A new look at the fractional Poisson problem via the Logarithmic Laplacian

Abstract

We analyze the $s$-dependence of solutions $u_s$ to the family of fractional Poisson problems $(-\Delta {)}^{s}u=f$ in $\Omega$, $u\equiv 0$ on ${\mathbb{R}}^N\setminus \Omega$ in an open bounded set $\Omega \subset {\mathbb{R}}^N$, $s\in (0,1)$. In the case where $\Omega$ is of class $C^2$ and $f\in C^{\alpha }(\bar{\Omega })$ for some $\alpha >0$, we show that the map $(0,1)\to L^{\infty }(\Omega )$, $s\mapsto u_s$ is of class $C^1$, and we characterize the derivative ${\partial }_s{u}_s$ in terms of the logarithmic Laplacian of $f$. As a corollary, we derive pointwise monotonicity properties of the solution map $s\mapsto u_s$ under suitable assumptions on $f$ and $\Omega$. Moreover, we derive explicit bounds for the corresponding Green operator on arbitrary bounded domains which are new even for the case $s=1$, i.e., for the local Dirichlet problem $-\Delta u=f$ in $\Omega$, $u\equiv 0$ on $\partial \Omega$.