Asymptotic expansion of a nonlocal phase transition energy
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909322806558720 |
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| author | Dipierro, Serena Patrizi, Stefania Valdinoci, Enrico Vaughan, Mary |
| author_facet | Dipierro, Serena Patrizi, Stefania Valdinoci, Enrico Vaughan, Mary |
| contents | We study the asymptotic behavior of the fractional Allen--Cahn energy functional in bounded domains with prescribed Dirichlet boundary conditions.
When the fractional power $s \in (0,\frac12)$, we establish establish the first-order asymptotic development up to the boundary in the sense of $Γ$-convergence. In particular, we prove that the first-order term is the nonlocal minimal surface functional. Also, we show that, in general, the second-order term is not properly defined and intermediate orders may have to be taken into account.
For $s \in [\frac12,1)$, we focus on the one-dimensional case and we prove that the first order term is the classical perimeter functional plus a penalization on the boundary. Towards this end, we establish existence of minimizers to a corresponding fractional energy in a half-line, which provides itself a new feature with respect to the existing literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06215 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic expansion of a nonlocal phase transition energy Dipierro, Serena Patrizi, Stefania Valdinoci, Enrico Vaughan, Mary Analysis of PDEs 82B26, 35R11, 49J45, 35A01 We study the asymptotic behavior of the fractional Allen--Cahn energy functional in bounded domains with prescribed Dirichlet boundary conditions. When the fractional power $s \in (0,\frac12)$, we establish establish the first-order asymptotic development up to the boundary in the sense of $Γ$-convergence. In particular, we prove that the first-order term is the nonlocal minimal surface functional. Also, we show that, in general, the second-order term is not properly defined and intermediate orders may have to be taken into account. For $s \in [\frac12,1)$, we focus on the one-dimensional case and we prove that the first order term is the classical perimeter functional plus a penalization on the boundary. Towards this end, we establish existence of minimizers to a corresponding fractional energy in a half-line, which provides itself a new feature with respect to the existing literature. |
| title | Asymptotic expansion of a nonlocal phase transition energy |
| topic | Analysis of PDEs 82B26, 35R11, 49J45, 35A01 |
| url | https://arxiv.org/abs/2409.06215 |