Minimizing solutions of degenerate vector Allen–Cahn equations with three wells in $\mathbb{R}^2$

Lia Bronsard; Étienne Sandier; Peter Sternberg

doi:10.5802/crmath.818

Article de recherche - Equations aux dérivées partielles

Minimizing solutions of degenerate vector Allen–Cahn equations with three wells in $\mathbb{R}^2$
[Solutions minimisantes d’équations vectorielles dégénérées d’Allen–Cahn en $\mathbb{R}^2$ avec trois puits]

Lia Bronsard ¹ ; Étienne Sandier ² ; Peter Sternberg ³

¹Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4L8, Canada
²Université Paris-Est Créteil, France
³Department of Mathematics, Indiana University, Bloomington, USA

Comptes Rendus. Mathématique, Volume 364 (2026), pp. 137-158

Abstract (VO)
Résumé

We characterize all minimizers of the vector-valued Allen–Cahn equation in $\mathbb{R}^2$ under the assumption that the potential $W$ has three wells and that the associated degenerate metric does not satisfy the usual strict triangle inequality. These minimizers depend on one variable only in a suitable coordinate system. In particular, we show that no minimizing solution to $\Delta u=\nabla W(u)$ on $\mathbb{R}^2$ can approach the three distinct values of the potential wells.

Nous caractérisons tous les minimiseurs de l’équation vectorielle d’Allen–Cahn dans $\mathbb{R}^2$ sous l’hypothèse que le potentiel $W$ a trois puits et que la métrique dégénérée associée ne satisfait pas l’inégalité triangulaire stricte usuelle. Ces minimiseurs dépendent d’une variable seulement dans un système de coordonnées adapté. En particulier, nous montrons qu’aucune solution minimisante de $\Delta u=\nabla W(u)$ sur $\mathbb{R}^2$ ne peut approcher les trois valeurs distinctes des puits du potentiel.

Reçu le : 2025-09-08
Révisé le : 2026-01-14
Accepté le : 2026-01-19
Publié le : 2026-03-13

DOI : 10.5802/crmath.818

Classification : 35J47, 35B08
Keywords: Elliptic systems, phase transitions, optimal partitions, entire solutions
Mots-clés : Systèmes elliptiques, transitions de phase, partitions optimales, solutions entières

Note : Article soumis sur invitation

Affiliations des auteurs :

Lia Bronsard ¹ ; Étienne Sandier ² ; Peter Sternberg ³

¹ Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4L8, Canada
² Université Paris-Est Créteil, France
³ Department of Mathematics, Indiana University, Bloomington, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Lia Bronsard and \'Etienne Sandier and Peter Sternberg},
     title = {Minimizing solutions of degenerate vector {Allen{\textendash}Cahn} equations with three wells in $\mathbb{R}^2$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {137--158},
     year = {2026},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {364},
     doi = {10.5802/crmath.818},
     language = {en},
}

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Lia Bronsard; Étienne Sandier; Peter Sternberg. Minimizing solutions of degenerate vector Allen–Cahn equations with three wells in $\mathbb{R}^2$. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 137-158. doi: 10.5802/crmath.818

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