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

Eleonora Cinti, Joaquim Serra, Enrico Valdinoci

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36 Citations (Scopus)
89 Downloads (Pure)

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 >= 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.

Original languageEnglish
Pages (from-to)447-504
Number of pages58
JournalJournal of Differential Geometry
Volume112
Issue number3
DOIs
Publication statusPublished - Jul 2019

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