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

TL;DR

This paper proves universal BV-estimates for stable nonlocal minimal surfaces in all dimensions and provides quantitative flatness results in low dimensions, advancing understanding of their geometric properties.

Contribution

It introduces new BV-estimates for stable sets and establishes quantitative flatness results for minimizers in low dimensions, extending classical minimal surface theory to nonlocal cases.

Findings

Stable sets in $B_1$ have finite classical perimeter with a universal bound.

Stable sets in low dimensions are close to half spaces in measure.

New classification results for stable sets in the plane.

Abstract

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.