Homeomorphic Sobolev extensions of parametrizations of Jordan curves
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913746427838464 |
|---|---|
| author | Bouchala, Ondrěj Jääskeläinen, Jarmo Koskela, Pekka Xu, Haiqing Zhou, Xilin |
| author_facet | Bouchala, Ondrěj Jääskeläinen, Jarmo Koskela, Pekka Xu, Haiqing Zhou, Xilin |
| contents | Each homeomorphic parametrization of a Jordan curve via the unit circle extends to a homeomorphism of the entire plane. It is a natural question to ask if such a homeomorphism can be chosen so as to have some Sobolev regularity. This prompts the simplified question: for a homeomorphic embedding of the unit circle into the plane, when can we find a homeomorphism from the unit disk that has the same boundary values and integrable first-order distributional derivatives?
We give the optimal geometric criterion for the interior Jordan domain so that there exists a Sobolev homeomorphic extension for any homeomorphic parametrization of the Jordan curve. The problem is partially motivated by trying to understand which boundary values can correspond to deformations of finite energy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00506 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homeomorphic Sobolev extensions of parametrizations of Jordan curves Bouchala, Ondrěj Jääskeläinen, Jarmo Koskela, Pekka Xu, Haiqing Zhou, Xilin Complex Variables 46E35 (Primary), 30C62, 58E20 (Secondary) Each homeomorphic parametrization of a Jordan curve via the unit circle extends to a homeomorphism of the entire plane. It is a natural question to ask if such a homeomorphism can be chosen so as to have some Sobolev regularity. This prompts the simplified question: for a homeomorphic embedding of the unit circle into the plane, when can we find a homeomorphism from the unit disk that has the same boundary values and integrable first-order distributional derivatives? We give the optimal geometric criterion for the interior Jordan domain so that there exists a Sobolev homeomorphic extension for any homeomorphic parametrization of the Jordan curve. The problem is partially motivated by trying to understand which boundary values can correspond to deformations of finite energy. |
| title | Homeomorphic Sobolev extensions of parametrizations of Jordan curves |
| topic | Complex Variables 46E35 (Primary), 30C62, 58E20 (Secondary) |
| url | https://arxiv.org/abs/2408.00506 |