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

Abstract

In this paper, we analyze the $ s $-dependence of the solution $ u_s $ to the free Poisson problem $ (-\Delta)^s_{\Omega}u_s = f $ in an open bounded set $ \Omega\subset\mathbb{R}^N $. Precisely, we show that the solution map $ (0, 1)\rightarrow L^2(\Omega) $, $ s\mapsto u_s $ is continuously differentiable. Moreover, when $ f = \lambda_su_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) $.