{
  "abstract": "We establish a priori interior $C^{1,α}$ and $C^{2,α}$ Schauder estimates for a class of $Q$-valued quasilinear elliptic systems which are perturbations of the Laplace system, for arbitrary multiplicity $Q$, domain dimension $n\\geq2$, and target dimension $k$. The $C^{1,α}$ estimate generalises the work of Simon and Wickramasekera in \\cite{SW16} on $2$-valued solutions of linear systems and concerns weak solutions of divergence-form systems, while the $C^{2,α}$ estimate concerns strong solutions. In dimension $n=2$, both estimates hold for every $0<α<1/Q$. In dimensions $n\\geq3$, there exists $δ=δ(n,k,Q)>0$ such that both estimates hold for every $0<α<δ$. As applications, we obtain a small-slope Schauder estimate and a small-slope Bernstein theorem for $Q$-valued maps whose graph varifolds are stationary. Combining the Schauder estimate with the recent branch-set stratification theory of Krummel--Minter--Wickramasekera in \\cite{KMW26}, we further prove that, for every $γ>0$, the branch set $\\mathcal B_u$ of any $C^{1,γ}$ $Q$-valued map $u$ whose graph varifold is stationary satisfies $\\dim_{\\mathcal H}\\mathcal B_u\\leq n-2$. This bound is sharp already in codimension one.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12950",
  "author_total": 1,
  "authors": [
    {
      "name": "Mattia Luchese"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/asmpkj79th238.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12950",
  "fulltext_url": null,
  "id": "asmpkj79th238",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12950",
  "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"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7212531801",
  "pdf_url": "https://arxiv.org/pdf/2609.12950",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/asmpkj79th238.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12950",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12950"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Perturbative Schauder Estimates for $Q$-Valued Quasilinear Elliptic Systems and a Sharp Dimension Bound for Branch Sets of Stationary Graphs",
  "title_text": "Perturbative Schauder Estimates for Q-Valued Quasilinear Elliptic Systems and a Sharp Dimension Bound for Branch Sets of Stationary Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/asmpkj79th238",
  "venue": "arXiv",
  "versions": []
}