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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 halfspace, the $s$minimal set is the same halfspace. On the other hand, if one removes (even from far away) some small set from the halfspace, 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):11111144, 2010. \[ \mbox{ } \] [2] Nicola Abatangelo and Enrico Valdinoci. \newblock A notion of nonlocal curvature. \newblock {\em Numer. Funct. Anal. Optim.}, 35(79):793815, 2014. \[ \mbox{ } \] [3] Serena Dipierro, Ovidiu Savin, and Enrico Valdinoci. Boundary behavior of nonlocal minimal surfaces. arXiv preprint arXiv:1506.04282, 2015.
Item Metadata
Title 
Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter

Creator  
Publisher 
Banff International Research Station for Mathematical Innovation and Discovery

Date Issued 
20161108T14:45

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 halfspace, the $s$minimal set is the same halfspace.
On the other hand, if one removes (even from far away) some small set from the halfspace, 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):11111144, 2010.
\[
\mbox{ }
\]
[2] Nicola Abatangelo and Enrico Valdinoci.
\newblock A notion of nonlocal curvature.
\newblock {\em Numer. Funct. Anal. Optim.}, 35(79):793815, 2014.
\[
\mbox{ }
\]
[3] Serena Dipierro, Ovidiu Savin, and Enrico Valdinoci.
Boundary behavior of nonlocal minimal surfaces.
arXiv preprint arXiv:1506.04282, 2015.

Extent 
35 minutes

Subject  
Type  
File Format 
video/mp4

Language 
eng

Notes 
Author affiliation: Università degli Studi di Milano

Series  
Date Available 
20170618

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0348333

URI  
Affiliation  
Peer Review Status 
Unreviewed

Scholarly Level 
Graduate

Rights URI  
Aggregated Source Repository 
DSpace

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Item Citations and Data
Rights
AttributionNonCommercialNoDerivatives 4.0 International