BIRS Workshop Lecture Videos

Banff International Research Station Logo

BIRS Workshop Lecture Videos

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

Description

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.

Item Media

Item Citations and Data

Rights

Attribution-NonCommercial-NoDerivatives 4.0 International