Stripe patterns and the eikonal equation

We study a new formulation for the Eikonal equation |∇u| = 1 on a bounded subset of R^2. Considering a field P of orthogonal projections onto 1-dimensional subspaces, with div P ∈ L2, we prove existence and uniqueness for solutions of the equation P div P = 0. We give a geometric description, comparable with the classical case, and we prove that such solutions exist only if the domain is a tubular neighbourhood of a regular closed curve.
This formulation provides a useful approach to the analysis of stripe pat- terns. It is specifically suited to systems where the physical properties of the pattern are invariant under rotation over 180 degrees, such as systems of block copolymers or liquid crystals.