# Sobolev Homeomorphisms and Brennan’s Conjecture Academic Article

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### abstract

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

### publication date

• January 1, 2014