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