Sobolev regularity of quasiconformal mappings on domains

Abstract

Consider a Lipschitz domain Ω and a measurable function μ supported in $$\overline{\Omega}$$ with ‖μ‖L∞ < 1. Then the derivatives of a quasiconformal solution of the Beltrami equation $$\overline{\partial}f=\mu\;\partial{f}$$ inherit the Sobolev regularity Wn,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, $$N\in{B}_{p,p}^{n-1/p}(\partial\Omega)$$ .