Fractional problems in thin domains

Abstract

In this paper we consider nonlocal fractional problems in thin domains. Given open bounded subsets U⊂Rn and V⊂Rm, we show that the solution uε to Δxsuε(x,y)+Δytuε(x,y)=f(x,ε−1y)in U×εV with uε(x,y)=0 if x⁄∈U and y∈εV, verifies that ũε(x,y)≔uε(x,εy)→u0 strongly in the natural fractional Sobolev space associated to this problem. We also identify the limit problem that is satisfied by u0 and estimate the rate of convergence in the uniform norm. Here Δxsu and Δytu are the fractional Laplacian in the 1st variable x (with a Dirichlet condition, u(x)=0 if x⁄∈U) and in the 2nd variable y (with a Neumann condition, integrating only inside V), respectively, that is, Δxsu(x,y)=∫Rnu(x,y)−u(w,y)|x−w|n+2sdw and Δytu(x,y)=∫Vu(x,y)−u(x,z)|y−z|m+2tdz.