Quasiconformal solutions to elliptic partial differential equations

Abstract

In this paper, we assume that $G$ and $\mathrm{\Omega }$ are two Jordan domains in ${\mathbb{R}}^n$ with ${\mathcal{C}}^2$ boundaries, where $n\ge 2$, and prove that every quasiconformal mapping $f\in {\mathcal{W}}_{\mathrm{loc}}^{2,1+ϵ}$ of $G$ onto $\mathrm{\Omega }$, satisfying the elliptic partial differential inequality $|L_A[f]|\lesssim (\Vert Df{\Vert }^2+|g|)$, with $g\in {\mathcal{L}}^p(G)$, where $p>n$, is Lipschitz continuous. The result is sharp since for $p=n$, the mapping $f$ is not necessarily Lipschitz continuous. This extends several results for harmonic quasiconformal mappings.