Geodesic
Approximation of sets of finite fractional perimeter by smooth sets and comparison of local and global s-minimal surfaces
Luca Lombardini1
1Università degli Studi di Milano, Italy and Université de Picardie Jules Verne, Amiens, France
Interfaces and free boundaries, Tome 20 (2018) no. 2, pp. 261-296

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DOI

This article is divided into two parts. In the first part we show that a set E has locally finite s-perimeter if and only if it can be approximated in an appropriate sense by smooth open sets. In the second part we prove some elementary properties of local and global s-minimal sets, such as existence and compactness. We also compare the two notions of minimizer (i.e., local and global), showing that in bounded open sets with Lipschitz boundary they coincide. Conversely, in general this is not true in unbounded open sets, where a global s-minimal set may fail to exist (we provide an example in the case of a cylinder Ω×R).
DOI : 10.4171/ifb/402
Classification : 49-XX, 35-XX
Mots-clés : Nonlocal minimal surfaces, smooth approximation, existence theory, subgraphs

Luca Lombardini  1

1 Università degli Studi di Milano, Italy and Université de Picardie Jules Verne, Amiens, France 
Luca Lombardini. Approximation of sets of finite fractional perimeter by smooth sets and comparison of local and global s-minimal surfaces. Interfaces and free boundaries, Tome 20 (2018) no. 2, pp. 261-296. doi: 10.4171/ifb/402
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     pages = {261--296},
     year = {2018},
     volume = {20},
     number = {2},
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     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/402/}
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