Steiner symmetrization for anisotropic quasilinear equations via partial discretization

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 $AU = (nomega^1/ns^1/n' )^p (- U_ss)^p-1$. 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 $-mathrmdiv (a(| abla_x u|) abla_x u)$ where $a$ is non-decreasing and behaves like $| cdot |^p-2$ at infinity.