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

Abstract

AbstractLet $$u_{s}$$ u s denote a solution of the fractional Poisson problem $$\begin{aligned} (-\Delta )^{s} u_{s} = f\quad \text { in }\Omega ,\qquad u_{s}=0\quad \text { on }{\mathbb {R}}^{N}\setminus \Omega , \end{aligned}$$ ( - Δ ) s u s = f in Ω , u s = 0 on R N \ Ω , where $$N\ge 2$$ N ≥ 2 and $$\Omega \subset {\mathbb {R}}^{N}$$ Ω ⊂ R N is a bounded domain of class $$C^{2}$$ C 2 . We show that the solution mapping $$s\mapsto u_{s}$$ s ↦ u s is differentiable in $$L^\infty (\Omega )$$ L ∞ ( Ω ) at s = 1, namely, at the nonlocal-to-local transition. Moreover, using the logarithmic Laplacian, we characterize the derivative $$\partial _{s} u_{s}$$ ∂ 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 $$\partial _{s} u_{s}$$ ∂ s u s for s$$\in (0,1)$$ ∈ ( 0 , 1 ) which allows us to refine previously obtained Green function estimates.