The Poisson equation from non-local to local

Abstract

We analyze the limiting behavior as s → 1¯ of the solution to the fractional Poisson equation (-∆)^su_s = ƒ_s, x ∈ Ω with homogeneous Dirichlet boundary conditions u_s ≡ 0, x ∈ Ω^c. We show that lim_s→1¯ u_s = u, with -∆u = ƒ, x ∈ Ω and u = 0, x ∈ ∂Ω. Our results are complemented by a discussion on the rate of convergence and on extensions to the parabolic setting.