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- Volume 362 (2024)

- p. 1357-1364

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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 .

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Antoine Detaille 1 ; Katarzyna Mazowiecka 2

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