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  "body": "\n## Summary\n\nThe eikonal equation $\\abs{\\nabla u} = f$ says that the gradient of the unknown function has a prescribed length at each point.\nIt is a stationary [[Hamilton–Jacobi equation|5TRXQKP7DY]].\nIts solutions measure travel times: with $f = 1$ and $u = 0$ on the boundary of a domain, the [[viscosity solution|H1ERJ3T75T]] is the distance to the boundary.\nIt is the basic equation of [[geometric optics|90F3NWRHVD]], where the level sets of $u$ are the wave fronts.\nSee the [[definition|X8CF1K8QB4]].\n\n## Explanations\n\n### L1 human\n\nLight a fire along the edge of a field, and suppose that the fire spreads with speed $1/f(x)$ at the point $x$.\nLet $u(x)$ be the time at which the fire reaches $x$.\nMoving one step in the best direction raises the arrival time at rate $f(x)$, and this is what $\\abs{\\nabla u} = f$ says.\nFor a disc of radius $1$ with $f = 1$, the fire burns inward from the circle, and $u(x) = 1 - \\abs{x}$.\nAt the centre the fronts from all sides meet, and $u$ has a corner there.\n",
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