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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 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.
|Number of pages||49|
|Journal||Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire|
|Early online date||4 Oct 2018|
|Publication status||Published - 1 May 2019|