On the definition and the properties of the principal eigenvalue of some nonlocal operators
Henri Berestycki, Jérôme Coville, Hoang‐Hung Vo · arXiv (Cornell University) · 2015
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 $\\varphi\\_{p,\\sigma}$.