The Poisson equation from non-local to local

Biccari U., Hernández-Santamaría V. The Poisson equation from non-local to local, Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 145, pp. 1-13. ISSN:1072-669

Abstract: We analyze the limit behavior as s\to 1^- of the solution to the fractional Poisson equation \fl{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.

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