Steiner symmetrization for anisotropic quasilinear equations via partial discretization

Abstract

In this paper we obtain comparison results for the quasilinear equation −_p,xu−uyy = f with homogeneous Dirichlet boundary conditions by Steiner rearrangement in variable x, thus solving a long open problem. In fact, we study a broader class of anisotropic problems. Our approach is based on a finite-differences discretization in y, and the proof of a comparison principle for the discrete version of the auxiliary problem AU − Uyy ≤ R s0 f, where AU = (n!1/nn s1/n′)p(−Uss)p−1. We show that this operator is T-accretive in L1. We extend our results for −_p,x to general operators of the form −div(a(|∇xu|)∇xu) where a is non-decreasing and behaves like | ・ |p−2 at infinity.