The Poisson equation from non-local to local

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.

References in corpus (2)

• Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators
• Interior regularity of solutions of non-local equations in Sobolev and Nikol'skii spaces

Cited by in corpus (6)

• Numerical methods for nonlocal and fractional models
• Symmetrization for fractional elliptic problems: a direct approach
• Nonradiality of second eigenfunctions of the fractional Laplacian in a ball
• Non-local to local transition for ground states of fractional Schrödinger equations on bounded domains
• Analysis of the controllability from the exterior of strong damping nonlocal wave equations
• Non-local to local transition for ground states of fractional Schrödinger equations on ${\mathbb{R}}^N$