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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 $\mathcal{L}\_Ω+a$ defined on the space $C(\barΩ)$ by :$$ \mathcal{L}\_Ω[φ] +aφ:=\int\_ΩK(x,y)φ(y)\,dy+a(x)φ(x)$$ where $Ω\subset \mathbb{R}^N$ 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 generalised principal eigenvalue $λ\_p(\mathcal{L}\_Ω+a)$ defined by $$λ\_p(\mathcal{L}\_Ω+a):= \sup\{λ\in \mathbb{R} \,|\, \exists φ\in C(\bar Ω), φ\textgreater{}0, \textit{such that}\, \mathcal{L}\_Ω[φ] +aφ+λφ\le 0 \, \text{in}\;Ω\}. $$ We establish some new properties of this generalised principal eigenvalue $λ\_p$. Namely, we prove the equivalence of different definitions of the principal eigenvalue. We also study the behaviour of $λ\_p(\mathcal{L}\_Ω+a)$ with respect to some scaling of $K$. For kernels $K$ of the type, $K(x,y)=J(x-y)$ with $J$ a compactly supported probability density, we also establish some asymptotic properties of $λ\_{p} \left(\mathcal{L}\_{σ,m,Ω} -\frac{1}{σ^m}+a\right)$ where $\mathcal{L}\_{σ,m,Ω}$ is defined by $\displaystyle{\mathcal{L}\_{σ,m,Ω}[φ]:=\frac{1}{σ^{2+N}}\int\_ΩJ\left(\frac{x-y}σ\right)φ(y)\, dy}$. In particular, we prove that $$\lim\_{σ\to 0}λ\_p\left(\mathcal{L}\_{σ,2,Ω}-\frac{1}{σ^{2}}+a\right)=λ\_1\left(\frac{D\_2(J)}{2N}Δ+a\right),$$where $D\_2(J):=\int\_{\mathbb{R}^N}J(z)|z|^2\,dz$ and $λ\_1$ denotes the Dirichlet principal eigenvalue of the elliptic operator. In addition, we obtain some convergence results for the corresponding eigenfunction $φ\_{p,σ}$.