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Stable s-minimal cones in ℝ3 are flat for s ~ 1

  • Xavier Cabré ORCID logo EMAIL logo , Eleonora Cinti and Joaquim Serra

Abstract

We prove that half spaces are the only stable nonlocal s-minimal cones in 3, for s(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. 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.

Funding statement: The authors have been supported by MINECO grants MTM2014-52402-C3-1-P and MTM2017-84214-C2-1-P, and are part of the Catalan research group 2014 SGR 1083. The first author is member of the Barcelona Graduate School of Mathematics.

References

[1] B. N. Barrios, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 (2014), no. 3, 609–639. 10.2422/2036-2145.201202_007Search in Google Scholar

[2] J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, Optimal control and partial differential equations, IOS, Amsterdam (2001), 439–455. Search in Google Scholar

[3] J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for Ws,p when s1 and applications, J. Anal. Math. 87 (2002), 77–101. 10.1007/BF02868470Search in Google Scholar

[4] X. Cabré, E. Cinti and J. Serra, Stable solutions to the fractional Allen–Cahn equation, preprint. Search in Google Scholar

[5] L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), no. 9, 1111–1144. 10.1002/cpa.20331Search in Google Scholar

[6] L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math. 248 (2013), 843–871. 10.1016/j.aim.2013.08.007Search in Google Scholar

[7] E. Cinti, J. Serra and E. Valdinoci, Quantitative flatness results and BV-estimates for stable nonlocal minimal surfaces, preprint (2016), https://arxiv.org/abs/1602.00540; to appear in J. Differential Geom. 10.4310/jdg/1563242471Search in Google Scholar

[8] J. Dávila, M. del Pino and J. Wei, Nonlocal s-minimal surfaces and Lawson cones, J. Differential Geom. 109 (2018), no. 1, 111–175. 10.4310/jdg/1525399218Search in Google Scholar

[9] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. 10.1016/j.bulsci.2011.12.004Search in Google Scholar

[10] S. Dipierro, J. Serra and E. Valdinoci, Improvement of flatness for nonlocal phase transitions, preprint (2016), https://arxiv.org/abs/1611.10105, to appear in Amer. J. Math. 10.1353/ajm.2020.0032Search in Google Scholar

[11] A. Figalli, N. Fusco, F. Maggi, V. Millot and M. Morini, Isoperimetry and stability properties of balls with respect to nonlocal energies, Comm. Math. Phys. 336 (2015), no. 1, 441–507. 10.1007/s00220-014-2244-1Search in Google Scholar

[12] A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces, J. reine angew. Math. 729 (2017), 263–273. 10.1515/crelle-2015-0006Search in Google Scholar

[13] R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal. 255 (2008), no. 12, 3407–3430. 10.1016/j.jfa.2008.05.015Search in Google Scholar

[14] O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations 48 (2013), no. 1–2, 33–39. 10.1007/s00526-012-0539-7Search in Google Scholar

[15] O. Savin and E. Valdinoci, Some monotonicity results for minimizers in the calculus of variations, J. Funct. Anal. 264 (2013), no. 10, 2469–2496. 10.1016/j.jfa.2013.02.005Search in Google Scholar

[16] J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. 10.2307/1970556Search in Google Scholar

Received: 2017-11-08
Revised: 2019-03-08
Published Online: 2019-04-16
Published in Print: 2020-07-01

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