Sobolev regularity of quasiconformal mappings on domains

TL;DR

This paper investigates how the Sobolev regularity of quasiconformal mappings is influenced by the boundary regularity of the domain and the Beltrami coefficient, establishing conditions under which derivatives inherit regularity.

Contribution

It establishes a precise boundary regularity condition involving the trace space of the boundary normal vector that ensures derivatives of quasiconformal solutions inherit Sobolev regularity.

Findings

Derivatives of quasiconformal solutions inherit Sobolev regularity under boundary conditions.

Normal vector in the trace space $B^{n-1/p}_{p,p}$ is sufficient for regularity inheritance.

Regularity transfer depends on boundary smoothness and Beltrami coefficient properties.

Abstract

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, $N\in B^{n-1/p}_{p,p}(\partial\Omega)$.