Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere
In this note, we study non-uniqueness for minimizing harmonic maps from $B^3$ to $§^2$. 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 au...
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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.648/ |
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| author | Detaille, Antoine Mazowiecka, Katarzyna |
| author_facet | Detaille, Antoine Mazowiecka, Katarzyna |
| author_sort | Detaille, Antoine |
| collection | DOAJ |
| description | In this note, we study non-uniqueness for minimizing harmonic maps from $B^3$ to $§^2$. 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 | Article |
| id | doaj-art-01e42eff0a2645c8a2a9a8a1d156e49c |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-01e42eff0a2645c8a2a9a8a1d156e49c2025-02-07T11:23:50ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G111357136410.5802/crmath.64810.5802/crmath.648Generic non-uniqueness of minimizing harmonic maps from a ball to a sphereDetaille, Antoine0Mazowiecka, Katarzyna1Universite Claude Bernard Lyon 1, ICJ UMR5208, CNRS, École Centrale de Lyon, INSA Lyon, Université Jean Monnet, 69622 Villeurbanne, France.Institute of Mathematics,University of Warsaw, Banacha 2, 02-097 Warszawa, PolandIn this note, we study non-uniqueness for minimizing harmonic maps from $B^3$ to $§^2$. 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.648/Harmonic mapshomotopy theory |
| spellingShingle | Detaille, Antoine Mazowiecka, Katarzyna Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere Comptes Rendus. Mathématique Harmonic maps homotopy theory |
| title | Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere |
| title_full | Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere |
| title_fullStr | Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere |
| title_full_unstemmed | Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere |
| title_short | Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere |
| title_sort | generic non uniqueness of minimizing harmonic maps from a ball to a sphere |
| topic | Harmonic maps homotopy theory |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.648/ |
| work_keys_str_mv | AT detailleantoine genericnonuniquenessofminimizingharmonicmapsfromaballtoasphere AT mazowieckakatarzyna genericnonuniquenessofminimizingharmonicmapsfromaballtoasphere |