{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $D \\subseteq \\R^n$ be a [[domain|093CM69AGS]], that is, a connected open set.\nLet $(u_k)$ be a sequence of [[harmonic functions|0W5NJQ8MKY]] on $D$ with $u_1 \\le u_2 \\le u_3 \\le \\cdots$ at each point of $D$.\nThen one of the following holds.\n\n1. $u_k \\to \\infty$ uniformly on each compact subset of $D$.\n2. $(u_k)$ converges uniformly on each compact subset of $D$ to a harmonic function $u$ on $D$.\n\n## Remarks\n\nProof idea: the functions $u_k - u_1$ are nonnegative and harmonic.\n[[Harnack's inequality|YNH832D50W]] compares the values of a nonnegative harmonic function at two points of a compact set by constants that do not depend on the function.\nSo the increasing limit is either infinite everywhere or finite everywhere, and in the finite case the convergence is uniform on compact sets.\nA uniform limit of harmonic functions is harmonic, by the mean value property.\nConnectedness is needed: on two disjoint discs, take $u_k = 0$ on one disc and $u_k = k$ on the other.\n\n## Explanations\n\n### L1 human\n\nAn increasing sequence of harmonic functions can do only two things.\nEither it goes to infinity everywhere at once, or it converges nicely everywhere to a harmonic function.\nIt cannot converge at one point and blow up at another.\n\n### L2 human\n\nThe principle is the harmonic analogue of the monotone convergence theorem, and it follows from Harnack's inequality.\nIt is used in Perron's method for the Dirichlet problem, where the solution is built as a limit of subharmonic functions.\nIt also appears in the construction of Green's functions and harmonic measures by exhaustion of a domain.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "0DEB0JMN0W"
    ],
    "kind": "statement",
    "msc": [
      "31B05",
      "31A05"
    ],
    "name": "Harnack's principle",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q1135706",
      "wikipedia": "Harnack's principle"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Harnack%27s_principle"
      }
    ],
    "tag": "P15K8RKHDV",
    "uses": [
      "0W5NJQ8MKY",
      "YNH832D50W",
      "093CM69AGS"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/P15K8RKHDV/",
  "links": [
    {
      "field": "about",
      "target": "0DEB0JMN0W"
    },
    {
      "field": "uses",
      "target": "0W5NJQ8MKY"
    },
    {
      "field": "uses",
      "target": "YNH832D50W"
    },
    {
      "field": "uses",
      "target": "093CM69AGS"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/P15K8RKHDV.md",
  "msc": [
    "31B05",
    "31A05"
  ],
  "name": "Harnack's principle",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/P15/P15K8RKHDV.md",
  "status": null,
  "tag": "P15K8RKHDV",
  "uri": "m2://P15K8RKHDV",
  "url": "https://math2.tamnd.com/wiki/harnacks-principle/",
  "writing": {
    "tag": "P15K8RKHDV",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}