Lipschitz continuity for harmonic functions and solutions of the $\bar{\alpha}$-Poisson equation

TL;DR

This paper investigates Lipschitz continuity of solutions to the $ar{eta}$-Poisson equation in the plane, reviewing recent results and applying them to harmonic functions and their gradients.

Contribution

It provides new insights into Lipschitz regularity for the $ar{eta}$-Poisson equation and extends these results to harmonic and gradient harmonic functions.

Findings

Lipschitz continuity holds for solutions of the $ar{eta}$-Poisson equation in planar cases.

Recent results on regularity are reviewed and contextualized.

Corollaries include Lipschitz properties for harmonic and gradient harmonic functions.

Abstract

We study Lipschitz continuity for solutions of the $\bar{\alpha}$-Poisson equation in planar cases. We also review some recently obtained results. As corolary we can restate results for harmonic and gradient harmonic functions.