Nonlocal perimeters and variations: Extremality and decomposability for finite and infinite horizons

TL;DR

This paper investigates the properties of nonlocal perimeters, introduces a notion of indecomposability, and establishes decomposition and extremality results, connecting nonlocal and classical perimeters through a limit process.

Contribution

It introduces a nonlocal indecomposability concept, characterizes it via the interaction horizon, and extends classical decomposition theorems to nonlocal perimeters.

Findings

Characterization of nonlocal indecomposability in terms of interaction range

Decomposition of sets into $ ext{ extmu}$-connected components

Identification of extreme points of nonlocal perimeter balls

Abstract

We analyze the extremality and decomposability properties with respect to two types of nonlocal perimeters available in the literature, the Gagliardo perimeter based on the eponymous seminorms and the nonlocal distributional Caccioppoli perimeter, both with finite and infinite interaction ranges. A nonlocal notion of indecomposability associated to these perimeters is introduced, and we prove that in both cases it can be characterized solely in terms of the interaction range or horizon $\varepsilon$. Utilizing this, we show that it is possible to uniquely decompose a set into its $\varepsilon$-connected components, establishing a nonlocal analogue of the decomposition theorem of Ambrosio, Caselles, Masnou and Morel. Moreover, the extreme points of the balls induced by the Gagliardo and nonlocal total variation seminorm are identified, which naturally correspond to the two nonlocal perimeters. Surprisingly, while the extreme points in the former case are normalized indicator functions of $\varepsilon$-simple sets, akin to the classical TV-ball, in the latter case they are instead obtained from a nonlocal transformation applied to the extreme points of the TV-ball. Finally, we explore the nonlocal-to-local transition via a $\Gamma$-limit as $\varepsilon \rightarrow 0$ for both perimeters, recovering the classical Caccioppoli perimeter.