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

Hardy Chan; Mattia Freguglia; Marco Inversi

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arXiv · 2407.06102

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

Hardy Chan·Mattia Freguglia·Marco Inversi

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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.

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Hardy Chan, Mattia Freguglia, Marco Inversi. 2024-07-08. $\Gamma$-Limsup estimate for a nonlocal approximation of the Willmore functional. https://arxiv.org/abs/2407.06102

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