Asymptotic analysis of a family of non-local functionals on sets

TL;DR

This paper investigates the asymptotic behavior of non-local functionals involving short-range interactions on sets with finite perimeter, providing pointwise limits and estimates for regular sets, with illustrative examples.

Contribution

It introduces a detailed asymptotic analysis of a new class of non-local functionals, including explicit limit computations and regular set estimates.

Findings

Computed pointwise limits of the functionals.

Provided lower bounds for regular sets.

Discussed specific examples illustrating the theory.

Abstract

We study the asymptotic behavior of a family of functionals which penalize a short-range interaction of convolution type between a finite perimeter set and its complement. We first compute the pointwise limit and we obtain a lower estimate on more regulars sets. Finally, some examples are discussed.