{
  "abstract": "In this paper, we investigate regularity properties for viscosity solutions to a general class of degenerate or singular fully nonlinear elliptic equations with Hamiltonian terms, \\[ \\left\\{ \\begin{alignedat}{2} Φ(|Du|,x)F(D^{2}u,x)+H(|Du|,x) & =f(x) \\quad && \\text{in } \\quad Ω,\\\\ u(x) & = g(x) \\quad && \\text{on } \\quad \\partialΩ. \\end{alignedat} \\right. \\] Here, $F$ is uniformly elliptic, while $Φ$ and $H$ satisfy suitable structural and growth conditions allowing for both degenerate and singular regimes. Our first result establishes sharp global $C^{1,α}$ regularity for this general class of equations, thereby providing a unified regularity framework for both degenerate and singular regimes. We next develop an oscillation-based approach for fully nonlinear equations with unbalanced variable degeneracy and Hamiltonian terms. Under a Hölder-type decay assumption on $ f $, we derive sharp boundary $C^{1,β'}$ estimates with an explicit exponent, and establish a quantitative non-degeneracy estimate in the singular region. Finally, as applications of the general theory, under a suitable viscosity curvature condition on the level sets of the solution, we establish global $ C^{1,\\frac{1}{3}} $ regularity for the infinity-Poisson and global $ C^{1,\\frac{1}{p-1}} $ regularity for the $ p $-Poisson with $ p>2 $. These results may provide a new perspective on these two long-standing open problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.11607",
  "author_total": 2,
  "authors": [
    {
      "name": "Jiangwen Wang"
    },
    {
      "name": "Feida Jiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dw450sp5zgqwj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.11607",
  "fulltext_url": null,
  "id": "dw450sp5zgqwj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.11607",
  "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|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7164353397",
  "pdf_url": "https://arxiv.org/pdf/2606.11607",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dw450sp5zgqwj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2606.11607",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.11607"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Regularity for degenerate or singular fully nonlinear elliptic equations with Hamiltonian terms and applications",
  "title_text": "Regularity for degenerate or singular fully nonlinear elliptic equations with Hamiltonian terms and applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/dw450sp5zgqwj",
  "venue": "arXiv",
  "versions": []
}