{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a real normed space and let $J \\colon X \\to (-\\infty, +\\infty]$ be a [[functional|1EQM0DJ4NC]].\nThe functional $J$ is *coercive* if $J(u) \\to +\\infty$ as $\\norm{u} \\to \\infty$.\nThat is, for every $M \\in \\R$ there is $R \u003e 0$ such that $J(u) \u003e M$ for all $u \\in X$ with $\\norm{u} \u003e R$.\n\n## Remarks\n\nEquivalently, every sublevel set $\\set{u \\in X : J(u) \\le t}$, $t \\in \\R$, is bounded.\nSo a minimising sequence of a coercive functional with $\\inf J \u003c +\\infty$ is bounded.\nIn a reflexive [[Banach space|06GZ956657]] it then has a weakly convergent subsequence.\n\nExample: on a Hilbert space $H$ with $f \\in H$, let $J(u) = \\frac{1}{2}\\norm{u}^2 - \\ip{f}{u}$.\nThen $J(u) \\ge \\frac{1}{2}\\norm{u}^2 - \\norm{f} \\norm{u}$, which tends to $+\\infty$, so $J$ is coercive.\n\nNon-example: on $X = \\R$, let $J(u) = e^{u}$.\nThe values stay below $1$ for all $u \u003c 0$, so $J$ is not coercive, and it has no minimiser.\n\nSome authors use a stronger, quantitative form: $J(u) \\ge c \\norm{u}^p - C$ for constants $c \u003e 0$, $p \\ge 1$ and $C$.\nFor integral functionals, this follows from a growth bound $L(x, u, \\xi) \\ge c \\abs{\\xi}^p - C$ on the integrand together with boundary conditions and the Poincaré inequality.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "NGT5YCVAMR",
    "kind": "definition",
    "msc": [
      "49J45",
      "58E30"
    ],
    "name": "Coercive functional",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Coercive_function"
      }
    ],
    "tag": "FH92FW76Q0",
    "uses": [
      "06GZ956657",
      "1EQM0DJ4NC"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/FH92FW76Q0/",
  "links": [
    {
      "field": "defines",
      "target": "NGT5YCVAMR"
    },
    {
      "field": "uses",
      "target": "06GZ956657"
    },
    {
      "field": "uses",
      "target": "1EQM0DJ4NC"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/FH92FW76Q0.md",
  "msc": [
    "49J45",
    "58E30"
  ],
  "name": "Coercive functional",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FH9/FH92FW76Q0.md",
  "status": null,
  "tag": "FH92FW76Q0",
  "uri": "m2://FH92FW76Q0",
  "url": "https://tamnd.github.io/math2/wiki/coercive-functional-definition/",
  "writing": {
    "tag": "FH92FW76Q0",
    "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
      }
    ]
  }
}