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

TL;DR

This paper studies how solutions to fractional Poisson problems depend on the fractional parameter s, showing smoothness, characterizing derivatives via the logarithmic Laplacian, and providing bounds for the Green operator.

Contribution

It introduces a detailed analysis of the s-dependence of solutions, including differentiability and explicit bounds, extending understanding even to the classical Laplacian case.

Findings

The solution map s ↦ u_s is C^1 in (0,1).

The derivative of u_s involves the logarithmic Laplacian of f.

New bounds for the Green operator are established for arbitrary domains.

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 $\partial \Omega$.