Sobolev regularity of quasiconformal mappings on domains

Abstract

Consider a Lipschitz domain $\Omega$ and a measurable function $\mu$ supported in $\overline\Omega$ with $\left\|{\mu}\right\|_{L^\infty}<1$. 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, $N\in B^{n-1/p}_{p,p}(\partial\Omega)$.