{
  "id": "2406.05471",
  "version": "v1",
  "published": "2024-06-08T13:21:49.000Z",
  "updated": "2024-06-08T13:21:49.000Z",
  "title": "Monge-Ampère equation with Guillemin boundary condition in high dimension",
  "authors": [
    "Genggeng Huang",
    "Weiming Shen"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "The Guillemin boundary condition appears naturally in the study of K\\\"ahler geometry of toric manifolds. In the present paper, the following Guillemin boundary value problem is investigated \\begin{align} \\label{eq1} &\\det D^2 u=\\frac{h(x)}{\\prod_{i=1}^N l_i(x)},\\quad\\text{in}\\quad\\quad P\\subset\\mathbb R^n, \\quad\\quad \\quad \\quad\\quad \\quad \\quad \\quad\\quad (1)\\\\ \\label{bdy1} &u(x)-\\sum_{i=1}^N l_i(x)\\ln l_i(x)\\in C^\\infty(\\overline{P}). \\quad\\quad\\quad\\quad \\quad \\quad\\quad \\quad \\quad \\quad\\quad\\quad (2) \\end{align} Here \\begin{equation*} 0<h(x)\\in C^\\infty(\\overline{P}),\\quad P=\\cap_{i=1}^N \\{l_i(x)>0\\} \\end{equation*} is a simple convex polytope in $\\mathbb R^n$. The solvability of (1)-(2) is given under the necessary and sufficient condition. The key issue in the proof is to obtain the boundary regularity of $u(x)-\\displaystyle \\sum_{i=1}^N l_i(x)\\ln l_i(x)$. Due to the difficulty caused by the structure of the equation itself and the singularity of $\\partial P$, we need to pay special attention to the influence of the difference of singularity types at different positions on $\\partial P$ on the behavior of $u$ in its vicinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-08T13:21:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J96",
      "35J75",
      "35J70",
      "58J60"
    ],
    "keywords": [
      "monge-ampère equation",
      "high dimension",
      "guillemin boundary condition appears",
      "guillemin boundary value problem",
      "simple convex polytope"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}