Quantitative flatness results and $BV$-estimates for stable nonlocal minimal surfaces
- Cinti, Eleonora
- Serra, Joaquim
- Valdinoci, Enrico
2010 Mathematics Subject Classification
- 49Q05 35R11 53A10
- Nonlocal minimal surfaces, existence and regularity results
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 $nge 2$ 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 $R^3$. 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.