Brennan's Conjecture and universal Sobolev inequalities

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Abstract

Brennan's Conjecture states integrability of derivatives of plane conformal homeomorphisms φ:Ω→D that map a simply connected plane domain with non-empty boundary Ω⊂C to the unit disc D⊂R2. We prove that Brennan's Conjecture leads to existence of compact embeddings of Sobolev spaces W°p1(Ω) into weighted Lebesgue spaces L q (Ω, h) with universal conformal weights h(z):=J(z,φ)=|φ'(z)|2. For p=2 the number q is an arbitrary number between 1 and ∞ (Gol'dshtein and Ukhlov, in press [12]), for p≠2 the number q depends on p and the integrability exponent for Brennan's Conjecture.

Original languageEnglish
Pages (from-to)253-269
Number of pages17
JournalBulletin des Sciences Mathematiques
Volume138
Issue number2
DOIs
StatePublished - 1 Mar 2014

Keywords

  • Brennan's Conjecture
  • Conformal mappings
  • Elliptic equations
  • Sobolev inequalities

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