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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.06102 |
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| _version_ | 1866918042327318528 |
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| author | Chan, Hardy Freguglia, Mattia Inversi, Marco |
| author_facet | Chan, Hardy Freguglia, Mattia Inversi, Marco |
| contents | 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 $Γ$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06102 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $Γ$-Limsup estimate for a nonlocal approximation of the Willmore functional Chan, Hardy Freguglia, Mattia Inversi, Marco Analysis of PDEs Differential Geometry Optimization and Control 49J45, 26A33, 35R11 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 $Γ$-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. |
| title | $Γ$-Limsup estimate for a nonlocal approximation of the Willmore functional |
| topic | Analysis of PDEs Differential Geometry Optimization and Control 49J45, 26A33, 35R11 |
| url | https://arxiv.org/abs/2407.06102 |