  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.
