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 LΩ+aLΩ+a defined on the space C(Ω¯) by:LΩ[φ]+aφ:=∫ΩK(x,y)φ(y)dy+a(x)φ(x) where Ω⊂RNΩ⊂RN 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 generalised principal eigenvalue λp(LΩ+a)λp(LΩ+a) defined byλp(LΩ+a):=sup⁡{λ∈R|∃φ∈C(Ω¯),φ>0,such that LΩ[φ]+aφ+λφ≤0 in Ω}.We establish some new properties of this generalised principal eigenvalue λpλp. Namely, we prove the equivalence of different definitions of the principal eigenvalue. We also study the behaviour of λp(LΩ+a)λp(LΩ+a) with respect to some scaling of K.For kernels K   of the type, K(x,y)=J(x−y)K(x,y)=J(x−y) with J   a compactly supported probability density, we also establish some asymptotic properties of λp(Lσ,m,Ω−1σm+a) where Lσ,m,ΩLσ,m,Ω is defined by Lσ,2,Ω[φ]:=1σ2+N∫ΩJ(x−yσ)φ(y)dy. In particular, we prove thatlimσ→0⁡λp(Lσ,2,Ω−1σ2+a)=λ1(D2(J)2NΔ+a), where D2(J):=∫RNJ(z)|z|2dz and λ1λ1 denotes the Dirichlet principal eigenvalue of the elliptic operator. In addition, we obtain some convergence results for the corresponding eigenfunction φp,σφp,σ.