Schwarz's Lemmas for mappings satisfying Poisson's equation

Abstract

For $n\geq3$, $m\geq1$ and a given continuous function $g:~\Omega\rightarrow\mathbb{R}^{m}$, we establish some Schwarz type lemmas for mappings $f$ of $\Omega$ into $\mathbb{R}^{m}$ satisfying the PDE: $\Delta f=g$, where $\Omega$ is a subset of $\mathbb{R}^{n}$. Then we apply these results to obtain a Landau type theorem.