On the principal eigenvalue of some nonlocal diffusion problems

In this paper we analyze some properties of the principal eigenvalue gimel(1) (Omega) of the nonlocal Dirichlet problem (J * u)(x) - u(x) = -lambda(x) in Omega with u(x) = 0 in R-N\Omega. Here Omega is a smooth bounded domain of RN and the kernel J is assumed to be a C-1 compactly supported, even, nonnegative function with unit integral. Among other properties, we show that lambda(1) (Omega) is continuous (or even differentiable) with respect to continuous (differentiable) perturbations of the domain Omega. We also provide an explicit formula for the derivative. Finally, we analyze the asymptotic behavior of the decreasing function Lambda(gamma) = lambda(1) (gamma Omega) when the dilatation parameter gamma > 0 tends to zero or to infinity. (c) 2008 Elsevier Inc. All rights reserved.