Differentiability of the Nonlocal-to-local Transition in Fractional Poisson Problems

Abstract Let uₒ u s denote a solution of the fractional Poisson problem aligned (-) ^s uₒ = f in, uₒ=0 on {R}^N, aligned (- Δ) s u s = f in Ω, u s = 0 on R N \ Ω, where N 2 N ≥ 2 and {R}^N Ω ⊂ R N is a bounded domain of class C^2 C 2. We show that the solution mapping s uₒ s ↦ u s is differentiable in L^ () L ∞ (Ω) at s = 1, namely, at the nonlocal-to-local transition. Moreover, using the logarithmic Laplacian, we characterize the derivative ₒ uₒ ∂ s u s as the solution to a boundary value problem. This complements the previously known differentiability results for s in the open interval (0, 1). Our proofs are based on an asymptotic analysis to describe the collapse of the nonlocality of the fractional Laplacian as s approaches 1. We also provide a new representation of ₒ uₒ ∂ s u s for s (0, 1) ∈ (0, 1) which allows us to refine previously obtained Green function estimates.