ON THE PRINCIPAL EIGENVALUE OF SOME NONLOCAL DIFFUSION PROBLEMS

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 re-spect 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. 1