The Poisson equation from non-local to local

Original · EN

We analyze the limit behavior as s→ 1- of the solution to the fractional Poisson equation (-Δ)ˢ uₛ=fₛ, x∈Ω with homogeneous Dirichlet boundary conditions uₛ≡ 0, x∈Ωᶜ. We show that ₛ→ ₁₋ uₛ =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.