Eikonal equation

Remarks

It is the stationary Hamilton–Jacobi equation with Hamiltonian $H(p, x) = \abs{p} - f(x)$. In optics $f$ is the refractive index, $u$ is the optical path length, and the level sets of $u$ are the wave fronts.

Example: on $\Omega = \R^n \setminus \set{0}$ with $f = 1$, the function $u(x) = \abs{x}$ is a classical solution. Check: $\nabla u(x) = x / \abs{x}$, which has length $1$.

Non-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. Indeed 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. In both cases $\nabla u = 0$ somewhere in $\Omega$, against $\abs{\nabla u} = 1$. The function $u(x) = 1 - \abs{x}$ solves the equation except at $0$, where it is not differentiable. Written as $\abs{\nabla u} - 1 = 0$, the problem has $1 - \abs{x}$, the distance to the boundary, as its unique viscosity solution.