On the definition and the properties of the principal eigenvalue of some nonlocal operators
In this article we study some spectral properties of the linear operator defined on the space by :
where is a domain, possibly unbounded, is a continuous bounded function and is a continuous, non negative kernel satisfying an integrability condition.
We focus our analysis on the properties of the generalized principal eigenvalue defined by
We establish some new properties of this generalized principal eigenvalue . Namely, we prove the equivalence of different definitions of the principal eigenvalue. We also study the behaviour of with respect to some scaling of .
For kernels of the type, with a compactly supported probability density, we also establish some asymptotic properties of where is defined by . In particular, we prove that
where and denotes the Dirichlet principal eigenvalue of the elliptic operator. In addition, we obtain some convergence results for the corresponding eigenfunction .
- 1 Introduction
- 2 Preliminaries
- 3 Relation between and
- 4 Asymptotic behaviour of the principal eigenvalue under scaling
- 5 Asymptotics of
The principal eigenvalue of an operator is a fundamental notion in modern analysis. In particular, this notion is widely used in PDE’s literature and is at the source of many profound results especially in the study of elliptic semi linear problems. For example, the principal eigenvalue is used to characterise the stability 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 principle properties satisfies by elliptic operators [12, 8] and to describe continuous semi-groups that preserve an order [1, 32, 46]. It is further used in obtaining Liouville type results for elliptic semi-linear equations [10, 6].
In this article we are interested in such notion for linear operators defined on the space of continuous functions by :
where is a domain, possibly unbounded, is a continuous bounded function and is a non negative kernel satisfying an integrability condition. The precise assumptions on and will be given later on.
To our knowledge, for most of positive operators, the principal eigenvalue is a notion related to the existence of an eigen-pair, namely an eigenvalue associated with a positive eigen-element. For the operator , when the function is not constant, for any real , neither nor its inverse are compact operators. Moreover, as noticed in [25, 30, 41, 60], the operator may not have any eigenvalues in the space or . For such operator, the existence of an eigenvalue associated with a positive eigenvector is then not guaranteed. Studying quantities that can be used as surrogates of a principal eigenvalue and establishing their most important properties are therefore of great interest for such operators.
In this perspective, we are interested in the properties of the following quantity:
which can be expressed equivalently by the formula:
This number was originally introduced in the Perron-Frobenius Theory to characterise the eigenvalues of an irreducible positive matrix [21, 63]. Namely, for a positive irreducible matrix the eigenvalue associated with a positive eigenvector can be characterised as follows:
also known as the Collatz-Wieldandt characterisation.
Numerous generalisation of these types of characterisation exist in the literature. Generalisations of 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 [42, 43, 57] and later for general positive operators that posses an eigen-pair .
In parallel with the generalisation of the Perron-Frobenius Theory, several inf sup formulas have been developed to characterise the spectral properties of elliptic operators satisfying a maximum principle, see the fundamental works of Donsker and Varadhan , Nussbaum, Pinchover , Berestycki, Nirenberg, Varadhan  and Pinsky [51, 52]. In particular, for an elliptic operator defined in a bounded domain and with bounded continuous coefficients, , several notions of principal eigenvalue have been introduced. On one hand, Donsker and Varadhan  have introduced a quantity , called principal eigenvalue of , that satisfies
where denotes the domain of definition of . On the other hand Berestycki, Nirenberg and Varadhan  have introduced defined by:
as another possible definition for the principal eigenvalue of . When is a smooth bounded domain and has smooth coefficients, both notions coincide (i.e. ). The equivalence of this two notions has been recently extended for more general elliptic operators, in particular the equivalence holds true in any bounded domains and in any domains when is an elliptic self-adjoint operator with bounded coefficients . It is worth mentioning that the quantity was originally introduced by Donsker and Varadhan  to obtain the following variational characterisation of in a bounded domain:
where is the set of all probability measure on . Such characterisation is still valid when is unbounded, see Nussbaum and Pinchover .
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 properties of and several other notions of principal eigenvalue have emerged. For instance, several new notions of principal eigenvalue have been introduced for general elliptic operators defined on (limit or almost) periodic media [6, 10, 49, 56]. For the interested reader, we refer to , for a review and a comparison of the different notions of principal eigenvalue for an elliptic operator defined in a unbounded domain.
For the operator , much less is known and only partial results have been obtained when is bounded [25, 28, 32, 37, 40, 41] or in a periodic media [29, 30, 60, 59]. More precisely, has been compared to one of the following definitions :
or when is a self-adjoint operator:
where denotes the scalar product of . For a bounded domain and for particular kernels , an equality similar to has been obtained in , provided that satisfies some non-degeneracy conditions. The author shows that
In this paper, we pursue the works begun in [25, 30, 28] by one of the present authors and we investigate more closely the properties of . Namely, we first look whether can be characterised by other notions of principal eigenvalue and under which conditions on and the equality (1.4) or (1.5) holds true. In particular, we introduce a new notion of principal eigenvalue, , defined by :
and we compare this new quantity with and .
Another natural question is to obtain a clear picture on the dependence of with respect to all the parameters involved. If the behaviour of with respect to or can be exhibited directly from the definition, the impact of scalings of the kernel is usually unknown and has been largely ignored in the literature except in some specific situations involving particular nonlocal dispersal operators defined in a bounded domain [2, 24, 41, 58].
For a particular type of and , we establish the asymptotic properties of with respect to some scaling parameter. More precisely, let and let us denote . When is a non negative function of unit mass, we study the properties of the principal eigenvalue of the operator , where the operator is defined by:
In this situation, the operator refers to a nonlocal version of the standard diffusion operator with a homogeneous Dirichlet boundary condition. Such type of operators has appeared recently in the literature to model a population that have a constrained dispersal [4, 34, 39, 41, 58]. In this context, the pre-factor is interpreted as a frequency at which the events of dispersal occur. For and a large class of , we obtain the asymptotic limits of as and as .
1.1 Motivations: nonlocal reaction diffusion equation
Our interest in studying the properties of stems from the recent studies of populations having a long range dispersal strategy [25, 28, 4, 41, 60]. For such a population, a commonly used model that integrates such long range dispersal is the following nonlocal reaction diffusion equation ([34, 38, 39, 47, 62]):
In this context, is the density of the considered population, is a dispersal kernel and is a KPP type non-linearity describing the growth rate of the population. When is a bounded domain [3, 25, 28, 36, 41, 59], an optimal persistence criteria has been obtained using the sign of , where stands for the operator:
In such model, a population will persists if and only if . We can easily check that .
When and in periodic media, adapted versions of have been recently used to define an optimal persitence criteria [29, 30, 60, 58]. The extension of such type of persistence criteria for more general environments is currently investigated by ourself  by means of our findings on the properties of .
The understanding of the effect of a dispersal process conditioned by a dispersal budget is another important question. The idea introduced by Hutson, Martinez, Mischaikow and Vickers , is simple and consists in introducing a cost function related to the amount of energy an individual has to use to produce offspring, that jumps on a long range. When a long range of dispersal is privileged, the energy consumed to disperse an individual is large and so very few offsprings are dispersed. On the contrary, when the population chooses to disperse on a short range, few energy is used and a large amount of the offsprings is dispersed. In to understand the impact of a dispersal budget on the range of dispersal, we are led to consider the family of dispersal operator :
where is the standard scaling of the probability density . For such family, the study of the dependence of with respect to and is a first step to analyse the impact of the range of the dispersal on the persistence of the population. In particular the asymptotic limits and are of primary interest.
1.2 Assumptions and Main Results
Let us now state the precise assumptions we are making on the domain , the kernel and the function . Here, throughout the paper, is a domain (open connected set of ) and for and we assume the following:
and is a non-negative Caratheodory function, that is and,
For our analysis, we also require that satisfies the following non-degeneracy condition:
There exist positive constants such that satisfies:
where denotes the characteristic function of the set and is the ball centred at of radius . These conditions are satisfied for example for kernels like with and positive and bounded in and , a compactly supported function such that . Note that when is bounded, any kernel which is positive on the diagonal, satisfies all theses assumptions. Under this assumptions, we can check that the operator is continuous in ,.
Let us now state our main results. We start by investigating the case of a bounded domain . In this situation, we prove that and represent the same quantity. Namely, we show the following
When is an unbounded domain, the equivalence of and is not clear for general kernels. Namely, let consider , with a density of probability with a compact support and such that . For the operator , which corresponds to the standard convolution by , by using and constants as test functions, we can easily check that . However some inequalities remain true in general and the equivalence of the three notions holds for self-adjoint operators. More precisely, we prove here the following
Another striking property of refers to the invariance of under a particular scaling of the kernel . More precisely, we show
Observe that no condition on the domain is imposed. Therefore, the invariance of is still valid for . In this case, since is invariant under the scaling, we get
Next, for particular type of kernel , we investigate the behaviour of with respect of some scaling parameter. More precisely, let and let . We consider the following operator
For is a non negative function of unit mass, we study the asymptotic properties of the principal eigenvalue of the operator when and .
To simplify the presentation of our results, let us introduce the following notation. We denote by , the following operator:
For any domains , we obtain the limits of when tends either to zero or to . Let us denote the second moment of by
the following statement describes the limiting behaviour of :
In addition, when and if is symmetric ( for all ) and the map is non increasing for all then is monotone non decreasing with respect to .
When and for some , then
Note that the results hold for any domains , so the results holds true in particular for . Having established the asymptotic limits of the principal eigenvalue , it is natural to ask whether similar results hold for the corresponding eigenfunction when it exists. In this direction, we prove that for , such convergence does occur :
Let be any domain and assume that and satisfy (1.7) – (1.9). Assume further that is even and of unit mass. Then there exists such that for all , there exists a positive principal eigenfunction associated to . In addition, when for all , we have
where is a positive principle eigenfunction associated to .
When is bounded, then the condition is always satisfied. Moreover, in this situation, the above limits as holds in instead of .
1.3 Comments and straightforward generalisation
First, we can notice that the quantity defined by Donsker and Varadhan  for elliptic operators can also be defined for the operator and is equivalent to the quantity . The equality (1.4) can then be seen as the nonlocal version of the equality where is the notion introduced by Berestycki-Nirenberg-Varadhan .
Next, we would like to emphasize, that unlike the classical elliptic operators, due to the lack of a regularising effect of the operator , the quantity may not be an eigenvalue, i.e. the spectral problem
may not have a solution in spaces of functions like [29, 27, 32, 41]. As a consequence, even in bounded domains, the relations between , and are quite delicate to obtain. Another difficulty inherent to the study of nonlocal operators in unbounded domains concerns the lack of natural a priori estimates for the positive eigenfunction thus making standard approximation schemes difficult to use in most case.
Lastly, we make some additional comments on the assumptions we have used on the dispersal kernel . The non-degeneracy assumption (1.9) we are using, is related to the existence of Local Uniform Estimates [22, 23] (Harnack type estimates) for a positive solution of a nonlocal equation:
Such type of estimates is a key tool in our analysis, in particular in unbounded domains, where we use it to obtain fundamental properties of the principal eigenvalue , such as the limit:
where is a sequence of set converging to . As observed in , some local uniform estimates can also be obtained for some particular kernels which does not satisfies the non-degeneracy condition (1.9). For example, for kernels of the form with satisfying (1.8) and (1.9) and a bounded function such that is a bounded set and with Lebesgue measure zero, 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 such that, for some positive integer , the kernel defined recursively by :
satisfies the non-degeneracy condition (1.9).
For a convolution operator, i.e. , this last condition is optimal. It is related to a geometric property of the convex hull of :
satisfies (1.9) for some if and only if the convex hull of contains .
Note that if a relaxed assumption on the lower bound of the non-degeneracy condition satisfied by appears simple to find, the condition on the support of seems quite tricky to relax. To tackle this problem, it is tempting to investigate the spectrum of linear operators involving the Fractional Laplacian, :
That is, to look for the properties of the principal eigenvalue of the spectral problem:
As for elliptic operators and , analogues of and can be defined for and the relations between all possible definitions can be investigated. When is bounded or is periodic, the different definitions are equivalent . However, in the situations considered in  the operator has a compact resolvent enabling the use of the Krein Rutmann Theory. Thus, the corresponding 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 yielding to the inequality
for any smooth domain .
Such Harnack type inequalities are not known for operators involving a continuous kernel with unbounded support. Furthermore, it seems that most of the tools used to establish these Harnack estimates in the case of the Fractional Laplacian [16, 61] do not apply when we consider an operator . Thus, obtaining the inequality
with a more general kernel requires a deeper understanding of Harnack type estimates and/or the development 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,
Outline of the paper: The paper is organised as follows. In Section 2, we recall some known results and properties of the principal eigenvalue . The relations between the different definitions of the principal eigenvalue, , and (Theorems 1.1, 1.2 and Proposition 1.6) are proved in Section 3. Finally, in Section 4 we derive the asymptotic behaviour of with respect to the different scalings of (Proposition 1.3 and Theorems 1.4 and 1.5 ).
To simplify the presentation of the proofs, we introduce some notations and various linear operator that we will use throughout this paper:
denotes the standard ball of radius centred at the point
will always refer to the characteristic function of the ball .
denotes the Schwartz space,
denotes the space of continuous function in ,
denotes the space of continuous function with compact support in .
For a positive integrable function , the constant will refer to
For a bounded set , will denotes its Lebesgue measure
For two functions , denotes the scalar product of and
We denote by the continuous linear operator
We denote by the operator
In this section, we recall some standard results on the principal eigenvalue of the operator . Since the early work  on the variational formulation of the principal eigenvalue, an intrinsic difficulty related to the study of these quantities comes from the possible non-existence of a positive continuous eigenfunction associated to the definition of or to . This means that there is not always a positive continuous eigenfunction associated to or . A simple illustration of this fact can be found in [25, 27]. Recently, some progress have been made in the understanding of . In particular, some flexible criteria have been found to guarantee the existence of a positive continuous eigenfunction [25, 41, 59]. More precisely,
Theorem 2.1 (Sufficient condition ).
Let be a domain, and non negative, satisfying the condition (1.9). Let us denote and assume further that the function satisfies for some bounded domain . Then there exists a principal eigen-pair solution of
Moreover, , and we have the following estimate
When is bounded, sharper results have been recently derived in  where it is proved that is always an eigenvalue in the Banach space of positive measure, that is, we can always find a positive measure that is solution in the sense of measure of
In addition, we have the following characterisation of :
We refer to  for a more complete description of the positive solution associated to when the domain is bounded.
Now, we recall some properties of that we constantly use throughout this paper:
Assume , then for the two operators and
respectively defined on and , we have :
For a fixed and assume that , for all . Then
is Lipschitz continuous with respect to . More precisely,
The following estimate always holds
Lastly, we prove some limit behaviour of