Sobolev homeomorphic extensions from two to three dimensions

Stanislav Hencl, Aleksis Koski*, Jani Onninen

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

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We study the basic question of characterizing which boundary homeomorphisms of the unit sphere can be extended to a Sobolev homeomorphism of the interior in 3D space. While the planar variants of this problem are well-understood, completely new and direct ways of constructing an extension are required in 3D. We prove, among other things, that a Sobolev homeomorphism φ:R2ontoR2 in Wloc1,p(R2,R2) for some p∈[1,∞) admits a homeomorphic extension h:R3ontoR3 in Wloc1,q(R3,R3) for [Formula presented]. Such an extension result is nearly sharp, as the bound [Formula presented] cannot be improved due to the Hölder embedding. The case q=3 gains an additional interest as it also provides an L1-variant of the celebrated Beurling-Ahlfors quasiconformal extension result.

Original languageEnglish
Article number110371
JournalJournal of Functional Analysis
Issue number9
Publication statusPublished - 1 May 2024
MoE publication typeA1 Journal article-refereed


  • L-Beurling-Ahlfors extension
  • Sobolev extensions
  • Sobolev homeomorphisms


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