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

Abstract

Let Ω ⊂ ℝN be an open bounded connected set. We consider the eigenvalue problem −Δpu = λρ|u|p−2u in Ω with homogeneous Dirichlet boundary condition, where Δp is the p-Laplacian operator and ρ is an arbitrary function that takes only two given values 0 < α < β and that is subject to the constraint ∫Ω ρdx = αγ +β(|Ω|−γ) for a fixed 0 < γ < |Ω|. The optimization of the map ρ ↦ λ1(ρ), where λ1 is the first eigenvalue, has been studied by Cuccu, Emamizadeh and Porru. In this paper we consider a Steiner symmetric domain Ω and we show that the minimizers inherit the same symmetry