Beltrami equations in the plane and Sobolev regularity

TL;DR

This paper investigates the Sobolev regularity of solutions to the Beltrami equation with discontinuous coefficients, revealing that the derivatives of solutions share similar smoothness properties as the coefficients, with some integrability loss.

Contribution

It introduces new Sobolev regularity results for Beltrami equation solutions with discontinuous coefficients using Kato-Ponce commutators, and proposes a conjecture on the method's limitations.

Findings

ar  belongs to a Sobolev space with the same smoothness as ta and  coefficients

The method shows a loss in integrability but preserves smoothness levels

A conjecture is raised about cases where the method's limitations do not apply

Abstract

New results regarding the Sobolev regularity of the principal solution of the linear Beltrami equation $\bar{\partial} f = \mu \partial f + \nu \overline{\partial f}$ for discontinuous Beltrami coefficients $\mu$ and $\nu$ are obtained, using Kato-Ponce commutators, obtaining that $\overline \partial f$ belongs to a Sobolev space with the same smoothness as the coefficients but some loss in the integrability parameter. A conjecture on the cases where the limitations of the method do not work is raised.