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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $\\Omega \\subseteq \\R^n$ be an open set, and let $f \\colon \\Omega \\times \\R \\times \\R^n \\to \\R$ be a given function.\nThe *Monge–Ampère equation* is the [[fully nonlinear PDE|2D3N1ZJWBW]]\n$$\\det D^2 u(x) = f\\big(x, u(x), \\nabla u(x)\\big), \\qquad x \\in \\Omega,$$\nfor a twice differentiable function $u \\colon \\Omega \\to \\R$, where $D^2 u$ is the [[Hessian matrix|0693Y2METT]] of $u$.\nThe most studied case is $\\det D^2 u = f(x)$ with $f \u003e 0$, for an unknown [[convex function|07GR3TNW4H]] $u$.\n\n## Remarks\n\nExample: $u(x) = \\tfrac12 \\abs{x}^2$ solves $\\det D^2 u = 1$ on $\\R^n$, since $D^2 u$ is the identity matrix.\nNon-example: in $\\R^2$, $u(x) = \\tfrac12 x_1^2$ does not solve $\\det D^2 u = 1$, since $D^2 u = \\operatorname{diag}(1, 0)$ has determinant $0$.\nThe equation is elliptic at a solution $u$ when $D^2 u$ is positive definite, which is why one looks for convex solutions with $f \u003e 0$.\nFor $n = 2$ it reads $u_{xx} u_{yy} - u_{xy}^2 = f$.\nThe graph of $u$ has Gauss curvature $K = \\det D^2 u / (1 + \\abs{\\nabla u}^2)^{(n+2)/2}$, so prescribing $K$ is a Monge–Ampère equation.\nFor convex functions that are not twice differentiable, the equation is understood in the sense of Aleksandrov or in the viscosity sense.\n",
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