On the s-derivative of weak solutions of the Poisson problem for the regional fractional Laplacian

TL;DR

This paper investigates how solutions to the fractional Poisson problem depend on the fractional order s, proving the continuous differentiability of the solution map and analyzing eigenvalue differentiability.

Contribution

It establishes the continuous differentiability of the solution with respect to s and explores the one-sided differentiability of the first eigenvalue for the regional fractional Laplacian.

Findings

Solution map s to u_s is continuously differentiable in L^2()

Eigenvalue of fractional Laplacian is one-sided differentiable in s

Provides new insights into parameter dependence of fractional PDE solutions

Abstract

In this paper, we analyze the $s$-dependence of the solution $u_s$ to the fractional Poisson equation $(-\Delta)^s_{\Omega} = f$ in an open bounded set $\Omega \subset \mathbb{R}^N$. Precisely, we show that the solution map $(0,1)\to L^2(\Omega)$, $s\mapsto u_s$ is continuously differentiable. Moreover, when $f = \lambda_s u_s$, we also analyze the one-sided differentiability of the first nontrivial eigenvalue of $(-\Delta)^s_{\Omega}$ regarded as a function of $s \in (0,1)$.