Sobolev homeomorphisms and Brennan's conjecture

2013/09/08 by Vladimir Gol’dshtein, Vladimir Gol'dshtein, Alexander Ukhlov +2
Mathematics · #Analytic and geometric function theory #math.FA

paper · pdf · doi:10.48550/arxiv.1309.1940

8 pages

arxiv created 2013/09/08 · arxiv updated 2013/09/10

Abstract

Let Ω⊂ ℝⁿ be a domain that supports the p-Poincaré inequality. Given a homeomorphism φ∈ L¹_p(Ω), for p>n we show the domain φ(Ω) 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 Ω' ⊂\mathbb C with nonempty boundary and for any conformal homeomorphism φ from the unit disc \mathbbD onto Ω' the complex derivative φ' is integrable in the degree s, -2<s<2/3. If Ω' is bounded than -2<s≤ 2. We prove that integrability in the degree s> 2 is not possible for domains Ω' with infinite geodesic diameter.

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