{
  "abstract": "On any bounded convex domain $Ω\\subset\\mathbb{R}^n$ with a $d$-dimensional flat boundary piece $F\\subset\\partialΩ$, we consider convex solutions with zero boundary values to the Monge-Ampère equations $$ \\det\\mathsf{D}^2 u= \\mathrm{dist}(\\,\\cdot\\,,\\mathrm{Aff}(F))^α,\\quad\\det\\mathsf{D}^2 u=\\mathrm{dist}(\\,\\cdot\\,,\\partialΩ)^α,\\quad \\det\\mathsf{D}^2 u=(-u)^α, $$ where $\\mathrm{dist}$ stands for Euclidean distance and $\\mathrm{Aff}(F)$ is the affine hull of $F$. We show that the threshold of the exponent $α\\in\\mathbb{R}$ for $u$ to have infinite slope at $F$ is $α\\leq 2d-n$. For the first two equations, we also show that no solution exists when $α\\leq 2d-2n$. Moreover, when $Ω$ satisfies the ``exterior cone condition with a $d$-dimensional ridge'' at $F$, both thresholds are sharp for the first equation, and the infinite-slope threshold is sharp for the last equation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12165",
  "author_total": 1,
  "authors": [
    {
      "name": "Xin Nie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f7gaxmtr55ekb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12165",
  "fulltext_url": null,
  "id": "f7gaxmtr55ekb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12165",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7221082477",
  "pdf_url": "https://arxiv.org/pdf/2610.12165",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f7gaxmtr55ekb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12165",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12165"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "(In)finiteness of slopes for Monge-Ampère equations at flat boundary pieces",
  "title_text": "(In)finiteness of slopes for Monge-Ampère equations at flat boundary pieces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/f7gaxmtr55ekb",
  "venue": "arXiv",
  "versions": []
}