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

We will prove a global estimate for the gradient of the solution to the {\it Poisson differential inequality} $\mathrm{\mid }\mathrm{\Delta }u(x)\mathrm{\mid }\le a\mathrm{\mid }\mathrm{\nabla }u(x){\mathrm{\mid }}^2+b$, $x\in B^{n}$,where$a,b<\infty$and$u|{S^{n-1}}\in C^{1,\alpha}(S^{n-1}, \Bbb R^m)$\mathrm{.}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.