The Poisson equation from non-local to local

2018/01/29 by Biccari, Umberto, Hernández-Santamaría, Víctor
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1801.09470

Abstract

We analyze the limit behavior as s→ 1^- of the solution to the fractional Poisson equation (-Δ)^s u_s=f_s, x∈Ω with homogeneous Dirichlet boundary conditions u_s≡ 0, x∈Ω^c. We show that lim_{s→ 1^-} u_s =u, with -Δu =f, x∈Ω and u=0, x∈∂Ω. Our results are complemented by a discussion on the rate of convergence and on extensions to the parabolic setting.

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