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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be an open set and let $f \\colon \\Omega \\to (0, \\infty)$ be a continuous function.\nThe *eikonal equation* with speed $1/f$ is\n$$\\abs{\\nabla u(x)} = f(x), \\qquad x \\in \\Omega,$$\nfor an unknown function $u \\colon \\Omega \\to \\R$.\nHere $\\nabla u$ is the [[gradient|0ABK9V9Y64]] of $u$ and $\\abs{\\cdot}$ is the Euclidean norm.\nIt is usually completed by a boundary condition such as $u = 0$ on $\\partial \\Omega$.\n\n## Remarks\n\nIt is the stationary [[Hamilton–Jacobi equation|5TRXQKP7DY]] with Hamiltonian $H(p, x) = \\abs{p} - f(x)$.\nIn optics $f$ is the refractive index, $u$ is the optical path length, and the level sets of $u$ are the wave fronts.\n\nExample: on $\\Omega = \\R^n \\setminus \\set{0}$ with $f = 1$, the function $u(x) = \\abs{x}$ is a classical solution.\nCheck: $\\nabla u(x) = x / \\abs{x}$, which has length $1$.\n\nNon-example: on the open unit disc $\\Omega$ in $\\R^2$ with $f = 1$ and $u = 0$ on the circle, there is no $C^1$ solution.\nIndeed a function that is continuous on the closed disc, $C^1$ in the open disc and zero on the circle is identically $0$ or has an interior maximum or minimum.\nIn both cases $\\nabla u = 0$ somewhere in $\\Omega$, against $\\abs{\\nabla u} = 1$.\nThe function $u(x) = 1 - \\abs{x}$ solves the equation except at $0$, where it is not differentiable.\nWritten as $\\abs{\\nabla u} - 1 = 0$, the problem has $1 - \\abs{x}$, the distance to the boundary, as its unique [[viscosity solution|H1ERJ3T75T]].\n",
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