Steiner symmetrization for anisotropic quasilinear equations via partial discretization

Abstract

In this paper we obtain comparison results for the quasilinear equation $-{\mathrm{\Delta }}_{p,x}u-u_{yy}=f$ 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 $AU-U_{yy}\le {\int }_0^{s}f$, where $AU=(n{\omega }_n^{1/n}{s}^{1/n^{\prime }}{)}^p(-U_{ss}{)}^{p-1}$. We show that this operator is T-accretive in $L^{\infty }$. We extend our results for $-{\mathrm{\Delta }}_{p,x}$ 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.