The Poisson equation from non-local to local

Abstract

We analyze the limit behavior as $s\to 1^{-}$ of the solution to the fractional Poisson equation $(-\Delta {)}^s{u}_s=f_s$, $x\in \Omega$ with homogeneous Dirichlet boundary conditions $u_s\equiv 0$, $x\in {\Omega }^c$. We show that ${\lim }_{s\to 1^{-}}{u}_s=u$, with $-\Delta u=f$, $x\in \Omega$ and $u=0$, $x\in \partial \Omega$. Our results are complemented by a discussion on the rate of convergence and on extensions to the parabolic setting.