On the principal eigenvalue of some nonlocal diffusion problems |
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Authors: | Jorge Garcí a-Meliá n |
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Affiliation: | a Departamento de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271 La Laguna, Spain b IMDEA Matemáticas, C-IX, Campus Cantoblanco, Universidad Autonoma de Madrid, Madrid, Spain |
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Abstract: | In this paper we analyze some properties of the principal eigenvalue λ1(Ω) of the nonlocal Dirichlet problem (J∗u)(x)−u(x)=−λu(x) in Ω with u(x)=0 in RN?Ω. Here Ω is a smooth bounded domain of RN and the kernel J is assumed to be a C1 compactly supported, even, nonnegative function with unit integral. Among other properties, we show that λ1(Ω) is continuous (or even differentiable) with respect to continuous (differentiable) perturbations of the domain Ω. We also provide an explicit formula for the derivative. Finally, we analyze the asymptotic behavior of the decreasing function Λ(γ)=λ1(γΩ) when the dilatation parameter γ>0 tends to zero or to infinity. |
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