Electron. J. Differential Equations, Vol. 2018 (2018), No. 145, pp. 1-13.

The Poisson equation from non-local to local

Umberto Biccari, Victor Hernandez-Santamaria

Abstract:
We analyze the limiting 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.

Submitted January 22, 2018. Published July 17, 2018.
Math Subject Classifications: 35B30, 35R11, 35S05.
Key Words: Fractional Laplacian; elliptic equations; weak solutions.

Show me the PDF file (670 KB), TEX file for this article.

Umberto Biccari
DeustoTech, University of Deusto
48007 Bilbao, Basque Country, Spain
email: umberto.biccari@deusto.es, u.biccari@gmail.com
Víctor Hernández-Santamaría
DeustoTech, University of Deusto
48007 Bilbao, Basque Country, Spain
email: victor.santamaria@deusto.es

Return to the EJDE web page