JournalsifbVol. 20, No. 2pp. 261–296

Approximation of sets of finite fractional perimeter by smooth sets and comparison of local and global s-minimal surfaces

  • Luca Lombardini

    Università degli Studi di Milano, Italy and Université de Picardie Jules Verne, Amiens, France
Approximation of sets of finite fractional perimeter by smooth sets and comparison of local and global s-minimal surfaces cover
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Abstract

This article is divided into two parts. In the first part we show that a set EE has locally finite ss-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 ss-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 \Omega \times \mathbb R).

Cite this article

Luca Lombardini, Approximation of sets of finite fractional perimeter by smooth sets and comparison of local and global s-minimal surfaces. Interfaces Free Bound. 20 (2018), no. 2, pp. 261–296

DOI 10.4171/IFB/402