We prove that unit area minimizers are disks for an isoperimetric problem in which the standard perimeter $P(E)$ is replaced by $P(E)−\gamma P_{\varepsilon}(E)$, where $\gamma\in(0,1)$ and $P_{\varepsilon}$ is a nonlocal energy such that $P_{\varepsilon}(E) \to P(E)$ as $\varepsilon$ vanishes.
More precisely, we show that in dimension 2, connected 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 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 so-called liquid drop model for the atomic nucleus introduced by Gamow, where the nonlocal repulsive potential is given by a radial kernel sufficiently integrable kernel. In that formulation our result states that above a critical mass $m_0$, the disk of area $m > m_0$ is the unique minimizer of area $m$, up to translations.