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

TL;DR

This work develops a comprehensive nonlocal variational framework for two main perimeters—Gagliardo-type and distributional Caccioppoli—under finite and infinite horizons. It introduces and analyzes an $\\varepsilon$-indecomposability concept, yielding a unique decomposition into $\\varepsilon$-connected components and a precise characterization of extreme points of the corresponding unit balls, with distinct behaviors in finite-horizon versus fractional settings. The authors establish Gamma-convergence links to local perimeters, including a convergence of constrained minimizers to balls, thereby providing a rigorous bridge between nonlocal and classical isoperimetric problems. Extending the theory to nonlocal BV spaces, they derive decomposition and extremality results that parallel and extend the classical Ambrosio–Caselles–Masnou–Morel framework to nonlocal contexts. Together, these results form a robust nonlocal analogue of indecomposability, connected components, and extremality, with clear implications for nonlocal isoperimetry and variational modeling.

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 $Γ$-limit as $\varepsilon \rightarrow 0$ for both perimeters, recovering the classical Caccioppoli perimeter.

Figures (4)

• Figure 1: Decomposition of a set in three different $\varepsilon$-connected components. The dotted lines depict the boundaries of the sets $\{y\,|\, \mathop{\mathrm{dist}}\nolimits(y, E^{x_i}) < \varepsilon\}$.
• Figure 2: The construction in the proof of Proposition \ref{['prop:connectedrelaxation']}. Adding the union of balls $F_n$ depicted in gray to the set $E$ with two $\varepsilon$-connected components depicted in green and blue makes $E \cup F_n$$\varepsilon$-indecomposable.
• Figure 3: Numerical illustration of the normalized nonlocal gradient $D_\rho \mathds{1}_E / |D_\rho \mathds{1}_E|$ with $E = (-5/4,5/4)^2$ and $\rho(\cdot)=\operatorname{exp}\left(\dfrac{1/10}{|\cdot|^2-(3/4)^2}\right)\mathds 1_{B_{3/4}(0)}(\cdot)\dfrac{1}{|\cdot|^{3/2}}.$ The computation was performed at locations $x$ on a grid with spacing $1/4$, with the integral for each of them approximated by a finite sum on a grid of $161 \times 161$ points. The locations where $D_\rho \mathds{1}_E$ vanishes are shown as dots.
• Figure 4: Plot of the one-dimensional extreme point of Remark \ref{['rem:ext1d']} (ii) with $\alpha=1/2$, $a=-1$ and $b=1$.

Theorems & Definitions (72)

• Definition 2.1: $\varepsilon$-decomposability
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Definition 2.5: $\varepsilon$-connected component
• Lemma 2.6
• proof