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

Let uₛ u s denote a solution of the fractional Poisson problem aligned (-Δ )ˢ uₛ = f in Ω , uₛ=0 on RN Ω , aligned ( - Δ ) s u s = f in Ω , u s = 0 on R N \ Ω , where N≥ 2 N ≥ 2 and Ω ⊂ RN Ω ⊂ R N is a bounded domain of class C² C 2 . We show that the solution mapping s↦ uₛ s ↦ u s is differentiable in L^∞ (Ω ) L ∞ ( Ω ) at s = 1, namely, at the nonlocal-to-local transition. Moreover, using the logarithmic Laplacian, we characterize the derivative ∂ ₛ uₛ ∂ 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 ∂ ₛ uₛ ∂ s u s for s ∈ (0,1) ∈ ( 0 , 1 ) which allows us to refine previously obtained Green function estimates.