Regularity of the Eikonal equation with two vanishing entropies
We study regularity of solutions to the Eikonal equation $|\nabla u|=1$ a.e. in a bounded simply-connected two dimensional domain. Solutions to this equation have very little regularity in general. However, with the help of two vanishing entropies, we are able to obtain strong regularity results for solutions of the Eikonal equation. The motivation of our problem comes from a classical problem in Calculus of Variations, called the Aviles-Giga functional in connection with the theory of smectic liquid crystals and thin film blisters. Our result for the first time uses only two entropies to characterize regularity properties in this direction. This is joint work with Andrew Lorent.