Uniform C1,α-Regularity for Almost-Minimizers of Some Nonlocal Perturbations of the Perimeter

In this paper, we establish a C^1,α-regularity theorem for almost-minimizers of the functional F_ε,γ=P-γP_ε, where γ∈(0,1) and P_ε is a nonlocal energy converging to the perimeter as ε vanishes. Our theorem provides a criterion for C^1,α-regularity at a point of the boundary which is uniform as the parameter ε goes to 0. Since the two terms in the energy are of the same order when ε 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 ε small enough, volume-constrained minimizers of F_ε,γ are balls. For small ε, 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.