{
  "abstract": "We prove the existence of admissible subsolutions to the Dirichlet problem for symmetric augmented $k$-Hessian type equations. An important sufficient condition is the uniform $(k-1)$-$A$-convexity of the domain $Ω,$ where $A(x, z, p)$ is the augmented symmetric matrix appearing in the equation. This condition was originally introduced by F. Jiang, N. S. Trudinger, and X.-P. Yang and we have chosen a special their case. The structural conditions on the matrix $A(x, z, p)$ include its growth with respect to the variables $z$ and $p,$ particularly requiring that some of its first and second derivatives are sufficiently small in a sufficiently small neighborhood of the boundary. Under certain structural conditions on $A(x, z, p),$ the uniform $(k-1)$-$A$-convexity of $Ω$ is also a necessary condition for the existence of admissible subsolutions of the equation in a neighborhood of the boundary. Our results extend the classic result by L. Caffarelli, L. Nirenberg, and J. Spruck from the case $A \\equiv 0$ to the general case $A \\neq 0.$ Our same theorems are valid also for augmented quotient Hessian type equations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18889",
  "author_total": 5,
  "authors": [
    {
      "name": "Quang Hong Dinh"
    },
    {
      "name": "Bang Van Tran"
    },
    {
      "name": "Ngoan Tien Ha"
    },
    {
      "name": "Tho Huu Nguyen"
    },
    {
      "name": "Tien Trong Phan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f729d65fzasyv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18889",
  "fulltext_url": null,
  "id": "f729d65fzasyv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18889",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7213530602",
  "pdf_url": "https://arxiv.org/pdf/2609.18889",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f729d65fzasyv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18889",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18889"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Existence of Admissible Subsolutions to the Dirichlet Problem for Symmetric Augmented $k$-Hessian Type Equations in Bounded Domains",
  "title_text": "Existence of Admissible Subsolutions to the Dirichlet Problem for Symmetric Augmented k-Hessian Type Equations in Bounded Domains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/f729d65fzasyv",
  "venue": "arXiv",
  "versions": []
}