Improved Hölder Continuity of Quasiconformal Maps

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.