{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "23RK8EZ20P"
    },
    {
      "field": "about",
      "target": "6TSMHGCXVM"
    },
    {
      "field": "uses",
      "target": "23RK8EZ20P"
    },
    {
      "field": "uses",
      "target": "6TSMHGCXVM"
    },
    {
      "field": "uses",
      "target": "RNDKD74V17"
    },
    {
      "field": "illustrates",
      "target": "4T72ZP5XHF"
    },
    {
      "field": "illustrates",
      "target": "ECHTH3P9RE"
    },
    {
      "field": "specializes",
      "target": "KH0V6QA2ZN"
    },
    {
      "field": "defines",
      "target": "NNAR1HN9BQ"
    }
  ],
  "body": "\n## Summary\n\nΓ-convergence is a notion of convergence for functionals that is designed for minimization problems.\nA sequence $F_n$ Γ-converges to $F$ when two things hold.\nA lower bound inequality holds along every convergent sequence of points, and the bound is reached along some sequence.\nUnder a compactness condition, minimum values and minimizers of $F_n$ then converge to those of $F$.\nIt was introduced by Ennio De Giorgi.\nIt is the standard tool to pass from a family of variational problems to a limit problem, as in homogenization and phase transitions.\nThe [[definition|NNAR1HN9BQ]] gives the precise conditions.\n\n## Explanations\n\n### L1 human\n\nPointwise convergence of functions does not control where their minima go.\nΓ-convergence asks instead for two things.\nThe limit $F$ is never larger than what the $F_n$ give near a point.\nAnd every value of $F$ can be approached by values of $F_n$ at nearby points.\nThink of a landscape with finer and finer ripples: the ripples have no pointwise limit, but the lowest points of the landscape settle down.\nFor example, $F_n(x) = \\sin(nx)$ on $\\R$ has no pointwise limit, yet it Γ-converges to the constant $-1$, because near every point there are points where $\\sin(nx) = -1$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "49J45"
    ],
    "name": "Γ-convergence",
    "same_as": {
      "wikidata": "Q278549",
      "wikipedia": "Γ-convergence"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Γ-convergence"
      }
    ],
    "tag": "NW9MM370A5"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/NW9MM370A5/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/NW9MM370A5.md",
  "msc": [
    "49J45"
  ],
  "name": "Γ-convergence",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NW9/NW9MM370A5.md",
  "status": null,
  "tag": "NW9MM370A5",
  "uri": "m2://NW9MM370A5",
  "url": "https://tamnd.github.io/math2/wiki/convergence/",
  "writing": {
    "tag": "NW9MM370A5",
    "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
      }
    ]
  }
}