{
  "abstract": "We study stable solutions of the Allen-Cahn equation with homogeneous Neumann boundary condition on Riemannian manifolds with boundary in the regime where the transition layers meet the boundary orthogonally. Under an a priori bound for the enhanced second fundamental form in the transition region, we establish uniform second-order Hölder estimates and quantitative mean curvature decay for the nodal hypersurfaces up to the boundary in ambient dimensions at most 10. The proof adapts the technique pioneered by Wang-Wei, adding a further correction determined by the geometry of the ambient boundary. Its leading contribution to mean curvature cancels on the nodal set but survives on non-zero level sets in the transition region, whose mean curvature need not decay uniformly. This phenomenon has no interior analogue and is exhibited by stable solutions constructed by Kowalczyk.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32829",
  "author_total": 4,
  "authors": [
    {
      "name": "Akashdeep Dey"
    },
    {
      "name": "Wenkui Du"
    },
    {
      "name": "Davide Parise"
    },
    {
      "name": "Lorenzo Sarnataro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/evream09qrm21.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32829",
  "fulltext_url": null,
  "id": "evream09qrm21",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32829",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214993252",
  "pdf_url": "https://arxiv.org/pdf/2609.32829",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/evream09qrm21.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.32829",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32829"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Second order estimates on transition layers for the Allen-Cahn equation with homogeneous Neumann boundary conditions: Part I - Boundary-orthogonal transition layers",
  "title_text": "Second order estimates on transition layers for the Allen-Cahn equation with homogeneous Neumann boundary conditions: Part I - Boundary-orthogonal transition layers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/evream09qrm21",
  "venue": "arXiv",
  "versions": []
}