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

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 $u{|}_{S^{n-1}}\in C^{1,\alpha }(S^{n-1},{\mathbb{R}}^m)$. If $m=1$ and $a\le (n+1)/(|u{|}_{\infty }4n\sqrt{n})$, 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, Mateljević and Pavlović from the complex plane to the space.

arXiv comment24 pages

References in corpus (1)

• On certain nonlinear elliptic PDE and quasiconfomal mapps between Euclidean surfaces