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

Abstract

In this paper, we analyze the $s$-dependence of the solution $u_s$ to the fractional Poisson equation $(-\Delta {)}_{\Omega }^s=f$ 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 $(-\Delta {)}_{\Omega }^s$ regarded as a function of $s\in (0,1)$.