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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.12662 |
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| _version_ | 1866914328580456448 |
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| author | Detaille, Antoine Mazowiecka, Katarzyna |
| author_facet | Detaille, Antoine Mazowiecka, Katarzyna |
| contents | In this note, we study non-uniqueness for minimizing harmonic maps from $B^3$ to $\mathbb{S}^2$. We show that every boundary map can be modified to a boundary map that admits multiple minimizers of the Dirichlet energy by a small $W^{1,p}$-change for $p<2$. This strengthens a remark by the second-named author and Strzelecki. The main novel ingredient is a homotopy construction, which is the answer to an easier variant of a challenging question regarding the existence of a norm control for homotopies between $ W^{1,p} $ maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_12662 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere Detaille, Antoine Mazowiecka, Katarzyna Analysis of PDEs 58E20, 46E35 In this note, we study non-uniqueness for minimizing harmonic maps from $B^3$ to $\mathbb{S}^2$. We show that every boundary map can be modified to a boundary map that admits multiple minimizers of the Dirichlet energy by a small $W^{1,p}$-change for $p<2$. This strengthens a remark by the second-named author and Strzelecki. The main novel ingredient is a homotopy construction, which is the answer to an easier variant of a challenging question regarding the existence of a norm control for homotopies between $ W^{1,p} $ maps. |
| title | Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere |
| topic | Analysis of PDEs 58E20, 46E35 |
| url | https://arxiv.org/abs/2403.12662 |