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 $-\mathrm{div}(a(|{\mathrm{\nabla }}_{x}u|){\mathrm{\nabla }}_{x}u)$ where a is non-decreasing and behaves like $|\cdot {|}^{p-2}$ at infinity.

在本文中，我们获得了拟线性方程的比较结果 $-{\mathrm{\Delta }}_{p，X}ü-{ü}_{ÿÿ}=F$通过变量x的Steiner重排实现具有齐次Dirichlet边界条件的条件，从而解决了一个长期开放的问题。实际上，我们研究了更广泛的各向异性问题。我们的方法基于y中的有限差分离散化，并且证明了辅助问题的离散版本的比较原理$\mathrm{一个}ü-{ü}_{ÿÿ}\le {\int }_0^{s}F$，在哪里 $\mathrm{一个}ü={（ñ{\omega }_{ñ}^{\mathrm{1个}/ñ}{s}^{\mathrm{1个}/{ñ}^{\prime }}）}^p{（-{ü}_{ss}）}^{p-\mathrm{1个}}$。我们证明这个算子在${\mathrm{大号}}^{\infty }$。我们将结果扩展到$-{\mathrm{\Delta }}_{p，X}$ 形式的一般运营商 $-\mathrm{div}（\mathrm{一个}（|{\mathrm{\nabla }}_Xü|）{\mathrm{\nabla }}_Xü）$其中a是非递减的且行为类似于$|\cdot {|}^{p-2}$ 在无穷大。