Steiner symmetrization for anisotropic quasilinear equations via partial discretization

© 2020 Elsevier Masson SAS In this paper we obtain comparison results for the quasilinear equation −Δp,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≤∫0sf, where AU=(nωn1/ns1/n′)p(−Uss)p−1. We show that this operator is T-accretive in L∞. We extend our results for −Δp,x to general operators of the form −div(a(|∇xu|)∇xu) where a is non-decreasing and behaves like |⋅|p−2 at infinity.