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July 2019 Quantitative flatness results and $BV$-estimates for stable nonlocal minimal surfaces
Eleonora Cinti, Joaquim Serra, Enrico Valdinoci
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J. Differential Geom. 112(3): 447-504 (July 2019). DOI: 10.4310/jdg/1563242471


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.


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Eleonora Cinti. Joaquim Serra. Enrico Valdinoci. "Quantitative flatness results and $BV$-estimates for stable nonlocal minimal surfaces." J. Differential Geom. 112 (3) 447 - 504, July 2019.


Received: 3 June 2016; Published: July 2019
First available in Project Euclid: 16 July 2019

zbMATH: 1420.53014
MathSciNet: MR3981295
Digital Object Identifier: 10.4310/jdg/1563242471

Primary: 35R11, 49Q05, 53A10

Rights: Copyright © 2019 Lehigh University


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Vol.112 • No. 3 • July 2019
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