# A priori estimate of gradient of a solution to certain differential inequality and quasiconformal mappings - Mathematics > Analysis of PDEs

A priori estimate of gradient of a solution to certain differential inequality and quasiconformal mappings - Mathematics > Analysis of PDEs - Download this document for free, or read online. Document in PDF available to download.

Abstract: We will prove a global estimate for the gradient of the solution to the {\itPoisson differential inequality} $|\Delta ux|\le a| abla ux|^2+b$, $x\inB^{n}$, where $a,b<\infty$ and $u| {S^{n-1}}\in C^{1,\alpha}S^{n-1}, \BbbR^m$. If $m=1$ and $a\le n+1-|u| \infty4n\sqrt n$, then $| abla u| $ is apriori bounded. This generalizes some similar results due to E. Heinz\cite{EH} and Bernstein \cite{BS} for the plane. An application of theseresults yields the theorem, which is the main result of the paper: Aquasiconformal mapping of the unit ball onto a domain with $C^2$ smoothboundary, satisfying the Poisson differential inequality, is Lipschitzcontinuous. This extends some results of the author, Mateljevi\-c andPavlovi\-c from the complex plane to the space.

Author: ** David Kalaj**

Source: https://arxiv.org/