Improved H\"older Continuity of Quasiconformal Maps

TL;DR

This paper improves the known lower bounds for the H"older continuity of quasiconformal maps in the complex plane by analyzing geometric distortion and characterizing extremizers, with applications to elliptic PDEs.

Contribution

It provides new lower bounds for H"older continuity of quasiconformal maps and characterizes extremizers, enhancing understanding of their regularity properties.

Findings

Improved lower bounds for H"older continuity

Characterization of extremizers for H"older regularity

Applications to elliptic partial differential equations

Abstract

Quasiconformal maps in the complex plane are homeomorphisms that satisfy certain geometric distortion inequalities; infinitesimally, they map circles to ellipses with bounded eccentricity. The local distortion properties of these maps give rise to a certain degree of global regularity and H\"older continuity. In this paper, we give improved lower bounds for the H\"older continuity of these maps; the analysis is based on combining the isoperimetric inequality with a study of the length of quasicircles. Furthermore, the extremizers for H\"older continuity are characterized, and some applications are given to solutions to elliptic partial differential equations.