$Γ$-convergence for higher order nonlocal phase transitions
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912672704888832 |
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| author | Chan, Hardy Dipierro, Serena Freguglia, Mattia Inversi, Marco Valdinoci, Enrico |
| author_facet | Chan, Hardy Dipierro, Serena Freguglia, Mattia Inversi, Marco Valdinoci, Enrico |
| contents | For every $0 < s <3/4$, we study the asymptotic behavior of the $\varepsilon$-rescaled sum of the $s$-fractional Allen-Cahn energy and the squared $L^2$-norm of its first variation. We prove that the contribution of the first variation vanishes as $\varepsilon \to 0$. This implies the Gamma-convergence of the initial sum to either the classical perimeter or to the $2s$-fractional perimeter, depending on whether $s \ge 1/2$ or not. This contradicts the expectation of finding curvature-dependent terms in the limit, as suggested by the regime $3/4 \le s < 1$, and as known to hold in low dimensions in the local case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23527 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $Γ$-convergence for higher order nonlocal phase transitions Chan, Hardy Dipierro, Serena Freguglia, Mattia Inversi, Marco Valdinoci, Enrico Analysis of PDEs For every $0 < s <3/4$, we study the asymptotic behavior of the $\varepsilon$-rescaled sum of the $s$-fractional Allen-Cahn energy and the squared $L^2$-norm of its first variation. We prove that the contribution of the first variation vanishes as $\varepsilon \to 0$. This implies the Gamma-convergence of the initial sum to either the classical perimeter or to the $2s$-fractional perimeter, depending on whether $s \ge 1/2$ or not. This contradicts the expectation of finding curvature-dependent terms in the limit, as suggested by the regime $3/4 \le s < 1$, and as known to hold in low dimensions in the local case. |
| title | $Γ$-convergence for higher order nonlocal phase transitions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.23527 |