On the definition and the properties of the principal eigenvalue of some nonlocal operators

Abstract

In this article we study some spectral properties of the linear operator $\mathcal{L}\_\Omega +a$ defined on the space $C(\bar{\Omega })$ by :$$\mathcal{L}\_\Omega [\phi ]+a\phi :=\int \_\Omega K(x,y)\phi (y)dy+a(x)\phi (x)$$ where $\Omega \subset {\mathbb{R}}^N$ 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 $\lambda \_p(\mathcal{L}\_\Omega +a)$ 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 $\lambda \_p$. Namely, we prove the equivalence of different definitions of the principal eigenvalue. We also study the behaviour of $\lambda \_p(\mathcal{L}\_\Omega +a)$ 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 $\lambda \_p\left(\mathcal{L}\_\sigma ,m,\Omega -\frac{1}{{\sigma }^m}+a\right)$ where $\mathcal{L}\_\sigma ,m,\Omega$ is defined by $\mathcal{L}\_\sigma ,m,\Omega [\phi ]:=\frac{1}{{\sigma }^{2+N}}\int \_\Omega J\left(\frac{x-y}{\sigma }\right)\phi (y)dy$. In particular, we prove that $$\lim \_\sigma \to 0\lambda \_p\left(\mathcal{L}\_\sigma ,2,\Omega -\frac{1}{{\sigma }^2}+a\right)=\lambda \_1\left(\frac{D\_2(J)}{2N}\Delta +a\right),$$where $D\_2(J):=\int \_{\mathbb{R}}^{N}J(z)|z{|}^{2}dz$ and $\lambda \_1$ denotes the Dirichlet principal eigenvalue of the elliptic operator. In addition, we obtain some convergence results for the corresponding eigenfunction $\phi \_p,\sigma$.