Geodesic
Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter
Bucur, Claudia1 ; Lombardini, Luca2, 3, 4 ; Valdinoci, Enrico2, 4, 5
1 School of Mathematics and Statistics, The University of Melbourne, 813 Swanston Street, Parkville, VIC 3010, Australia
2 Dipartimento di Matematica, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy
3 Faculté des Sciences, Université de Picardie Jules Verne, 33 Rue Saint Leu, 80039 Amiens CEDEX 1, France
4 Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009, Australia
5 Istituto di Matematica Applicata e Tecnologie Informatiche, Consiglio Nazionale delle Ricerche, Via Ferrata 1, 27100 Pavia, Italy
Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 3, pp. 655-703

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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 C2 boundary Ω⊂Rn. 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.

DOI : 10.1016/j.anihpc.2018.08.003
Classification : 49Q05, 35R11, 58E12
Keywords: Nonlocal minimal surfaces, Stickiness phenomena, Loss of regularity, Strongly nonlocal regime

Bucur, Claudia 1 ; Lombardini, Luca 2, 3, 4 ; Valdinoci, Enrico 2, 4, 5

1 School of Mathematics and Statistics, The University of Melbourne, 813 Swanston Street, Parkville, VIC 3010, Australia
2 Dipartimento di Matematica, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy
3 Faculté des Sciences, Université de Picardie Jules Verne, 33 Rue Saint Leu, 80039 Amiens CEDEX 1, France
4 Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009, Australia
5 Istituto di Matematica Applicata e Tecnologie Informatiche, Consiglio Nazionale delle Ricerche, Via Ferrata 1, 27100 Pavia, Italy 
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     author = {Bucur, Claudia and Lombardini, Luca and Valdinoci, Enrico},
     title = {Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {655--703},
     publisher = {Elsevier},
     volume = {36},
     number = {3},
     year = {2019},
     doi = {10.1016/j.anihpc.2018.08.003},
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Bucur, Claudia; Lombardini, Luca; Valdinoci, Enrico. Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 3, pp. 655-703. doi: 10.1016/j.anihpc.2018.08.003

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