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

Bucur, Claudia; Lombardini, Luca; Valdinoci, Enrico

doi:10.1016/j.anihpc.2018.08.003

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

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 3, pp. 655-703.

Résumé

In this paper, we consider the asymptotic behavior of the fractional mean curvature when $s \to 0^{+}$ . Moreover, we deal with the behavior of s-minimal surfaces when the fractional parameter $s \in (0, 1)$ is small, in a bounded and connected open set with $C^{2}$ boundary $Ω \subset R^{n}$ . 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 $s \in [0, 1]$ . Using this, we see that as the parameter s varies, the fractional mean curvature may change sign.

MR Zbl

DOI : 10.1016/j.anihpc.2018.08.003

Classification : 49Q05, 35R11, 58E12
Mots-clés : Nonlocal minimal surfaces, Stickiness phenomena, Loss of regularity, Strongly nonlocal regime

Affiliations des auteurs :

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

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     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},
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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. http://www.numdam.org/articles/10.1016/j.anihpc.2018.08.003/

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