Steiner symmetry in the minimization of the first eigenvalue of a fractional eigenvalue problem with indefinite weight

Let Ω ⊂ R^N , N ≥ 2, be an open bounded connected set. We consider the fractional weighted eigenvalue problem (−∆)^s u = λρu in Ω with homogeneous Dirichlet boundary condition, where (−∆)^s, s ∈ (0, 1), is the fractional Laplacian operator, λ ∈ R and ρ ∈ L^∞(Ω). We study weak* continuity, convexity and Gateaux differentiability of the map ρ → 1/λ_1(ρ), where λ_1(ρ) is the first positive eigenvalue. Moreover, denoting by 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 G (ρ_0 ). Finally, we show that, if Ω is Steiner symmetric, then every minimizer shares the same symmetry.