Quantitative flatness results and $BV$-estimates for stable nonlocal minimal surfaces

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 2 for stable sets. Namely, we prove that any stable set in B₁ has finite classical perimeter in B1/2, 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 R³. 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 BR, with R large, is very close in measure to being a half space in B₁ – 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.