I am interested in equations of the form $|\nabla d|= F(x)$, where $F(x)$ is piecewise constant and $d(x) = 0$ on $\Gamma_D$, a subset of the boundary. In particular, like in the figure, one can consider $F(x)$ taking two values, delimited by an interface $\Sigma$ (which can be considered smooth).

Snell's law gives a relation between the incidence angles of a ray passing through the interface: $\sin \theta_+/\sin \theta_- = V_+/V_-$. I cannot find any clear references proving that Snell's law holds for the eikonal equation described above. Therefore I arrive at my question:

- Does Snell's law hold across the interface $\Sigma$?

- If the answer to the first question is affirmative, can you indicate references where a proof can be found?