{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Problem\n\nLet $H$ be a real [[Hilbert space|HM2HHE3J51]] with inner product $\\ip{\\cdot}{\\cdot}$.\nLet $K \\subseteq H$ be a nonempty closed [[convex set|02AS45T3H9]], and let $F \\colon K \\to H$ be a map.\nFind $u \\in K$ such that\n$$\\ip{F(u)}{v - u} \\ge 0 \\quad \\text{for all } v \\in K.$$\n\n## Remarks\n\nIf $K = H$, then taking $v = u \\pm w$ shows that the inequality holds exactly when $F(u) = 0$, so the problem contains the equation $F(u) = 0$.\nIf $F = \\nabla f$ for a convex differentiable function $f$, then $u$ solves the problem exactly when $u$ minimizes $f$ on $K$.\nStampacchia's theorem treats the linear case.\nLet $a$ be a bounded [[coercive bilinear form|86KRHS6X22]] on $H$ and let $\\ell$ be a continuous linear functional on $H$.\nThen there is exactly one $u \\in K$ with $a(u, v - u) \\ge \\ell(v - u)$ for all $v \\in K$.\nWhen $a$ is symmetric, this $u$ minimizes $\\tfrac12 a(v, v) - \\ell(v)$ over $K$.\nFor $H = \\R^n$, $K$ compact and $F$ continuous, a solution always exists, by the Hartman–Stampacchia theorem, whose proof uses the Brouwer fixed point theorem.\nFor a continuous [[monotone|Y1M03JQKKR]] $F$ the solution set is closed and convex, and a strictly monotone $F$ has at most one solution.\n\n## Explanations\n\n### L1 human\n\nA variational inequality describes an equilibrium under a constraint.\nThink of a ball resting on a curved surface inside a box.\nIf the lowest point lies inside the box, the force there vanishes; if it lies on a wall, the force need not vanish but must push into the wall.\nThe inequality $\\ip{F(u)}{v - u} \\ge 0$ says exactly this: from $u$, no allowed direction $v - u$ goes \"downhill\".\nObstacle problems in elasticity, contact problems and many equilibrium models in economics take this form.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "02AS45T3H9"
    ],
    "history": {
      "posed_by": "Antonio Signorini"
    },
    "kind": "problem",
    "msc": [
      "47J20",
      "49J40"
    ],
    "name": "Variational inequality",
    "same_as": {
      "wikidata": "Q4556320",
      "wikipedia": "Variational inequality"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Variational_inequality"
      }
    ],
    "tag": "6QS7X5GVMY",
    "uses": [
      "HM2HHE3J51",
      "02AS45T3H9",
      "Y1M03JQKKR",
      "86KRHS6X22"
    ]
  },
  "kind": "problem",
  "link": "https://math2.tamnd.com/node/6QS7X5GVMY/",
  "links": [
    {
      "field": "about",
      "target": "02AS45T3H9"
    },
    {
      "field": "uses",
      "target": "HM2HHE3J51"
    },
    {
      "field": "uses",
      "target": "02AS45T3H9"
    },
    {
      "field": "uses",
      "target": "Y1M03JQKKR"
    },
    {
      "field": "uses",
      "target": "86KRHS6X22"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/6QS7X5GVMY.md",
  "msc": [
    "47J20",
    "49J40"
  ],
  "name": "Variational inequality",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6QS/6QS7X5GVMY.md",
  "status": null,
  "tag": "6QS7X5GVMY",
  "uri": "m2://6QS7X5GVMY",
  "url": "https://math2.tamnd.com/wiki/variational-inequality/",
  "writing": {
    "tag": "6QS7X5GVMY",
    "kind": "problem",
    "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
      }
    ]
  }
}