Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter |
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Authors: | Claudia Bucur Luca Lombardini Enrico Valdinoci |
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Affiliation: | 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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Abstract: | In this paper, we consider the asymptotic behavior of the fractional mean curvature when . Moreover, we deal with the behavior of s-minimal surfaces when the fractional parameter is small, in a bounded and connected open set with boundary . 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 . Using this, we see that as the parameter s varies, the fractional mean curvature may change sign. |
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Keywords: | 49Q05 35R11 58E12 Nonlocal minimal surfaces Stickiness phenomena Loss of regularity Strongly nonlocal regime |
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