Uniform \(C^{1,\alpha }\)-Regularity for Almost-Minimizers of Some Nonlocal Perturbations of the Perimeter

In this paper, we establish a \(C^{1,\alpha }\) $C^{1,\alpha }$ -regularity theorem for almost-minimizers of the functional \(\mathcal {F}_{\varepsilon ,\gamma }=P-\gamma P_{\varepsilon }\) ${\mathcal{F}}_{\epsilon ,\gamma }=P-\gamma P_{\epsilon }$ , where \(\gamma \in (0,1)\) $\gamma \in (0,1)$ and \(P_{\varepsilon }\) $P_{\epsilon }$ is a nonlocal energy converging to the perimeter as \(\varepsilon \) $\epsilon$ vanishes. Our theorem provides a criterion for \(C^{1,\alpha }\) $C^{1,\alpha }$ -regularity at a point of the boundary which is uniform as the parameter \(\varepsilon \) $\epsilon$ goes to 0. Since the two terms in the energy are of the same order when \(\varepsilon \) $\epsilon$ is small, we are considering here much stronger nonlocal interactions than those considered in most related works. As a consequence of our regularity result, we obtain that, for \(\varepsilon \) $\epsilon$ small enough, volume-constrained minimizers of \(\mathcal {F}_{\varepsilon ,\gamma }\) ${\mathcal{F}}_{\epsilon ,\gamma }$ are balls. For small \(\varepsilon \) $\epsilon$ , this minimization problem corresponds to the large mass regime for a Gamow-type problem where the nonlocal repulsive term is given by an integrable kernel G with sufficiently fast decay at infinity.