On Lebesgue points of entropy solutions to the 2D eikonal equation

Entropy solutions of the 2D eikonal equation arise as limits of sequences of bounded Aviles-Giga energy. They might be much less regular than viscosity solutions, which have BV gradient, nevertheless their gradients share several structural properties with BV maps, as discovered by De Lellis and Otto. Fine estimates on their Lebesgue points are however still open. In a joint work with Elio Marconi, we obtain a bound on local oscillations in terms of the entropy production, implying in particular that non-Lebesgue points have codimension 1 (as they would for BV maps).