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


Nonlocal minimal surfaces are introduced in [1] as boundary of sets that minimize the fractional perimeter in a bounded and open set $\Omega \subset \mathbb{R}^n$, among sets with fixed exterior data. It is a known result that when $\Omega$ has a smooth boundary and the exterior data is a half-space, the $s$-minimal set is the same half-space. On the other hand, if one removes (even from far away) some small set from the half-space, for $s$ small enough the $s$-minimal set completely sticks to the boundary, that is, the $s$-minimal set is empty inside $\Omega$. In the paper [3], it is proved indeed that fixing the first quadrant of the plane as boundary data, the $s$-minimal set in $B_1\subset\mathbb{R}^2$ is empty in $B_1$ for $s$ small enough. In this talk, we will present 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 $\Omega$. We classify the behavior of $s$-minimal surfaces with respect to the fixed exterior data. So, for $s$ small and depending on the data at infinity, the $s$-minimal set can be either empty in $\Omega$, fill all $\Omega$, or possibly develop a wildly oscillating boundary. Also, we will present the asymptotic behavior of the fractional mean curvature (see [2]) when $s\to 0^+$. In particular, as $s$ gets smaller, the fractional mean curvature at any point on the boundary of a $C^{1,\gamma}$ set (for $\gamma\in(0,1)$) takes into account only the nonlocal contribution. The results in this talk are obtained in a preprint by myself, Luca Lombardini and Enrico Valdinoci. \[ \mbox{ } \] [1] L.Caffarelli, J.-M. Roquejoffre, and O.Savin. Nonlocal minimal surfaces. Comm. Pure Appl. Math., 63(9):1111--1144, 2010. \[ \mbox{ } \] [2] Nicola Abatangelo and Enrico Valdinoci. \newblock A notion of nonlocal curvature. \newblock {\em Numer. Funct. Anal. Optim.}, 35(7-9):793--815, 2014. \[ \mbox{ } \] [3] Serena Dipierro, Ovidiu Savin, and Enrico Valdinoci. Boundary behavior of nonlocal minimal surfaces. arXiv preprint arXiv:1506.04282, 2015.

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