$\Gamma$-Limsup estimate for a nonlocal approximation of the Willmore functional
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2024
- DOI:
- 10.48550/arXiv.2407.06102
- arXiv:
- arXiv:2407.06102
- Bibcode:
- 2024arXiv240706102C
- Keywords:
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- Mathematics - Analysis of PDEs;
- Mathematics - Differential Geometry;
- Mathematics - Optimization and Control;
- 49J45;
- 26A33;
- 35R11