Sobolev homeomorphisms and Brennan's conjecture

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.