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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2405.18037 |
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| _version_ | 1866909681975296000 |
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| author | Martinazzi, Luca Hyder, Ali |
| author_facet | Martinazzi, Luca Hyder, Ali |
| contents | Given a half-harmonic map $u\in \dot H^{\frac{1}{2},2}(\mathbb{R},\mathbb{S}^1)$ minimizing the fractional Dirichlet energy under Dirichlet boundary conditions in $\mathbb{R}\setminus I$, we show the existence of a second half-harmonic map, minimizing the fractional Dirichlet energy in a different homotopy class. This is based on the study of the degree of fractional Sobolev maps and a sharp estimate à la Brezis-Coron. We give examples showing that it is in general not possible to minimize in every homotopy class and show a contrast with the 2-dimensional case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_18037 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | One-dimensional half-harmonic maps into the circle and their degree Martinazzi, Luca Hyder, Ali Analysis of PDEs Algebraic Topology 35J60 Given a half-harmonic map $u\in \dot H^{\frac{1}{2},2}(\mathbb{R},\mathbb{S}^1)$ minimizing the fractional Dirichlet energy under Dirichlet boundary conditions in $\mathbb{R}\setminus I$, we show the existence of a second half-harmonic map, minimizing the fractional Dirichlet energy in a different homotopy class. This is based on the study of the degree of fractional Sobolev maps and a sharp estimate à la Brezis-Coron. We give examples showing that it is in general not possible to minimize in every homotopy class and show a contrast with the 2-dimensional case. |
| title | One-dimensional half-harmonic maps into the circle and their degree |
| topic | Analysis of PDEs Algebraic Topology 35J60 |
| url | https://arxiv.org/abs/2405.18037 |