{
  "abstract": "We study the uniqueness and boundary behavior of nonzero convex Aleksandrov solutions to $\\det D^2 u=M|u|^pν$ with zero boundary values on bounded convex domains in $\\mathbb{R}^n (n \\geq 2)$. For $0<p<n$, we prove the uniqueness of nonzero convex solutions in the finite-energy class when $ν$ is a locally finite Borel measure with positive mass and $\\int_Ω\\text{dist }(\\cdot,\\partialΩ)\\,dν<\\infty$. For $p>n$, we construct an explicit two-shell measure on the unit ball for which the problem has at least three radial solutions that are globally Lipschitz and have finite energy. In the case of $ν=\\text{dist }(\\cdot,\\partialΩ)^{-α}\\,dL^n$, $0\\leqα<2$, we prove global Lipschitz continuity when $p-α>n-2$ and obtain sharp upper and lower estimates on domains with a flat boundary part when $n(α-1)-2<p-α\\leq n-2$. When $α=0$ and $p=n-2$, our log-Lipschitz lower estimate has the same exponent as the known upper estimate. This answers the question raised by Le (Global Lipschitz and Sobolev estimates for the Monge-Ampère eigenfunctions of general bounded convex domains. Ann. Fac. Sci. Toulouse Math. (6) 35 (2026)). We also give a sufficient condition for finite Monge-Ampère energy on every bounded convex domain, prove its necessity when the boundary contains a flat part, and apply it to prove the uniqueness of the Monge-Ampère eigenvalue among all nonzero convex solutions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27341",
  "author_total": 1,
  "authors": [
    {
      "name": "Chong Gu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4c6gv88s9y66g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27341",
  "fulltext_url": null,
  "id": "4c6gv88s9y66g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27341",
  "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": "W7214148428",
  "pdf_url": "https://arxiv.org/pdf/2609.27341",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4c6gv88s9y66g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27341",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27341"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Uniqueness and sharp boundary estimates for degenerate Monge-Ampère equations with singular measures",
  "title_text": "Uniqueness and sharp boundary estimates for degenerate Monge-Ampère equations with singular measures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/4c6gv88s9y66g",
  "venue": "arXiv",
  "versions": []
}