Maximum and antimaximum principles for some nonlocal diffusion operators

In this work we consider the maximum and antimaximum principles for the nonlocal Dirichlet problem J * u - u + lambda u + h = integral(RN)J(x - y)u(y) dy - u(x) + lambda u(x) + h(x) = 0 in a bounded domain Omega, with u(x) = 0 in R-N \ Omega. The kernel J in the convolution is assumed to be a continuous, compactly supported nonnegative function with unit integral. We prove that for lambda < lambda(1)(Omega), the solution verifies u > 0 in (Omega) over bar if h is an element of L-2(Omega), h >= 0, while for lambda > lambda(1)(Omega), and lambda close to lambda(1)(Omega), the solution verifies u < 0 in <(Omega)over bar> provided f(Omega) h(x) phi(x) dx > 0, h is an element of L-infinity(Omega). This last assumption is also shown to be optimal. The Neumann version of the problem is also analyzed. (C) 2009 Elsevier Ltd. All rights reserved.