Large mass rigidity for a liquid drop model in 2D with kernels of finite moments
[Rigidité pour un modèle de goutte liquide en 2D avec des noyaux de moments finis et dans le régime des grandes masses]
Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 63-100.

Motivé par l’étude du modèle de goutte liquide de Gamow dans le régime des grandes masses, nous considérons un problème isopérimétrique dans lequel le périmètre classique P(E) est remplacé par P(E)-γP ε (E), où 0<γ<1 et P ε est une énergie non locale telle que P ε (E)P(E) lorsque ε tend vers zéro. Nous montrons que pour ε assez petit les minimiseurs à aire fixée sont les disques. Pour cela, nous établissons d’abord qu’en dimension 2, les minimiseurs sont convexes dès que ε est suffisamment petit. Ceci implique que le bord d’un minimiseur est une petite perturbation Lipschitz d’un cercle. Puis, par un argument à la Fuglede, nous prouvons (en dimension arbitraire n2) que si un minimiseur à volume fixé est une perturbation d’une boule au sens précédent, alors c’est une boule. Ce problème isopérimétrique est équivalent à une généralisation du modèle de goutte liquide pour le noyau atomique introduit par Gamow lorsque le potentiel répulsif non local est donné par un noyau suffisamment intégrable. Dans cette formulation, notre résultat principal indique que si le premier moment du noyau est inférieur à un seuil explicite, il existe une masse critique m 0 telle que les minimiseurs de masse prescrite m>m 0 sont les disques. Ceci contraste fortement avec le cas classique des noyaux de Riesz, où le problème n’admet pas de minimiseur au-delà d’une masse critique.

Motivated by Gamow’s liquid drop model in the large mass regime, we consider an isoperimetric problem in which the standard perimeter P(E) is replaced by P(E)-γP ε (E), with 0<γ<1 and P ε a nonlocal energy such that P ε (E)P(E) as ε vanishes. We prove that unit area minimizers are disks for ε>0 small enough. More precisely, we first show that in dimension 2, minimizers are necessarily convex, provided that ε is small enough. In turn, this implies that minimizers have nearly circular boundaries, that is, their boundary is a small Lipschitz perturbation of the circle. Then, using a Fuglede-type argument, we prove that (in arbitrary dimension n2) the unit ball in n is the unique unit-volume minimizer of the problem among centered nearly spherical sets. As a consequence, up to translations, the unit disk is the unique minimizer. This isoperimetric problem is equivalent to a generalization of the liquid drop model for the atomic nucleus introduced by Gamow, where the nonlocal repulsive potential is given by a radial, sufficiently integrable kernel. In that formulation, our main result states that if the first moment of the kernel is smaller than an explicit threshold, there exists a critical mass m 0 such that for any m>m 0 , the disk is the unique minimizer of area m up to translations. This is in sharp contrast with the usual case of Riesz kernels, where the problem does not admit minimizers above a critical mass.

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Accepté le :
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DOI : https://doi.org/10.5802/jep.178
Classification : 28A75,  49Q05,  49Q10,  49Q20,  52A10
Mots clés : Problèmes variationnels géométriques, problèmes isopérimétriques non locaux, périmètres non locaux, régularité, modèle de goutte liquide
Benoit Merlet 1 ; Marc Pegon 1

1. Univ. Lille, CNRS, Inria, UMR 8524 - Laboratoire Paul Painlevé F-59000 Lille, France
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Benoit Merlet; Marc Pegon. Large mass rigidity for a liquid drop model in $2$D with kernels of finite moments. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 63-100. doi : 10.5802/jep.178. https://jep.centre-mersenne.org/articles/10.5802/jep.178/

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