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.

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

Bucur C.;
2019-01-01

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.
2019
Nonlocal minimal surfaces; Stickiness phenomena; Strongly nonlocal regime; Loss of regularity; Nonlocal minimal surfaces; Stickiness phenomena; Strongly nonlocal regime
Bucur, C.; Lombardini, L.; Valdinoci, E.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2105135
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