Consider a Lipschitz domain \Omega and the Beurling transform of its characteristic function \mathcal{B} \chi_\Omega(z)= - {\rm p.v.}\frac1{\pi z^2}*\chi_\Omega (z) . It is shown that if the outward unit normal vector N of the boundary of the domain is in the trace space of W^{n,p}(\Omega) (i.e., the Besov space B^{n-1/p}_{p,p}(\partial\Omega) ) then \mathcal{B} \chi_\Omega \in W^{n,p}(\Omega) . Moreover, when p>2 the boundedness of the Beurling transform on W^{n,p}(\Omega) follows. This fact has far-reaching consequences in the study of the regularity of quasiconformal solutions of the Beltrami equation.