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arXiv · 1801.09470

The Poisson equation from non-local to local

Umberto Biccari·Víctor Hernández-Santamaría
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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.

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Umberto Biccari, Víctor Hernández-Santamaría. 2018-01-29. The Poisson equation from non-local to local. https://arxiv.org/abs/1801.09470

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