A regularizing property of the $2D$-eikonal equation
  We prove that any $2$-dimensional solution $\psi\in W_{loc}^{1+\frac 1 3, 3}$ of the eikonal equation has locally Lipschitz gradient $\nabla \psi$ except at a locally finite number of vortices. 
Camillo De Lellis and Radu Ignat
math.AP