Steiner symmetrization for anisotropic quasilinear equations via partial discretization

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 $AU-U_{yy}\le {\int }_0^s{f}^{\ast }$, where $AU=(n{\omega }^{1/n}{s}^{1/n^{\prime }}{)}^p(-U_{ss}{)}^{p-1}$. 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.