We prove that half spaces are the only stable nonlocal s-minimal cones in ℝ 3, for s ϵ (0, 1) sufficiently close to 1. This is the first classification result of stable s-minimal cones in dimension higher than two. Its proof cannot rely on a compactness argument perturbing from s = 1 s=1. In fact, our proof gives a quantifiable value for the required closeness of s to 1. We use the geometric formula for the second variation of the fractional s-perimeter, which involves a squared nonlocal second fundamental form, as well as the recent BV estimates for stable nonlocal minimal sets.
Cabre X., Cinti E., Serra J. (2020). Stable s-minimal cones in ℝ 3 are flat for s ∼ 1. JOURNAL FÜR DIE REINE UND ANGEWANDTE MATHEMATIK, 764(0), 157-180 [10.1515/crelle-2019-0005].
Stable s-minimal cones in ℝ 3 are flat for s ∼ 1
Cabre X.Conceptualization
;Cinti E.;
2020
Abstract
We prove that half spaces are the only stable nonlocal s-minimal cones in ℝ 3, for s ϵ (0, 1) sufficiently close to 1. This is the first classification result of stable s-minimal cones in dimension higher than two. Its proof cannot rely on a compactness argument perturbing from s = 1 s=1. In fact, our proof gives a quantifiable value for the required closeness of s to 1. We use the geometric formula for the second variation of the fractional s-perimeter, which involves a squared nonlocal second fundamental form, as well as the recent BV estimates for stable nonlocal minimal sets.File | Dimensione | Formato | |
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