{
  "abstract": "Let $K\\subset\\R^m$ be the closure of a bounded domain whose boundary is a compact embedded hypersurface, and let \\( \\overline M=\\R^m\\setminus\\operatorname{int}K \\) be the allowed target. We prove that if $\\partial K$ is of class $C^{1,1}$, then every $\\overline M$-valued local minimizer of the Dirichlet energy $E(u;B):=\\int_B|Du|^2\\dd x$ is locally $W^{2,\\infty}$, and hence locally $C^{1,1}$, on its continuity set. For every $0<α<1$, we also construct a convex body with $C^{1,α}$ boundary and an everywhere continuous global minimizer that is not $C^{1,1}$. Thus $C^{1,1}$ is the sharp target regularity threshold in the Hölder scale.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08952",
  "author_total": 1,
  "authors": [
    {
      "name": "Yilu Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c8sdxdh1hr998.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08952",
  "fulltext_url": null,
  "id": "c8sdxdh1hr998",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08952",
  "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|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7212074793",
  "pdf_url": "https://arxiv.org/pdf/2609.08952",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c8sdxdh1hr998.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.08952",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08952"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Sharp $C^{1,1}$ Target Threshold for Minimizing Constraint Maps",
  "title_text": "The Sharp C^1,1 Target Threshold for Minimizing Constraint Maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/c8sdxdh1hr998",
  "venue": "arXiv",
  "versions": []
}