On the Lipschitz continuity of certain quasiregular mappings between smooth Jordan domains

TL;DR

This paper investigates the Lipschitz continuity of certain quasiregular mappings between smooth Jordan domains, providing new results that extend previous work and applying them to harmonic and quasiconformal mappings.

Contribution

It establishes Lipschitz continuity for $(K,K')$-quasiregular mappings under PDE inequalities and applies these results to harmonic and quasiconformal mappings, extending prior findings.

Findings

Lipschitz continuity of $(K,K')$-quasiregular mappings between smooth Jordan domains.

Lipschitz continuity of $ ho$-harmonic $(K,K')$-quasiregular mappings.

Lipschitz continuity of quasiconformal self-mappings of the unit disk solving the Poisson equation.

Abstract

We first investigate the Lipschitz continuity of $(K, K')$-quasiregular $C^2$ mappings between two Jordan domains with smooth boundaries, satisfying certain partial differential inequalities concerning Laplacian. Then two applications of the obtained result are given: As a direct consequence, we get the Lipschitz continuity of $\rho$-harmonic $(K, K')$-quasiregular mappings, and as the other application, we study the Lipschitz continuity of $(K,K')$-quasiconformal self-mappings of the unit disk, which are the solutions of the Poisson equation $\Delta w=g$. These results generalize and extend several recently obtained results by Kalaj, Mateljevi\'{c} and Pavlovi\'{c}.