The Hodge Star Operator and the Beltrami Equation

TL;DR

This paper introduces an alternative approach to solving the Beltrami equation using the Hodge star operator and elliptic PDE theory, offering new insights into solution regularity.

Contribution

It presents a novel method leveraging the Hodge star operator for solving the Beltrami equation, differing from traditional Calderón-Zygmund techniques.

Findings

Successful application of Hodge star operator to the Beltrami equation

New proof of solution regularity using elliptic PDE theory

Comparison with existing methods highlights advantages of the new approach

Abstract

An essentially unique homeomorphic solution to the Beltrami equation was found in the 1960s using the theory of Calder\'{o}n-Zygmund and singular integral operators in $L^p(\mathbb C)$. We will present an alternative method to solve the Beltrami equation using the Hodge star operator and standard elliptic PDE theory. We will also discuss a different method to prove the regularity of the solution. This approach is partially based on work by Dittmar.