Nonlocal problems in thin domains

Abstract

In this paper we consider nonlocal problems in thin domains. First, we deal with a nonlocal Neumann problem, that is, we study the behavior of the solutions to f(x)=∫Ω1×Ω2Jϵ(x−y)(uϵ(y)−uϵ(x))dy with Jϵ(z)=J(z1,ϵz2) and Ω=Ω1×Ω2⊂RN=RN1×RN2 a bounded domain. We find that there is a limit problem, that is, we show that uϵ→u0 as ϵ→0 in Ω and this limit function verifies ∫Ω2f(x1,x2)dx2=|Ω2|∫Ω1J(x1−y1,0)(U0(y1)−U0(x1))dy1, with U0(x1)=∫Ω2u0(x1,x2)dx2. In addition, we deal with a double limit when we add to this model a rescale in the kernel with a parameter that controls the size of the support of J. We show that this double limit exhibits some interesting features.We also study a nonlocal Dirichlet problem f(x)=∫RNJϵ(x−y)(uϵ(y)−uϵ(x))dy, x∈Ω, with uϵ(x)≡0, x∈RN∖Ω, and deal with similar issues. In this case the limit as ϵ→0 is u0=0 and the double limit problem commutes and also gives v≡0 at the end.