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  "body": "\n## Problem\n\nLet $\\Gamma$ be a Jordan curve in $\\R^3$, that is, a subset homeomorphic to a circle.\nIs there a surface of least area among all surfaces with boundary $\\Gamma$?\n\n## Remarks\n\nThe problem is named after Joseph Plateau, who studied soap films spanning wire frames by experiment.\nJesse Douglas and Tibor Radó solved it independently for surfaces of the type of a disc, around 1930.\nTheir result applies to a rectifiable $\\Gamma$ and gives a continuous map $X$ from the closed unit disc $\\overline{D}$ to $\\R^3$.\nThe map sends the boundary circle homeomorphically onto $\\Gamma$.\nIt is [[harmonic|0DEB0JMN0W]] in the open disc and [[conformal|0YYT36R4YD]] away from isolated branch points.\nIt has least area among all such disc-type surfaces spanning $\\Gamma$.\nThe image of $X$ is then a [[minimal surface|41X0YW2R9N]], possibly with branch points.\nDouglas received one of the first two Fields Medals in 1936, in part for this work.\nLater, geometric measure theory treated the problem in higher dimensions and without a fixed topological type, with solutions such as integral currents.\nThe least-area surface need not be unique, and it need not be a disc: some curves are spanned by surfaces of other types with smaller area.\n\n## Explanations\n\n### L1 human\n\nBend a wire into a closed loop, dip it in soapy water, and a film appears that spans the loop with as little area as it can.\nPlateau's problem asks whether such a least-area surface always exists, as a mathematical object, for every loop.\nThe answer is yes for disc-shaped surfaces, and the proof was one of the great successes of analysis in the twentieth century.\n",
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