{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "FF8P46NH5S"
    },
    {
      "field": "uses",
      "target": "GFJE824E93"
    },
    {
      "field": "uses",
      "target": "SKJB3ANDF4"
    }
  ],
  "body": "\n## Summary\n\nA minimal harmonic function is a [[positive harmonic function|EYMQ82Z8XJ]] that has no positive harmonic minorants other than its own constant multiples.\nMinimal functions play the role of extreme points in the convex cone of positive harmonic functions on a domain.\nBy Martin's theory, every positive harmonic function is an integral of minimal ones.\nThe minimal ones correspond to the points of the minimal Martin boundary.\n\n## Explanations\n\n### L1 human\n\nPositive harmonic functions can be added and scaled by positive numbers.\nA minimal one cannot be split into two genuinely different positive harmonic pieces.\nOn the unit disc, the Poisson kernel with its pole at one boundary point is minimal.\nIt is the response to a unit mass at a single point of the circle.\nThe constant function $1$ on the disc is not minimal, since it is the average of the Poisson kernels over the circle.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "31C35",
      "31B05"
    ],
    "name": "Minimal harmonic function",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Poisson_boundary"
      }
    ],
    "specializes": [
      "EYMQ82Z8XJ"
    ],
    "tag": "NA4BTZ6WRB"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/NA4BTZ6WRB/",
  "links": [
    {
      "field": "specializes",
      "target": "EYMQ82Z8XJ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/NA4BTZ6WRB.md",
  "msc": [
    "31C35",
    "31B05"
  ],
  "name": "Minimal harmonic function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NA4/NA4BTZ6WRB.md",
  "status": null,
  "tag": "NA4BTZ6WRB",
  "uri": "m2://NA4BTZ6WRB",
  "url": "https://tamnd.github.io/math2/wiki/minimal-harmonic-function/",
  "writing": {
    "tag": "NA4BTZ6WRB",
    "kind": "concept",
    "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
      }
    ]
  }
}