Differentiability of the nonlocal-to-local transition in fractional Poisson problems

Let $u_s$ denote a solution of the fractional Poisson problem $$ (-\Delta)^s u_s = f\quad\text{ in }\Omega,\qquad u_s=0\quad \text{ on }\mathbb{R}^N\setminus \Omega, $$ where $N\geq 2$ 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.