Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere

Antoine Detaille; Katarzyna Mazowiecka

doi:10.5802/crmath.648

Research article - Partial differential equations

Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere

Antoine Detaille ¹ ; Katarzyna Mazowiecka ²

¹ Universite Claude Bernard Lyon 1, ICJ UMR5208, CNRS, École Centrale de Lyon, INSA Lyon, Université Jean Monnet, 69622 Villeurbanne, France.
² Institute of Mathematics,University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1357-1364.

Abstract
Résumé

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.

Dans cette note, nous étudions la non-unicité pour les applications harmoniques de $B^{3}$ dans $𝕊^{2}$ . Nous montrons que toute application au bord peut être modifiée en une application au bord qui admet plusieurs minimiseurs de l’énergie de Dirichlet au moyen d’un petit changement $W^{1, p}$ pour $p < 2$ . Ceci renforce la conclusion d’une remarque du second auteur avec Strzelecki. L’ingrédient nouveau principal est une construction d’homotopie, qui répond à une variante simplifiée d’une question difficile concernant l’existence d’un contrôle sur la norme des homotopies entre applications $W^{1, p}$ .

Received: 2024-04-04
Accepted: 2024-05-03
Revised after acceptance: 2024-05-05
Published online: 2024-11-14

MR Zbl

DOI: 10.5802/crmath.648

Classification: 58E20, 46E35
Keywords: Harmonic maps, homotopy theory
Mots-clés : Applications harmoniques, théorie de l’homotopie

Author's affiliations:

Antoine Detaille ¹; Katarzyna Mazowiecka ²

¹ Universite Claude Bernard Lyon 1, ICJ UMR5208, CNRS, École Centrale de Lyon, INSA Lyon, Université Jean Monnet, 69622 Villeurbanne, France.
² Institute of Mathematics,University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Antoine Detaille and Katarzyna Mazowiecka},
     title = {Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1357--1364},
     publisher = {Acad\'emie des sciences, Paris},
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Antoine Detaille; Katarzyna Mazowiecka. Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1357-1364. doi : 10.5802/crmath.648. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.648/

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