In the standard version of Snell's law, there are rays, which cannot extend beyond a certain point (while satisfying the law), hence are reflected (total internal reflection). I wonder if this is also the case for the differential form of the law (e.g. when assuming that the optical density function is smooth or Lipschitz continuous on $\mathbf{R}^3$ or $\mathbf{R}^n$).
If not, how does the approximation of a non-continuous optical density function (where total internal reflection takes place) by a sufficiently smooth one look like, in terms of solutions? Can solutions be bounded, e.g. a ray going in a circle?