Non-oriented solutions of the eikonal equation

We study a new formulation for the eikonal equation |∇u| = 1 on a bounded subset of R^2. Instead of a vector field ∇u, we consider a field P of orthogonal projections on one-dimensional subspaces, with div P ∈ L^2. We prove that solutions of this equation propagate direction as in the classical eikonal equation. We also show that solutions exist if and only if the domain is a tubular neighborhood of a regular closed curve.