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  "body": "\n## Problem\n\nLet $\\Omega \\subseteq \\R^n$ be a bounded open set with Lipschitz boundary.\nLet $\\psi \\colon \\overline{\\Omega} \\to \\R$ be a continuous function in the [[Sobolev space|M4CEZ5K9HP]] $H^1(\\Omega)$ with $\\psi \\le 0$ on $\\partial \\Omega$, the obstacle.\nLet\n$$K = \\set{v \\in H^1_0(\\Omega) : v \\ge \\psi \\text{ almost everywhere in } \\Omega}.$$\nFind $u \\in K$ that minimizes the Dirichlet energy\n$$D(v) = \\frac{1}{2} \\int_\\Omega \\abs{\\nabla v}^2 \\, dx$$\nover all $v \\in K$.\n\n## Remarks\n\nThe set $K$ is nonempty, closed and [[convex|02AS45T3H9]] in $H^1_0(\\Omega)$, since it contains $\\max(\\psi, 0)$.\nSo the problem has exactly one solution, by the Stampacchia theorem for [[variational inequalities|6QS7X5GVMY]] and the Poincaré inequality.\nEquivalently, $u \\in K$ satisfies $\\int_\\Omega \\nabla u \\cdot \\nabla (v - u) \\, dx \\ge 0$ for all $v \\in K$.\nWhen $u$ is regular enough, this is the complementarity system $u \\ge \\psi$, $\\Delta u \\le 0$, and $\\Delta u = 0$ where $u \u003e \\psi$.\nSo $u$ is [[harmonic|0DEB0JMN0W]] in the open set $\\set{u \u003e \\psi}$ and equals $\\psi$ on the contact set $\\set{u = \\psi}$.\nThe boundary of the contact set inside $\\Omega$ is not known in advance, so this is a [[free boundary problem|YR42E2W19Q]].\nIf $\\psi$ is smooth, the solution has bounded second derivatives locally, and this is the best regularity possible in general, since $\\Delta u$ jumps across the free boundary.\nIn dimension one with $\\Omega = (-1, 1)$, the solution is the smallest concave function above $\\psi$ that vanishes at $\\pm 1$: it is linear off the contact set.\n\n## Explanations\n\n### L1 human\n\nStretch an elastic membrane over a frame and push an obstacle up from below.\nThe membrane rests on the obstacle in some places and is free elsewhere.\nWhere it is free, it takes the shape of least tension, a harmonic function.\nThe problem is to find the shape of the membrane and the region where it touches the obstacle.\nThat region is not known at the start: finding its edge is part of the problem.\n",
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      "first_stated": "1967",
      "notes": "Lions and Stampacchia set the obstacle problem as a variational inequality in Variational inequalities, Comm. Pure Appl. Math. 20 (1967).",
      "posed_by": "Jacques-Louis Lions and Guido Stampacchia"
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