{
  "abstract": "We study the sharp-interface limit of the fractional Allen--Cahn equation in $\\R^n$, $n\\geq 2$, in the strongly nonlocal regime $s\\in(0,\\frac12)$, for initial data consisting of finitely many nested transition layers. We identify a coupled geometric law for the motion of the resulting fronts: each velocity contains fractional mean curvature and a nonlocal interaction potential generated by the other fronts. These interactions persist while the fronts remain separated, in contrast to the independent motion in the critical regime $s=\\frac12$. Assuming that the coupled law admits a smooth evolution with strictly nested sets on a given time interval, we prove that the Allen--Cahn solutions converge locally uniformly away from the moving fronts to the corresponding integer-valued phases.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35666",
  "author_total": 2,
  "authors": [
    {
      "name": "Erisa Hasani"
    },
    {
      "name": "Stefania Patrizi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bvbxgw3g3zqqf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35666",
  "fulltext_url": null,
  "id": "bvbxgw3g3zqqf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35666",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7214794188",
  "pdf_url": "https://arxiv.org/pdf/2609.35666",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bvbxgw3g3zqqf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.35666",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35666"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Interacting fronts in the strongly nonlocal Allen--Cahn equation",
  "title_text": "Interacting fronts in the strongly nonlocal Allen--Cahn equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/bvbxgw3g3zqqf",
  "venue": "arXiv",
  "versions": []
}