Comparison results for solutions of elliptic problems via Steiner symmetrization

Abstract

We consider the Dirichlet problem for a class of linear elliptic equations, whose model is \begin{align*} & -\Delta u-\sum _{i=1}^{n}\left( b_{i}(y)u\right) _{x_{i}}-\sum _{j=1}^{m}\left( \widetilde{b}_{j}(y)u\right) _{y_{j}}+ \sum _{i=1}^{n}d_{i}(y)u_{x_{i}} \\ & +\sum\limits_{j=1}^{m}\widetilde{d} _{j}(y)u_{y_{j}}+c(y)u=f(x,y) \ \ \text{ in }G, \end{align*} where $G=G^{\prime }\times G^{\prime \prime }$ is an open, bounded and connected subset of ${\mathbb R}^{N}={\mathbb R}^{n}\times {\mathbb R}^{m}$, the coefficients $b_{i}(y),$ $\widetilde{b}_{j}(y),$ $d_{i}(y)$, $\widetilde{d} _{j}(y)$ and $c(y)$ are in $L^{\infty }(G)$ and the datum $f(x,y)$ belongs to $ L^{p}(G)$ with $p>\frac{N}{2}$. We prove some comparison results by using Steiner symmetrization.