{
  "abstract": "Nirenberg's theorem gives interior $C^{2,α}$ regularity for continuous viscosity solutions of fully nonlinear uniformly elliptic equations $F(D^2u)=0$ in dimension two. In dimensions five and higher, the work of Nadirashvili--Tkachev--Vlăduţ shows that such solutions need not be $C^2$. We construct a family of $C^{1,1}\\setminus C^2$ functions $\\{w_m\\}_{m\\geq3}$ and uniformly elliptic operators $F_m$ such that $F_m(D^2 w_m)=0$ in $\\mathbb{R}^{d_m}$, where $d_m=m(m+1)/2-1$. These solutions are given by $\\operatorname{tr}(X^3)/\\sqrt{\\operatorname{tr}(X^2)}$ on the space of traceless real symmetric $m\\times m$ matrices. When $m=3,4$, this family recovers the known singular solutions associated with the five-dimensional Cartan isoparametric cubic and the nine-dimensional Hsiang minimal cubic, respectively. Our construction treats all $m\\geq3$ in a unified way by direct analysis on traceless symmetric matrices. We also prove a geometric criterion for pointwise $C^{2,β}$ regularity of continuous viscosity solutions $u$ of uniformly elliptic equations $F(D^2u)=0$ in every dimension $n\\geq 3$. Suppose that $n-2$ smooth hypersurfaces pass through a point $p$, each carrying a smooth vector field along which the directional derivative of $u$ is constant on that hypersurface. If these vector fields are linearly independent at $p$, then $u$ is twice differentiable at $p$ and admits a pointwise $C^{2,β}$ expansion for some $β\\in(0,1)$. Our criterion is quantitative and yields interior $C^{2,β}$ regularity under uniform geometric hypotheses. As an application, we extend Nirenberg's theorem to a general class of solutions in dimensions three and higher. As a consequence, we establish interior $C^{2,β}$ regularity for solutions of Dirichlet problems with certain group symmetries. This extends a result of Nadirashvili--Vlăduţ for axially symmetric problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25416",
  "author_total": 1,
  "authors": [
    {
      "name": "BaoZhi Chu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bvzjzbv90yfhh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25416",
  "fulltext_url": null,
  "id": "bvzjzbv90yfhh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25416",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.25416",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bvzjzbv90yfhh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25416",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25416"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Nirenberg's Theorem and Singular Solutions from Traceless Symmetric Matrices",
  "title_text": "Nirenberg's Theorem and Singular Solutions from Traceless Symmetric Matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/bvzjzbv90yfhh",
  "venue": "arXiv",
  "versions": []
}