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\\Ω+a defined on the space C(\\Ω) by :\\Ω[\φ]+a\φ:=\∫\\ΩK(x,y)\φ(y)\\,dy+a(x)\φ(x) where\Ω\⊂ N is a domain, possibly unbounded, a is a bounded function and K is a continuous, non negative kernel an integrability condition. We focus our analysis on the properties the generalised principal eigenvalue \λ\ₚ(\\Ω+a) by $$\λ\_p(\\Ω+a):= \\\{\λ \∈ \\,|\\, \∃ \φ \∈ C(\ \Ω), \φ\{}0, \\, \\Ω[\φ] +a\φ +\λ\φ \≤ 0 \\,∈\\;\Ω\\}. We establish some new properties of this generalised eigenvalue\λ\_p. Namely, we prove the equivalence of definitions of the principal eigenvalue. We also study the behaviour\λ\_p(\\Ω+a)$ with respect to some scaling of $K. kernelsK$ of the type, $K(x,y)=J(x-y)$ with $Ja compactly supported density, we also establish some asymptotic properties of\λ\ₚ \(\\σ,m,\Ω-\1/\σ^m+a\)$ where $\\σ,m,\Ωis defined\{\\σ,m,\Ω[\φ]:=\{1}{\σ2+N}\∫\\ΩJ\(-y/\σ\)\φ(y)\\,}$. In particular, we prove that $$\lim\\σ\→0\λ\_p\(\\σ,2,\Ω-\{1}{\σ²}+a\)=\λ\_1\(\_2(J)/2N\Δ+a\),$$where $D\_2(J):=\∫\^NJ(z)|z|^2\\,dz$ and $\λ\_1the Dirichlet principal eigenvalue of the elliptic operator. In, we obtain some convergence results for the corresponding\φ\p,\σ$.