A priori estimate of gradient of a solution to certain differential inequality and quasiconformal mappings

TL;DR

This paper establishes a priori gradient estimates for solutions to a Poisson differential inequality and demonstrates that certain quasiconformal mappings satisfying this inequality are Lipschitz continuous, extending classical results to higher dimensions.

Contribution

It generalizes known results on gradient bounds and Lipschitz continuity of quasiconformal mappings from the plane to higher-dimensional spaces.

Findings

Gradient of solutions is bounded under specific conditions.

Quasiconformal mappings satisfying the inequality are Lipschitz continuous.

Extends classical planar results to higher dimensions.

Abstract

We will prove a global estimate for the gradient of the solution to the {\it Poisson differential inequality} $|\Delta u(x)|\le a|\nabla u(x)|^2+b$, $x\in B^{n}$, where $a,b<\infty$ and $u|_{S^{n-1}}\in C^{1,\alpha}(S^{n-1}, \Bbb R^m)$. If $m=1$ and $a\le (n+1)/(|u|_\infty4n\sqrt n)$, then $|\nabla u| $ is a priori bounded. This generalizes some similar results due to E. Heinz (\cite{EH}) and Bernstein (\cite{BS}) for the plane. An application of these results yields the theorem, which is the main result of the paper: A quasiconformal mapping of the unit ball onto a domain with $C^2$ smooth boundary, satisfying the Poisson differential inequality, is Lipschitz continuous. This extends some results of the author, Mateljevi\'c and Pavlovi\'c from the complex plane to the space.