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

TL;DR

This paper characterizes sets with finite fractional perimeter through smooth approximations and compares local and global $s$-minimal surfaces, establishing conditions for their equivalence and existence.

Contribution

It proves approximation of $s$-perimeter sets by smooth sets and analyzes the properties and differences of local and global $s$-minimal surfaces.

Findings

Sets with finite $s$-perimeter can be approximated by smooth open sets.

Local and global $s$-minimal sets coincide in bounded Lipschitz domains.

Global $s$-minimal sets may not exist in unbounded domains, as shown by a cylinder example.

Abstract

In the first part of this paper 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. However, 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 $\Omega\times\mathbb{R}$).