Sobolev homeomorphic extensions

Aleksis Koski, Jani Onninen

Research output: Contribution to journalArticlepeer-review

Abstract

Let X and Y be ℓ-connected Jordan domains, ℓ 2 N, with rectifiable boundaries in the complex plane. We prove that any boundary homeomorphism ' W ∂X onto -→ ∂Y admits a Sobolev homeomorphic extension hW X onto -→Y inW1;1.X;C/. If instead X has s-hyperbolic growth with s > p - 1, we show the existence of such an extension in the Sobolev classW1;p.X;C/ for p 2 .1; 2/. Our examples show that the assumptions of rectifiable boundary and hyperbolic growth cannot be relaxed. We also consider the existence of W1;2-homeomorphic extensions with given boundary data.

Original languageEnglish (US)
Pages (from-to)4065-4089
Number of pages25
JournalJournal of the European Mathematical Society
Volume23
Issue number12
DOIs
StatePublished - 2021

Keywords

  • Douglas condition
  • Sobolev extensions
  • Sobolev homeomorphisms

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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