# On the convergence rate of some nonlocal energies

## TL;DR

This paper investigates how certain nonlocal energy functionals converge to a second-order limit functional, establishing the rate of convergence and $ ext{Gamma}$-convergence for these functionals.

## Contribution

It provides the first rigorous analysis of the convergence rate of nonlocal energies to a second-order limit functional, extending previous work on $ ext{Gamma}$-convergence.

## Key findings

- Established $ ext{Gamma}$-convergence of rescaled nonlocal functionals
- Identified the limit functional as a second-order functional
- Quantified the convergence rate of the nonlocal energies

## Abstract

We study the rate of convergence of some nonlocal functionals recently considered by Bourgain, Brezis and Mironescu. In particular we establish the $\Gamma$-convergence of the corresponding rate functionals, suitably rescaled, to a limit functional of second order.