Bi-Sobolev Extensions

Aleksis Koski; Jani Onninen

doi:10.1007/s12220-023-01363-1

Bi-Sobolev Extensions

Aleksis Koski, Jani Onninen

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We give a full characterization of circle homeomorphisms which admit a homeomorphic extension to the unit disk with finite bi-Sobolev norm. As a special case, a bi-conformal variant of the famous Beurling–Ahlfors extension theorem is obtained. Furthermore we show that the existing extension techniques such as applying either the harmonic or the Beurling–Ahlfors operator work poorly in the degenerated setting. This also gives an affirmative answer to a question of Karafyllia and Ntalampekos.

Original language	English (US)
Article number	301
Journal	Journal of Geometric Analysis
Volume	33
Issue number	9
DOIs	https://doi.org/10.1007/s12220-023-01363-1
State	Published - Sep 2023

Keywords

Beurling–Ahlfors extension
Harmonic extension
Quasiconformal mapping and mapping of finite distortion
Sobolev extensions
Sobolev homeomorphisms

ASJC Scopus subject areas

Geometry and Topology

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10.1007/s12220-023-01363-1

Cite this

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title = "Bi-Sobolev Extensions",

abstract = "We give a full characterization of circle homeomorphisms which admit a homeomorphic extension to the unit disk with finite bi-Sobolev norm. As a special case, a bi-conformal variant of the famous Beurling–Ahlfors extension theorem is obtained. Furthermore we show that the existing extension techniques such as applying either the harmonic or the Beurling–Ahlfors operator work poorly in the degenerated setting. This also gives an affirmative answer to a question of Karafyllia and Ntalampekos.",

keywords = "Beurling–Ahlfors extension, Harmonic extension, Quasiconformal mapping and mapping of finite distortion, Sobolev extensions, Sobolev homeomorphisms",

author = "Aleksis Koski and Jani Onninen",

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AU - Onninen, Jani

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AB - We give a full characterization of circle homeomorphisms which admit a homeomorphic extension to the unit disk with finite bi-Sobolev norm. As a special case, a bi-conformal variant of the famous Beurling–Ahlfors extension theorem is obtained. Furthermore we show that the existing extension techniques such as applying either the harmonic or the Beurling–Ahlfors operator work poorly in the degenerated setting. This also gives an affirmative answer to a question of Karafyllia and Ntalampekos.

KW - Beurling–Ahlfors extension

KW - Harmonic extension

KW - Quasiconformal mapping and mapping of finite distortion

KW - Sobolev extensions

KW - Sobolev homeomorphisms

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Bi-Sobolev Extensions

Abstract

Keywords

ASJC Scopus subject areas

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