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- W2375119037 abstract "In this article we study some spectral properties of the linear operator $mathcal{L}_{Omega}+a$ defined on the space $C(barOmega)$ by :$$ mathcal{L}_{Omega}[varphi] +avarphi:=int_{Omega}K(x,y)varphi(y),dy+a(x)varphi(x)$$ where $Omegasubset mathbb{R}^N$ 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 $$lambda_p(mathcal{L}_{Omega}+a):= sup{lambda in mathbb{R} ,|, exists varphi in C(bar Omega), varphitextgreater{}0, textit{such that}, mathcal{L}_{Omega}[varphi] +avarphi +lambdavarphi le 0 , text{in};Omega}. $$ 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 $lambda_{p} left(mathcal{L}_{sigma,m,Omega} -frac{1}{sigma^m}+aright)$ where $mathcal{L}_{sigma,m,Omega}$ is defined by $displaystyle{mathcal{L}_{sigma,m,Omega}[varphi]:=frac{1}{sigma^{2+N}}int_{Omega}Jleft(frac{x-y}{sigma}right)varphi(y), dy}$. In particular, we prove that $$lim_{sigmato 0}lambda_pleft(mathcal{L}_{sigma,2,Omega}-frac{1}{sigma^{2}}+aright)=lambda_1left(frac{D_2(J)}{2N}Delta +aright),$$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 $varphi_{p,sigma}$." @default.
- W2375119037 created "2016-06-24" @default.
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- W2375119037 date "2015-12-21" @default.
- W2375119037 modified "2023-10-06" @default.
- W2375119037 title "On the definition and the properties of the principal eigenvalue of some nonlocal operators" @default.
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