Published Paper

Inserted: 8 jun 2021
Last Updated: 12 nov 2021

Journal: Journal de l’École polytechnique — Mathématiques
Volume: 9
Pages: 63-100
Year: 2021
Doi: https://doi.org/10.5802/jep.178

Abstract:

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 $P(E)-\gamma P_\varepsilon(E)$, with $0<\gamma<1$ and $P_\varepsilon$ a nonlocal energy such that $P_\varepsilon(E)\to P(E)$ as $\varepsilon$ vanishes. We prove that unit area minimizers are disks for $\varepsilon>0$ small enough. More precisely, we first show that in dimension $2$, minimizers are necessarily convex, provided that $\varepsilon$ 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 $n\geq 2$) the unit ball in $\mathbb{R}^n$ 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.

Keywords: nonlocal perimeters, Geometric variational problems problems, Nonlocal isoperimetric problems, Liquid drop model