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  "body": "\n## Problem\n\nA free boundary problem is a problem for a [[partial differential equation|GC5BDZ6QBC]] in which part of the domain is unknown.\nA typical stationary form is the following.\nLet $D \\subseteq \\R^n$ be a bounded open set and let $g$ be given boundary data on $\\partial D$.\nFind an open set $\\Omega \\subseteq D$ and a function $u$ on $\\overline{\\Omega}$ such that:\n\n1. $u$ solves a given partial differential equation in $\\Omega$;\n2. $u = g$ on the known part $\\partial \\Omega \\cap \\partial D$ of the boundary;\n3. $u$ satisfies two conditions on the unknown part $\\Gamma = \\partial \\Omega \\cap D$, the free boundary.\n\nOne condition on $\\Gamma$ alone would fix $u$ for each choice of $\\Omega$.\nThe second condition is what determines $\\Gamma$.\nA standard example is the one-phase Bernoulli problem: find $u \\ge 0$ with $\\Delta u = 0$ in $\\Omega = \\set{u \u003e 0}$, $u = 0$ on $\\Gamma$ and $\\abs{\\nabla u} = 1$ on $\\Gamma$.\nWhen the free boundary moves in time, one also speaks of a moving boundary problem.\n\n## Remarks\n\nFree boundary problems are nonlinear even when the equation in clause 1 is linear, since the domain depends on the solution.\nThe [[Stefan problem|3B0HB3DHW1]] for melting ice is the classical evolution example.\nThe heat equation holds in the liquid, and the melting front moves with a speed fixed by the heat flux.\nThe obstacle problem is the classical stationary example.\nThere the free boundary separates the region where a membrane touches an obstacle from the region where it is [[harmonic|0DEB0JMN0W]].\nA common method is to recast the problem as a minimization or a variational inequality on a fixed domain, solve it there in a weak sense, and recover $\\Gamma$ afterwards.\nThe main questions are then existence and uniqueness of a solution, the regularity of $u$, and the regularity of the free boundary $\\Gamma$.\nLamé and Clapeyron studied the solidification of a liquid sphere in 1831, a problem of this type.\n\n## Explanations\n\n### L1 human\n\nIn most problems for differential equations the region is given, and only the function is unknown.\nIn a free boundary problem the edge of the region is also unknown and must be found together with the function.\nThink of ice melting in water: the temperature in the water is unknown, and so is the position of the ice front.\nBecause the region has an extra unknown, the problem needs one extra condition on its edge.\n",
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      "first_stated": "1831",
      "posed_by": "Gabriel Lamé and Émile Clapeyron"
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