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. fact, study broader class of anisotropic problems. Our approach is based on finite-differences discretization $y$, and proof principle discrete version 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 operator T-accretive $L^\infty$. extend our $-\Delta_{p,x}$ to general operators form $-\mathrm{div} (a(|\nabla_x u|) \nabla_x u)$ $a$ non-decreasing behaves like $| \cdot |^{p-2}$ at infinity.