U. Biccari, V. Hernández-Santamaría The Poisson equation from non-local to local, Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 145, pp. 1-13. DOI: arXiv:1801.09470
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.
