Steiner symmetrization for anisotropic quasilinear equations via partial discretization

TL;DR

This paper establishes comparison results for anisotropic quasilinear equations using Steiner symmetrization and partial discretization, solving a longstanding open problem and extending to broader classes of operators.

Contribution

It introduces a novel discretization-based approach to prove comparison principles for anisotropic quasilinear equations, including general operators beyond the p-Laplacian.

Findings

Comparison results for anisotropic quasilinear equations

Extension to general divergence-form operators

Use of finite-differences discretization in y

Abstract

In this paper we obtain comparison results for the quasilinear equation $-\Delta_{p,x} u - u_{yy} = f$ 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 $A U - U_{yy} \le \int_0^s f^*$, where $AU = (n\omega^{1/n}s^{1/n'} )^p (- U_{ss})^{p-1}$. We show that this operator is T-accretive in $L^\infty$. We extend our results for $-\Delta_{p,x}$ to general operators of the form $-\mathrm{div} (a(|\nabla_x u|) \nabla_x u)$ where $a$ is non-decreasing and behaves like $| \cdot |^{p-2}$ at infinity.