Calculus of Variations and Geometric Measure Theory

E. Cinti - J. Serra - E. Valdinoci

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

created by cinti on 01 Feb 2016
modified on 26 Oct 2017


Accepted Paper

Inserted: 1 feb 2016
Last Updated: 26 oct 2017

Journal: J. Diff. Geom.
Year: 2016


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.