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Improved Hölder Continuity of Quasiconformal Maps

Rosemarie Bongers

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 Hölder continuity. In this paper, we give improved lower bounds for the Hölder 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 Hölder continuity are characterized, and some applications are given to solutions to elliptic partial differential equations.

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 Hölder continuity. In this paper, we give improved lower bounds for the Hölder 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 Hölder continuity are characterized, and some applications are given to solutions to elliptic partial differential equations.

Paper Structure

This paper contains 5 sections, 12 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Estimate of Hölder Exponent
  3. Extremizers for Hölder Continuity
  4. Applications to Elliptic PDEs
  5. References

Key Result

Theorem 1.1

Let $f : \Omega \to \Omega'$ be a continuous and $W^{1, 2}_{\text{loc}}$ solution to the Beltrami equation $\partial_{\overline{z}} f = \mu(z) \partial_z f$ with $|\mu(z)| \le \frac{K - 1}{K + 1}$. Then $f$ is $\alpha$-Hölder continuous for some exponent $\alpha$ satisfying where $S_{\rho, x}$ is the circle centered at $x$ with radius $\rho$, $\eta$ is the outward unit normal, and $\sigma$ is arc

Theorems & Definitions (20)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 10 more