On Lebesgue points of entropy solutions to the eikonal equation

We consider entropy solutions to the eikonal equation $\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }=1$ in two space dimensions. These solutions are motivated by a class of variational problems and fail in general to have bounded variation. Nevertheless they share with BV functions, several of their fine properties: we show in particular that the set of non-Lebesgue points has co-dimension at least one.