆ α + a, Ω)for any smooth domain Ω.Such Harnack type inequalities are not known for operators L Ω + a involving a continuous kernel K with unbounded support. Furthermore, it seems that most of the tools used to establish theseHarnack estimates in the case of the Fractional Laplacian [16, 61] do not apply when we consider anoperator L Ω + a . Thus, obtaining the inequality λ ′ p ( L Ω + a ) ≤ λ p ( L Ω + a )10ith a more general kernel requires a deeper understanding of Harnack type estimates and/or thedevelopment of new analytical tools for such type of nonlocal operators.Nevertheless, in this direction and in dimension one, for some kernels with unbounded support,we could obtain some inequalities between the different notions of principal eigenvalue. Namely, Proposition 1.6.
Assume N = 1 and let Ω ⊂ R be a unbounded domain. Assume that K and a satisfy (1.7) – (1.8) . Assume further that K is symmetric and there exists C > and α > such that K ( x, y ) ≤ C (1 + | x − y | ) − α . Then we have λ p ( L Ω + a ) ≤ λ v ( L Ω + a ) ≤ λ ′ p ( L Ω + a ) ≤ λ ′′ p ( L Ω + a ) . Outline of the paper:
The paper is organised as follows. In Section 2, we recall some knownresults and properties of the principal eigenvalue λ p ( L Ω + a ). The relations between the differentdefinitions of the principal eigenvalue, λ p , λ ′ p , λ ′′ p and λ v (Theorems 1.1, 1.2 and Proposition 1.6) areproved in Section 3. Finally, in Section 4 we derive the asymptotic behaviour of λ p with respect tothe different scalings of K (Proposition 1.3 and Theorems 1.4 and 1.5 ). To simplify the presentation of the proofs, we introduce some notations and various linear operatorthat we will use throughout this paper: • B R ( x ) denotes the standard ball of radius R centred at the point x • R will always refer to the characteristic function of the ball B R (0). • S ( R N ) denotes the Schwartz space,[15] • C (Ω) denotes the space of continuous function in Ω, • C c (Ω) denotes the space of continuous function with compact support in Ω. • For a positive integrable function J ∈ S ( R N ), the constant R R N J ( z ) | z | dz will refer to Z R N J ( z ) | z | dz := Z R N J ( z ) N X i =1 z i ! dz • For a bounded set ω ⊂ R N , | ω | will denotes its Lebesgue measure • For two L functions ϕ, ψ , h ϕ, ψ i denotes the L scalar product of ψ and ϕ • For J ∈ L ( R N ), J σ ( z ) := σ N J (cid:0) zσ (cid:1) We denote by L σ,m, Ω the continuous linear operator L σ,m, Ω : C ( ¯Ω) → C ( ¯Ω) ϕ σ m Z Ω J σ ( x − y ) u ( y ) dy, (1.13)where Ω ⊂ R N . • We denote by M σ,m, Ω the operator M σ,m, Ω := L σ,m, Ω − σ m In this section, we recall some standard results on the principal eigenvalue of the operator L Ω + a .Since the early work [32] on the variational formulation of the principal eigenvalue, an intrinsicdifficulty related to the study of these quantities comes from the possible non-existence of a positivecontinuous eigenfunction associated to the definition of λ p , λ ′ p , λ ′′ p or to λ v . This means that there isnot always a positive continuous eigenfunction associated to λ p , λ ′ p , λ ′′ p or λ v . A simple illustrationof this fact can be found in [25, 27]. Recently, some progress have been made in the understandingof λ p . In particular, some flexible criteria have been found to guarantee the existence of a positivecontinuous eigenfunction [25, 41, 59]. More precisely, Theorem 2.1 (Sufficient condition [25]) . Let Ω ⊂ R N be a domain, a ∈ C (Ω) ∩ L ∞ (Ω) and K ∈ C ( ¯Ω × ¯Ω) non negative, satisfying the condition (1.9) . Let us denote ν := sup ¯Ω a and assume furtherthat the function a satisfies ν − a L (Ω ) for some bounded domain Ω ⊂ ¯Ω . Then there exists aprincipal eigen-pair ( λ p , ϕ p ) solution of L Ω [ ϕ ]( x ) + ( a ( x ) + λ ) ϕ ( x ) = 0 in Ω . Moreover, ϕ p ∈ C ( ¯Ω) , ϕ p > and we have the following estimate − ν ′ < λ p < − ν, where ν ′ := sup x ∈ Ω (cid:20) a ( x ) + Z Ω K ( x, y ) dy (cid:21) . This criteria is almost optimal, in the sense that we can construct example of operator L Ω + a with Ω bounded and a such that ν − a ∈ L (Ω) and where λ p ( L Ω + a ) is not an eigenvalue in C ( ¯Ω),see [25, 41, 59].When Ω is bounded, sharper results have been recently derived in [27] where it is proved that λ p ( L Ω + a ) is always an eigenvalue in the Banach space of positive measure, that is, we can alwaysfind a positive measure dµ p that is solution in the sense of measure of L Ω [ dµ p ]( x ) + a ( x ) dµ p ( x ) + λ p dµ p ( x ) = 0 . (2.1)In addition, we have the following characterisation of λ p : Theorem 2.2 ([30, 27]) . λ p ( L Ω + a ) is an eigenvalue in C ( ¯Ω) if and only if λ p ( L Ω + a ) < − sup x ∈ Ω a ( x ) .
12e refer to [27] for a more complete description of the positive solution associated to λ p whenthe domain Ω is bounded.Now, we recall some properties of λ p that we constantly use throughout this paper: Proposition 2.3. (i) Assume Ω ⊂ Ω , then for the two operators L Ω1 + a and L Ω2 + a respectively defined on C (Ω ) and C (Ω ) , we have : λ p ( L Ω1 + a ) ≥ λ p ( L Ω2 + a ) . (ii) For a fixed Ω and assume that a ( x ) ≥ a ( x ) , for all x ∈ Ω . Then λ p ( L Ω + a ) ≥ λ p ( L Ω + a ) . (iii) λ p ( L Ω + a ) is Lipschitz continuous with respect to a . More precisely, | λ p ( L Ω + a ) − λ p ( L Ω + b ) | ≤ k a − b k ∞ (iv) The following estimate always holds − sup Ω (cid:18) a ( x ) + Z Ω K ( x, y ) dy (cid:19) ≤ λ p ( L Ω + a ) ≤ − sup Ω a. We refer to [25, 28] for the proofs of ( i ) − ( iv ).Lastly, we prove some limit behaviour of λ p ( L Ω + a ) with respect to the domain Ω. Namely, weshow Lemma 2.4.
Let Ω be a domain and assume that a and K satisfy (1.7) – (1.9) . Let (Ω n ) n ∈ N be asequence of subset of Ω so that lim n →∞ Ω n = Ω , Ω n ⊂ Ω n +1 . Then we have lim n →∞ λ p ( L Ω n + a ) = λ p ( L Ω + a ) Proof.
By a straightforward application of the monotone properties of λ p with respect to the domain((i) of Proposition 2.3) we get the inequality λ p ( L Ω + a ) ≤ lim n →∞ λ p ( L Ω n + a ) . (2.2)To prove the equality, we argue by contradiction. So, let us assume λ p ( L Ω + a ) < lim n →∞ λ p ( L Ω n + a ) , (2.3)and choose λ ∈ R such that λ p ( L Ω + a ) < λ < lim n →∞ λ p ( L Ω n + a ) . (2.4)We claim 13 laim 2.5. There exists ϕ > ϕ ∈ C (Ω) so that ( λ, ϕ ) is an adequate test function. That is, ϕ satisfies L Ω [ ϕ ]( x ) + ( a ( x ) + λ ) ϕ ( x ) ≤ . Assume for the moment that the above claim holds. By definition of λ p ( L Ω + a ), we get astraightforward contradiction λ p ( L Ω + a ) < λ ≤ λ p ( L Ω + a ) . Hence, lim n →∞ λ p ( L Ω n + a ) = λ p ( L Ω + a )Let us now prove Claim 2.5 Proof of Claim 2.5.
By definition of ν := sup Ω a , there exists a sequence of points ( x k ) k ∈ N such that x k ∈ Ω and | a ( x k ) − ν | < k . By continuity of a , for each k , there exists η k > B η k ( x k ) ⊂ Ω , and sup B ηk ( x k ) | a − ν | ≤ k . Now, let χ k be the following cut-off” functions : χ k ( x ) := χ (cid:16) k x k − x k ε k (cid:17) where ε k > χ is a smooth function such that 0 ≤ χ ≤ χ ( z ) = 0 for | z | ≥ χ ( z ) = 1 for | z | ≤
1. Finally, let us consider the continuous functions a k ( · ), defined by a k ( x ) :=sup { a, ( ν − inf Ω a ) χ k ( x ) + inf Ω a } . By taking a sequence ( ε k ) k ∈ N so that ε k ≤ η k , ε k →
0, we have a k ( x ) = ( a for x ∈ Ω \ B ε k ( x k ) ν for x ∈ Ω ∩ B ε k ( x k )and therefore k a − a k k ∞ ≤ sup B ηk ( x k ) | ν − a | → k → ∞ . By construction, for k large enough, say k ≥ k , we get for all k ≥ k k a − a k k ∞ ≤ inf (cid:26) | λ p ( L Ω + a ) − λ | , | lim n →∞ λ p ( L Ω n + a ) − λ | (cid:27) . Since Ω n → Ω when n → ∞ , there exists n := n ( k ) so that B η k ( x k ) ⊂ Ω n for all n ≥ n . On the othre hand, from the Lipschitz continuity of λ p ( L Ω + a ) with respect to a ((iii) Proposition2.3), inequality (2.4) yields λ p ( L Ω + a k ) < λ < lim n →∞ λ p ( L Ω n + a k ) . (2.5)14ow, by construction we see that for n ≥ n , sup Ω n a k = sup Ω a k = ν and since a k ≡ ν in B εk ( x k ), for all n ≥ n the function ν − a k L loc ( ¯Ω n ). Therefore, by Theorem 2.1, for all n ≥ n there exists ϕ n ∈ C ( ¯Ω n ), ϕ n > λ p ( L Ω n + a k ).Moreover, since x k ∈ T n ≥ n Ω n , for all n ≥ n , we can normalize ϕ n by ϕ n ( x k ) = 1. Recall thatfor all n ≥ n , ϕ n satisfies L Ω n [ ϕ n ]( x ) + ( a k ( x ) + λ p ( L Ω n + a k ( x ))) ϕ n ( x ) = 0 in Ω n , so from (2.5), it follows that ( ϕ n , λ ) satisfies L Ω n [ ϕ n ]( x ) + ( a k ( x ) + λ ) ϕ n ( x ) < L Ω n [ ϕ n ]( x ) + ( a k ( x ) + λ p ( L Ω n + a k )) ϕ n ( x ) = 0 in Ω n . (2.6)Let us now define b n ( x ) := − λ p ( L Ω n + a k ( x )) − a k ( x ), then for all n ≥ n , ϕ n satisfies L Ω n [ ϕ n ]( x ) = b n ( x ) ϕ n ( x ) in Ω n . (2.7)By construction, for n ≥ n , we have b n ( x ) ≥ − λ p ( L Ω n + a k ( x )) − ν >
0. Therefore, since K satisfies the condition (1.9), the Harnack inequality (Theorem 1.4 in [26]) applies to ϕ n . Thus, for n ≥ n fixed and for any compact set ω ⊂⊂ Ω n there exists a constant C n ( ω ) such that ϕ n ( x ) ≤ C n ( ω ) ϕ n ( y ) ∀ x, y ∈ ω. Moreover, the constant C n ( ω ) only depends on δ < d ( ω,∂ Ω)4 , c , S x ∈ ω B δ ( x ) and inf Ω n b n . Fur-thermore, this constant is decreasing with respect to inf Ω n b n . Notice that for all n ≥ n , thefunction b n ( x ) being uniformly bounded from below by a constant independent of n , the constant C n is bounded from above independently of n by a constant C ( ω ). Thus, we have ϕ n ( x ) ≤ C ( ω ) ϕ n ( y ) ∀ x, y ∈ ω. From a standard argumentation, using the normalization ϕ n ( x k ) = 1, we deduce that the se-quence ( ϕ n ) n ≥ n is uniformly bounded in C loc (Ω) topology and is locally uniformly equicontinuous.Therefore, from a standard diagonal extraction argument, there exists a subsequence, still denoted( ϕ n ) n ≥ n , such that ( ϕ n ) n ≥ n converges locally uniformly to a continuous function ϕ which is non-negative, non trivial function and satisfies ϕ ( x k ) = 1.Since K satisfies the condition (1.9), we can pass to the limit in the Equation (2.6) using theLebesgue monotone convergence theorem and we get L Ω [ ϕ ] + ( a k ( x ) + λ ) ϕ ( x ) ≤ . Hence, we have L Ω [ ϕ ]( x ) + ( a ( x ) + λ ) ϕ ( x ) ≤ , since a ≤ a k . 15 Relation between λ p , λ ′ p , λ ′′ p and λ v In this section, we investigate the relations between the quantities λ p , λ ′ p , λ ′′ p and λ v and prove The-orems 1.1 and 1.2.First, remark that, as consequences of the definitions, the monotone and Lipschitz continuityproperties satisfied by λ p (( i ) − ( iii ) of Proposition (2.3)) are still true for λ ′ p and λ v . We investigatenow the relation between λ ′ p and λ p : Lemma 3.1.
Let Ω ⊂ R N be a domain and assume that K and a satisfy (1.7) – (1.9) . Then, λ ′ p ( L Ω + a ) ≤ λ p ( L Ω + a ) . Proof.
Observe that to get inequality λ ′ p ( L Ω + a ) ≤ λ p ( L Ω + a ) , it is sufficient to show that for any δ > λ ′ p ( L Ω + a ) ≤ λ p ( L Ω + a ) + δ. For δ >
0, let us consider the operator L Ω + b δ where b δ := a + λ p ( L Ω + a ) + δ . We claim that Claim 3.2.
For all δ >
0, there exists ϕ δ ∈ C c (Ω) such that ϕ δ ≥ ϕ δ satisfies L Ω [ ϕ δ ]( x ) + b δ ( x ) ϕ δ ( x ) ≥ . By proving the claim, we prove the Lemma. Indeed, assume for the moment that the claim holds.Then, by construction, ( ϕ δ , λ p ( L Ω + a ) + δ ) satisfies L Ω [ ϕ δ ]( x ) + [ a ( x ) + λ p ( L Ω + a ) + δ ] ϕ δ ( x ) ≥ . Thus, by definition of λ ′ p ( L Ω + a ), we have λ ′ p ( L Ω + a ) ≤ λ p ( L Ω + a ) + δ . The constant δ beingarbitrary, we get for all δ > λ ′ p ( L Ω + a ) ≤ λ p ( L Ω + a ) + δ. Proof of the Claim.
Let δ > λ p ( L Ω + b δ ) <
0, so by Lemma 2.4, thereexists a bounded open set ω such that λ p ( L ω + b δ ) <
0. For any ε > ω larger if necessary, arguing as in the proof of Claim 2.5, we can find b ε such that k b δ − b ε k ∞ ,ω = k b δ − b ε k ∞ , Ω ≤ ε,λ p ( L ω + b ε ( x )) + ε < , and there is ϕ p ∈ C (¯ ω ), ϕ p > λ p ( L ω + b ε ( x )). That is ϕ p satisfies L ω [ ϕ p ]( x ) + b ε ( x ) ϕ p ( x ) = − λ p ( L ω + b ε ( x )) ϕ p ( x ) in ω. (3.1)Without loss of generality, assume that ϕ p ≤ ν denotes the maximum of b ε in ¯ ω , then by Proposition 2.3, there exists τ > − λ p ( L ω + b ε ( x )) − ε − ν ≥ τ > . ϕ p satisfies (3.1), there exists d > ω ϕ p ≥ d .Let us choose ω ′ ⊂⊂ ω such that | ω \ ω ′ | ≤ d inf { τ, − λ p ( L ω + b ε ) − ε } k K k ∞ , where for a set A, | A | denotes the Lebesgue measure of A .Since ¯ ω ′ ⊂⊂ ω and ∂ω are two disjoint closed sets, by the Urysohn’s Lemma there exists acontinuous function η such that 0 ≤ η ≤ η = 1 in ω ′ , η = 0 in ∂ω . Consider now ϕ p η and let uscompute L ω [ ϕ p η ] + b δ ϕ p η. Then, we have L ω [ ϕ p η ] + b δ ϕ p η ≥ − λ p ( L ω + b ε ) ϕ p − k K k| ω \ ω ′ | − b ε ϕ p (1 − η ) − ( b ε − b δ ) ϕ p η, ≥ − ( λ p ( L ω + b ε ) + k b δ − b ε k ∞ ,ω ) ϕ p − d inf { τ, − λ p ( L ω + b ε ) − ε } − b ε ( x ) ϕ p (1 − η ) , ≥ − ( λ p ( L ω + b ε ) + ε ) ϕ p − d inf { τ, − λ p ( L ω + b ε ) − ε } − max { ν, } ϕ p , ≥ − ( λ p ( L ω + b ε ) + ε + max { ν, } ) ϕ p − d inf { τ, − λ p ( L ω + b ε ) − ε } . Since − λ p ( L ω + b ε ) − ε > − λ p ( L ω + b ε ) − ε − ν ≥ τ >
0, from the above inequality, we inferthat L ω [ ϕ p η ] + b δ ϕ p η ≥ − ( λ p ( L ω + b ε ) + ε + max { ν, } ) d − d inf { τ, − λ p ( L ω + b ε ) − ε } , ≥ d inf { τ, − λ p ( L ω + b ε ) − ε } ≥ . By construction, we have ϕ p η ∈ C ( ω ) satisfying L ω [ ϕ p η ] + b δ ϕ p η ≥ ω,ϕ p η = 0 on ∂ω. By extending ϕ p η by 0 outside ω and denoting ϕ δ this extension, we get L Ω [ ϕ δ ]( x ) + b δ ( x ) ϕ δ ( x ) = L ω [ ϕ δ ]( x ) + b δ ( x ) ϕ ( x ) ≥ ω, L Ω [ ϕ δ ]( x ) + b δ ( x ) ϕ δ ( x ) = L ω [ ϕ δ ]( x ) ≥ \ ω. Hence, ϕ δ ≥ , ϕ ∈ C c (Ω) is the desired test function. Remark 2.
The assumption (1.9) on K is only needed to reduce the problem on unbounded domainsto problem on bounded domains. In addition, the above construction shows that the inequality isstill valid if we replace λ ′ p by λ ′′ p ( L Ω + a ). Thus we have for any domain Ω, λ ′′ p ( L Ω + a ) ≤ λ p ( L Ω + a ) . .1 The bounded case: Assume for the moment that Ω is a bounded domain and let us show that the three definitions λ p , λ ′ p and λ ′′ p are equivalent and if in addition K is symmetric, λ v is equivalent to λ p . We start by the case λ ′ p = λ p . Namely, we show Lemma 3.3.
Let Ω be a bounded domain of R N and assume that a and K satisfy (1.7) – (1.9) . Then, λ p ( L Ω + a ) = λ ′ p ( L Ω + a ) . In addition, when L Ω + a is self adjoined, we have λ p ( L Ω + a ) = λ v ( L Ω + a ) . The proof of Theorem 1.1 is a straightforward consequence of the above Lemma. Indeed, byRemark 2 and the definition of λ ′′ p we have λ ′ p ( L Ω + a ) ≤ λ ′′ p ( L Ω + a ) ≤ λ p ( L Ω + a ) . Thus, from the above Lemma we get λ p ( L Ω + a ) = λ ′ p ( L Ω + a ) ≤ λ ′′ p ( L Ω + a ) ≤ λ p ( L Ω + a ) = λ ′ p ( L Ω + a ) . (cid:3) Let us now turn to the proof of Lemma 3.3
Proof of Lemma 3.3.
By Lemma 3.1, we already have λ ′ p ( L Ω + a ) ≤ λ p ( L Ω + a ) . So, it remains to prove the converse inequality. Let us assume by contradiction that λ ′ p ( L Ω + a ) < λ p ( L Ω + a ) . Pick now λ ∈ ( λ ′ p ( L Ω + a ) , λ p ( L Ω + a )), then, by definition of λ p and λ ′ p , there exists ϕ and ψ nonnegative continuous functions such that L Ω [ ϕ ]( x ) + ( a ( x ) + λ ) ϕ ( x ) ≤ , L Ω [ ψ ]( x ) + ( a ( x ) + λ ) ψ ( x ) ≥ . Moreover, ϕ > λ smaller if necessary, we can assume that ϕ satisfies L Ω [ ϕ ]( x ) + ( a ( x ) + λ ) ϕ ( x ) < . A direct computation yields Z Ω K ( x, y ) ϕ ( y ) (cid:18) ψ ( y ) ϕ ( y ) − ψ ( x ) ϕ ( x ) (cid:19) dy > . ψϕ ∈ C ( ¯Ω), the function ψϕ achieves a maximum at some point x ∈ ¯Ω, evidencing thus thecontradiction: 0 < Z Ω K ( x , y ) ϕ ( y ) (cid:18) ψ ( y ) ϕ ( y ) − ψ ( x ) ϕ ( x ) (cid:19) dy ≤ . Thus, λ ′ p ( L Ω + a ) = λ p ( L Ω + a ) . In the self-adjoined case, it is enough to prove that λ ′ p ( L Ω + a ) = λ v ( L Ω + a ) . From the definitions of λ ′ p and λ v , we easily obtain that λ v ≤ λ ′ p . Indeed, let λ > λ ′ p ( L Ω + a ), thenby definition of λ ′ p there exists ψ ≥ ψ ∈ C (Ω) ∩ L ∞ (Ω) and L Ω [ ψ ]( x ) + ( a ( x ) + λ ) ψ ( x ) ≥ . (3.2)Since Ω is bounded and ψ ∈ L ∞ (Ω), ψ ∈ L (Ω). So, multiplying (3.2) by − ψ and integrating over Ωwe get − Z Ω Z Ω K ( x, y ) ψ ( x ) ψ ( y ) dxdy − Z Ω a ( x ) ψ ( x ) dx ≤ λ Z Ω ψ ( x ) dx, Z Ω Z Ω K ( x, y ) ( ψ ( x ) − ψ ( y )) dxdy − Z Ω ( a ( x ) + k ( x )) ψ ( x ) dx ≤ λ Z Ω ψ ( x ) dx,λ v ( L Ω + a ) Z Ω ψ ( x ) dx ≤ λ Z Ω ψ ( x ) dx. Therefore, λ v ( L Ω + a ) ≤ λ ′ p ( L Ω + a ).Let us prove now the converse inequality. Again, we argue by contradiction and let us assume that λ v ( L Ω + a ) < λ ′ p ( L Ω + a ) . (3.3)Observe first that by density of C ( ¯Ω) in L (Ω), we easily check that − λ v ( L Ω + a ) = − inf ϕ ∈ L (Ω) ,ϕ R Ω R Ω K ( x, y )( ϕ ( x ) − ϕ ( y )) dydx − R Ω ( a ( x ) + k ( x )) ϕ ( x ) dx k ϕ k L (Ω) , = − inf ϕ ∈ L (Ω) ,ϕ − R Ω R Ω K ( x, y ) ϕ ( x ) ϕ ( y ) dydx − R Ω a ( x ) ϕ ( x ) dx k ϕ k L (Ω) , = sup ϕ ∈ L (Ω) ,ϕ hL Ω [ ϕ ] + aϕ, ϕ ik ϕ k L (Ω) , = sup ϕ ∈ C (¯Ω) ,ϕ hL Ω [ ϕ ] + aϕ, ϕ ik ϕ k L (Ω) . By (iv) of Proposition 2.3, since λ ′ p ( L Ω + a ) = λ p ( L Ω + a ), from (3.3) we infer that λ + defined by λ + = sup ϕ ∈ C (Ω) hL Ω [ ϕ ] + aϕ, ϕ i R Ω ϕ (3.4)19atisfies λ + > − λ p ( L Ω + a ) ≥ max Ω a. (3.5)Now, using the same arguments as in [29, 39], we infer that the supremum in (3.4) is achieved.Indeed, it is a standard fact [15] that the spectrum of L Ω + a is at the left of λ + and that thereexists a sequence ϕ n ∈ C (Ω) such that k ϕ n k L (Ω) = 1 and k ( L Ω + a − λ + ) ϕ n k L (Ω) → n → + ∞ .By compactness of L Ω : L (Ω) → C (Ω), for a subsequence, lim n → + ∞ L Ω [ ϕ n ] exists in C (Ω). Then,using (3.5), we see that ϕ n → ϕ in L (Ω) for some ϕ and ( L Ω + a ) ϕ = λ + ϕ . This equation implies ϕ ∈ C (Ω), and λ + is an eigenvalue for the operator L Ω + a . Moreover, ϕ ≥
0, since ϕ + is also aminimizer. Indeed, we have λ + = R Ω [ L Ω [ ϕ ]( x ) + a ( x ) ϕ ( x )] ϕ + ( x ) dx k ϕ + k L (Ω) , = R Ω [ L Ω [ ϕ + ]( x ) + a ( x ) ϕ + ( x )] ϕ + ( x ) dx k ϕ + k L (Ω) + R Ω R Ω K ( x, y ) ϕ − ( x ) ϕ + ( y ) dydx k ϕ + k L (Ω) , ≤ R Ω [ L Ω [ ϕ + ] + aϕ + ( x )] ϕ + ( x ) dx k ϕ + k L (Ω) ≤ λ + . Thus, there exists a non-negative continuous ϕ so that L Ω [ ϕ ]( x ) + ( a ( x ) + λ v ) ϕ ( x ) = 0 in Ω . Since λ v < λ p , we can argue as above and get the desired contradiction. Hence, λ v = λ + = λ p = λ ′ p . Now let Ω be an unbounded domain. From Lemma 3.1 and Remark 2, we already know that λ ′ p ( L Ω + a ) ≤ λ ′′ p ( L Ω + a ) ≤ λ p ( L Ω + a ) . To complete the proof of Theorem 1.2, we are then left to prove that λ ′ p ( L Ω + a ) = λ ′′ p ( L Ω + a ) = λ p ( L Ω + a ) = λ v ( L Ω + a ) , when L Ω + a is self-adjoined and the kernel K is such that p ( x ) := R Ω K ( x, y ) dy is a bounded functionin Ω. To do so, we prove the following inequality : Lemma 3.4.
Let Ω be an unbounded domain and assume that a and K satisfies (1.7) – (1.9) . Assumefurther that K is symmetric and p ( x ) := R Ω K ( x, y ) dy ∈ L ∞ (Ω) . Then, we have λ p ( L Ω + a ) ≤ lim inf n → + ∞ λ v ( L Ω n + a ) ≤ λ ′ p ( L Ω + a ) , where Ω n := (Ω ∩ B n ) n ∈ N and B n is the ball of radius n centred at . Proof of Theorem 1.2 :
From Lemma 3.1 and 3.4, we get the inequalities:lim n →∞ λ v ( L Ω n + a ) ≤ λ ′ p ( L Ω + a ) ≤ λ ′′ p ( L Ω + a ) ≤ λ p ( L Ω + a ) ,λ p ( L Ω + a ) ≤ lim n →∞ λ v ( L Ω n + a ) ≤ λ ′ p ( L Ω + a ) ≤ λ ′′ p ( L Ω + a ) , with Ω n := Ω ∩ B n (0). Therefore,lim n →∞ λ v ( L Ω n + a ) = λ ′ p ( L Ω + a ) = λ ′′ p ( L Ω + a ) = λ ′ p ( L Ω + a ) = λ p ( L Ω + a ) . It remains to prove that λ v ( L Ω + a ) = λ p ( L Ω + a ).By definition of λ ′′ p ( L Ω + a ), we check that λ v ( L Ω + a ) ≤ λ ′′ p ( L Ω + a ) = λ p ( L Ω + a ) . On the other hand, by definition of λ v ( L Ω + a ), for any δ > ϕ δ ∈ L (Ω) such that RR Ω × Ω K ( x, y )( ϕ δ ( x ) − ϕ δ ( y )) dydx − R Ω ( a ( x ) + p ( x )) ϕ δ ( x ) dx k ϕ δ k L (Ω) ≤ λ v ( L Ω + a ) + δ. Define I R ( ϕ δ ) := RR Ω R × Ω R K ( x, y )( ϕ δ ( x ) − ϕ δ ( y )) dydx − R Ω R ( a ( x ) + p R ( x )) ϕ δ ( x ) dx k ϕ δ k L (Ω R ) , with p R ( x ) := R Ω R K ( x, y ) dy . Since lim R →∞ p R ( x ) = p ( x ) for all x ∈ Ω, a ∈ L ∞ and ϕ δ ∈ L (Ω), byLebesgue’s monotone convergence Theorem we get for R large enough − Z Ω R ( a ( x ) + p R ( x )) ϕ δ ( x ) dx ≤ δ k ϕ δ k L (Ω R ) − Z Ω ( a ( x ) + p ( x )) ϕ δ ( x ) dx. Thus, we have for R large enough I R ( ϕ δ ) ≤ RR Ω × Ω K ( x, y )( ϕ δ ( x ) − ϕ δ ( y )) dydx − R Ω ( a ( x ) + p ( x )) ϕ δ ( x ) dx k ϕ δ k L (Ω) , ≤ k ϕ δ k L (Ω) k ϕ δ k L (Ω R ) ( λ v ( L Ω + a ) + δ ) + δ, ≤ λ v ( L Ω + a ) + Cδ, for some universal constant
C >
0. 21y definition of λ v ( L Ω R + a ), we then get λ v ( L Ω R + a ) ≤ I R ( ϕ δ ) ≤ λ v ( L Ω + a ) + Cδ for R large enough . Therefore, lim R →∞ λ v ( L Ω R + a ) ≤ λ v ( L Ω + a ) + Cδ. (3.6)Since (3.6) holds true for any δ , we getlim R →∞ λ v ( L Ω R + a ) ≤ λ v ( L Ω + a ) . As a consequence, we obtain λ p ( L Ω + a ) = lim n →∞ λ v ( L Ω n + a ) ≤ λ v ( L Ω + a ) ≤ λ ′′ p ( L Ω + a ) = λ p ( L Ω + a ) , which enforces λ v ( L Ω + a ) = λ p ( L Ω + a ) . We can now turn to the proof of Lemma 3.4. But before proving this Lemma, we start by showingsome technical Lemma in the spirit of Lemma 2.6 in [9]. Namely, we prove
Lemma 3.5.
Assume Ω is unbounded and let g ∈ L ∞ (Ω) be a non negative function, then for any R > , we have lim R →∞ R Ω ∩ ( B R R \ B R ) g R Ω ∩ B R g = 0 . Proof.
Without loss of generality, by extending g by 0 outside Ω we can assume that Ω = R N . Forany R , R > C R ,R := B R + R \ B R . Assume by contradiction thatlim R →∞ R C R ,R g R B R g > . Then there exists ε > R ε > ∀ R ≥ R ε , R C R ,R g R B R g ≥ ε. Consider the sequence ( R n ) n ∈ N defined by R n := R ε + nR and set a n := R C R ,Rn g . For all n , wehave C R ,R n = B R n +1 \ B R n and B R n +1 = B R ε ∪ n [ k =0 C R ,R k ! . n ≥ a n ≥ ε Z B Rn g ≥ ε n − X k =0 a k . Arguing now as in [9],by a recursive argument, the last inequality yields ∀ n ≥ , a n ≥ εa (1 + ε ) n − . (3.7)On the other hand, we have a n = Z C R ,Rn g ≤ k g k ∞ | C R ,R n | ≤ d n N , with d a positive constant, contradicting thus (3.7).We are now in a position to prove Lemma 3.4. Proof of Lemma 3.4 :
The proof follows some ideas developed in [11, 9, 29, 31]. To simplify thepresentation, let us call λ p = λ p ( L Ω + a ) and λ ′ p = λ ′ p ( L Ω + a ).First recall that for a bounded domain Ω, we have λ p = λ ′ p = λ v . Let ( B n ) n ∈ N be the increasing sequence of balls of radius n centred at 0 and let Ω n := Ω ∩ B n . Bymonotonicity of λ p with respect to the domain, we have λ p ( L Ω + a ) ≤ λ p ( L Ω n + a ) = λ v ( L Ω n + a )Therefore λ p ( L Ω + a ) ≤ lim inf n →∞ λ v ( L Ω n + a ) . Thanks to the last inequality, we obtain the inequality λ p ( L Ω + a ) ≤ λ ′ p ( L Ω + a ) by proving thatlim inf n →∞ λ v ( L Ω n + a ) ≤ λ ′ p ( L Ω + a ) . (3.8)To prove (3.8), it is enough to show that for any δ > n →∞ λ v ( L Ω n + a ) ≤ λ ′ p ( L Ω + a ) + δ. (3.9)Let us fix δ > µ := λ ′ p ( L Ω + a ) + δ . By definition of λ ′ p ( L Ω + a ) there exists afunction ϕ ∈ C (Ω) ∩ L ∞ (Ω), ϕ ≥ L Ω [ ϕ ]( x ) + a ( x ) ϕ ( x ) + µϕ ( x ) ≥ . (3.10)Without loss of generality, we can also assume that k ϕ k L ∞ (Ω) = 1.23et Ω n be the characteristic function of Ω n = Ω ∩ B n and let w n = ϕ Ω n . By definition of λ v ( L Ω n + a ) and since w n ∈ L (Ω n ), we have λ v ( L Ω n + a ) k w n k L (Ω n ) ≤ Z Ω n (cid:0) −L Ω n [ w n ]( x ) − a ( x ) w n ( x ) (cid:1) w n ( x ) dx. (3.11)Since L Ω [ ϕ ] w n ∈ L (Ω n ), from (3.11) and by using (3.10) we get λ v ( L Ω n + a ) k w n k L (Ω n ) ≤ Z Ω n (cid:0) −L Ω n [ w n ]( x ) − a ( x ) w n ( x ) − µw n + µw n (cid:1) w n ( x ) dx, ≤ µ k w n k L (Ω n ) + Z Ω n (cid:0) −L Ω n [ w n ]( x ) + L Ω [ ϕ ]( x ) (cid:1) w n ( x ) dx, ≤ µ k w n k L (Ω n ) + Z Ω n Z Ω \ Ω n K ( x, y ) ϕ ( y ) dy ! w n ( x ) dx, ≤ µ k w n k L (Ω n ) + I n , where I n denotes I n := Z Ω n Z Ω \ Ω n K ( x, y ) ϕ ( y ) dy ! ϕ ( x ) dx. Observe that we achieve (3.9) by provinglim inf n →∞ I n k ϕ k L (Ω n ) = 0 . (3.12)Recall that K satisfies (1.9), therefore there exists C > R > K ( x, y ) ≤ C R ( | x − y | )). So, we get I n ≤ Z Ω n Z Ω ∩ ( B R n \ B n ) K ( x, y ) ϕ ( y ) dy ! w n ( x ) dx. (3.13)By Fubini’s Theorem, Jensen’s inequality and Cauchy-Schwarz’s inequality, it follows that I n ≤ Z Ω ∩ ( B R n \ B n ) ϕ ( y ) dy ! / Z Ω ∩ ( B R n \ B n ) (cid:18)Z Ω n K ( x, y ) ϕ ( x ) dx (cid:19) dy ! / , ≤ k ϕ k L (Ω ∩ ( B R n \ B n )) Z Ω ∩ ( B R n \ B n ) (cid:18)Z Ω ∩ B n K ( x, y ) ϕ ( x ) dx (cid:19) dy ! / , ≤ k ϕ k L (Ω ∩ ( B R n \ B n )) Z Ω ∩ B n Z Ω ∩ B R n \ B n ) K ( x, y ) dy ! ϕ ( x ) dx ! / . Since K and p are bounded functions, we obtain I n ≤ k K k ∞ k p k ∞ k ϕ k L (Ω ∩ ( B R n \ B n ) k ϕ k L (Ω ∩ B n ) . (3.14)24ividing (3.14) by k ϕ k L (Ω n ) , we then get I n k ϕ k L (Ω n ) ≤ C k ϕ k L (Ω ∩ ( B R n \ B n )) k ϕ k L (Ω ∩ B n ) . Thanks to Lemma 3.5, the right hand side of the above inequality tends to 0 as n → ∞ . Hence, weget lim inf n →∞ λ v ( L Ω n + a ) ≤ µ + lim inf n →∞ I n k ϕ n k L (Ω n ) = λ ′ p ( L Ω + a ) + δ. (3.15)Since the above arguments holds true for any arbitrary δ >
0, the Lemma is proved.
Remark 3.
In the above proof, since w = ϕ Ω n ∈ L (Ω) and L Ω [ w n ] = L Ω n [ w n ], the inequality(3.11) is true with λ v ( L Ω + a ) instead of λ v ( L Ω n + a ). Thus, we get immediately λ v ( L Ω + a ) ≤ λ ′ p ( L Ω + a ) . When N = 1, the decay restriction imposed on the kernel can be weakened, see [31]. In particular,we have Lemma 3.6.
Let Ω be an unbounded domain and assume that a and K satisfy (1.7) – (1.8) . Assumefurther that K is symmetric and K satisfies ≤ K ( x, y ) ≤ C (1 + | x − y | ) − α for some α > . Thenone has λ p ( L Ω + a ) ≤ lim inf n → + ∞ λ v ( L Ω n + a ) ≤ λ ′ p ( L Ω + a ) , where Ω n := Ω ∩ ( − n, n ) .Proof. By arguing as in the above proof, for any δ > ϕ ∈ C (Ω) ∩ L ∞ (Ω) such that L Ω [ ϕ ] + ( a + λ p ( L Ω + a ) + δ ) ϕ ( x ) ≥ . and λ v ( L Ω n + a ) k w n k L (Ω n ) ≤ µ k w n k L (Ω n ) + I n , where µ := λ p ( L Ω + a ) + δ ) , w n := ϕ ( − n,n ) and I n denotes I n := Z Ω n Z Ω \ Ω n K ( x, y ) ϕ ( y ) dy ! ϕ ( x ) dx. (3.16)As above, we end our proof by showinglim inf n →∞ I n k ϕ k L (Ω n ) = 0 . (3.17)Let us now treat two cases independently: 25 ase 1: ϕ ∈ L (Ω)In this situation, again by using Cauchy-Schwarz’s inequality, Jensen’s inequality and Fubini’s The-orem, the inequality (3.16) yields I n ≤ k ϕ k L (Ω n ) "Z Ω \ Ω n (cid:18)Z Ω n K ( x, y ) dx (cid:19) ϕ ( y ) dy . Recall that K satisfies K ( x, y ) ≤ C (1 + | x − y | ) − α for some C > α > /
2, therefore p ( y ) := R Ω K ( x, y ) dx is bounded and from the latter inequality we enforce I n ≤ C k ϕ k L (Ω \ Ω n ) k ϕ k L (Ω n ) . Thus, lim inf n →∞ I n k ϕ k L (Ω n ) ≤ lim inf n →∞ k ϕ k L (Ω \ Ω n ) k ϕ k L (Ω n ) = 0 . Case 2: ϕ L (Ω)Assume now that ϕ L (Ω), then we argue as follows. Again, applying Fubini’s Theorem andCauchy-Schwarz’s inequality in the inequality (3.16) yields I n ≤ k ϕ k L (Ω n ) "Z (Ω ∩ R − ) \ Ω n (cid:18)Z Ω n K ( x, y ) dx (cid:19) ! ϕ ( y ) dy + Z (Ω ∩ R + ) \ Ω n (cid:18)Z Ω n K ( x, y ) dx (cid:19) ! ϕ ( y ) dy , ≤ k ϕ k L (Ω n ) h ˜ I − n + ˜ I + n i . (3.18)Recall that by assumption there exists C > K ( x, y ) ≤ C (1 + | x − y | ) − α with α > .So, we have ˜ I − n ≤ C Z (Ω ∩ R − ) \ Ω n (cid:18)Z Ω n (1 + | x − y | ) − α dx (cid:19) ! ϕ ( y ) dy, ˜ I + n ≤ C Z (Ω ∩ R + ) \ Ω n (cid:18)Z n − n (1 + | x − y | ) − α dx (cid:19) ! ϕ ( y ) dy. To complete our proof, we have to show that ˜ I ± n k ϕ k L n ) →
0. The proof being similar in both cases,so we only prove that ˜ I + n k ϕ k L n ) →
0. We claim that
Claim 3.7.
There exists
C > n ∈ N , Z (Ω ∩ R + ) \ Ω n (cid:18)Z Ω n (1 + | x − y | ) − α dx (cid:19) ! ϕ ( y ) dy ≤ C. I n k ϕ k L (Ω n ) ≤ C k ϕ k L (Ω n ) → n → ∞ . Hence, in both situation, we getlim inf n →∞ λ v ( L Ω n + a ) ≤ µ + lim inf n →∞ I n k ϕ n k L (Ω n ) = λ ′ p ( L Ω + a ) + δ Since δ > δ >
Proof of the Claim.
Since ϕ ∈ L ∞ (Ω) and y ≥ n then x ≤ y and we have˜ I + n ≤ k ϕ k ∞ Z Ω ∩ R + \ Ω n (cid:18)Z Ω n (1 + y − x ) − α dx (cid:19) ! dy, ≤ k ϕ k ∞ Z + ∞ n (cid:18)Z n − n (1 + y − x ) − α dx (cid:19) ! dy, ≤ k ϕ k ∞ √ α − Z + ∞ n (1 + y − n ) − α + dy, ≤ C Z + ∞ (1 + z ) − α + dz. In this section, we investigate further the properties of the principal eigenvalue λ p ( L Ω + a ) and inparticular its behaviour with respect to some scaling of the kernel K ((Proposition 1.3) and Theorem1.4). For simplicity, we split this section into two subsections, one dedicated to the the proof ofProposition 1.3 and the other one dealing with the proof of Theorem 1.4. Let us start with thescaling invariance of L Ω + a , (Proposition 1.3) This invariance is a consequence of the following observation. By definition of λ p ( L Ω + a ), we havefor all λ < λ p ( L Ω + a ), L Ω [ ϕ ]( x ) + ( a ( x ) + λ ) ϕ ( x ) ≤ , ϕ ∈ C (Ω). Let X = σx , Ω σ := σ Ω and ψ ( X ) := ϕ ( σX ) then we can rewrite theabove inequality as follows Z Ω K (cid:18) Xσ , y (cid:19) ϕ ( y ) dy + ( a (cid:18) Xσ (cid:19) + λ ) ϕ (cid:18) Xσ (cid:19) ≤ X ∈ Ω σ , Z Ω K (cid:18) Xσ , y (cid:19) ϕ ( y ) dy + ( a σ ( X ) + λ ) ψ ( X ) ≤ X ∈ Ω σ , Z Ω σ K σ ( X, Y ) ψ ( Y ) dY + ( a σ ( X ) + λ ) ψ ( X ) ≤ X ∈ Ω σ , with K σ ( x, y ) := σ N K ( xσ , yσ ) and a σ ( x ) := a (cid:0) xσ (cid:1) . Thus ψ is a positive continuous function thatsatisfies L σ, Ω σ [ ψ ]( x ) + ( a σ ( x ) + λ ) ψ ( x ) ≤ σ . Therefore, λ ≤ λ p ( L σ, Ω σ + a σ ) and as a consequence λ p ( L Ω + a ) ≤ λ p ( L σ, Ω σ + a σ ) . Interchanging the role of λ p ( L Ω + a ) and λ p ( L σ, Ω σ + a σ ) in the above argument yields λ p ( L Ω + a ) ≥ λ p ( L σ, Ω σ + a σ ) . Hence, we get λ p ( L Ω + a ) = λ p ( L σ, Ω σ + a σ ) . (cid:3) λ p (cid:18) L σ,m, Ω − σ m + a (cid:19) Let us focus on the behaviour of the principal eigenvalue of the spectral problem M σ,m, Ω [ ϕ ] + ( a + λ ) ϕ = 0 in Ω , where M σ,m, Ω [ ϕ ] := 1 σ m (cid:18)Z Ω J σ ( x − y ) ϕ ( y ) dy − ϕ ( x ) (cid:19) , with J σ ( z ) := σ N J (cid:0) zσ (cid:1) . Assuming that 0 ≤ m ≤
2, we obtain here the limits of λ p ( M σ,m + a ) when σ → σ → ∞ . But before going to the study of these limits, we recall a known inequality. Lemma 4.1.
Let J ∈ C ( R N ) , J ≥ , J symmetric with unit mass, such that | z | J ( z ) ∈ L ( R N ) .Then for all ϕ ∈ H (Ω) we have − Z Ω (cid:18)Z Ω J ( x − y ) ϕ ( y ) dy − ϕ ( x ) (cid:19) ϕ ( x ) dx ≤ Z R N J ( z ) | z | dz k∇ ϕ k L (Ω) . roof. Let ϕ ∈ C ∞ c , then by applying the standard Taylor expansion we have ϕ ( x + z ) − ϕ ( x ) = Z z i ∂ i ϕ ( x + tz ) dt (4.1)= z i ∂ i ϕ ( x ) + Z t (cid:18)Z z i z j ∂ ij ϕ ( x + tsz ) ds (cid:19) dt (4.2)where use the Einstein summation convention a i b i = P Ni =1 a i b i .Let us denote I ( ϕ ) := − Z Ω (cid:18)Z Ω J ( x − y ) ϕ ( y ) dy − ϕ ( x ) (cid:19) ϕ ( x ) dx. Then, for any ϕ ∈ C c (Ω), ϕ ∈ C c ( R N ) and we can easily see that I ( ϕ ) = 12 Z Z R N J ( x − y )( ϕ ( x ) − ϕ ( y )) dxdy. By plugging the Taylor expansion of ϕ (4.1) in the above equality we see that12 Z Z R N J ( z )( ϕ ( x + z ) − ϕ ( x )) dzdx = 12 Z R N Z R N J ( z ) (cid:18)Z z i ∂ i ϕ ( x + tz ) dt (cid:19) dzdx, ≤ Z Z R N J ( z ) | z i | (cid:20)Z | ∂ i ϕ ( x + tz ) | dt (cid:21) ! dzdx, ≤ Z Z R N J ( z ) N X i =1 z i ! " N X i =1 Z | ∂ i ϕ ( x + tz ) | dt dzdx where we use in the last inequality the standard inequality ( P i a i b i ) ≤ ( P Ni =1 a i )( P Ni =1 b i ).So, by Fubini’s Theorem and by rearranging the terms in the above inequality, it follows that I ( ϕ ) ≤ (cid:18)Z R N J ( z ) | z | dz (cid:19) k∇ ϕ k L (Ω) . By density of C ∞ c (Ω) in H (Ω), the above inequality holds true for ϕ ∈ H (Ω), since obviously thefunctional I ( ϕ ) is continuous in L (Ω). 29et us also introduce the following notation J σ ( z ) := 1 σ N J (cid:16) zσ (cid:17) , p σ ( x ) := Z Ω J σ ( x − y ) dy, D ( J ) := Z R N J ( z ) | z | dz, A ( ϕ ) := R Ω aϕ ( x ) dx k ϕ k L (Ω) , R σ,m ( ϕ ) := 1 σ m R Ω ( p σ ( x ) − ϕ ( x ) dx k ϕ k L (Ω) , I σ,m ( ϕ ) := σ m (cid:0) − R Ω (cid:0)R Ω J ( x − y ) ϕ ( y ) dy − ϕ ( x ) (cid:1) ϕ ( x ) dx (cid:1) k ϕ k L (Ω) − A ( ϕ ) J ( ϕ ) := D ( J )2 R Ω |∇ ϕ | ( x ) dx k ϕ k L (Ω) . With this notation, we see that λ v ( M σ,m, Ω + a ) = inf ϕ ∈ L (Ω) I σ,m ( ϕ ) , and by Lemma 4.1, for any ϕ ∈ H (Ω) we get I σ,m ( ϕ ) ≤ σ − m J ( ϕ ) − A ( ϕ ) . (4.3)We are now in position to obtain the different limits of λ p ( M σ,m, Ω + a ) as σ → σ → ∞ .For simplicity, we analyse three distinct situations: m = 0 , < m < m = 2. We will see that m = 0 and m = 2 are ,indeed, two critical situations.Let us first deal with the easiest case, that is, when 0 < m < < m < : In this situation, we claim that
Claim 4.2.
Let Ω be any domain and let J ∈ C ( R N ) be positive, symmetric and such that | z | J ( z ) ∈ L ( R N ). Assume further that J satisfies (1.7)–(1.9) and 0 < m < σ → λ p ( M σ,m, Ω + a ) = − sup Ω a lim σ → + ∞ λ p ( M σ,m, Ω + a ) = − sup Ω a Proof.
First, let us look at the limit of λ p when σ →
0. Up to adding a large positive constant to thefunction a , without any loss of generality, we can assume that the function a is positive somewherein Ω.Since M σ,m, Ω + a is a self-adjoined operator, by Theorem 1.2 and (4.3), for any ϕ ∈ H (Ω) wehave λ p ( M σ,m, Ω + a ) = λ v ( M σ,m, Ω + a ) ≤ I σ,m ( ϕ ) ≤ σ − m J ( ϕ ) − A ( ϕ ) . Define ν := sup Ω a , and let ( x n ) n ∈ N be a sequence of point such that | ν − a ( x n ) | < n . Since a ispositive somewhere, we can also assume that for all n , x n ∈ Γ := { x ∈ Ω | a ( x ) > } .30y construction, for any n >
0, there exists ρ n such that B ρ ( x n ) ⊂ Γ for any positive ρ ≤ ρ n .Fix now n , for any 0 < ρ ≤ ρ n there exists ϕ ρ ∈ H (Ω) such that supp ( ϕ ρ ) ⊂ B ρ ( x n ) and therefore,lim sup σ → λ p ( M σ,m, Ω + a ) ≤ −A ( ϕ ρ ) = − R B ρ ( x n ) aϕ ρ ( x ) dx k ϕ ρ k L (Ω) ≤ − min B ρ ( x n ) a + ( x ) . By taking the limit ρ → σ → λ p ( M σ,m, Ω + a ) ≤ − a ( x n ) . Thus, lim sup σ → λ p ( M σ,m, Ω + a ) ≤ − ν + 1 n . By sending now n → ∞ in the above inequality, we obtainlim sup σ → λ p ( M σ,m, Ω + a ) ≤ − ν. On the other hand, by using the test function ( ϕ, λ ) = (1 , − ν ) we can easily check that for any σ > λ p ( M σ,m, Ω + a ) ≥ − ν. Hence, − ν ≤ lim inf σ → λ p ( M σ,m, Ω + a ) ≤ lim sup σ → λ p ( M σ,m, Ω + a ) ≤ − ν. Now, let us look at the limit of λ p ( M σ,m, Ω + a ) when σ → + ∞ . This limit is a straightforwardconsequence of (iv) of the Proposition 2.3. Indeed, as remarked above, for any σ by using the testfunction ( ϕ, λ ) = (1 , − ν ), we have − ν ≤ λ p ( M σ,m, Ω + a )whereas from (iv) of the Proposition 2.3 we have λ p ( M σ,m, Ω + a ) ≤ − sup Ω (cid:18) − σ m + a (cid:19) . Therefore, since m > − ν ≤ lim σ → + ∞ λ p ( M σ,m, Ω + a ) ≤ − ν. Remark 4. . From the proof, we obtain also some of the limits in the cases m = 0 and m = 2.Indeed, the analysis of the limit of λ p ( M σ,m, Ω + a ) when σ → m <
2. Thus, λ p ( M σ, , Ω + a ) → − sup Ω a as σ → . On the other hand, the analysis of the limit of λ p ( M σ,m, Ω + a ) when σ → + ∞ holds true as soon as m >
0. Therefore, λ p ( M σ, , Ω + a ) → − sup Ω a as σ → + ∞ . .2.2 The case m = 0In this situation, one of the above argument fails and one of the expected limits is not − ν any more.Indeed, we have Lemma 4.3.
Let Ω be any domain and let J ∈ C ( R N ) be positive, symmetric and such that | z | J ( z ) ∈ L ( R N ) . Assume further that J satisfies (1.7) – (1.9) and m = 0 then lim σ → λ p ( M σ, , Ω + a ) = − sup Ω a lim σ → + ∞ λ p ( M σ, , Ω + a ) = 1 − sup Ω a Proof.
As already noticed in Remark 4, the limit of λ p ( M σ, , Ω + a ) when σ → < m < λ p ( M σ, , Ω + a ) when σ → ∞ .As above, up to adding a large positive constant to a , without any loss of generality, we canassume that a is positive somewhere in Ω and we denote ν := sup Ω a >
0. By using constant testfunctions and (iv) of the Proposition 2.3, we observe that − ν ≤ λ p ( M σ, , Ω + a ) ≤ − ν, for all σ > . So, we have lim sup σ →∞ λ p ( M σ, , Ω + a ) ≤ − ν. On the other hand, for any ϕ ∈ C c (Ω) we have for all σ , I σ, ( ϕ ) Z Ω ϕ ( x ) = − Z Ω (cid:18)Z Ω J σ ( x − y ) ϕ ( y ) dy − ϕ ( x ) (cid:19) ϕ ( x ) dx − Z Ω aϕ ( x ) dx, = − Z Z Ω × Ω J σ ( x − y ) ϕ ( x ) ϕ ( y ) dxdy + Z Ω ϕ ( x ) dx − Z Ω aϕ ( x ) dx, ≥ −k ϕ k L (Ω) Z Ω (cid:18)Z Ω J σ ( x − y ) ϕ ( x ) dx (cid:19) dy ! / + Z Ω ϕ ( x ) dx − sup Ω a Z Ω ϕ ( x ) dx, ≥ − p k J σ k ∞ k ϕ k L (Ω) + Z Ω ϕ ( x ) dx − ν Z Ω ϕ ( x ) dx, ≥ − p k J k ∞ σ N/ + 1 − ν ! Z Ω ϕ ( x ) dx. Thus, for all σ we have I σ, ( ϕ ) ≥ − p k J k ∞ σ N/ + 1 − ν ! . By density of C c (Ω) in L (Ω), the above inequality holds for any ϕ ∈ L (Ω).32herefore, by Theorem 1.2 for all σλ p ( M σ, , Ω + a ) = λ v ( M σ, , Ω + a ) ≥ − p k J k ∞ σ N/ + 1 − ν, and lim inf σ → + ∞ λ p ( M σ + a ) ≥ − ν. Hence, 1 − ν ≤ lim inf σ → + ∞ λ p ( M σ, , Ω + a ) ≤ lim sup σ → + ∞ λ p ( M σ, , Ω + a ) ≤ − ν. To conclude this subsection, we analyse the monotonic behaviour of λ p ( M σ, , Ω + a ) with respectto σ in the particular case Ω = R N . More precisely, Proposition 4.4.
Let
Ω = R N , a ∈ C ( R N ) and J ∈ C ( R N ) be positive, symmetric and suchthat | z | J ( z ) ∈ L ( R N ) . Assume further that J satisfies (1.7) – (1.9) , m = 0 and a is symmetric( a ( x ) = a ( − x ) for all x ) and the map t → a ( tx ) is non increasing for all x, t > . Then the map σ → λ p ( σ ) is monotone non decreasing.Proof. When Ω = R N , thanks to Proposition 1.3, we have λ p ( M σ, , R N + a ) = λ p ( M , , R N + a σ ( x )) . Since the function a σ ( x ) is monotone non increasing with respect to σ , by (i) of Proposition 2.3, forall σ ≥ σ ∗ we have λ p ( M σ ∗ , , R N + a ) = λ p ( M , , R N + a σ ∗ ( x )) ≤ λ p ( M , , R N + a σ ( x )) = λ p ( M σ, , R N + a ) . m = 2Finally, let us study the case m = 2 and end the proof of Theorem 1.4. In this situation, we claimthat Lemma 4.5.
Let Ω be a domain, a ∈ C (Ω) and let J ∈ C ( R N ) be positive, symmetric and such that | z | J ( z ) ∈ L ( R N ) . Assume further that J satisfies (1.7) – (1.9) , a ∈ C ,α (Ω) with α > and m = 2 then lim σ → + ∞ λ p ( M σ, , Ω + a ) = − sup Ω a, lim σ → λ p ( M σ, , Ω + a ) = λ (cid:18) D ( J ) K ,N a, Ω (cid:19) , (4.4) where K ,N := 1 | S N − | Z S N − ( s.e ) ds = 1 N nd λ ( K ,N D ( J )∆ + a, Ω) := inf ϕ ∈ H (Ω) ,ϕ K ,N J ( ϕ ) − A ( ϕ ) . Proof.
In this situation, as already noticed in Remark 4, by following the arguments used in the case2 > m >
0, we can obtain the limit of λ p ( M σ, , Ω + a ) as σ → ∞ . So, it remains to prove (4.4).Let us rewrite I σ, ( ϕ ) in a more convenient way. Let ρ σ ( z ) := σ D ( J ) J σ ( z ) | z | , then for ϕ ∈ H (Ω), we have I σ, ( ϕ ) = 1 k ϕ k L (Ω) (cid:18) σ Z Z Ω × Ω J σ ( x − y )( ϕ ( x ) − ϕ ( y )) dxdy (cid:19) − R σ ( ϕ ) − A ( ϕ ) , (4.5)= 1 k ϕ k L (Ω) (cid:18) D ( J )2 Z Z Ω × Ω ρ σ ( x − y ) ( ϕ ( x ) − ϕ ( y )) | x − y | dxdy (cid:19) − R σ ( ϕ ) − A ( ϕ ) . (4.6)We are now is position to prove (4.4). Let us first show thatlim sup σ → λ p ( M σ, , Ω + a ) ≤ λ (cid:18) K ,N D ( J )2 ∆ + a, Ω (cid:19) . (4.7)This inequality follows from the two following observations.First, for any ω ⊂ Ω compact subset of Ω, we have for σ small enough p σ ( x ) = Z Ω J σ ( x − y ) dy = 1 for all x ∈ ω. Therefore, for ϕ ∈ C ∞ c (Ω) and σ small enough, R σ, ( ϕ ) = 1 σ k ϕ k L (Ω) Z Ω ( p σ ( x ) − ϕ ( x ) dx = 0 . (4.8)Secondly, by definition, ρ σ is a continuous mollifier such that ρ σ ≥ R N , R R N ρ σ ( z ) dz = 1 , ∀ σ > , lim σ → R | z |≥ δ ρ σ ( z ) dz = 0 , ∀ δ > , which, from the characterisation of Sobolev spaces in [13, 14, 55], enforces thatlim σ → Z Z Ω × Ω ρ σ ( x − y ) ( ϕ ( x ) − ϕ ( y )) | x − y | dxdy = K ,N k∇ ϕ k L (Ω) , for any ϕ ∈ H (Ω) . (4.9)Thus, for any ϕ ∈ C ∞ c (Ω)lim sup σ → λ p ( M σ, , Ω + a ) ≤ lim σ → I σ, ( ϕ ) = K ,N J ( ϕ ) − A ( ϕ ) . λ (cid:16) K ,N D ( J )2 ∆ + a, Ω (cid:17) , it is then standard to obtainlim sup σ → λ p ( M σ, , Ω + a ) ≤ λ (cid:18) K ,N D ( J )2 ∆ + a, Ω (cid:19) . To complete our proof, it remains to establish the following inequality λ (cid:18) K ,N D ( J )2 ∆ + a, Ω (cid:19) ≤ lim inf σ → λ p ( M σ, , Ω + a ) . Observe that to obtain the above inequality, it is sufficient to prove that λ (cid:18) K ,N D ( J )2 ∆ + a, Ω (cid:19) ≤ lim inf σ → λ p ( M σ, , Ω + a ) + 2 δ for all δ > . (4.10)Let us fix δ >
0. Now, to obtain (4.10), we construct adequate smooth test functions ϕ σ andestimate K ,N J ( ϕ σ ) − A ( ϕ σ ) in terms of λ p ( M σ, , Ω + a ) , δ and some reminder R ( σ ) that convergesto 0 as σ →
0. Since our argument is rather long, we decompose it into three steps.
Step One: Construction of a good the test function
We first claim that, for all σ >
0, there exists ϕ σ ∈ C ∞ c (Ω) such that M σ, , Ω [ ϕ σ ]( x ) + ( a ( x ) + λ p ( M σ, , Ω + a ) + 2 δ ) ϕ σ ( x ) ≥ x ∈ Ω . Indeed, by Theorem 1.2, we have λ p ( M σ, , Ω + a ) = λ ′′ p ( M σ, , Ω + a ), therefore for all σ , there exists ψ σ ∈ C c (Ω) such that M σ, , Ω [ ψ σ ]( x ) + ( a ( x ) + λ p ( M σ, , Ω + a ) + δ ) ψ σ ( x ) ≥ x ∈ Ω . Since ψ σ ∈ C c (Ω), we can easily check that M σ, , R N [ ψ σ ]( x ) + ( a ( x ) + λ p ( M σ, , Ω + a ) + δ ) ψ σ ( x ) ≥ x ∈ R N . Now, let η be a smooth mollifier of unit mass and with support in the unit ball and consider η τ := τ N η (cid:0) zτ (cid:1) for τ > ϕ σ := η τ ⋆ ψ σ and observing that M σ, , R N [ ˜ ϕ σ ]( x ) = η τ ⋆ ( M σ, , R N [ ψ σ ])( x ) for any x ∈ R N , we deduce that η τ ⋆ (cid:16) M σ, , R N [ ψ σ ] + ( a ( x ) + λ p ( M σ, , Ω + a ) + δ ) ψ σ (cid:17) ≥ x ∈ R N , M σ, , R N [ ˜ ϕ σ ]( x ) + ( λ p ( M σ, , Ω + a ) + δ ) ˜ ϕ σ ( x ) + η τ ⋆ ( aψ σ )( x ) ≥ x ∈ R N . By adding and subtracting a , we then have, for all x ∈ R N , M σ, , R N [ ˜ ϕ σ ]( x ) + ( a ( x ) + λ p ( M σ, , Ω + a ) + δ ) ˜ ϕ σ ( x ) + Z R N η τ ( x − y ) ψ σ ( y )( a ( y ) − a ( x )) dy ≥ . τ small enough, say τ ≤ τ , the function ˜ ϕ σ ∈ C ∞ c (Ω) and for all x ∈ Ω we have M σ, , R N [ ˜ ϕ σ ]( x ) = 1 σ (cid:18)Z R N J σ ( x − y ) ˜ ϕ σ ( y ) dy − ˜ ϕ σ ( x ) (cid:19) , = 1 σ (cid:18)Z Ω J σ ( x − y ) ˜ ϕ σ ( y ) dy − ˜ ϕ σ ( x ) (cid:19) = M σ, , Ω [ ˜ ϕ σ ]( x ) . Thus, from the above inequalities, for τ ≤ τ , we get for all x ∈ Ω , M σ, , Ω [ ˜ ϕ σ ]( x ) + ( a ( x ) + λ p ( M σ, , Ω + a ) + δ ) ˜ ϕ σ ( x ) + Z R N η τ ( x − y ) ψ σ ( y )( a ( y ) − a ( x )) dy ≥ . Since a is H¨older continuous, we can estimate the integral by (cid:12)(cid:12)(cid:12)(cid:12)Z R N η τ ( x − y ) ψ σ ( y )( a ( y ) − a ( x )) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R N η τ ( x − y ) ψ σ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) a ( y ) − a ( x ) | y − x | α (cid:12)(cid:12)(cid:12)(cid:12) | x − y | α dy, ≤ κτ α ˜ ϕ σ ( x ) , where κ is the H¨older semi-norm of a . Thus, for τ small, says τ ≤ inf { (cid:0) δ κ (cid:1) /α , τ } , we have M σ, , Ω [ ˜ ϕ σ ]( x ) + ( a ( x ) + λ p ( M σ, , Ω + a ) + 2 δ ) ˜ ϕ σ ( x ) ≥ x ∈ Ω . (4.11)Let us consider now ϕ σ := γ ˜ ϕ σ , where γ is a positive constant to be chosen. From (4.11), weobviously have M σ, , Ω [ ϕ σ ]( x ) + ( a ( x ) + λ p ( M σ, , Ω + a ) + 2 δ ) ϕ σ ( x ) ≥ x ∈ Ω . (4.12)By taking γ := R R N ψ σ ( x ) dx R R N ˜ ϕ σ ( x ) dx , we get R R N ψ σ ( x ) dx R R N ϕ σ ( x ) dx = 1 . (4.13) Step Two: A first estimate of λ Now, by multiplying M σ, , Ω [ ϕ σ ] by − ϕ σ and integrating over Ω, we then get − Z Ω M σ, , Ω [ ϕ σ ] ϕ σ ( x ) dx = − Z Z R N × R N σ J σ ( x − y )( ϕ σ ( y ) − ϕ σ ( x )) ϕ σ ( x ) dydx, (4.14)= 12 σ Z Z R N × R N J σ ( x − y )( ϕ σ ( y ) − ϕ σ ( x )) dxdy, (4.15)= D ( J )2 Z Z R N × R N ρ σ ( z ) ( ϕ σ ( x + z ) − ϕ σ ( x )) | z | dzdx. (4.16)36y combining (4.11) and (4.16) we therefore obtain D ( J )2 Z Z R N × R N ρ σ ( z ) ( ϕ σ ( x + z ) − ϕ σ ( x )) | z | dzdx − Z R N a ( x ) ϕ σ ( x ) dx ≤ ( λ p ( M σ, , Ω + a ) + 2 δ ) Z R N ϕ σ ( x ) dx. (4.17)On the other hand, inspired by the proof of Theorem 2 in [14], since ϕ σ ∈ C ∞ c ( R N ), by Taylor’sexpansion, for all x, z ∈ R N , we have | ϕ σ ( x + z ) − ϕ σ ( x ) − z · ∇ ϕ σ ( x ) | ≤ X i,j | z i z j | Z t (cid:18)Z | ∂ ij ϕ σ ( x + tsz ) | ds (cid:19) dt. Therefore, | z · ∇ ϕ σ ( x ) | ≤ X i,j | z i z j | Z t (cid:18)Z | ∂ ij ϕ σ ( x + tsz ) | ds (cid:19) dt + | ϕ σ ( x + z ) − ϕ σ ( x ) | , and for every θ > | z · ∇ ϕ σ ( x ) | ≤ C θ X i,j | z i z j | Z t (cid:18)Z | ∂ ij ϕ σ ( x + tsz ) | ds (cid:19) dt + (1 + θ ) | ϕ σ ( x + z ) − ϕ σ ( x ) | , ≤ C θ X i,j | z i z j | Z Z [0 , t | ∂ ij ϕ σ ( x + tsz ) | dsdt + (1 + θ ) | ϕ σ ( x + z ) − ϕ σ ( x ) | . Thus, by integrating in x and z over R N × R N , we get Z Z ρ σ ( | z | ) | z | | z · ∇ ϕ σ ( x ) | dzdx ≤ C θ Z Z ρ σ ( | z | ) X i,j | z i z j | | z | Z Z [0 , t | ∂ ij ϕ σ ( x + tsz ) | dsdt ! dzdx +(1 + θ ) Z Z ρ σ ( | z | ) | ϕ σ ( x + z ) − ϕ σ ( x ) | | z | dzdx. For σ small, supp ( ρ σ ) ⊂ B (0), and we have for all x ∈ R N , Z R N ρ σ ( | z | ) | z | | z · ∇ ϕ σ ( x ) | dz = K ,N |∇ ϕ σ ( x ) | , whence, K ,N Z R N |∇ ϕ σ ( x ) | dx ≤ C θ Z Z ρ σ ( | z | ) X i,j | z i z j | | z | Z Z [0 , t | ∂ ij ϕ σ ( x + tsz ) | dsdt ! dzdx +(1 + θ ) Z Z ρ σ ( | z | ) | ϕ σ ( x + z ) − ϕ σ ( x ) | | z | dzdx. (4.18)37ividing (4.18) by k ϕ σ k L (Ω) and then subtracting A ( ϕ σ ) on both side, we get K ,N J ( ϕ σ ) − A ( ϕ σ ) ≤ R ( σ ) + (1 + θ ) I σ, ( ϕ σ ) + θ A ( ϕ σ ) , (4.19)where R ( σ ) is defined by R ( σ ) := C θ k ϕ σ k L (Ω) Z Z ρ σ ( | z | ) X i,j | z i z j | | z | Z Z [0 , t | ∂ ij ϕ σ ( x + tsz ) | dsdt ! dzdx. By combining now (4.19) with (4.17), by definition of λ (cid:16) K ,N D ( J )2 ∆ + a, Ω (cid:17) , we obtain λ (cid:18) K ,N D ( J )2 ∆ + a, Ω (cid:19) ≤ R ( σ ) + (1 + θ )[ λ p ( M σ, , Ω + a ) + 2 δ ] + θ k a k L ∞ (Ω) . (4.20) Step Three: Estimates of R ( σ ) and conclusion Let us now estimate R ( σ ) and finish our argument.By construction, we have ∂ ij ϕ σ = ∂ ij η τ ⋆ ψ σ . So, by Fubini’s Theorem and standard convolutionestimates, we get for σ small R ( σ ) ≤ X i,j Z | z |≤ Z Z [0 , ρ σ ( | z | ) | z i z j | | z | t (cid:18)Z R N | ∂ ij η τ ⋆ ψ σ ( x + tsz ) | dx (cid:19) dtdsdz, ≤ Z | z |≤ Z [0 , ρ σ ( | z | ) X i,j | z i z j | | z | t dtdz k∇ η τ k L ( R N ) k ψ σ k L ( R N ) , ≤ k∇ η τ k L ( R N ) k ψ σ k L ( R N ) Z | z |≤ ρ σ ( | z | ) | z | dz. Combining this inequality with (4.20), we get λ (cid:18) K ,N D ( J )2 ∆ + a, Ω (cid:19) ≤ (1 + θ )[ λ p ( M σ, , Ω + a ) + 2 δ ] + θ k a k L ∞ (Ω) + 2 C θ k∇ η τ k L ( R N ) k ψ σ k L ( R N ) k ϕ σ k L (Ω) Z | z |≤ ρ σ ( | z | ) | z | dz. Since ϕ σ ∈ C ∞ c (Ω), k ϕ σ k L (Ω) = k ϕ σ k L ( R N ) and thanks to (4.13), the above inequality reduces to λ (cid:18) K ,N D ( J )2 ∆ + a, Ω (cid:19) ≤ (1 + θ )[ λ p ( M σ, , Ω + a ) + 2 δ ] + θ k a k L ∞ (Ω) + 2 C θ k∇ η τ k L ( R N ) Z | z |≤ ρ σ ( | z | ) | z | dz. (4.21)38ow, since R | z |≤ ρ σ ( | z | ) | z | dz ≤ σ , letting σ → λ (cid:18) K ,N D ( J )2 ∆ + a, Ω (cid:19) ≤ (1 + θ )[2 δ + lim inf σ → λ p ( M σ, , Ω + a )] + θ k a k L ∞ (Ω) . (4.22)Since (4.22) holds for every θ , we obtain λ (cid:18) K ,N D ( J )2 ∆ + a, Ω (cid:19) ≤ lim inf σ → λ p ( M σ, , Ω + a ) + 2 δ. ϕ p,σ In this last section, we investigate the existence of a positive continuous eigenfunction ϕ p,σ associatedto the principal eigenvalue λ p ( M σ, , Ω + a ).The existence of such a ϕ p,σ is a straightforward consequence of the existence criteria in boundeddomain (Theorem 2.2) and the asymptotic behaviour of the principal eigenvalue (Theorem 1.4).Indeed, assume first that Ω is bounded, then since a ∈ L ∞ ( ¯Ω), there exists σ such that for all σ ≤ σ , 1 σ − sup Ω a > (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:18) K ,N D ( J )2 ∆ + a, Ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . Now, thanks to λ p ( M σ, , Ω + a ) → λ (cid:16) K ,N D ( J )2 ∆ + a, Ω (cid:17) , for σ small enough, says σ ≤ σ , weget λ p ( M σ, , Ω + a ) ≤ (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:18) K ,N D ( J )2 ∆ + a, Ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . Thus, for σ ≤ inf { σ , σ } , λ p ( M σ, , Ω + a ) < σ − sup Ω a, which, thanks to Theorem 2.2, enforces the existence of a principal positive continuous eigenfunction ϕ p,σ associated with λ p ( M σ, , Ω + a ).From the above argument, we can easily obtain the existence of eigenfunction when Ω is un-bounded. Indeed, let Ω be a bounded sub-domain of Ω and let γ := sup {| λ (Ω ) | ; | λ (Ω) |} . Since a is bounded in Ω, there exists σ such that for all σ ≤ σ ,1 σ − sup Ω a > γ. As above, since λ p ( M σ, , Ω + a ) → λ (Ω ), there exists σ such that for all σ ≤ σ we have λ p ( M σ, , Ω + a ) ≤ γ. For any bounded domain Ω ′ such that Ω ⊂ Ω ′ ⊂ Ω, by monotonicity of λ p ( M σ, , Ω ′ + a ) withrespect to Ω ′ , for all σ ≤ σ we have λ p ( M σ, , Ω ′ + a ) ≤ γ. σ ≤ σ := inf { σ , σ } , we have λ p ( M σ, , Ω ′ + a ) + 1 ≤ σ − sup Ω ′ a, and thus, thanks to Theorem 2.2, for all σ ≤ σ there exists ϕ p,σ associated to λ p ( M σ, , Ω ′ + a ).To construct a positive eigenfunction ϕ p,σ associated to λ p ( M σ, , Ω + a ), we then argue as follows.Let (Ω n ) n ∈ N be an increasing sequence of bounded sub-domain of Ω that converges to Ω. Then, forall σ ≤ σ , for each n there exists a continuous positive function ϕ n,σ associated to λ p ( M σ, , Ω n + a ).Without any loss of generality, we can assume that ϕ n is normalised by ϕ n ( x ) = 1 for some fixed x ∈ Ω . Since for all n, λ p ( M σ, , Ω n + a ) + 1 ≤ σ − sup Ω n a , the Harnack inequality applies to ϕ n and thus the sequence ( ϕ n ) n ∈ N is locally uniformly bounded in C topology. By a standard diagonalargument, there exists a subsequence, still denoted ( ϕ ) n ∈ N , that converges point-wise to some non-negative function ϕ . Thanks to the Harnack inequality, ϕ is positive. Passing to the limit in theequation satisfied by ϕ n , thanks to the Lebesgue dominated convergence Theorem, ϕ satisfies M σ, , Ω [ ϕ ]( x ) + ( a ( x ) + λ p,σ ( M σ, , Ω + a )) ϕ ( x ) = 0 for all x ∈ Ω . Since a is continuous and (( a ( x ) + λ p,σ ( M σ, , Ω + a )) − σ ) <
0, we deduce that ϕ is also continuous.Hence, ϕ is a positive continuous eigenfunction associated with λ p,σ ( M σ, , Ω + a ). Remark 5.
We observe that such arguments hold also for the operators M σ,m, Ω + a with 0 2, since in such cases, λ p ( σ ) < + ∞ for all σ and − sup Ω ( − σ + a ) → + ∞ . Thus, when0 < m < 2, for σ ( m ) small enough, there exists always a positive function ϕ p,σ ∈ C ( ¯Ω) associatedwith λ p ( M σ,m, Ω + a ).Finally, let us complete the proof of Theorem 1.5 by obtaining the asymptotic behaviour of ϕ p,σ when σ → ϕ p,σ ∈ L (Ω). We first recall the following useful identity : Proposition 5.1. Let ρ ∈ C c ( R N ) be a radial function, then for all u ∈ L ( R N ) , ϕ ∈ C ∞ c ( R N ) wehave Z Z R N × R N ρ ( z )[ u ( x + z ) − u ( x )] ϕ ( x ) dzdx = 12 Z Z R N × R N ρ ( z ) u ( x )∆ z [ ϕ ]( x ) dzdx, where ∆ z [ ϕ ]( x ) := ϕ ( x + z ) − ϕ ( x ) + ϕ ( x − z ) . Proof. Set I := Z Z R N × R N ρ ( z )[ u ( x + z ) − u ( x )] ϕ ( x ) dzdx. 40y standard change of variable, thanks to the symmetry of ρ , we get I = 12 Z Z R N × R N ρ ( z )[ u ( x + z ) − u ( x )] ϕ ( x ) + 12 Z Z R N × R N ρ ( − z )[ u ( x − z ) − u ( x )] ϕ ( x ) , = 12 Z Z R N × R N ρ ( z )[ u ( x + z ) − u ( x )] ϕ ( x ) + 12 Z Z R N × R N ρ ( z )[ u ( x ) − u ( x + z )] ϕ ( x + z ) , = − Z Z R N × R N ρ ( z )[ u ( x + z ) − u ( x )][ ϕ ( x + z ) − ϕ ( x )] , = − Z Z R N × R N ρ ( z ) u ( x )[ ϕ ( x ) − ϕ ( x − z )] + 12 Z Z R N × R N ρ ( z ) u ( x )[ ϕ ( x + z ) − ϕ ( x )] , = 12 Z Z R N × R N ρ ( z ) u ( x )[ ϕ ( x + z ) − ϕ ( x ) + ϕ ( x − z )] . Consider now σ ≤ σ (Ω) and let ϕ p,σ be a positive eigenfunction associated with λ p,σ . That is ϕ p,σ satisfies M σ, , Ω [ ϕ p,σ ]( x ) + ( a ( x ) + λ p,σ ) ϕ p,σ ( x ) = 0 for all x ∈ Ω . (5.1)Let us normalize ϕ p,σ by k ϕ p,σ k L (Ω) = 1.Multiplying (5.1) by ϕ p,σ and integrating over Ω, we get D ( J )2 Z Z Ω × Ω ρ σ ( x − y ) | ϕ p,σ ( y ) − ϕ p,σ ( x ) | | x − y | dxdy ≤ Z Ω ( a ( x ) + λ p,σ ) ϕ p,σ ( x ) dx ≤ C. Since a and λ p,σ are bounded independently of σ ≤ σ (Ω), the constant C stands for all σ ≤ σ (Ω).Therefore for any bounded sub-domain Ω ′ ⊂ Ω, Z Z Ω ′ × Ω ′ ρ σ ( x − y ) ( ϕ p,σ ( y ) − ϕ p,σ ( x )) | x − y | dxdy < C. Therefore by the characterisation of Sobolev space in [55, 54], for any bounded sub-domainΩ ′ ⊂ Ω, along a sequence, ϕ p,σ → ϕ in L (Ω ′ ). Moreover, by extending ϕ p,σ by 0 outside Ω, we have ϕ p,σ ∈ L ( R N ) and for any ψ ∈ C c (Ω) by Proposition 5.1 it follows that D ( J )2 Z Z Ω × R N ρ σ ( z ) | z | ϕ p,σ ( x )∆ z [ ψ ]( x ) dxdz = − Z Ω ( a ( x ) + λ p,σ − p σ ( x )) ψϕ p,σ dx. (5.2)Recall that ψ ∈ C ∞ c ( R N ), so there exists C ( ψ ) and R ( ψ ) such that for all x ∈ R N | ∆ z [ ψ ]( x ) − t z ( ∇ ψ ( x )) z | < C ( ψ ) | z | B R ( ψ ) ( x ) . Therefore, since ϕ p,σ is bounded uniformly in L (Ω), D ( J )2 Z Z Ω × R N ρ σ ( z ) | z | ϕ p,σ ( x )[∆ z [ ψ ]( x ) − t z ( ∇ ψ ( x )) z ] dxdz ≤ CC ( ψ ) Z R N ρ n ( z ) | z | → . (5.3)41n the other hand, ψ ∈ C c (Ω) enforces that for σ small enough supp (1 − p σ ( x )) ∩ supp ( ψ ) = ∅ .Thus passing to the limit along a sequence in (5.2),thanks to (5.3), we get D ( J ) K ,N Z Ω ϕ ( x )∆ ψ ( x ) dx + Z Ω ϕ ( x ) ψ ( x )( a ( x ) + λ ) dx = 0 . (5.4)(5.4) being true for any ψ , it follows that ϕ is the smooth positive eigenfunction associated to λ normalised by k ϕ k L (Ω) = 1 = lim σ → k ϕ p,σ k L (Ω) . 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