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On Courant and Pleijel theorems for sub-Riemannian Laplacians
[Sur les théorèmes de Courant et de Pleijel pour des laplaciens sous-riemanniens]
Rupert L. Frank1, 2, 3 ; Bernard Helffer4
1 Mathematisches Institut, Ludwig-Maximilians Universität München, Theresienstr. 39, 80333 München, Germany
2 & Munich Center for Quantum Science and Technology, Schellingstr. 4, 80799 München, Germany
3 & Mathematics 253-37, Caltech, Pasadena, CA 91125, USA
4 Laboratoire de Mathématiques Jean Leray, CNRS, Nantes Université, F44000. Nantes, France
Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1083-1160.

We are interested in the number of nodal domains of eigenfunctions of sub-Laplacians on sub-Riemannian manifolds. Specifically, we investigate the validity of Pleijel’s theorem, which states that, as soon as the dimension is strictly larger than $1$, the number of nodal domains of an eigenfunction corresponding to the $k$-th eigenvalue is strictly (and uniformly, in a certain sense) smaller than $k$ for large $k$. In the first part of this paper we reduce this question from the case of general sub-Riemannian manifolds to that of nilpotent groups. In the second part, we analyze in detail the case where the nilpotent group is a Heisenberg group times a Euclidean space. Along the way, we improve known bounds on the optimal constants in the Faber–Krahn and isoperimetric inequalities on these groups.

Nous nous intéressons au comptage des ensembles nodaux des fonctions propres des sous-laplaciens dans le cadre des variétés sous-riemanniennes. Plus précisément, nous discutons la validité du théorème de Pleijel qui énonce qu’en dimension supérieure à $1$, le nombre d’ensembles nodaux d’une fonction propre associée à la $k$-ième valeur propre est strictement plus petit que $k$ pour $k$ assez grand. Dans la première partie de cet article, nous ramenons le cas général de cette question dans le cas sous-riemannien au cas des groupes nilpotents. Dans la deuxième partie, nous analysons en détail le cas où le groupe nilpotent est le produit du groupe de Heisenberg par un espace euclidien. En chemin, nous améliorons pour ces groupes certaines bornes connues des constantes optimales pour les inégalités isopérimétriques ou de Faber-Krahn.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.307
Classification : 35P15, 35P20, 53C17, 58C40
Keywords: Nodal domains, Pleijel theorem, Courant theorem, sub-Laplacians, Faber-Krahn inequality, Nilpotent groups, Heisenberg group, Weyl formula
Mots-clés : Ensembles nodaux, théorème de Courant, théorème de Pleijel, sous-laplaciens, inégalité de Faber-Krahn, groupes nilpotents, groupe de Heisenberg, formule de Weyl

Rupert L. Frank 1, 2, 3 ; Bernard Helffer 4

1 Mathematisches Institut, Ludwig-Maximilians Universität München, Theresienstr. 39, 80333 München, Germany
2 & Munich Center for Quantum Science and Technology, Schellingstr. 4, 80799 München, Germany
3 & Mathematics 253-37, Caltech, Pasadena, CA 91125, USA
4 Laboratoire de Mathématiques Jean Leray, CNRS, Nantes Université, F44000. Nantes, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Rupert L. Frank; Bernard Helffer. On Courant and Pleijel theorems for sub-Riemannian Laplacians. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1083-1160. doi : 10.5802/jep.307. https://jep.centre-mersenne.org/articles/10.5802/jep.307/

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