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

Let \(u_{s}\) denote a solution of the fractional Poisson problem

$$\begin{aligned} (-\Delta )^{s} u_{s} = f\quad \text { in }\Omega ,\qquad u_{s}=0\quad \text { on }{\mathbb {R}}^{N}\setminus \Omega , \end{aligned}$$

where \(N\ge 2\) and \(\Omega \subset {\mathbb {R}}^{N}\) is a bounded domain of class \(C^{2}\). We show that the solution mapping \(s\mapsto u_{s}\) is differentiable in \(L^\infty (\Omega )\) at s = 1, namely, at the nonlocal-to-local transition. Moreover, using the logarithmic Laplacian, we characterize the derivative \(\partial _{s} u_{s}\) 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 \(\partial _{s} u_{s}\) for s \(\in (0,1)\) which allows us to refine previously obtained Green function estimates.