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

Abstract

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

20 pages