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

In this paper, we establish a \(C^{1,\alpha }\)-regularity theorem for almost-minimizers of the functional \(\mathcal {F}_{\varepsilon ,\gamma }=P-\gamma P_{\varepsilon }\), where \(\gamma \in (0,1)\) and \(P_{\varepsilon }\) is a nonlocal energy converging to the perimeter as \(\varepsilon \) vanishes. Our theorem provides a criterion for \(C^{1,\alpha }\)-regularity at a point of the boundary which is uniform as the parameter \(\varepsilon \) goes to 0. Since the two terms in the energy are of the same order when \(\varepsilon \) 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 \(\varepsilon \) small enough, volume-constrained minimizers of \(\mathcal {F}_{\varepsilon ,\gamma }\) are balls. For small \(\varepsilon \), 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.