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Motivated by Gamow's liquid drop model in the large mass regime, we consideran isoperimetric problem in which the standard perimeter $P(E)$ is replaced by$P(E)-\gamma P_\varepsilon(E)$, with $0<\gamma<1$ and $P_\varepsilon$ anonlocal 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 necessarilyconvex, provided that $\varepsilon$ is small enough. In turn, this implies thatminimizers have nearly circular boundaries, that is, their boundary is a smallLipschitz perturbation of the circle. Then, using a Fuglede-type argument, weprove 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 nearlyspherical sets. As a consequence, up to translations, the unit disk is theunique minimizer. This isoperimetric problem is equivalent to a generalizationof the liquid drop model for the atomic nucleus introduced by Gamow, where thenonlocal repulsive potential is given by a radial, sufficiently integrablekernel. In that formulation, our main result states that if the first moment ofthe 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 Rieszkernels, where the problem does not admit minimizers above a critical mass.

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