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) .