Sobolev regularity of quasiconformal mappings on domains

Consider a Lipschitz domain $Ω$ and a measurable function $μ$ supported in $\overlineΩ$ with $\left\|μ\right\|_{L^\infty}<1$. Then the derivatives of a quasiconformal solution of the Beltrami equation $\overline{\partial} f =μ\partial f$ inherit the Sobolev regularity $W^{n,p}(Ω)$ of the Beltrami coefficient $μ$ as long as $Ω$ 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, $N\in B^{n-1/p}_{p,p}(\partialΩ)$.

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