Sobolev homeomorphisms and Brennan's conjecture

Vladimir Gol'dshtein, Alexander Ukhlov

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Abstract

Let Ω ⊂ Rn be a domain that supports the p-Poincaré inequality. Given a homeomorphism ϕ ∈ L1p(Ω), for p > n we show that the domain ϕ() 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 Ω' ⊂ C with non-empty boundary and for any conformal homeomorphism ϕ from the unit disc D onto Ω' the complex derivative ϕ is integrable in the degree s, −2 < s < 2/3. If Ω' is bounded then −2 < s ≤ 2. We prove that integrability in the degree s > 2 is not possible for domains Ω' with infinite geodesic diameter.
Original languageEnglish GB
JournalComputational Methods and Function Theory
StatePublished - 1 Sep 2013

Keywords

  • Mathematics - Functional Analysis

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