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 $(-\mathrm{\Delta }{)}^s{u}_s=f_s$, $x\in \mathrm{\Omega }$ with homogeneous Dirichlet boundary conditions $u_s\equiv 0$, $x\in {\mathrm{\Omega }}^c$. 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.