{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\R^n$ be a domain.\nLet $h$ be a [[positive harmonic function|EYMQ82Z8XJ]] on $\\Omega$.\nThe function $h$ is *minimal* if every [[harmonic function|0DEB0JMN0W]] $v$ on $\\Omega$ with $0 \\le v \\le h$ equals $c h$ for a constant $c \\in [0, 1]$.\n\n## Remarks\n\nSome authors also normalise $h(x_0) = 1$ at a fixed reference point $x_0$.\nExample: on the unit disc $\\mathbb{D}$, let $h(z) = \\frac{1 - \\abs{z}^2}{\\abs{1 - z}^2}$, the [[Poisson kernel|TCYSWCTYYM]] with pole $1$.\nIf $0 \\le v \\le h$ is harmonic, the [[Herglotz representation theorem|9ZBAX4ZPX0]] writes $v$ and $h - v$ as Poisson integrals of positive measures $\\mu$ and $\\nu$ with $\\mu + \\nu = \\delta_1$, the measure of $h$.\nThen $\\mu$ is a multiple of $\\delta_1$, so $v$ is a multiple of $h$, and $h$ is minimal.\nNon-example: $h = 1$ on the unit disc is not minimal.\nThe function $v(z) = \\frac{1 + \\operatorname{Re} z}{2}$ is harmonic with $0 \\le v \\le 1$ on $\\mathbb{D}$, but it is not constant.\nOn $\\R^n$ every positive harmonic function is constant, so $h = 1$ is minimal there.\nThe minimal functions with $h(x_0) = 1$ are the extreme points of the convex set of positive harmonic functions with value $1$ at $x_0$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "NA4BTZ6WRB",
    "kind": "definition",
    "msc": [
      "31C35",
      "31B05"
    ],
    "name": "Minimal harmonic function",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Poisson_boundary"
      }
    ],
    "tag": "FF8P46NH5S",
    "uses": [
      "EYMQ82Z8XJ",
      "0DEB0JMN0W",
      "9ZBAX4ZPX0",
      "TCYSWCTYYM"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/FF8P46NH5S/",
  "links": [
    {
      "field": "defines",
      "target": "NA4BTZ6WRB"
    },
    {
      "field": "uses",
      "target": "EYMQ82Z8XJ"
    },
    {
      "field": "uses",
      "target": "0DEB0JMN0W"
    },
    {
      "field": "uses",
      "target": "9ZBAX4ZPX0"
    },
    {
      "field": "uses",
      "target": "TCYSWCTYYM"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/FF8P46NH5S.md",
  "msc": [
    "31C35",
    "31B05"
  ],
  "name": "Minimal harmonic function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FF8/FF8P46NH5S.md",
  "status": null,
  "tag": "FF8P46NH5S",
  "uri": "m2://FF8P46NH5S",
  "url": "https://tamnd.github.io/math2/wiki/minimal-harmonic-function-definition/",
  "writing": {
    "tag": "FF8P46NH5S",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}