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

Claudia Bucur, Luca Lombardini, Enrico Valdinoci

Research output: Contribution to journalArticle

5 Citations (Scopus)

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 languageEnglish
Pages (from-to)655-703
Number of pages49
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume36
Issue number3
Early online date4 Oct 2018
DOIs
Publication statusPublished - 1 May 2019

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Minimal surface
Fractional
Mean Curvature
Minimal Set
Connected Set
Sign Change
Open set
Asymptotic Behavior
Classify
Infinity
Vary

Cite this

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

In: Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, Vol. 36, No. 3, 01.05.2019, p. 655-703.

Research output: Contribution to journalArticle

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

AU - Bucur, Claudia

AU - Lombardini, Luca

AU - Valdinoci, Enrico

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