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

André  Guerra; Xavier Lamy; Konstantinos Zemas

doi:10.5802/crmath.731

Research article - Partial differential equations

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

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.

Abstract (Original language)
Résumé

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

\begin{equation*} \Delta \omega =2\omega _x\wedge \omega _y\, \end{equation*}

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.

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

\begin{equation*} \Delta \omega =2\omega _x\wedge \omega _y, \end{equation*}

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.

Received: 2024-10-14
Revised: 2025-02-05
Accepted: 2025-02-05
Published online: 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

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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},
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     publisher = {Acad\'emie des sciences, Paris},
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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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