Distributional solutions of the Beltrami equation

TL;DR

This paper investigates distributional solutions to the Beltrami equation with Sobolev coefficients, demonstrating they are quasiregular maps with higher regularity than previously known.

Contribution

It proves that distributional solutions under Sobolev assumptions are genuine quasiregular maps with enhanced smoothness, specifically second order derivatives in $L^{1+ ext{epsilon}}_{loc}$.

Findings

Distributional solutions are quasiregular maps.

Solutions have second order derivatives in $L^{1+ ext{epsilon}}_{loc}$.

Enhanced regularity of solutions under Sobolev conditions.

Abstract

We study the distributional solutions to the (generalized) Beltrami equation under Sobolev assumptions on the Beltrami coefficients. In this setting, we prove that these distributional solutions are true quasiregular maps and they are smoother than expected, that is, they have second order derivatives in $L^{1+\varepsilon}_{loc}$, for some $\varepsilon >0$.