$\Gamma$-Limsup estimate for a nonlocal approximation of the Willmore functional

TL;DR

This paper introduces a nonlocal approximation of the Willmore functional using fractional Allen-Cahn energies, establishing a Gamma-limsup estimate through detailed analysis of fractional Laplacian expansions.

Contribution

It provides the first Gamma-convergence based nonlocal approximation of the Willmore functional, extending previous local phase-field models to a fractional setting.

Findings

Established Gamma-limsup estimate for the nonlocal approximation.

Analyzed fractional Laplacian expansion in Fermi coordinates.

Derived decay estimates for nonlocal optimal profiles.

Abstract

We propose a possible nonlocal approximation of the Willmore functional, in the sense of Gamma-convergence, based on the first variation ot the fractional Allen-Cahn energies, and we prove the corresponding $\Gamma$-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.