Fractional problems in thin domains. (April 2020) - Pereira, Marcone C.; Rossi, Julio D.; Saintier, Nicolas
Abstract: In this paper we consider nonlocal fractional problems in thin domains. Given open bounded subsets U ⊂ R n and V ⊂ R m, we show that the solution u ε to Δ x s u ε ( x, y ) + Δ y t u ε ( x, y ) = f ( x, ε − 1 y ) in U × ε V with u ε ( x, y ) = 0 if x ⁄ ∈ U and y ∈ ε V, verifies that u ̃ ε ( x, y ) ≔ u ε ( x, ε y ) → u 0 strongly in the natural fractional Sobolev space associated to this problem. We also identify the limit problem that is satisfied by u 0 and estimate the rate of convergence in the uniform norm. Here Δ x s u and Δ y t u 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, Δ x s u ( x, y ) = ∫ R n u ( x, y ) − u ( w, y ) | x − w | n + 2 s d w and Δ y t u ( x, y ) = ∫ V u ( x, y ) − u ( x, z ) | y − z | m + 2 t d z .