We establish quantitative properties of minimizers and stable sets for nonlocal interaction functionals, including the $s$-fractional perimeter as a particular case. On the one hand, we establish universal $BV$-estimates in every dimension $n\ge 2$ for stable sets. Namely, we prove that any stable set in $B_1$ has finite classical perimeter in $B_{1/2}$, with a universal bound. This nonlocal result is new even in the case of $s$-perimeters and its local counterpart (for classical stable minimal surfaces) was known only for simply connected two-dimensional surfaces immersed in $\mathbb R^3$. On the other hand, we prove quantitative flatness estimates for minimizers and stable sets in low dimensions $n=2,3$. More precisely, we show that a stable set in $B_R$, with $R$ large, is very close in measure to being a half space in $B_1$ ---with a quantitative estimate on the measure of the symmetric difference. As a byproduct, we obtain new classification results for stable sets in the whole plane.

中文翻译：

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稳定非局部最小表面的定量平坦度结果和 $BV$ 估计

我们为非局部相互作用泛函建立了最小化器和稳定集的定量特性，包括作为特殊情况的 $s$-分数周长。一方面，我们在每个维度 $n\ge 2$ 中为稳定集建立通用的 $BV$-估计。也就是说，我们证明 $B_1$ 中的任何稳定集在 $B_{1/2}$ 中都具有有限的经典周长，并具有普遍界限。即使在 $s$-perimeters 的情况下，这种非局部结果也是新的，并且它的局部对应物（对于经典稳定的最小表面）仅因沉浸在 $\mathbb R^3$ 中的简单连接的二维表面而为人所知。另一方面，我们证明了低维 $n=2,3$ 中最小化器和稳定集的定量平坦度估计。更准确地说，我们证明了 $B_R$ 中的一个稳定集，其中 $R$ 很大，在度量上非常接近 $B_1$ 中的一半空间——对对称差异的度量进行定量估计。作为副产品，我们获得了整个平面稳定集的新分类结果。