{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "P15K8RKHDV"
    }
  ],
  "body": "\n## Statement\n\nLet $a \\in \\C$, let $R \u003e 0$, and let $u$ be a [[harmonic function|0DEB0JMN0W]] on the disc $\\set{z \\in \\C : \\abs{z - a} \u003c R}$ with $u \\ge 0$.\nThen for every $z$ with $r = \\abs{z - a} \u003c R$,\n$$\\frac{R - r}{R + r}\\, u(a) \\le u(z) \\le \\frac{R + r}{R - r}\\, u(a).$$\n\n## Remarks\n\nFor $\\rho$ with $r \u003c \\rho \u003c R$, the Poisson formula on the disc of radius $\\rho$ gives $u(z)$ as an average of $u$ on the circle $\\abs{w - a} = \\rho$ against the [[Poisson kernel|W19567AVYG]] $(\\rho^2 - r^2)/\\abs{w - z}^2$.\nSince $\\rho - r \\le \\abs{w - z} \\le \\rho + r$, this kernel lies between $(\\rho - r)/(\\rho + r)$ and $(\\rho + r)/(\\rho - r)$, and the average of $u$ over the circle is $u(a)$.\nLetting $\\rho \\to R$ gives the inequality.\nThe hypothesis $u \\ge 0$ is necessary: $u(z) = \\operatorname{Re} z$ on the unit disc has $u(0) = 0$ but takes nonzero values.\nIn $\\R^n$ with $n \\ge 2$, the same argument gives $\\dfrac{R^{n-2}(R - r)}{(R + r)^{n-1}}\\, u(a) \\le u(x) \\le \\dfrac{R^{n-2}(R + r)}{(R - r)^{n-1}}\\, u(a)$ for $\\abs{x - a} = r \u003c R$.\n\n## Explanations\n\n### L1 human\n\nA nonnegative harmonic function cannot change too fast.\nIf you know its value at the centre of a disc, you know its values at other points up to a factor.\nThis factor depends only on how close the points are to the edge.\nFor example, at half the radius the value is between one third and three times the value at the centre.\n\n### L2 human\n\nA chain of discs gives the general form on a connected open set $\\Omega$ with a compact subset $K$.\nThere is a constant $C$ such that $\\sup_K u \\le C \\inf_K u$ for every nonnegative harmonic $u$ on $\\Omega$.\nThis is the key estimate behind [[Harnack's principle|P15K8RKHDV]] on increasing sequences of harmonic functions.\nIt is used in Perron's method for the [[Dirichlet problem|8D56GQ4AG8]] and in the study of positive harmonic functions, such as the [[Herglotz representation theorem|9ZBAX4ZPX0]].\nMoser extended Harnack's inequality to solutions of elliptic equations in divergence form with measurable coefficients.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "0DEB0JMN0W"
    ],
    "kind": "statement",
    "msc": [
      "31A05",
      "31B05"
    ],
    "name": "Harnack's inequality",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q868279",
      "wikipedia": "Harnack's inequality"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Harnack%27s_inequality"
      }
    ],
    "tag": "YNH832D50W",
    "uses": [
      "0DEB0JMN0W",
      "W19567AVYG"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/YNH832D50W/",
  "links": [
    {
      "field": "about",
      "target": "0DEB0JMN0W"
    },
    {
      "field": "uses",
      "target": "0DEB0JMN0W"
    },
    {
      "field": "uses",
      "target": "W19567AVYG"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/YNH832D50W.md",
  "msc": [
    "31A05",
    "31B05"
  ],
  "name": "Harnack's inequality",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YNH/YNH832D50W.md",
  "status": null,
  "tag": "YNH832D50W",
  "uri": "m2://YNH832D50W",
  "url": "https://math2.tamnd.com/wiki/harnacks-inequality/",
  "writing": {
    "tag": "YNH832D50W",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}