On the existence of degenerate solutions of the two-dimensional $H$-system

André  Guerra; Xavier Lamy; Konstantinos Zemas

doi:10.5802/crmath.731

Article de recherche - Équations aux dérivées partielles

On the existence of degenerate solutions of the two-dimensional

H

-system
[Sur l’existence de solutions dégénérées du

H

-système en dimension deux]

André Guerra ¹ ; Xavier Lamy ² ; Konstantinos Zemas ³

¹ Institute for Theoretical Studies, ETH Zürich, CLV, Clausiusstrasse 47, 8006 Zürich, Switzerland
² Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse Cedex 8, France
³ Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 271-281.

Résumé
Abstract

Nous considérons des solutions entières $ω \in {\dot{H}}^{1} (R^{2}; R^{3})$ du $H$ -système

Δ ω = 2 ω_{x} \land ω_{y},

appelées bulles. De manière surprenante, et contrairement à des conjectures exprimées dans la littérature, nous montrons que les bulles de degré au moins trois peuvent être dégénérées : le $H$ -système linéarisé autour d’une bulle peut admettre des solutions qui ne soient pas tangentes à la famille lisse des bulles. Nous donnons de plus une caractérisation algébrique complète des bulles dégénérées.

We consider entire solutions $ω \in {\dot{H}}^{1} (R^{2}; R^{3})$ of the $H$ -system

Δ ω = 2 ω_{x} \land ω_{y}

which we refer to as bubbles. Surprisingly, and contrary to conjectures raised in the literature, we find that bubbles with degree at least three can be degenerate: the linearized $H$ -system around a bubble can admit solutions that are not tangent to the smooth family of bubbles. We then give a complete algebraic characterization of degenerate bubbles.

Reçu le : 2024-10-14
Révisé le : 2025-02-05
Accepté le : 2025-02-05
Publié le : 2025-04-08

DOI : 10.5802/crmath.731

Classification : 35J47, 53A10
Keywords:

H

-system, harmonic maps, conformal maps
Mots-clés :

H

-système, applications harmoniques, applications conformes

Affiliations des auteurs :

André Guerra ¹ ; Xavier Lamy ² ; Konstantinos Zemas ³

¹ Institute for Theoretical Studies, ETH Zürich, CLV, Clausiusstrasse 47, 8006 Zürich, Switzerland
² Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse Cedex 8, France
³ Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Andr\'e  Guerra and Xavier Lamy and Konstantinos Zemas},
     title = {On the existence of degenerate solutions of the two-dimensional $H$-system},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {271--281},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.731},
     language = {en},
}

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André  Guerra; Xavier Lamy; Konstantinos Zemas. On the existence of degenerate solutions of the two-dimensional $H$-system. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 271-281. doi : 10.5802/crmath.731. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.731/

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