Minimization and Steiner symmetry of the first eigenvalue for a fractional eigenvalue problem with indefinite weight

Abstract

Let $\Omega\subset\mathbb{R}^N$, $N\geq 2$, be an open bounded connected set. We consider the fractional weighted eigenvalue problem $(-\Delta)^s u =\lambda \rho u$ in $\Omega$ with homogeneous Dirichlet boundary condition, where $(-\Delta)^s$, $s\in (0,1)$, is the fractional Laplacian operator, $\lambda \in \mathbb{R}$ and $ \rho\in L^\infty(\Omega)$. We study weak* continuity, convexity and G\^ateaux differentiability of the map $\rho\mapsto1/\lambda_1(\rho)$, where $\lambda_1(\rho)$ is the first positive eigenvalue. Moreover, denoting by $\mathcal{G}(\rho_0)$ the class of rearrangements of $\rho_0$, we prove the existence of a minimizer of $\lambda_1(\rho)$ when $\rho$ varies on $\mathcal{G}(\rho_0)$. Finally, we show that, if $\Omega$ is Steiner symmetric, then every minimizer shares the same symmetry.