{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\R^n$ be a connected [[open set|02BH3MCBQB]].\nA *potential* on $\\Omega$ is a function $u$ on $\\Omega$ such that:\n\n1. $u$ is [[superharmonic|X4V2S56YQQ]] on $\\Omega$.\n2. $u \\ge 0$ on $\\Omega$.\n3. The greatest harmonic minorant of $u$ is $0$; equivalently, every [[harmonic function|0DEB0JMN0W]] $h$ on $\\Omega$ with $h \\le u$ satisfies $h \\le 0$.\n\n## Remarks\n\nSince $u \\ge 0$, the function $0$ is a harmonic minorant, so the greatest harmonic minorant exists; it is $-h_0$, where $h_0$ is the [[least harmonic majorant|G0H97JZRS9]] of $-u$.\nLet $\\Omega$ have a Green function $G$.\nThen the [[Riesz decomposition theorem|CBZ20SS8RQ]] shows that the potentials are exactly the functions $x \\mapsto \\int G(x, y)\\, d\\mu(y)$.\nHere $\\mu$ runs over the positive measures for which this function is not identically $+\\infty$.\nEvery nonnegative superharmonic function is uniquely the sum of a potential and a nonnegative harmonic function.\n\nExample: for $n \\ge 3$ let $u(x) = \\abs{x}^{2-n}$ on $\\Omega = \\R^n$, with $u(0) = +\\infty$.\nIt is superharmonic and positive.\nLet $h$ be harmonic with $h \\le u$, and fix $a$.\nBy the [[mean value property|0HYPR64JM2]], $h(a)$ is the mean of $h$ over the sphere $S(a, R)$, which is at most the mean of $u$ there, namely $\\max(\\abs{a}, R)^{2-n}$.\nLetting $R \\to \\infty$ gives $h(a) \\le 0$, so $u$ is a potential.\n\nNon-example: on a bounded domain the constant $u = 1$ is superharmonic and nonnegative, but $h = 1$ is a harmonic minorant with $h \u003e 0$, so clause 3 fails.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "AQ5RN6FFT0",
    "kind": "definition",
    "msc": [
      "31B05",
      "31B15"
    ],
    "name": "Superharmonic potential",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Subharmonic_function"
      }
    ],
    "tag": "EQ23YX9CFG",
    "uses": [
      "X4V2S56YQQ",
      "G0H97JZRS9",
      "0DEB0JMN0W",
      "02BH3MCBQB",
      "CBZ20SS8RQ",
      "0HYPR64JM2"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/EQ23YX9CFG/",
  "links": [
    {
      "field": "defines",
      "target": "AQ5RN6FFT0"
    },
    {
      "field": "uses",
      "target": "X4V2S56YQQ"
    },
    {
      "field": "uses",
      "target": "G0H97JZRS9"
    },
    {
      "field": "uses",
      "target": "0DEB0JMN0W"
    },
    {
      "field": "uses",
      "target": "02BH3MCBQB"
    },
    {
      "field": "uses",
      "target": "CBZ20SS8RQ"
    },
    {
      "field": "uses",
      "target": "0HYPR64JM2"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/EQ23YX9CFG.md",
  "msc": [
    "31B05",
    "31B15"
  ],
  "name": "Superharmonic potential",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/EQ2/EQ23YX9CFG.md",
  "status": null,
  "tag": "EQ23YX9CFG",
  "uri": "m2://EQ23YX9CFG",
  "url": "https://tamnd.github.io/math2/wiki/superharmonic-potential-definition/",
  "writing": {
    "tag": "EQ23YX9CFG",
    "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
      }
    ]
  }
}