Sobolev homeomorphisms and Brennan's conjecture

TL;DR

This paper investigates Sobolev homeomorphisms and their implications for Brennan's conjecture, establishing conditions under which the domain's geodesic diameter is finite and analyzing the integrability of conformal derivatives.

Contribution

It proves that Sobolev homeomorphisms in certain spaces imply finite geodesic diameter and provides new insights into the inverse Brennan's conjecture regarding conformal maps.

Findings

Domains with Sobolev homeomorphisms have finite geodesic diameter for p > n.

Integrability of conformal derivatives is limited to certain degrees depending on domain diameter.

Domains with infinite geodesic diameter cannot have derivatives integrable in degrees greater than 2.

Abstract

Let $\Omega \subset \mathbb{R}^n$ be a domain that supports the $p$-Poincar\'e inequality. Given a homeomorphism $\varphi \in L^1_p(\Omega)$, for $p>n$ we show the domain $\varphi(\Omega)$ has finite geodesic diameter. This result has a direct application to Brennan's conjecture and quasiconformal homeomorphisms. {\bf The Inverse Brennan's conjecture} states that for any simply connected plane domain $\Omega' \subset\mathbb C$ with nonempty boundary and for any conformal homeomorphism $\varphi$ from the unit disc $\mathbb{D}$ onto $\Omega'$ the complex derivative $\varphi'$ is integrable in the degree $s$, $-2<s<2/3$. If $\Omega'$ is bounded than $-2<s\leq 2$. We prove that integrability in the degree $s> 2$ is not possible for domains $\Omega'$ with infinite geodesic diameter.