# Γ-Limsup estimate for a nonlocal approximation of the Willmore functional

## TL;DR

The paper introduces a nonlocal approximation of the Willmore functional using Gamma-convergence and proves a Γ-limsup estimate.

## Contribution

The novelty is proposing a nonlocal counterpart to a known local phase-field approximation of the Willmore functional.

## Key findings

- A Γ-limsup estimate is proven for the nonlocal approximation of the Willmore functional.
- The analysis uses Fermi coordinates and decay estimates of higher-order derivatives of a nonlocal optimal profile.

## Abstract

We propose a possible nonlocal approximation of the Willmore functional, in the sense of Gamma-convergence, based on the first variation of the fractional Allen–Cahn energies, and we prove the corresponding \documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}Γ-limsup estimate. Our analysis is based on the expansion of the fractional Laplacian in Fermi coordinates and fine estimates on the decay of higher order derivatives of the one-dimensional nonlocal optimal profile. This result is the nonlocal counterpart of that obtained by Bellettini and Paolini, where they proposed a phase-field approximation of the Willmore functional based on the first variation of the (local) Allen–Cahn energies.