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