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, $N\in B^{ n-1/p}_{ p,p}(\partial\Omega)$.
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