Uniform $C^{1,\alpha}$-regularity for almost-minimizers of some nonlocal perturbations of the perimeter

TL;DR

This paper proves $C^{1,eta}$-regularity for almost-minimizers of a nonlocal perturbation of the perimeter functional, leading to the conclusion that small-volume minimizers are spherical, with results uniform as the nonlocal parameter vanishes.

Contribution

It establishes a uniform $C^{1,eta}$-regularity criterion for almost-minimizers of a nonlocal perimeter perturbation, extending regularity results to stronger nonlocal interactions.

Findings

Regularity criterion holds uniformly as nonlocal parameter tends to zero.

Small-volume minimizers are spherical (balls).

Results apply to large mass regimes in Gamow-type problems.

Abstract

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 $G$ with sufficiently fast decay at infinity.