On the definition and the properties of the principal eigenvalue of some nonlocal operators (1512.06529v2)

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 :$$ \mathcal{L}_{\Omega}[\varphi] +a\varphi:=\int_{\Omega}K(x,y)\varphi(y)\,dy+a(x)\varphi(x)$$ 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 $$\lambda_p(\mathcal{L}_{\Omega}+a):= \sup{\lambda \in \mathbb{R} \,|\, \exists \varphi \in C(\bar \Omega), \varphi\textgreater{}0, \textit{such that}\, \mathcal{L}_{\Omega}[\varphi] +a\varphi +\lambda\varphi \le 0 \, \text{in}\;\Omega}. $$ 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 $\displaystyle{\mathcal{L}_{\sigma,m,\Omega}[\varphi]:=\frac{1}{\sigma^{2+N}}\int_{\Omega}J\left(\frac{x-y}{\sigma}\right)\varphi(y)\, dy}$. 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 $\varphi_{p,\sigma}$.