Journal of Differential Geometry

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

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 $n \geqslant 2$ 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.

Article information

Source
J. Differential Geom., Volume 112, Number 3 (2019), 447-504.

Dates
First available in Project Euclid: 16 July 2019

https://projecteuclid.org/euclid.jdg/1563242471

Digital Object Identifier
doi:10.4310/jdg/1563242471

Mathematical Reviews number (MathSciNet)
MR3981295

Zentralblatt MATH identifier
1420.53014

Citation

Cinti, Eleonora; Serra, Joaquim; Valdinoci, Enrico. Quantitative flatness results and $BV$-estimates for stable nonlocal minimal surfaces. J. Differential Geom. 112 (2019), no. 3, 447--504. doi:10.4310/jdg/1563242471. https://projecteuclid.org/euclid.jdg/1563242471