Sobolev regularity of quasiconformal mappings on domains

Abstract

Consider a Lipschitz domain $\Omega$ and a measurable function $\mu$ supported in $\overline{\Omega }$ with ${\Vert \mu \Vert }_{L^{\infty }}<1$. Then the derivatives of a quasiconformal solution of the Beltrami equation $\overline{\partial }f=\mu \partial f$ inherit the Sobolev regularity $W^{n,p}(\Omega )$ of the Beltrami coefficient $\mu$ as long as $\Omega$ is regular enough. The condition obtained is that the outward unit normal vector $N$ of the boundary of the domain is in the trace space, that is, $N\in B_{p,p}^{n-1/p}(\partial \Omega )$.

arXiv comment38 pages, 2 figures. Accepted in Journal d'Analyse Mathematique