### Abstract

In this paper, we consider the asymptotic behavior of the fractional mean curvature when s→0 ^{+} . Moreover, we deal with the behavior of s-minimal surfaces when the fractional parameter s∈(0,1) is small, in a bounded and connected open set with C ^{2} boundary Ω⊂R ^{n} . We classify the behavior of s-minimal surfaces with respect to the fixed exterior data (i.e. the s-minimal set fixed outside of Ω). So, for s small and depending on the data at infinity, the s-minimal set can be either empty in Ω fill all Ω or possibly develop a wildly oscillating boundary. Also, we prove the continuity of the fractional mean curvature in all variables, for s∈[0,1]. Using this, we see that as the parameter s varies, the fractional mean curvature may change sign.

Original language | English |
---|---|

Pages (from-to) | 655-703 |

Number of pages | 49 |

Journal | Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire |

Volume | 36 |

Issue number | 3 |

Early online date | 4 Oct 2018 |

DOIs | |

Publication status | Published - 1 May 2019 |

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*Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire*,

*36*(3), 655-703. https://doi.org/10.1016/j.anihpc.2018.08.003

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*Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire*, vol. 36, no. 3, pp. 655-703. https://doi.org/10.1016/j.anihpc.2018.08.003

**Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter.** / Bucur, Claudia; Lombardini, Luca; Valdinoci, Enrico.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter

AU - Bucur, Claudia

AU - Lombardini, Luca

AU - Valdinoci, Enrico

PY - 2019/5/1

Y1 - 2019/5/1

N2 - In this paper, we consider the asymptotic behavior of the fractional mean curvature when s→0 + . Moreover, we deal with the behavior of s-minimal surfaces when the fractional parameter s∈(0,1) is small, in a bounded and connected open set with C 2 boundary Ω⊂R n . We classify the behavior of s-minimal surfaces with respect to the fixed exterior data (i.e. the s-minimal set fixed outside of Ω). So, for s small and depending on the data at infinity, the s-minimal set can be either empty in Ω fill all Ω or possibly develop a wildly oscillating boundary. Also, we prove the continuity of the fractional mean curvature in all variables, for s∈[0,1]. Using this, we see that as the parameter s varies, the fractional mean curvature may change sign.

AB - In this paper, we consider the asymptotic behavior of the fractional mean curvature when s→0 + . Moreover, we deal with the behavior of s-minimal surfaces when the fractional parameter s∈(0,1) is small, in a bounded and connected open set with C 2 boundary Ω⊂R n . We classify the behavior of s-minimal surfaces with respect to the fixed exterior data (i.e. the s-minimal set fixed outside of Ω). So, for s small and depending on the data at infinity, the s-minimal set can be either empty in Ω fill all Ω or possibly develop a wildly oscillating boundary. Also, we prove the continuity of the fractional mean curvature in all variables, for s∈[0,1]. Using this, we see that as the parameter s varies, the fractional mean curvature may change sign.

KW - Loss of regularity

KW - Nonlocal minimal surfaces

KW - Stickiness phenomena

KW - Strongly nonlocal regime

UR - http://www.scopus.com/inward/record.url?scp=85054457986&partnerID=8YFLogxK

U2 - 10.1016/j.anihpc.2018.08.003

DO - 10.1016/j.anihpc.2018.08.003

M3 - Article

VL - 36

SP - 655

EP - 703

JO - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

SN - 0294-1449

IS - 3

ER -