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  "body": "\n## Definition\n\nA [[partial differential equation|GC5BDZ6QBC]] $F\\left(D^m u, \\dots, D u, u, x\\right) = 0$ of [[order|E7JDFDPM1X]] $m$ is *fully nonlinear* if it is not [[quasilinear|7X9N9H3RK6]].\nThat is, $F$ cannot be written as an affine function of the derivatives $D^m u$ of order $m$ whose coefficients depend only on $D^{m-1} u, \\dots, D u, u, x$.\n\n## Remarks\n\nExample: the Monge–Ampère equation $\\det D^2 u = f(x)$ in $\\R^n$ with $n \\ge 2$ is fully nonlinear of order $2$.\nFor $n = 2$ it reads $u_{xx} u_{yy} - u_{xy}^2 = f$, which contains the product $u_{xx} u_{yy}$ of two second derivatives.\nNon-example: the minimal surface equation is not fully nonlinear, because it is affine in the second derivatives with coefficients depending on $\\nabla u$ only.\nFor first-order equations, the eikonal equation $\\abs{\\nabla u}^2 = 1$ is fully nonlinear.\n\n## Explanations\n\n### L1 human\n\nA PDE is fully nonlinear when its highest derivatives enter in a genuinely nonlinear way.\n",
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