A threshold for higher-order asymptotic development of genuinely nonlocal phase transition energies
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912095885328384 |
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| author | Dipierro, Serena Valdinoci, Enrico Vaughan, Mary |
| author_facet | Dipierro, Serena Valdinoci, Enrico Vaughan, Mary |
| contents | We study the higher-order asymptotic development of a nonlocal phase transition energy in bounded domains and with prescribed external boundary conditions. The energy under consideration has fractional order $2s \in (0,1)$ and a first-order asymptotic development in the $Γ$-sense as described by the fractional perimeter functional.
We prove that there is no meaningful second-order asymptotic expansion and, in fact, no asymptotic expansion of fractional order $μ> 2-2s$. In view of this range value for $μ$, it would be interesting to develop a new asymptotic development for the $Γ$-convergence of our energy functional which takes into account fractional orders.
The results obtained here are also valid in every space dimension and with mild assumptions on the exterior data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22873 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A threshold for higher-order asymptotic development of genuinely nonlocal phase transition energies Dipierro, Serena Valdinoci, Enrico Vaughan, Mary Analysis of PDEs We study the higher-order asymptotic development of a nonlocal phase transition energy in bounded domains and with prescribed external boundary conditions. The energy under consideration has fractional order $2s \in (0,1)$ and a first-order asymptotic development in the $Γ$-sense as described by the fractional perimeter functional. We prove that there is no meaningful second-order asymptotic expansion and, in fact, no asymptotic expansion of fractional order $μ> 2-2s$. In view of this range value for $μ$, it would be interesting to develop a new asymptotic development for the $Γ$-convergence of our energy functional which takes into account fractional orders. The results obtained here are also valid in every space dimension and with mild assumptions on the exterior data. |
| title | A threshold for higher-order asymptotic development of genuinely nonlocal phase transition energies |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.22873 |