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 ≤ s 0 f , where AU = (nω 1/n n s 1/n ′ ) p (−Uss) p−1 . We show that this operator is T-accretive in L ∞ . 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.