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

Abstract

Let $Ω\subset\mathbb{R}^N$, $N\geq 2$, be an open bounded connected set. We consider the fractional weighted eigenvalue problem $(-Δ)^s u =λρu$ in $Ω$ with homogeneous Dirichlet boundary condition, where $(-Δ)^s$, $s\in (0,1)$, is the fractional Laplacian operator, $λ\in \mathbb{R}$ and $ ρ\in L^\infty(Ω)$. We study weak* continuity, convexity and Gâteaux differentiability of the map $ρ\mapsto1/λ_1(ρ)$, where $λ_1(ρ)$ is the first positive eigenvalue. Moreover, denoting by $\mathcal{G}(ρ_0)$ the class of rearrangements of $ρ_0$, we prove the existence of a minimizer of $λ_1(ρ)$ when $ρ$ varies on $\mathcal{G}(ρ_0)$. Finally, we show that, if $Ω$ is Steiner symmetric, then every minimizer shares the same symmetry.