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

Abstract

In this article we study some spectral properties of the linear operator\n$\\mathcal{L}\\_{\\Omega}+a$ defined on the space $C(\\bar\\Omega)$ by :$$\n\\mathcal{L}\\_{\\Omega}[\\varphi]\n+a\\varphi:=\\int\\_{\\Omega}K(x,y)\\varphi(y)\\,dy+a(x)\\varphi(x)$$ where\n$\\Omega\\subset \\mathbb{R}^N$ is a domain, possibly unbounded, $a$ is a\ncontinuous bounded function and $K$ is a continuous, non negative kernel\nsatisfying an integrability condition. We focus our analysis on the properties\nof the generalised principal eigenvalue $\\lambda\\_p(\\mathcal{L}\\_{\\Omega}+a)$\ndefined by $$\\lambda\\_p(\\mathcal{L}\\_{\\Omega}+a):= \\sup\\{\\lambda \\in \\mathbb{R}\n\\,|\\, \\exists \\varphi \\in C(\\bar \\Omega), \\varphi\\textgreater{}0, \\textit{such\nthat}\\, \\mathcal{L}\\_{\\Omega}[\\varphi] +a\\varphi +\\lambda\\varphi \\le 0 \\,\n\\text{in}\\;\\Omega\\}. $$ We establish some new properties of this generalised\nprincipal eigenvalue $\\lambda\\_p$. Namely, we prove the equivalence of\ndifferent definitions of the principal eigenvalue. We also study the behaviour\nof $\\lambda\\_p(\\mathcal{L}\\_{\\Omega}+a)$ with respect to some scaling of $K$.\nFor kernels $K$ of the type, $K(x,y)=J(x-y)$ with $J$ a compactly supported\nprobability density, we also establish some asymptotic properties of\n$\\lambda\\_{p} \\left(\\mathcal{L}\\_{\\sigma,m,\\Omega}\n-\\frac{1}{\\sigma^m}+a\\right)$ where $\\mathcal{L}\\_{\\sigma,m,\\Omega}$ is defined\nby\n$\\displaystyle{\\mathcal{L}\\_{\\sigma,m,\\Omega}[\\varphi]:=\\frac{1}{\\sigma^{2+N}}\\int\\_{\\Omega}J\\left(\\frac{x-y}{\\sigma}\\right)\\varphi(y)\\,\ndy}$. In particular, we prove that $$\\lim\\_{\\sigma\\to\n0}\\lambda\\_p\\left(\\mathcal{L}\\_{\\sigma,2,\\Omega}-\\frac{1}{\\sigma^{2}}+a\\right)=\\lambda\\_1\\left(\\frac{D\\_2(J)}{2N}\\Delta\n+a\\right),$$where $D\\_2(J):=\\int\\_{\\mathbb{R}^N}J(z)|z|^2\\,dz$ and $\\lambda\\_1$\ndenotes the Dirichlet principal eigenvalue of the elliptic operator. In\naddition, we obtain some convergence results for the corresponding\neigenfunction $\\varphi\\_{p,\\sigma}$.\n