STEINER SYMMETRY IN THE MINIMIZATION OF THE FIRST EIGENVALUE IN PROBLEMS INVOLVING THE p-LAPLACIAN

Let Omega subset of R-N be an open bounded connected set. We consider the eigenvalue problem -Delta(p)u, = lambda rho vertical bar u vertical bar(p-2)u in Omega with homogeneous Dirichlet boundary condition, where Delta(p) is the rho-Laplacian operator and rho is an arbitrary function that takes only two given values 0 < alpha < beta and that is subject to the constraint integral(Omega)rho dx = alpha gamma + beta(vertical bar Omega vertical bar-gamma) for a fixed 0 < gamma < vertical bar Omega vertical bar. The optimization of the map rho -> lambda(1)(rho), where lambda(1) is the first eigenvalue, has been studied by Cuccu, Emamizadeh and Porru. In this paper we consider a Steiner symmetric domain Omega and we show that the minimizers inherit the same symmetry.