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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2511.04818 |
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| _version_ | 1866918190124105728 |
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| author | Hasani, Erisa Patrizi, Stefania |
| author_facet | Hasani, Erisa Patrizi, Stefania |
| contents | We study the sharp interface limit of the fractional Allen-Cahn equation $$ \varepsilon \partial_t u^{\varepsilon} = \mathcal{I}^s_n [u^{\varepsilon}] -\frac{1}{\varepsilon ^{2s}} W'(u^\varepsilon) \quad \hbox{in}~(0,\infty)\times\mathbb{R}^n, ~n \geq 2, $$ where $\varepsilon >0$, $\mathcal{I}^s_n=-c_{n,s}(-Δ)^s$ is the fractional Laplacian of order $2s\in(0,1)$ in $\mathbb{R}^n$, and $W$ is a smooth double-well potential with minima at 0 and 1. We focus on the singular regime $s\in(0,\frac{1}{2})$, corresponding to strongly nonlocal diffusion. For suitably prepared initial data, we prove that the solution
$ u^\varepsilon $ converges, as $\varepsilon\to0$, to the minima of $W$ with the interface evolving by fractional mean curvature flow.
This establishes the first rigorous convergence result in this regime, complementing and completing previous work for $s\geq \frac{1}{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_04818 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The strongly nonlocal Allen-Cahn problem Hasani, Erisa Patrizi, Stefania Analysis of PDEs We study the sharp interface limit of the fractional Allen-Cahn equation $$ \varepsilon \partial_t u^{\varepsilon} = \mathcal{I}^s_n [u^{\varepsilon}] -\frac{1}{\varepsilon ^{2s}} W'(u^\varepsilon) \quad \hbox{in}~(0,\infty)\times\mathbb{R}^n, ~n \geq 2, $$ where $\varepsilon >0$, $\mathcal{I}^s_n=-c_{n,s}(-Δ)^s$ is the fractional Laplacian of order $2s\in(0,1)$ in $\mathbb{R}^n$, and $W$ is a smooth double-well potential with minima at 0 and 1. We focus on the singular regime $s\in(0,\frac{1}{2})$, corresponding to strongly nonlocal diffusion. For suitably prepared initial data, we prove that the solution $ u^\varepsilon $ converges, as $\varepsilon\to0$, to the minima of $W$ with the interface evolving by fractional mean curvature flow. This establishes the first rigorous convergence result in this regime, complementing and completing previous work for $s\geq \frac{1}{2}$. |
| title | The strongly nonlocal Allen-Cahn problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.04818 |