Large mass rigidity for a liquid drop model in $2$D with kernels of finite moments

Merlet, Benoit; Pegon, Marc

Journal de l’École polytechnique - Mathématiques, Volume 9 (2022), pp. 63-100.

Motivated by Gamow’s liquid drop model in the large mass regime, we consider an isoperimetric problem in which the standard perimeter $P\left(E\right)$ is replaced by $P\left(E\right)-\gamma P_{\epsilon }\left(E\right)$, with $0<\gamma <1$ and $P_{\epsilon }$ a nonlocal energy such that $P_{\epsilon }\left(E\right)\to P\left(E\right)$ as $\epsilon$ vanishes. We prove that unit area minimizers are disks for $\epsilon >0$ small enough. More precisely, we first show that in dimension $2$, minimizers are necessarily convex, provided that $\epsilon$ 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 $n\ge 2$) the unit ball in ${ℝ}^n$ 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.

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 $P\left(E\right)$ est remplacé par $P\left(E\right)-\gamma P_{\epsilon }\left(E\right)$, où $0<\gamma <1$ et $P_{\epsilon }$ est une énergie non locale telle que $P_{\epsilon }\left(E\right)\to P\left(E\right)$ lorsque $\epsilon$ tend vers zéro. Nous montrons que pour $\epsilon$ 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 $\epsilon$ 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 $n\ge 2$) 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.

Classification: 28A75, 49Q05, 49Q10, 49Q20, 52A10
Keywords: Geometric variational problems, nonlocal isoperimetric problems, nonlocal perimeters, regularity, liquid drop model
Mot clés : Problèmes variationnels géométriques, problèmes isopérimétriques non locaux, périmètres non locaux, régularité, modèle de goutte liquide

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