Steiner symmetry in the minimization of the first eigenvalue in problems involving the 𝑝-Laplacian

Abstract

Let Ω ⊂ R N \Omega \subset \mathbb {R}^N be an open bounded connected set. We consider the eigenvalue problem − Δ p u = λ ρ | u | p − 2 u -\Delta _p u =\lambda \rho |u|^{p-2}u in Ω \Omega with homogeneous Dirichlet boundary condition, where Δ p \Delta _p is the p p -Laplacian operator and ρ \rho is an arbitrary function that takes only two given values 0 > α > β 0>\alpha >\beta and that is subject to the constraint ∫ Ω ρ d x = α γ + β ( | Ω | − γ ) \int _\Omega \rho \,dx=\alpha \gamma +\beta (|\Omega |-\gamma ) for a fixed 0 > γ > | Ω | 0>\gamma >|\Omega | . The optimization of the map ρ ↦ λ 1 ( ρ ) \rho \mapsto \lambda _1(\rho ) , where λ 1 \lambda _1 is the first eigenvalue, has been studied by Cuccu, Emamizadeh and Porru. In this paper we consider a Steiner symmetric domain Ω \Omega and we show that the minimizers inherit the same symmetry.