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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Γ</mml:mi></mml:math> -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.