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  "body": "\n## Summary\n\nThe Monge–Ampère equation prescribes the determinant of the [[Hessian matrix|0693Y2METT]] of an unknown function: $\\det D^2 u = f$.\nIt is the basic example of a [[fully nonlinear PDE|2D3N1ZJWBW]], since it is nonlinear in the highest derivatives.\nIt is elliptic when $u$ is convex and $f \u003e 0$.\nIt appears in the problem of prescribing the Gauss curvature of a convex surface, in optimal transport and in complex geometry.\n\n## Explanations\n\n### L1 human\n\nThe Hessian of a function records how it curves in every direction, and its determinant multiplies these curvatures together.\nThe Monge–Ampère equation asks for a function whose total curvature, in this sense, is given at every point.\nIn optimal transport, the gradient of a convex solution moves one mass distribution onto another, and the determinant measures how much volume is stretched.\nThe function $u(x) = \\abs{x}^2 / 2$ has Hessian the identity matrix, so it solves $\\det D^2 u = 1$.\n",
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