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+λu+h=∫RNJ(x−y)u(y)dy−u(x)+λu(x)+h(x)=0 in a bounded domain ΩΩ, with u(x)=0u(x)=0 in RN∖ΩRN∖Ω. The kernel JJ in the convolution is assumed to be a continuous, compactly supported nonnegative function with unit integral. We prove that for λ<λ1(Ω)λ<λ1(Ω), the solution verifies u>0u>0 in Ω¯ if h∈L2(Ω)h∈L2(Ω), h≥0h≥0, while for λ>λ1(Ω)λ>λ1(Ω), and λλ close to λ1(Ω)λ1(Ω), the solution verifies u<0u<0 in Ω¯, provided ∫Ωh(x)ϕ(x)dx>0, h∈L∞(Ω)h∈L∞(Ω). This last assumption is also shown to be optimal. The “Neumann” version of the problem is also analyzed.