{
  "abstract": "We study elliptic superlevel geometry and return dynamics for the class $\\mathcal{O}_{C}$. Let $C \\subset \\mathbb{R}^{N}$ be a compact strictly convex core, and $Ω\\in \\mathcal{O}_{C}$ a bounded domain whose boundary is a global normal graph over $\\partial C$ with inward normals returning to $\\operatorname{int} C$. For $-Δu = f$ in $Ω$ ($u=0$ on $\\partialΩ$), with $f \\geq 0$, $f \\not\\equiv 0$, $\\operatorname{supp} f \\Subset \\operatorname{int} C$, we analyze the superlevel sets $Ω_{t} = \\{u > t\\}$ for $t \\downarrow 0$. We prove that for small $t > 0$, $\\partialΩ_{t}$ is a global normal graph given by $\\partialΩ_{t} = \\{c + d_{t}(c)ν(c) : c \\in \\partial C\\}$, satisfying $\\|d_{t} - d_{0}\\|_{C^{1}} = O(t)$ and $d_{t} = d_{0} - t b_{0} + o_{C^{1}}(t)$, where $b_{0}$ depends on the Hopf derivative and transversality. Using normal-graph formulas, we show $Ω_{t} \\in \\mathcal{O}_{C}$. The return maps $F_{t} : \\partial C \\to \\partial C$ are well-defined, satisfying $\\|F_{t} - F_{0}\\|_{L^{\\infty}} = O(t)$ and $F_{t} = F_{0} + t \\mathcal{A}_{0} + o_{L^{\\infty}}(t)$. For any sequence $t_{n} \\downarrow 0$, the non-autonomous system $x_{n+1} = F_{t_{n}}(x_{n})$ is asymptotically autonomous with background map $F_{0}$. These results bridge elliptic boundary-layer deformation and return-map dynamics relative to a convex core.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.16912",
  "author_total": 2,
  "authors": [
    {
      "name": "Mohamed El Morsalani"
    },
    {
      "name": "Mohammd Barkatou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bkj101sc8hak0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.16912",
  "fulltext_url": null,
  "id": "bkj101sc8hak0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.16912",
  "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"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7169884945",
  "pdf_url": "https://arxiv.org/pdf/2607.16912",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bkj101sc8hak0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2607.16912",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.16912"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Elliptic Superlevel Sets in the Class $\\OO_C$: Boundary-Layer Geometry and Stability of the Return Map",
  "title_text": "Elliptic Superlevel Sets in the Class OO_C: Boundary-Layer Geometry and Stability of the Return Map",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/bkj101sc8hak0",
  "venue": "arXiv",
  "versions": []
}