Maximum and antimaximum principles for some nonlocal diffusion operators Original Research Article

In this work we consider the maximum and antimaximum principles for the nonlocal Dirichlet problem View the MathML sourceJ∗u−u+λu+h=∫RNJ(x−y)u(y)dy−u(x)+λu(x)+h(x)=0 Turn MathJax on 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 View the MathML sourceΩ¯ 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 View the MathML sourceΩ¯, provided View the MathML source∫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.