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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/







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