The Poisson equation from non-local to local

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.