论文标题
均匀的$ c^{1，α} $ - 几乎适用于某些非本地扰动的最小化的规律性
Uniform $C^{1,α}$-regularity for almost-minimizers of some nonlocal perturbations of the perimeter
论文作者
论文摘要
在本文中，我们建立了一个$ c^{1，α} $ - 定期定理，用于几乎少量的$ \ Mathcal {f} _ {\ varepsilon，γ} = p-γp_ {\ varepsilon} $，其中$γ\ in（0,1）$ p _非局部能量将$ \ varepsilon $融合到外围。我们的定理提供了$ c^{1，α} $的标准 - 在边界点处的规律性，该点是统一的，因为参数$ \ varepsilon $转至$ 0 $。由于当$ \ varepsilon $很小时，能量中的两个术语是相同的顺序，因此我们在这里考虑的非本地互动要比大多数相关工作中所考虑的要强得多。由于我们的规律性结果，我们可以获得$ \ varepsilon $的$ \ Mathcal {f} _ {\ varepsilon，γ} $的$ \ varepsilon $。对于小$ \ varepsilon $，这个最小化问题对应于游戏类型问题的巨大质量政权，其中非本地排斥术语由可集成的$ g $给出，而无限的衰减很快。
In this paper, we establish a $C^{1,α}$-regularity theorem for almost-minimizers of the functional $\mathcal{F}_{\varepsilon,γ}=P-γP_{\varepsilon}$, where $γ\in(0,1)$ and $P_{\varepsilon}$ is a nonlocal energy converging to the perimeter as $\varepsilon$ vanishes. Our theorem provides a criterion for $C^{1,α}$-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 $\mathcal{F}_{\varepsilon,γ}$ 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.
