Schwarz's Lemmas for mappings satisfying Poisson's equation

TL;DR

This paper develops Schwarz lemmas for mappings satisfying Poisson's equation in higher dimensions and applies these results to derive a Landau type theorem, advancing the understanding of PDE-constrained mappings.

Contribution

It introduces new Schwarz type lemmas for solutions of Poisson's equation in higher dimensions and uses them to establish a Landau type theorem.

Findings

Schwarz lemmas for PDE solutions in higher dimensions

Bounds on mappings satisfying Poisson's equation

A Landau type theorem derived from these bounds

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.